Parameterization of Stillinger-Weber Potential for Two- Dimensional Atomic Crystals

Abstract

We parametrize the Stillinger-Weber potential for 156 two-dimensional atomic crystals (TDACs). Parameters for the Stillinger-Weber potential are obtained from the valence force field (VFF) model following the analytic approach (Nanotechnology. 2015;26:315706), in which the valence force constants are determined by the phonon spectrum. The Stillinger-Weber potential is an efficient nonlinear interaction and is applicable for numerical simulations of nonlinear physical or mechanical processes. The supplemental resources for all simulations in the present work are available online in http://jiangjinwu.org/sw, including a Fortran code to generate crystals’ structures, files for molecular dynamics simulations using LAMMPS, files for phonon calculations with the Stillinger-Weber potential using GULP, and files for phonon calculations with the valence force field model using GULP.

1. Introduction

The atomic interaction is of essential importance in the numerical investigation of most physical or mechanical processes [1]. The present work provides parameters for the Stillinger-Weber (SW) empirical potential for 156 two-dimensional atomic crystals (TDACs). In practical applications, these layered materials are usually played as Lego on atomic scale to construct the van der Waals heterostructures with comprehensive properties [2]. The computational cost of ab initio for the heterostructure will be substantially increased as compared with one individual atomic layer, because the unit cell for the heterostructure is typically very large resulting from the mismatch of the lattice constants of different layered components. The empirical potential will be a competitive alternative to help out this difficult situation, considering their high efficiency.

In the early stage before 1980s, the computation ability of the scientific community was quite limited. At that time, the valence force field (VFF) model was one popular empirical potential for the description of the atomic interaction, since the VFF model is linear and can be applied in the analytic derivation of most elastic quantities [3]. In this model, each VFF term corresponds to a particular motion style in the crystal. Hence, each parameter in the VFF model usually has clear physical essence, which is beneficial for the parameterization of this model. For instance, the bond stretching term in the VFF model is directly related to the frequency of the longitudinal optical phonon modes, so the force constant of the bond stretching term can be determined from the frequencies of the longitudinal optical phonon modes. The VFF model can thus serve as the starting point for developing atomic empirical potentials for different crystals.

While the VFF model is beneficial for the fastest numerical simulation, its strong limitation is the absence of nonlinear effect. Due to this limitation, the VFF model is not applicable to nonlinear phenomena, for which other potential models with nonlinear components are required. Some representative potential models are (in the order of their simulation costs) SW potential [4], Tersoff potential [5], Brenner potential [6], ab initio approaches, etc. The SW potential is one of the simplest potential forms with nonlinear effects included. An advanced feature for the SW potential is that it includes the nonlinear effect, and keeps the numerical simulation at a very fast level.

Considering its distinct advantages, the present article aims at providing the SW potential for 156 TDACs. We will determine parameters for the SW potential from the VFF model, following the analytic approach proposed by one of the present authors (JWJ) [7]. The VFF constants are fitted to the phonon spectrum or the elastic properties in the TDACs.

In this paper, we parametrize the SW potential for 156 TDACs. All structures discussed in the present work are listed in Tables 1 – 9 . The supplemental materials are freely available online in [1], including a Fortran code to generate crystals’ structures, files for molecular dynamics simulations using LAMMPS, files for phonon calculations with the SW potential using GULP, and files for phonon calculations with the valence force field model using GULP.

1H-ScO2	1H-ScS2	1H-ScSe2	1H-ScTe2	1H-TiTe2	1H-VO2	1H-VS2	1H-VSe2	1H-VTe2
1H-CrO2	1H-CrS2	1H-CrSe2	1H-CrTe2	1H-MnO2	1H-FeO2	1H-FeS2	1H-FeSe2	1H-FeTe2
1H-CoTe2	1H-NiS2	1H-NiSe2	1H-NiTe2	1H-NbS2	1H-NbSe2	1H-MoO2	1H-MoS2	1H-MoSe2
1H-MoTe2	1H-TaS2	1H-TaSe2	1H-WO2	1H-WS2	1H-WSe2	1H-WTe2

Table 1.

1H-MX₂, with M as the transition metal and X as oxygen or dichalcogenide.

The structure is shown in Figure 1 .

1T-ScO2	1T-ScS2	1T-ScSe2	1T-ScTe2	1T-TiS2	1T-TiSe2	1T-TiTe2	1T-VS2	1T-VSe2
1T-VTe2	1T-MnO2	1T-MnS2	1T-MnSe2	1T-MnTe2	1T-CoTe2	1T-NiO2	1T-NiS2	1T-NiSe2
1T-NiTe2	1T-ZrS2	1T-ZrSe2	1T-ZrTe2	1T-NbS2	1T-NbSe2	1T-NbTe2	1T-MoS2	1T-MoSe2
1T-MoTe2	1T-TcS2	1T-TcSe2	1T-TcTe2	1T-RhTe2	1T-PdS2	1T-PdSe2	1T-PdTe2	1T-SnS2
1T-SnSe2	1T-HfS2	1T-HfSe2	1T-HfTe2	1T-TaS2	1T-TaSe2	1T-TaTe2	1T-WS2	1T-WSe2
1T-WTe2	1T-ReS2	1T-ReSe2	1T-ReTe2	1T-IrTe2	1T-PtS2	1T-PtSe2	1T-PtTe2

Table 2.

1T-MX₂, with M as the transition metal and X as oxygen or dichalcogenide.

The structure is shown in Figure 71 .

Black phosphorus

p-Arsenene

p-Antimonene

p-Bismuthene

Table 3.

Puckered (p-) M, with M from group V.

The structure is shown in Figures 178 or 183 .

	p-SiO	p-GeO	p-SnO
p-CS	p-SiS	p-GeS	p-SnS
p-CSe	p-SiSe	p-GeSe	p-SnSe
p-CTe	p-SiTe	p-GeTe	p-SnTe

Table 4.

Puckered MX, with M from group IV and X from group VI.

The structure is shown in Figure 189 , and particularly Figure 191 for p-MX with X = O.

Silicene	Germanene	Stanene	Indiene
Blue phosphorus	b-Arsenene	b-Antimonene	b-Bismuthene

Table 5.

Buckled (b-) M, with M from group IV or V.

The structure is shown in Figure 222 .

b-CO	b-SiO	b-GeO	b-SnO
b-CS	b-SiS	b-GeS	b-SnS
b-CSe	b-SiSe	b-GeSe	b-SnSe
b-CTe	b-SiTe	b-GeTe	b-SnTe

Table 6.

Buckled MX, with M from group IV and X from group VI.

The structure is shown in Figure 239 .

b-SnGe

b-SiGe

b-SnSi

b-InP

b-InAs

b-InSb

b-GaAs

b-GaP

b-AlSb

Table 7.

Buckled MX, with both M and X from group IV or M from group III and X from group V.

The structure is shown in Figure 239 .

BO	AlO	GaO	InO
BS	AlS	GaS	InS
BSe	AlSe	GaSe	InSe
BTe	AlTe	GaTe	InTe

Table 8.

Bi-buckled MX, with M from group III and X from group VI.

The structure is shown in Figure 290 .

Borophene

Table 9.

The structure is shown in Figure 323 .

2. VFF model and SW potential

2.1. VFF model

The VFF model is one of the most widely used linear models for the description of atomic interactions [3]. The bond stretching and the angle bending are two typical motion styles for most covalent bonding materials. The bond stretching describes the energy variation for a bond due to a bond variation $\text{Δ}r=r-r_0$ , with r ₀ as the initial bond length. The angle bending gives the energy increment for an angle resulting from an angle variation $\text{Δ}\theta =\theta -{\theta }_0$ , with θ ₀ as the initial angle. In the VFF model, the energy variations for the bond stretching and the angle bending are described by the following quadratic forms,

$V_r=\frac{1}{2}{K}_r{\left(\text{Δ}r\right)}^2,$ E1

$V_{\theta }=\frac{1}{2}{K}_{\theta }{\left(\text{Δ}\theta \right)}^2,$ E2

where K _r and K _θ are two force constant parameters. These two potential expressions in Eqs. (1) and (2) are directly related to the optical phonon modes in the crystal. Hence, their force constant parameters K _r and K _θ are usually determined by fitting to the phonon dispersion.

2.2. SW potential

In the SW potential, energy increments for the bond stretching and angle bending are described by the following two-body and three-body forms,

$V_2\left(r_{ij}\right)=A\left(B/r_{ij}-1\right)e^{\left[\rho /\left(r_{ij}-r_{max}\right)\right]},$ E3

$V_3\left({\theta }_{ijk}\right)=K{e}^{\left[{\rho }_1/\left(r_{ij}-r_{max12}\right)+{\rho }_2/\left(r_{ik}-r_{max13}\right)\right]}{(\text{cos}{\theta }_{ijk}-cos{\theta }_0)}^2$ E4

where V ₂ corresponds to the bond stretching and V ₃ associates with the angle bending. The cut-offs r _max , r _max12, and r _max13 are geometrically determined by the material’s structure. There are five unknown geometrical parameters, i.e., ρ and B in the two-body V ₂ term and ρ ₁, ρ ₂, and θ ₀ in the three-body V ₃ term, and two energy parameters A and K. There is a constraint among these parameters due to the equilibrium condition [7],

$\rho =\frac{-4B{\left(d-r_{max}\right)}^2}{\left(Bd-d^5\right)},$ E5

where d is the equilibrium bond length from experiments. Eq. (5) ensures that the bond has an equilibrium length d and the V ₂ interaction for this bond is at the energy minimum state at the equilibrium configuration.

The energy parameters A and K in the SW potential can be analytically derived from the VFF model as follows,

$A=\frac{K_r}{\alpha e^{[\rho /\left(d-r_{max}\right)]}},$ E6

$K=\frac{K_{\theta }}{2{sin}^2{\theta }_0{e}^{[{\rho }_1/\left(d_1-r_{\text{m}ax12}\right)+{\rho }_2/\left(d_2-r_{\text{m}ax13}\right)]}},$ E7

where the coefficient α in Eq. (6) is,

$\begin{array}{cc}\alpha & ={\left[\frac{\rho }{{\left(d-r_{max}\right)}^2}\right]}^2\left(B/d^4-1\right)\\ & +\left[\frac{2\rho }{{\left(d-r_{max}\right)}^3}\right]\left(B/d^4-1\right)\\ & +\left[\frac{\rho }{{\left(d-r_{max}\right)}^2}\right]\left(\frac{8B}{d^5}\right)+\left(\frac{20B}{d^6}\right)\mathrm{.}\end{array}$ E8

In some situations, the SW potential is also written into the following form,

$V_2\left(r_{ij}\right)=ϵA_L\left(B_L{\sigma }^p{r}_{ij}^{-p}-{\sigma }^q{r}_{ij}^{-q}\right)e^{\left[\sigma /\left(r_{ij}-a\sigma \right)\right]},$ E9

$V_3\left({\theta }_{ijk}\right)=ϵ\lambda e^{\left[\gamma \sigma /\left(r_{ij}-a\sigma \right)+\gamma \sigma /\left(r_{jk}-a\sigma \right)\right]}{\left(\text{cos}{\theta }_{ijk}-\text{cos}{\theta }_0\right)}^2\mathrm{.}$ E10

The parameters here can be determined by comparing the SW potential forms in Eqs. (9) and (10) with Eqs. (3) and (4). It is obvious that p = 4 and q = 0. Eqs. (9) and (10) have two more parameters than Eqs. (3) and (4), so we can set ϵ = 1.0 eV and γ = 1.0. The other parameters in Eqs. (9) and (10) are related to these parameters in Eqs. (3) and (4) by the following equations

$A_L=A,$ E11

$\sigma =\rho ,$ E12

$B_L=B/{\rho }^4,$ E13

$a=r_{max}/\rho ,$ E14

$\lambda =K\mathrm{.}$ E15

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{Sc}-\text{O}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{Sc}-\text{O}-\text{O}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{Sc}-\text{O}-{\text{O}}^{\prime }}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{O}-\text{Sc}-\text{Sc}}{\left(\text{Δ}\theta \right)}^2$
Parameter	9.417	4.825	4.825	4.825
r 0 or θ 0	2.090	98.222	58.398	98.222

Table 10.

The VFF model for single-layer 1H-ScO₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of ev/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle θ _ijk has atom i as the apex.

	A (eV)	ρ (Å)	B (Å^4)	r min (Å)	r max(Å)
Sc-O	7.506	1.380	9.540	0.0	2.939

Table 11.

Two-body SW potential parameters for single-layer 1H-ScO₂ used by GULP [8] as expressed in Eq. (3).

The SW potential is implemented in GULP using Eqs. (3) and (4). The SW potential is implemented in LAMMPS using Eqs. (9) and (10).

In the rest of this article, we will develop the VFF model and the SW potential for layered crystals. The VFF model will be developed by fitting to the phonon dispersion from experiments or first-principles calculations. The SW potential will be developed following the above analytic parameterization approach. In this work, GULP [8] is used for the calculation of phonon dispersion and the fitting process, while LAMMPS [9] is used for molecular dynamics simulations. The OVITO [10] and XCRYSDEN [11] packages are used for visualization. All simulation scripts for GULP and LAMMPS are available online in [1].

3. 1H-SCO₂

Most existing theoretical studies on the single-layer 1H-ScO₂ are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1H-ScO₂.

Figure 1.

Configuration of the 1H-MX₂ in the 1H phase. (a) Top view. The unit cell is highlighted by a red parallelogram. (b) Enlarged view of atoms in the blue box in (a). Each M atom is surrounded by six X atoms, which are categorized into the top and bottom groups. Atoms X 1, 3, and 5 are from the top group, while atoms X 2, 4, and 6 are from the bottom group. M atoms are represented by larger gray balls. X atoms are represented by smaller yellow balls.

The structure for the single-layer 1H-ScO₂ is shown in Figure 1 (with M = Sc and X = O). Each Sc atom is surrounded by six O atoms. These O atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each O atom is connected to three Sc atoms. The structural parameters are from the first-principles calculations [12], including the lattice constant a = 3.16 Å and the bond length $d_{\text{Sc}-\text{O}}=2.09$ Å. The resultant angles are ${\theta }_{\text{ScOO}}={\theta }_{\text{OScSc}}=98.222°$ and ${\theta }_{\text{ScO}{\text{O}}^{\prime }}=58.398°$ , in which atoms O and O′ are from different (top or bottom) groups.

Figure 2.

Phonon spectrum for single-layer 1H-ScO₂. (a) Phonon dispersion along the direction ΓM in the Brillouin zone. The results from the VFF model (lines) are comparable with the ab initio results (pentagons) from [12]. (b) The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

	K (eV)	θ 0 (°)	ρ 1 (Å)	ρ 2 (Å)	r min 12 (Å)	r max 12 (Å)	r min 13 (Å)	r max 13 (Å)	r min 23 (Å)	r max 23 (Å)
${\theta }_{\text{Sc}-\text{O}-\text{O}}$	63.576	98.222	1.380	1.380	0.0	2.939	0.0	2.939	0.0	3.460
${\theta }_{\text{Sc}-\text{O}-{\text{O}}^{\prime }}$	85.850	58.398	1.380	1.380	0.0	2.939	0.0	2.939	0.0	3.460
${\theta }_{\text{O}-\text{Sc}-\text{Sc}}$	63.576	98.222	1.380	1.380	0.0	2.939	0.0	2.939	0.0	3.460

Table 12.

Three-body SW potential parameters for single-layer 1H-ScO₂ used by GULP [8] as expressed in Eq. (4).

The angle θ _ijk in the first line indicates the bending energy for the angle with atom i as the apex.

Table 10 shows four VFF terms for the single-layer 1H-ScO₂; one of which is the bond stretching interaction shown by Eq. (1), while the other three terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along ГM as shown in Figure 2(a) . The ab initio calculations for the phonon dispersion are from [12]. Figure 2(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 11 . The parameters for the three-body SW potential used by GULP are shown in Table 12 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 13 . We note that 12 atom types have been introduced for the simulation of the single-layer 1H-ScO₂ using LAMMPS, because the angles around atom Sc in Figure 1 (with M = Sc and X = O) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS [13, 14]. According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Figure 3 (with M = Sc and X = O) shows that, for 1H-ScO₂, we can differentiate these angles around the Sc atom by assigning these six neighboring O atoms with different atom types. It can be found that twelve atom types are necessary for the purpose of differentiating all six neighbors around one Sc atom.

Figure 3.

Twelve atom types are introduced to distinguish angles around each M atom for the single-layer 1H-MX₂. Atoms X₁, X₃, X₅, and X₇ are from the top layer. The other four atoms X₂, X₄, X₆, and X₈ are from the bottom layer, which are not displayed in the figure.

Figure 4.

Stress-strain for single-layer 1H-ScO₂ of dimension 100 × 100 Å along the armchair and zigzag directions.

	ϵ (eV)	σ (Å)	a	λ	γ	cos θ 0	A L	B L	p	q	Tol
Sc1─O1─O1	1.000	1.380	2.129	0.000	1.000	0.000	7.506	2.627	4	0	0.0
Sc1─O1─O3	1.000	0.000	0.000	63.576	1.000	−0.143	0.000	0.000	4	0	0.0
Sc1─O1─O2	1.000	0.000	0.000	85.850	1.000	0.524	0.000	0.000	4	0	0.0
O1─Sc1─Sc3	1.000	0.000	0.000	63.576	1.000	−0.143	0.000	0.000	4	0	0.0

Table 13.

SW potential parameters for single-layer 1H-ScO₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-ScO₂ under uniaxial tension at 1 and 300 K. Figure 4 shows the stress-strain curve for the tension of a single-layer 1H-ScO₂ of dimension 100 × 100 Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-ScO₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-ScO₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 126.3 and 125.4 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}={\nu }_{yx}=0.16$ .

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{Sc}-\text{S}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{Sc}-\text{S}-\text{S}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{Sc}-\text{S}-{\text{S}}^{\prime }}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{S}-\text{Sc}-\text{Sc}}{\left(\text{Δ}\theta \right)}^2$
Parameter	5.192	2.027	2.027	2.027
r 0 or θ 0	2.520	94.467	64.076	94.467

Table 14.

The VFF model for single-layer 1H-ScS₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of ev/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle θ _ijk has atom i as the apex.

	A (eV)	ρ (Å)	B (Å^4)	r min (Å)	r max (Å)
Sc─S	5.505	1.519	20.164	0.0	3.498

Table 15.

Two-body SW potential parameters for single-layer 1H-ScS₂ used by GULP [8] as expressed in Eq. (3).

There is no available value for nonlinear quantities in the single-layer 1H-ScO₂. We have thus used the nonlinear parameter B = 0.5d ⁴ in Eq. (5), which is close to the value of B in most materials. The value of the third-order nonlinear elasticity D can be extracted by fitting the stress-strain relation to the function $\sigma =Eϵ+\frac{1}{2}D{ϵ}^2$ with E as the Young’s modulus. The values of D from the present SW potential are −652.8 and −683.3 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 12.2 N/m at the ultimate strain of 0.19 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 11.7 N/m at the ultimate strain of 0.23 in the zigzag direction at the low temperature of 1 K.

Figure 5.

Phonon spectrum for single-layer 1H-ScS₂. (a) Phonon dispersion along the ΓM direction in the Brillouin zone. The results from the VFF model (lines) are comparable with the ab initio results (pentagons) from [12]. (b) The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

4. 1H-SCS₂

Most existing theoretical studies on the single-layer 1H-ScS₂ are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1H-ScS₂.

The structure for the single-layer 1H-ScS₂ is shown in Figure 1 (with M = Sc and X = S). Each Sc atom is surrounded by six S atoms. These S atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each S atom is connected to three Sc atoms. The structural parameters are from the first-principles calculations [12], including the lattice constant a = 3.70 Å and the bond length $d_{\text{Sc}-\text{S}}=2.52$ Å. The resultant angles are ${\theta }_{\text{ScSS}}={\theta }_{\text{SScSc}}{=94.467}^{\circ }$ and ${\theta }_{\text{ScS}{\text{S}}^{\prime }}={64.076}^{\circ },$ in which atoms S and S′ are from different (top or bottom) groups.

Figure 6.

Stress-strain for single-layer 1H-ScS₂ of dimension 100 × 100 Å along the armchair and zigzag directions.

	K (eV)	θ 0 (°)	ρ 1 (Å)	ρ 2 (Å)	r min 12 (Å)	r max 12 (Å)	r min 13 (Å)	r max 13 (Å)	r min 23 (Å)	r max 23 (Å)
${\theta }_{\text{Sc}-\text{S}-\text{S}}$	22.768	94.467	1.519	1.519	0.0	3.498	0.0	3.498	0.0	4.132
${\theta }_{\text{Sc}-\text{S}-{\text{S}}^{\prime }}$	27.977	64.076	1.519	1.519	0.0	3.498	0.0	3.498	0.0	4.132
${\theta }_{\text{S}-\text{Sc}-\text{Sc}}$	22.768	94.467	1.519	1.519	0.0	3.498	0.0	3.498	0.0	4.132

Table 16.

Three-body SW potential parameters for single-layer 1H-ScS₂ used by GULP [8] as expressed in Eq. (4).

The angle θ _ijk in the first line indicates the bending energy for the angle with atom i as the apex.

Table 14 shows four VFF terms for the single-layer 1H-ScS₂; one of which is the bond stretching interaction shown by Eq. (1), while the other three terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the ΓM as shown in Figure 5(a) . The ab initio calculations for the phonon dispersion are from [12]. Figure 5(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 15 . The parameters for the three-body SW potential used by GULP are shown in Table 16 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 17 . We note that 12 atom types have been introduced for the simulation of the single-layer 1H-ScS₂ using LAMMPS, because the angles around atom Sc in Figure 1 (with M = Sc and X = S) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS [13, 14]. According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Figure 2 (with M = Sc and X = S) shows that, for 1H-ScS₂, we can differentiate these angles around the Sc atom by assigning these six neighboring S atoms with different atom types. It can be found that twelve atom types are necessary for the purpose of differentiating all six neighbors around one Sc atom.

	ϵ (eV)	σ (Å)	a	λ	γ	cos θ 0	A L	B L	p	q	Tol
Sc1─S1─S1	1.000	1.519	2.303	0.000	1.000	0.000	5.505	3.784	4	0	0.0
Sc1─S1─S3	1.000	0.000	0.000	22.768	1.000	−0.078	0.000	0.000	4	0	0.0
Sc1─S1─S2	1.000	0.000	0.000	27.977	1.000	0.437	0.000	0.000	4	0	0.0
S1─Sc1─Sc3	1.000	0.000	0.000	22.768	1.000	−0.078	0.000	0.000	4	0	0.0

Table 17.

SW potential parameters for single-layer 1H-ScS₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-ScS₂ under uniaxial tension at 1 and 300 K. Figure 6 shows the stress-strain curve for the tension of a single-layer 1H-ScS₂ of dimension 100 × 100 Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-ScS₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-ScS₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 43.8 and 43.2 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}={\nu }_{yx}=0.30$ .

There is no available value for nonlinear quantities in the single-layer 1H-ScS₂. We have thus used the nonlinear parameter B = 0.5d ⁴ in Eq. (5), which is close to the value of B in most materials. The value of the third-order nonlinear elasticity D can be extracted by fitting the stress-strain relation to the function $\sigma =Eϵ+\frac{1}{2}D{ϵ}^2$ with E as the Young’s modulus. The values of D from the present SW potential are −146.9 and −159.1 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 6.1 N/m at the ultimate strain of 0.25 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 5.9 N/m at the ultimate strain of 0.32 in the zigzag direction at the low temperature of 1 K.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{Sc}-\text{Se}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{Sc}-\text{Se}-\text{Se}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{Sc}-\text{Se}-\text{S}{\text{e}}^{\prime }}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{Se}-\text{Sc}-\text{Sc}}{\left(\text{Δ}\theta \right)}^2$
Parameter	5.192	2.027	2.027	2.027
$r_0$ or ${\theta }_0$	2.650	92.859	66.432	92.859

Table 18.

The VFF model for single-layer 1H-ScSe₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle θ _ijk has atom i as the apex.

	A (eV)	ρ (Å)	B (Å^4)	r min (Å)	r max(Å)
Sc-Se	5.853	1.533	24.658	0.0	3.658

Table 19.

Two-body SW potential parameters for single-layer 1H-ScSe₂ used by GULP [8] as expressed in Eq. (3).

5. 1H-SCSE₂

Most existing theoretical studies on the single-layer 1H-ScSe₂ are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1H-ScSe₂.

The structure for the single-layer 1H-ScSe₂ is shown in Figure 1 (with M = Sc and X = Se). Each Sc atom is surrounded by six Se atoms. These Se atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each Se atom is connected to three Sc atoms. The structural parameters are from the first-principles calculations [12], including the lattice constant a = 3.38 Å and the bond length $d_{\text{S}c-Se}=2.65$ Å. The resultant angles are ${\theta }_{\text{ScSeSe}}={\theta }_{\text{SeScSc}}{=92.859}^{\circ }$ and ${\theta }_{\text{ScSeS}{\text{e}}^{\prime }}{=66.432}^{\circ },$ in which atoms Se and Se′ are from different (top or bottom) groups.

Figure 7.

Phonon spectrum for single-layer 1H-ScSe₂. (a) Phonon dispersion along the ΓM direction in the Brillouin zone. The results from the VFF model (lines) are comparable with the ab initio results (pentagons) from [12]. (b) The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

Table 18 shows four VFF terms for the single-layer 1H-ScSe₂; one of which is the bond stretching interaction shown by Eq. (1), while the other three terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the ΓM as shown in Figure 7(a) . The ab initio calculations for the phonon dispersion are from [12]. Figure 7(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

Figure 8.

Stress-strain for single-layer 1H-ScSe₂ of dimension 100 × 100 Å along the armchair and zigzag directions.

	K (eV)	θ 0 (°)	ρ 1 (Å)	ρ 2 (Å)	r min 12 (Å)	r max 12 (Å)	r min 13 (Å)	r max 13 (Å)	r min 23 (Å)	r max 23 (Å)
${\theta }_{\text{Sc}-\text{Se}-\text{Se}}$	21.292	92.859	1.533	1.533	0.0	3.658	0.0	3.658	0.0	4.327
${\theta }_{\text{Sc}-\text{Se}-\text{S}{\text{e}}^{\prime }}$	25.280	66.432	1.533	1.533	0.0	3.658	0.0	3.658	0.0	4.327
${\theta }_{\text{Se}-\text{Sc}-\text{Sc}}$	21.292	92.859	1.533	1.533	0.0	3.658	0.0	3.658	0.0	4.327

Table 20.

Three-body SW potential parameters for single-layer 1H-ScSe₂ used by GULP [8] as expressed in Eq. (4).

The angle θ _ijk in the first line indicates the bending energy for the angle with atom i as the apex.

The parameters for the two-body SW potential used by GULP are shown in Table 19 . The parameters for the three-body SW potential used by GULP are shown in Table 20 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 21 . We note that 12 atom types have been introduced for the simulation of the single-layer 1H-ScSe₂ using LAMMPS, because the angles around atom Sc in Figure 1 (with M = Sc and X = Se) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS [13, 14]. According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Figure 2 (with M = Sc and X = Se) shows that, for 1H-ScSe₂, we can differentiate these angles around the Sc atom by assigning these six neighboring Se atoms with different atom types. It can be found that twelve atom types are necessary for the purpose of differentiating all six neighbors around one Sc atom.

	ϵ (eV)	σ (Å)	a	λ	γ	cos θ 0	A L	B L	p	q	Tol
Sc1─Se1-Se1	1.000	1.533	2.386	0.000	1.000	0.000	5.853	4.464	4	0	0.0
Sc1─Se1─Se3	1.000	0.000	0.000	21.292	1.000	−0.050	0.000	0.000	4	0	0.0
Sc1─Se1─Se2	1.000	0.000	0.000	25.280	1.000	0.400	0.000	0.000	4	0	0.0
Se1─Sc1─Sc3	1.000	0.000	0.000	21.292	1.000	−0.050	0.000	0.000	4	0	0.0

Table 21.

SW potential parameters for single-layer 1H-ScSe₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-ScSe₂ under uniaxial tension at 1 and 300 K. Figure 8 shows the stress-strain curve for the tension of a single-layer 1H-ScSe₂ of dimension 100 × 100 Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-ScSe₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-ScSe₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 39.4 and 39.9 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}={\nu }_{yx}=0.32$ .

There is no available value for nonlinear quantities in the single-layer 1H-ScSe₂. We have thus used the nonlinear parameter B = 0.5d ⁴ in Eq. (5), which is close to the value of B in most materials. The value of the third-order nonlinear elasticity D can be extracted by fitting the stress-strain relation to the function $\sigma =Eϵ+\frac{1}{2}D{ϵ}^2$ with E as the Young’s modulus. The values of D from the present SW potential are −115.7 and −135.7 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 6.1 N/m at the ultimate strain of 0.27 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 5.9 N/m at the ultimate strain of 0.35 in the zigzag direction at the low temperature of 1 K.

Figure 9.

Phonon spectrum for single-layer 1H-ScTe₂. (a) Phonon dispersion along the ΓM direction in the Brillouin zone. The results from the VFF model (lines) are comparable with the ab initio results (pentagons) from [12]. (b) The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

6. 1H-SCTE₂

Most existing theoretical studies on the single-layer 1H-ScTe₂ are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1H-ScTe₂.

The structure for the single-layer 1H-ScTe₂ is shown in Figure 1 (with M = Sc and X = Te). Each Sc atom is surrounded by six Te atoms. These Te atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each Te atom is connected to three Sc atoms. The structural parameters are from the first-principles calculations [12], including the lattice constant a = 3.62 Å and the bond length $d_{\text{Sc}-\text{Te}}=2.89$ Å. The resultant angles are ${\theta }_{\text{ScTeTe}}={\theta }_{\text{TeScSc}}{=77.555}^{\circ }$ and ${\theta }_{\text{ScTeT}{\text{e}}^{\prime }}{=87.364}^{\circ }$ , in which atoms Te and Te′ are from different (top or bottom) groups.

Figure 10.

Stress-strain for single-layer 1H-ScTe₂ of dimension 100 × 100 Å along the armchair and zigzag directions.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{Sc}-\text{Te}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{Sc}-\text{Te}-\text{Te}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{Sc}-\text{Te}-\text{T}{\text{e}}^{\prime }}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{Te}-\text{Sc}-\text{Sc}}{\left(\text{Δ}\theta \right)}^2$
Parameter	5.192	2.027	2.027	2.027
$r_0$ or ${\theta }_0$	2.890	77.555	87.364	87.364

Table 22.

The VFF model for single-layer 1H-ScTe₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle θ _ijk has atom i as the apex.

Table 22 shows four VFF terms for the single-layer 1H-ScTe₂; one of which is the bond stretching interaction shown by Eq. (1), while the other three terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the ΓM as shown in Figure 9(a) . The ab initio calculations for the phonon dispersion are from [12]. There is only one (longitudinal) acoustic branch available. We find that the VFF parameters can be chosen to be the same as that of the 1H-ScSe₂, from which the longitudinal acoustic branch agrees with the ab initio results as shown in Figure 9(a) . It has also been shown that the VFF parameters can be the same for TaSe₂ and NbSe₂ of similar structure [15]. Figure 9(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

	$A\left(\text{eV}\right)$	$\rho \left(\mathbf{\AA }\right)$	$B({\mathbf{\AA }}^4)$	$r_{\text{min}}(\mathrm{\AA })$	$r_{\text{max}}\left(\mathbf{\AA }\right)$
Sc-Te	4.630	1.050	34.879	0.0	3.761

Table 23.

Two-body SW potential parameters for single-layer 1H-ScTe₂ used by GULP [8] as expressed in Eq. (3).

	$K$ (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{Sc}-\text{Te}-\text{Te}}$	11.848	77.555	1.050	1.050	0.0	3.761	0.0	3.761	0.0	4.504
${\theta }_{\text{Sc}-\text{Te}-\text{T}{\text{e}}^{\prime }}$	11.322	87.364	1.050	1.050	0.0	3.761	0.0	3.761	0.0	4.504
${\theta }_{\text{Te}-\text{Sc}-\text{Sc}}$	11.848	77.555	1.050	1.050	0.0	3.761	0.0	3.761	0.0	4.504

Table 24.

Three-body SW potential parameters for single-layer 1H-ScTe₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

	$\mathit{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Sc1─Te1─Te1	1.000	1.050	3.581	0.000	1.000	0.000	4.630	28.679	4	0	0.0
Sc1─Te1─Te3	1.000	0.000	0.000	11.848	1.000	0.216	0.000	0.000	4	0	0.0
Sc1─Te1─Te2	1.000	0.000	0.000	11.322	1.000	0.046	0.000	0.000	4	0	0.0
Te1─Sc1─Sc3	1.000	0.000	0.000	11.848	1.000	0.216	0.000	0.000	4	0	0.0

Table 25.

SW potential parameters for single-layer 1H-ScTe₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

The parameters for the two-body SW potential used by GULP are shown in Table 23 . The parameters for the three-body SW potential used by GULP are shown in Table 24 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 25 . We note that 12 atom types have been introduced for the simulation of the single-layer 1H-ScTe₂ using LAMMPS, because the angles around atom Sc in Figure 1 (with M = Sc and X = Te) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS [13, 14]. According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Figure 2 (with M = Sc and X = Te) shows that, for 1H-ScTe₂, we can differentiate these angles around the Sc atom by assigning these six neighboring Te atoms with different atom types. It can be found that twelve atom types are necessary for the purpose of differentiating all six neighbors around one Sc atom.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-ScTe₂ under uniaxial tension at 1 and 300 K. Figure 10 shows the stress-strain curve for the tension of a single-layer 1H-ScTe₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-ScTe₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-ScTe₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 29.3 and 28.8 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}={\nu }_{yx}=0.38$ .

There is no available value for nonlinear quantities in the single-layer 1H-ScTe₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −43.2 and −59.3 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 6.7 N/m at the ultimate strain of 0.33 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 6.7 N/m at the ultimate strain of 0.45 in the zigzag direction at the low temperature of 1 K.

Figure 11.

Phonon dispersion for single-layer 1H-TiTe₂. (a) The VFF model is fitted to the three acoustic branches in the long wave limit along the $\Gamma \text{M}$ direction. The ab initio results (gray pentagons) are from Ref. [12]. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for single-layer 1H-TiTe₂ along $\text{Γ}\text{MK}\text{Γ}$ .

7. 1H-TITE₂

Most existing theoretical studies on the single-layer 1H-TiTe₂ are based on the first-principles calculations. In this section, we will develop both VFF model and the SW potential for the single-layer 1H-TiTe₂.

The structure for the single-layer 1H-TiTe₂ is shown in Figure 1 (with M = Ti and X = Se). Each Ti atom is surrounded by six Te atoms. These Te atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each Te atom is connected to three Ti atoms. The structural parameters are from [12], including the lattice constant $a=3.62$ Å and the bond length $d_{\text{Ti}-\text{Te}}=2.75$ Å. The resultant angles are ${\theta }_{\text{TiTeTe}}={\theta }_{\text{TeTiTi}}=82.323°$ and ${\theta }_{\text{TiTeT}{\text{e}}^{\prime }}=81.071°$ , in which atoms Te and Te′ are from different (top or bottom) groups.

Figure 12.

Stress-strain for single-layer 1H-TiTe₂ of dimension $100\times 100$ Å along the armchair and zigzag directions.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{Ti}-\text{Te}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{Ti}-\text{Te}-\text{Te}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{Ti}-\text{Te}-\text{T}{\text{e}}^{\prime }}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{Te}-\text{Ti}-\text{Ti}}{\left(\text{Δ}\theta \right)}^2$
Parameter	4.782	3.216	3.216	3.216
$r_0$ or ${\theta }_0$	2.750	82.323	81.071	82.323

Table 26.

The VFF model for single-layer 1H-TiTe₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of ${\text{eV/}\mathrm{\AA }}^{\text{2}}$ for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of $\mathrm{\AA }$ ) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

Table 26 shows the VFF terms for the 1H-TiTe₂; one of which is the bond stretching interaction shown by Eq. (1), while the other terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the $\Gamma \text{M}$ as shown in Figure 11(a) . The ab initio calculations for the phonon dispersion are from [12]. Figure 11(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

	$A\left(\text{eV}\right)$	$\rho \left(\mathbf{\AA }\right)$	$B({\mathbf{\AA }}^4)$	$r_{\text{min}}(\mathbf{\AA })$	$r_{\text{max}}\left(\mathbf{\AA }\right)$
Ti-Te	4.414	1.173	28.596	0.0	3.648

Table 27.

Two-body SW potential parameters for single-layer 1H-TiTe₂ used by GULP [8] as expressed in Eq. (3).

	$K$ (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{Ti}-\text{Te}-\text{Te}}$	22.321	82.323	1.173	1.173	0.0	3.648	0.0	3.648	0.0	4.354
${\theta }_{\text{Ti}-\text{Te}-\text{T}{\text{e}}^{\prime }}$	22.463	81.071	1.173	1.173	0.0	3.648	0.0	3.648	0.0	4.354
${\theta }_{\text{Te}-\text{Ti}-\text{Ti}}$	11.321	82.323	1.173	1.173	0.0	3.648	0.0	3.648	0.0	4.354

Table 28.

Three-body SW potential parameters for single-layer 1H-TiTe₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

The parameters for the two-body SW potential used by GULP are shown in Table 27 . The parameters for the three-body SW potential used by GULP are shown in Table 28 . Parameters for the SW potential used by LAMMPS are listed in Table 29 . We note that 12 atom types have been introduced for the simulation of the single-layer 1H-TiTe₂ using LAMMPS, because the angles around atom Ti in Figure 1 (with M = Ti and X = Te) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS [13, 14]. According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Figure 2 (with M = Ti and X = Te) shows that, for 1H-TiTe₂, we can differentiate these angles around the Ti atom by assigning these six neighboring Te atoms with different atom types. It can be found that twelve atom types are necessary for the purpose of differentiating all six neighbors around one Ti atom.

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	p	q	Tol
Ti1-Te1-Te1	1.000	1.173	3.110	0.000	1.000	0.000	4.414	15.100	4	0	0.0
Ti1-Te1-Te3	1.000	0.000	0.000	22.321	1.000	0.134	0.000	0.000	4	0	0.0
Ti1-Te1-Te2	1.000	0.000	0.000	22.463	1.000	0.155	0.000	0.000	4	0	0.0
Te1-Ti1-Ti3	1.000	0.000	0.000	22.321	1.000	0.134	0.000	0.000	4	0	0.0

Table 29.

SW potential parameters for single-layer 1H-TiTe₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Atom types in the first column are displayed in Figure 2 (with M = Ti and X = Te).

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-TiTe₂ under uniaxial tension at 1 and 300 K. Figure 12 shows the stress-strain curve for the tension of a single-layer 1H-TiTe₂ of dimension 100 × 100 Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-TiTe₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-TiTe₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 47.9 and 47.1 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}={\nu }_{yx}=0.29$ .

There is no available value for the nonlinear quantities in the single-layer 1H-TiTe₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of B in most materials. The value of the third-order nonlinear elasticity D can be extracted by fitting the stress-strain relation to the function $\sigma =Eϵ+\frac{1}{2}D{ϵ}^2$ with E as the Young’s modulus. The values of D from the present SW potential are −158.6 and −176.3 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 6.6 N/m at the ultimate strain of 0.24 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 6.3 N/m at the ultimate strain of 0.28 in the zigzag direction at the low temperature of 1 K.

Figure 13.

Phonon spectrum for single-layer 1H-VO₂. (a) Phonon dispersion along the ΓM direction in the Brillouin zone. The results from the VFF model (lines) are comparable with the ab initio results (pentagons) from [12]. (b) The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

8. 1H-VO₂

Most existing theoretical studies on the single-layer 1H-VO₂ are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1H-VO₂.

The structure for the single-layer 1H-VO₂ is shown in Figure 1 (with M = V and X = O). Each V atom is surrounded by six O atoms. These O atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each O atom is connected to three V atoms. The structural parameters are from the first-principles calculations [12], including the lattice constant $a=2.70$ Å and the bond length $d_{\text{V}-\text{O}}=1.92$ Å. The resultant angles are ${\theta }_{\text{VOO}}={\theta }_{\text{OVV}}=89.356°$ and ${\theta }_{\text{VO}{\text{O}}^{\prime }}=71.436°$ , in which atoms O and O′ are from different (top or bottom) groups.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{V}-\text{O}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{V}-\text{O}-\text{O}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{V}-\text{O}-{\text{O}}^{\prime }}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{O}-\text{V}-\text{V}}{\left(\text{Δ}\theta \right)}^2$
Parameter	9.417	4.825	4.825	4.825
r 0 or θ 0	1.920	89.356	71.436	89.356

Table 30.

The VFF model for single-layer 1H-VO₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of ${\text{eV/}\mathrm{\AA }}^{\text{2}}$ for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle θ _ijk has atom i as the apex.

	$A\left(\text{eV}\right)$	$\rho \left(\mathbf{\AA }\right)$	$B({\mathbf{\AA }}^4)$	$r_{\text{min}}(\mathrm{\AA })$	$r_{\text{max}}\left(\mathbf{\AA }\right)$
V-O	5.105	1.011	6.795	0.0	2.617

Table 31.

Two-body SW potential parameters for single-layer 1H-VO₂ used by GULP [8] as expressed in Eq. (3).

	$K$ (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{V}-\text{O}-\text{O}}$	43.951	89.356	1.011	1.011	0.0	2.617	0.0	2.617	0.0	3.105
${\theta }_{\text{V}-\text{O}-{\text{O}}^{\prime }}$	48.902	71.436	1.011	1.011	0.0	2.617	0.0	2.617	0.0	3.105
${\theta }_{\text{O}-\text{V}-\text{V}}$	43.951	89.356	1.011	1.011	0.0	2.617	0.0	2.617	0.0	3.105

Table 32.

Three-body SW potential parameters for single-layer 1H-VO₂ used by GULP [8] as expressed in Eq. (4).

The angle θ _ijk in the first line indicates the bending energy for the angle with atom i as the apex.

	ϵ (eV)	σ (Å)	a	λ	γ	cos θ 0	A L	B L	p	q	Tol
V1-O1-O1	1.000	1.011	2.589	0.000	1.000	0.000	5.105	6.509	4	0	0.0
V1-O1-O3	1.000	0.000	0.000	43.951	1.000	0.011	0.000	0.000	4	0	0.0
V1-O1-O2	1.000	0.000	0.000	48.902	1.000	0.318	0.000	0.000	4	0	0.0
O1-V1-V3	1.000	0.000	0.000	43.951	1.000	0.011	0.000	0.000	4	0	0.0

Table 33.

SW potential parameters for single-layer 1H-VO₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Figure 14.

Stress-strain for single-layer 1H-VO₂ of dimension 100 × 100 Å along the armchair and zigzag directions.

Table 30 shows four VFF terms for the single-layer 1H-VO₂; one of which is the bond stretching interaction shown by Eq. (1), while the other three terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the ΓM as shown in Figure 13(a) . The ab initio calculations for the phonon dispersion are from [12]. Figure 13(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 31 . The parameters for the three-body SW potential used by GULP are shown in Table 32 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 33 . We note that 12 atom types have been introduced for the simulation of the single-layer 1H-VO₂ using LAMMPS, because the angles around atom V in Figure 1 (with M = V and X = O) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS [13, 14]. According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Figure 2 (with M = V and X = O) shows that, for 1H-VO₂, we can differentiate these angles around the V atom by assigning these six neighboring O atoms with different atom types. It can be found that twelve atom types are necessary for the purpose of differentiating all six neighbors around one V atom.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-VO₂ under uniaxial tension at 1 and 300 K. Figure 14 shows the stress-strain curve for the tension of a single-layer 1H-VO₂ of dimension 100 × 100 Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-VO₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-VO₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 133.0 and 132.9 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}={\nu }_{yx}=0.17$ .

There is no available value for nonlinear quantities in the single-layer 1H-VO₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of B in most materials. The value of the third-order nonlinear elasticity D can be extracted by fitting the stress-strain relation to the function $\sigma =Eϵ+\frac{1}{2}D{ϵ}^2$ with E as the Young’s modulus. The values of D from the present SW potential are −652.3 and −705.8 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 13.3 N/m at the ultimate strain of 0.19 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 12.7 N/m at the ultimate strain of 0.22 in the zigzag direction at the low temperature of 1 K.

9. 1H-VS₂

Most existing theoretical studies on the single-layer 1H-VS₂ are based on the first-principles calculations. In this section, we will develop both VFF model and the SW potential for the single-layer 1H-VS₂.

The structure for the single-layer 1H-VS₂ is shown in Figure 1 (with M = V and X = S). Each V atom is surrounded by six S atoms. These S atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each S atom is connected to three V atoms. The structural parameters are from [12], including the lattice constant $a=3.09$ Å and the bond length $d_{\text{V}-\text{S}}=2.31$ Å. The resultant angles are ${\theta }_{\text{VSS}}={\theta }_{\text{SVV}}=83.954°$ and ${\theta }_{\text{VS}{\text{S}}^{\prime }}=78.878°$ , in which atoms S and S′ are from different (top or bottom) groups.

Figure 15.

Phonon dispersion for single-layer 1H-VS₂. (a) The VFF model is fitted to the three acoustic branches in the long wave limit along the ΓM direction. The ab initio results (gray pentagons) are from [16]. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for single-layer 1H-VS₂ along ΓMKΓ.

Table 34 shows the VFF terms for the 1H-VS₂; one of which is the bond stretching interaction shown by Eq. (1), while the other terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the ΓM as shown in Figure 15(a) . The ab initio calculations for the phonon dispersion are from [16]. The phonon dispersion can also be found in other ab initio calculations [12]. Figure 15(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

Figure 16.

Stress-strain for single-layer 1H-VS₂ of dimension 100 × 100 Å along the armchair and zigzag directions.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{V}-\text{S}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{V}-\text{S}-\text{S}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{V}-\text{S}-{\text{S}}^{\prime }}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{S}-\text{V}-\text{V}}{\left(\text{Δ}\theta \right)}^2$
Parameter	8.392	4.074	4.074	4.074
r 0 or θ 0	2.310	83.954	78.878	83.954

Table 34.

The VFF model for single-layer 1H-VS₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle θ _ijk has atom i as the apex.

The parameters for the two-body SW potential used by GULP are shown in Table 35 . The parameters for the three-body SW potential used by GULP are shown in Table 36 . Parameters for the SW potential used by LAMMPS are listed in Table 37 . We note that 12 atom types have been introduced for the simulation of the single-layer 1H-VS₂ using LAMMPS, because the angles around atom V in Figure 1 (with M = V and X = S) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS [13, 14]. According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Figure 2 (with M = V and X = S) shows that, for 1H-VS₂, we can differentiate these angles around the V atom by assigning these six neighboring S atoms with different atom types. It can be found that twelve atom types are necessary for the purpose of differentiating all six neighbors around one V atom.

	A (eV)	ρ (Å)	B (Å^4)	r min (Å)	r max(Å)
V─S	5.714	1.037	14.237	0.0	3.084

Table 35.

Two-body SW potential parameters for single-layer 1H-VS₂ used by GULP [8] as expressed in Eq. (3).

	K (eV)	θ 0 (°)	ρ 1 (Å)	ρ 2 (Å)	r min 12 (Å)	r max 12 (Å)	r min 13 (Å)	r max 13 (Å)	r min 23 (Å)	r max 23 (Å)
${\theta }_{\text{V}-\text{S}-\text{S}}$	30.059	83.954	1.037	1.037	0.0	3.084	0.0	3.084	0.0	3.676
${\theta }_{\text{V}-\text{S}-{\text{S}}^{\prime }}$	30.874	78.878	1.037	1.037	0.0	3.084	0.0	3.084	0.0	3.676
${\theta }_{\text{S}-\text{V}-\text{V}}$	30.059	83.954	1.037	1.037	0.0	3.084	0.0	3.084	0.0	3.676

Table 36.

Three-body SW potential parameters for single-layer 1H-VS₂ used by GULP [8] as expressed in Eq. (4).

The angle θ _ijk in the first line indicates the bending energy for the angle with atom i as the apex.

	ϵ (eV)	σ (Å)	a	λ	γ	cos θ 0	A L	B L	p	q	Tol
V1─S1─S1	1.000	1.037	2.973	0.000	1.000	0.000	5.714	12.294	4	0	0.0
V1─S1─S3	1.000	0.000	0.000	30.059	1.000	0.105	0.000	0.000	4	0	0.0
V1─S1─S2	1.000	0.000	0.000	30.874	1.000	0.193	0.000	0.000	4	0	0.0
S1─V1─V3	1.000	0.000	0.000	30.059	1.000	0.105	0.000	0.000	4	0	0.0

Table 37.

SW potential parameters for single-layer 1H-VS₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Atom types in the first column are displayed in Figure 2 (with M = V and X = S).

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-VS₂ under uniaxial tension at 1 and 300 K. Figure 16 shows the stress-strain curve for the tension of a single-layer 1H-VS₂ of dimension 100 × 100 Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-VS₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-VS₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 86.5 and 85.3 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}={\nu }_{yx}=0.28$ .

There is no available value for the nonlinear quantities in the single-layer 1H-VS₂. We have thus used the nonlinear parameter B = 0.5d ⁴ in Eq. (5), which is close to the value of B in most materials. The value of the third-order nonlinear elasticity D can be extracted by fitting the stress-strain relation to the function $\sigma =Eϵ+\frac{1}{2}D{ϵ}^2$ with E as the Young’s modulus. The values of D from the present SW potential are −302.0 and −334.7 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 11.5 N/m at the ultimate strain of 0.23 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 10.9 N/m at the ultimate strain of 0.27 in the zigzag direction at the low temperature of 1 K.

10. 1H-VSe₂

Most existing theoretical studies on the single-layer 1H-VSe₂ are based on the first-principles calculations. In this section, we will develop both VFF model and the SW potential for the single-layer 1H-VSe₂.

The structure for the single-layer 1H-VSe₂ is shown in Figure 1 (with M = V and X = Se). Each V atom is surrounded by six Se atoms. These Se atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each Se atom is connected to three V atoms. The structural parameters are from [12], including the lattice constant a = 3.24 Å and the bond length $d_{\text{V}─\text{Se}}=2.45$ Å. The resultant angles are ${\theta }_{\text{VSeSe}}=\mathrm{}{\theta }_{\text{SeVV}}=82.787°$ and ${\theta }_{\text{VSeS}{\text{e}}^{\prime }}=80.450°$ , in which atoms Se and Se′ are from different (top or bottom) groups.

Figure 17.

Phonon dispersion for single-layer 1H-VSe₂. (a) The VFF model is fitted to the three acoustic branches in the long wave limit along the ΓM direction. The ab initio results (gray pentagons) are from [12]. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for single-layer 1H-VSe₂ along ΓMKT.

Table 38 shows the VFF terms for the 1H-VSe₂; one of which is the bond stretching interaction shown by Eq. (1), while the other terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the ΓM as shown in Figure 17(a) . The ab initio calculations for the phonon dispersion are from [12]. Figure 17(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 39 . The parameters for the three-body SW potential used by GULP are shown in Table 40 . Parameters for the SW potential used by LAMMPS are listed in Table 41 . We note that 12 atom types have been introduced for the simulation of the single-layer 1H-VSe₂ using LAMMPS, because the angles around atom V in Figure 1 (with M = V and X = Se) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS [13, 14]. According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Figure 2 (with M = V and X = Se) shows that, for 1H-Vse₂, we can differentiate these angles around the V atom by assigning these six neighboring Se atoms with different atom types. It can be found that 12 atom types are necessary for the purpose of differentiating all 6 neighbors around 1 V atom.

Figure 18.

Stress-strain for single-layer 1H-VSe₂ of dimension 100×100 Å along the armchair and zigzag directions.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{V}─\text{Se}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\text{V}─\text{Se}─\text{Se}}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\text{V}─\text{Se}─\text{S}{\text{e}}^{\prime }}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\text{Se}─\text{V}─\text{V}}{\left(\Delta \theta \right)}^2$
Parameter	6.492	4.716	4.716	4.716
r 0 or θ 0	2.450	82.787	80.450	82.787

Table 38.

The VFF model for single-layer 1H-VSe₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle θ _ijk has atom i as the apex.

	A(eV)	ρ(Å)	B(Å^4)	r min(Å)	r max(Å)
V─Se	4.817	1.061	18.015	0.0	3.256

Table 39.

Two-body SW potential parameters for single-layer 1H-VSe₂ used by GULP [8] as expressed in Eq. (3).

	K (eV)	θ 0 (°)	ρ 1 (Å)	ρ 2 (Å)	r min12 (Å)	r max12 (Å)	r min13 (Å)	r max13 (Å)	r min23 (Å)	r max23 (Å)
${\theta }_{\text{V}─\text{Se}─\text{Se}}$	33.299	82.787	1.061	1.061	0.0	3.256	0.0	3.256	0.0	3.884
${\theta }_{\text{V}─\text{Se}─\text{S}{\text{e}}^{\prime }}$	33.702	80.450	1.061	1.061	0.0	3.256	0.0	3.256	0.0	3.884
${\theta }_{\text{Se}─\text{V}─\text{V}}$	33.299	82.787	1.061	1.061	0.0	3.256	0.0	3.256	0.0	3.884

Table 40.

Three-body SW potential parameters for single-layer 1H-VSe₂ used by GULP [8] as expressed in Eq. (4).

The angle θ _ijk in the first line indicates the bending energy for the angle with atom i as the apex.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-VSe₂ under uniaxial tension at 1 and 300 K. Figure 18 shows the stress-strain curve for the tension of a single-layer 1H-VSe₂ of dimension 100×100 Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-VSe₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-VSe₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 81.7 and 80.6 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.23$ .

	ϵ (eV)	σ (Å)	α	λ	γ	cos θ 0	A L	B L	p	q	Tol
V1─Se1─Se1	1.000	1.061	3.070	0.000	1.000	0.000	4.817	14.236	4	0	0.0
V1─Se1─Se3	1.000	0.000	0.000	33.299	1.000	0.126	0.000	0.000	4	0	0.0
V1─Se1─Se2	1.000	0.000	0.000	33.702	1.000	0.166	0.000	0.000	4	0	0.0
Se1─V1─V3	1.000	0.000	0.000	33.299	1.000	0.126	0.000	0.000	4	0	0.0

Table 41.

SW potential parameters for single-layer 1H-VSe₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Atom types in the first column are displayed in Figure 2 (with M = V and X = Se).

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{V}─\text{Te}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\text{V}─\text{Te}─\text{Te}}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\text{V}─\text{Te}─\text{T}{\text{e}}^{\prime }}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\text{Te}─\text{V}─\text{V}}{\left(\Delta \theta \right)}^2$
Parameter	6.371	4.384	4.384	4.384
r 0 or θ 0	2.660	81.708	81.891	81.708

Table 42.

The VFF model for single-layer 1H-VTe₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle θ _ijk has atom i as the apex.

There is no available value for the nonlinear quantities in the single-layer 1H-VSe₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of B in most materials. The value of the third-order nonlinear elasticity D can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with E as the Young’s modulus. The values of D from the present SW potential are −335.2 and −363.3 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 9.5 N/m at the ultimate strain of 0.21 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 9.0 N/m at the ultimate strain of 0.24 in the zigzag direction at the low temperature of 1 K.

11. 1H-VTe₂

Most existing theoretical studies on the single-layer 1H-VTe₂ are based on the first-principles calculations. In this section, we will develop both VFF model and the SW potential for the single-layer 1H-VTe₂.

Figure 19.

Phonon dispersion for single-layer 1H-VTe₂. (a) The VFF model is fitted to the three acoustic branches in the long wave limit along the ΓM direction. The ab initio results (gray pentagons) are from [12]. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for single-layer 1H-VTe₂ along ΓMKΓ.

	$A\left(\text{eV}\right)$	$\rho \left(\mathbf{\AA }\right)$	$B({\mathbf{\AA }}^4)$	$r_{\text{min}}(\mathbf{\AA })$	$r_{\text{max}}\left(\mathbf{\AA }\right)$
V─Te	5.410	1.112	25.032	0.0	3.520

Table 43.

Two-body SW potential parameters for single-layer 1H-VTe₂ used by GULP [8] as expressed in Eq. (3).

The structure for the single-layer 1H-VTe₂ is shown in Figure 1 (with M = V and X = Te). Each V atom is surrounded by six Te atoms. These Te atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each Te atom is connected to three V atoms. The structural parameters are from [12], including the lattice constant $a=3.48$ Å and the bond length $d_{\text{V}─\text{Te}}=2.66$ Å. The resultant angles are ${\theta }_{\text{VTeTe}}=\mathrm{}{\theta }_{\text{TeVV}}=81.708°$ and ${\theta }_{\text{VTeT}{\text{e}}^{\prime }}=81.891°$ , in which atoms Te and Te’ are from different (top or bottom) groups.

Figure 20.

Stress-strain for single-layer 1H-VTe₂ of dimension 100×100 Å along the armchair and zigzag directions.

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{V}─\text{Te}─\text{Te}}$	29.743	81.708	1.112	1.112	0.0	3.520	0.0	3.520	0.0	4.203
${\theta }_{\text{V}─\text{Te}─\text{T}{\text{e}}^{\prime }}$	29.716	81.891	1.112	1.112	0.0	3.520	0.0	3.520	0.0	4.203
${\theta }_{\text{Te}─\text{V}─\text{V}}$	29.743	81.708	1.112	1.112	0.0	3.520	0.0	3.520	0.0	4.203

Table 44.

Three-body SW potential parameters for single-layer 1H-VTe₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

Table 42 shows the VFF terms for the 1H-VTe₂; one of which is the bond stretching interaction shown by Eq. (1), while the other terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the ΓM as shown in Figure 19(a) . The ab initio calculations for the phonon dispersion are from [12]. Figure 19(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 43 . The parameters for the three-body SW potential used by GULP are shown in Table 44 . Parameters for the SW potential used by LAMMPS are listed in Table 45 . We note that 12 atom types have been introduced for the simulation of the single-layer 1H-VTe₂ using LAMMPS, because the angles around atom V in Figure 1 (with M = V and X = Te) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS [13, 14]. According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Figure 2 (with M = V and X = Te) shows that, for 1H-VTe₂, we can differentiate these angles around the V atom by assigning these six neighboring Te atoms with different atom types. It can be found that 12 atom types are necessary for the purpose of differentiating all 6 neighbors around 1 V atom.

	$\mathit{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
V1─Te1─Te1	1.000	1.112	3.164	0.000	1.000	0.000	5.410	16.345	4	0	0.0
V1─Te1─Te3	1.000	0.000	0.000	29.743	1.000	0.144	0.000	0.000	4	0	0.0
V1─Te1─Te2	1.000	0.000	0.000	29.716	1.000	0.141	0.000	0.000	4	0	0.0
Te1─V1─V3	1.000	0.000	0.000	29.743	1.000	0.144	0.000	0.000	4	0	0.0

Table 45.

SW potential parameters for single-layer 1H-VTe₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Atom types in the first column are displayed in Figure 2 (with M = V and X = Te).

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-VTe₂ under uniaxial tension at 1 and 300 K. Figure 20 shows the stress-strain curve for the tension of a single-layer 1H-VTe₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-VTe₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-VTe₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 68.1 and 66.8 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.28$ .

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{Cr}─\text{O}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\text{Cr}─\text{O}─\text{O}}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\text{Cr}─\text{O}─{\text{O}}^{\prime }}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\text{O}─\text{Cr}─\text{Cr}}{\left(\Delta \theta \right)}^2$
Parameter	12.881	8.039	8.039	8.039
$r_0$ or ${\theta }_0$	1.880	86.655	75.194	86.655

Table 46.

The VFF model for single-layer 1H-CrO₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of ${\text{eV/}\mathrm{\AA }}^{\text{2}}$ for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of $\mathrm{\AA }$ ) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

	$A\left(\text{eV}\right)$	$\rho \left(\mathbf{\AA }\right)$	$B({\mathbf{\AA }}^4)$	$r_{\text{min}}(\mathbf{\AA })$	$r_{\text{max}}\left(\mathbf{\AA }\right)$
Cr─O	6.343	0.916	6.246	0.0	2.536

Table 47.

Two-body SW potential parameters for single-layer 1H-CrO₂ used by GULP [8] as expressed in Eq. (3).

There is no available value for the nonlinear quantities in the single-layer 1H-VTe₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −237.4 and −260.4 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 9.0 N/m at the ultimate strain of 0.23 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 8.6 N/m at the ultimate strain of 0.27 in the zigzag direction at the low temperature of 1 K.

12. 1H-CrO₂

Most existing theoretical studies on the single-layer 1H-CrO₂ are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1H-CrO₂.

Figure 21.

Phonon spectrum for single-layer 1H-CrO₂. (a) Phonon dispersion along the $\text{Γ}\text{M}$ direction in the Brillouin zone. The results from the VFF model (lines) are comparable with the ab initio results (pentagons) from [12]. (b) The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

The structure for the single-layer 1H-CrO₂ is shown in Figure 1 (with M = Cr and X = O). Each Cr atom is surrounded by six O atoms. These O atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each O atom is connected to three Cr atoms. The structural parameters are from the first-principles calculations [12], including the lattice constant $a=2.58$ Å and the bond length $d_{\text{Cr}─\text{O}}=1.88$ Å. The resultant angles are ${\theta }_{\text{CrOO}}=\mathrm{}{\theta }_{\text{OCrCr}}=86.655°$ and ${\theta }_{\text{CrO}{\text{O}}^{\prime }}=75.194°$ , in which atoms O and O′ are from different (top or bottom) groups.

Table 46 shows four VFF terms for the single-layer 1H-CrO₂; one of which is the bond stretching interaction shown by Eq. (1), while the other three terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the $\text{Γ}\text{M}$ as shown in Figure 21(a) . The ab initio calculations for the phonon dispersion are from [12]. Figure 21(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

Figure 22.

Stress-strain for single-layer 1H-CrO₂ of dimension $100\times 100$ Å along the armchair and zigzag directions.

	$K$ (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{Cr}─\text{O}─\text{O}}$	65.805	86.655	0.916	0.916	0.0	2.536	0.0	2.536	0.0	3.016
${\theta }_{\text{Cr}─\text{O}─{\text{O}}^{\prime }}$	70.163	75.194	0.916	0.916	0.0	2.536	0.0	2.536	0.0	3.016
${\theta }_{\text{O}─\text{Cr}─\text{Cr}}$	65.805	86.655	0.916	0.916	0.0	2.536	0.0	2.536	0.0	3.016

Table 48.

Three-body SW potential parameters for single-layer 1H-CrO₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

The parameters for the two-body SW potential used by GULP are shown in Table 47 . The parameters for the three-body SW potential used by GULP are shown in Table 48 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 49 . We note that 12 atom types have been introduced for the simulation of the single-layer 1H-CrO₂ using LAMMPS, because the angles around atom Cr in Figure 1 (with M = Cr and X = O) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS [13, 14]. According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Figure 2 (with M = Cr and X = O) shows that, for 1H-CrO₂, we can differentiate these angles around the Cr atom by assigning these six neighboring O atoms with different atom types. It can be found that 12 atom types are necessary for the purpose of differentiating all 6 neighbors around 1 Cr atom.

	$\mathit{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Cr1─O1─O1	1.000	0.916	2.769	0.000	1.000	0.000	6.242	8.871	4	0	0.0
Cr1─O1─O3	1.000	0.000	0.000	65.805	1.000	0.058	0.000	0.000	4	0	0.0
Cr1─O1─O2	1.000	0.000	0.000	70.163	1.000	0.256	0.000	0.000	4	0	0.0
O1─Cr1─Cr3	1.000	0.000	0.000	65.805	1.000	0.058	0.000	0.000	4	0	0.0

Table 49.

SW potential parameters for single-layer 1H-CrO₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-CrO₂ under uniaxial tension at 1 and 300 K. Figure 22 shows the stress-strain curve for the tension of a single-layer 1H-CrO₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-CrO₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-CrO₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 210.6 and 209.0 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.13$ .

There is no available value for nonlinear quantities in the single-layer 1H-CrO₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −1127.7 and −1185.8 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 19.4 N/m at the ultimate strain of 0.18 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 18.7 N/m at the ultimate strain of 0.20 in the zigzag direction at the low temperature of 1 K.

Figure 23.

Phonon dispersion for single-layer 1H-CrS₂. (a) The VFF model is fitted to the three acoustic branches in the long wave limit along the $\text{Γ}\text{M}$ direction. The ab initio results (gray pentagons) are from [17]. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for single-layer 1H-CrS₂ along $\text{Γ}\text{MK}\text{Γ}$ .

13. 1H-CrS₂

Most existing theoretical studies on the single-layer 1H-CrS₂ are based on the first-principles calculations. In this section, we will develop both VFF model and the SW potential for the single-layer 1H-CrS₂.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{Cr}─\text{S}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\text{Cr}─\text{S}─\text{S}}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\text{Cr}─\text{S}─{\text{S}}^{\prime }}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\text{S}─\text{Cr}─\text{Cr}}{\left(\Delta \theta \right)}^2$
Parameter	8.752	4.614	4.614	4.614
$r_0$ or ${\theta }_0$	2.254	83.099	80.031	83.099

Table 50.

The VFF model for single-layer 1H-CrS₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of ${\text{eV/}\mathrm{\AA }}^{\text{2}}$ for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of $\mathrm{\AA }$ ) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

	$A\left(\text{eV}\right)$	$\mathit{\rho }\left(\mathbf{\AA }\right)$	$B({\mathbf{\AA }}^4)$	$r_{\text{min}}(\mathbf{\AA })$	$r_{\text{max}}\left(\mathbf{\AA }\right)$
Cr─S	5.544	0.985	12.906	0.0	2.999

Table 51.

Two-body SW potential parameters for single-layer 1HCrS₂ used by GULP [8] as expressed in Eq. (3).

	$K$ (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{Cr}─\text{S}─\text{S}}$	32.963	83.099	0.985	0.985	0.0	2.999	0.0	2.999	0.0	3.577
${\theta }_{\text{Cr}─\text{S}─{\text{S}}^{\prime }}$	33.491	80.031	0.985	0.985	0.0	2.999	0.0	2.999	0.0	3.577
${\theta }_{\text{S}─\text{Cr}─\text{Cr}}$	32.963	83.099	0.985	0.985	0.0	2.999	0.0	2.999	0.0	3.577

Table 52.

Three-body SW potential parameters for single-layer 1H-CrS₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

	$\mathit{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Cr1─S1─S1	1.000	0.985	3.043	0.000	1.000	0.000	5.544	13.683	4	0	0.0
Cr1─S1─S3	1.000	0.000	0.000	32.963	1.000	0.120	0.000	0.000	4	0	0.0
Cr1─S1─S2	1.000	0.000	0.000	33.491	1.000	0.173	0.000	0.000	4	0	0.0
S1─Cr1─Cr3	1.000	0.000	0.000	32.963	1.000	0.120	0.000	0.000	4	0	0.0

Table 53.

SW potential parameters for single-layer 1H-CrS₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Atom types in the first column are displayed in Figure 2 (with M = Cr and X = S).

Figure 24.

Stress-strain for single-layer 1H-CrS₂ of dimension $100\times 100$ Å along the armchair and zigzag directions.

The structure for the single-layer 1H-CrS₂ is shown in Figure 1 (with M = Cr and X = S). Each Cr atom is surrounded by six S atoms. These S atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each S atom is connected to three Cr atoms. The structural parameters are from [17], including the lattice constant $a=2.99$ Å and the bond length $d_{\text{Cr}─\text{S}}=2.254$ Å. The resultant angles are ${\theta }_{\text{CrSS}}=\mathrm{}{\theta }_{\text{SCrCr}}=83.099°$ and ${\theta }_{\text{CrS}{\text{S}}^{\prime }}=80.031°$ , in which atoms S and S’ are from different (top or bottom) groups.

Table 50 shows four VFF terms for the 1H-CrS₂; one of which is the bond stretching interaction shown by Eq. (1), while the other three terms are the angle bending interaction shown by while the other three terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the $\text{Γ}\text{M}$ as shown in Figure 23(a) . The ab initio calculations for the phonon dispersion are from [17]. Similar phonon dispersion can also be found in other ab initio calculations [12]. Figure 23(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 51 . The parameters for the three-body SW potential used by GULP are shown in Table 52 . Parameters for the SW potential used by LAMMPS are listed in Table 53 . We note that 12 atom types have been introduced for the simulation of the single-layer 1H-CrS₂ using LAMMPS, because the angles around atom Cr in Figure 1 (with M = Cr and X = S) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS [13, 14] According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Figure 2 (with M = Cr and X = S) shows that, for 1H-CrS₂, we can differentiate these angles around the Cr atom by assigning these six neighboring S atoms with different atom types. It can be found that 12 atom types are necessary for the purpose of differentiating all 6 neighbors around 1 Cr atom.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-CrS₂ under uniaxial tension at 1 and 300 K. Figure 24 shows the stress-strain curve for the tension of a single-layer 1H-CrS₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-CrS₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-CrS₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 98.4 and 97.8 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. These values are in reasonably agreement with the ab initio results, e.g., 112.0 N/m from [18], or 111.9 N/m from [19]. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.26$ , which agrees with the ab initio value of 0.27 [18, 19].

There is no available value for the nonlinear quantities in the single-layer 1H-CrS₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −364.8 and −409.3 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 12.4 N/m at the ultimate strain of 0.22 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 11.8 N/m at the ultimate strain of 0.26 in the zigzag direction at the low temperature of 1 K.

Figure 25.

Phonon dispersion for single-layer 1H-CrSe₂. (a) The VFF model is fitted to the three acoustic branches in the long wave limit along the ΓM direction. The ab initio results (gray pentagons) are from [21]. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for single-layer 1H-CrSe₂ along ΓMKΓ.

14. 1H-CrSe₂

Most existing theoretical studies on the single-layer 1H-CrSe₂ are based on the first-principles calculations. In this section, we will develop both VFF model and the SW potential for the single-layer 1H-CrSe₂.

The structure for the single-layer 1H-CrSe₂ is shown in Figure 1 (with M = Cr and X = Se). Each Cr atom is surrounded by six Se atoms. These Se atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each Se atom is connected to three Cr atoms. The structural parameters are from [12], including the lattice constant $a=3.13$ Å and the bond length $d_{\text{Cr}─\text{Se}}=2.38$ Å. The resultant angles are ${\theta }_{\text{CrSeSe}}=\mathrm{}{\theta }_{\text{SeCrCr}}=82.229°$ and ${\theta }_{\text{CrSeS}{\text{e}}^{\prime }}=81.197°$ , in which atoms Se and Se’ are from different (top or bottom) groups.

Figure 26.

Stress-strain for single-layer 1H-CrSe₂ of dimension $100\times 100$ Å along the armchair and zigzag directions.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{Cr}─\text{Se}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\text{Cr}─\text{Se}─\text{Se}}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\text{Cr}─\text{Se}─\text{S}{\text{e}}^{\prime }}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\text{Se}─\text{Cr}─\text{Cr}}{\left(\Delta \theta \right)}^2$
Parameter	9.542	4.465	4.465	4.465
$r_0$ or ${\theta }_0$	2.380	82.229	81.197	82.229

Table 54.

The VFF model for single-layer 1H-CrSe₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of ${\text{eV/}\mathrm{\AA }}^{\text{2}}$ for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of $\mathrm{\AA }$ ) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

Table 54 shows four VFF terms for the 1H-CrSe₂; one of which is the bond stretching interaction shown by Eq. (1), while the other three terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the ΓM as shown in Figure 25(a) . The ab initio calculations for the phonon dispersion are from [12]. Figure 25(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Cr─Se	6.581	1.012	16.043	0.0	3.156

Table 55.

Two-body SW potential parameters for single-layer 1H-CrSe₂ used by GULP [8] as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (°)	${\mathit{\rho }}_1$ (Å)	${\mathit{\rho }}_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{Cr}─\text{Se}─\text{Se}}$	30.881	82.229	1.012	1.012	0.0	3.156	0.0	3.156	0.0	3.767
${\theta }_{\text{Cr}─\text{Se}─\text{S}{\text{e}}^{\prime }}$	31.044	81.197	1.012	1.012	0.0	3.156	0.0	3.156	0.0	3.767
${\theta }_{\text{Se}─\text{Cr}─\text{Cr}}$	30.881	82.229	1.012	1.012	0.0	3.156	0.0	3.156	0.0	3.767

Table 56.

Three-body SW potential parameters for single-layer 1H-CrSe₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

The parameters for the two-body SW potential used by GULP are shown in Table 55 . The parameters for the three-body SW potential used by GULP are shown in Table 56 . Parameters for the SW potential used by LAMMPS are listed in Table 57 . We note that 12 atom types have been introduced for the simulation of the single-layer 1H-CrSe₂ using LAMMPS, because the angles around atom Cr in Figure 1 (with M = Cr and X = Se) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS [13, 14]. According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Figure 2 (with M = Cr and X = Se) shows that, for 1H-CrSe₂, we can differentiate these angles around the Cr atom by assigning these six neighboring Se atoms with different atom types. It can be found that 12 atom types are necessary for the purpose of differentiating all 6 neighbors around 1 Cr atom.

	$\mathit{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Cr1─Se1─Se1	1.000	1.012	3.118	0.000	1.000	0.000	6.581	15.284	4	0	0.0
Cr1─Se1─Se3	1.000	0.000	0.000	30.881	1.000	0.135	0.000	0.000	4	0	0.0
Cr1─Se1─Se2	1.000	0.000	0.000	31.044	1.000	0.153	0.000	0.000	4	0	0.0
Se1─Cr1─Cr3	1.000	0.000	0.000	30.881	1.000	0.135	0.000	0.000	4	0	0.0

Table 57.

SW potential parameters for single-layer 1H-CrSe₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Atom types in the first column are displayed in Figure 2 (with M = Cr and X = Se).

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-CrSe₂ under uniaxial tension at 1 and 300 K. Figure 26 shows the stress-strain curve for the tension of a single-layer 1H-CrSe₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-CrSe₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-CrSe₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 90.0 and 89.0 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. These values are in reasonably agreement with the ab initio results, e.g., 88.0 N/m from [18], or 87.9 N/m from [19]. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.30$ , which agrees with the ab initio value of 0.30 [18, 19].

There is no available value for the nonlinear quantities in the single-layer 1H-CrSe₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most two-dimensional atomic layered materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −279.6 and −318.8 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 13.0 N/m at the ultimate strain of 0.25 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 12.4 N/m at the ultimate strain of 0.29 in the zigzag direction at the low temperature of 1 K.

Figure 27.

Phonon dispersion for single-layer 1H-CrTe₂. (a) The VFF model is fitted to the three acoustic branches in the long wave limit along the ΓM direction. The ab initio results (gray pentagons) are from [12]. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for single-layer 1H-CrTe₂ along ΓMKΓ.

15. 1H-CrTe₂

Most existing theoretical studies on the single-layer 1H-CrTe₂ are based on the first-principles calculations. In this section, we will develop both VFF model and the SW potential for the single-layer 1H-CrTe₂.

The structure for the single-layer 1H-CrTe₂ is shown in Figure 1 (with M = Cr and X = Te). Each Cr atom is surrounded by six Te atoms. These Te atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each Te atom is connected to three Cr atoms. The structural parameters are from [12], including the lattice constant $a=3.39$ Å and the bond length $d_{\text{Cr}─\text{Te}}=2.58$ Å. The resultant angles are ${\theta }_{\text{CrTeTe}}=\mathrm{}{\theta }_{\text{TeCrCr}}=82.139°$ and ${\theta }_{\text{CrTeT}{\text{e}}^{\prime }}=81.316°$ , in which atoms Te and Te’ are from different (top or bottom) groups.

Figure 28.

Stress-strain for single-layer 1H-CrTe₂ of dimension $100\times 100$ Å along the armchair and zigzag directions.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{Cr}─\text{Te}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\text{Cr}─\text{Te}─\text{Te}}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\text{Cr}─\text{Te}─\text{T}{\text{e}}^{\prime }}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\text{Te}─\text{Cr}─\text{Cr}}{\left(\Delta \theta \right)}^2$
Parameter	8.197	4.543	4.543	4.543
$r_0$ or ${\theta }_0$	2.580	82.139	81.316	82.139

Table 58.

The VFF model for single-layer 1H-CrTe₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of ${\text{eV/}\mathrm{\AA }}^{\text{2}}$ for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of $\mathrm{\AA }$ ) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

Table 58 shows three VFF terms for the 1H-CrTe₂; one of which is the bond stretching interaction shown by Eq. (1), while the other two terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the ΓM as shown in Figure 27(a) . The ab initio calculations for the phonon dispersion are from [12]. Figure 27(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Cr─Te	6.627	1.094	22.154	0.0	3.420

Table 59.

Two-body SW potential parameters for single-layer 1H-CrTe₂ used by GULP [8] as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (°)	${\mathit{\rho }}_1$ (Å)	${\mathit{\rho }}_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{Cr}─\text{Te}─\text{Te}}$	31.316	82.139	1.094	1.094	0.0	3.420	0.0	3.420	0.0	4.082
${\theta }_{\text{Cr}─\text{Te}─\text{T}{\text{e}}^{\prime }}$	31.447	81.316	1.094	1.094	0.0	3.420	0.0	3.420	0.0	4.082
${\theta }_{\text{Te}─\text{Cr}─\text{Cr}}$	31.316	82.139	1.094	1.094	0.0	3.420	0.0	3.420	0.0	4.082

Table 60.

Three-body SW potential parameters for single-layer 1H-CrTe₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

The parameters for the two-body SW potential used by GULP are shown in Table 59 . The parameters for the three-body SW potential used by GULP are shown in Table 60 . Parameters for the SW potential used by LAMMPS are listed in Table 61 . We note that 12 atom types have been introduced for the simulation of the single-layer 1H-CrTe₂ using LAMMPS, because the angles around atom Cr in Figure 1 (with M = Cr and X = Te) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS [13, 14]. According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Figure 2 (with M = Cr and X = Te) shows that, for 1H-CrTe₂, we can differentiate these angles around the Cr atom by assigning these six neighboring Te atoms with different atom types. It can be found that 12 atom types are necessary for the purpose of differentiating all 6 neighbors around 1 Cr atom.

	$\mathit{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Cr1─Te1─Te1	1.000	1.094	3.126	0.000	1.000	0.000	6.627	15.461	4	0	0.0
Cr1─Te1─Te3	1.000	0.000	0.000	31.316	1.000	0.137	0.000	0.000	4	0	0.0
Cr1─Te1─Te2	1.000	0.000	0.000	31.447	1.000	0.151	0.000	0.000	4	0	0.0
Te1─Cr1─Cr3	1.000	0.000	0.000	31.316	1.000	0.137	0.000	0.000	4	0	0.0

Table 61.

SW potential parameters for single-layer 1H-CrTe₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Atom types in the first column are displayed in Figure 2 (with M = Cr and X = Te).

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-CrTe₂ under uniaxial tension at 1 and 300 K. Figure 28 shows the stress-strain curve for the tension of a single-layer 1H-CrTe₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-CrTe₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-CrTe₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 77.2 and 76.4 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. These values are in reasonably agreement with the ab initio results, e.g., 63.9 N/m from [18, 19]. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.30$ , which agrees with the ab initio value of 0.30 [18, 19].

There is no available value for the nonlinear quantities in the single-layer 1H-CrTe₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most two-dimensional atomic layered materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −237.1 and −280.8 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 11.2 N/m at the ultimate strain of 0.25 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 10.7 N/m at the ultimate strain of 0.29 in the zigzag direction at the low temperature of 1 K.

Figure 29.

Phonon spectrum for single-layer 1H-MnO₂. (a) Phonon dispersion along the ΓM direction in the Brillouin zone. The results from the VFF model (lines) are comparable with the ab initio results (pentagons) from [12]. (b) The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

16. 1H-MnO₂

Most existing theoretical studies on the single-layer 1H-MnO₂ are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1H-MnO₂.

The structure for the single-layer 1H-MnO₂ is shown in Figure 1 (with M = Mn and X = O). Each Mn atom is surrounded by six O atoms. These O atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each O atom is connected to three Mn atoms. The structural parameters are from the first-principles calculations [12], including the lattice constant $a=2.61$ Å and the bond length $d_{\text{Mn}─\text{O}}=1.87$ Å. The resultant angles are ${\theta }_{\text{MnOO}}=\mathrm{}{\theta }_{\text{OMnMn}}=88.511°$ and ${\theta }_{\text{MnO}{\text{O}}^{\prime }}=72.621°$ , in which atoms O and O′ are from different (top or bottom) groups.

Figure 30.

Stress-strain for single-layer 1H-MnO₂ of dimension $100\times 100$ Å along the armchair and zigzag directions.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{Mn}─\text{O}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\text{Mn}─\text{O}─\text{O}}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\text{Mn}─\text{O}─{\text{O}}^{\prime }}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\text{O}─\text{Mn}─\text{Mn}}{\left(\Delta \theta \right)}^2$
Parameter	9.382	6.253	6.253	6.253
$r_0$ or ${\theta }_0$	1.870	88.511	72.621	88.511

Table 62.

The VFF model for single-layer 1H-MnO₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of ${\text{eV/}\mathrm{\AA }}^{\text{2}}$ for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of $\mathrm{\AA }$ ) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

Table 62 shows four VFF terms for the single-layer 1H-MnO₂; one of which is the bond stretching interaction shown by Eq. (1), while the other three terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the ΓM as shown in Figure 29(a) . The ab initio calculations for the phonon dispersion are from [12]. Typically, the transverse acoustic branch has a linear dispersion, so is higher than the flexural branch. However, it can be seen that the transverse acoustic branch is close to the flexural branch, which should be due to the underestimation from the ab initio calculations. Figure 29(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Mn─O	4.721	0.961	6.114	0.0	2.540

Table 63.

Two-body SW potential parameters for single-layer 1H-MnO₂ used by GULP [8] as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{Mn}─\text{O}─\text{O}}$	55.070	88.511	0.961	0.961	0.0	2.540	0.0	2.540	0.0	3.016
${\theta }_{\text{Mn}─\text{O}─{\text{O}}^{\prime }}$	60.424	72.621	0.961	0.961	0.0	2.540	0.0	2.540	0.0	3.016
${\theta }_{\text{O}─\text{Mn}─\text{Mn}}$	55.070	88.511	0.961	0.961	0.0	2.540	0.0	2.540	0.0	3.016

Table 64.

Three-body SW potential parameters for single-layer 1H-MnO₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

The parameters for the two-body SW potential used by GULP are shown in Table 63 . The parameters for the three-body SW potential used by GULP are shown in Table 64 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 65 . We note that 12 atom types have been introduced for the simulation of the single-layer 1H-MnO₂ using LAMMPS, because the angles around atom Mn in Figure 1 (with M = Mn and X = O) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS [13, 14]. According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Figure 2 (with M = Mn and X = O) shows that, for 1H-MnO₂, we can differentiate these angles around the Mn atom by assigning these six neighboring O atoms with different atom types. It can be found that 12 atom types are necessary for the purpose of differentiating all 6 neighbors around 1 Mn atom.

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Mn1─O1─O1	1.000	0.961	2.643	0.000	1.000	0.000	4.721	7.158	4	0	0.0
Mn1─O1─O3	1.000	0.000	0.000	55.070	1.000	0.026	0.000	0.000	4	0	0.0
Mn1─O1─O2	1.000	0.000	0.000	60.424	1.000	0.299	0.000	0.000	4	0	0.0
O1─Mn1─Mn3	1.000	0.000	0.000	55.070	1.000	0.026	0.000	0.000	4	0	0.0

Table 65.

SW potential parameters for single-layer 1H-MnO₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-MnO₂ under uniaxial tension at 1 and 300 K. Figure 30 shows the stress-strain curve for the tension of a single-layer 1H-MnO₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-MnO₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-MnO₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 161.1 and 160.2 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.10$ .

There is no available value for nonlinear quantities in the single-layer 1H-MnO₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −915.9 and −957.1 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 14.1 N/m at the ultimate strain of 0.17 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 13.7 N/m at the ultimate strain of 0.20 in the zigzag direction at the low temperature of 1 K.

Figure 31.

Phonon spectrum for single-layer 1H-FeO₂. (a) Phonon dispersion along the ΓM direction in the Brillouin zone. The results from the VFF model (lines) are comparable with the ab initio results (pentagons) from [12]. (b) The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

17. 1H-FeO₂

Most existing theoretical studies on the single-layer 1H-FeO₂ are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1H-FeO₂.

The structure for the single-layer 1H-FeO₂ is shown in Figure 1 (with M = Fe and X = O). Each Fe atom is surrounded by six O atoms. These O atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each O atom is connected to three Fe atoms. The structural parameters are from the first-principles calculations [12], including the lattice constant $a=2.62$ Å and the bond length $d_{\text{Fe}─\text{O}}=1.88$ Å. The resultant angles are ${\theta }_{\text{FeOO}}=\mathrm{}{\theta }_{\text{OFeFe}}=88.343°$ and ${\theta }_{\text{FeO}{\text{O}}^{\prime }}=72.856°$ , in which atoms O and O′ are from different (top or bottom) groups.

Figure 32.

Stress-strain for single-layer 1H-FeO₂ of dimension $100\times 100$ Å along the armchair and zigzag directions.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{Fe}─\text{O}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\text{Fe}─\text{O}─\text{O}}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\text{Fe}─\text{O}─{\text{O}}^{\prime }}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\text{O}─\text{Fe}─\text{Fe}}{\left(\Delta \theta \right)}^2$
Parameter	8.377	3.213	3.213	3.213
$r_0$ or ${\theta }_0$	1.880	88.343	72.856	88.343

Table 66.

The VFF model for single-layer 1H-FeO₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of ${\text{eV/}\mathrm{\AA }}^{\text{2}}$ for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of $\mathrm{\AA }$ ) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

Table 66 shows four VFF terms for the single-layer 1H-FeO₂; one of which is the bond stretching interaction shown by Eq. (1), while the other three terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the ΓM as shown in Figure 31(a) . The ab initio calculations for the phonon dispersion are from [12]. Figure 31(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Fe─O	4.242	0.962	6.246	0.0	2.552

Table 67.

Two-body SW potential parameters for single-layer 1H-FeO₂ used by GULP [8] as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{Fe}─\text{O}─\text{O}}$	28.105	88.343	0.962	0.962	0.0	2.552	0.0	2.552	0.0	3.031
${\theta }_{\text{Fe}─\text{O}─{\text{O}}^{\prime }}$	30.754	72.856	0.962	0.962	0.0	2.552	0.0	2.552	0.0	3.031
${\theta }_{\text{O}─\text{Fe}─\text{Fe}}$	28.105	88.343	0.962	0.962	0.0	2.552	0.0	2.552	0.0	3.031

Table 68.

Three-body SW potential parameters for single-layer 1H-FeO₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

The parameters for the two-body SW potential used by GULP are shown in Table 67 . The parameters for the three-body SW potential used by GULP are shown in Table 68 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 69 . We note that 12 atom types have been introduced for the simulation of the single-layer 1H-FeO₂ using LAMMPS, because the angles around atom Fe in Figure 1 (with M = Fe and X = O) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS [13, 14]. According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Figure 2 (with M = Fe and X = O) shows that, for 1H-FeO₂, we can differentiate these angles around the Fe atom by assigning these six neighboring O atoms with different atom types. It can be found that 12 atom types are necessary for the purpose of differentiating all 6 neighbors around 1 Fe atom.

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	p	q	Tol
Fe1─O1─O1	1.000	0.962	2.654	0.000	1.000	0.000	4.242	7.298	4	0	0.0
Fe1─O1─O3	1.000	0.000	0.000	28.105	1.000	0.029	0.000	0.000	4	0	0.0
Fe1─O1─O2	1.000	0.000	0.000	30.754	1.000	0.295	0.000	0.000	4	0	0.0
O1─Fe1─Fe3	1.000	0.000	0.000	28.105	1.000	0.029	0.000	0.000	4	0	0.0

Table 69.

SW potential parameters for single-layer 1H-FeO₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-FeO₂ under uniaxial tension at 1 and 300 K. Figure 32 shows the stress-strain curve for the tension of a single-layer 1H-FeO₂ of dimension 100×100 Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-FeO₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-FeO₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 100.2 and 99.3 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.23$ .

There is no available value for nonlinear quantities in the single-layer 1H-FeO₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of B in most materials. The value of the third-order nonlinear elasticity D can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with E as the Young’s modulus. The values of D from the present SW potential are −423.4 and −460.2 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 11.4 N/m at the ultimate strain of 0.21 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 10.9 N/m at the ultimate strain of 0.25 in the zigzag direction at the low temperature of 1 K.

18. 1H-FES₂

Most existing theoretical studies on the single-layer 1H-FeS₂ are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1H-FeS₂.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{Fe}-\text{S}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{Fe}-\text{S}-\text{S}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{Fe}-\text{S}-{\text{S}}^{\prime }}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{S}-\text{Fe}-\text{Fe}}{\left(\text{Δ}\theta \right)}^2$
Parameter	6.338	3.964	3.964	3.964
$r_0$ or ${\theta }_0$	2.220	87.132	74.537	87.132

Table 70.

The VFF model for single-layer 1H-FeS₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Fe─S	4.337	1.097	12.145	0.0	3.000

Table 71.

Two-body SW potential parameters for single-layer 1H-FeS₂ used by GULP [8] as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{Fe}-\text{S}-\text{S}}$	33.060	87.132	1.097	1.097	0.0	3.000	0.0	3.000	0.0	3.567
${\theta }_{\text{Fe}-\text{S}-{\text{S}}^{\prime }}$	35.501	74.537	1.097	1.097	0.0	3.000	0.0	3.000	0.0	3.567
${\theta }_{\text{S}-\text{Fe}-\text{Fe}}$	33.060	87.132	1.097	1.097	0.0	3.000	0.0	3.000	0.0	3.567

Table 72.

Three-body SW potential parameters for single-layer 1H-FeS₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Fe1─S1─S1	1.000	1.097	2.735	0.000	1.000	0.000	4.337	8.338	4	0	0.0
Fe1─S1─S3	1.000	0.000	0.000	33.060	1.000	0.050	0.000	0.000	4	0	0.0
Fe1─S1─S2	1.000	0.000	0.000	35.501	1.000	0.267	0.000	0.000	4	0	0.0
S1─Fe1─Fe3	1.000	0.000	0.000	33.060	1.000	0.050	0.000	0.000	4	0	0.0

Table 73.

SW potential parameters for single-layer 1H-FeS₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Figure 33.

Phonon spectrum for single-layer 1H-FeS₂. (a) Phonon dispersion along the ΓM direction in the Brillouin zone. The results from the VFF model (lines) are comparable with the ab initio results (pentagons) from Ref. [12]. (b) The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

The structure for the single-layer 1H-FeS₂ is shown in Figure 1 (with M=Fe and X=S). Each Fe atom is surrounded by six S atoms. These S atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each S atom is connected to three Fe atoms. The structural parameters are from the first-principles calculations [12], including the lattice constant $a=3.06$ Å and the bond length $d_{\text{Fe}-\text{S}}=2.22$ Å. The resultant angles are ${\theta }_{\text{FeSS}}={\theta }_{\text{SFeFe}}=87.132°$ and ${\theta }_{\text{FeS}{\text{S}}^{\prime }}=74.537°$ , in which atoms S and S’ are from different (top or bottom) groups.

Table 70 shows four VFF terms for the single-layer 1H-FeS₂; one of which is the bond stretching interaction shown by Eq. (1), while the other three terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the ΓM as shown in Figure 33(a) . The ab initio calculations for the phonon dispersion are from Ref. [12]. Figure 33(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

Figure 34.

Stress-strain for single-layer 1H-FeS₂ of dimension $100\times 100$ Å along the armchair and zigzag directions.

The parameters for the two-body SW potential used by GULP are shown in Table 71 . The parameters for the three-body SW potential used by GULP are shown in Table 72 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 73 . We note that 12 atom types have been introduced for the simulation of the single-layer 1H-FeS₂ using LAMMPS, because the angles around atom Fe in Figure 1 (with M=Fe and X=S) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS [13, 14]. According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Figure 2 (with M=Fe and X=S) shows that, for 1H-FeS₂, we can differentiate these angles around the Fe atom by assigning these six neighboring S atoms with different atom types. It can be found that 12 atom types are necessary for the purpose of differentiating all 6 neighbors around 1 Fe atom.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-FeS₂ under uniaxial tension at 1 and 300 K. Figure 34 shows the stress-strain curve for the tension of a single-layer 1H-FeS₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-FeS₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-FeS₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 83.6 and 83.4 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}={\nu }_{yx}=0.20$ .

There is no available value for nonlinear quantities in the single-layer 1H-FeS₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −377.5 and −412.7 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 9.0 N/m at the ultimate strain of 0.20 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 8.6 N/m at the ultimate strain of 0.23 in the zigzag direction at the low temperature of 1 K.

19. 1H-FESE₂

Most existing theoretical studies on the single-layer 1H-FeSe₂ are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1H-FeSe₂.

The structure for the single-layer 1H-FeSe₂ is shown in Figure 1 (with M=Fe and X=Se). Each Fe atom is surrounded by six Se atoms. These Se atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each Se atom is connected to three Fe atoms. The structural parameters are from the first-principles calculations [12], including the lattice constant $a=3.22$ Å and the bond length $d_{\text{Fe}-\text{Se}}=2.35$ Å. The resultant angles are ${\theta }_{\text{FeSeSe}}={\theta }_{\text{SeFeFe}}=86.488°$ and ${\theta }_{\text{FeSeS}{\text{e}}^{\prime }}=75.424°$ , in which atoms Se and Se’ are from different (top or bottom) groups.

Figure 35.

Phonon spectrum for single-layer 1H-FeSe₂. (a) Phonon dispersion along the ΓM direction in the Brillouin zone. The results from the VFF model (lines) are comparable with the ab initio results (pentagons) from Ref. [12]. (b) The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

Table 74 shows four VFF terms for the single-layer 1H-FeSe₂; one of which is the bond stretching interaction shown by Eq. (1), while the other three terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the ΓM as shown in Figure 35(a) . The ab initio calculations for the phonon dispersion are from Ref. [12]. Figure 35(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 75 . The parameters for the three-body SW potential used by GULP are shown in Table 76 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 77 . We note that 12 atom types have been introduced for the simulation of the single-layer 1H-FeSe₂ using LAMMPS, because the angles around atom Fe in Figure 1 (with M=Fe and X=Se) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS [13, 14]. According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Figure 2 (with M=Fe and X=Se) shows that, for 1H-FeSe₂, we can differentiate these angles around the Fe atom by assigning these six neighboring Se atoms with different atom types. It can be found that 12 atom types are necessary for the purpose of differentiating all 6 neighbors around 1 Fe atom.

Figure 36.

Stress-strain for single-layer 1H-FeSe₂ of dimension $100\times 100$ Å along the armchair and zigzag directions.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{Fe}-\text{Se}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{Fe}-\text{Se}-\text{Se}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{Fe}-\text{Se}-\text{S}{\text{e}}^{\prime }}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{Se}-\text{Fe}-\text{Fe}}{\left(\text{Δ}\theta \right)}^2$
Parameter	6.338	3.964	3.964	3.964
$r_0$ or ${\theta }_0$	2.350	86.488	75.424	86.488

Table 74.

The VFF model for single-layer 1H-FeSe₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Fe-Se	4.778	1.139	15.249	0.0	3.168

Table 75.

Two-body SW potential parameters for single-layer 1H-FeSe₂ used by GULP [8] as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{Fe}-\text{Se}-\text{Se}}$	32.235	86.488	1.139	1.139	0.0	3.168	0.0	3.168	0.0	3.768
${\theta }_{\text{Fe}-\text{Se}-\text{S}{\text{e}}^{\prime }}$	34.286	75.424	1.139	1.139	0.0	3.168	0.0	3.168	0.0	3.768
${\theta }_{\text{Se}-\text{Fe}-\text{Fe}}$	32.235	86.488	1.139	1.139	0.0	3.168	0.0	3.168	0.0	3.768

Table 76.

Three-body SW potential parameters for single-layer 1H-FeSe₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-FeSe₂ under uniaxial tension at 1 and 300 K. Figure 36 shows the stress-strain curve for the tension of a single-layer 1H-FeSe₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-FeSe₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-FeSe₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 77.3 and 77.4 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}={\nu }_{yx}=0.23$ .

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Fe1─Se1─Se1	1.000	1.139	2.781	0.000	1.000	0.000	4.778	9.049	4	0	0.0
Fe1─Se1─Se3	1.000	0.000	0.000	32.235	1.000	0.061	0.000	0.000	4	0	0.0
Fe1─Se1─Se2	1.000	0.000	0.000	34.286	1.000	0.252	0.000	0.000	4	0	0.0
Se1─Fe1─Fe3	1.000	0.000	0.000	32.235	1.000	0.061	0.000	0.000	4	0	0.0

Table 77.

SW potential parameters for single-layer 1H-FeSe₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{Fe}-\text{Te}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{Fe}-\text{Te}-\text{Te}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{Fe}-\text{Te}-\text{T}{\text{e}}^{\prime }}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{Te}-\text{Fe}-\text{Fe}}{\left(\text{Δ}\theta \right)}^2$
Parameter	6.338	3.964	3.964	3.964
$r_0$ or ${\theta }_0$	2.530	86.904	74.851	86.904

Table 78.

The VFF model for single-layer 1H-FeTe₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

There is no available value for nonlinear quantities in the single-layer 1H-FeSe₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −323.8 and −360.8 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 8.8 N/m at the ultimate strain of 0.21 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 8.4 N/m at the ultimate strain of 0.25 in the zigzag direction at the low temperature of 1 K.

Figure 37.

Phonon spectrum for single-layer 1H-FeTe₂. (a) Phonon dispersion along the ΓM direction in the Brillouin zone. The results from the VFF model (lines) are comparable with the ab initio results (pentagons) from Ref. [12]. (b) The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Fe─Te	5.599	1.242	20.486	0.0	3.416

Table 79.

Two-body SW potential parameters for single-layer 1H-FeTe₂ used by GULP [8] as expressed in Eq. (3).

20. 1H-FETE₂

Most existing theoretical studies on the single-layer 1H-FeTe₂ are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1H-FeTe₂.

Figure 38.

Stress-strain for single-layer 1H-FeTe₂ of dimension $100\times 100$ Å along the armchair and zigzag directions.

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{Fe}-\text{Te}-\text{Te}}$	32.766	86.904	1.242	1.242	0.0	3.416	0.0	3.416	0.0	4.062
${\theta }_{\text{Fe}-\text{Te}-\text{T}{\text{e}}^{\prime }}$	35.065	74.851	1.242	1.242	0.0	3.416	0.0	3.416	0.0	4.062
${\theta }_{\text{Te}-\text{Fe}-\text{Fe}}$	32.766	86.904	1.242	1.242	0.0	3.416	0.0	3.416	0.0	4.062

Table 80.

Three-body SW potential parameters for single-layer 1H-FeTe₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

The structure for the single-layer 1H-FeTe₂ is shown in Figure 1 (with M=Fe and X=Te). Each Fe atom is surrounded by six Te atoms. These Te atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each Te atom is connected to three Fe atoms. The structural parameters are from the first-principles calculations [12], including the lattice constant $a=3.48$ Å and the bond length $d_{\text{Fe}-\text{Te}}=2.53$ Å. The resultant angles are ${\theta }_{\text{FeTeTe}}={\theta }_{\text{TeFeFe}}=86.904°$ and ${\theta }_{\text{FeTeT}{\text{e}}^{\prime }}=74.851°$ , in which atoms Te and Te’ are from different (top or bottom) groups.

Table 78 shows four VFF terms for the single-layer 1H-FeTe₂; one of which is the bond stretching interaction shown by Eq. (1), while the other three terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the two in-plane acoustic branches in the phonon dispersion along the ΓM as shown in Figure 37(a) . The ab initio calculations for the phonon dispersion are from Ref. [12]. Figure 37(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Fe1─Te1─Te1	1.000	1.242	2.751	0.000	1.000	0.000	5.599	8.615	4	0	0.0
Fe1─Te1─Te3	1.000	0.000	0.000	32.766	1.000	0.054	0.000	0.000	4	0	0.0
Fe1─Te1─Te2	1.000	0.000	0.000	35.065	1.000	0.261	0.000	0.000	4	0	0.0
Te1─Fe1─Fe3	1.000	0.000	0.000	32.766	1.000	0.054	0.000	0.000	4	0	0.0

Table 81.

SW potential parameters for single-layer 1H-FeTe₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

The parameters for the two-body SW potential used by GULP are shown in Table 79 . The parameters for the three-body SW potential used by GULP are shown in Table 80 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 81 . We note that 12 atom types have been introduced for the simulation of the single-layer 1H-FeTe₂ using LAMMPS, because the angles around atom Fe in Figure 1 (with M=Fe and X=Te) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS [13, 14]. According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Figure 2 (with M=Fe and X=Te) shows that, for 1H-FeTe₂, we can differentiate these angles around the Fe atom by assigning these six neighboring Te atoms with different atom types. It can be found that 12 atom types are necessary for the purpose of differentiating all 6 neighbors around 1 Fe atom.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-FeTe₂ under uniaxial tension at 1 and 300 K. Figure 38 shows the stress-strain curve for the tension of a single-layer 1H-FeTe₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-FeTe₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-FeTe₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 69.6 and 69.8 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}={\nu }_{yx}=0.25$ .

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{Co}-\text{Te}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{Co}-\text{Te}-\text{Te}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{Co}-\text{Te}-\text{T}{\text{e}}^{\prime }}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{Te}-\text{Co}-\text{Co}}{\left(\text{Δ}\theta \right)}^2$
Parameter	6.712	2.656	2.656	2.656
$r_0$ or ${\theta }_0$	2.510	89.046	71.873	89.046

Table 82.

The VFF model for single-layer 1H-CoTe₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Ų) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

There is no available value for nonlinear quantities in the single-layer 1H-FeTe₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −267.5 and −302.8 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 8.6 N/m at the ultimate strain of 0.22 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 8.1 N/m at the ultimate strain of 0.26 in the zigzag direction at the low temperature of 1 K.

21. 1H-COTE₂

Most existing theoretical studies on the single-layer 1H-CoTe₂ are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1H-CoTe₂.

Figure 39.

Phonon spectrum for single-layer 1H-CoTe₂. (a) Phonon dispersion along the ΓM direction in the Brillouin zone. The results from the VFF model (lines) are comparable with the ab initio results (pentagons) from Ref. [12]. (b) The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Co─Te	6.169	1.310	19.846	0.0	3.417

Table 83.

Two-body SW potential parameters for single-layer 1H-CoTe₂ used by GULP [8] as expressed in Eq. (3).

The structure for the single-layer 1H-CoTe₂ is shown in Figure 1 (with M=Co and X=Te). Each Co atom is surrounded by six Te atoms. These Te atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each Te atom is connected to three Co atoms. The structural parameters are from the first-principles calculations [12], including the lattice constant $a=3.52$ Å and the bond length $d_{\text{Co}-\text{Te}}=2.51$ Å. The resultant angles are ${\theta }_{\text{CoTeTe}}={\theta }_{\text{TeCoCo}}=89.046°$ and ${\theta }_{\text{CoTeT}{\text{e}}^{\prime }}=71.873°$ , in which atoms Te and Te’ are from different (top or bottom) groups.

Figure 40.

Stress-strain for single-layer 1H-CoTe₂ of dimension $100\times 100$ Å along the armchair and zigzag directions.

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{Co}-\text{Te}-\text{Te}}$	23.895	89.046	1.310	1.310	0.0	3.417	0.0	3.417	0.0	4.055
${\theta }_{\text{Co}-\text{Te}-\text{T}{\text{e}}^{\prime }}$	26.449	71.873	1.310	1.310	0.0	3.417	0.0	3.417	0.0	4.055
${\theta }_{\text{Te}-\text{Co}-\text{Co}}$	23.895	89.046	1.310	1.310	0.0	3.417	0.0	3.417	0.0	4.055

Table 84.

Three-body SW potential parameters for single-layer 1H-CoTe₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

Table 82 shows four VFF terms for the single-layer 1H-CoTe₂; one of which is the bond stretching interaction shown by Eq. (1), while the other three terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the ΓM as shown in Figure 39(a) . The ab initio calculations for the phonon dispersion are from Ref. [12]. Figure 39(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 83 . The parameters for the three-body SW potential used by GULP are shown in Table 84 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 85 . We note that 12 atom types have been introduced for the simulation of the single-layer 1H-CoTe₂ using LAMMPS, because the angles around atom Co in Figure 1 (with M=Co and X=Te) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS [13, 14]. According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Figure 2 (with M=Co and X=Te) shows that, for 1H-CoTe₂, we can differentiate these angles around the Co atom by assigning these six neighboring Te atoms with different atom types. It can be found that 12 atom types are necessary for the purpose of differentiating all 6 neighbors around 1 Co atom.

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Co1─Te1─Te1	1.000	1.310	2.608	0.000	1.000	0.000	6.169	6.739	4	0	0.0
Co1─Te1─Te3	1.000	0.000	0.000	23.895	1.000	0.017	0.000	0.000	4	0	0.0
Co1─Te1─Te2	1.000	0.000	0.000	26.449	1.000	0.311	0.000	0.000	4	0	0.0
Te1─Co1─Co3	1.000	0.000	0.000	23.895	1.000	0.017	0.000	0.000	4	0	0.0

Table 85.

SW potential parameters for single-layer 1H-CoTe₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-CoTe₂ under uniaxial tension at 1 and 300 K. Figure 40 shows the stress-strain curve for the tension of a single-layer 1H-CoTe₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-CoTe₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-CoTe₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 53.7 and 54.3 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}={\nu }_{yx}=0.32$ .

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{Ni}-\text{S}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{Ni}-\text{S}-\text{S}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{Ni}-\text{S}-{\text{S}}^{\prime }}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{S}-\text{Ni}-\text{Ni}}{\left(\text{Δ}\theta \right)}^2$
Parameter	6.933	3.418	3.418	3.418
$r_0$ or ${\theta }_0$	2.240	98.740	57.593	98.740

Table 86.

The VFF model for single-layer 1H-NiS₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Ni-S	6.425	1.498	12.588	0.0	3.156

Table 87.

Two-body SW potential parameters for single-layer 1H-NiS₂ used by GULP [8] as expressed in Eq. (3).

There is no available value for nonlinear quantities in the single-layer 1H-CoTe₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −157.2 and −187.9 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 8.2 N/m at the ultimate strain of 0.26 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 7.9 N/m at the ultimate strain of 0.33 in the zigzag direction at the low temperature of 1 K.

22. 1H-NIS₂

Most existing theoretical studies on the single-layer 1H-NiS₂ are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1H-NiS₂.

Figure 41.

Phonon spectrum for single-layer 1H-NiS₂. (a) Phonon dispersion along the ΓM direction in the Brillouin zone. The results from the VFF model (lines) are comparable with the ab initio results (pentagons) from Ref. [12]. (b) The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

The structure for the single-layer 1H-NiS₂ is shown in Figure 1 (with M=Ni and X=S). Each Ni atom is surrounded by six S atoms. These S atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each S atom is connected to three Ni atoms. The structural parameters are from the first-principles calculations [12], including the lattice constant $a=3.40$ Å and the bond length $d_{\text{Ni}-\text{S}}=2.24$ Å. The resultant angles are ${\theta }_{\text{NiSS}}={\theta }_{\text{SNiNi}}=98.740°$ and ${\theta }_{\text{NiS}{\text{S}}^{\prime }}=57.593°$ , in which atoms S and S’ are from different (top or bottom) groups.

Table 86 shows four VFF terms for the single-layer 1H-NiS₂; one of which is the bond stretching interaction shown by Eq. (1), while the other three terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the ΓM as shown in Figure 41(a) . The ab initio calculations for the phonon dispersion are from Ref. [12]. Figure 41(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

Figure 42.

Stress-strain for single-layer 1H-NiS₂ of dimension $100\times 100$ Å along the armchair and zigzag directions.

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{Ni}-\text{S}-\text{S}}$	46.062	98.740	1.498	1.498	0.0	3.156	0.0	3.156	0.0	3.713
${\theta }_{\text{Ni}-\text{S}-{\text{S}}^{\prime }}$	63.130	57.593	1.498	1.498	0.0	3.156	0.0	3.156	0.0	3.713
${\theta }_{\text{S}-\text{Ni}-\text{Ni}}$	46.062	98.740	1.498	1.498	0.0	3.156	0.0	3.156	0.0	3.713

Table 88.

Three-body SW potential parameters for single-layer 1H-NiS₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

The parameters for the two-body SW potential used by GULP are shown in Table 87 . The parameters for the three-body SW potential used by GULP are shown in Table 88 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 89 . We note that 12 atom types have been introduced for the simulation of the single-layer 1H-NiS₂ using LAMMPS, because the angles around atom Ni in Figure 1 (with M=Ni and X=S) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS [13, 14]. According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Figure 2 (with M=Ni and X=S) shows that, for 1H-NiS₂, we can differentiate these angles around the Ni atom by assigning these six neighboring S atoms with different atom types. It can be found that 12 atom types are necessary for the purpose of differentiating all 6 neighbors around 1 Ni atom.

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Ni1─S1─S1	1.000	1.498	2.107	0.000	1.000	0.000	6.425	2.502	4	0	0.0
Ni1─S1─S3	1.000	0.000	0.000	46.062	1.000	−0.152	0.000	0.000	4	0	0.0
Ni1─S1─S2	1.000	0.000	0.000	63.130	1.000	0.536	0.000	0.000	4	0	0.0
S1─Ni1─Ni3	1.000	0.000	0.000	46.062	1.000	−0.152	0.000	0.000	4	0	0.0

Table 89.

SW potential parameters for single-layer 1H-NiS₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-NiS₂ under uniaxial tension at 1 and 300 K. Figure 42 shows the stress-strain curve for the tension of a single-layer 1H-NiS₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-NiS₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-NiS₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 84.0 and 82.5 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}={\nu }_{yx}=0.19$ .

There is no available value for nonlinear quantities in the single-layer 1H-NiS₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −403.2 and −414.8 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 8.7 N/m at the ultimate strain of 0.20 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 8.3 N/m at the ultimate strain of 0.24 in the zigzag direction at the low temperature of 1 K.

Figure 43.

Phonon spectrum for single-layer 1H-NiSe₂. (a) Phonon dispersion along the ΓM direction in the Brillouin zone. The results from the VFF model (lines) are comparable with the ab initio results (pentagons) from Ref. [12]. (b) The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

23. 1H-NISE₂

Most existing theoretical studies on the single-layer 1H-NiSe₂ are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1H-NiSe₂.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{Ni}-\text{Se}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{Ni}-\text{Se}-\text{Se}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{Ni}-\text{Se}-\text{S}{\text{e}}^{\prime }}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{Se}-\text{Ni}-\text{Ni}}{\left(\text{Δ}\theta \right)}^2$
Parameter	4.823	2.171	2.171	2.171
$r_0$ or ${\theta }_0$	2.350	90.228	70.206	90.228

Table 90.

The VFF model for single-layer 1H-NiSe₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/c² for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of eV/Ų) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Ni─Se	4.004	1.267	15.249	0.0	3.213

Table 91.

Two-body SW potential parameters for single-layer 1H-NiSe₂ used by GULP [8] as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{Ni}-\text{Se}-\text{Se}}$	20.479	90.228	1.267	1.267	0.0	3.213	0.0	3.213	0.0	3.809
${\theta }_{\text{Ni}-\text{Se}-\text{S}{\text{e}}^{\prime }}$	23.132	70.206	1.267	1.267	0.0	3.213	0.0	3.213	0.0	3.809
${\theta }_{\text{Se}-\text{Ni}-\text{Ni}}$	20.479	90.228	1.267	1.267	0.0	3.213	0.0	3.213	0.0	3.809

Table 92.

Three-body SW potential parameters for single-layer 1H-NiSe₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Ni1─Se1─Se1	1.000	1.267	2.535	0.000	1.000	0.000	4.004	5.913	4	0	0.0
Ni1─Se1─Se3	1.000	0.000	0.000	20.479	1.000	−0.004	0.000	0.000	4	0	0.0
Ni1─Se1─Se2	1.000	0.000	0.000	23.132	1.000	0.339	0.000	0.000	4	0	0.0
Se1─Ni1─Ni3	1.000	0.000	0.000	20.479	1.000	−0.004	0.000	0.000	4	0	0.0

Table 93.

SW potential parameters for single-layer 1H-NiSe₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Figure 44.

Stress-strain for single-layer 1H-NiSe₂ of dimension $100\times 100$ Å along the armchair and zigzag directions.

The structure for the single-layer 1H-NiSe₂ is shown in Figure 1 (with M=Ni and X=Se). Each Ni atom is surrounded by six Se atoms. These Se atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each Se atom is connected to three Ni atoms. The structural parameters are from the first-principles calculations [12], including the lattice constant $a=3.33$ Å and the bond length $d_{\text{Ni}-\text{Se}}=2.35$ Å. The resultant angles are ${\theta }_{\text{NiSeSe}}={\theta }_{\text{SeNiNi}}=90.228°$ and ${\theta }_{\text{NiSeS}{\text{e}}^{\prime }}=70.206°$ , in which atoms Se and Se’ are from different (top or bottom) groups.

Table 90 shows four VFF terms for the single-layer 1H-NiSe₂; one of which is the bond stretching interaction shown by Eq. (1), while the other three terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the ΓM as shown in Figure 43(a) . The ab initio calculations for the phonon dispersion are from Ref. [12]. The lowest acoustic branch (flexural mode) is almost linear in the ab initio calculations, which may be due to the violation of the rigid rotational invariance [20]. Figure 43(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 91 . The parameters for the three-body SW potential used by GULP are shown in Table 92 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 93 . We note that 12 atom types have been introduced for the simulation of the single-layer 1H-NiSe₂ using LAMMPS, because the angles around atom Ni in Figure 1 (with M=Ni and X=Se) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS [13, 14]. According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Figure 2 (with M=Ni and X=Se) shows that, for 1H-NiSe₂, we can differentiate these angles around the Ni atom by assigning these six neighboring Se atoms with different atom types. It can be found that 12 atom types are necessary for the purpose of differentiating all 6 neighbors around 1 Ni atom.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-NiSe₂ under uniaxial tension at 1 and 300 K. Figure 44 shows the stress-strain curve for the tension of a single-layer 1H-NiSe₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-NiSe₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-NiSe₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 47.6 and 47.8 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}={\nu }_{yx}=0.27$ .

There is no available value for nonlinear quantities in the single-layer 1H-NiSe₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −173.9 and −197.6 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 6.1 N/m at the ultimate strain of 0.23 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 5.9 N/m at the ultimate strain of 0.28 in the zigzag direction at the low temperature of 1 K.

Figure 45.

Phonon spectrum for single-layer 1H-NiTe₂. (a) Phonon dispersion along the ΓM direction in the Brillouin zone. The results from the VFF model (lines) are comparable with the ab initio results (pentagons) from Ref. [12]. (b) The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

24. 1H-NITE₂

Most existing theoretical studies on the single-layer 1H-NiTe₂ are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1H-NiTe₂.

The structure for the single-layer 1H-NiTe₂ is shown in Figure 1 (with M=Ni and X=Te). Each Ni atom is surrounded by six Te atoms. These Te atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each Te atom is connected to three Ni atoms. The structural parameters are from the first-principles calculations [12], including the lattice constant $a=3.59$ Å and the bond length $d_{\text{Ni}-\text{Te}}=2.54$ Å. The resultant angles are ${\theta }_{\text{NiTeTe}}={\theta }_{\text{TeNiNi}}=89.933°$ and ${\theta }_{\text{NiTeT}{\text{e}}^{\prime }}=70.624°$ , in which atoms Te and Te’ are from different (top or bottom) groups.

Figure 46.

Stress-strain for single-layer 1H-NiTe₂ of dimension $100\times 100$ Å along the armchair and zigzag directions.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{Ni}-\text{Te}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{Ni}-\text{Te}-\text{Te}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{Ni}-\text{Te}-\text{T}{\text{e}}^{\prime }}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{Te}-\text{Ni}-\text{Ni}}{\left(\text{Δ}\theta \right)}^2$
Parameter	6.712	2.656	2.656	2.656
$r_0$ or ${\theta }_0$	2.540	89.933	70.624	89.933

Table 94.

The VFF model for single-layer 1H-NiTe₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

Table 94 shows four VFF terms for the single-layer 1H-NiTe₂; one of which is the bond stretching interaction shown by Eq. (1), while the other three terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the ΓM as shown in Figure 45(a) . The ab initio calculations for the phonon dispersion are from Ref. [12]. The lowest acoustic branch (flexural mode) is almost linear in the ab initio calculations, which may be due to the violation of the rigid rotational invariance [20]. The transverse acoustic branch is very close to the longitudinal acoustic branch in the ab initio calculations. Figure 45(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Ni─Te	6.461	1.359	20.812	0.0	3.469

Table 95.

Two-body SW potential parameters for single-layer 1H-NiTe₂ used by GULP [8] as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{Ni}-\text{Te}-\text{Te}}$	24.759	89.933	1.359	1.359	0.0	3.469	0.0	3.469	0.0	4.114
${\theta }_{\text{Ni}-\text{Te}-\text{T}{\text{e}}^{\prime }}$	27.821	70.624	1.359	1.359	0.0	3.469	0.0	3.469	0.0	4.114
${\theta }_{\text{Te}-\text{Ni}-\text{Ni}}$	24.759	89.933	1.359	1.359	0.0	3.469	0.0	3.469	0.0	4.114

Table 96.

Three-body SW potential parameters for single-layer 1H-NiTe₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

The parameters for the two-body SW potential used by GULP are shown in Table 95 . The parameters for the three-body SW potential used by GULP are shown in Table 96 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 97 . We note that 12 atom types have been introduced for the simulation of the single-layer 1H-NiTe₂ using LAMMPS, because the angles around atom Ni in Figure 1 (with M=Ni and X=Te) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS [13, 14]. According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Figure 2 (with M=Ni and X=Te) shows that, for 1H-NiTe₂, we can differentiate these angles around the Ni atom by assigning these six neighboring Te atoms with different atom types. It can be found that 12 atom types are necessary for the purpose of differentiating all 6 neighbors around 1 Ni atom.

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Ni1─Te1─Te1	1.000	1.359	2.553	0.000	1.000	0.000	6.461	6.107	4	0	0.0
Ni1─Te1─Te3	1.000	0.000	0.000	24.759	1.000	0.001	0.000	0.000	4	0	0.0
Ni1─Te1─Te2	1.000	0.000	0.000	27.821	1.000	0.332	0.000	0.000	4	0	0.0
Te1─Ni1─Ni3	1.000	0.000	0.000	24.759	1.000	0.001	0.000	0.000	4	0	0.0

Table 97.

SW potential parameters for single-layer 1H-NiTe₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-NiTe₂ under uniaxial tension at 1 and 300 K. Figure 46 shows the stress-strain curve for the tension of a single-layer 1H-NiTe₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-NiTe₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-NiTe₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 53.2 and 53.6 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}={\nu }_{yx}=0.32$ .

There is no available value for nonlinear quantities in the single-layer 1H-NiTe₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −156.6 and −184.8 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 8.1 N/m at the ultimate strain of 0.26 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 7.8 N/m at the ultimate strain of 0.33 in the zigzag direction at the low temperature of 1 K.

Figure 47.

Phonon dispersion for single-layer 1H-NbS₂. (a) The VFF model is fitted to the three acoustic branches in the long wave limit along the ΓM direction. The theoretical results (gray pentagons) are from Ref. [21]. The blue lines are from the present VFF model. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for single-layer 1H-NbS₂ along ΓMKΓ.

25. 1H-NBS₂

In 1983, the VFF model was developed to investigate the lattice dynamical properties in the bulk 2H-NbS₂ [21]. In this section, we will develop the SW potential for the single-layer 1H-NbS₂.

The structure for the single-layer 1H-NbS₂ is shown in Figure 1 (with M=Nb and X=S). Each Nb atom is surrounded by six S atoms. These S atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each S atom is connected to three Nb atoms. The structural parameters are from Ref. [21], including the lattice constant $a=3.31$ Å and the bond length $d_{\text{Nb}-\text{S}}=2.47$ Å. The resultant angles are ${\theta }_{\text{NbSS}}={\theta }_{\text{SNbNb}}=84.140°$ and ${\theta }_{\text{NbS}{\text{S}}^{\prime }}=78.626°$ , in which atoms S and S’ are from different (top or bottom) groups.

Figure 48.

Stress-strain for single-layer 1H-NbS₂ of dimension $100\times 100$ Å along the armchair and zigzag directions.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{Nb-S}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{Nb-S-S}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{Nb-S-S^{\prime }}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{S-Nb-Nb}{\left(\text{Δ}\theta \right)}^2$
Parameter	8.230	4.811	4.811	4.811
$r_0$ or ${\theta }_0$	2.470	84.140	78.626	84.140

Table 98.

The VFF model for single-layer 1H-NbS₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

Table 98 shows four VFF terms for the 1H-NbS₂; one of which is the bond stretching interaction shown by Eq. (1), while the other three terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the ΓM as shown in Figure 47(a) . The theoretical phonon frequencies (gray pentagons) are from Ref. [21], which are the phonon dispersion of bulk 2H-NbS₂. We have used these phonon frequencies as the phonon dispersion of the single-layer 1H-NbS₂, as the interlayer interaction in the bulk 2H-NbS₂ only induces weak effects on the two in-plane acoustic branches. The interlayer coupling will strengthen the out-of-plane acoustic branch (flexural branch), so the flexural branch from the present VFF model (blue line) is lower than the theoretical results for bulk 2H-NbS₂ (gray pentagons). Figure 47(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Nb─S	6.439	1.116	18.610	0.0	3.300

Table 99.

Two-body SW potential parameters for single-layer 1H-NbS₂ used by GULP [8] as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{Nb}-\text{S}-\text{S}}$	35.748	84.140	1.116	1.116	0.0	3.300	0.0	3.300	0.0	3.933
${\theta }_{\text{Nb}-\text{S}-{\text{S}}^{\prime }}$	36.807	78.626	1.116	1.116	0.0	3.300	0.0	3.300	0.0	3.933
${\theta }_{\text{S}-\text{Nb}-\text{Nb}}$	35.748	84.140	1.116	1.116	0.0	3.300	0.0	3.300	0.0	3.933

Table 100.

Three-body SW potential parameters for single-layer 1H-NbS₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

The parameters for the two-body SW potential used by GULP are shown in Table 99 . The parameters for the three-body SW potential used by GULP are shown in Table 100 . Parameters for the SW potential used by LAMMPS are listed in Table 101 . We note that 12 atom types have been introduced for the simulation of the single-layer 1H-NbS₂ using LAMMPS, because the angles around atom Nb in Figure 1 (with M=Nb and X=S) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS [13, 14]. According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Figure 2 (with M=Nb and X=S) shows that, for 1H-NbS₂, we can differentiate these angles around the Nb atom by assigning these six neighboring S atoms with different atom types. It can be found that 12 atom types are necessary for the purpose of differentiating all 6 neighbors around 1 Nb atom.

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Nb1─S1─S1	1.000	1.116	2.958	0.000	1.000	0.000	6.439	12.014	4	0	0.0
Nb1─S1─S3	1.000	0.000	0.000	35.748	1.000	0.102	0.000	0.000	4	0	0.0
Nb1─S1─S2	1.000	0.000	0.000	36.807	1.000	0.197	0.000	0.000	4	0	0.0
S1─Nb1─Nb3	1.000	0.000	0.000	35.748	1.000	0.102	0.000	0.000	4	0	0.0

Table 101.

SW potential parameters for single-layer 1H-NbS₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Atom types in the first column are displayed in Figure 2 (with M=Nb and X=S).

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-NbS₂ under uniaxial tension at 1 and 300 K. Figure 48 shows the stress-strain curve for the tension of a single-layer 1H-NbS₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-NbS₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-NbS₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 87.7 and 87.2 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}={\nu }_{yx}=0.27$ .

There is no available value for the nonlinear quantities in the single-layer 1H-NbS₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −315.3 and −355.1 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 11.4 N/m at the ultimate strain of 0.23 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 10.8 N/m at the ultimate strain of 0.27 in the zigzag direction at the low temperature of 1 K.

Figure 49.

Phonon dispersion for single-layer 1H-NbSe₂. (a) The VFF model is fitted to the three acoustic branches in the long wave limit along the ΓM direction. The theoretical results (gray pentagons) are from Ref. [15]. The blue lines are from the present VFF model. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for single-layer 1H-NbSe₂ along ΓMKΓ.

26. 1H-NBSE₂

In 1983, the VFF model was developed to investigate the lattice dynamical properties in the bulk 2H-NbSe₂ [15, 21]. In this section, we will develop the SW potential for the single-layer 1H-NbSe₂.

The structure for the single-layer 1H-NbSe₂ is shown in Figure 1 (with M=Nb and X=Se). Each Nb atom is surrounded by six Se atoms. These Se atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each Se atom is connected to three Nb atoms. The structural parameters are from Ref. [21], including the lattice constant $a=3.45$ Å and the bond length $d_{\text{Nb}-\text{Se}}=2.60$ Å. The resultant angles are ${\theta }_{\text{NbSeSe}}={\theta }_{\text{SNbNb}}=83.129°$ and ${\theta }_{\text{NbSeS}{\text{e}}^{\prime }}=79.990°$ , in which atoms Se and Se’ are from different (top or bottom) groups.

Figure 50.

Stress-strain for single-layer 1H-NbSe₂ of dimension $100\times 100$ Å along the armchair and zigzag directions.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{Nb}-\text{Se}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{Nb}-\text{Se}-\text{Se}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{Nb}-\text{Se}-\text{S}{\text{e}}^{\prime }}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{Se}-\text{Nb}-\text{Nb}}{\left(\text{Δ}\theta \right)}^2$
Parameter	8.230	4.811	4.811	4.811
$r_0$ or ${\theta }_0$	2.600	83.129	79.990	83.129

Table 102.

The VFF model for single-layer 1H-NbSe₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

Table 102 shows four VFF terms for the 1H-NbSe₂; one of which is the bond stretching interaction shown by Eq. (1), while the other three terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the ΓM as shown in Figure 49(a) . The theoretical phonon frequencies (gray pentagons) are from Ref. [21], which are the phonon dispersion of bulk 2H-NbSe₂. We have used these phonon frequencies as the phonon dispersion of the single-layer 1H-NbSe₂, as the interlayer interaction in the bulk 2H-NbSe₂ only induces weak effects on the two in-plane acoustic branches. The interlayer coupling will strengthen the out-of-plane acoustic branch (flexural branch), so the flexural branch from the present VFF model (blue line) is lower than the theoretical results for bulk 2H-NbSe₂ (gray pentagons). It turns out that the VFF parameters for the single-layer 1H-NbSe₂ are the same as the single-layer NbS₂. The phonon dispersion for single-layer 1H-NbSe₂ was also shown in Ref. [12]. Figure 49(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Nb-Se	6.942	1.138	22.849	0.0	3.460

Table 103.

Two-body SW potential parameters for single-layer 1H-NbSe₂ used by GULP [8] as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{Nb}-\text{Se}-\text{Se}}$	34.409	83.129	1.138	1.138	0.0	3.460	0.0	3.460	0.0	4.127
${\theta }_{\text{Nb}-\text{Se}-\text{S}{\text{e}}^{\prime }}$	34.973	79.990	1.138	1.138	0.0	3.460	0.0	3.460	0.0	4.127
${\theta }_{\text{Se}-\text{Nb}-\text{Nb}}$	34.409	83.129	1.138	1.138	0.0	3.460	0.0	3.460	0.0	4.127

Table 104.

Three-body SW potential parameters for single-layer 1H-NbSe₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Nb1─Se1─Se1	1.000	1.138	3.041	0.000	1.000	0.000	6.942	13.631	4	0	0.0
Nb1─Se1─Se3	1.000	0.000	0.000	34.409	1.000	0.120	0.000	0.000	4	0	0.0
Nb1─Se1─Se2	1.000	0.000	0.000	34.973	1.000	0.174	0.000	0.000	4	0	0.0
Se1─Nb1─Nb3	1.000	0.000	0.000	34.409	1.000	0.120	0.000	0.000	4	0	0.0

Table 105.

SW potential parameters for single-layer 1H-NbSe₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Atom types in the first column are displayed in Figure 2 (with M=Nb and X=Se).

The parameters for the two-body SW potential used by GULP are shown in Table 103 . The parameters for the three-body SW potential used by GULP are shown in Table 104 . Parameters for the SW potential used by LAMMPS are listed in Table 105 . We note that 12 atom types have been introduced for the simulation of the single-layer 1H-NbSe₂ using LAMMPS, because the angles around atom Nb in Figure 1 (with M=Nb and X=Se) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS [13, 14]. According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Figure 2 (with M=Nb and X=Se) shows that, for 1H-NbSe₂, we can differentiate these angles around the Nb atom by assigning these six neighboring Se atoms with different atom types. It can be found that 12 atom types are necessary for the purpose of differentiating all 6 neighbors around 1 Nb atom.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-NbSe₂ under uniaxial tension at 1 and 300 K. Figure 50 shows the stress-strain curve for the tension of a single-layer 1H-NbSe₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-NbSe₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-NbSe₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 80.2 and 80.7 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}={\nu }_{yx}=0.29$ .

There is no available value for the nonlinear quantities in the single-layer 1H-NbSe₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −258.8 and −306.1 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 11.2 N/m at the ultimate strain of 0.24 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 10.7 N/m at the ultimate strain of 0.28 in the zigzag direction at the low temperature of 1 K.

Figure 51.

Phonon spectrum for single-layer 1H-MoO₂. (a) Phonon dispersion along the ΓM direction in the Brillouin zone. The results from the VFF model (lines) are comparable with the ab initio results (pentagons) from Ref. [12]. (b) The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

27. 1H-MoO₂

Most existing theoretical studies on the single-layer 1H-MoO₂ are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1H-MoO₂.

The structure for the single-layer 1H-MoO₂ is shown in Figure 1 (with M = Mo and X = O). Each Mo atom is surrounded by six O atoms. These O atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each O atom is connected to three Mo atoms. The structural parameters are from the first-principles calculations [12], including the lattice constant $a=2.78$ Å and the bond length $d_{\text{Mo}-\text{O}}=2.00$ Å. The resultant angles are ${\theta }_{\text{MoOO}}=\mathrm{}{\theta }_{\text{OMoMo}}=88.054°$ and ${\theta }_{\text{MoO}{\text{O}}^{\prime }}=73.258°$ , in which atoms O and O′ are from different (top or bottom) groups.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{Mo}-\text{O}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{Mo}-\text{O}-\text{O}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{Mo}-\text{O}-{\text{O}}^{\prime }}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{O}-\text{Mo}-\text{Mo}}{\left(\text{Δ}\theta \right)}^2$
Parameter	14.622	8.410	8.410	8.410
$r_0$ or ${\theta }_0$	2.000	88.054	73.258	88.054

Table 106.

The VFF model for single-layer 1H-MoO₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of $\text{eV}/{\mathrm{\AA }}^2$ for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of $\mathrm{\AA }$ ) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Mo-O	8.317	1.015	8.000	0.0	2.712

Table 107.

Two-body SW potential parameters for single-layer 1H-MoO₂ used by GULP [8] as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{Mo}-\text{O}-\text{O}}$	72.735	88.054	1.015	1.015	0.0	2.712	0.0	2.712	0.0	3.222
${\theta }_{\text{Mo}-\text{O}-{\text{O}}^{\prime }}$	79.226	73.258	1.015	1.015	0.0	2.712	0.0	2.712	0.0	3.222
${\theta }_{\text{O}-\text{Mo}-\text{Mo}}$	72.735	88.054	1.015	1.015	0.0	2.712	0.0	2.712	0.0	3.222

Table 108.

Three-body SW potential parameters for single-layer 1H-MoO₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Mo1─O1─O1	1.000	1.015	2.673	0.000	1.000	0.000	8.317	7.541	4	0	0.0
Mo1─O1─O3	1.000	0.000	0.000	72.735	1.000	0.034	0.000	0.000	4	0	0.0
Mo1─O1─O2	1.000	0.000	0.000	79.226	1.000	0.288	0.000	0.000	4	0	0.0
O1─Mo1─Mo3	1.000	0.000	0.000	72.735	1.000	0.034	0.000	0.000	4	0	0.0

Table 109.

SW potential parameters for single-layer 1H-MoO₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Figure 52.

Stress-strain for single-layer 1H-MoO₂ of dimension $100\times 100$ Å along the armchair and zigzag directions.

Table 106 shows four VFF terms for the single-layer 1H-MoO₂; one of which is the bond stretching interaction shown by Eq. (1), while the other three terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the ΓM as shown in Figure 51(a) . The ab initio calculations for the phonon dispersion are from Ref. [12]. Figure 51(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 107 . The parameters for the three-body SW potential used by GULP are shown in Table 108 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 109 . We note that 12 atom types have been introduced for the simulation of the single-layer 1H-MoO₂ using LAMMPS, because the angles around atom Mo in Figure 1 (with M = Mo and X = O) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS [13, 14]. According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Figure 2 (with M = Mo and X = O) shows that, for 1H-MoO₂, we can differentiate these angles around the Mo atom by assigning these six neighboring O atoms with different atom types. It can be found that 12 atom types are necessary for the purpose of differentiating all 6 neighbors around 1 Mo atom.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-MoO₂ under uniaxial tension at 1 and 300 K. Figure 52 shows the stress-strain curve for the tension of a single-layer 1H-MoO₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-MoO₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-MoO₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 210.0 and 209.3 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.17$ .

There is no available value for nonlinear quantities in the single-layer 1H-MoO₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −1027.8 and −1106.8 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 21.0 N/m at the ultimate strain of 0.19 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 20.1 N/m at the ultimate strain of 0.22 in the zigzag direction at the low temperature of 1 K.

28. 1H-MoS₂

Several potentials have been proposed to describe the interaction for the single-layer 1H-MoS₂. In 1975, Wakabayashi et al. developed a VFF model to calculate the phonon spectrum of the bulk 2H-MoS₂ [22]. In 2009, Liang et al. parameterized a bond-order potential for 1H-MoS₂ [23], which is based on the bond order concept underlying the Brenner potential [6]. A separate force field model was parameterized in 2010 for MD simulations of bulk 2H-MoS₂ [24]. The present author (J.W.J.) and his collaborators parameterized the SW potential for 1H-MoS₂ in 2013 [13], which was improved by one of the present authors (J.W.J.) in 2015 [7]. Recently, another set of parameters for the SW potential were proposed for the single-layer 1H-MoS₂ [25].

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{Mo-S}{\left(\text{Δ}{r}_{Mo-S}\right)}^2$	$\frac{1}{2}{K}_{MoSS}{\left(\text{Δ}{\theta }_{MoSS}\right)}^2$	$\frac{1}{2}{K}_{SMoMo}{\left(\text{Δ}{\theta }_{SMoMo}\right)}^2$
Parameter	8.640	5.316	4.891
$r_0$ or ${\theta }_0$	2.382	80.581	80.581

Table 110.

The VFF model parameters for single-layer 1H-MoS₂ from Ref. [22].

The second line gives the expression for each VFF term. Parameters are in the unit of $\text{eV}/{\mathrm{\AA }}^2$ for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of $\mathrm{\AA }$ ) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
$r_{\text{Mo}-\text{S}}$	6.918	1.252	17.771	0.0	3.16

Table 111.

Two-body SW potential parameters for single-layer 1H-MoS₂ used by GULP [8] as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{MoSS}}$	67.883	81.788	1.252	1.252	0.0	3.16	0.0	3.16	0.0	3.78
${\theta }_{\text{SMoMo}}$	62.449	81.788	1.252	1.252	0.0	3.16	0.0	3.16	0.0	4.27

Table 112.

Three-body SW potential parameters for single-layer 1H-MoS₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Mo1─S1─S1	1.000	1.252	2.523	0.000	1.000	0.000	6.918	7.223	4	0	0.0
Mo1─S1─S3	1.000	0.000	0.000	67.883	1.000	0.143	0.000	0.000	4	0	0.0
S1─Mo1─Mo3	1.000	0.000	0.000	62.449	1.000	0.143	0.000	0.000	4	0	0.0

Table 113.

SW potential parameters for single-layer 1H-MoS2 used by LAMMPS 9 as expressed in Eqs. (9) and 10.

Atom types in the first column are displayed in Figure 2 (with M = Mo and X = S).

Figure 53.

Phonon spectrum for single-layer 1H-MoS₂. Phonon dispersion along the ΓM direction in the Brillouin zone. The results from the VFF model (lines) are comparable with the experiment data (pentagons) from Ref. [22]. The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

Figure 54.

Stress-strain for single-layer 1H-MoS₂ of dimension $100\times 100$ Å along the armchair and zigzag directions.

We show the VFF model and the SW potential for single-layer 1H-MoS₂ in this section. These potentials have been developed in previous works. The VFF model presented here is from Ref. [22], while the SW potential presented in this section is from Ref. [7].

The structural parameters for the single-layer 1H-MoS₂ are from the first-principles calculations as shown in Figure 1 (with M = Mo and X = S) [26]. The Mo atom layer in the single-layer 1H-MoS₂ is sandwiched by two S atom layers. Accordingly, each Mo atom is surrounded by six S atoms, while each S atom is connected to three Mo atoms. The bond length between neighboring Mo and S atoms is $d=2.382$ Å, and the angles are ${\theta }_{\text{MoSS}}=80.581°$ and ${\theta }_{\text{SMoMo}}=80.581°$ .

The VFF model for single-layer 1H-MoS₂ is from Ref. [22], which is able to describe the phonon spectrum and the sound velocity accurately. We have listed the first three leading force constants for single-layer 1H-MoS₂ in Table 110 , neglecting other weak interaction terms. The SW potential parameters for single-layer 1H-MoS₂ used by GULP are listed in Tables 111 and 112 . The SW potential parameters for single-layer 1H-MoS₂ used by LAMMPS [9] are listed in Table 113 . We note that 12 atom types have been introduced for the simulation of the single-layer 1H-MoS₂ using LAMMPS, because the angles around atom Mo in Figure 1 (with M = Mo and X = S) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS [13, 14]. According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Figure 2 (with M = Mo and X = S) shows that, for 1H-MoS₂, we can differentiate these angles around the Mo atom by assigning these six neighboring S atoms with different atom types. It can be found that 12 atom types are necessary for the purpose of differentiating all 6 neighbors around 1 Mo atom.

We use GULP to compute the phonon dispersion for the single-layer 1H-MoS₂ as shown in Figure 53 . The results from the VFF model are quite comparable with the experiment data. The phonon dispersion from the SW potential is the same as that from the VFF model, which indicates that the SW potential has fully inherited the linear portion of the interaction from the VFF model.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{Mo-Se}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{Mo-Se-Se}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{Mo-Se-S{e}^{\prime }}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{Se-Mo-Mo}{\left(\text{Δ}\theta \right)}^2$
Parameter	7.928	6.945	6.945	5.782
$r_0$ or ${\theta }_0$	2.528	82.119	81.343	82.119

Table 114.

The VFF model for single-layer 1H-MoSe₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of $\text{eV}/{\mathrm{\AA }}^2$ for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of $\mathrm{\AA }$ ) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-MoS₂ under uniaxial tension at 1 and 300 K. Figure 54 shows the stress-strain curve during the tension of a single-layer 1H-MoS₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-MoS₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-MoS₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 97 and 96 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is isotropic in the armchair and zigzag directions. These values are in considerable agreement with the experimental results, e.g., $120\pm 30$ N/m from Refs [27, 28], or $180\pm 60$ N/m from Ref. [29]. The third-order nonlinear elastic constant $D$ can be obtained by fitting the stress-strain relation to $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with E as the Young’s modulus. The values of $D$ are −418 and −461 N/m along the armchair and zigzag directions, respectively. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.27$ .

29. 1H-MoSe₂

There is a recent parameter set for the SW potential in the single-layer 1H-MoSe₂ [25]. In this section, we will develop both VFF model and the SW potential for the single-layer 1H-MoSe₂.

Figure 55.

Phonon dispersion for single-layer 1H-MoSe₂. (a) The VFF model is fitted to the three acoustic branches in the long wave limit along the ΓM direction. The ab initio results (gray pentagons) are from Ref. [30]. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for single-layer 1H-MoSe₂ along ΓMKΓ.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Mo-Se	5.737	0.913	18.787	0.0	3.351

Table 115.

Two-body SW potential parameters for single-layer 1H-MoSe₂ used by GULP [8] as expressed in Eq. (3).

The structure for the single-layer 1H-MoSe₂ is shown in Figure 1 (with M = Mo and X = Se). Each Mo atom is surrounded by six Se atoms. These Se atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each Se atom is connected to three Mo atoms. The structural parameters are from Ref. [30], including the lattice constant $a=3.321$ Å and the bond length $d_{\text{Mo}-\text{Se}}=2.528$ Å. The resultant angles are ${\theta }_{\text{MoSeSe}}=\mathrm{}{\theta }_{\text{SeMoMo}}=82.119°$ and ${\theta }_{\text{MoSeS}{\text{e}}^{\prime }}=81.343°$ , in which atoms Se and Se′ are from different (top or bottom) groups.

Figure 56.

Stress-strain for single-layer 1H-MoSe₂ of dimension $100\times 100$ Å along the armchair and zigzag directions.

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{Mo}-\text{Se}-\text{Se}}$	32.526	82.119	0.913	0.913	0.0	3.351	0.0	3.351	0.0	4.000
${\theta }_{\text{Mo}-\text{Se}-\text{S}{\text{e}}^{\prime }}$	32.654	81.343	0.913	0.913	0.0	3.351	0.0	3.351	0.0	4.000
${\theta }_{\text{Se}-\text{Mo}-\text{Mo}}$	27.079	82.119	0.913	0.913	0.0	3.351	0.0	3.351	0.0	4.000

Table 116.

Three-body SW potential parameters for single-layer 1H-MoSe₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

Table 114 shows four VFF terms for the 1H-MoSe₂; one of which is the bond stretching interaction shown by Eq. (1), while the other three terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the ΓM as shown in Figure 55(a) . The ab initio calculations for the phonon dispersion are from Ref. [30]. Similar phonon dispersion can also be found in other ab initio calculations [12, 31–34]. Figure 55(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 115 . The parameters for the three-body SW potential used by GULP are shown in Table 116 . Parameters for the SW potential used by LAMMPS are listed in Table 117 . We note that 12 atom types have been introduced for the simulation of the single-layer 1H-MoSe₂ using LAMMPS, because the angles around atom Mo in Figure 1 (with M = Mo and X = Se) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS [13, 14]. According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Figure 2 (with M = Mo and X = Se) shows that, for 1H-MoSe₂, we can differentiate these angles around the Mo atom by assigning these six neighboring Se atoms with different atom types. It can be found that 12 atom types are necessary for the purpose of differentiating all 6 neighbors around 1 Mo atom.

	$\sigma$ (eV)	$a$ (Å)	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	$100\times 100$	Tol
Mo1─Se1─Se1	1.000	0.913	3.672	0.000	1.000	0.000	5.737	27.084	4	0	0.0
Mo1─Se1─Se3	1.000	0.000	0.000	32.526	1.000	0.137	0.000	0.000	4	0	0.0
Mo1─Se1─Se2	1.000	0.000	0.000	32.654	1.000	0.151	0.000	0.000	4	0	0.0
Se1─Mo1─Mo3	1.000	0.000	0.000	27.079	1.000	0.137	0.000	0.000	4	0	0.0

Table 117.

SW potential parameters for single-layer 1H-MoSe₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Atom types in the first column are displayed in Figure 2 (with M = Mo and X = Se).

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-MoSe₂ under uniaxial tension at 1 and 300 K. Figure 56 shows the stress-strain curve during the tension of a single-layer 1H-MoSe₂ of dimension $100\times 100\mathrm{\AA }$ . Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-MoSe₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-MoSe₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 103.0 and 101.8 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. These values are in considerable agreement with the experimental results, e.g., 103.9 N/m from Ref. [18], or 113.9 N/m from Ref. [35]. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.24$ , which agrees quite well with the ab initio value of 0.23 [18].

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{Mo}-\text{Te}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{Mo}-\text{Te}-\text{Te}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{Mo}-\text{Te}-\text{T}{\text{e}}^{\prime }}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{Te}-\text{Mo}-\text{Mo}}{\left(\text{Δ}\theta \right)}^2$
Parameter	6.317	6.184	6.184	5.225
$r_0$ or ${\theta }_0$	2.730	81.111	82.686	81.111

Table 118.

The VFF model for single-layer 1H-MoTe₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of $\text{eV}/{\mathrm{\AA }}^2$ for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of $\mathrm{\AA }$ ) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

We have determined the nonlinear parameter to be $B=0.46d^4$ in Eq. (5) by fitting to the third-order nonlinear elastic constant $D$ from the ab initio calculations [35]. We have extracted the value of $D=\mathrm{}-383.7$ N/m by fitting the stress-strain relation along the armchair direction in the ab initio calculations to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −365.4 and −402.4 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 13.6 N/m at the ultimate strain of 0.23 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 13.0 N/m at the ultimate strain of 0.27 in the zigzag direction at the low temperature of 1 K.

30. 1H-MoTe₂

Most existing theoretical studies on the single-layer 1H-MoTe₂ are based on the first-principles calculations. In this section, we will develop both VFF model and the SW potential for the single-layer 1H-MoTe₂.

The structure for the single-layer 1H-MoTe₂ is shown in Figure 1 (with M = Mo and X = Te). Each Mo atom is surrounded by six Te atoms. These Te atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each Te atom is connected to three Mo atoms. The structural parameters are from Ref. [36], including the lattice constant $a=3.55\mathrm{\AA }$ and the bond length $d_{\text{Mo}-\text{Te}}=2.73\mathrm{\AA }$ . The resultant angles are ${\theta }_{\text{MoTeTe}}=\mathrm{}{\theta }_{\text{TeMoMo}}=81.111°$ and ${\theta }_{\text{MoTeT}{\text{e}}^{\prime }}=82.686°$ , in which atoms Te and Te′ are from different (top or bottom) groups.

Figure 57.

Phonon dispersion for single-layer 1H-MoTe₂. (a) The VFF model is fitted to the three acoustic branches in the long wave limit along the ΓM direction. The ab initio results (gray pentagons) are from Ref. [36]. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for single-layer 1H-MoTe₂ along ΓMKΓ.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Mo-Te	5.086	0.880	24.440	0.0	3.604

Table 119.

Two-body SW potential parameters for single-layer 1H-MoTe₂ used by GULP [8] as expressed in Eq. (3).

Figure 58.

Stress-strain for single-layer 1H-MoTe₂ of dimension ${\theta }_0$ Å along the armchair and zigzag directions.

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{Mo}-\text{Te}-\text{Te}}$	23.705	81.111	0.880	0.880	0.0	3.604	0.0	3.604	0.0	4.305
${\theta }_{\text{Mo}-\text{Te}-\text{T}{\text{e}}^{\prime }}$	23.520	82.686	0.880	0.880	0.0	3.604	0.0	3.604	0.0	4.305
${\theta }_{\text{Te}-\text{Mo}-\text{Mo}}$	20.029	81.111	0.880	0.880	0.0	3.604	0.0	3.604	0.0	4.305

Table 120.

Three-body SW potential parameters for single-layer 1H-MoTe₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

Table 118 shows four VFF terms for the 1H-MoTe₂; one of which is the bond stretching interaction shown by Eq. (1), while the other three terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the ΓM as shown in Figure 57(a) . The ab initio calculations for the phonon dispersion are from Ref. [36]. Similar phonon dispersion can also be found in other ab initio calculations [12, 34, 37]. Figure 57(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 119 . The parameters for the three-body SW potential used by GULP are shown in Table 120 . Parameters for the SW potential used by LAMMPS are listed in Table 121 . We note that 12 atom types have been introduced for the simulation of the single-layer 1H-MoTe₂ using LAMMPS, because the angles around atom Mo in Figure 1 (with M = Mo and X = Te) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS [13, 14]. According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Figure 2 (with M = Mo and X = Te) shows that, for 1H-MoTe₂, we can differentiate these angles around the Mo atom by assigning these six neighboring Te atoms with different atom types. It can be found that 12 atom types are necessary for the purpose of differentiating all 6 neighbors around 1 Mo atom.

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Mo1─Te1─Te1	1.000	0.900	4.016	0.000	1.000	0.000	5.169	37.250	4	0	0.0
Mo1─Te1─Te3	1.000	0.000	0.000	24.163	1.000	0.143	0.000	0.000	4	0	0.0
Te1─Mo1─Mo3	1.000	0.000	0.000	20.416	1.000	0.143	0.000	0.000	4	0	0.0

Table 121.

SW potential parameters for single-layer 1H-MoTe₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Atom types in the first column are displayed in Figure 2 (with M = Mo and X = Te).

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-MoTe₂ under uniaxial tension at 1 and 300 K. Figure 58 shows the stress-strain curve for the tension of a single-layer 1H-MoTe₂ of dimension $100\times 100\mathrm{\AA }$ . Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-MoTe₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-MoTe₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 79.8 and 78.5 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. These values are in considerable agreement with the experimental results, e.g., 79.4 N/m from Ref. [18], or 87.0 N/m from Ref. [35]. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.25$ , which agrees with the ab initio value of 0.24 [18].

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{Ta}-\text{S}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{Ta}-\text{S}-\text{S}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{Ta}-\text{S}-{\text{S}}^{\prime }}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{S}-\text{Ta}-\text{Ta}}{\left(\text{Δ}\theta \right)}^2$
Parameter	8.230	4.811	4.811	4.811
$r_0$ or ${\theta }_0$	2.480	83.879	78.979	83.879

Table 122.

The VFF model for single-layer 1H-TaS₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of $\text{eV}/{\mathrm{\AA }}^2$ for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of $\mathrm{\AA }$ ) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Ta-S	6.446	1.111	18.914	0.0	3.310

Table 123.

Two-body SW potential parameters for single-layer 1H-TaS₂ used by GULP [8] as expressed in Eq. (3).

We have determined the nonlinear parameter to be $B=0.44d^4$ in Eq. (5) by fitting to the third-order nonlinear elastic constant $D$ from the ab initio calculations [35]. We have extracted the value of $D=-278.2$ N/m by fitting the stress-strain relation along the armchair direction in the ab initio calculations to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −250.5 and −276.6 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 11.7 N/m at the ultimate strain of 0.25 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 11.1 N/m at the ultimate strain of 0.29 in the zigzag direction at the low temperature of 1 K.

31. 1H-TaS₂

In 1983, the VFF model was developed to investigate the lattice dynamical properties in the bulk 2H-TaS₂ [21]. In this section, we will develop the SW potential for the single-layer 1H-TaS₂.

Figure 59.

Phonon dispersion for single-layer 1H-TaS₂. (a) The VFF model is fitted to the three acoustic branches in the long wave limit along the $\text{Γ}\text{M}$ direction. The theoretical results (gray pentagons) are from Ref. [21]. The blue lines are from the present VFF model. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for single-layer 1H-TaS₂ along $\text{Γ}\text{MK}\text{Γ}$ .

The structure for the single-layer 1H-TaS₂ is shown in Figure 1 (with M = Ta and X = S). Each Ta atom is surrounded by six S atoms. These S atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each S atom is connected to three Ta atoms. The structural parameters are from Ref. [21], including the lattice constant $a=3.315\mathrm{\AA }$ and the bond length $d_{\text{Ta}-\text{S}}=2.48\mathrm{\AA }$ . The resultant angles are ${\theta }_{\text{TaSS}}=\mathrm{}{\theta }_{\text{STaTa}}=83.879°$ and ${\theta }_{\text{TaS}{\text{S}}^{\prime }}=78.979°$ , in which atoms S and S′ are from different (top or bottom) groups.

Figure 60.

Stress-strain for single-layer 1H-TaS₂ of dimension $100\times 100\mathrm{\AA }$ along the armchair and zigzag directions.

	$K$ (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{Ta}-\text{S}-\text{S}}$	35.396	83.879	1.111	1.111	0.0	3.310	0.0	3.310	0.0	3.945
${\theta }_{\text{Ta}-\text{S}-{\text{S}}^{\prime }}$	36.321	78.979	1.111	1.111	0.0	3.310	0.0	3.310	0.0	3.945
${\theta }_{\text{S}-\text{Ta}-\text{Ta}}$	35.396	83.879	1.111	1.111	0.0	3.310	0.0	3.310	0.0	3.945

Table 124.

Three-body SW potential parameters for single-layer 1H-TaS₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

Table 122 shows the VFF terms for the 1H-TaS₂; one of which is the bond stretching interaction shown by Eq. (1), while the others are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the $ϵ$ as shown in Figure 59(a) . The theoretical phonon frequencies (gray pentagons) are from Ref. [21], which are the phonon dispersion of bulk 2H-TaS₂. We have used these phonon frequencies as the phonon dispersion of the single-layer 1H-TaS₂, as the interlayer interaction in the bulk 2H-TaS₂ only induces weak effects on the two in-plane acoustic branches. The interlayer coupling will strengthen the out-of-plane acoustic (flexural) branch, so the flexural branch from the present VFF model (blue line) is lower than the theoretical results for bulk 2H-TaS₂ (gray pentagons). Figure 59(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Ta1─S1─S1	1.000	1.111	2.979	0.000	1.000	0.000	6.446	12.408	4	0	0.0
Ta1─S1─S3	1.000	0.000	0.000	35.396	1.000	0.107	0.000	0.000	4	0	0.0
Ta1─S1─S2	1.000	0.000	0.000	36.321	1.000	0.191	0.000	0.000	4	0	0.0
S1─Ta1─Ta3	1.000	0.000	0.000	35.396	1.000	0.107	0.000	0.000	4	0	0.0

Table 125.

SW potential parameters for single-layer 1H-TaS₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Atom types in the first column are displayed in Figure 2 (with M = Ta and X = S).

The parameters for the two-body SW potential used by GULP are shown in Table 123 . The parameters for the three-body SW potential used by GULP are shown in Table 124 . Parameters for the SW potential used by LAMMPS are listed in Table 125 . We note that 12 atom types have been introduced for the simulation of the single-layer 1H-TaS₂ using LAMMPS, because the angles around atom Ta in Figure 1 (with M = Ta and X = S) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS [13, 14]. According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Figure 2 (with M = Ta and X = S) shows that, for 1H-TaS₂, we can differentiate these angles around the Ta atom by assigning these six neighboring S atoms with different atom types. It can be found that 12 atom types are necessary for the purpose of differentiating all 6 neighbors around 1 Ta atom.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-TaS₂ under uniaxial tension at 1 and 300 K. Figure 60 shows the stress-strain curve for the tension of a single-layer 1H-TaS₂ of dimension $100\times 100\mathrm{\AA }$ . Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-TaS₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-TaS₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 87.4 and 86.6 N/m along the armchair and zigzag directions, respectively. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.27$ .

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{Ta}-\text{Se}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{Ta}-\text{Se}-\text{Se}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{Ta}-\text{Se}-\text{S}{\text{e}}^{\prime }}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{Se}-\text{Ta}-\text{Ta}}{\left(\text{Δ}\theta \right)}^2$
Parameter	8.230	4.811	4.811	4.811
$r_0$ or ${\theta }_0$	2.590	83.107	80.019	83.107

Table 126.

The VFF model for single-layer 1H-TaSe₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of $\text{eV}/{\mathrm{\AA }}^2$ for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of $\mathrm{\AA }$ ) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

	$A$ (eV)	$\rho$ (Å)	$B$ (Å^4)	$r_{\text{min}}(\mathrm{\AA })$	$r_{\text{max}}$ (Å)
Ta─Se	6.885	1.133	22.499	0.0	3.446

Table 127.

Two-body SW potential parameters for single-layer 1H-TaSe₂ used by GULP [8] as expressed in Eq. (3).

There is no available value for the nonlinear quantities in the single-layer 1H-TaS₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −313.0 and −349.3 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 11.4 N/m at the ultimate strain of 0.23 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 10.8 N/m at the ultimate strain of 0.27 in the zigzag direction at the low temperature of 1 K.

32. 1H-TaSe₂

The VFF model was developed to investigate the lattice dynamical properties in the bulk 2H-TaSe₂ [15, 21]. In this section, we will develop the SW potential for the single-layer 1H-TaSe₂.

Figure 61.

Phonon dispersion for single-layer 1H-TaSe₂. (a) The VFF model is fitted to the three acoustic branches in the long wave limit along the $\text{Γ}\text{M}$ direction. The theoretical results (gray pentagons) are from Ref. [15]. The blue lines are from the present VFF model. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for single-layer 1H-TaSe₂ along $\text{Γ}\text{MK}\text{Γ}$ .

The structure for the single-layer 1H-TaSe₂ is shown in Figure 1 (with M = Ta and X = Se). Each Ta atom is surrounded by six Se atoms. These Se atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each Se atom is connected to three Ta atoms. The structural parameters are from Ref. [21], including the lattice constant $a=3.436\mathrm{\AA }$ and the bond length $d_{\text{Ta}-\text{Se}}=2.59\mathrm{\AA }$ . The resultant angles are ${\theta }_{\text{TaSeSe}}=\mathrm{}{\theta }_{\text{SeTaTa}}=83.107°$ and ${\theta }_{\text{TaSeS}{\text{e}}^{\prime }}=80.019°$ , in which atoms Se and Se′ are from different (top or bottom) groups.

Figure 62.

Stress-strain for single-layer 1H-TaSe₂ of dimension $100\times 100\mathrm{\AA }$ along the armchair and zigzag directions.

	$K$ (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{Ta}-\text{Se}-\text{Se}}$	34.381	83.107	1.133	1.133	0.0	3.446	0.0	3.446	0.0	4.111
${\theta }_{\text{Ta}-\text{Se}-\text{S}{\text{e}}^{\prime }}$	34.936	80.019	1.133	1.133	0.0	3.446	0.0	3.446	0.0	4.111
${\theta }_{\text{Se}-\text{Ta}-\text{Ta}}$	34.381	83.107	1.133	1.133	0.0	3.446	0.0	3.446	0.0	4.111

Table 128.

Three-body SW potential parameters for single-layer 1H-TaSe₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

Table 126 shows the VFF terms for the 1H-TaSe₂; one of which is the bond stretching interaction shown by Eq. (1), while the others are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the $\text{Γ}\text{M}$ as shown in Figure 61(a) . The theoretical phonon frequencies (gray pentagons) are from Ref. [21], which are the phonon dispersion of bulk 2H-TaSe₂. We have used these phonon frequencies as the phonon dispersion of the single-layer 1H-TaSe₂, as the interlayer interaction in the bulk 2H-TaSe₂ only induces weak effects on the two in-plane acoustic branches. The interlayer coupling will strengthen the out-of-plane acoustic branch (flexural branch), so the flexural branch from the present VFF model (blue line) is lower than the theoretical results for bulk 2H-TaSe₂ (gray pentagons). Figure 61(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Ta1─Se1─Se1	1.000	1.133	3.043	0.000	1.000	0.000	6.885	13.668	4	0	0.0
Ta1─Se1─Se3	1.000	0.000	0.000	34.381	1.000	0.120	0.000	0.000	4	0	0.0
Ta1─Se1─Se2	1.000	0.000	0.000	34.936	1.000	0.173	0.000	0.000	4	0	0.0
Se1─Ta1─Ta3	1.000	0.000	0.000	34.381	1.000	0.120	0.000	0.000	4	0	0.0

Table 129.

SW potential parameters for single-layer 1H-TaSe₂ used by LAMMPS [16] as expressed in Eqs. (9) and (10).

Atom types in the first column are displayed in Figure 2 (with M = Ta and X = Se).

The parameters for the two-body SW potential used by GULP are shown in Table 127 . The parameters for the three-body SW potential used by GULP are shown in Table 128 . Parameters for the SW potential used by LAMMPS are listed in Table 129 . We note that 12 atom types have been introduced for the simulation of the single-layer 1H-TaSe₂ using LAMMPS, because the angles around atom Ta in Figure 1 (with M = Ta and X = Se) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS [13, 14]. According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Figure 2 (with M = Ta and X = Se) shows that, for 1H-TaSe₂, we can differentiate these angles around the Ta atom by assigning these six neighboring Se atoms with different atom types. It can be found that 12 atom types are necessary for the purpose of differentiating all 6 neighbors around 1 Ta atom.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-TaSe₂ under uniaxial tension at 1 and 300 K. Figure 62 shows the stress-strain curve for the tension of a single-layer 1H-TaSe₂ of dimension $100\times 100\mathrm{\AA }$ . Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-TaSe₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-TaSe₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 80.8 and 81.1 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.29$ .

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{W}-\text{O}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{W}-\text{O}-\text{O}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{W}-\text{O}-{\text{O}}^{\prime }}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{O}-\text{W}-\text{W}}{\left(\text{Δ}\theta \right)}^2$
Parameter	15.318	10.276	10.276	10.276
$r_0$ or ${\theta }_0$	2.030	87.206	74.435	87.206

Table 130.

The VFF model for single-layer 1H-WO₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of $\text{eV}/{\mathrm{\AA }}^2$ for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of $\mathrm{\AA }$ ) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

	$A$ (eV)	$\rho$ (Å)	$B$ (Å^4)	$r_{\text{min}}(\mathrm{\AA })$	$r_{\text{max}}$ (Å)
W─O	8.781	1.005	8.491	0.0	2.744

Table 131.

Two-body SW potential parameters for single-layer 1H-WO₂ used by GULP [8] as expressed in Eq. (3).

There is no available value for the nonlinear quantities in the single-layer 1H-TaSe₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −263.3 and −308.6 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 11.3 N/m at the ultimate strain of 0.24 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 10.7 N/m at the ultimate strain of 0.28 in the zigzag direction at the low temperature of 1 K.

33. 1H-WO₂

Most existing theoretical studies on the single-layer 1H-WO₂ are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1H-WO₂.

Figure 63.

Phonon spectrum for single-layer 1H-WO₂. (a) Phonon dispersion along the $a=2.80$ direction in the Brillouin zone. The results from the VFF model (lines) are comparable with the ab initio results (pentagons) from Ref. [12]. (b) The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

The structure for the single-layer 1H-WO₂ is shown in Figure 1 (with M = W and X = O). Each W atom is surrounded by six O atoms. These O atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each O atom is connected to three W atoms. The structural parameters are from the first-principles calculations [12], including the lattice constant $a=2.80\mathrm{\AA }$ and the bond length $d_{\text{W}-\text{O}}=2.03\mathrm{\AA }$ . The resultant angles are ${\theta }_{\text{WOO}}=\mathrm{}{\theta }_{\text{OWW}}=87.206°$ and ${\theta }_{\text{WO}{\text{O}}^{\prime }}=74.435°$ , in which atoms O and O′ are from different (top or bottom) groups.

Table 130 shows four VFF terms for the single-layer 1H-WO₂; one of which is the bond stretching interaction shown by Eq. (1), while the other three terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the $\text{Γ}\text{M}$ as shown in Figure 63(a) . The ab initio calculations for the phonon dispersion are from Ref. [12]. Figure 63(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

Figure 64.

Stress-strain for single-layer 1H-WO₂ of dimension $100\times 100\mathrm{\AA }$ along the armchair and zigzag directions.

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{W}-\text{O}-\text{O}}$	85.955	87.206	1.005	1.005	0.0	2.744	0.0	2.744	0.0	3.262
${\theta }_{\text{W}-\text{O}-{\text{O}}^{\prime }}$	92.404	74.435	1.005	1.005	0.0	2.744	0.0	2.744	0.0	3.262
${\theta }_{\text{O}-\text{W}-\text{W}}$	85.955	87.206	1.005	1.005	0.0	2.744	0.0	2.744	0.0	3.262

Table 132.

Three-body SW potential parameters for single-layer 1H-WO₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

The parameters for the two-body SW potential used by GULP are shown in Table 131 . The parameters for the three-body SW potential used by GULP are shown in Table 132 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 133 . We note that 12 atom types have been introduced for the simulation of the single-layer 1H-WO₂ using LAMMPS, because the angles around atom W in Figure 1 (with M = W and X = O) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS [13, 14]. According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Figure 2 (with M = W and X = O) shows that, for 1H-WO₂, we can differentiate these angles around the W atom by assigning these six neighboring O atoms with different atom types. It can be found that 12 atom types are necessary for the purpose of differentiating all 6 neighbors around 1 W atom.

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
W1─O1─O1	1.000	1.005	2.730	0.000	1.000	0.000	8.781	8.316	4	0	0.0
W1─O1─O3	1.000	0.000	0.000	85.955	1.000	0.049	0.000	0.000	4	0	0.0
W1─O1─O2	1.000	0.000	0.000	92.404	1.000	0.268	0.000	0.000	4	0	0.0
O1─W1─W3	1.000	0.000	0.000	85.955	1.000	0.049	0.000	0.000	4	0	0.0

Table 133.

SW potential parameters for single-layer 1H-WO₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-WO₂ under uniaxial tension at 1 and 300 K. Figure 64 shows the stress-strain curve for the tension of a single-layer 1H-WO₂ of dimension $100\times 100\mathrm{\AA }$ . Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-WO₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-WO₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 237.1 and 237.2 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.15$ .

Figure 65.

Phonon dispersion for single-layer 1H-WS₂. (a) The VFF model is fitted to three acoustic branches in the long wave limit along the $\Gamma \text{MK}\Gamma$ direction. The ab initio results (gray pentagons) are from Ref. [31]. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for single-layer 1H-WS₂ along $B=0.5d^4$ .

There is no available value for nonlinear quantities in the single-layer 1H-WO₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −1218.0 and −1312.9 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 22.6 N/m at the ultimate strain of 0.18 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 21.8 N/m at the ultimate strain of 0.21 in the zigzag direction at the low temperature of 1 K

34. 1H-WS₂

Most existing theoretical studies on the single-layer 1H-WS₂ are based on the first-principles calculations. In this section, we will develop both VFF model and the SW potential for the single-layer 1H-WS₂.

Figure 66.

Stress-strain for single-layer 1H-WS₂ of dimension $\frac{1}{2}{K}_{\text{W}-\text{S}}{\left(\text{Δ}r\right)}^2$ Å along the armchair and zigzag directions.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{W}-\text{S}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{W}-\text{S}-\text{S}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{W}-\text{S}-{\text{S}}^{\prime }}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{S}-\text{W}-\text{W}}{\left(\text{Δ}\theta \right)}^2$
Parameter	8.701	7.421	7.421	6.607
$r_0$ or ${\theta }_0$	2.390	81.811	81.755	81.811

Table 134.

The VFF model for single-layer 1H-WS₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of $\text{eV}/{\mathrm{\AA }}^2$ for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of $\mathrm{\AA }$ ) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

Table 134 shows the VFF terms for the 1H-WS₂; one of which is the bond stretching interaction shown by Eq. (1), while the other terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the $\text{Γ}\text{M}$ as shown in Figure 65(a) . The ab initio calculations for the phonon dispersion are from Ref. [31]. Similar phonon dispersion can also be found in other ab initio calculations [12, 26, 34, 38, 39]. Figure 65(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

	$A$ (eV)	$\rho$ (Å)	$B$ (Å^4)	$r_{\text{min}}(\mathrm{\AA })$	$r_{\text{max}}$ (Å)
W─S	5.664	0.889	15.335	0.0	3.164

Table 135.

Two-body SW potential parameters for single-layer 1H-WS₂ used by GULP [8] as expressed in Eq. (3).

	$K$ (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max13}}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max23}}$ (Å)
${\theta }_{\text{W}-\text{S}-\text{S}}$	37.687	81.811	0.889	0.889	0.0	3.164	0.0	3.164	0.0	3.778
${\theta }_{\text{W}-\text{S}-{\text{S}}^{\prime }}$	37.697	81.755	0.889	0.889	0.0	3.164	0.0	3.164	0.0	3.778
${\theta }_{\text{S}-\text{W}-\text{W}}$	33.553	81.811	0.889	0.889	0.0	3.164	0.0	3.164	0.0	3.778

Table 136.

Three-body SW potential parameters for single-layer 1H-WS₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

The parameters for the two-body SW potential used by GULP are shown in Table 135 . The parameters for the three-body SW potential used by GULP are shown in Table 136 . Parameters for the SW potential used by LAMMPS are listed in Table 137 . We note that 12 atom types have been introduced for the simulation of the single-layer 1H-WS₂ using LAMMPS, because the angles around atom W in Figure 1 (with M = W and X = S) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS [13, 14]. According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Figure 2 (with M = W and X = S) shows that, for 1H-WS₂, we can differentiate these angles around the W atom by assigning these six neighboring S atoms with different atom types. It can be found that 12 atom types are necessary for the purpose of differentiating all 6 neighbors around 1 W atom.

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
W1─S1─S1	1.000	0.889	3.558	0.000	1.000	0.000	5.664	24.525	4	0	0.0
W1─S1─S3	1.000	0.000	0.000	37.687	1.000	0.142	0.000	0.000	4	0	0.0
W1─S1─S2	1.000	0.000	0.000	37.697	1.000	0.143	0.000	0.000	4	0	0.0
S1─W1─W3	1.000	0.000	0.000	33.553	1.000	0.142	0.000	0.000	4	0	0.0

Table 137.

SW potential parameters for single-layer 1H-WS₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Atom types in the first column are displayed in Figure 2 (with M = W and X = S).

The parameters for the two-body SW potential used by GULP are shown in Table 131 . The parameters for the three-body SW potential used by GULP are shown in Table 132 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 133 . We note that 12 atom types have been introduced for the simulation of the single-layer 1H-WO₂ using LAMMPS, because the angles around atom W in Figure 1 (with M = W and X = O) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS [13, 14]. According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Figure 2 (with M = W and X = O) shows that, for 1H-WO₂, we can differentiate these angles around the W atom by assigning these six neighboring O atoms with different atom types. It can be found that 12 atom types are necessary for the purpose of differentiating all 6 neighbors around 1 W atom.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-WS₂ under uniaxial tension at 1 and 300 K. Figure 66 shows the stress-strain curve for the tension of a single-layer 1H-WS₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-WS₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-WS₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 121.5 N/m along both armchair and zigzag directions. These values are in reasonable agreement with the ab initio results, e.g., 139.6 N/m from Ref. [18], or 148.5 N/m from Ref. [35]. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.21$ , which agrees with the ab initio value of 0.22 [18].

We have determined the nonlinear parameter to be $B=0.47d^4$ in Eq. (5) by fitting to the third-order nonlinear elastic constant $D$ from the ab initio calculations [35]. We have extracted the value of $D=\mathrm{}-502.9$ N/m by fitting the stress-strain relation along the armchair direction in the ab initio calculations to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −472.8 and −529.6 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 14.7 N/m at the ultimate strain of 0.22 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 14.1 N/m at the ultimate strain of 0.25 in the zigzag direction at the low temperature of 1 K.

Figure 67.

Phonon dispersion for single-layer 1H-WSe₂. (a) The VFF model is fitted to the three acoustic branches in the long wave limit along the $\Gamma \text{M}$ direction. The ab initio results (gray pentagons) are from Ref. [31]. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for single-layer 1H-WSe₂ along $\Gamma \text{MK}\Gamma$ .

35. 1H-WSe₂

Most existing theoretical studies on the single-layer 1H-WSe₂ are based on the first-principles calculations. Norouzzadeh and Singh provided one set of parameters for the SW potential for the single-layer 1H-WSe₂ [40]. In this section, we will develop both VFF model and the SW potential for the single-layer 1H-WSe₂.

The structure for the single-layer 1H-WSe₂ is shown in Figure 1 (with M = W and X = Se). Each W atom is surrounded by six Se atoms. These Se atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each Se atom is connected to three W atoms. The structural parameters are from [12], including the lattice constant $a=3.25$ Å and the bond length $d_{\text{W}-\text{Se}}=2.51$ Å. The resultant angles are ${\theta }_{\text{WSeSe}}=\mathrm{}{\theta }_{\text{SeWW}}=80.693°$ and ${\theta }_{\text{WSeS}{\text{e}}^{\prime }}=83.240°$ , in which atoms Se and Se′ are from different (top or bottom) groups.

Figure 68.

Stress-strain for single-layer 1H-WSe₂ of dimension $100\times 100$ Å along the armchair and zigzag directions.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{W}-\text{Se}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{W}-\text{Se}-\text{Se}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{W}-\text{Se}-\text{S}{\text{e}}^{\prime }}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{Se}-\text{W}-\text{W}}{\left(\text{Δ}\theta \right)}^2$
Parameter	8.286	8.513	8.513	7.719
$r_0$ or ${\theta }_0$	2.510	80.693	83.140	80.693

Table 138.

The VFF model for single-layer 1H-WSe₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

Table 138 shows three VFF terms for the 1H-WSe₂; one of which is the bond stretching interaction shown by Eq. (1), while the other two terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the $\text{Γ}\text{M}$ as shown in Figure 67(a) . The ab initio calculations for the phonon dispersion are from [31]. Similar phonon dispersion can also be found in other ab initio calculations [12, 33, 34, 39, 41]. Figure 67(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

	$A$ (eV)	$\rho$ (Å)	$B$ (Å^4)	$r_{\text{min}}(\mathbf{\AA })$	$r_{\text{max}}$ (Å)
W─Se	5.476	0.706	16.273	0.0	3.308

Table 139.

Two-body SW potential parameters for single-layer 1H-WSe₂ used by GULP [8] as expressed in Eq. (3).

	$K$ (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\min 12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{W}-\text{Se}-\text{Se}}$	25.607	80.693	0.706	0.706	0.0	3.308	0.0	3.308	0.0	3.953
${\theta }_{\text{W}-\text{Se}-\text{S}{\text{e}}^{\prime }}$	25.287	83.240	0.706	0.706	0.0	3.308	0.0	3.308	0.0	3.953
${\theta }_{\text{Se}-\text{W}-\text{W}}$	23.218	80.693	0.706	0.706	0.0	3.308	0.0	3.308	0.0	3.953

Table 140.

Three-body SW potential parameters for single-layer 1H-WSe₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

The parameters for the two-body SW potential used by GULP are shown in Table 139 . The parameters for the three-body SW potential used by GULP are shown in Table 140 . Parameters for the SW potential used by LAMMPS are listed in Table 141 . We note that 12 atom types have been introduced for the simulation of the single-layer 1H-WSe₂ using LAMMPS, because the angles around atom W in Figure 1 (with M = W and X = Se) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS [13, 14]. According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Figure 2 (with M = W and X = Se) shows that, for 1H-WSe₂, we can differentiate these angles around the W atom by assigning these six neighboring Se atoms with different atom types. It can be found that 12 atom types are necessary for the purpose of differentiating all 6 neighbors around 1 W atom.

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
W1─Se1─Se1	1.000	0.706	4.689	0.000	1.000	0.000	5.476	65.662	4	0	0.0
W1─Se1─Se3	1.000	0.000	0.000	25.607	1.000	0.162	0.000	0.000	4	0	0.0
W1─Se1─Se2	1.000	0.000	0.000	25.287	1.000	0.118	0.000	0.000	4	0	0.0
Se1─W1─W3	1.000	0.000	0.000	23.218	1.000	0.162	0.000	0.000	4	0	0.0

Table 141.

SW potential parameters for single-layer 1H-WSe₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Atom types in the first column are displayed in Figure 2 (with M = W and X = Se).

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-WSe₂ under uniaxial tension at 1 and 300 K. Figure 68 shows the stress-strain curve for the tension of a single-layer 1H-WSe₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-WSe₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-WSe₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 124.1 and 123 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. These values are in reasonably agreement with the ab initio results, e.g., 116 N/m from [18], or 126.2 N/m from [35]. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.20$ , which agrees with the ab initio value of 0.19 [18].

We have determined the nonlinear parameter to be $B=0.41d^4$ in Eq. (5) by fitting to the third-order nonlinear elastic constant $D$ from the ab initio calculations [17]. We have extracted the value of $D=\mathrm{}-413.1$ N/m by fitting the stress-strain relation along the armchair direction in the ab initio calculations to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −400.4 and −444.3 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about $\text{17}.\text{8 N/m}$ at the ultimate strain of 0.25 in the armchair direction at the low temperature of 1 K. The ultimate stress is about $\text{17}.\text{1 N/m}$ at the ultimate strain of 0.29 in the zigzag direction at the low temperature of 1 K.

Figure 69.

Phonon dispersion for single-layer 1H-WTe₂. (a) The VFF model is fitted to the three acoustic branches in the long wave limit along the $\text{Γ}\text{M}$ direction. The ab initio results (gray pentagons) are from [42]. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for single-layer 1H-WTe₂ along $\text{Γ}\text{MK}\text{Γ}$ .

36. 1H-WTe₂

Most existing theoretical studies on the single-layer 1H-WTe₂ are based on the first-principles calculations. In this section, we will develop both VFF model and the SW potential for the single-layer 1H-WTe₂.

The bulk WTe₂ has the trigonally coordinated H phase structure [43]. However, it has been predicted that the structure of the single-layer WTe₂ can be either the trigonally coordinated H phase [12] or the octahedrally coordinated $T_d$ phase [44–47], with $T_d$ phase as the more stable structure [42]. We will thus consider both phases in the present paper. This section is devoted to the H phase for the WTe₂ (1H-WTe₂), while the SW potential for the $T_d$ -WTe₂ (1T-WTe₂) is presented in another section.

Figure 70.

Stress-strain for single-layer 1H-WTe₂ of dimension $100\times 100$ Å along the armchair and zigzag directions.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{W}-\text{Te}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{W}-\text{Te}-\text{Te}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{W}-\text{Te}-\text{T}{\text{e}}^{\prime }}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{Te}-\text{W}-\text{W}}{\left(\text{Δ}\theta \right)}^2$
Parameter	5.483	7.016	7.016	5.718
$r_0$ or ${\theta }_0$	2.730	81.111	82.686	81.111

Table 142.

The VFF model for single-layer 1H-WTe₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

The structure for the single-layer 1H-WTe₂ is shown in Figure 1 (with M = W and X = Te). Each W atom is surrounded by six Te atoms. These Te atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each Te atom is connected to three W atoms. The structural parameters are from [42], including the lattice constant $a=3.55$ Å and the bond length $d_{\text{W}-\text{Te}}=2.73$ Å. The resultant angles are ${\theta }_{\text{WTeTe}}=\mathrm{}{\theta }_{\text{TeWW}}=81.111°$ and ${\theta }_{\text{WTeT}{\text{e}}^{\prime }}=82.686°$ , in which atoms Te and Te′ are from different (top or bottom) groups.

	$A$ (eV)	$\rho$ (Å)	$B$ (Å^4)	$r_{\text{min}}(\mathrm{\AA })$	$r_{\text{max}}$ (Å)
W─Te	4.326	0.778	22.774	0.0	3.604

Table 143.

Two-body SW potential parameters for single-layer 1H-WTe₂ used by GULP [8] as expressed in Eq. (3).

	$K$ (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{W}-\text{Te}-\text{Te}}$	21.313	81.111	0.778	0.778	0.0	3.604	0.0	3.604	0.0	4.305
${\theta }_{\text{W}-\text{Te}-\text{T}{\text{e}}^{\prime }}$	21.147	82.686	0.778	0.778	0.0	3.604	0.0	3.604	0.0	4.305
${\theta }_{\text{Te}-\text{W}-\text{W}}$	17.370	81.111	0.778	0.778	0.0	3.604	0.0	3.604	0.0	4.305

Table 144.

Three-body SW potential parameters for single-layer 1H-WTe₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

Table 142 shows the VFF terms for the 1H-WTe₂; one of which is the bond stretching interaction shown by Eq. (1), while the other terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the $\text{Γ}\text{M}$ as shown in Figure 69(a) . The ab initio calculations for the phonon dispersion are from [42]. Similar phonon dispersion can also be found in other ab initio calculations [12]. Figure 69(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\text{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
W1─Te1─Te1	1.000	0.778	4.632	0.000	1.000	0.000	4.326	62.148	4	0	0.0
W1─Te1─Te3	1.000	0.000	0.000	21.313	1.000	0.155	0.000	0.000	4	0	0.0
W1─Te1─Te2	1.000	0.000	0.000	21.147	1.000	0.127	0.000	0.000	4	0	0.0
Te1─W1─W3	1.000	0.000	0.000	17.370	1.000	0.155	0.000	0.000	4	0	0.0

Table 145.

SW potential parameters for single-layer 1H-WTe₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Atom types in the first column are displayed in Figure 2 (with M = W and X = Te).

The parameters for the two-body SW potential used by GULP are shown in Table 143 . The parameters for the three-body SW potential used by GULP are shown in Table 144 . Parameters for the SW potential used by LAMMPS are listed in Table 145 . We note that 12 atom types have been introduced for the simulation of the single-layer 1H-WTe₂ using LAMMPS, because the angles around atom W in Figure 1 (with M = W and X = Te) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS [13, 14]. According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Figure 2 (with M = W and X = Te) shows that, for 1H-WTe₂, we can differentiate these angles around the W atom by assigning these six neighboring Te atoms with different atom types. It can be found that 12 atom types are necessary for the purpose of differentiating all 6 neighbors around 1 W atom.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-WTe₂ under uniaxial tension at 1 and 300 K. Figure 70 shows the stress-strain curve for the tension of a single-layer 1H-WTe₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-WTe₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-WTe₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 82.7 and 81.9 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. These values are in reasonably agreement with the ab initio results, e.g., 86.4 N/m from [18] or 93.9 N/m from [35]. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.20$ , which agrees with the ab initio value of 0.18 [18].

We have determined the nonlinear parameter to be $B=0.41d^4$ in Eq. (5) by fitting to the third-order nonlinear elastic constant $D$ from the ab initio calculations [35]. We have extracted the value of $D=\mathrm{}-280.3$ N/m by fitting the stress-strain relation along the armchair direction in the ab initio calculations to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −269.4 and −297.9 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about $\text{11}.\text{8 N/m}$ at the ultimate strain of 0.25 in the armchair direction at the low temperature of 1 K. The ultimate stress is about $\text{11}.\text{3 N/m}$ at the ultimate strain of 0.29 in the zigzag direction at the low temperature of 1 K.

Figure 71.

Configuration of the 1T-MX₂ in the 1T phase. (a) Top view. The unit cell is highlighted by a red parallelogram. The armchair direction is defined to be along the horizontal direction. The zigzag direction is along the vertical direction. (b) Enlarged view of atoms in the blue box in (a). Each M atom is surrounded by six X atoms, which are categorized into the top and bottom groups. Atoms X 1, 3, and 5 are from the top group, while atoms X 2, 4, and 6 are from the bottom group. M atoms are represented by larger gray balls. X atoms are represented by smaller yellow balls.

Figure 72.

Phonon spectrum for single-layer 1T-ScO₂. (a) Phonon dispersion along the $\text{Γ}\text{M}$ direction in the Brillouin zone. The results from the VFF model (lines) are comparable with the ab initio results (pentagons) from Ref. [12]. (b) The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

37. 1T-ScO₂

Most existing theoretical studies on the single-layer 1T-ScO₂ are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-ScO₂.

The structure for the single-layer 1T-ScO₂ is shown in Figure 71 (with M = Sc and X = O). Each Ni atom is surrounded by six O atoms. These O atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each O atom is connected to three Sc atoms. The structural parameters are from the first-principles calculations [12], including the lattice constant $a=3.22$ Å and the bond length $d_{\text{Sc}-\text{O}}=2.07$ Å. The resultant angles are ${\theta }_{\text{ScOO}}=102.115°$ with O atoms from the same (top or bottom) group and ${\theta }_{\text{OScSc}}=102.115°$ .

Figure 73.

Stress-strain for single-layer 1T-ScO₂ of dimension $100\times 100$ Å along the armchair and zigzag directions.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{Sc}-\text{O}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{Sc}-\text{O}-\text{O}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{O}-\text{Sc}-\text{Sc}}{\left(\text{Δ}\theta \right)}^2$
Parameter	11.926	3.258	3.258
$r_0$ or ${\theta }_0$	2.07	102.115	102.115

Table 146.

The VFF model for single-layer 1T-ScO₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

Table 146 shows three VFF terms for the single-layer 1T-ScO₂; one of which is the bond stretching interaction shown by Eq. (1), while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\text{Sc}-\text{O}-\text{O}}$ is for the angle ${\theta }_{\text{Sc}-\text{O}-\text{O}}$ with both O atoms from the same (top or bottom) group. These force constant parameters are determined by fitting to the two in-plane acoustic branches in the phonon dispersion along the ΓM as shown in Figure 72(a) . The ab initio calculations for the phonon dispersion are from [12]. Figure 72(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

	A (eV)	ρ(Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Sc─O	10.187	1.493	9.180	0.0	2.949

Table 147.

Two-body SW potential parameters for single-layer 1T-ScO₂ used by GULP [8] as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{Sc}-\text{O}-\text{O}}$	50.913	102.115	1.493	1.493	0.0	2.949	0.0	2.949	0.0	4.399
${\theta }_{\text{O}-\text{Sc}-\text{Sc}}$	50.913	102.115	1.493	1.493	0.0	2.949	0.0	2.949	0.0	4.399

Table 148.

Three-body SW potential parameters for single-layer 1T-ScO₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

The parameters for the two-body SW potential used by GULP are shown in Table 147 . The parameters for the three-body SW potential used by GULP are shown in Table 148 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 149 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-ScO₂ under uniaxial tension at 1 and 300 K. Figure 73 shows the stress-strain curve for the tension of a single-layer 1T-ScO₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-ScO₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-ScO₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 100.9 and 100.4 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.15$ .

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Sc─O1─O1	1.000	1.493	1.975	50.913	1.000	−0.210	10.187	1.847	4	0	0.0

Table 149.

SW potential parameters for single-layer 1T-ScO₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{Sc}-\text{S}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{Sc}-\text{S}-\text{S}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{S}-\text{Sc}-\text{Sc}}{\left(\text{Δ}\theta \right)}^2$
Parameter	3.512	1.593	1.593
$r_0$ or ${\theta }_0$	2.50	92.771	92.771

Table 150.

The VFF model for single-layer 1T-ScS₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Sc─S	3.516	1.443	19.531	0.0	3.450

Table 151.

Two-body SW potential parameters for single-layer 1T-ScS₂ used by GULP [8] as expressed in Eq. (3).

There is no available value for nonlinear quantities in the single-layer 1T-ScO₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −422.4 and −453.7 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 11.7 N/m at the ultimate strain of 0.22 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 11.3 N/m at the ultimate strain of 0.25 in the zigzag direction at the low temperature of 1 K.

Figure 74.

Phonon spectrum for single-layer 1T-ScS₂. (a) Phonon dispersion along the ΓM direction in the Brillouin zone. The results from the VFF model (lines) are comparable with the ab initio results (pentagons) from [12]. (b) The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

38. 1T-ScS₂

Most existing theoretical studies on the single-layer 1T-ScS₂ are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-ScS₂.

The structure for the single-layer 1T-ScS₂ is shown in Figure 71 (with M = Sc and X = S). Each Sc atom is surrounded by six S atoms. These S atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each S atom is connected to three Sc atoms. The structural parameters are from the first-principles calculations [12], including the lattice constant $a=3.62$ Å and the bond length $d_{\text{Sc}-\text{S}}=2.50$ Å. The resultant angle is ${\theta }_{\text{SScSc}}=92.771°$ and ${\theta }_{\text{ScSS}}=92.771°$ with S atoms from the same (top or bottom) group.

Figure 75.

Stress-strain for single-layer 1T-ScS₂ of dimension $100\times 100$ Å along the armchair and zigzag directions.

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{Sc}-\text{S}-\text{S}}$	16.674	92.771	1.443	1.443	0.0	3.450	0.0	3.450	0.0	4.945
${\theta }_{\text{S}-\text{Sc}-\text{Sc}}$	16.674	92.771	1.443	1.443	0.0	3.450	0.0	3.450	0.0	4.945

Table 152.

Three-body SW potential parameters for single-layer 1T-ScS₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

Table 150 shows three VFF terms for the single-layer 1T-ScS₂; one of which is the bond stretching interaction shown by Eq. (1), while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\text{Sc}-\text{S}-\text{S}}$ is for the angle ${\theta }_{\text{Sc}-\text{S}-\text{S}}$ with both S atoms from the same (top or bottom) group. These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the ΓM as shown in Figure 74(a) . The ab initio calculations for the phonon dispersion are from [12]. Figure 74(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Sc─S1─S1	1.000	1.443	2.390	16.674	1.000	−0.048	3.516	4.504	4	0	0.0

Table 153.

SW potential parameters for single-layer 1T-ScS₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

The parameters for the two-body SW potential used by GULP are shown in Table 151 . The parameters for the three-body SW potential used by GULP are shown in Table 152 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 153 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-ScS₂ under uniaxial tension at 1 and 300 K. Figure 75 shows the stress-strain curve for the tension of a single-layer 1T-ScS₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-ScS₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-ScS₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 30 and 29.9 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.17$ .

There is no available value for nonlinear quantities in the single-layer 1T-ScS₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −113.7 and −124.6 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 3.8 N/m at the ultimate strain of 0.23 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 3.6 N/m at the ultimate strain of 0.27 in the zigzag direction at the low temperature of 1 K.

39. 1T-ScSe₂

Most existing theoretical studies on the single-layer 1T-ScSe₂ are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-ScSe₂.

Figure 76.

Phonon spectrum for single-layer 1T-ScSe₂. (a) Phonon dispersion along the ΓM direction in the Brillouin zone. The results from the VFF model (lines) are comparable with the ab initio results (pentagons) from [12]. (b) The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

The structure for the single-layer 1T-ScSe₂ is shown in Figure 71 (with M = Sc and X = Se). Each Sc atom is surrounded by six Se atoms. These Se atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each Se atom is connected to three Sc atoms. The structural parameters are from the first-principles calculations [12], including the lattice constant $a=3.52$ Å and the bond length $d_{\text{Sc}-\text{Se}}=2.64$ Å. The resultant angle is ${\theta }_{\text{SeScSc}}=83.621°$ and ${\theta }_{\text{ScSeSe}}=83.621°$ with Se atoms from the same (top or bottom) group.

Table 154 shows three VFF terms for the single-layer 1T-ScSe₂, one of which is the bond stretching interaction shown by Eq. (1), while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\text{Sc}-\text{Se}-\text{Se}}$ is for the angle ${\theta }_{\text{Sc}-\text{Se}-\text{Se}}$ with both Se atoms from the same (top or bottom) group. These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the ΓM as shown in Figure 76(a) . The ab initio calculations for the phonon dispersion are from [12]. We note that the lowest-frequency branch around the Γ point from the VFF model is lower than the ab initio results. This branch is the flexural branch, which should be a quadratic dispersion. However, the ab initio calculations give a linear dispersion for the flexural branch due to the violation of the rigid rotational invariance in the first-principles package [20], so ab initio calculations typically overestimate the frequency of this branch. Figure 76(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

Figure 77.

Stress-strain for single-layer 1T-ScSe₂ of dimension $100\times 100$ Å along the armchair and zigzag directions.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{Sc}-\text{Se}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{Sc}-\text{Se}-\text{Se}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{Se}-\text{Sc}-\text{Sc}}{\left(\text{Δ}\theta \right)}^2$
Parameter	4.407	2.399	2.399
$r_0$ or ${\theta }_0$	2.64	83.621	83.621

Table 154.

The VFF model for single-layer 1T-ScSe₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Sc─Se	3.884	1.173	24.288	0.0	3.520

Table 155.

Two-body SW potential parameters for single-layer 1T-ScSe₂ used by GULP [8] as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{Sc}-\text{Se}-\text{Se}}$	17.479	83.621	1.173	1.173	0.0	3.520	0.0	3.520	0.0	4.808
${\theta }_{\text{Se}-\text{Sc}-\text{Sc}}$	17.479	83.621	1.173	1.173	0.0	3.520	0.0	3.520	0.0	4.808

Table 156.

Three-body SW potential parameters for single-layer 1T-ScSe₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

The parameters for the two-body SW potential used by GULP are shown in Table 155 . The parameters for the three-body SW potential used by GULP are shown in Table 156 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 157 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-ScSe₂ under uniaxial tension at 1 and 300 K. Figure 77 shows the stress-strain curve for the tension of a single-layer 1T-ScSe₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-ScSe₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-ScSe₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 36.4 and 36.3 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.20$ .

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Sc─Se1─Se1	1.000	1.173	3.000	17.479	1.000	0.111	3.884	12.814	4	0	0.0

Table 157.

SW potential parameters for single-layer 1T-ScSe₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{Sc}-\text{Te}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{Sc}-\text{Te}-\text{Te}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{Te}-\text{Sc}-\text{Sc}}{\left(\text{Δ}\theta \right)}^2$
Parameter	4.407	2.399	2.399
$r_0$ or ${\theta }_0$	2.85	81.481	81.481

Table 158.

The VFF model for single-layer 1T-ScTe₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Sc─Te	4.269	1.183	32.988	0.0	3.768

Table 159.

Two-body SW potential parameters for single-layer 1T-ScTe₂ used by GULP [8] as expressed in Eq. (3).

There is no available value for nonlinear quantities in the single-layer 1T-ScSe₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −113.7 and −130.3 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 5.3 N/m at the ultimate strain of 0.25 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 5 N/m at the ultimate strain of 0.29 in the zigzag direction at the low temperature of 1 K.

Figure 78.

Phonon spectrum for single-layer 1T-ScTe₂. (a) Phonon dispersion along the ΓM direction in the Brillouin zone. The results from the VFF model (lines) are comparable with the ab initio results (pentagons) from [12]. (b) The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

40. 1T-ScTe₂

Most existing theoretical studies on the single-layer 1T-ScTe₂ are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-ScTe₂.

The structure for the single-layer 1T-ScTe₂ is shown in Figure 71 (with M = Sc and X = Te). Each Sc atom is surrounded by six Te atoms. These Te atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each Te atom is connected to three Sc atoms. The structural parameters are from the first-principles calculations [12], including the lattice constant $a=3.72$ Å and the bond length $d_{\text{Sc}-\text{Te}}=2.85$ Å. The resultant angle is ${\theta }_{\text{TeScSc}}=81.481°$ and ${\theta }_{\text{ScTeTe}}=81.481°$ with Se atoms from the same (top or bottom) group.

Figure 79.

Stress-strain for single-layer 1T-ScTe₂ of dimension $100\times 100$ Å along the armchair and zigzag directions.

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{Sc}-\text{Te}-\text{Te}}$	16.139	81.481	1.183	1.183	0.0	3.768	0.0	3.768	0.0	5.082
${\theta }_{\text{Te}-\text{Sc}-\text{Sc}}$	16.139	81.481	1.183	1.183	0.0	3.768	0.0	3.768	0.0	5.082

Table 160.

Three-body SW potential parameters for single-layer 1T-ScTe₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

Table 158 shows three VFF terms for the single-layer 1T-ScTe₂; one of which is the bond stretching interaction shown by Eq. (1), while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\text{Sc}-\text{Te}-\text{Te}}$ is for the angle ${\theta }_{\text{Sc}-\text{Te}-\text{Te}}$ with both Te atoms from the same (top or bottom) group. These force constant parameters are determined by fitting to the two in-plane acoustic branches in the phonon dispersion along the ΓM as shown in Figure 78(a) . The ab initio calculations for the phonon dispersion are from [12]. Figure 78(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Sc─Te1─Te1	1.000	1.183	3.185	16.139	1.000	0.148	4.269	16.841	4	0	0.0

Table 161.

SW potential parameters for single-layer 1T-ScTe₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

The parameters for the two-body SW potential used by GULP are shown in Table 159 . The parameters for the three-body SW potential used by GULP are shown in Table 160 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 161 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-ScTe₂ under uniaxial tension at 1 and 300 K. Figure 79 shows the stress-strain curve for the tension of a single-layer 1T-ScTe₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-ScTe₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-ScTe₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 31.4 and 31.3 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.22$ .

There is no available value for nonlinear quantities in the single-layer 1T-ScTe₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −81.2 and −96.7 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 5.2 N/m at the ultimate strain of 0.27 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 5 N/m at the ultimate strain of 0.31 in the zigzag direction at the low temperature of 1 K.

Figure 80.

Stress-strain for single-layer 1T-TiS₂ of dimension $100\times 100$ Å along the armchair and zigzag directions.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{Ti}-\text{S}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{Ti}-\text{S}-\text{S}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{S}-\text{Ti}-\text{Ti}}{\left(\text{Δ}\theta \right)}^2$
Parameter	9.815	3.754	3.754
$r_0$ or ${\theta }_0$	2.390	87.984	87.984

Table 162.

The VFF model for single-layer 1T-TiS₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

41. 1T-TiS₂

Most existing theoretical studies on the single-layer 1T-TiS₂ are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-TiS₂.

The structure for the single-layer 1T-TiS₂ is shown in Figure 71 (with M = Ti and X = S). Each Ti atom is surrounded by six S atoms. These S atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each S atom is connected to three Ti atoms. The structural parameters are from the first-principles calculations [12], including the lattice constant $a=3.32$ Å and the bond length $d_{\text{Ti}-\text{S}}=2.39$ Å. The resultant angles are ${\theta }_{\text{TiSS}}=87.984°$ with S atoms from the same (top or bottom) group and ${\theta }_{\text{STiTi}}=87.984°$ .

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Ti-S	7.958	1.210	16.314	0.0	3.240

Table 163.

Two-body SW potential parameters for single-layer 1T-TiS₂ used by GULP [8] as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{Ti}-\text{S}-\text{S}}$	32.377	87.984	1.210	1.210	0.0	3.240	0.0	3.240	0.0	4.535
${\theta }_{\text{S}-\text{Ti}-\text{Ti}}$	32.377	87.984	1.210	1.210	0.0	3.240	0.0	3.240	0.0	4.535

Table 164.

Three-body SW potential parameters for single-layer 1T-TiS₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

Table 162 shows three VFF terms for the single-layer 1T-TiS₂; one of which is the bond stretching interaction shown by Eq. (1), while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\text{Ti}-\text{S}-\text{S}}$ is for the angle ${\theta }_{\text{Ti}-\text{S}-\text{S}}$ with both S atoms from the same (top or bottom) group. We find that there are actually only two parameters in the VFF model, so we can determine their value by fitting to the Young’s modulus and the Poisson’s ratio of the system. The ab initio calculations have predicted the Young’s modulus to be 85 N/m and the Poisson’s ratio as 0.20 [48].

The parameters for the two-body SW potential used by GULP are shown in Table 163 . The parameters for the three-body SW potential used by GULP are shown in Table 164 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 165 .

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Ti─S─S	1.000	1.210	2.677	32.377	1.000	0.035	7.958	7.602	4	0	0.0

Table 165.

SW potential parameters for single-layer 1T-TiS₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Figure 81.

Phonon spectrum for single-layer 1T-TiS₂ along the ΓMKΓ direction in the Brillouin zone. The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-TiS₂ under uniaxial tension at 1 and 300 K. Figure 80 shows the stress-strain curve for the tension of a single-layer 1T-TiS₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-TiS₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-TiS₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 75 and 74.6 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.20$ . The fitted Young’s modulus value is about 10% smaller than the ab initio result of 85 N/m [48], as only short-range interactions are considered in the present work. The long-range interactions are ignored, which typically leads to about 10% underestimation for the value of the Young’s modulus.

There is no available value for nonlinear quantities in the single-layer 1T-TiS₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −220.8 and −264.4 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 10.8 N/m at the ultimate strain of 0.25 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 10.4 N/m at the ultimate strain of 0.29 in the zigzag direction at the low temperature of 1 K.

Figure 82.

Stress-strain for single-layer 1T-TiSe₂ of dimension $100\times 100$ Å along the armchair and zigzag directions.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{Ti}-\text{Se}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{Ti}-\text{Se}-\text{Se}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{Se}-\text{Ti}-\text{Ti}}{\left(\text{Δ}\theta \right)}^2$
Parameter	7.712	3.363	3.363
$r_0$ or ${\theta }_0$	2.510	86.199	86.199

Table 166.

The VFF model for single-layer 1T-TiSe₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

Figure 81 shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Ti─Se	6.582	1.207	19.846	0.0	3.380

Table 167.

Two-body SW potential parameters for single-layer 1T-TiSe₂ used by GULP [8] as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{Ti}-\text{Se}-\text{Se}}$	27.044	86.199	1.207	1.207	0.0	3.380	0.0	3.380	0.0	4.685
${\theta }_{\text{Se}-\text{Ti}-\text{Ti}}$	27.044	86.199	1.207	1.207	0.0	3.380	0.0	3.380	0.0	4.685

Table 168.

Three-body SW potential parameters for single-layer 1T-TiSe₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

42. 1T-TiSe₂

Most existing theoretical studies on the single-layer 1T-TiSe₂ are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-TiSe₂.

The structure for the single-layer 1T-TiSe₂ is shown in Figure 71 (with M = Ti and X = Se). Each Ti atom is surrounded by six Se atoms. These Se atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each Se atom is connected to three Ti atoms. The structural parameters are from the first-principles calculations [12], including the lattice constant $a=3.43$ Å and the bond length $d_{\text{Ti}-\text{Se}}=2.51$ Å. The resultant angles are ${\theta }_{\text{TiSeSe}}=86.199°$ with Se atoms from the same (top or bottom) group and ${\theta }_{\text{SeTiTi}}=86.199°$ .

Table 166 shows three VFF terms for the single-layer 1T-TiSe₂; one of which is the bond stretching interaction shown by Eq. (1), while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\text{Ti}-\text{Se}-\text{Se}}$ is for the angle ${\theta }_{\text{Ti}-\text{Se}-\text{Se}}$ with both Se atoms from the same (top or bottom) group. We find that there are actually only two parameters in the VFF model, so we can determine their value by fitting to the Young’s modulus and the Poisson’s ratio of the system. The ab initio calculations have predicted the Young’s modulus to be 70 N/m and the Poisson’s ratio as 0.20 [48].

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Ti─Se─Se	1.000	1.207	2.801	27.044	1.000	0.066	6.582	9.362	4	0	0.0

Table 169.

SW potential parameters for single-layer 1T-TiSe₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Figure 83.

Phonon spectrum for single-layer 1T-TiSe₂ along the ΓMKΓ direction in the Brillouin zone. The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 167 . The parameters for the three-body SW potential used by GULP are shown in Table 168 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 169 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-TiSe₂ under uniaxial tension at 1 and 300 K. Figure 82 shows the stress-strain curve for the tension of a single-layer 1T-TiSe₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-TiSe₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-TiSe₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 59.2 and 58.9 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.20$ . The fitted Young’s modulus value is about 10% smaller than the ab initio result of 70 N/m [48], as only short-range interactions are considered in the present work. The long-range interactions are ignored, which typically leads to about 10% underestimation for the value of the Young’s modulus.

Figure 84.

Stress-strain for single-layer 1T-TiTe₂ of dimension $100\times 100$ Å along the armchair and zigzag directions.

There is no available value for nonlinear quantities in the single-layer 1T-TiSe₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −166.5 and −201.6 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 8.7 N/m at the ultimate strain of 0.25 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 8.3 N/m at the ultimate strain of 0.29 in the zigzag direction at the low temperature of 1 K.

Figure 83 shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

43. 1T-TiTe₂

Most existing theoretical studies on the single-layer 1T-TiTe₂ are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-TiTe₂.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{Ti}-\text{Te}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{Ti}-\text{Te}-\text{Te}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{Te}-\text{Ti}-\text{Ti}}{\left(\text{Δ}\theta \right)}^2$
Parameter	3.758	3.217	3.217
$r_0$ or ${\theta }_0$	2.730	83.621	83.621

Table 170.

The VFF model for single-layer 1T-TiTe₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Ti─Te	3.542	1.213	27.773	0.0	3.640

Table 171.

Two-body SW potential parameters for single-layer 1T-TiTe₂ used by GULP [8] as expressed in Eq. (3).

The structure for the single-layer 1T-TiTe₂ is shown in Figure 71 (with M = Ti and X = Te). Each Ti atom is surrounded by six Te atoms. These Te atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each Te atom is connected to three Ti atoms. The structural parameters are from the first-principles calculations [12], including the lattice constant $a=3.64$ Å and the bond length $d_{\text{Ti}-\text{Te}}=2.73$ Å. The resultant angles are ${\theta }_{\text{TiTeTe}}=83.621°$ with Te atoms from the same (top or bottom) group and ${\theta }_{\text{TeTiTi}}=83.621°$ .

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{Ti}-\text{Te}-\text{Te}}$	23.439	83.621	1.213	1.213	0.0	3.640	0.0	3.640	0.0	4.972
${\theta }_{\text{Te}-\text{Ti}-\text{Ti}}$	23.439	83.621	1.213	1.213	0.0	3.640	0.0	3.640	0.0	4.972

Table 172.

Three-body SW potential parameters for single-layer 1T-TiTe₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Ti─Te─Te	1.000	1.213	3.000	23.439	1.000	0.111	3.542	12.814	4	0	0.0

Table 173.

SW potential parameters for single-layer 1T-TiTe₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Figure 85.

Phonon spectrum for single-layer 1T-TiTe₂ along the ΓMKΓ direction in the Brillouin zone. The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

Table 170 shows three VFF terms for the single-layer 1T-TiTe₂; one of which is the bond stretching interaction shown by Eq. (1), while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\text{Ti}-\text{Te}-\text{Te}}$ is for the angle ${\theta }_{\text{Ti}-\text{Te}-\text{Te}}$ with both Te atoms from the same (top or bottom) group. We find that there are actually only two parameters in the VFF model, so we can determine their value by fitting to the Young’s modulus and the Poisson’s ratio of the system. The ab initio calculations have predicted the Young’s modulus to be 46 N/m and the Poisson’s ratio as 0.15 [48].

The parameters for the two-body SW potential used by GULP are shown in Table 171 . The parameters for the three-body SW potential used by GULP are shown in Table 172 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 173 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-TiTe₂ under uniaxial tension at 1 and 300 K. Figure 84 shows the stress-strain curve for the tension of a single-layer 1T-TiTe₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-TiTe₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-TiTe₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 41.4 and 41.2 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.15$ . The fitted Young’s modulus value is about 10% smaller than the ab initio result of 46 N/m [48], as only short-range interactions are considered in the present work. The long-range interactions are ignored, which typically leads to about 10% underestimation for the value of the Young’s modulus.

There is no available value for nonlinear quantities in the single-layer 1T-TiTe₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −161.3 and −181.4 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 4.9 N/m at the ultimate strain of 0.22 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 4.7 N/m at the ultimate strain of 0.25 in the zigzag direction at the low temperature of 1 K.

Figure 85 shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

44. 1T-VS₂

Most existing theoretical studies on the single-layer 1T-VS₂ are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-VS₂.

The structure for the single-layer 1T-VS₂ is shown in Figure 71 (with M = V and X = S). Each V atom is surrounded by six S atoms. These S atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each S atom is connected to three V atoms. The structural parameters are from the first-principles calculations [12], including the lattice constant $a=3.10$ Å and the bond length $d_{\text{V}-\text{S}}=2.31$ Å. The resultant angles are ${\theta }_{\text{VSS}}=84.288°$ with S atoms from the same (top or bottom) group and ${\theta }_{\text{SVV}}=84.288°$ .

Figure 86.

Phonon spectrum for single-layer 1T-VS₂. (a) Phonon dispersion along the ΓM direction in the Brillouin zone. The results from the VFF model (lines) are comparable with the ab initio results (pentagons) from [12]. (b) The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

Table 174 shows three VFF terms for the single-layer 1T-VS₂; one of which is the bond stretching interaction shown by Eq. (1), while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\text{V}-\text{S}-\text{S}}$ is for the angle ${\theta }_{\text{V}-\text{S}-\text{S}}$ with both S atoms from the same (top or bottom) group. These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the ΓM as shown in Figure 86(a) . The ab initio calculations for the phonon dispersion are from [12]. The lowest acoustic branch (flexural mode) is linear and very close to the in-plane transverse acoustic branch in the ab initio calculations, which may be due to the violation of the rigid rotational invariance [20]. Figure 86(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

Figure 87.

Stress-strain for single-layer 1T-VS₂ of dimension $100\times 100$ Å along the armchair and zigzag directions.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{V}-\text{S}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{V}-\text{S}-\text{S}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{S}-\text{V}-\text{V}}{\left(\text{Δ}\theta \right)}^2$
Parameter	11.562	4.237	4.237
$r_0$ or ${\theta }_0$	2.310	84.288	84.288

Table 174.

The VFF model for single-layer 1T-VS₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

The parameters for the two-body SW potential used by GULP are shown in Table 175 . The parameters for the three-body SW potential used by GULP are shown in Table 176 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 177 .

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
V─S	7.943	1.048	14.237	0.0	3.088

Table 175.

Two-body SW potential parameters for single-layer 1T-VS₂ used by GULP [8] as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{V}-\text{S}-\text{S}}$	31.659	84.288	1.048	1.048	0.0	3.088	0.0	3.088	0.0	4.235
${\theta }_{\text{S}-\text{V}-\text{V}}$	31.659	84.288	1.048	1.048	0.0	3.088	0.0	3.088	0.0	4.235

Table 176.

Three-body SW potential parameters for single-layer 1T-VS₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-VS₂ under uniaxial tension at 1 and 300 K. Figure 87 shows the stress-strain curve for the tension of a single-layer 1T-VS₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-VS₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-VS₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 87.1 and 86.8 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.21$ .

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
V─S1─S1	1.000	1.048	2.946	31.659	1.000	0.100	7.943	11.797	4	0	0.0

Table 177.

SW potential parameters for single-layer 1T-VS₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Figure 88.

Phonon spectrum for single-layer 1T-VSe₂. (a) Phonon dispersion along the ΓM direction in the Brillouin zone. The results from the VFF model (lines) are comparable with the experiment data (pentagons) from [12]. (b) The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

There is no available value for nonlinear quantities in the single-layer 1T-VS₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with E as the Young’s modulus. The values of D from the present SW potential are −230.5 and −283.6 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 13.3 N/m at the ultimate strain of 0.26 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 12.7 N/m at the ultimate strain of 0.30 in the zigzag direction at the low temperature of 1 K.

Figure 89.

Stress-strain for single-layer 1T-VSe₂ of dimension $100\times 100$ Å along the armchair and zigzag directions.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{V}-\text{Se}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{V}-\text{Se}-\text{Se}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{Se}-\text{V}-\text{V}}{\left(\text{Δ}\theta \right)}^2$
Parameter	11.562	4.237	4.237
$r_0$ or ${\theta }_0$	2.440	83.201	83.201

Table 178.

The VFF model for single-layer 1T-VSe₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

45. 1T-VSe₂

Most existing theoretical studies on the single-layer 1T-VSe₂ are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-VSe₂.

The structure for the single-layer 1T-VSe₂ is shown in Figure 71 (with M = V and X = Se). Each V atom is surrounded by six Se atoms. These Se atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each Se atom is connected to three V atoms. The structural parameters are from the first-principles calculations [12], including the lattice constant $a=3.24$ Å and the bond length $d_{\text{V}-\text{Se}}=2.44$ Å. The resultant angles are ${\theta }_{\text{VSeSe}}=83.201°$ with Se atoms from the same (top or bottom) group and ${\theta }_{\text{SeVV}}=83.201°$ .

	A (eV)	ρ (Å)	B (Å	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
V─Se	8.606	1.070	17.723	0.0	3.248

Table 179.

Two-body SW potential parameters for single-layer 1T-VSe₂ used by GULP [8] as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{V}-\text{Se}-\text{Se}}$	30.387	83.201	1.070	1.070	0.0	3.248	0.0	3.248	0.0	4.426
${\theta }_{\text{Se}-\text{V}-\text{V}}$	30.387	83.201	1.070	1.070	0.0	3.248	0.0	3.248	0.0	4.426

Table 180.

Three-body SW potential parameters for single-layer 1T-VSe₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

Table 178 shows three VFF terms for the single-layer 1T-VSe₂; one of which is the bond stretching interaction shown by Eq. (1), while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\text{V}-\text{Se}-\text{Se}}$ is for the angle ${\theta }_{\text{V}-\text{Se}-\text{Se}}$ with both Se atoms from the same (top or bottom) group. These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the ΓM as shown in Figure 88(a) . The ab initio calculations for the phonon dispersion are from [12]. The lowest acoustic branch (flexural mode) is almost linear in the ab initio calculations, which may be due to the violation of the rigid rotational invariance [20]. Figure 88(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 179 . The parameters for the three-body SW potential used by GULP are shown in Table 180 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 181 .

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
V─Se1─Se1	1.000	1.070	3.035	30.387	1.000	0.118	8.606	13.507	4	0	0.0

Table 181.

SW potential parameters for single-layer 1T-VSe₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-VSe₂ under uniaxial tension at 1 and 300 K. Figure 89 shows the stress-strain curve for the tension of a single-layer 1T-VSe₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-VSe₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-VSe₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 78.4 and 78.1 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.22$ .

There is no available value for nonlinear quantities in the single-layer 1T-VSe₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −168.5 and −218.6 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 13.1 N/m at the ultimate strain of 0.27 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 12.5 N/m at the ultimate strain of 0.32 in the zigzag direction at the low temperature of 1 K.

46. 1T-VTe₂

Most existing theoretical studies on the single-layer 1T-VTe₂ are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-VTe₂.

The structure for the single-layer 1T-VTe₂ is shown in Figure 71 (with M = V and X = Te). Each V atom is surrounded by six Te atoms. These Te atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each Te atom is connected to three V atoms. The structural parameters are from the first-principles calculations [12], including the lattice constant $a=3.46$ Å and the bond length $d_{\text{V}-\text{Te}}=2.64$ Å. The resultant angles are ${\theta }_{\text{VTeTe}}=81.885°$ with Te atoms from the same (top or bottom) group and ${\theta }_{\text{TeVV}}=81.885°$ .

Figure 90.

Stress-strain for single-layer 1T-VTe₂ of dimension $100\times 100$ Å along the armchair and zigzag directions.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{V}-\text{Te}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{V}-\text{Te}-\text{Te}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{Te}-\text{V}-\text{V}}{\left(\text{Δ}\theta \right)}^2$
Parameter	10.476	3.814	3.814
$r_0$ or ${\theta }_0$	2.640	81.885	81.885

Table 182.

The VFF model for single-layer 1T-VTe₂. The second line gives an explicit expression for each VFF term.

The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

Table 182 shows three VFF terms for the single-layer 1T-VTe₂; one of which is the bond stretching interaction shown by Eq. (1), while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\text{V}-\text{Te}-\text{Te}}$ is for the angle ${\theta }_{\text{V}-\text{Te}-\text{Te}}$ with both Te atoms from the same (top or bottom) group. We find that there are actually only two parameters in the VFF model, so we can determine their value by fitting to the Young’s modulus and the Poisson’s ratio of the system. The ab initio calculations have predicted the Young’s modulus to be 67 N/m and the Poisson’s ratio as 0.24 [48].

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
V─Te	8.805	1.110	24.288	0.0	3.496

Table 183.

Two-body SW potential parameters for single-layer 1T-VTe₂ used by GULP [8] as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{V}-\text{Te}-\text{Te}}$	26.043	81.885	1.110	1.110	0.0	3.496	0.0	3.496	0.0	4.726
${\theta }_{\text{Te}-\text{V}-\text{V}}$	26.043	81.885	1.110	1.110	0.0	3.496	0.0	3.496	0.0	4.726

Table 184.

Three-body SW potential parameters for single-layer 1T-VTe₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

The parameters for the two-body SW potential used by GULP are shown in Table 183 . The parameters for the three-body SW potential used by GULP are shown in Table 184 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 185 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-VTe₂ under uniaxial tension at 1 and 300 K. Figure 90 shows the stress-strain curve for the tension of a single-layer 1T-VTe₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-VTe₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-VTe₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 61.2 and 61.0 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.24$ . The fitted Young’s modulus value is about 10% smaller than the ab initio result of 67 N/m [48], as only short-range interactions are considered in the present work. The long-range interactions are ignored, which typically leads to about 10% underestimation for the value of the Young’s modulus.

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
V─Te1─Te1	1.000	1.110	3.149	26.043	1.000	0.141	8.805	15.980	4	0	0.0

Table 185.

SW potential parameters for single-layer 1T-VTe₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Figure 91.

Phonon spectrum for single-layer 1T-VTe₂ along the ΓMKΓ direction in the Brillouin zone. The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

There is no available value for nonlinear quantities in the single-layer 1T-VTe₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −95.8 and −135.6 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 11.5 N/m at the ultimate strain of 0.30 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 11.0 N/m at the ultimate strain of 0.34 in the zigzag direction at the low temperature of 1 K.

Figure 91 shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

Figure 92.

Phonon spectrum for single-layer 1T-MnO₂. (a) Phonon dispersion along the ΓM direction in the Brillouin zone. The results from the VFF model (lines) are comparable with the ab initio results (pentagons) from [12]. (b) The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

47. 1T-MnO₂

Most existing theoretical studies on the single-layer 1T-MnO₂ are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-MnO₂.

The structure for the single-layer 1T-MnO₂ is shown in Figure 71 (with M = Mn and X = O). Each Mn atom is surrounded by six O atoms. These O atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each O atom is connected to three Mn atoms. The structural parameters are from the first-principles calculations [12], including the lattice constant $a=2.82$ Å and the bond length $d_{\text{Mn}-\text{O}}=1.88$ Å. The resultant angles are ${\theta }_{\text{MnOO}}=97.181°$ with O atoms from the same (top or bottom) group and ${\theta }_{\text{OMnMn}}=97.181°$ .

Figure 93.

Stress-strain for single-layer 1T-MnO₂ of dimension $100\times 100$ Å along the armchair and zigzag directions.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{Mn}-\text{O}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{Mn}-\text{O}-\text{O}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{O}-\text{Mn}-\text{Mn}}{\left(\text{Δ}\theta \right)}^2$
Parameter	15.371	4.822	4.822
$r_0$ or ${\theta }_0$	1.88	97.181	97.181

Table 186.

The VFF model for single-layer 1T-MnO₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

Table 186 shows three VFF terms for the single-layer 1T-MnO₂; one of which is the bond stretching interaction shown by Eq. (1), while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\text{Mn}-\text{O}-\text{O}}$ is for the angle ${\theta }_{\text{Mn}-\text{O}-\text{O}}$ with both O atoms from the same (top or bottom) group. These force constant parameters are determined by fitting to the two in-plane acoustic branches in the phonon dispersion along the ΓM as shown in Figure 92(a) . The ab initio calculations for the phonon dispersion are from [12]. Figure 92(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Mn─O	9.675	1.212	6.246	0.0	2.635

Table 187.

Two-body SW potential parameters for single-layer 1T-MnO₂ used by GULP [8] as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{Mn}-\text{O}-\text{O}}$	60.755	97.181	1.212	1.212	0.0	2.635	0.0	2.635	0.0	3.852
${\theta }_{\text{O}-\text{Mn}-\text{Mn}}$	60.755	97.181	1.212	1.212	0.0	2.635	0.0	2.635	0.0	3.852

Table 188.

Three-body SW potential parameters for single-layer 1T-MnO₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

The parameters for the two-body SW potential used by GULP are shown in Table 187 . The parameters for the three-body SW potential used by GULP are shown in Table 188 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 189 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-MnO₂ under uniaxial tension at 1 and 300 K. Figure 93 shows the stress-strain curve for the tension of a single-layer 1T-MnO₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-MnO₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-MnO₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 156.3 and 155.4 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.12$ .

	$\mathbf{\epsilon }$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Mn─O1─O1	1.000	1.212	2.175	60.755	1.000	-0.125	9.675	2.899	4	0	0.0

Table 189.

SW potential parameters for single-layer 1T-MnO₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{Mn}-\text{S}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{Mn}-\text{S}-\text{S}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{S}-\text{Mn}-\text{Mn}}{\left(\text{Δ}\theta \right)}^2$
Parameter	4.407	2.399	2.399
$r_0$ or ${\theta }_0$	2.27	86.822	86.822

Table 190.

The VFF model for single-layer 1T-MnS₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

There is no available value for nonlinear quantities in the single-layer 1T-MnO₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −711.7 and −756.1 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 16.8 N/m at the ultimate strain of 0.21 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 16.2 N/m at the ultimate strain of 0.24 in the zigzag direction at the low temperature of 1 K.

48. 1T-MnS₂

Most existing theoretical studies on the single-layer 1T-MnS₂ are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-MnS₂.

The structure for the single-layer 1T-MnS₂ is shown in Figure 71 (with M = Mn and X = S). Each Mn atom is surrounded by six S atoms. These S atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each S atom is connected to three Mn atoms. The structural parameters are from the first-principles calculations [12], including the lattice constant $a=3.12$ Å and the bond length $d_{\text{Mn}-\text{S}}=2.27$ Å. The resultant angles are ${\theta }_{\text{MnSS}}=86.822°$ with S atoms from the same (top or bottom) group and ${\theta }_{\text{SMnMn}}=86.822°$ .

Figure 94.

Phonon spectrum for single-layer 1T-MnS₂. (a) Phonon dispersion along the ΓM direction in the Brillouin zone. The results from the VFF model (lines) are comparable with the ab initio results (pentagons) from [12]. (b) The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Mn─S	3.127	1.111	13.276	0.0	3.064

Table 191.

Two-body SW potential parameters for single-layer 1T-MnS₂ used by GULP [8] as expressed in Eq. (3).

Figure 95.

Stress-strain for single-layer 1T-MnS₂ of dimension $100\times 100$ Å along the armchair and zigzag directions.

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{Mn}-\text{S}-\text{S}}$	19.765	86.822	1.111	1.111	0.0	3.064	0.0	3.064	0.0	4.262
${\theta }_{\text{S}-\text{Mn}-\text{Mn}}$	19.765	86.822	1.111	1.111	0.0	3.064	0.0	3.064	0.0	4.262

Table 192.

Three-body SW potential parameters for single-layer 1T-MnS₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

Table 190 shows three VFF terms for the single-layer 1T-MnS₂; one of which is the bond stretching interaction shown by Eq. (1), while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\text{Mn}-\text{S}-\text{S}}$ is for the angle ${\theta }_{\text{Mn}-\text{S}-\text{S}}$ with both S atoms from the same (top or bottom) group. These force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the ΓM as shown in Figure 94(a) . The ab initio calculations for the phonon dispersion are from [12]. Figure 94(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Mn─S1─S1	1.000	1.111	2.757	19.765	1.000	0.055	3.127	8.700	4	0	0.0

Table 193.

SW potential parameters for single-layer 1T-MnS₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

The parameters for the two-body SW potential used by GULP are shown in Table 191 . The parameters for the three-body SW potential used by GULP are shown in Table 192 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 193 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-MnS₂ under uniaxial tension at 1 and 300 K. Figure 95 shows the stress-strain curve for the tension of a single-layer 1T-MnS₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-MnS₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-MnS₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 47.1 and 46.8 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.15$ .

There is no available value for nonlinear quantities in the single-layer 1T-MnS₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −193.8 and −210.1 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 5.5 N/m at the ultimate strain of 0.21 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 5.3 N/m at the ultimate strain of 0.25 in the zigzag direction at the low temperature of 1 K.

49. 1T-MnSe₂

Most existing theoretical studies on the single-layer 1T-MnSe₂ are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-MnSe₂.

Figure 96.

Phonon spectrum for single-layer 1T-MnSe₂. (a) Phonon dispersion along the ΓM direction in the Brillouin zone. The results from the VFF model (lines) are comparable with the ab initio results (pentagons) from [12]. (b) The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

The structure for the single-layer 1T-MnSe₂ is shown in Figure 71 (with M = Mn and X = Se). Each Mn atom is surrounded by six Se atoms. These Se atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each Se atom is connected to three Mn atoms. The structural parameters are from the first-principles calculations [12], including the lattice constant $a=3.27$ Å and the bond length $d_{\text{Mn}-\text{Se}}=2.39$ Å. The resultant angles are ${\theta }_{\text{MnSeSe}}=86.330°$ with Se atoms from the same (top or bottom) group and ${\theta }_{\text{SeMnMn}}=86.330°$ .

Table 194 shows three VFF terms for the single-layer 1T-MnSe₂; one of which is the bond stretching interaction shown by Eq. (1), while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\text{Mn}-\text{Se}-\text{Se}}$ is for the angle ${\theta }_{\text{Mn}-\text{Se}-\text{Se}}$ with both Se atoms from the same (top or bottom) group. These force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the ΓM as shown in Figure 96(a) . The ab initio calculations for the phonon dispersion are from [12]. Figure 96(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

Figure 97.

Stress-strain for single-layer 1T-MnSe₂ of dimension $100\times 100$ Å along the armchair and zigzag directions.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{Mn}-\text{Se}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{Mn}-\text{Se}-\text{Se}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{Se}-\text{Mn}-\text{Mn}}{\left(\text{Δ}\theta \right)}^2$
Parameter	4.407	2.399	2.399
$r_0$ or ${\theta }_0$	2.39	86.330	86.330

Table 194.

The VFF model for single-layer 1T-MnSe₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

The parameters for the two-body SW potential used by GULP are shown in Table 195 . The parameters for the three-body SW potential used by GULP are shown in Table 196 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 197 .

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Mn─Se	3.422	1.153	16.314	0.0	3.220

Table 195.

Two-body SW potential parameters for single-layer 1T-MnSe₂ used by GULP [8] as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{Mn}-\text{Se}-\text{Se}}$	19.390	86.330	1.153	1.153	0.0	3.220	0.0	3.220	0.0	4.467
${\theta }_{\text{Se}-\text{Mn}-\text{Mn}}$	19.390	86.330	1.153	1.153	0.0	3.220	0.0	3.220	0.0	4.467

Table 196.

Three-body SW potential parameters for single-layer 1T-MnSe₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-MnSe₂ under uniaxial tension at 1 and 300 K. Figure 97 shows the stress-strain curve for the tension of a single-layer 1T-MnSe₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-MnSe₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-MnSe₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 43.2 and 42.9 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.17$ .

There is no available value for nonlinear quantities in the single-layer 1T-MnSe₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −163.4 and −179.4 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 5.4 N/m at the ultimate strain of 0.22 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 5.2 N/m at the ultimate strain of 0.26 in the zigzag direction at the low temperature of 1 K.

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Mn─Se1─Se1	1.000	1.153	2.792	19.390	1.000	0.064	3.422	9.219	4	0	0.0

Table 197.

SW potential parameters for single-layer 1T-MnSe₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Figure 98.

Phonon spectrum for single-layer 1T-MnTe₂. (a) Phonon dispersion along the ΓM direction in the Brillouin zone. The results from the VFF model (lines) are comparable with the ab initio results (pentagons) from [12]. (b) The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

Figure 99.

Stress-strain for single-layer 1T-MnTe₂ of dimension $100\times 100$ Å along the armchair and zigzag directions.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{Mn}-\text{Te}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{Mn}-\text{Te}-\text{Te}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{Te}-\text{Mn}-\text{Mn}}{\left(\text{Δ}\theta \right)}^2$
Parameter	4.407	2.399	2.399
$r_0$ or ${\theta }_0$	2.59	86.219	86.219

Table 198.

The VFF model for single-layer 1T-MnTe₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

50. 1T-MnTe₂

Most existing theoretical studies on the single-layer 1T-MnTe₂ are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-MnTe₂.

The structure for the single-layer 1T-MnTe₂ is shown in Figure 71 (with M = Mn and X = Te). Each Mn atom is surrounded by six Te atoms. These Te atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each Te atom is connected to three Mn atoms. The structural parameters are from the first-principles calculations [12], including the lattice constant $a=3.54$ Å and the bond length $d_{\text{Mn}-\text{Te}}=2.59$ Å. The resultant angles are ${\theta }_{\text{MnTeTe}}=86.219°$ with Te atoms from the same (top or bottom) group and ${\theta }_{\text{TeMnMn}}=86.219°$ .

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Mn─Te	4.007	1.246	22.499	0.0	3.488

Table 199.

Two-body SW potential parameters for single-layer 1T-MnTe₂ used by GULP [8] as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{Mn}-\text{Te}-\text{Te}}$	19.307	86.219	1.246	1.246	0.0	3.488	0.0	3.488	0.0	4.836
${\theta }_{\text{Te}-\text{Mn}-\text{Mn}}$	19.307	86.219	1.246	1.246	0.0	3.488	0.0	3.488	0.0	4.836

Table 200.

Three-body SW potential parameters for single-layer 1T-MnTe₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

Table 198 shows three VFF terms for the single-layer 1T-MnTe₂; one of which is the bond stretching interaction shown by Eq. (1), while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\text{Mn}-\text{Te}-\text{Te}}$ is for the angle ${\theta }_{\text{Mn}-\text{Te}-\text{Te}}$ with both Se atoms from the same (top or bottom) group. These force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the ΓM as shown in Figure 98(a) . The ab initio calculations for the phonon dispersion are from [12]. Figure 98(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 199 . The parameters for the three-body SW potential used by GULP are shown in Table 200 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 201 .

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Mn─Te1─Te1	1.000	1.246	2.800	19.307	1.000	0.066	4.007	9.340	4	0	0.0

Table 201.

SW potential parameters for single-layer 1T-MnTe₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-MnTe₂ under uniaxial tension at 1 and 300 K. Figure 99 shows the stress-strain curve for the tension of a single-layer 1T-MnTe₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-MnTe₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-MnTe₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 38.5 and 38.4 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.19$ .

There is no available value for nonlinear quantities in the single-layer 1T-MnTe₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −133.5 and −149.5 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 5.2 N/m at the ultimate strain of 0.24 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 5.0 N/m at the ultimate strain of 0.28 in the zigzag direction at the low temperature of 1 K.

51. 1T-CoTe₂

Most existing theoretical studies on the single-layer 1T-CoTe₂ are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-CoTe₂.

The structure for the single-layer 1T-CoTe₂ is shown in Figure 71 (with M = Co and X = Te). Each Co atom is surrounded by six Te atoms. These Te atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each Te atom is connected to three Co atoms. The structural parameters are from the first-principles calculations [48], including the lattice constant $a=3.5983$ Å and the bond length $d_{\text{Co}-\text{Te}}=2.5117$ Å, which are derived from the angle ${\theta }_{\text{TeCoCo}}=91.5°$ . The other angle is ${\theta }_{\text{CoTeTe}}=91.5°$ with Te atoms from the same (top or bottom) group.

Figure 100.

Stress-strain for single-layer 1T-CoTe₂ of dimension $100\times 100$ Å along the armchair and zigzag directions.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{Co}-\text{Te}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{Co}-\text{Te}-\text{Te}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{Te}-\text{Co}-\text{Co}}{\left(\text{Δ}\theta \right)}^2$
Parameter	4.726	3.035	3.035
$r_0$ or ${\theta }_0$	2.512	91.501	91.501

Table 202.

The VFF model for single-layer 1T-CoTe₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

Table 202 shows three VFF terms for the single-layer 1T-CoTe₂; one of which is the bond stretching interaction shown by Eq. (1), while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\text{Co}-\text{Te}-\text{Te}}$ is for the angle ${\theta }_{\text{Co}-\text{Te}-\text{Te}}$ with both Te atoms from the same (top or bottom) group. We find that there are actually only two parameters in the VFF model, so we can determine their value by fitting to the Young’s modulus and the Poisson’s ratio of the system. The ab initio calculations have predicted the Young’s modulus to be 59 N/m and the Poisson’s ratio as 0.14 [48].

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Co─Te	4.628	1.402	19.899	0.0	3.450

Table 203.

Two-body SW potential parameters for single-layer 1T-CoTe₂ used by GULP [8] as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{Co}-\text{Te}-\text{Te}}$	30.149	91.501	1.402	1.402	0.0	3.450	0.0	3.450	0.0	4.915
${\theta }_{\text{Te}-\text{Co}-\text{Co}}$	30.149	91.501	1.402	1.402	0.0	3.450	0.0	3.450	0.0	4.915

Table 204.

Three-body SW potential parameters for single-layer 1T-CoTe₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

The parameters for the two-body SW potential used by GULP are shown in Table 203 . The parameters for the three-body SW potential used by GULP are shown in Table 204 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 205 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-CoTe₂ under uniaxial tension at 1 and 300 K. Figure 100 shows the stress-strain curve for the tension of a single-layer 1T-CoTe₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-CoTe₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-CoTe₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 50.5 and 50.3 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.13$ . The fitted Young’s modulus value is about 10% smaller than the ab initio result of 59 N/m [48], as only short-range interactions are considered in the present work. The long-range interactions are ignored, which typically leads to about 10% underestimation for the value of the Young’s modulus.

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Co─Te1─Te1	1.000	1.402	2.461	30.149	1.000	-0.026	4.628	5.151	4	0	0.0

Table 205.

SW potential parameters for single-layer 1T-CoTe₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Figure 101.

Phonon spectrum for single-layer 1T-CoTe₂ along the ΓMKΓ direction in the Brillouin zone. The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

Figure 102.

Phonon spectrum for single-layer 1T-NiO₂. (a) Phonon dispersion along the ΓM direction in the Brillouin zone. The results from the VFF model (lines) are comparable with the ab initio results (pentagons) from [12]. (b) The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

There is no available value for nonlinear quantities in the single-layer 1T-CoTe₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −221.5 and −238.3 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 5.6 N/m at the ultimate strain of 0.21 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 5.4 N/m at the ultimate strain of 0.24 in the zigzag direction at the low temperature of 1 K.

Figure 101 shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

Figure 103.

Stress-strain for single-layer 1T-NiO₂ of dimension $100\times 100$ Å along the armchair and zigzag directions.

52. 1T-NiO₂

Most existing theoretical studies on the single-layer 1T-NiO₂ are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-NiO₂.

The structure for the single-layer 1T-NiO₂ is shown in Figure 71 (with M = Ni and X = O). Each Ni atom is surrounded by six O atoms. These O atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each O atom is connected to three Ni atoms. The structural parameters are from the first-principles calculations [12], including the lattice constant $a=2.77$ Å and the bond length $d_{\text{Ni}-\text{O}}=1.84$ Å. The resultant angles are ${\theta }_{\text{NiOO}}=97.653°$ with O atoms from the same (top or bottom) group and ${\theta }_{\text{ONiNi}}=97.653°$ .

Figure 104.

Phonon spectrum for single-layer 1T-NiS₂. (a) Phonon dispersion along the ΓM direction in the Brillouin zone. The results from the VFF model (lines) are comparable with the ab initio results (pentagons) from [12]. (b) The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{Ni}-\text{O}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{Ni}-\text{O}-\text{O}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{O}-\text{Ni}-\text{Ni}}{\left(\text{Δ}\theta \right)}^2$
Parameter	15.925	4.847	4.847
$r_0$ or ${\theta }_0$		97.653	97.653

Table 206.

The VFF model for single-layer 1T-NiO₂. The second line gives an explicit expression for each VFF term.

The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of $\mathrm{\AA }$ ) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

Table 206 shows three VFF terms for the single-layer 1T-NiO₂; one of which is the bond stretching interaction shown by Eq. (1), while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\text{Ni}-\text{O}-\text{O}}$ is for the angle ${\theta }_{\text{Ni}-\text{O}-\text{O}}$ with both O atoms from the same (top or bottom) group. These force constant parameters are determined by fitting to the two in-plane acoustic branches in the phonon dispersion along the ΓM as shown in Figure 102(a) . The ab initio calculations for the phonon dispersion are from [12]. Figure 102(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

	A (eV)	ρ(Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Ni-O	9.709	1.199	5.731	0.0	2.583

Table 207.

Two-body SW potential parameters for single-layer 1T-NiO₂ used by GULP [8] as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{Ni}-\text{O}-\text{O}}$	62.317	97.653	1.199	1.199	0.0	2.583	0.0	2.583	0.0	3.784
${\theta }_{\text{O}-\text{Ni}-\text{Ni}}$	62.317	97.653	1.199	1.199	0.0	2.583	0.0	2.583	0.0	3.784

Table 208.

Three-body SW potential parameters for single-layer 1T-NiO₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

The parameters for the two-body SW potential used by GULP are shown in Table 207 . The parameters for the three-body SW potential used by GULP are shown in Table 208 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 209 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-NiO₂ under uniaxial tension at 1 and 300 K. Figure 103 shows the stress-strain curve for the tension of a single-layer 1T-NiO₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-NiO₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-NiO₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 163.3 and 162.4 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.12$ .

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Ni─O1─O1	1.000	1.199	2.154	62.317	1.000	-0.133	9.709	2.772	4	0	0.0

Table 209.

SW potential parameters for single-layer 1T-NiO₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{Ni}-\text{S}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{Ni}-\text{S}-\text{S}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{S}-\text{Ni}-\text{Ni}}{\left(\text{Δ}\theta \right)}^2$
Parameter	9.385	2.952	2.952
$r_0$ or ${\theta }_0$	2.232	96.000	96.000

Table 210.

The VFF model for single-layer 1T-NiS₂. The second line gives an explicit expression for each VFF term.

The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of $\mathrm{\AA }$ ) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

There is no available value for nonlinear quantities in the single-layer 1T-NiO₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −748.7 and −796.0 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 17.4 N/m at the ultimate strain of 0.20 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 16.8 N/m at the ultimate strain of 0.24 in the zigzag direction at the low temperature of 1 K.

53. 1T-NiS₂

Most existing theoretical studies on the single-layer 1T-NiS₂ are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-NiS₂.

The structure for the single-layer 1T-NiS₂ is shown in Figure 71 (with M = Ni and X = S). Each Ni atom is surrounded by six S atoms. These S atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each S atom is connected to three Ni atoms. The structural parameters are from the first-principles calculations [48], including the lattice constant $a=3.3174$ Å and the bond length $d_{\text{Ni}-\text{S}}=2.2320$ Å, which are derived from the angle ${\theta }_{\text{SNiNi}}=96°$ . The other angle is ${\theta }_{\text{NiSS}}=96°$ with S atoms from the same (top or bottom) group.

Figure 105.

Stress-strain for single-layer 1T-NiS₂ of dimension $100\times 100$ Å along the armchair and zigzag directions.

	A (eV)	ρ(Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Ni-S	8.098	1.398	12.409	0.0	3.115

Table 211.

Two-body SW potential parameters for single-layer 1T-NiS₂ used by GULP [8] as expressed in Eq. (3).

Figure 106.

Phonon spectrum for single-layer 1T-NiSe₂. (a) Phonon dispersion along the ΓM direction in the Brillouin zone. The results from the VFF model (lines) are comparable with the ab initio results (pentagons) from [12]. (b) The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{Ni}-\text{S}-\text{S}}$	35.372	96.000	1.398	1.398	0.0	3.115	0.0	3.115	0.0	4.532
${\theta }_{\text{S}-\text{Ni}-\text{Ni}}$	35.372	96.000	1.398	1.398	0.0	3.115	0.0	3.115	0.0	4.532

Table 212.

Three-body SW potential parameters for single-layer 1T-NiS₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

Table 210 shows three VFF terms for the single-layer 1T-NiS₂; one of which is the bond stretching interaction shown by Eq. (1), while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\text{Ni}-\text{S}-\text{S}}$ is for the angle ${\theta }_{\text{Ni}-\text{S}-\text{S}}$ with both S atoms from the same (top or bottom) group. These force constant parameters are determined by fitting to the two in-plane acoustic branches in the phonon dispersion along the ΓM as shown in Figure 104(a) . The ab initio calculations for the phonon dispersion are from [12]. Figure 104(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Ni─S1─S1	1.000	1.398	2.228	35.372	1.000	-0.105	8.098	3.249	4	0	0.0

Table 213.

SW potential parameters for single-layer 1T-NiS₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

The parameters for the two-body SW potential used by GULP are shown in Table 211 . The parameters for the three-body SW potential used by GULP are shown in Table 212 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 213 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-NiS₂ under uniaxial tension at 1 and 300 K. Figure 105 shows the stress-strain curve for the tension of a single-layer 1T-NiS₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-NiS₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-NiS₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 74.2 and 73.9 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.17$ .

There is no available value for nonlinear quantities in the single-layer 1T-NiS₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −274.5 and −301.4 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 9.5 N/m at the ultimate strain of 0.23 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 9.2 N/m at the ultimate strain of 0.27 in the zigzag direction at the low temperature of 1 K.

54. 1T-NiSe₂

Most existing theoretical studies on the single-layer 1T-NiSe₂ are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-NiSe₂.

Figure 107.

Stress-strain for single-layer 1T-NiSe₂ of dimension $100\times 100$ Å along the armchair and zigzag directions.

The structure for the single-layer 1T-NiSe₂ is shown in Figure 71 (with M = Ni and X = Se). Each Ni atom is surrounded by six Se atoms. These Se atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each Se atom is connected to three Ni atoms. The structural parameters are from the first-principles calculations [48], including the lattice constant $a=3.4712$ Å and the bond length $d_{\text{Ni}-\text{Se}}=2.3392$ Å, which are derived from the angle ${\theta }_{\text{SeNiNi}}=95.8°$ . The other angle is ${\theta }_{\text{NiSeSe}}=95.8°$ with Se atoms from the same (top or bottom) group.

Table 214 shows three VFF terms for the single-layer 1T-NiSe₂; one of which is the bond stretching interaction shown by Eq. (1), while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\text{Ni}-\text{Se}-\text{Se}}$ is for the angle ${\theta }_{\text{Ni}-\text{Se}-\text{Se}}$ with both Se atoms from the same (top or bottom) group. These force constant parameters are determined by fitting to the two in-plane acoustic branches in the phonon dispersion along the ΓM as shown in Figure 106(a) . The ab initio calculations for the phonon dispersion are from [12]. Figure 106(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

Figure 108.

Stress-strain for single-layer 1T-NiTe₂ of dimension $100\times 100$ Å along the armchair and zigzag directions.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{Ni}-\text{Se}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{Ni}-\text{Se}-\text{Se}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{Se}-\text{Ni}-\text{Ni}}{\left(\text{Δ}\theta \right)}^2$
Parameter	8.814	3.149	3.149
$r_0$ or ${\theta }_0$	2.339	95.798	95.798

Table 214.

The VFF model for single-layer 1T-NiSe₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

The parameters for the two-body SW potential used by GULP are shown in Table 215 . The parameters for the three-body SW potential used by GULP are shown in Table 216 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 217 .

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Ni─Se	8.313	1.458	14.971	0.0	3.263

Table 215.

Two-body SW potential parameters for single-layer 1T-NiSe₂ used by GULP [8] as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{Ni}-\text{Se}-\text{Se}}$	37.407	95.798	1.458	1.458	0.0	3.263	0.0	3.263	0.0	4.742
${\theta }_{\text{Se}-\text{Ni}-\text{Ni}}$	37.407	95.798	1.458	1.458	0.0	3.263	0.0	3.263	0.0	4.742

Table 216.

Three-body SW potential parameters for single-layer 1T-NiSe₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-NiSe₂ under uniaxial tension at 1 and 300 K. Figure 107 shows the stress-strain curve for the tension of a single-layer 1T-NiSe₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-NiSe₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-NiSe₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 70.9 and 70.6 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.17$ .

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Ni─Se1─Se1	1.000	1.458	2.238	37.407	1.000	-0.101	8.313	3.315	4	0	0.0

Table 217.

SW potential parameters for single-layer 1T-NiSe₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Figure 109.

Phonon spectrum for single-layer 1T-NiTe₂ along the ΓMKΓ direction in the Brillouin zone. The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

There is no available value for nonlinear quantities in the single-layer 1T-NiSe₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −263.7 and −289.5 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 9.0 N/m at the ultimate strain of 0.23 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 8.7 N/m at the ultimate strain of 0.27 in the zigzag direction at the low temperature of 1 K.

55. 1T-NiTe₂

Most existing theoretical studies on the single-layer 1T-NiTe₂ are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-NiTe₂.

The structure for the single-layer 1T-NiTe₂ is shown in Figure 71 (with M = Ni and X = Te). Each Ni atom is surrounded by six Te atoms. These Te atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each Te atom is connected to three Ni atoms. The structural parameters are from the first-principles calculations [48], including the lattice constant $a=3.7248$ Å and the bond length $d_{\text{Ni}-\text{Te}}=2.5321$ Å, which are derived from the angle ${\theta }_{\text{TeNiNi}}=94.7°$ . The other angle is ${\theta }_{\text{NiTeTe}}=94.7°$ with Te atoms from the same (top or bottom) group.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{Ni}-\text{Te}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{Ni}-\text{Te}-\text{Te}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{Te}-\text{Ni}-\text{Ni}}{\left(\text{Δ}\theta \right)}^2$
Parameter	4.230	2.429	2.429
$r_0$ or ${\theta }_0$	2.532	94.702	94.702
or ${\theta }_0$	2.635	95.999	95.999

Table 218.

The VFF model for single-layer 1T-NiTe₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Ni─Te	4.554	1.536	20.554	0.0	3.518

Table 219.

Two-body SW potential parameters for single-layer 1T-NiTe₂ used by GULP [8] as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{Ni}-\text{Te}-\text{Te}}$	27.553	94.702	1.536	1.536	0.0	3.518	0.0	3.518	0.0	5.088
${\theta }_{\text{Te}-\text{Ni}-\text{Ni}}$	27.553	94.702	1.536	1.536	0.0	3.518	0.0	3.518	0.0	5.088

Table 220.

Three-body SW potential parameters for single-layer 1T-NiTe₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Ni─Te1─Te1	1.000	1.536	2.291	27.553	1.000	−0.082	4.554	3.696	4	0	0.0

Table 221.

SW potential parameters for single-layer 1T-NiTe₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Table 218 shows three VFF terms for the single-layer 1T-NiTe₂; one of which is the bond stretching interaction shown by Eq. (1), while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\text{Ni}-\text{Te}-\text{Te}}$ is for the angle ${\theta }_{\text{Ni}-\text{Te}-\text{Te}}$ with both Te atoms from the same (top or bottom) group. We find that there are actually only two parameters in the VFF model, so we can determine their value by fitting to the Young’s modulus and the Poisson’s ratio of the system. The ab initio calculations have predicted the Young’s modulus to be 44 N/m and the Poisson’s ratio as 0.14 [48].

The parameters for the two-body SW potential used by GULP are shown in Table 219 . The parameters for the three-body SW potential used by GULP are shown in Table 220 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 221 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-NiTe₂ under uniaxial tension at 1 and 300 K. Figure 108 shows the stress-strain curve for the tension of a single-layer 1T-NiTe₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-NiTe₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-NiTe₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 42.6 and 42.4 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.14$ .

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{Zr}-\text{S}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{Zr}-\text{S}-\text{S}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{S}-\text{Zr}-\text{Zr}}{\left(\text{Δ}\theta \right)}^2$
Parameter	7.930	4.283	4.283
$r_0$ or ${\theta }_0$	2.580	91.305	91.305

Table 222.

The VFF model for single-layer 1T-ZrS₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

There is no available value for nonlinear quantities in the single-layer 1T-NiTe₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of D from the present SW potential are −187.6 and −200.6 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 4.7 N/m at the ultimate strain of 0.21 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 4.6 N/m at the ultimate strain of 0.24 in the zigzag direction at the low temperature of 1 K.

Figure 109 shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

Figure 110.

Phonon spectrum for single-layer 1T-ZrS₂. (a) Phonon dispersion along the ΓM direction in the Brillouin zone. The results from the VFF model (lines) are comparable with the experiment data (pentagons) from [38]. (b) The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Zr─S	8.149	1.432	22.154	0.0	3.541

Table 223.

Two-body SW potential parameters for single-layer 1T-ZrS₂ used by GULP [8] as expressed in Eq. (3).

56. 1T-ZrS₂

Most existing theoretical studies on the single-layer 1T-ZrS₂ are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-ZrS₂.

Figure 111.

Stress-strain for single-layer 1T-ZrS₂ of dimension $100\times 100$ Å along the armchair and zigzag directions.

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{Zr}-\text{S}-\text{S}}$	42.170	91.305	1.432	1.432	0.0	3.541	0.0	3.541	0.0	5.041
${\theta }_{\text{S}-\text{Zr}-\text{Zr}}$	42.170	91.305	1.432	1.432	0.0	3.541	0.0	3.541	0.0	5.041

Table 224.

Three-body SW potential parameters for single-layer 1T-ZrS₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

The structure for the single-layer 1T-ZrS₂ is shown in Figure 71 (with M = Zr and X = S). Each Zr atom is surrounded by six S atoms. These S atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each S atom is connected to three Zr atoms. The structural parameters are from the first-principles calculations [49], including the lattice constant $a=3.690$ Å and the bond length $d_{\text{Zr}-\text{S}}=2.58$ Å. The resultant angles are ${\theta }_{\text{ZrSS}}=91.305°$ with S atoms from the same (top or bottom) group and ${\theta }_{\text{SZrZr}}=91.305°$ .

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Zr─S1─S1	1.000	1.432	2.473	42.177	1.000	−0.023	8.149	5.268	4	0	0.0
S1─Zr─Zr	1.000	1.432	2.473	42.177	1.000	−0.023	8.149	5.268	4	0	0.0

Table 225.

SW potential parameters for single-layer 1T-ZrS₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Table 222 shows three VFF terms for the single-layer 1T-ZrS₂; one of which is the bond stretching interaction shown by Eq. (1), while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\text{Zr}-\text{S}-\text{S}}$ is for the angle ${\theta }_{\text{Zr}-\text{S}-\text{S}}$ with both S atoms from the same (top or bottom) group. These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the ΓM as shown in Figure 110(a) . The ab initio calculations for the phonon dispersion are from [38]. Similar phonon dispersion can also be found in other ab initio calculations [34]. Figure 110(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 223 . The parameters for the three-body SW potential used by GULP are shown in Table 224 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 225 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-ZrS₂ under uniaxial tension at 1 and 300 K. Figure 111 shows the stress-strain curve for the tension of a single-layer 1T-ZrS₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-ZrS₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-ZrS₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 71.8 and 71.5 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. These values are close to the ab initio results at 0 K temperature, e.g., 75.74 N/m in [49]. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.16$ , which are comparable with the ab initio result [49] of 0.22.

There is no available value for nonlinear quantities in the single-layer 1T-ZrS₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −268.9 and −305.2 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 8.9 N/m at the ultimate strain of 0.23 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 8.5 N/m at the ultimate strain of 0.26 in the zigzag direction at the low temperature of 1 K.

Figure 112.

Phonon spectrum for single-layer 1T-ZrSe₂. (a) Phonon dispersion along the ΓM direction in the Brillouin zone. The results from the VFF model (lines) are comparable with the experiment data (pentagons) from [50]. (b) The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

57. 1T-ZrSe₂

Most existing theoretical studies on the single-layer 1T-ZrSe₂ are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-ZrSe₂.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{Zr}-\text{Se}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{Zr}-\text{Se}-\text{Se}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{Se}-\text{Zr}-\text{Zr}}{\left(\text{Δ}\theta \right)}^2$
Parameter	7.930	4.283	4.283
$r_0$ or ${\theta }_0$	2.667	88.058	88.058

Table 226.

The VFF model for single-layer 1T-ZrSe₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Zr─Se	8.022	1.354	25.297	0.0	3.617

Table 227.

Two-body SW potential parameters for single-layer 1T-ZrSe₂ used by GULP [8] as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{Zr}-\text{Se}-\text{Se}}$	37.051	88.058	1.354	1.354	0.0	3.617	0.0	3.617	0.0	5.064
${\theta }_{\text{Se}-\text{Zr}-\text{Zr}}$	37.051	88.058	1.354	1.354	0.0	3.617	0.0	3.617	0.0	5.064

Table 228.

Three-body SW potential parameters for single-layer 1T-ZrSe₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Zr─Se─Se	1.000	1.354	2.671	37.051	1.000	0.034	8.022	7.527	4	0	0.0
Se─Zr─Zr	1.000	1.354	2.671	37.051	1.000	0.034	8.022	7.527	4	0	0.0

Table 229.

SW potential parameters for single-layer 1T-ZrSe₂ used by LAMMPS⁹ as expressed in Eqs. (9) and (10).

Figure 113.

Stress-strain for single-layer 1T-ZrSe₂ of dimension $100\times 100$ Å along the armchair and zigzag directions.

The structure for the single-layer 1T-ZrSe₂ is shown in Figure 71 (with M = Zr and X = Se). Each Zr atom is surrounded by six Se atoms. These Se atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each Se atom is connected to three Zr atoms. The structural parameters are from the first-principles calculations [51], including the lattice constant $a=3.707$ Å and the position of the Se atom with respect to the Zr atomic plane $h=1.591$ Å. The resultant angles are ${\theta }_{\text{ZrSeSe}}=88.058°$ with Se atoms from the same (top or bottom) group and ${\theta }_{\text{SeZrZr}}=88.058°$ .

Table 226 shows three VFF terms for the single-layer 1T-ZrSe₂; one of which is the bond stretching interaction shown by Eq. (1), while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\text{Zr}-\text{Se}-\text{Se}}$ is for the angle ${\theta }_{\text{Zr}-\text{Se}-\text{Se}}$ with both Se atoms from the same (top or bottom) group. These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the ΓM as shown in Figure 112(a) . The ab initio calculations for the phonon dispersion are from [50]. Similar phonon dispersion can also be found in other ab initio calculations [34]. Figure 112(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 227 . The parameters for the three-body SW potential used by GULP are shown in Table 228 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 229 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-ZrSe₂ under uniaxial tension at 1 and 300 K. Figure 113 shows the stress-strain curve for the tension of a single-layer 1T-ZrSe₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-ZrSe₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-ZrSe₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 66.7 and 66.4 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.19$ .

There is no available value for nonlinear quantities in the single-layer 1T-ZrSe₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −219.6 and −256.6 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 9.0 N/m at the ultimate strain of 0.24 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 8.6 N/m at the ultimate strain of 0.28 in the zigzag direction at the low temperature of 1 K.

58. 1T-ZrTe₂

Most existing theoretical studies on the single-layer 1T-ZrTe₂ are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-ZrTe₂.

The structure for the single-layer 1T-ZrTe₂ is shown in Figure 71 (with M=Zr and X=Te). Each Zr atom is surrounded by six Te atoms. These Te atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each Te atom is connected to three Zr atoms. The structural parameters are from the first-principles calculations [48], including the lattice constant $a=4.0064$ Å and the bond length $d_{\text{Zr}-\text{Te}}=2.9021$ Å, which are derived from the angle ${\theta }_{\text{TeZrZr}}=87.3°$ . The other angle is ${\theta }_{\text{ZrTeTe}}=87.3°$ with Te atoms from the same (top or bottom) group.

Figure 114.

Stress-strain for single-layer 1T-ZrTe₂ of dimension $100\times 100$ Å along the armchair and zigzag directions.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{Zr}-\text{Te}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{Zr}-\text{Te}-\text{Te}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{Te}-\text{Zr}-\text{Zr}}{\left(\text{Δ}\theta \right)}^2$
Parameter	2.974	3.681	3.681
$r_0$ or ${\theta }_0$	2.902	87.301	87.301

Table 230.

The VFF model for single-layer 1T-ZrTe₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

Table 230 shows three VFF terms for the single-layer 1T-ZrTe₂; one of which is the bond stretching interaction shown by Eq. (1), while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\text{Zr}-\text{Te}-\text{Te}}$ is for the angle ${\theta }_{\text{Zr}-\text{Te}-\text{Te}}$ with both Te atoms from the same (top or bottom) group. We find that there are actually only two parameters in the VFF model, so we can determine their value by fitting to the Young’s modulus and the Poisson’s ratio of the system. The ab initio calculations have predicted the Young’s modulus to be 44 N/m and the Poisson’s ratio as 0.13 [48].

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Zr─Te	3.493	1.441	35.467	0.0	3.925

Table 231.

Two-body SW potential parameters for single-layer 1T-ZrTe₂ used by GULP [8] as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{Zr}-\text{Te}-\text{Te}}$	30.905	87.301	1.441	1.441	0.0	3.925	0.0	3.925	0.0	5.473
${\theta }_{\text{Te}-\text{Zr}-\text{Zr}}$	30.905	87.301	1.441	1.441	0.0	3.925	0.0	3.925	0.0	5.473

Table 232.

Three-body SW potential parameters for single-layer 1T-ZrTe₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

The parameters for the two-body SW potential used by GULP are shown in Table 231 . The parameters for the three-body SW potential used by GULP are shown in Table 232 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 233 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-ZrTe₂ under uniaxial tension at 1 and 300 K. Figure 114 shows the stress-strain curve for the tension of a single-layer 1T-ZrTe₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-ZrTe₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-ZrTe₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 39.2 and 39.1 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.10$ . The fitted Young’s modulus value is about 10% smaller than the ab initio result of 44 N/m [48], as only short-range interactions are considered in the present work. The long-range interactions are ignored, which typically leads to about 10% underestimation for the value of the Young’s modulus.

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Zr─Te1─Te1	1.000	1.441	2.723	30.905	1.000	0.047	3.493	8.225	4	0	0.0

Table 233.

SW potential parameters for single-layer 1T-ZrTe₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Figure 115.

Phonon spectrum for single-layer 1T-ZrTe₂ along the ΓMKΓ direction in the Brillouin zone. The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

There is no available value for nonlinear quantities in the single-layer 1T-ZrTe₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −187.2 and −201.1 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 4.0 N/m at the ultimate strain of 0.19 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 3.9 N/m at the ultimate strain of 0.22 in the zigzag direction at the low temperature of 1 K.

Figure 115 shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{Nb}-\text{S}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{Nb}-\text{S}-\text{S}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{S}-\text{Nb}-\text{Nb}}{\left(\text{Δ}\theta \right)}^2$
Parameter	7.930	4.283	4.283
$r_0$ or ${\theta }_0$	2.450	84.671	84.671

Table 234.

The VFF model for single-layer 1T-NbS₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Nb─S	6.192	1.125	18.015	0.0	3.280

Table 235.

Two-body SW potential parameters for single-layer 1T-NbS₂ used by GULP [8] as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{Nb}-\text{S}-\text{S}}$	32.472	84.671	1.125	1.125	0.0	3.280	0.0	3.280	0.0	4.508
${\theta }_{\text{S}-\text{Nb}-\text{Nb}}$	32.472	84.671	1.125	1.125	0.0	3.280	0.0	3.280	0.0	4.508

Table 236.

Three-body SW potential parameters for single-layer 1T-NbS₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Nb─S1─S1	1.000	1.125	2.916	32.472	1.000	0.093	6.192	11.247	4	0	0.0
S1─Nb─Nb	1.000	1.125	2.916	32.472	1.000	0.093	6.192	11.247	4	0	0.0

Table 237.

SW potential parameters for single-layer 1T-NbS₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Figure 116.

Phonon spectrum for single-layer 1T-NbS₂. (a) Phonon dispersion along the ΓM direction in the Brillouin zone. The results from the VFF model (lines) are comparable with the experiment data (pentagons) from [12]. (b) The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

59. 1T-NbS₂

Most existing theoretical studies on the single-layer 1T-NbS₂ are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-NbS₂.

The structure for the single-layer 1T-NbS₂ is shown in Figure 71 (with M=Nb and X=S). Each Nb atom is surrounded by six S atoms. These S atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each S atom is connected to three Nb atoms. The structural parameters are from the first-principles calculations [12], including the lattice constant $a=3.30$ Å and the bond length $d_{\text{Nb}-\text{S}}=2.45$ Å. The resultant angles are ${\theta }_{\text{NbSS}}=84.671°$ with S atoms from the same (top or bottom) group and ${\theta }_{\text{SNbNb}}=84.671°$ .

Figure 117.

Stress-strain for single-layer 1T-NbS₂ of dimension $100\times 100$ Å along the armchair and zigzag directions.

Table 234 shows three VFF terms for the single-layer 1T-NbS₂; one of which is the bond stretching interaction shown by Eq. (1), while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\text{Nb}-\text{S}-\text{S}}$ is for the angle ${\theta }_{\text{Nb}-\text{S}-\text{S}}$ with both S atoms from the same (top or bottom) group. These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the ΓM as shown in Figure 116(a) . The ab initio calculations for the phonon dispersion are from [12]. The lowest acoustic branch (flexural mode) is linear and very close to the in-plane transverse acoustic branch in the ab initio calculations, which may be due to the violation of the rigid rotational invariance [20]. Figure 116(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 235 . The parameters for the three-body SW potential used by GULP are shown in Table 236 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 237 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-NbS₂ under uniaxial tension at 1 and 300 K. Figure 117 shows the stress-strain curve for the tension of a single-layer 1T-NbS₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-NbS₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-NbS₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 73.8 and 73.4 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.18$ .

Figure 118.

Stress-strain for single-layer 1T-NbSe₂ of dimension $100\times 100$ Å along the armchair and zigzag directions.

There is no available value for nonlinear quantities in the single-layer 1T-NbS₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −250.5 and −290.4 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 9.7 N/m at the ultimate strain of 0.23 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 9.4 N/m at the ultimate strain of 0.27 in the zigzag direction at the low temperature of 1 K.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{Nb}-\text{Se}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{Nb}-\text{Se}-\text{Se}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{Se}-\text{Nb}-\text{Nb}}{\left(\text{Δ}\theta \right)}^2$
Parameter	7.930	4.283	4.283
$r_0$ or ${\theta }_0$	2.570	82.529	82.529

Table 238.

The VFF model for single-layer 1T-NbSe₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Nb─Se	6.430	1.104	21.812	0.0	3.412

Table 239.

Two-body SW potential parameters for single-layer 1T-NbSe₂ used by GULP [8] as expressed in Eq. (3).

60. 1T-NbSe₂

Most existing theoretical studies on the single-layer 1T-NbSe₂ are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-NbSe₂.

The structure for the single-layer 1T-NbSe₂ is shown in Figure 71 (with M=Nb and X=Se). Each Nb atom is surrounded by six Se atoms. These Se atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each Se atom is connected to three Nb atoms. The structural parameters are from the first-principles calculations [12], including the lattice constant $a=3.39$ Å and the bond length $d_{\text{Nb}-\text{Se}}=2.57$ Å. The resultant angles are ${\theta }_{\text{NbSeSe}}=82.529°$ with Se atoms from the same (top or bottom) group and ${\theta }_{\text{SeNbNb}}=82.529°$ .

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{Nb}-\text{Se}-\text{Se}}$	29.956	82.528	1.104	1.104	0.0	3.412	0.0	3.412	0.0	4.631
${\theta }_{\text{Se}-\text{Nb}-\text{Nb}}$	29.956	82.528	1.104	1.104	0.0	3.412	0.0	3.412	0.0	4.631

Table 240.

Three-body SW potential parameters for single-layer 1T-NbSe₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Nb─Se1─Se1	1.000	1.104	3.092	29.956	1.000	0.130	6.430	14.706	4	0	0.0

Table 241.

SW potential parameters for single-layer 1T-NbSe₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Figure 119.

Phonon spectrum for single-layer 1T-NbSe₂ along the ΓMKΓ direction in the Brillouin zone. The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

Table 238 shows three VFF terms for the single-layer 1T-NbSe₂; one of which is the bond stretching interaction shown by Eq. (1), while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\text{Nb}-\text{Se}-\text{Se}}$ is for the angle ${\theta }_{\text{Nb}-\text{Se}-\text{Se}}$ with both Se atoms from the same (top or bottom) group. We find that there are actually only two parameters in the VFF model, so we can determine their value by fitting to the Young’s modulus and the Poisson’s ratio of the system. The ab initio calculations have predicted the Young’s modulus to be 73 N/m and the Poisson’s ratio as 0.20 [48].

The parameters for the two-body SW potential used by GULP are shown in Table 239 . The parameters for the three-body SW potential used by GULP are shown in Table 240 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 241 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-NbSe₂ under uniaxial tension at 1 and 300 K. Figure 118 shows the stress-strain curve for the tension of a single-layer 1T-NbSe₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-NbSe₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-NbSe₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 67.1 and 66.8 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.20$ . The fitted Young’s modulus value is about 10% smaller than the ab initio result of 73 N/m [48], as only short-range interactions are considered in the present work. The long-range interactions are ignored, which typically lead to about 10% underestimation for the Young’s modulus value.

There is no available value for nonlinear quantities in the single-layer 1T-NbSe₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −193.5 N/m and −233.4 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 9.7 N/m at the ultimate strain of 0.25 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 9.3 N/m at the ultimate strain of 0.29 in the zigzag direction at the low temperature of 1 K.

Figure 119 shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

61. 1T-NbTe₂

Most existing theoretical studies on the single-layer 1T-NbTe₂ are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-NbTe₂.

Figure 120.

Stress-strain for single-layer 1T-NbTe₂ of dimension $100\times 100$ Å along the armchair and zigzag directions.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{Nb}-\text{Te}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{Nb}-\text{Te}-\text{Te}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{Te}-\text{Nb}-\text{Nb}}{\left(\text{Δ}\theta \right)}^2$
Parameter	3.559	4.863	4.863
$r_0$ or ${\theta }_0$	2.770	79.972	79.972

Table 242.

The VFF model for single-layer 1T-NbTe₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

The structure for the single-layer 1T-NbTe₂ is shown in Figure 71 (with M=Nb and X=Te). Each Nb atom is surrounded by six Te atoms. These Te atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each Te atom is connected to three Nb atoms. The structural parameters are from the first-principles calculations [12], including the lattice constant $a=3.56$ Å and the bond length $d_{\text{Nb}-\text{Te}}=2.77$ Å. The resultant angles are ${\theta }_{\text{NbTeTe}}=79.972°$ with Te atoms from the same (top or bottom) group and ${\theta }_{\text{TeNbNb}}=79.972°$ .

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Nb─Te	3.123	1.094	29.437	0.0	3.640

Table 243.

Two-body SW potential parameters for single-layer 1T-NbTe₂ used by GULP [8] as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{Nb}-\text{Te}-\text{Te}}$	30.968	79.972	1.094	1.094	0.0	3.640	0.0	3.640	0.0	4.863
${\theta }_{\text{Te}-\text{Nb}-\text{Nb}}$	30.968	79.972	1.094	1.094	0.0	3.640	0.0	3.640	0.0	4.863

Table 244.

Three-body SW potential parameters for single-layer 1T-NbTe₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

Table 242 shows three VFF terms for the single-layer 1T-NbTe₂; one of which is the bond stretching interaction shown by Eq. (1), while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\text{Nb}-\text{Te}-\text{Te}}$ is for the angle ${\theta }_{\text{Nb}-\text{Te}-\text{Te}}$ with both Te atoms from the same (top or bottom) group. We find that there are actually only two parameters in the VFF model, so we can determine their value by fitting to the Young’s modulus and the Poisson’s ratio of the system. The ab initio calculations have predicted the Young’s modulus to be 56 N/m and the Poisson’s ratio as 0.11 [48].

The parameters for the two-body SW potential used by GULP are shown in Table 243 . The parameters for the three-body SW potential used by GULP are shown in Table 244 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 245 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-NbTe₂ under uniaxial tension at 1 and 300 K. Figure 120 shows the stress-strain curve for the tension of a single-layer 1T-NbTe₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-NbTe₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-NbTe₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 52.2 and 51.9 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.11$ . The fitted Young’s modulus value is about 10% smaller than the ab initio result of 56 N/m [48], as only short-range interactions are considered in the present work. The long-range interactions are ignored, which typically leads to about 10% underestimation for the value of the Young’s modulus.

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Nb─Te1─Te1	1.000	1.094	3.328	30.968	1.000	0.174	3.123	20.560	4	0	0.0

Table 245.

SW potential parameters for single-layer 1T-NbTe₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Figure 121.

Phonon spectrum for single-layer 1T-NbTe₂ along the ΓMKΓ direction in the Brillouin zone. The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

There is no available value for nonlinear quantities in the single-layer 1T-NbTe₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −237.7 and −265.0 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 5.4 N/m at the ultimate strain of 0.19 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 5.2 N/m at the ultimate strain of 0.22 in the zigzag direction at the low temperature of 1 K.

Figure 122.

Stress-strain for single-layer 1T-MoS₂ of dimension $100\times 100$ Å along the armchair and zigzag directions.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{Mo}-\text{S}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{Mo}-\text{S}-\text{S}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{S}-\text{Mo}-\text{Mo}}{\left(\text{Δ}\theta \right)}^2$
Parameter	3.523	10.394	10.394
$r_0$ or ${\theta }_0$	2.419	82.799	82.799

Table 246.

The VFF model for single-layer 1T-MoS₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

Figure 121 shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

62. 1T-MoS₂

Most existing theoretical studies on the single-layer 1T-MoS₂ are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-MoS₂.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Mo─S	2.550	1.048	17.129	0.0	3.215

Table 247.

Two-body SW potential parameters for single-layer 1T-MoS₂ used by GULP [8] as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{Mo}-\text{S}-\text{S}}$	73.436	82.799	1.048	1.048	0.0	3.215	0.0	3.215	0.0	4.371
${\theta }_{\text{S}-\text{Mo}-\text{Mo}}$	73.436	82.799	1.048	1.048	0.0	3.215	0.0	3.215	0.0	4.371

Table 248.

Three-body SW potential parameters for single-layer 1T-MoS₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

The structure for the single-layer 1T-MoS₂ is shown in Figure 71 (with M=Mo and X=S). Each Mo atom is surrounded by six S atoms. These S atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each S atom is connected to three Mo atoms. The structural parameters are from the first-principles calculations [48], including the lattice constant $a=3.1998$ Å and the bond length $d_{\text{Mo}-\text{S}}=2.4193$ Å, which are derived from the angle ${\theta }_{\text{SMoMo}}=82.8°$ . The other angle is ${\theta }_{\text{MoSS}}=82.8°$ with S atoms from the same (top or bottom) group.

Table 246 shows three VFF terms for the single-layer 1T-MoS₂; one of which is the bond stretching interaction shown by Eq. (1), while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\text{Mo}-\text{S}-\text{S}}$ is for the angle ${\theta }_{\text{Mo}-\text{S}-\text{S}}$ with both S atoms from the same (top or bottom) group. We find that there are actually only two parameters in the VFF model, so we can determine their value by fitting to the Young’s modulus and the Poisson’s ratio of the system. The ab initio calculations have predicted the Young’s modulus to be 103 N/m and the Poisson’s ratio as −0.07 [48]. The ab initio calculations have predicted a negative Poisson’s ratio in the 1T-MoS₂, which was attributed to the orbital coupling in this material. The orbital coupling enhances the angle bending interaction in the VFF model. As a result, the value of the angle bending parameter is much larger than the bond stretching force constant parameter, which is typical in auxetic materials with negative Poisson’s ratio [52].

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Mo─S1─S1	1.000	1.048	3.069	73.436	1.000	0.125	2.550	14.207	4	0	0.0

Table 249.

SW potential parameters for single-layer 1T-MoS₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Figure 123.

Phonon spectrum for single-layer 1T-MoS₂ along the ΓMKΓ direction in the Brillouin zone. The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 247 . The parameters for the three-body SW potential used by GULP are shown in Table 248 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 249 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-MoS₂ under uniaxial tension at 1 and 300 K. Figure 122 shows the stress-strain curve for the tension of a single-layer 1T-MoS₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-MoS₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-MoS₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 88.7 and 88.3 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=\mathrm{}-0.07$ . The fitted Young’s modulus value is about 10% smaller than the ab initio result of 103 N/m [48], as only short-range interactions are considered in the present work. The long-range interactions are ignored, which typically leads to about 10% underestimation for the value of the Young’s modulus.

Figure 124.

Stress-strain for single-layer 1T-MoSe₂ of dimension $100\times 100$ Å along the armchair and zigzag directions.

There is no available value for nonlinear quantities in the single-layer 1T-MoS₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −595.2 and −624.1 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 6.5 N/m at the ultimate strain of 0.14 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 6.5 N/m at the ultimate strain of 0.16 in the zigzag direction at the low temperature of 1 K.

Figure 123 shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{Mo}-\text{Se}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{Mo}-\text{Se}-\text{Se}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{Se}-\text{Mo}-\text{Mo}}{\left(\text{Δ}\theta \right)}^2$
Parameter	2.964	14.753	14.753
$r_0$ or ${\theta }_0$	2.529	80.501	80.501

Table 250.

The VFF model for single-layer 1T-MoSe₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Mo─Se	2.201	1.017	20.463	0.0	3.331

Table 251.

Two-body SW potential parameters for single-layer 1T-MoSe₂ used by GULP [8] as expressed in Eq. (3).

63. 1T-MoSe₂

Most existing theoretical studies on the single-layer 1T-MoSe₂ are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-MoSe₂.

The structure for the single-layer 1T-MoSe₂ is shown in Figure 71 (with M=Mo and X=Se). Each Mo atom is surrounded by six Se atoms. These Se atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each Se atom is connected to three Mo atoms. The structural parameters are from the first-principles calculations [48], including the lattice constant $a=3.2685$ Å and the bond length $d_{\text{Mo}-\text{Se}}=2.5293$ Å, which are derived from the angle ${\theta }_{\text{SeMoMo}}=80.5°$ . The other angle is ${\theta }_{\text{MoSeSe}}=80.5°$ with Se atoms from the same (top or bottom) group.

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{Mo}-\text{Se}-\text{Se}}$	95.770	80.501	1.017	1.017	0.0	3.331	0.0	3.331	0.0	4.465
${\theta }_{\text{Se}-\text{Mo}-\text{Mo}}$	95.770	80.501	1.017	1.017	0.0	3.331	0.0	3.331	0.0	4.465

Table 252.

Three-body SW potential parameters for single-layer 1T-MoSe₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Mo─Se1─Se1	1.000	1.017	3.276	95.770	1.000	0.165	2.201	19.152	4	0	0.0

Table 253.

SW potential parameters for single-layer 1T-MoSe₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Figure 125.

Phonon spectrum for single-layer 1T-MoSe₂ along the ΓMKΓ direction in the Brillouin zone. The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

Table 250 shows three VFF terms for the single-layer 1T-MoSe₂; one of which is the bond stretching interaction shown by Eq. (1), while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\text{Mo}-\text{Se}-\text{Se}}$ is for the angle ${\theta }_{\text{Mo}-\text{Se}-\text{Se}}$ with both Se atoms from the same (top or bottom) group. We find that there are actually only two parameters in the VFF model, so we can determine their value by fitting to the Young’s modulus and the Poisson’s ratio of the system. The ab initio calculations have predicted the Young’s modulus to be 104 N/m and the Poisson’s ratio as −0.13 [48]. The ab initio calculations have predicted a negative Poisson’s ratio in the 1T-MoSe₂, which was attributed to the orbital coupling in this material. The orbital coupling enhances the angle bending interaction in the VFF model. As a result, the value of the angle bending parameter is much larger than the bond stretching force constant parameter, which is typical in auxetic materials with negative Poisson’s ratio [52].

The parameters for the two-body SW potential used by GULP are shown in Table 251 . The parameters for the three-body SW potential used by GULP are shown in Table 252 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 253 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-MoSe₂ under uniaxial tension at 1 and 300 K. Figure 124 shows the stress-strain curve for the tension of a single-layer 1T-MoSe₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-MoSe₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-MoSe₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 88.2 and 87.9 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=\mathrm{}-0.13$ . The fitted Young’s modulus value is about 10% smaller than the ab initio result of 104 N/m [48], as only short-range interactions are considered in the present work. The long-range interactions are ignored, which typically leads to about 10% underestimation for the value of the Young’s modulus.

There is no available value for nonlinear quantities in the single-layer 1T-MoSe₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −632.6 and −629.7 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 6.1 N/m at the ultimate strain of 0.13 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 6.2 N/m at the ultimate strain of 0.15 in the zigzag direction at the low temperature of 1 K.

Figure 125 shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

Figure 126.

Stress-strain for single-layer 1T-MoTe₂ of dimension $100\times 100$ Å along the armchair and zigzag directions.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{Mo}-\text{Te}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{Mo}-\text{Te}-\text{Te}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{Te}-\text{Mo}-\text{Mo}}{\left(\text{Δ}\theta \right)}^2$
Parameter	3.074	12.516	12.516
$r_0$ or ${\theta }_0$	2.729	79.700	79.700

Table 254.

The VFF model for single-layer 1T-MoTe₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

64. 1T-MoTe₂

Most existing theoretical studies on the single-layer 1T-MoTe₂ are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-MoTe₂.

The structure for the single-layer 1T-MoTe₂ is shown in Figure 71 (with M=Mo and X=Te). Each Mo atom is surrounded by six Te atoms. These Te atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each Te atom is connected to three Mo atoms. The structural parameters are from the first-principles calculations [48], including the lattice constant $a=3.4970$ Å and the bond length $d_{\text{Mo}-\text{Te}}=2.7287$ Å, which are derived from the angle ${\theta }_{\text{TeMoMo}}=79.7°$ . The other angle is ${\theta }_{\text{MoTeTe}}=79.7°$ with Te atoms from the same (top or bottom) group.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Mo─Te	2.597	1.068	27.720	0.0	3.582

Table 255.

Two-body SW potential parameters for single-layer 1T-MoTe₂ used by GULP [8] as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{Mo}-\text{Te}-\text{Te}}$	78.925	79.700	1.068	1.068	0.0	3.582	0.0	3.582	0.0	4.777
${\theta }_{\text{Te}-\text{Mo}-\text{Mo}}$	78.925	79.700	1.068	1.068	0.0	3.582	0.0	3.582	0.0	4.777

Table 256.

Three-body SW potential parameters for single-layer 1T-MoTe₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

Table 254 shows three VFF terms for the single-layer 1T-MoTe₂; one of which is the bond stretching interaction shown by Eq. (1), while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\text{Mo}-\text{Te}-\text{Te}}$ is for the angle ${\theta }_{\text{Mo}-\text{Te}-\text{Te}}$ with both Te atoms from the same (top or bottom) group. We find that there are actually only two parameters in the VFF model, so we can determine their value by fitting to the Young’s modulus and the Poisson’s ratio of the system. The ab initio calculations have predicted the Young’s modulus to be 92 N/m and the Poisson’s ratio as −0.07 [48]. The ab initio calculations have predicted a negative Poisson’s ratio in the 1T-MoTe₂, which was attributed to the orbital coupling in this material. The orbital coupling enhances the angle bending interaction in the VFF model. As a result, the value of the angle bending parameter is much larger than the bond stretching force constant parameter, which is typical in auxetic materials with negative Poisson’s ratio [52].

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Mo─Te1─Te1	1.000	1.068	3.355	78.925	1.000	0.179	2.597	21.328	4	0	0.0

Table 257.

SW potential parameters for single-layer 1T-MoTe₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Figure 127.

Phonon spectrum for single-layer 1T-MoTe₂ along the ΓMKΓ direction in the Brillouin zone. The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 255 . The parameters for the three-body SW potential used by GULP are shown in Table 256 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 257 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-MoTe₂ under uniaxial tension at 1 and 300 K. Figure 126 shows the stress-strain curve for the tension of a single-layer 1T-MoTe₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-MoTe₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-MoTe₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 81.6 and 81.2 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=\mathrm{}-0.07$ . The fitted Young’s modulus value is about 10% smaller than the ab initio result of 92 N/m [48], as only short-range interactions are considered in the present work. The long-range interactions are ignored, which typically leads to about 10% underestimation for the value of the Young’s modulus.

Figure 128.

Stress-strain for single-layer 1T-TcS₂ of dimension $100\times 100$ Å along the armchair and zigzag directions.

There is no available value for nonlinear quantities in the single-layer 1T-MoTe₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −543.1 and −558.6 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 6.0 N/m at the ultimate strain of 0.14 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 6.0 N/m at the ultimate strain of 0.16 in the zigzag direction at the low temperature of 1 K.

Figure 127 shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

65. 1T-TcS₂

Most existing theoretical studies on the single-layer 1T-TcS₂ are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-TcS₂.

The structure for the single-layer 1T-TcS₂ is shown in Figure 71 (with M = Tc and X = S). Each Tc atom is surrounded by six S atoms. These S atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each S atom is connected to three Tc atoms. The structural parameters are from the first-principles calculations [48] including the lattice constant $a=3.0692$ Å and the bond length $d_{\text{Tc}-\text{S}}=2.3924$ Å, which are derived from the angle ${\theta }_{\text{STcTc}}=79.8°$ . The other angle is ${\theta }_{\text{TcSS}}=79.8°$ with S atoms from the same (top or bottom) group.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{Tc}-\text{S}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{Tc}-\text{S}-\text{S}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{S}-\text{Tc}-\text{Tc}}{\left(\text{Δ}\theta \right)}^2$
Parameter	2.986	11.141	11.141
$r_0$ or ${\theta }_0$	2.392	79.800	79.800

Table 258.

The VFF model for single-layer 1T-TcS₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of ev/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Tc─S	1.945	0.939	16.380	0.0	3.142

Table 259.

Two-body SW potential parameters for single-layer 1T-TcS₂ used by GULP [8] as expressed in Eq. (3).

Table 258 shows three VFF terms for the single-layer 1T-TcS₂; one of which is the bond stretching interaction shown by Eq. (1), while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\text{Tc}-\text{S}-\text{S}}$ is for the angle ${\theta }_{\text{Tc}-\text{S}-\text{S}}$ with both S atoms from the same (top or bottom) group. We find that there are actually only two parameters in the VFF model, so we can determine their value by fitting to the Young’s modulus and the Poisson’s ratio of the system. The ab initio calculations have predicted the Young’s modulus to be 94 N/m and the Poisson’s ratio as −0.10 [48]. The ab initio calculations have predicted a negative Poisson’s ratio in the 1T-TcS₂, which was attributed to the orbital coupling in this material. The orbital coupling enhances the angle bending interaction in the VFF model. As a result, the value of the angle bending parameter is much larger than the bond stretching force constant parameter, which is typical in auxetic materials with negative Poisson’s ratio [52].

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{Tc}-\text{S}-\text{S}}$	70.512	79.800	0.939	0.939	0.0	3.142	0.0	3.142	0.0	4.193
${\theta }_{\text{S}-\text{Tc}-\text{Tc}}$	70.512	79.800	0.939	0.939	0.0	3.142	0.0	3.142	0.0	4.193

Table 260.

Three-body SW potential parameters for single-layer 1T-TcS₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Tc─S1─S1	1.000	0.939	3.345	70.512	1.000	0.177	1.945	21.038	4	0	0.0

Table 261.

SW potential parameters for single-layer 1T-TcS₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Figure 129.

Phonon spectrum for single-layer 1T-TcS₂ along the ΓMKΓ direction in the Brillouin zone. The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 259 . The parameters for the three-body SW potential used by GULP are shown in Table 260 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 261 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-TcS₂ under uniaxial tension at 1 and 300 K. Figure 128 shows the stress-strain curve for the tension of a single-layer 1T-TcS₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-TcS₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-TcS₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 84.3 and 84.0 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=\mathrm{}-0.10$ . The fitted Young’s modulus value is about 10% smaller than the ab initio result of 94 N/m [48], as only short-range interactions are considered in the present work. The long-range interactions are ignored, which typically leads to about 10% underestimation for the value of the Young’s modulus.

There is no available value for nonlinear quantities in the single-layer 1T-TcS₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −572.0 and −588.6 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 6.0 N/m at the ultimate strain of 0.13 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 6.1 N/m at the ultimate strain of 0.16 in the zigzag direction at the low temperature of 1 K.

Figure 129 shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

66. 1T-TcSe₂

Most existing theoretical studies on the single-layer 1T-TcSe₂ are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-TcSe₂.

The structure for the single-layer 1T-TcSe₂ is shown in Figure 71 (with M=Tc and X=Se). Each Tc atom is surrounded by six Se atoms. These Se atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each Se atom is connected to three Tc atoms. The structural parameters are from the first-principles calculations [48], including the lattice constant $a=3.1543$ Å and the bond length $d_{\text{Tc}-\text{Se}}=2.5061$ Å, which are derived from the angle ${\theta }_{\text{SeTcTc}}=78°$ . The other angle is ${\theta }_{\text{TcSeSe}}=78°$ with Se atoms from the same (top or bottom) group.

Figure 130.

Stress-strain for single-layer 1T-TcSe₂ of dimension $100\times 100$ Å along the armchair and zigzag directions.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{Tc}-\text{Se}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{Tc}-\text{Se}-\text{Se}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{Se}-\text{Tc}-\text{Tc}}{\left(\text{Δ}\theta \right)}^2$
Parameter	3.467	10.636	10.636
$r_0$ or ${\theta }_0$	2.506	78.001	78.001

Table 262.

The VFF model for single-layer 1T-TcSe₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of ev/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

Table 262 shows three VFF terms for the single-layer 1T-TcSe₂; one of which is the bond stretching interaction shown by Eq. (1), while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\text{Tc}-\text{Se}-\text{Se}}$ is for the angle ${\theta }_{\text{Tc}-\text{Se}-\text{Se}}$ with both Se atoms from the same (top or bottom) group. We find that there are actually only two parameters in the VFF model, so we can determine their value by fitting to the Young’s modulus and the Poisson’s ratio of the system. The ab initio calculations have predicted the Young’s modulus to be 104 N/m and the Poisson’s ratio as −0.04 [48]. The ab initio calculations have predicted a negative Poisson’s ratio in the 1T-TcSe₂, which was attributed to the orbital coupling in this material. The orbital coupling enhances the angle bending interaction in the VFF model. As a result, the value of the angle bending parameter is much larger than the bond stretching force constant parameter, which is typical in auxetic materials with negative Poisson’s ratio [52].

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Tc─Se	2.355	0.925	19.723	0.0	3.267

Table 263.

Two-body SW potential parameters for single-layer 1T-TcSe₂ used by GULP [8] as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{Tc}-\text{Se}-\text{Se}}$	63.150	78.001	0.925	0.925	0.0	3.267	0.0	3.267	0.0	4.309
${\theta }_{\text{Se}-\text{Tc}-\text{Tc}}$	63.150	78.001	0.925	0.925	0.0	3.267	0.0	3.267	0.0	4.309

Table 264.

Three-body SW potential parameters for single-layer 1T-TcSe₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

The parameters for the two-body SW potential used by GULP are shown in Table 263 . The parameters for the three-body SW potential used by GULP are shown in Table 264 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 265 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-TcSe₂ under uniaxial tension at 1 and 300 K. Figure 130 shows the stress-strain curve for the tension of a single-layer 1T-TcSe₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-TcSe₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-TcSe₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 88.8 and 88.3 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=\mathrm{}-0.04$ . The fitted Young’s modulus value is about 10% smaller than the ab initio result of 104 N/m [48], as only short-range interactions are considered in the present work. The long-range interactions are ignored, which typically leads to about 10% underestimation for the value of the Young’s modulus.

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Tc─Se1─Se1	1.000	0.925	3.532	63.150	1.000	0.208	2.355	26.932	4	0	0.0

Table 265.

SW potential parameters for single-layer 1T-TcSe₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Figure 131.

Phonon spectrum for single-layer 1T-TcSe₂ along the ΓMKΓ direction in the Brillouin zone. The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

There is no available value for nonlinear quantities in the single-layer 1T-TcSe₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −565.7 and −587.3 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 6.8 N/m at the ultimate strain of 0.14 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 6.8 N/m at the ultimate strain of 0.17 in the zigzag direction at the low temperature of 1 K.

Figure 132.

Stress-strain for single-layer 1T-TcTe₂ of dimension $100\times 100$ Å along the armchair and zigzag directions.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{Tc}-\text{Te}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{Tc}-\text{Te}-\text{Te}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{Te}-\text{Tc}-\text{Tc}}{\left(\text{Δ}\theta \right)}^2$
Parameter	0.785	8.894	8.894
$r_0$ or ${\theta }_0$	2.690	78.801	78.801

Table 266.

The VFF model for single-layer 1T-TcTe₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of ev/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

Figure 131 shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Tc─Te	0.628	1.021	26.181	0.0	3.519

Table 267.

Two-body SW potential parameters for single-layer 1T-TcTe₂ used by GULP [8] as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{Tc}-\text{Te}-\text{Te}}$	54.313	78.801	1.021	1.021	0.0	3.519	0.0	3.519	0.0	4.665
${\theta }_{\text{Te}-\text{Tc}-\text{Tc}}$	54.313	78.801	1.021	1.021	0.0	3.519	0.0	3.519	0.0	4.665

Table 268.

Three-body SW potential parameters for single-layer 1T-TcTe₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

67. 1T-TcTe₂

Most existing theoretical studies on the single-layer 1T-TcTe₂ are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-TcTe₂.

The structure for the single-layer 1T-TcTe₂ is shown in Figure 71 (with M=Tc and X=Te). Each Tc atom is surrounded by six Te atoms. These Te atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each Te atom is connected to three Tc atoms. The structural parameters are from the first-principles calculations [48], including the lattice constant $a=3.4149$ Å and the bond length $d_{\text{Tc}-\text{Te}}=2.6900$ Å, which are derived from the angle ${\theta }_{\text{TeTcTc}}=78.8°$ . The other angle is ${\theta }_{\text{TcTeTe}}=78.8°$ with Te atoms from the same (top or bottom) group.

Table 266 shows three VFF terms for the single-layer 1T-TcTe₂; one of which is the bond stretching interaction shown by Eq. (1), while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\text{Tc}-\text{Te}-\text{Te}}$ is for the angle ${\theta }_{\text{Tc}-\text{Te}-\text{Te}}$ with both Te atoms from the same (top or bottom) group. We find that there are actually only two parameters in the VFF model, so we can determine their value by fitting to the Young’s modulus and the Poisson’s ratio of the system. The ab initio calculations have predicted the Young’s modulus to be 34 N/m and the Poisson’s ratio as −0.36 [48]. The ab initio calculations have predicted a negative Poisson’s ratio in the 1T-TcTe₂, which was attributed to the orbital coupling in this material. The orbital coupling enhances the angle bending interaction in the VFF model. As a result, the value of the angle bending parameter is much larger than the bond stretching force constant parameter, which is typical in auxetic materials with negative Poisson’s ratio [52].

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Tc─Te1─Te1	1.000	1.021	3.447	54.313	1.000	0.194	0.628	24.110	4	0	0.0

Table 269.

SW potential parameters for single-layer 1T-TcTe₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Figure 133.

Phonon spectrum for single-layer 1T-TcTe₂ along the ΓMKΓ direction in the Brillouin zone. The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 267 . The parameters for the three-body SW potential used by GULP are shown in Table 268 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 269 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-TcTe₂ under uniaxial tension at 1 and 300 K. Figure 132 shows the stress-strain curve for the tension of a single-layer 1T-TcTe₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-TcTe₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-TcTe₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 28.6 N/m along the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=\mathrm{}-0.21$ . The fitted Young’s modulus value is about 10% smaller than the ab initio result of 34 N/m [48], as only short-range interactions are considered in the present work. The long-range interactions are ignored, which typically leads to about 10% underestimation for the value of the Young’s modulus.

There is no available value for nonlinear quantities in the single-layer 1T-TcTe₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −207.8 and −208.7 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 1.9 N/m at the ultimate strain of 0.11 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 2.0 N/m at the ultimate strain of 0.14 in the zigzag direction at the low temperature of 1 K. The ultimate strain decreases to be about 0.01 at 300 K, so the single-layer 1T-TcTe₂ is not very stable at higher temperature. It is because this material is very soft and the Poisson’s ratio is very small (negative value).

Figure 133 shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

68. 1T-RhTe₂

Most existing theoretical studies on the single-layer 1T-RhTe₂ are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-RhTe₂.

The structure for the single-layer 1T-RhTe₂ is shown in Figure 71 (with M=Rh and X=Te). Each Rh atom is surrounded by six Te atoms. These Te atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each Te atom is connected to three Rh atoms. The structural parameters are from the first-principles calculations [48], including the lattice constant $a=3.7563$ Å and the bond length $d_{\text{Rh}-\text{Te}}=2.6332$ Å, which are derived from the angle ${\theta }_{\text{TeRhRh}}=91°$ . The other angle is ${\theta }_{\text{RhTeTe}}=91°$ with Te atoms from the same (top or bottom) group.

Figure 134.

Stress-strain for single-layer 1T-RhTe₂ of dimension $100\times 100$ Å along the armchair and zigzag directions.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{Rh}-\text{Te}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{Rh}-\text{Te}-\text{Te}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{Te}-\text{Rh}-\text{Rh}}{\left(\text{Δ}\theta \right)}^2$
Parameter	4.366	1.869	1.869
$r_0$ or ${\theta }_0$	2.633	91.001	91.001

Table 270.

The VFF model for single-layer 1T-RhTe₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of ev/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

Table 270 shows three VFF terms for the single-layer 1T-RhTe₂; one of which is the bond stretching interaction shown by Eq. (1), while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\text{Rh}-\text{Te}-\text{Te}}$ is for the angle ${\theta }_{\text{Rh}-\text{Te}-\text{Te}}$ with both Te atoms from the same (top or bottom) group. We find that there are actually only two parameters in the VFF model, so we can determine their value by fitting to the Young’s modulus and the Poisson’s ratio of the system. The ab initio calculations have predicted the Young’s modulus to be 37 N/m and the Poisson’s ratio as 0.20 [48].

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Rh─Te	4.640	1.450	24.038	0.0	3.610

Table 271.

Two-body SW potential parameters for single-layer 1T-RhTe₂ used by GULP [8] as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{Rh}-\text{Te}-\text{Te}}$ 18.192	18.192	91.001	1.450	1.450	0.0	3.610	0.0	3.610	0.0	5.131
${\theta }_{\text{Te}-\text{Rh}-\text{Rh}}$ 18.192	18.192	91.001	1.450	1.450	0.0	3.610	0.0	3.610	0.0	5.131

Table 272.

Three-body SW potential parameters for single-layer 1T-RhTe₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

The parameters for the two-body SW potential used by GULP are shown in Table 271 . The parameters for the three-body SW potential used by GULP are shown in Table 272 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 273 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-RhTe₂ under uniaxial tension at 1 and 300 K. Figure 134 shows the stress-strain curve for the tension of a single-layer 1T-RhTe₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-RhTe₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-RhTe₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 32.1 and 32.0 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.20$ . The fitted Young’s modulus value is about 10% smaller than the ab initio result of 37 N/m [48], as only short-range interactions are considered in the present work. The long-range interactions are ignored, which typically leads to about 10% underestimation for the value of the Young’s modulus.

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Rh─Te1─Te1	1.000	1.450	2.490	18.192	1.000	−0.017	4.640	5.436	4	0	0.0

Table 273.

SW potential parameters for single-layer 1T-RhTe₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Figure 135.

Phonon spectrum for single-layer 1T-RhTe₂ along the ΓMKΓ direction in the Brillouin zone. The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

There is no available value for nonlinear quantities in the single-layer 1T-RhTe₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −103.1 and −116.5 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 4.6 N/m at the ultimate strain of 0.25 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 4.4 N/m at the ultimate strain of 0.29 in the zigzag direction at the low temperature of 1 K.

Figure 135 shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

Figure 136.

Stress-strain for single-layer 1T-PdS₂ of dimension $100\times 100$ Å along the armchair and zigzag directions.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{Pd}-\text{S}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{Pd}-\text{S}-\text{S}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{S}-\text{Pd}-\text{Pd}}{\left(\text{Δ}\theta \right)}^2$
Parameter	10.374	3.122	3.122
$r_0$ or ${\theta }_0$	2.401	94.998	94.998

Table 274.

The VFF model for single-layer 1T-PdS₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of ev/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

69. 1T-PdS₂

Most existing theoretical studies on the single-layer 1T-PdS₂ are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-PdS₂.

The structure for the single-layer 1T-PdS₂ is shown in Figure 71 (with M=Pd and X=S). Each Pd atom is surrounded by six S atoms. These S atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each S atom is connected to three Pd atoms. The structural parameters are from the first-principles calculations [48], including the lattice constant $a=3.5408$ Å and the bond length $d_{\text{Pd}-\text{S}}=2.4013$ Å, which are derived from the angle ${\theta }_{\text{SPdPd}}=95°$ . The other angle is ${\theta }_{\text{PdSS}}=95°$ with S atoms from the same (top or bottom) group.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Pd─S	10.116	1.467	16.625	0.0	3.340

Table 275.

Two-body SW potential parameters for single-layer 1T-PdS₂ used by GULP [8] as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{Pd}-\text{S}-\text{S}}$	35.859	94.998	1.467	1.467	0.0	3.340	0.0	3.340	0.0	4.837
${\theta }_{\text{S}-\text{Pd}-\text{Pd}}$	35.859	94.998	1.467	1.467	0.0	3.340	0.0	3.340	0.0	4.837

Table 276.

Three-body SW potential parameters for single-layer 1T-PdS₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

Table 274 shows three VFF terms for the single-layer 1T-PdS₂; one of which is the bond stretching interaction shown by Eq. (1), while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\text{Pd}-\text{S}-\text{S}}$ is for the angle ${\theta }_{\text{Pd}-\text{S}-\text{S}}$ with both S atoms from the same (top or bottom) group. We find that there are actually only two parameters in the VFF model, so we can determine their value by fitting to the Young’s modulus and the Poisson’s ratio of the system. The ab initio calculations have predicted the Young’s modulus to be 77 N/m and the Poisson’s ratio as 0.53 [48].

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Pd─S1─S1	1.000	1.467	2.276	35.859	1.000	−0.087	10.116	3.588	4	0	0.0

Table 277.

SW potential parameters for single-layer 1T-PdS₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Figure 137.

Phonon spectrum for single-layer 1T-PdS₂ along the ΓMKΓ direction in the Brillouin zone. The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 275 . The parameters for the three-body SW potential used by GULP are shown in Table 276 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 277 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-PdS₂ under uniaxial tension at 1 and 300 K. Figure 136 shows the stress-strain curve for the tension of a single-layer 1T-PdS₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-PdS₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-PdS₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 69.9 and 69.5 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.20$ . The fitted Young’s modulus value is about 10% smaller than the ab initio result of 77 N/m [48], as only short-range interactions are considered in the present work. The long-range interactions are ignored, which typically leads to about 10% underestimation for the value of the Young’s modulus.

There is no available value for nonlinear quantities in the single-layer 1T-PdS₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −222.0 and −248.8 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 10.1 N/m at the ultimate strain of 0.25 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 9.7 N/m at the ultimate strain of 0.30 in the zigzag direction at the low temperature of 1 K.

Figure 138.

Stress-strain for single-layer 1T-PdSe₂ of dimension $100\times 100$ Å along the armchair and zigzag directions.

Figure 137 shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

70. 1T-PdSe₂

Most existing theoretical studies on the single-layer 1T-PdSe₂ are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-PdSe₂.

The structure for the single-layer 1T-PdSe₂ is shown in Figure 71 (with M=Pd and X=Se). Each Pd atom is surrounded by six Se atoms. These Se atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each Se atom is connected to three Pd atoms. The structural parameters are from the first-principles calculations [48], including the lattice constant $a=3.6759$ Å and the bond length $d_{\text{Pd}-\text{Se}}=2.4929$ Å, which are derived from the angle ${\theta }_{\text{SePdPd}}=95°$ . The other angle is ${\theta }_{\text{PdSeSe}}=95°$ with Se atoms from the same (top or bottom) group.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{Pd}-\text{Se}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{Pd}-\text{Se}-\text{Se}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{Se}-\text{Pd}-\text{Pd}}{\left(\text{Δ}\theta \right)}^2$
Parameter	10.374	3.122	3.122
$r_0$ or ${\theta }_0$	2.493	94.999	94.999

Table 278.

The VFF model for single-layer 1T-PdSe₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of ev/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Pd─Se	10.902	1.523	19.310	0.0	3.467

Table 279.

Two-body SW potential parameters for single-layer 1T-PdSe₂ used by GULP [8] as expressed in Eq. (3).

Table 278 shows three VFF terms for the single-layer 1T-PdSe₂; one of which is the bond stretching interaction shown by Eq. (1), while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\text{Pd}-\text{Se}-\text{Se}}$ is for the angle ${\theta }_{\text{Pd}-\text{Se}-\text{Se}}$ with both Se atoms from the same (top or bottom) group. We find that there are actually only two parameters in the VFF model, so we can determine their value by fitting to the Young’s modulus and the Poisson’s ratio of the system. The ab initio calculations have predicted the Young’s modulus to be 66 N/m and the Poisson’s ratio as 0.45 [48].

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{Pd}-\text{Se}-\text{Se}}$	35.859	94.999	1.523	1.523	0.0	3.467	0.0	3.467	0.0	5.021
${\theta }_{\text{Se}-\text{Pd}-\text{Pd}}$	35.859	94.999	1.523	1.523	0.0	3.467	0.0	3.467	0.0	5.021

Table 280.

Three-body SW potential parameters for single-layer 1T-PdSe₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Pd─Se1─Se1	1.000	1.523	2.276	35.859	1.000	−0.087	10.902	3.588	4	0	0.0

Table 281.

SW potential parameters for single-layer 1T-PdSe₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Figure 139.

Phonon spectrum for single-layer 1T-PdSe₂ along the ΓMKΓ direction in the Brillouin zone. The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 279 . The parameters for the three-body SW potential used by GULP are shown in Table 280 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 281 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-PdSe₂ under uniaxial tension at 1 and 300 K. Figure 138 shows the stress-strain curve for the tension of a single-layer 1T-PdSe₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-PdSe₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-PdSe₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 65.5 and 65.3 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.21$ .

Figure 140.

Stress-strain for single-layer 1T-PdTe₂ of dimension $100\times 100$ Å along the armchair and zigzag directions.

There is no available value for nonlinear quantities in the single-layer 1T-PdSe₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −194.7 and −222.8 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 9.9 N/m at the ultimate strain of 0.26 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 9.5 N/m at the ultimate strain of 0.31 in the zigzag direction at the low temperature of 1 K.

Figure 139 shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

71. 1T-PdTe₂

Most existing theoretical studies on the single-layer 1T-PdTe₂ are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-PdTe₂.

The structure for the single-layer 1T-PdTe₂ is shown in Figure 71 (with M=Pd and X=Te). Each Pd atom is surrounded by six Te atoms. These Te atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each Te atom is connected to three Pd atoms. The structural parameters are from the first-principles calculations [48], including the lattice constant $a=3.9162$ Å and the bond length $d_{\text{Pd}-\text{Te}}=2.6349$ Å, which are derived from the angle ${\theta }_{\text{TePdPd}}=96°$ . The other angle is ${\theta }_{\text{PdTeTe}}=96°$ with Te atoms from the same (top or bottom) group.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{Pd}-\text{Te}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{Pd}-\text{Te}-\text{Te}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{Te}-\text{Pd}-\text{Pd}}{\left(\text{Δ}\theta \right)}^2$
Parameter	10.374	3.122	3.122
$r_0$ or ${\theta }_0$	2.635	95.999	95.999

Table 282.

The VFF model for single-layer 1T-PdTe₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of ev/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Pd─Te	12.474	1.650	24.101	0.0	3.678

Table 283.

Two-body SW potential parameters for single-layer 1T-PdTe₂ used by GULP [8] as expressed in Eq. (3).

Table 282 shows three VFF terms for the single-layer 1T-PdTe₂; one of which is the bond stretching interaction shown by Eq. (1), while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\text{Pd}-\text{Te}-\text{Te}}$ is for the angle ${\theta }_{\text{Pd}-\text{Te}-\text{Te}}$ with both Te atoms from the same (top or bottom) group. We find that there are actually only two parameters in the VFF model, so we can determine their value by fitting to the Young’s modulus and the Poisson’s ratio of the system. The ab initio calculations have predicted the Young’s modulus to be 63 N/m and the Poisson’s ratio as 0.35 [48].

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{Pd}-\text{Te}-\text{Te}}$	37.406	95.999	1.650	1.650	0.0	3.678	0.0	3.678	0.0	5.350
${\theta }_{\text{Te}-\text{Pd}-\text{Pd}}$	37.406	95.999	1.650	1.650	0.0	3.678	0.0	3.678	0.0	5.350

Table 284.

Three-body SW potential parameters for single-layer 1T-PdTe₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Pd─Te1─Te1	1.000	1.650	2.229	37.406	1.000	−0.105	12.474	3.250	4	0	0.0

Table 285.

SW potential parameters for single-layer 1T-PdTe₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Figure 141.

Phonon spectrum for single-layer 1T-PdTe₂ along the ΓMKΓ direction in the Brillouin zone. The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 283 . The parameters for the three-body SW potential used by GULP are shown in Table 284 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 285 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-PdTe₂ under uniaxial tension at 1 and 300 K. Figure 140 shows the stress-strain curve for the tension of a single-layer 1T-PdTe₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-PdTe₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-PdTe₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 61.6 and 61.4 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.22$ .

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{Sn}-\text{S}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{Sn}-\text{S}-\text{S}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{S}-\text{Sn}-\text{Sn}}{\left(\text{Δ}\theta \right)}^2$
Parameter	7.872	5.817	5.817
$r_0$ or ${\theta }_0$	2.570	90.173	90.173

Table 286.

The VFF model for single-layer 1T-SnS₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of ev/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

There is no available value for nonlinear quantities in the single-layer 1T-PdTe₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −178.8 and −203.8 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 9.6 N/m at the ultimate strain of 0.27 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 9.2 N/m at the ultimate strain of 0.32 in the zigzag direction at the low temperature of 1 K.

Figure 141 shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

Figure 142.

Phonon spectrum for single-layer 1T-SnS₂. (a) Phonon dispersion along the ΓM direction in the Brillouin zone. The results from the VFF model (lines) are comparable with the ab initio results (pentagons) from Ref. [34]. (b) The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Sn─S	7.805	1.384	21.812	0.0	3.513

Table 287.

Two-body SW potential parameters for single-layer 1T-SnS₂ used by GULP [8] as expressed in Eq. (3).

72. 1T-SnS₂

Most existing theoretical studies on the single-layer 1T-SnS₂ are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-SnS₂.

Figure 143.

Stress-strain for single-layer 1T-SnS₂ of dimension $100\times 100$ Å along the armchair and zigzag directions.

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{Sn}-\text{S}-\text{S}}$	54.748	90.173	1.384	1.384	0.0	3.513	0.0	3.513	0.0	4.972
${\theta }_{\text{S}-\text{Sn}-\text{Sn}}$	54.748	90.173	1.384	1.384	0.0	3.513	0.0	3.513	0.0	4.972

Table 288.

Three-body SW potential parameters for single-layer 1T-SnS₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

The structure for the single-layer 1T-SnS₂ is shown in Figure 71 (with M=Sn and X=S). Each Sn atom is surrounded by six S atoms. These S atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each S atom is connected to three Sn atoms. The structural parameters are from the first-principles calculations [34], including the lattice constant $a=3.640$ Å and the bond length $d_{\text{Sn}-\text{S}}=2.570$ Å. The resultant angles are ${\theta }_{\text{SSnSn}}=90.173°$ and ${\theta }_{\text{SnSS}}=90.173°$ with S atoms from the same (top or bottom) group.

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Sn─S1─S1	1.000	1.384	2.539	54.748	1.000	−0.003	7.805	5.949	4	0	0.0

Table 289.

SW potential parameters for single-layer 1T-SnS₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Table 286 shows three VFF terms for the single-layer 1T-SnS₂; one of which is the bond stretching interaction shown by Eq. (1), while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\text{Sn}-\text{S}-\text{S}}$ is for the angle ${\theta }_{\text{Sn}-\text{S}-\text{S}}$ with both S atoms from the same (top or bottom) group. These force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the ΓM as shown in Figure 142(a) . The ab initio calculations for the phonon dispersion are from Ref. [34]. The lowest acoustic branch (flexural mode) is almost linear in the ab initio calculations, which may be due to the violation of the rigid rotational invariance [20]. Figure 142(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 287 . The parameters for the three-body SW potential used by GULP are shown in Table 288 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 289 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-SnS₂ under uniaxial tension at 1 and 300 K. Figure 143 shows the stress-strain curve for the tension of a single-layer 1T-SnS₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-SnS₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-SnS₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 88.4 and 87.9 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.13$ .

There is no available value for nonlinear quantities in the single-layer 1T-SnS₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −392.8 and −421.6 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 9.6 N/m at the ultimate strain of 0.20 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 9.3 N/m at the ultimate strain of 0.24 in the zigzag direction at the low temperature of 1 K.

Figure 144.

Phonon spectrum for single-layer 1T-SnSe₂. (a) Phonon dispersion along the ΓM direction in the Brillouin zone. The results from the VFF model (lines) are comparable with the ab initio results (pentagons) from Ref. [34]. (b) The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

73. 1T-SnSe₂

Most existing theoretical studies on the single-layer 1T-SnSe₂ are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-SnSe₂.

The structure for the single-layer 1T-SnSe₂ is shown in Figure 71 (with M=Sn and X=Se). Each Sn atom is surrounded by six Se atoms. These Se atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each Se atom is connected to three Sn atoms. The structural parameters are from the first-principles calculations [34] including the lattice constant $a=3.792$ Å and the bond length $d_{\text{Sn}-\text{Se}}=2.704$ Å. The resultant angles are ${\theta }_{\text{SeSnSn}}=89.044°$ and ${\theta }_{\text{SnSeSe}}=89.044°$ with Se atoms from the same (top or bottom) group.

Figure 145.

Stress-strain for single-layer 1T-SnSe₂ of dimension $100\times 100$ Å along the armchair and zigzag directions.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{Sn}-\text{Se}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{Sn}-\text{Se}-\text{Se}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{Se}-\text{Sn}-\text{Sn}}{\left(\text{Δ}\theta \right)}^2$
Parameter	7.872	5.817	5.817
$r_0$ or ${\theta }_0$	2.704	89.044	89.044

Table 290.

The VFF model for single-layer 1T-SnSe₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of ev/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

Table 290 shows three VFF terms for the single-layer 1T-SnSe₂; one of which is the bond stretching interaction shown by Eq. (1), while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\text{Sn}-\text{Se}-\text{Se}}$ is for the angle ${\theta }_{\text{Sn}-\text{Se}-\text{Se}}$ with both Se atoms from the same (top or bottom) group. These force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the ΓM as shown in Figure 144(a) . The ab initio calculations for the phonon dispersion are from Ref. [34]. The lowest acoustic branch (flexural mode) is almost linear in the ab initio calculations, which may be due to the violation of the rigid rotational invariance [20]. Figure 144(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Sn─Se	8.395	1.411	26.730	0.0	3.681

Table 291.

Two-body SW potential parameters for single-layer 1T-SnSe₂ used by GULP [8] as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{Sn}-\text{Se}-\text{Se}}$	52.322	89.044	1.411	1.411	0.0	3.681	0.0	3.681	0.0	5.180
${\theta }_{\text{Se}-\text{Sn}-\text{Sn}}$	52.322	89.044	1.411	1.411	0.0	3.681	0.0	3.681	0.0	5.180

Table 292.

Three-body SW potential parameters for single-layer 1T-SnSe₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

The parameters for the two-body SW potential used by GULP are shown in Table 291 . The parameters for the three-body SW potential used by GULP are shown in Table 292 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 293 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-SnSe₂ under uniaxial tension at 1 and 300 K. Figure 145 shows the stress-strain curve for the tension of a single-layer 1T-SnSe₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-SnSe₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-SnSe₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 82.0 and 81.6 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.15$ .

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Sn─Se1─Se1	1.000	1.411	2.609	52.322	1.000	0.017	8.395	6.743	4	0	0.0

Table 293.

SW potential parameters for single-layer 1T-SnSe₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{Hf}-\text{S}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{Hf}-\text{S}-\text{S}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{S}-\text{Hf}-\text{Hf}}{\left(\text{Δ}\theta \right)}^2$
Parameter	7.930	4.283	4.283
$r_0$ or ${\theta }_0$	2.550	91.078	91.078

Table 294.

The VFF model for single-layer 1T-HfS₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of ev/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

There is no available value for nonlinear quantities in the single-layer 1T-SnSe₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −339.2 and −368.3 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 9.5 N/m at the ultimate strain of 0.21 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 9.1 N/m at the ultimate strain of 0.25 in the zigzag direction at the low temperature of 1 K.

Figure 146.

Phonon spectrum for single-layer 1T-HfS₂. (a) Phonon dispersion along the ΓM direction in the Brillouin zone. The results from the VFF model (lines) are comparable with the experiment data (pentagons) from Ref. [38]. (b) The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Hf─S	7.917	1.407	21.141	0.0	3.497

Table 295.

Two-body SW potential parameters for single-layer 1T-HfS2 used by GULP [8] as expressed in Eq. (3).

74. 1T-HfS₂

Most existing theoretical studies on the single-layer 1T-HfS₂ are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-HfS₂.

Figure 147.

Stress-strain for single-layer 1T-HfS₂ of dimension $100\times 100$ Å along the armchair and zigzag directions.

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{Hf}-\text{S}-\text{S}}$ (Å)	41.798	91.078	1.407	1.407	0.0	3.497	0.0	3.497	0.0	4.973
${\theta }_{\text{S}-\text{Hf}-\text{Hf}}$ (Å)	41.798	91.078	1.407	1.407	0.0	3.497	0.0	3.497	0.0	4.973

Table 296.

Three-body SW potential parameters for single-layer 1T-HfS₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

The structure for the single-layer 1T-HfS₂ is shown in Figure 71 (with M=Hf and X=S). Each Hf atom is surrounded by six S atoms. These S atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each S atom is connected to three Hf atoms. The structural parameters are from the first-principles calculations [53], including the lattice constant $a=3.64$ Å and the bond length $d_{\text{Hf}-\text{S}}=2.55$ Å. The resultant angles are ${\theta }_{\text{HfSS}}=91.078°$ with S atoms from the same (top or bottom) group and ${\theta }_{\text{SHfHf}}=91.078°$ .

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Hf─S─S	1.000	1.407	2.485	41.798	1.000	−0.019	7.917	5.394	4	0	0.0
S─Hf─Hf	1.000	1.407	2.485	41.798	1.000	−0.019	7.917	5.394	4	0	0.0

Table 297.

SW potential parameters for single-layer 1T-HfS2 used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Table 294 shows three VFF terms for the single-layer 1T-HfS₂; one of which is the bond stretching interaction shown by Eq. (1), while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\text{Hf}-\text{S}-\text{S}}$ is for the angle ${\theta }_{\text{Hf}-\text{S}-\text{S}}$ with both S atoms from the same (top or bottom) group. These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the ΓM as shown in Figure 146(a) . The ab initio calculations for the phonon dispersion are from Ref. [38]. Similar phonon dispersion can also be found in other ab initio calculations [34, 35]. Figure 146(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 295 . The parameters for the three-body SW potential used by GULP are shown in Table 296 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 297 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-HfS₂ under uniaxial tension at 1 and 300 K. Figure 147 shows the stress-strain curve for the tension of a single-layer 1T-HfS₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-HfS₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-HfS₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 73.3 and 72.9 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. These values are close to the ab initio results at 0 K temperature, e.g., 79.86 N/m in Ref. [53]. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.16$ , which agrees reasonably with the ab initio result [54] of 0.19.

There is no available value for nonlinear quantities in the single-layer 1T-HfS₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −280.9 and −317.2 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 8.9 N/m at the ultimate strain of 0.22 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 8.6 N/m at the ultimate strain of 0.26 in the zigzag direction at the low temperature of 1 K.

Figure 148.

Phonon spectrum for single-layer 1T-HfSe₂. (a) Phonon dispersion along the ΓM direction in the Brillouin zone. The results from the VFF model (lines) are comparable with the experiment data (pentagons) from Ref. [50]. (b) The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

75. 1T-HfSe₂

Most existing theoretical studies on the single-layer 1T-HfSe₂ are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-HfSe₂.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{Hf}-\text{Se}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{Hf}-\text{Se}-\text{Se}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{Se}-\text{Hf}-\text{Hf}}{\left(\text{Δ}\theta \right)}^2$
Parameter	7.930	4.283	4.283
$r_0$ or ${\theta }_0$	2.642	88.093	88.093

Table 298.

The VFF model for single-layer 1T-HfSe₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of ev/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Hf─Se	7.871	1.341	24.361	0.0	3.583

Table 299.

Two-body SW potential parameters for single-layer 1T-HfSe₂ used by GULP [8] as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{Hf}-\text{Se}-\text{Se}}$	37.039	88.093	1.341	1.341	0.0	3.583	0.0	3.583	0.0	5.018
${\theta }_{\text{Se}-\text{Hf}-\text{Hf}}$	37.039	88.093	1.341	1.341	0.0	3.583	0.0	3.583	0.0	5.018

Table 300.

Three-body SW potential parameters for single-layer 1T-HfSe₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Hf─Se─Se	1.000	1.341	2.672	37.039	1.000	0.033	7.871	7.533	4	0	0.0
Se─Hf─Hf	1.000	1.341	2.672	37.039	1.000	0.033	7.871	7.533	4	0	0.0

Table 301.

SW potential parameters for single-layer 1T-HfSe₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Figure 149.

Stress-strain for single-layer 1T-HfSe₂ of dimension $100\times 100$ Å along the armchair and zigzag directions.

The structure for the single-layer 1T-HfSe₂ is shown in Figure 71 (with M=Hf and X=Se). Each Hf atom is surrounded by six Se atoms. These Se atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each Se atom is connected to three Hf atoms. The structural parameters are from the first-principles calculations [51], including the lattice constant $a=3.673$ Å and the position of the Se atom with respect to the Hf atomic plane $h=1.575$ Å. The resultant angles are ${\theta }_{\text{HfSeSe}}=88.093°$ with S atoms from the same (top or bottom) group and ${\theta }_{\text{SeHfHf}}=88.093°$ .

Table 298 shows three VFF terms for the single-layer 1T-HfSe₂; one of which is the bond stretching interaction shown by Eq. (1), while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\text{Hf}-\text{Se}-\text{Se}}$ is for the angle ${\theta }_{\text{Hf}-\text{Se}-\text{Se}}$ with both Se atoms from the same (top or bottom) group. These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the ΓM as shown in Figure 148(a) . The ab initio calculations for the phonon dispersion are from Ref. [50]. Similar phonon dispersion can also be found in other ab initio calculations [34]. Figure 148(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 299 . The parameters for the three-body SW potential used by GULP are shown in Table 300 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 301 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-HfSe₂ under uniaxial tension at 1 and 300 K. Figure 149 shows the stress-strain curve for the tension of a single-layer 1T-HfSe₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-HfSe₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-HfSe₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 67.3 and 67.0 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.18$ .

There is no available value for nonlinear quantities in the single-layer 1T-HfSe₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −221.5 and −258.6 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 9.0 N/m at the ultimate strain of 0.24 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 8.7 N/m at the ultimate strain of 0.28 in the zigzag direction at the low temperature of 1 K.

76. 1T-HfTe₂

Most existing theoretical studies on the single-layer 1T-HfTe₂ are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-HfTe₂.

The structure for the single-layer 1T-HfTe₂ is shown in Figure 71 (with M = Hf and X = Te). Each Hf atom is surrounded by six Te atoms. These Te atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each Te atom is connected to three Hf atoms. The structural parameters are from the first-principles calculations [48], including the lattice constant $a=3.9606$ Å and the bond length $d_{\text{Hf}-\text{Te}}=2.8559$ Å, which are derived from the angle ${\theta }_{\text{TeHfHf}}=87.8°$ . The other angle is ${\theta }_{\text{HfTeTe}}=87.8°$ with Te atoms from the same (top or bottom) group.

Figure 150.

Stress-strain for single-layer 1T-HfTe₂ of dimension $100\times 100$ Å along the armchair and zigzag directions.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{Hf}-\text{Te}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{Hf}-\text{Te}-\text{Te}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{Te}-\text{Hf}-\text{Hf}}{\left(\text{Δ}\theta \right)}^2$
Parameter	3.328	3.877	3.877
$r_0$ or ${\theta }_0$	2.856	87.801	87.801

Table 302.

The VFF model for single-layer 1T-HfTe₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of ev/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of $\mathrm{\AA }$ ) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

Table 302 shows three VFF terms for the single-layer 1T-HfTe₂; one of which is the bond stretching interaction shown by Eq. (1), while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\text{Hf}-\text{Te}-\text{Te}}$ is for the angle ${\theta }_{\text{Hf}-\text{Te}-\text{Te}}$ with both Te atoms from the same (top or bottom) group. We find that there are actually only two parameters in the VFF model, so we can determine their value by fitting to the Young’s modulus and the Poisson’s ratio of the system. The ab initio calculations have predicted the Young’s modulus to be 50 N/m and the Poisson’s ratio as 0.10 [48].

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Hf─Te	3.835	1.439	33.262	0.0	3.869

Table 303.

Two-body SW potential parameters for single-layer 1T-HfTe₂ used by GULP [8] as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{Hf}-\text{Te}-\text{Te}}$	33.196	87.801	1.439	1.439	0.0	3.869	0.0	3.869	0.0	5.410
${\theta }_{\text{Te}-\text{Hf}-\text{Hf}}$	33.196	87.801	1.439	1.439	0.0	3.869	0.0	3.869	0.0	5.410

Table 304.

Three-body SW potential parameters for single-layer 1T-HfTe₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

The parameters for the two-body SW potential used by GULP are shown in Table 303 . The parameters for the three-body SW potential used by GULP are shown in Table 304 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 305 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-HfTe₂ under uniaxial tension at 1 and 300 K. Figure 150 shows the stress-strain curve for the tension of a single-layer 1T-HfTe₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-HfTe₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-HfTe₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 43.1 N/m along the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.10$ . The fitted Young’s modulus value is about 10% smaller than the ab initio result of 50 N/m [48], as only short-range interactions are considered in the present work. The long-range interactions are ignored, which typically lead to about 10% underestimation for the value of the Young’s modulus.

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Hf─Te1─Te1	1.000	1.439	2.690	33.196	1.000	0.038	3.835	7.764	4	0	0.0

Table 305.

SW potential parameters for single-layer 1T-HfTe₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Figure 151.

Phonon spectrum for single-layer 1T-HfTe₂ along the ΓMKΓ direction in the Brillouin zone. The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

There is no available value for nonlinear quantities in the single-layer 1T-HfTe₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −204.3 and −220.7 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 4.4 N/m at the ultimate strain of 0.19 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 4.3 N/m at the ultimate strain of 0.22 in the zigzag direction at the low temperature of 1 K.

Figure 151 shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

Figure 152.

Stress-strain for single-layer 1T-TaS₂ of dimension $100\times 100$ Å along the armchair and zigzag directions.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{Ta}-\text{S}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{Ta}-\text{S}-\text{S}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{S}-\text{Ta}-\text{Ta}}{\left(\text{Δ}\theta \right)}^2$
Parameter	11.192	4.774	4.774
$r_0$ or ${\theta }_0$	2.458	85.999	85.999

Table 306.

The VFF model for single-layer 1T-TaS₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of ev/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of $\mathrm{\AA }$ ) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

77. 1T-TaS₂

Most existing theoretical studies on the single-layer 1T-TaS₂ are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-TaS₂.

The structure for the single-layer 1T-TaS₂ is shown in Figure 71 (with M=Ta and X=S). Each Ta atom is surrounded by six S atoms. These S atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each S atom is connected to three Ta atoms. The structural parameters are from the first-principles calculations [48], including the lattice constant $a=3.3524$ Å and the bond length $d_{\text{Ta}-\text{S}}=2.4578$ Å, which are derived from the angle ${\theta }_{\text{STaTa}}=86°$ . The other angle is ${\theta }_{\text{TaSS}}=86°$ with S atoms from the same (top or bottom) group.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Ta─S	9.110	1.174	18.246	0.0	3.307

Table 307.

Two-body SW potential parameters for single-layer 1T-TaS₂ used by GULP [8] as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{Ta}-\text{S}-\text{S}}$	38.092	85.999	1.174	1.174	0.0	3.307	0.0	3.307	0.0	4.579
${\theta }_{\text{S}-\text{Ta}-\text{Ta}}$	38.092	85.999	1.174	1.174	0.0	3.307	0.0	3.307	0.0	4.579

Table 308.

Three-body SW potential parameters for single-layer 1T-TaS₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

Table 306 shows three VFF terms for the single-layer 1T-TaS₂; one of which is the bond stretching interaction shown by Eq. (1), while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\text{Ta}-\text{S}-\text{S}}$ is for the angle ${\theta }_{\text{Ta}-\text{S}-\text{S}}$ with both S atoms from the same (top or bottom) group. We find that there are actually only two parameters in the VFF model, so we can determine their value by fitting to the Young’s modulus and the Poisson’s ratio of the system. The ab initio calculations have predicted the Young’s modulus to be 101 N/m and the Poisson’s ratio as 0.20 [48].

The parameters for the two-body SW potential used by GULP are shown in Table 307 . The parameters for the three-body SW potential used by GULP are shown in Table 308 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 309 .

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Ta─S1─S1	1.000	1.174	2.816	38.092	1.000	0.070	9.110	9.589	4	0	0.0

Table 309.

SW potential parameters for single-layer 1T-TaS₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Figure 153.

Phonon spectrum for single-layer 1T-TaS₂ along the ΓMKΓ direction in the Brillouin zone. The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-TaS₂ under uniaxial tension at 1 and 300 K. Figure 152 shows the stress-strain curve for the tension of a single-layer 1T-TaS₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-TaS₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-TaS₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 87.8 and 87.4 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.20$ . The fitted Young’s modulus value is about 10% smaller than the ab initio result of 101 N/m [48], as only short-range interactions are considered in the present work. The long-range interactions are ignored, which typically lead to about 10% underestimation for the value of the Young’s modulus.

There is no available value for nonlinear quantities in the single-layer 1T-TaS₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −276.3 and −313.0 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 12.7 N/m at the ultimate strain of 0.25 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 12.2 N/m at the ultimate strain of 0.29 in the zigzag direction at the low temperature of 1 K.

Figure 154.

Stress-strain for single-layer 1T-TaSe₂ of dimension $100\times 100$ Å along the armchair and zigzag directions.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{Ta}-\text{Se}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{Ta}-\text{Se}-\text{Se}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{Se}-\text{Ta}-\text{Ta}}{\left(\text{Δ}\theta \right)}^2$
Parameter	9.348	4.535	4.535
$r_0$ or ${\theta }_0$	2.561	84.999	84.999

Table 310.

The VFF model for single-layer 1T-TaSe₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of ev/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of $\mathrm{\AA }$ ) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

Figure 153 shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Ta─Se	8.045	1.188	21.505	0.0	3.433

Table 311.

Two-body SW potential parameters for single-layer 1T-TaSe₂ used by GULP [8] as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{Ta}-\text{Se}-\text{Se}}$	34.820	84.999	1.188	1.188	0.0	3.433	0.0	3.3433	0.0	4.727
${\theta }_{\text{Se}-\text{Ta}-\text{Ta}}$	34.820	84.999	1.188	1.188	0.0	3.433	0.0	3.3433	0.0	4.727

Table 312.

Three-body SW potential parameters for single-layer 1T-TaSe₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

78. 1T-TaSe₂

Most existing theoretical studies on the single-layer 1T-TaSe₂ are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-TaSe₂.

The structure for the single-layer 1T-TaSe₂ is shown in Figure 71 (with M=Ta and X=Se). Each Ta atom is surrounded by six Se atoms. These Se atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each Se atom is connected to three Ta atoms. The structural parameters are from the first-principles calculations [48], including the lattice constant $a=3.4602$ Å and the bond length $d_{\text{Ta}-\text{Se}}=2.5609$ Å, which are derived from the angle ${\theta }_{\text{SeTaTa}}=85°$ . The other angle is ${\theta }_{\text{TaSeSe}}=85°$ with Se atoms from the same (top or bottom) group.

Table 310 shows three VFF terms for the single-layer 1T-TaSe₂; one of which is the bond stretching interaction shown by Eq. (1), while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\text{Ta}-\text{Se}-\text{Se}}$ is for the angle ${\theta }_{\text{Ta}-\text{Se}-\text{Se}}$ with both Se atoms from the same (top or bottom) group. We find that there are actually only two parameters in the VFF model, so we can determine their value by fitting to the Young’s modulus and the Poisson’s ratio of the system. The ab initio calculations have predicted the Young’s modulus to be 85 N/m and the Poisson’s ratio as 0.20 [48].

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Ta─Se1─Se1	1.000	1.188	2.891	34.820	1.000	0.087	8.045	10.813	4	0	0.0

Table 313.

SW potential parameters for single-layer 1T-TaSe₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Figure 155.

Phonon spectrum for single-layer 1T-TaSe₂ along the ΓMKΓ direction in the Brillouin zone. The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 311 . The parameters for the three-body SW potential used by GULP are shown in Table 312 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 313 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-TaSe₂ under uniaxial tension at 1 and 300 K. Figure 154 shows the stress-strain curve for the tension of a single-layer 1T-TaSe₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-TaSe₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-TaSe₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 74.6 and 74.4 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.20$ . The fitted Young’s modulus value is about 10% smaller than the ab initio result of 85 N/m [48], as only short-range interactions are considered in the present work. The long-range interactions are ignored, which typically leads to about 10% underestimation for the value of the Young’s modulus.

Figure 156.

Stress-strain for single-layer 1T-TaTe₂ of dimension $100\times 100$ Å along the armchair and zigzag directions.

There is no available value for nonlinear quantities in the single-layer 1T-TaSe₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −231.7 and −265.4 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 10.8 N/m at the ultimate strain of 0.25 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 10.4 N/m at the ultimate strain of 0.29 in the zigzag direction at the low temperature of 1 K.

Figure 155 shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

79. 1T-TaTe₂

Most existing theoretical studies on the single-layer 1T-TaTe₂ are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-TaTe₂.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{Ta}-\text{Te}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{Ta}-\text{Te}-\text{Te}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{Te}-\text{Ta}-\text{Ta}}{\left(\text{Δ}\theta \right)}^2$
Parameter	3.442	4.516	4.516
$r_0$ or ${\theta }_0$	2.770	82.999	82.999

Table 314.

The VFF model for single-layer 1T-TaTe₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of ev/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of $\mathrm{\AA }$ ) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Ta─Te	3.283	1.207	29.415	0.0	3.684

Table 315.

Two-body SW potential parameters for single-layer 1T-TaTe₂ used by GULP [8] as expressed in Eq. (3).

The structure for the single-layer 1T-TaTe₂ is shown in Figure 71 (with M=Ta and X=Te). Each Ta atom is surrounded by six Te atoms. These Te atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each Te atom is connected to three Ta atoms. The structural parameters are from the first-principles calculations [48], including the lattice constant $a=3.6702$ Å and the bond length $d_{\text{Ta}-\text{Te}}=2.7695$ Å, which are derived from the angle ${\theta }_{\text{TeTaTa}}=83°$ . The other angle is ${\theta }_{\text{TaTeTe}}=83°$ with Te atoms from the same (top or bottom) group.

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{Ta}-\text{Te}-\text{Te}}$	32.144	82.999	1.207	1.207	0.0	3.684	0.0	3.684	0.0	5.014
${\theta }_{\text{Te}-\text{Ta}-\text{Ta}}$	32.144	82.999	1.207	1.207	0.0	3.684	0.0	3.684	0.0	5.014

Table 316.

Three-body SW potential parameters for single-layer 1T-TaTe₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Ta─Se1─Se1	1.000	1.188	2.891	34.820	1.000	0.087	8.045	10.813	4	0	0.0

Table 317.

SW potential parameters for single-layer 1T-TaTe₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Figure 157.

Phonon spectrum for single-layer 1T-TaTe₂ along the ΓMKΓ direction in the Brillouin zone. The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

Table 314 shows three VFF terms for the single-layer 1T-TaTe₂; one of which is the bond stretching interaction shown by Eq. (1), while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\text{Ta}-\text{Te}-\text{Te}}$ is for the angle ${\theta }_{\text{Ta}-\text{Te}-\text{Te}}$ with both Te atoms from the same (top or bottom) group. We find that there are actually only two parameters in the VFF model, so we can determine their value by fitting to the Young’s modulus and the Poisson’s ratio of the system. The ab initio calculations have predicted the Young’s modulus to be 57 N/m and the Poisson’s ratio as 0.10 [48].

The parameters for the two-body SW potential used by GULP are shown in Table 315 . The parameters for the three-body SW potential used by GULP are shown in Table 316 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 317 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-TaTe₂ under uniaxial tension at 1 and 300 K. Figure 156 shows the stress-strain curve for the tension of a single-layer 1T-TaTe₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-TaTe₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-TaTe₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 50.3 and 50.0 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.10$ . The fitted Young’s modulus value is about 10% smaller than the ab initio result of 57 N/m [48], as only short-range interactions are considered in the present work. The long-range interactions are ignored, which typically lead to about 10% underestimation for the value of the Young’s modulus.

There is no available value for nonlinear quantities in the single-layer 1T-TaTe₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −247.1 and −262.2 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 5.0 N/m at the ultimate strain of 0.19 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 4.9 N/m at the ultimate strain of 0.22 in the zigzag direction at the low temperature of 1 K.

Figure 157 shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

80. 1T-WS₂

Most existing theoretical studies on the single-layer 1T-WS₂ are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-WS₂.

The structure for the single-layer 1T-WS₂ is shown in Figure 71 (with M=W and X=S). Each W atom is surrounded by six S atoms. These S atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each S atom is connected to three W atoms. The structural parameters are from the first-principles calculations [48], including the lattice constant $a=3.1908$ Å and the bond length $d_{\text{W}-\text{S}}=2.4125$ Å, which are derived from the angle ${\theta }_{\text{SWW}}=82.8°$ . The other angle is ${\theta }_{\text{WSS}}=82.8°$ with S atoms from the same (top or bottom) group.

Figure 158.

Stress-strain for single-layer 1T-WS₂ of dimension $100\times 100$ Å along the armchair and zigzag directions.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{W}-\text{S}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{W}-\text{S}-\text{S}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{S}-\text{W}-\text{W}}{\left(\text{Δ}\theta \right)}^2$
Parameter	4.395	10.087	10.087
$r_0$ or ${\theta }_0$	2.413	82.799	82.799

Table 318.

The VFF model for single-layer 1T-WS₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of ev/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of $\mathrm{\AA }$ ) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

Table 318 shows three VFF terms for the single-layer 1T-WS₂; one of which is the bond stretching interaction shown by Eq. (1), while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\text{W}-\text{S}-\text{S}}$ is for the angle ${\theta }_{\text{W}-\text{S}-\text{S}}$ with both S atoms from the same (top or bottom) group. We find that there are actually only two parameters in the VFF model, so we can determine their value by fitting to the Young’s modulus and the Poisson’s ratio of the system. The ab initio calculations have predicted the Young’s modulus to be 113 N/m and the Poisson’s ratio as −0.03 [48]. The ab initio calculations have predicted a negative Poisson’s ratio in the 1T-WS₂, which was attributed to the orbital coupling in this material. The orbital coupling enhances the angle bending interaction in the VFF model. As a result, the value of the angle bending parameter is much larger than the bond stretching force constant parameter, which is typical in auxetic materials with negative Poisson’s ratio [52].

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
W─S	3.163	1.045	16.937	0.0	3.206

Table 319.

Two-body SW potential parameters for single-layer 1T-WS₂ used by GULP [8] as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{W}-\text{S}-\text{S}}$	71.264	82.799	1.045	1.045	0.0	3.206	0.0	3.206	0.0	4.359
${\theta }_{\text{S}-\text{W}-\text{W}}$	71.264	82.799	1.045	1.045	0.0	3.206	0.0	3.206	0.0	4.359

Table 320.

Three-body SW potential parameters for single-layer 1T-WS₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

The parameters for the two-body SW potential used by GULP are shown in Table 319 . The parameters for the three-body SW potential used by GULP are shown in Table 320 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 321 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-WS₂ under uniaxial tension at 1 and 300 K. Figure 158 shows the stress-strain curve for the tension of a single-layer 1T-WS₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-WS₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-WS₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 100.2 and 99.5 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=\mathrm{}-0.03$ . The fitted Young’s modulus value is about 10% smaller than the ab initio result of 113 N/m [48], as only short-range interactions are considered in the present work. The long-range interactions are ignored, which typically lead to about 10% underestimation for the value of the Young’s modulus.

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
W─S1─S1	1.000	1.045	3.069	71.264	1.000	0.125	3.163	14.209	4	0	0.0

Table 321.

SW potential parameters for single-layer 1T-WS₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Figure 159.

Phonon spectrum for single-layer 1T-WS₂ along the ΓMKΓ direction in the Brillouin zone. The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

There is no available value for nonlinear quantities in the single-layer 1T-WS₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −666.6 and −660.6 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 7.7 N/m at the ultimate strain of 0.15 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 7.7 N/m at the ultimate strain of 0.17 in the zigzag direction at the low temperature of 1 K.

Figure 160.

Stress-strain for single-layer 1T-WSe₂ of dimension $100\times 100$ Å along the armchair and zigzag directions.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{W}-\text{Se}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{W}-\text{Se}-\text{Se}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{Se}-\text{W}-\text{W}}{\left(\text{Δ}\theta \right)}^2$
Parameter	2.556	15.375	15.375
$r_0$ or ${\theta }_0$	2.521	80.501	80.501

Table 322.

The VFF model for single-layer 1T-WSe₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of ev/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of $\mathrm{\AA }$ ) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

Figure 159 shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
W─Se	1.885	1.013	20.186	0.0	3.320

Table 323.

Two-body SW potential parameters for single-layer 1T-WSe₂ used by GULP [8] as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{W}-\text{Se}-\text{Se}}$	99.800	80.501	1.013	1.013	0.0	3.320	0.0	3.320	0.0	4.450
${\theta }_{\text{Se}-\text{W}-\text{W}}$	99.800	80.501	1.013	1.013	0.0	3.320	0.0	3.320	0.0	4.450

Table 324.

Three-body SW potential parameters for single-layer 1T-WSe₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

81. 1T-WSe₂

Most existing theoretical studies on the single-layer 1T-WSe₂ are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-WSe₂.

The structure for the single-layer 1T-WSe₂ is shown in Figure 71 (with M=W and X=Se). Each W atom is surrounded by six Se atoms. These Se atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each Se atom is connected to three W atoms. The structural parameters are from the first-principles calculations [48], including the lattice constant $a=3.2574$ Å and the bond length $d_{\text{W}-\text{Se}}=2.5207$ Å, which are derived from the angle ${\theta }_{\text{SeWW}}=80.5°$ . The other angle is ${\theta }_{\text{WSeSe}}=80.5°$ with Se atoms from the same (top or bottom) group.

Table 322 shows three VFF terms for the single-layer 1T-WSe₂; one of which is the bond stretching interaction shown by Eq. (1), while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\text{W}-\text{Se}-\text{Se}}$ is for the angle ${\theta }_{\text{W}-\text{Se}-\text{Se}}$ with both Se atoms from the same (top or bottom) group. We find that there are actually only two parameters in the VFF model, so we can determine their value by fitting to the Young’s modulus and the Poisson’s ratio of the system. The ab initio calculations have predicted the Young’s modulus to be 94 N/m and the Poisson’s ratio as −0.15 [48]. The ab initio calculations have predicted a negative Poisson’s ratio in the 1T-WSe₂, which was attributed to the orbital coupling in this material. The orbital coupling enhances the angle bending interaction in the VFF model. As a result, the value of the angle bending parameter is much larger than the bond stretching force constant parameter, which is typical in auxetic materials with negative Poisson’s ratio [52].

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
W─Se1─Se1	1.000	1.013	3.277	99.800	1.000	0.165	1.885	19.156	4	0	0.0

Table 325.

SW potential parameters for single-layer 1T-WSe₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Figure 161.

Phonon spectrum for single-layer 1T-WSe₂ along the ΓMKΓ direction in the Brillouin zone. The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 323 . The parameters for the three-body SW potential used by GULP are shown in Table 324 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 325 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-WSe₂ under uniaxial tension at 1 and 300 K. Figure 160 shows the stress-strain curve for the tension of a single-layer 1T-WSe₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-WSe₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-WSe₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 80.5 and 80.3 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=\mathrm{}-0.15$ . The fitted Young’s modulus value is about 10% smaller than the ab initio result of 94 N/m [48], as only short-range interactions are considered in the present work. The long-range interactions are ignored, which typically lead to about 10% underestimation for the value of the Young’s modulus.

There is no available value for nonlinear quantities in the single-layer 1T-WSe₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −666.1 and −580.1 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 5.5 N/m at the ultimate strain of 0.13 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 5.6 N/m at the ultimate strain of 0.15 in the zigzag direction at the low temperature of 1 K.

Figure 161 shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

82. 1T-WTe₂

Most existing theoretical studies on the single-layer 1T-WTe₂ are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-WTe₂.

The structure for the single-layer 1T-WTe₂ is shown in Figure 71 (with M=W and X=Te). Each W atom is surrounded by six Te atoms. These Te atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each Te atom is connected to three W atoms. The structural parameters are from the first-principles calculations [48], including the lattice constant $a=3.4970$ Å and the bond length $d_{\text{W}-\text{Te}}=2.7202$ Å, which are derived from the angle ${\theta }_{\text{TeWW}}=80.0°$ . The other angle is ${\theta }_{\text{WTeTe}}=80.0°$ with Te atoms from the same (top or bottom) group.

Figure 162.

Stress-strain for single-layer 1T-WTe₂ of dimension $100\times 100$ Å along the armchair and zigzag directions.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{W}-\text{Te}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{W}-\text{Te}-\text{Te}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{Te}-\text{W}-\text{W}}{\left(\text{Δ}\theta \right)}^2$
Parameter	2.272	19.437	19.437
$r_0$ or ${\theta }_0$	2.720	79.999	79.999

Table 326.

The VFF model for single-layer 1T-WTe₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of ev/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of $\mathrm{\AA }$ ) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

Table 326 shows three VFF terms for the single-layer 1T-WTe₂; one of which is the bond stretching interaction shown by Eq. (1), while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\text{W}-\text{Te}-\text{Te}}$ is for the angle ${\theta }_{\text{W}-\text{Te}-\text{Te}}$ with both Te atoms from the same (top or bottom) group. We find that there are actually only two parameters in the VFF model, so we can determine their value by fitting to the Young’s modulus and the Poisson’s ratio of the system. The ab initio calculations have predicted the Young’s modulus to be 88 N/m and the Poisson’s ratio as −0.18 [48]. The ab initio calculations have predicted a negative Poisson’s ratio in the 1T-WTe₂, which was attributed to the orbital coupling in this material. The orbital coupling enhances the angle bending interaction in the VFF model. As a result, the value of the angle bending parameter is much larger than the bond stretching force constant parameter, which is typical in auxetic materials with negative Poisson’s ratio [52].

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
W─Te	1.924	1.075	27.376	0.0	3.575

Table 327.

Two-body SW potential parameters for single-layer 1T-WTe₂ used by GULP [8] as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{W}-\text{Te}-\text{Te}}$	123.899	79.999	1.075	1.075	0.0	3.575	0.0	3.575	0.0	4.777
${\theta }_{\text{Te}-\text{W}-\text{W}}$	123.899	79.999	1.075	1.075	0.0	3.575	0.0	3.575	0.0	4.777

Table 328.

Three-body SW potential parameters for single-layer 1T-WTe₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

The parameters for the two-body SW potential used by GULP are shown in Table 327 . The parameters for the three-body SW potential used by GULP are shown in Table 328 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 329 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-WTe₂ under uniaxial tension at 1 and 300 K. Figure 162 shows the stress-strain curve for the tension of a single-layer 1T-WTe₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-WTe₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-WTe₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 75.9 and 75.8 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=\mathrm{}-0.18$ . The fitted Young’s modulus value is about 10% smaller than the ab initio result of 88 N/m [48], as only short-range interactions are considered in the present work. The long-range interactions are ignored, which typically leads to about 10% underestimation for the value of the Young’s modulus.

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
W─Te1─Te1	1.000	1.075	3.325	123.899	1.000	0.174	1.924	20.483	4	0	0.0

Table 329.

SW potential parameters for single-layer 1T-WTe₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Figure 163.

Phonon spectrum for single-layer 1T-WTe₂ along the ΓMKΓ direction in the Brillouin zone. The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

There is no available value for nonlinear quantities in the single-layer 1T-WTe₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −546.0 and −551.5 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 5.1 N/m at the ultimate strain of 0.12 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 5.2 N/m at the ultimate strain of 0.14 in the zigzag direction at the low temperature of 1 K.

Figure 163 shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

Figure 164.

Stress-strain for single-layer 1T-ReS₂ of dimension $100\times 100$ Å along the armchair and zigzag directions.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{Re}-\text{S}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{Re}-\text{S}-\text{S}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{S}-\text{Re}-\text{Re}}{\left(\text{Δ}\theta \right)}^2$
Parameter	2.684	10.829	10.829
$r_0$ or ${\theta }_0$	2.405	79.498	79.498

Table 330.

The VFF model for single-layer 1T-ReS₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of ev/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of $\mathrm{\AA }$ ) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

83. 1T-ReS₂

Most existing theoretical studies on the single-layer 1T-ReS₂ are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-ReS₂.

The structure for the single-layer 1T-ReS₂ is shown in Figure 71 (with M=Re and X=S). Each Re atom is surrounded by six S atoms. These S atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each S atom is connected to three Re atoms. The structural parameters are from the first-principles calculations [48], including the lattice constant $a=3.0750$ Å and the bond length $d_{\text{Re}-\text{S}}=2.4045$ Å, which are derived from the angle ${\theta }_{\text{SReRe}}=79.5°$ . The other angle is ${\theta }_{\text{ReSS}}=79.5°$ with S atoms from the same (top or bottom) group.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Re─S	1.751	0.934	16.714	0.0	3.154

Table 331.

Two-body SW potential parameters for single-layer 1T-ReS₂ used by GULP [8] as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{Re}-\text{S}-\text{S}}$	67.797	79.498	0.934	0.934	0.0	3.154	0.0	3.154	0.0	4.201
${\theta }_{\text{S}-\text{Re}-\text{Re}}$	67.797	79.498	0.934	0.934	0.0	3.154	0.0	3.154	0.0	4.201

Table 332.

Three-body SW potential parameters for single-layer 1T-ReS₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

Table 330 shows three VFF terms for the single-layer 1T-ReS₂; one of which is the bond stretching interaction shown by Eq. (1), while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\text{Re}-\text{S}-\text{S}}$ is for the angle ${\theta }_{\text{Re}-\text{S}-\text{S}}$ with both S atoms from the same (top or bottom) group. We find that there are actually only two parameters in the VFF model, so we can determine their value by fitting to the Young’s modulus and the Poisson’s ratio of the system. The ab initio calculations have predicted the Young’s modulus to be 90 N/m and the Poisson’s ratio as −0.11 [48]. The ab initio calculations have predicted a negative Poisson’s ratio in the 1T-ReS₂, which was attributed to the orbital coupling in this material. The orbital coupling enhances the angle bending interaction in the VFF model. As a result, the value of the angle bending parameter is much larger than the bond stretching force constant parameter, which is typical in auxetic materials with negative Poisson’s ratio [52].

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Re─S1─S1	1.000	0.934	3.375	67.797	1.000	0.182	1.751	21.916	4	0	0.0

Table 333.

SW potential parameters for single-layer 1T-ReS₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Figure 165.

Phonon spectrum for single-layer 1T-ReS₂ along the ΓMKΓ direction in the Brillouin zone. The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 331 . The parameters for the three-body SW potential used by GULP are shown in Table 332 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 333 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-ReS₂ under uniaxial tension at 1 and 300 K. Figure 164 shows the stress-strain curve for the tension of a single-layer 1T-ReS₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-ReS₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-ReS₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 78.1 and 77.8 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=\mathrm{}-0.11$ . The fitted Young’s modulus value is about 10% smaller than the ab initio result of 90 N/m [48], as only short-range interactions are considered in the present work. The long-range interactions are ignored, which typically leads to about 10% underestimation for the value of the Young’s modulus.

Figure 166.

Stress-strain for single-layer 1T-ReSe₂ of dimension $100\times 100$ Å along the armchair and zigzag directions.

There is no available value for nonlinear quantities in the single-layer 1T-ReS₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −537.1 and −550.7 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 5.5 N/m at the ultimate strain of 0.13 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 5.6 N/m at the ultimate strain of 0.15 in the zigzag direction at the low temperature of 1 K.

Figure 165 shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

84. 1T-ReSe₂

Most existing theoretical studies on the single-layer 1T-ReSe₂ are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-ReSe₂.

The structure for the single-layer 1T-ReSe₂ is shown in Figure 71 (with M=Re and X=Se). Each Re atom is surrounded by six Se atoms. These Se atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each Se atom is connected to three Re atoms. The structural parameters are from the first-principles calculations [48], including the lattice constant $a=3.1311$ Å and the bond length $d_{\text{Re}-\text{Se}}=2.5149$ Å, which are derived from the angle ${\theta }_{\text{SeReRe}}=77°$ . The other angle is ${\theta }_{\text{ReSeSe}}=77°$ with Se atoms from the same (top or bottom) group.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{Re}-\text{Se}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{Re}-\text{Se}-\text{Se}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{Se}-\text{Re}-\text{Re}}{\left(\text{Δ}\theta \right)}^2$
Parameter	4.313	12.674	12.674
$r_0$ or ${\theta }_0$	2.515	76.999	76.999

Table 334.

The VFF model for single-layer 1T-ReSe₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of ev/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of $\mathrm{\AA }$ ) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Re─Se	2.866	0.896	20.001	0.0	3.265

Table 335.

Two-body SW potential parameters for single-layer 1T-ReSe₂ used by GULP [8] as expressed in Eq. (3).

Table 334 shows three VFF terms for the single-layer 1T-ReSe₂; one of which is the bond stretching interaction shown by Eq. (1), while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\text{Re}-\text{Se}-\text{Se}}$ is for the angle ${\theta }_{\text{Re}-\text{Se}-\text{Se}}$ with both Se atoms from the same (top or bottom) group. We find that there are actually only two parameters in the VFF model, so we can determine their value by fitting to the Young’s modulus and the Poisson’s ratio of the system. The ab initio calculations have predicted the Young’s modulus to be 123 N/m and the Poisson’s ratio as −0.03 [48]. The ab initio calculations have predicted a negative Poisson’s ratio in the 1T-ReSe₂, which was attributed to the orbital coupling in this material. The orbital coupling enhances the angle bending interaction in the VFF model. As a result, the value of the angle bending parameter is much larger than the bond stretching force constant parameter, which is typical in auxetic materials with negative Poisson’s ratio [52].

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{Re}-\text{Se}-\text{Se}}$	72.666	76.999	0.896	0.896	0.0	3.265	0.0	3.265	0.0	4.277
${\theta }_{\text{Se}-\text{Re}-\text{Re}}$	72.666	76.999	0.896	0.896	0.0	3.265	0.0	3.265	0.0	4.277

Table 336.

Three-body SW potential parameters for single-layer 1T-ReSe₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Re─Se1─Se1	1.000	0.896	3.645	72.666	1.000	0.225	2.866	31.036	4	0	0.0

Table 337.

SW potential parameters for single-layer 1T-ReSe₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Figure 167.

Phonon spectrum for single-layer 1T-ReSe₂ along the ΓMKΓ direction in the Brillouin zone. The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 335 . The parameters for the three-body SW potential used by GULP are shown in Table 336 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 337 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-ReSe₂ under uniaxial tension at 1 and 300 K. Figure 166 shows the stress-strain curve for the tension of a single-layer 1T-ReSe₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-ReSe₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-ReSe₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 108.2 and 107.7 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=\mathrm{}-0.03$ . The fitted Young’s modulus value is about 10% smaller than the ab initio result of 123 N/m [48], as only short-range interactions are considered in the present work. The long-range interactions are ignored, which typically lead to about 10% underestimation for the value of the Young’s modulus.

There is no available value for nonlinear quantities in the single-layer 1T-ReSe₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −669.3 and −699.9 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 8.5 N/m at the ultimate strain of 0.14 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 8.4 N/m at the ultimate strain of 0.17 in the zigzag direction at the low temperature of 1 K.

Figure 167 shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

85. 1T-ReTe₂

Most existing theoretical studies on the single-layer 1T-ReTe₂ are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-ReTe₂.

Figure 168.

Stress-strain for single-layer 1T-ReTe₂ of dimension $100\times 100$ Å along the armchair and zigzag directions.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{Re}-\text{Te}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{Re}-\text{Te}-\text{Te}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{Te}-\text{Re}-\text{Re}}{\left(\text{Δ}\theta \right)}^2$
Parameter	1.724	14.812	14.812
$r_0$ or ${\theta }_0$	2.703	77.501	77.501

Table 338.

The VFF model for single-layer 1T-ReTe₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of ev/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of $\mathrm{\AA }$ ) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

The structure for the single-layer 1T-ReTe₂ is shown in Figure 71 (with M=Re and X=Te). Each Re atom is surrounded by six Te atoms. These Te atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each Te atom is connected to three Re atoms. The structural parameters are from the first-principles calculations [48], including the lattice constant $a=3.3834$ Å and the bond length $d_{\text{Re}-\text{Te}}=2.7027$ Å, which are derived from the angle ${\theta }_{\text{TeReRe}}=77.5°$ . The other angle is ${\theta }_{\text{ReTeTe}}=77.5°$ with Te atoms from the same (top or bottom) group.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Re─Te	1.343	0.980	26.678	0.0	3.517

Table 339.

Two-body SW potential parameters for single-layer 1T-ReTe₂ used by GULP [8] as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{Re}-\text{Te}-\text{Te}}$	86.424	77.501	0.980	0.980	0.0	3.517	0.0	3.517	0.0	4.622
${\theta }_{\text{Te}-\text{Re}-\text{Re}}$	86.424	77.501	0.980	0.980	0.0	3.517	0.0	3.517	0.0	4.622

Table 340.

Three-body SW potential parameters for single-layer 1T-ReTe₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

Table 338 shows three VFF terms for the single-layer 1T-ReTe₂; one of which is the bond stretching interaction shown by Eq. (1), while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\text{Re}-\text{Te}-\text{Te}}$ is for the angle ${\theta }_{\text{Re}-\text{Te}-\text{Te}}$ with both Te atoms from the same (top or bottom) group. We find that there are actually only two parameters in the VFF model, so we can determine their value by fitting to the Young’s modulus and the Poisson’s ratio of the system. The ab initio calculations have predicted the Young’s modulus to be 71 N/m and the Poisson’s ratio as −0.22 [48]. The ab initio calculations have predicted a negative Poisson’s ratio in the 1T-ReTe₂, which was attributed to the orbital coupling in this material. The orbital coupling enhances the angle bending interaction in the VFF model. As a result, the value of the angle bending parameter is much larger than the bond stretching force constant parameter, which is typical in auxetic materials with negative Poisson’s ratio [52].

The parameters for the two-body SW potential used by GULP are shown in Table 339 . The parameters for the three-body SW potential used by GULP are shown in Table 340 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 341 .

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Re─Te1─Te1	1.000	0.980	3.587	86.424	1.000	0.216	1.343	28.891	4	0	0.0

Table 341.

SW potential parameters for single-layer 1T-ReTe₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Figure 169.

Phonon spectrum for single-layer 1T-ReTe₂ along the ΓMKΓ direction in the Brillouin zone. The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-ReTe₂ under uniaxial tension at 1 and 300 K. Figure 168 shows the stress-strain curve for the tension of a single-layer 1T-ReTe₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-ReTe₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-ReTe₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 59.4 and 59.3 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=\mathrm{}-0.17$ . The fitted Young’s modulus value is about 10% smaller than the ab initio result of 71 N/m [48], as only short-range interactions are considered in the present work. The long-range interactions are ignored, which typically lead to about 10% underestimation for the value of the Young’s modulus.

There is no available value for nonlinear quantities in the single-layer 1T-ReTe₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −416.1 and −425.1 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 4.0 N/m at the ultimate strain of 0.12 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 4.1 N/m at the ultimate strain of 0.14 in the zigzag direction at the low temperature of 1 K.

Figure 170.

Stress-strain for single-layer 1T-IrTe₂ of dimension 100×100 Å along the armchair and zigzag directions.

Figure 169 shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

86. 1T-IrTe₂

Most existing theoretical studies on the single-layer 1T-IrTe₂ are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-IrTe₂.

The structure for the single-layer 1T-IrTe₂ is shown in Figure 71 (with M = Ir and X = Te). Each Ir atom is surrounded by six Te atoms. These Te atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each Te atom is connected to three Ir atoms. The structural parameters are from the first-principles calculations [48], including the lattice constant $a=3.8431$ Å and the bond length $d_{\text{Ir}-\text{Te}}=2.6490$ Å, which are derived from the angle ${\theta }_{\text{TeIrIr}}=93°$ . The other angle is ${\theta }_{\text{IrTeTe}}=93°$ with Te atoms from the same (top or bottom) group.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{Ir}-\text{Te}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{Ir}-\text{Te}-\text{Te}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{Te}-\text{Ir}-\text{Ir}}{\left(\text{Δ}\theta \right)}^2$
Parameter	5.334	2.182	2.182
$r_0$ or ${\theta }_0$	2.649	93.002	93.002

Table 342.

The VFF model for single-layer 1T-IrTe₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Ir─Te	6.030	1.538	24.621	0.0	3.658

Table 343.

Two-body SW potential parameters for single-layer 1T-IrTe₂ used by GULP [8] as expressed in Eq. (3).

Table 342 shows three VFF terms for the single-layer 1T-IrTe₂; one of which is the bond stretching interaction shown by Eq. (1), while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\text{Ir}-\text{Te}-\text{Te}}$ is for the angle ${\theta }_{\text{Ir}-\text{Te}-\text{Te}}$ with both Te atoms from the same (top or bottom) group. We find that there are actually only two parameters in the VFF model, so we can determine their value by fitting to the Young’s modulus and the Poisson’s ratio of the system. The ab initio calculations have predicted the Young’s modulus to be 45 N/m and the Poisson’s ratio as 0.22 [48].

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{Ir}-\text{Te}-\text{Te}}$	23.056	93.002	1.538	1.538	0.0	3.658	0.0	3.658	0.0	5.250
${\theta }_{\text{Te}-\text{Ir}-\text{Ir}}$	23.056	93.002	1.538	1.538	0.0	3.658	0.0	3.658	0.0	5.250

Table 344.

Three-body SW potential parameters for single-layer 1T-IrTe₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Ir─Te1─Te1	1.000	1.538	2.370	23.056	1.000	−0.052	6.030	4.398	4	0	0.0

Table 345.

SW potential parameters for single-layer 1T-IrTe₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Figure 171.

Phonon spectrum for single-layer 1T-IrTe₂ along the ΓMKΓ direction in the Brillouin zone. The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 343 . The parameters for the three-body SW potential used by GULP are shown in Table 344 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 345 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-IrTe₂ under uniaxial tension at 1 and 300 K. Figure 170 shows the stress-strain curve for the tension of a single-layer 1T-IrTe₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-IrTe₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-IrTe₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 38.6 and 38.4 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.20$ . The fitted Young’s modulus value is about 10% smaller than the ab initio result of 45 N/m [48], as only short-range interactions are considered in the present work. The long-range interactions are ignored, which typically leads to about 10% underestimation for the value of the Young’s modulus.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{Pt}-\text{S}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{Pt}-\text{S}-\text{S}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{S}-\text{Pt}-\text{Pt}}{\left(\text{Δ}\theta \right)}^2$
Parameter	12.128	4.975	4.975
$r_0$ or ${\theta }_0$	2.371	96.00	96.00

Table 346.

The VFF model for single-layer 1T-PtS₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

There is no available value for nonlinear quantities in the single-layer 1T-IrTe₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of B in most materials. The value of the third-order nonlinear elasticity D can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with E as the Young’s modulus. The values of D from the present SW potential are −127.7 and −142 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 5.4 N/m at the ultimate strain of 0.25 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 5.2 N/m at the ultimate strain of 0.29 in the zigzag direction at the low temperature of 1 K.

Figure 171 shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

87. 1T-PtS₂

Most existing theoretical studies on the single-layer 1T-PtS₂ are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-PtS₂.

Figure 172.

Phonon spectrum for single-layer 1T-PtS₂. (a) Phonon dispersion along the ΓM direction in the Brillouin zone. The results from the VFF model (lines) are comparable with the ab initio results (pentagons) from [34]. (b) The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Pt─S	11.806	1.485	15.796	0.0	3.309

Table 347.

Two-body SW potential parameters for single-layer 1T-PtS₂ used by GULP [8] as expressed in Eq. (3).

The structure for the single-layer 1T-PtS₂ is shown in Figure 71 (with M = Pt and X = S). Each Pt atom is surrounded by six S atoms. These S atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each S atom is connected to three Pt atoms. The structural parameters are from the first-principles calculations [48], including the lattice constant $a=3.5237$ Å and the bond length $d_{\text{Pt}-\text{S}}=2.3708$ Å, which are derived from the angle ${\theta }_{\text{SPtPt}}=96°$ . The other angle is ${\theta }_{\text{PtSS}}=96°$ with S atoms from the same (top or bottom) group.

Figure 173.

Stress-strain for single-layer 1T-PtS₂ of dimension 100×100 Å along the armchair and zigzag directions.

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{Pt}-\text{S}-\text{S}}$	59.607	96.00	1.485	1.485	0.0	3.309	0.0	3.309	0.0	4.813
${\theta }_{\text{S}-\text{Pt}-\text{Pt}}$	59.607	96.00	1.485	1.485	0.0	3.309	0.0	3.309	0.0	4.813

Table 348.

Three-body SW potential parameters for single-layer 1T-PtS₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

Table 346 shows three VFF terms for the single-layer 1T-PtS₂; one of which is the bond stretching interaction shown by Eq. (1), while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\text{Pt}-\text{S}-\text{S}}$ is for the angle ${\theta }_{\text{Pt}-\text{S}-\text{S}}$ with both S atoms from the same (top or bottom) group. These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the ΓM as shown in Figure 172(a) . The ab initio calculations for the phonon dispersion are from [34]. The lowest acoustic branch (flexural mode) is almost linear in the ab initio calculations, which may be due to the violation of the rigid rotational invariance [20]. Figure 172(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Pt─S1─S1	1.000	1.485	2.229	59.607	1.000	−0.105	11.806	3.250	4	0	0.0

Table 349.

SW potential parameters for single-layer 1T-PtS₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

The parameters for the two-body SW potential used by GULP are shown in Table 347 . The parameters for the three-body SW potential used by GULP are shown in Table 348 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 349 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-PtS₂ under uniaxial tension at 1 and 300 K. Figure 173 shows the stress-strain curve for the tension of a single-layer 1T-PtS₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-PtS₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-PtS₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 105.9 and 105.4 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.16$ .

There is no available value for nonlinear quantities in the single-layer 1T-PtS₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −420.6 and −457.1 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 12.8 N/m at the ultimate strain of 0.22 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 12.3 N/m at the ultimate strain of 0.26 in the zigzag direction at the low temperature of 1 K.

Figure 174.

Phonon spectrum for single-layer 1T-PtSe₂. (a) Phonon dispersion along the ΓM direction in the Brillouin zone. The results from the VFF model (lines) are comparable with the ab initio results (pentagons) from [34]. (b) The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

88. 1T-PtSe₂

Most existing theoretical studies on the single-layer 1T-PtSe₂ are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-PtSe₂.

The structure for the single-layer 1T-PtSe₂ is shown in Figure 71 (with M = Pt and X = Se). Each Pt atom is surrounded by six Se atoms. These Se atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each Se atom is connected to three Pt atoms. The structural parameters are from the first-principles calculations [48], including the lattice constant $a=3.6662$ Å and the bond length $d_{\text{Pt}-\text{Se}}=2.4667$ Å, which are derived from the angle ${\theta }_{\text{SePtPt}}=96°$ . The other angle is ${\theta }_{\text{PtSeSe}}=96°$ with Se atoms from the same (top or bottom) group.

Figure 175.

Stress-strain for single-layer 1T-PtSe₂ of dimension $100\times 100$ Å along the armchair and zigzag directions.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{Pt}-\text{Se}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{Pt}-\text{Se}-\text{Se}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{Se}-\text{Pt}-\text{Pt}}{\left(\text{Δ}\theta \right)}^2$
Parameter	12.128	4.975	4.975
$r_0$ or ${\theta }_0$	2.467	95.999	95.999

Table 350.

The VFF model for single-layer 1T-PtSe₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

Table 350 shows three VFF terms for the single-layer 1T-PtSe₂; one of which is the bond stretching interaction shown by Eq. (1), while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\text{Pt}-\text{Se}-\text{Se}}$ is for the angle ${\theta }_{\text{Pt}-\text{Se}-\text{Se}}$ with both Se atoms from the same (top or bottom) group. These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the ΓM as shown in Figure 174(a) . The ab initio calculations for the phonon dispersion are from [34]. The lowest acoustic branch (flexural mode) is almost linear in the ab initio calculations, which may be due to the violation of the rigid rotational invariance [20]. Figure 174(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Pt─Se	12.781	1.545	18.511	0.0	3.443

Table 351.

Two-body SW potential parameters for single-layer 1T-PtSe₂ used by GULP [8] as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{Pt}-\text{Se}-\text{Se}}$	59.608	95.999	1.545	1.545	0.0	3.443	0.0	3.443	0.0	5.008
${\theta }_{\text{Se}-\text{Pt}-\text{Pt}}$	59.608	95.999	1.545	1.545	0.0	3.443	0.0	3.443	0.0	5.008

Table 352.

Three-body SW potential parameters for single-layer 1T-PtSe₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

The parameters for the two-body SW potential used by GULP are shown in Table 351 . The parameters for the three-body SW potential used by GULP are shown in Table 352 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 353 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-PtSe₂ under uniaxial tension at 1 and 300 K. Figure 175 shows the stress-strain curve for the tension of a single-layer 1T-PtSe₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-PtSe₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-PtSe₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 101.1 and 100.5 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.17$ .

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Pt─Se1─Se1	1.000	1.545	2.229	59.608	1.000	−0.105	12.781	3.250	4	0	0.0

Table 353.

SW potential parameters for single-layer 1T-PtSe₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{\text{Pt}-\text{Te}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{Pt}-\text{Te}-\text{Te}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{Te}-\text{Pt}-\text{Pt}}{\left(\text{Δ}\theta \right)}^2$
Parameter	12.128	4.975	4.975
$r_0$ or ${\theta }_0$	2.661	95.998	95.998

Table 354.

The VFF model for single-layer 1T-PtTe₂.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

There is no available value for nonlinear quantities in the single-layer 1T-PtSe₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −391.4 and −424 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 12.5 N/m at the ultimate strain of 0.23 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 12.1 N/m at the ultimate strain of 0.27 in the zigzag direction at the low temperature of 1 K.

89. 1T-PtTe₂

Most existing theoretical studies on the single-layer 1T-PtTe₂ are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-PtTe₂.

Figure 176.

Phonon spectrum for single-layer 1T-PtTe₂. (a) Phonon dispersion along the ΓM direction in the Brillouin zone. The results from the VFF model (lines) are comparable with the ab initio results (pentagons) from [34]. (b) The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Pt─Te	14.877	1.667	25.081	0.0	3.714

Table 355.

Two-body SW potential parameters for single-layer 1T-PtTe₂ used by GULP [8] as expressed in Eq. (3).

The structure for the single-layer 1T-PtTe₂ is shown in Figure 71 (with M = Pt and X = Te). Each Pt atom is surrounded by six Te atoms. These Te atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each Te atom is connected to three Pt atoms. The structural parameters are from the first-principles calculations [48], including the lattice constant $a=3.9554$ Å and the bond length $d_{\text{Pt}-\text{Te}}=2.6613$ Å, which are derived from the angle ${\theta }_{\text{TePtPt}}=96°$ . The other angle is ${\theta }_{\text{PtTeTe}}=96°$ with Te atoms from the same (top or bottom) group.

Figure 177.

Stress-strain for single-layer 1T-PtTe₂ of dimension $100\times 100$ Å along the armchair and zigzag directions.

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{\text{Pt}-\text{Te}-\text{Te}}$	59.607	95.998	1.667	1.667	0.0	3.714	0.0	3.714	0.0	5.403
${\theta }_{\text{Te}-\text{Pt}-\text{Pt}}$	59.607	95.998	1.667	1.667	0.0	3.714	0.0	3.714	0.0	5.403

Table 356.

Three-body SW potential parameters for single-layer 1T-PtTe₂ used by GULP [8] as expressed in Eq. (4).

The angle ${\theta }_{ijk}$ in the first line indicates the bending energy for the angle with atom i as the apex.

Table 354 shows three VFF terms for the single-layer 1T-PtTe₂; one of which is the bond stretching interaction shown by Eq. (1), while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\text{Pt}-\text{Te}-\text{Te}}$ is for the angle ${\theta }_{\text{Pt}-\text{Te}-\text{Te}}$ with both Te atoms from the same (top or bottom) group. These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the ΓM as shown in Figure 176(a) . The ab initio calculations for the phonon dispersion are from [34]. Figure 176(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Pt─Te1─Te1	1.000	1.667	2.229	59.607	1.000	−0.104	14.877	3.250	4	0	0.0

Table 357.

SW potential parameters for single-layer 1T-PtTe₂ used by LAMMPS [9] as expressed in Eqs. (9) and (10).

The parameters for the two-body SW potential used by GULP are shown in Table 355 . The parameters for the three-body SW potential used by GULP are shown in Table 356 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 357 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-PtTe₂ under uniaxial tension at 1 and 300 K. Figure 177 shows the stress-strain curve for the tension of a single-layer 1T-PtTe₂ of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-PtTe₂ is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-PtTe₂. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 89.1 and 88.7 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.19$ .

There is no available value for nonlinear quantities in the single-layer 1T-PtTe₂. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −306.8 and −340.9 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 12.1 N/m at the ultimate strain of 0.24 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 11.6 N/m at the ultimate strain of 0.28 in the zigzag direction at the low temperature of 1 K.

Figure 178.

Structure for single-layer black phosphorus. (a) Top view. The armchair direction is along the x-axis, while the zigzag direction is along the y-axis. Inset shows the first Brillouin zone. (b) Perspective view illustrates the puckered configuration. The pucker is perpendicular to the x-axis and is parallel with the y-axis. Atoms are divided into the top (denoted by P₁) and the bottom (denoted by P₂) groups. (c) Atomic configuration.

90. Black phosphorus

The black phosphorus is also named the $\alpha$ phosphorus. There are several empirical potentials available for the atomic interaction in the black phosphorus. A VFF model was proposed for the single-layer black phosphorus in 1982 [56]. One of the present authors (J.W.J.) simplified this VFF model by ignoring some angle-angle crossing terms and using the simplified VFF model to develop the SW potential for the black phosphorus [7]. However, the mechanical properties from this SW potential are smaller than first-principles calculations, as some angle-angle crossing VFF terms cannot be implemented in the SW potential. We will thus propose a new set of SW potential for the single-layer black phosphorus in this section.

Figure 179.

Phonon dispersion for the single-layer black phosphorus. (a) The VFF model is fitted to the three acoustic branches in the long wave limit along the ΓX direction. The ab initio results (gray pentagons) are from [55]. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for the black phosphorus along ΓXSYΓ.

The structure of the single-layer black phosphorus is shown in Figure 178 , with structural parameters from the ab initio calculations [57]. The black phosphorus has a puckered configuration as shown in Figure 178(b) , where the pucker is perpendicular to the x-direction. The bases for the rectangular unit cell are $a_1=4.422$ Å and $a_2=3.348$ Å. For bulk black phosphorus, the basis lattice vector in the third direction is $a_3=10.587$ Å. There are four phosphorus atoms in the basic unit cell, and their relative coordinates are $(-u\mathrm{,0,}-v)$ , $(u\mathrm{,0,}v)$ , $(0.5-u\mathrm{,0.5,}v)$ , and $(0.5+u\mathrm{,0.5,}-v)$ with $u=0.0821$ and $v=0.1011$ . Atoms are categorized into the top and bottom groups. Atoms in the top group are denoted by P₁, while atoms in the bottom group are denoted by P₂.

Figure 180.

Stress-strain relations for the black phosphorus of size $100\times 100$ Å. The black phosphorus is uniaxially stretched along the armchair or zigzag directions at temperatures 1 and 300 K.

VFF type	Bond stretching		Angle bending
Expression	$\frac{1}{2}{K}_{12}{\left(\text{Δ}{r}_{12}\right)}^2$	$\frac{1}{2}{K}_{14}{\left(\text{Δ}{r}_{14}\right)}^2$	$\frac{1}{2}{K}_{123}{\left(\text{Δ}{\theta }_{123}\right)}^2$	$\frac{1}{2}{K}_{134}{\left(\text{Δ}{\theta }_{134}\right)}^2$
Parameter	10.542	10.542	7.048	7.048
$r_0$ or ${\theta }_0$	2.238	2.260	96.581	102.307

Table 358.

The VFF model for black phosphorus.

The second line gives an explicit expression for each VFF term, where atom indexes are from Figure 178(c) . The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

Table 358 shows four VFF terms for the single-layer black phosphorus; two of which are the bond stretching interactions shown by Eq. (1), while the other two terms are the angle bending interaction shown by Eq. (2). The force constant parameters are reasonably chosen to be the same for the two bond stretching terms denoted by $r_{12}$ and $r_{14}$ , as these two bonds have very close bond length value. The force constant parameters are the same for the two angle bending terms ${\theta }_{123}$ and ${\theta }_{134}$ , which have very similar chemical environment. These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the ΓX as shown in Figure 179(a) . The ab initio calculations for the phonon dispersion are from [55]. Similar phonon dispersion can also be found in other ab initio calculations [58–64]. Figure 179(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
$r_{12}$	4.172	0.551	12.543	0.0	2.793
$r_{14}$	4.976	0.685	13.044	0.0	2.882

Table 359.

Two-body SW potential parameters for black phosphorus used by GULP [8] as expressed in Eq. (3).

The quantity ( $r_{ij}$ ) in the first line lists one representative term for the two-body SW potential between atoms i and j. Atom indexes are from Figure 178(c) .

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{123}$	25.965	96.581	0.551	0.551	0.0	2.793	0.0	2.793	2.793	3.365
${\theta }_{134}$	29.932	102.307	0.551	0.685	0.0	2.793	0.0	2.882	2.882	3.772

Table 360.

Three-body SW potential parameters for black phosphorus used by GULP [8] as expressed in Eq. (4).

The first line ( ${\theta }_{ijk}$ ) presents one representative term for the three-body SW potential. The angle ${\theta }_{ijk}$ has the atom i as the apex. Atom indexes are from Figure 178(c) .

The parameters for the two-body SW potential used by GULP are shown in Table 359 . The parameters for the three-body SW potential used by GULP are shown in Table 360 . Parameters for the SW potential used by LAMMPS are listed in Table 361 .

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
P1─P1─P1	1.000	0.551	5.069	25.965	1.000	−0.115	4.172	136.080	4	0	0.0
P1─P2─P2	1.000	0.685	4.207	0.000	1.000	0.000	4.976	59.245	4	0	0.0
P1─P1─P2	1.000	0.000	0.000	29.932	1.000	−0.213	0.000	0.000	4	0	0.0

Table 361.

SW potential parameters for black phosphorus used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Atom types (P₁ and P₂) in the first column are displayed in Figure 178 .

Figure 180 shows the stress-strain relations for the black phosphorus of size $100\times 100$ Å. The structure is uniaxially stretched in the armchair or zigzag directions at 1 and 300 K. The Young’s modulus is 24.3 and 90.5 N/m in the armchair and zigzag directions, respectively, at 1 K, which is obtained by linear fitting of the stress-strain relations in [0, 0.01]. These values agree quite well with previously reported ab initio results, e.g., 28.9 and 101.6 N/m from [65] or 24.4 and 92.1 N/m from [66] or 24.3 and 80.2 N/m from [58]. The ultimate stress is about 4.27 N/m at the critical strain of 0.33 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 8 N/m at the critical strain of 0.19 in the zigzag direction at the low temperature of 1 K. These values agree quite well with the ab initio results at 0 K [66].

It should be noted that the Poisson’s ratios from the VFF model and the SW potential are ${\nu }_{xy}=0.058$ and ${\nu }_{yx}=0.22$ . These values are obviously smaller than first-principles calculations results, e.g., 0.4 and 0.93 from [67] or 0.17 and 0.62 from [58] or 0.24 and 0.81 from [59]. The Poisson’s ratio cannot be obtained correctly by the VFF model proposed in 1982 [56] and the SW potential [7] either [68]. These failures are due to the missing of one angle-angle crossing term [69], which has not been implemented in the package LAMMPS and is not included in the present work.

91. p-Arsenene

Present studies on the puckered (p-) arsenene, also named $\alpha$ arsenene, are based on first-principles calculations, and no empirical potential has been proposed for the p-arsenene. We will thus parametrize a set of VFF model for the single-layer p-arsenene in this section. We will also derive the SW potential based on the VFF model for the single-layer p-arsenene.

The structure of the single-layer p-arsenene is exactly the same as that of the black phosphorus as shown in Figure 178 . Structural parameters for p-arsenene are from the ab initio calculations [70]. The pucker of the p-arsenene is perpendicular to the x (armchair)-direction. The bases for the rectangular unit cell are $a_1=4.77$ Å and $a_2=3.68$ Å. There are four As atoms in the basic unit cell, and their relative coordinates are $(-u\mathrm{,0,}-v)$ , $(u\mathrm{,,}v)$ , $(0.5-u\mathrm{,0.5,}v)$ , and $(0.5+u\mathrm{,0.5,}-v)$ with $u=0.0714$ and $v=0.108$ . The value of the dimensionless parameter $u$ is extracted from the geometrical parameters provided in [70]. The other dimensionless parameter $v$ is a ratio based on the lattice constant in the out-of-plane z-direction, so the other basis $a_3=11.11$ Å from [71] is also adopted in extracting the value of $v$ . We note that the main purpose of the usage of $u$ and $v$ in representing atomic coordinates is to follow the same convention for all puckered structures. The resultant atomic coordinates are the same as that in [70].

Figure 181.

Phonon dispersion for the single-layer p-arsenene. (a) The VFF model is fitted to the three acoustic branches in the long wave limit along the ΓX direction. The ab initio results (gray pentagons) are from [70]. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for the p-arsenene along ΓXSYΓ.

Table 362 shows four VFF terms for the single-layer p-arsenene; two of which are the bond stretching interactions shown by Eq. (1), while the other two terms are the angle bending interaction shown by Eq. (2). The force constant parameters are reasonably chosen to be the same for the two bond stretching terms denoted by $r_{12}$ and $r_{14}$ , as these two bonds have very close bond length value. The force constant parameters happen to be the same for the two angle bending terms ${\theta }_{123}$ and ${\theta }_{134}$ . These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the ΓX as shown in Figure 181(a) . The ab initio calculations for the phonon dispersion are from [70]. Similar phonon dispersion can also be found in other ab initio calculations [64, 72–74]. We note that the lowest frequency branch around the Γ point from the VFF model is lower than the ab initio results. This branch is the flexural branch, which should be a quadratic dispersion. However, the ab initio calculations give a linear dispersion for the flexural branch due to the violation of the rigid rotational invariance in the first-principles package [20] so ab initio calculations typically overestimate the frequency of this branch. Figure 181(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

Figure 182.

Stress-strain relations for the p-arsenene of size $100\times 100$ Å. The p-arsenene is uniaxially stretched along the armchair or zigzag directions at temperatures 1 and 300 K.

VFF type	Bond stretching		Angle bending
Expression	$\frac{1}{2}{K}_{12}{\left(\text{Δ}{r}_{12}\right)}^2$	$\frac{1}{2}{K}_{14}{\left(\text{Δ}{r}_{14}\right)}^2$	$\frac{1}{2}{K}_{123}{\left(\text{Δ}{\theta }_{123}\right)}^2$	$\frac{1}{2}{K}_{134}{\left(\text{Δ}{\theta }_{134}\right)}^2$
Parameter	7.936	7.936	7.456	7.456
$r_0$ or ${\theta }_0$	2.508	2.495	94.400	100.692

Table 362.

The VFF model for p-arsenene.

The second line gives an explicit expression for each VFF term, where atom indexes are from Figure 178(c) . The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
$r_{12}$	3.180	0.455	19.782	0.0	3.042
$r_{14}$	4.477	0.737	19.375	0.0	3.173

Table 363.

Two-body SW potential parameters for p-arsenene used by GULP [8] as expressed in Eq. (3).

The quantity ( $r_{ij}$ ) in the first line lists one representative term for the two-body SW potential between atoms i and j. Atom indexes are from Figure 178(c) .

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{123}$	20.597	94.400	0.455	0.455	0.0	3.042	0.0	3.042	3.628	4.225
${\theta }_{134}$	26.831	100.692	0.455	0.737	0.0	3.042	0.0	3.173	3.173	4.149

Table 364.

Three-body SW potential parameters for p-arsenene used by GULP [8] as expressed in Eq. (4).

The first line ( ${\theta }_{ijk}$ ) presents one representative term for the three-body SW potential. The angle ${\theta }_{ijk}$ has the atom i as the apex. Atom indexes are from Figure 178(c) .

The parameters for the two-body SW potential used by GULP are shown in Table 363 . The parameters for the three-body SW potential used by GULP are shown in Table 364 . Parameters for the SW potential used by LAMMPS are listed in Table 365 .

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
As1─As1─As1	1.000	0.455	6.686	20.597	1.000	−0.077	3.180	461.556	4	0	0.0
As1─As2─As2	1.000	0.737	4.305	0.000	1.000	0.000	4.477	65.671	4	0	0.0
As1─As1─As2	1.000	0.000	0.000	26.831	1.000	−0.186	0.000	0.000	4	0	0.0

Table 365.

SW potential parameters for p-arsenene used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Atom types in the first column are displayed in Figure 178(b) , with element symbol P substituted by As.

VFF type	Bond stretching		Angle bending
Expression	$\frac{1}{2}{K}_{12}{\left(\text{Δ}{r}_{12}\right)}^2$	$\frac{1}{2}{K}_{14}{\left(\text{Δ}{r}_{14}\right)}^2$	$\frac{1}{2}{K}_{123}{\left(\text{Δ}{\theta }_{123}\right)}^2$	$\frac{1}{2}{K}_{134}{\left(\text{Δ}{\theta }_{134}\right)}^2$	$\frac{1}{2}{K}_{415}{\left(\text{Δ}{\theta }_{415}\right)}^2$
Parameter	7.675	7.675	6.534	12.068	12.068
$r_0$ or ${\theta }_0$	2.950	2.870	95.380	88.300	102.800

Table 366.

The VFF model for p-antimonene.

The second line gives an explicit expression for each VFF term, where atom indexes are from Figure 183(c) . The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

Figure 182 shows the stress-strain relations for the p-arsenene of size $100\times 100$ Å. The structure is uniaxially stretched in the armchair or zigzag directions at 1 and 300 K. The Young’s modulus is 20.7 and 73 N/m in the armchair and zigzag directions, respectively, at 1 K, which is obtained by linear fitting of the stress-strain relations in [0, 0.01]. The third-order nonlinear elastic constant $D$ can be obtained by fitting the stress-strain relation to $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with E as the Young’s modulus. The values of $D$ are −56.4 and −415.5 N/m at 1 K along the armchair and zigzag directions, respectively. The ultimate stress is about 3.5 N/m at the critical strain of 0.31 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 6.5 N/m at the critical strain of 0.18 in the zigzag direction at the low temperature of 1 K.

92. p-Antimonene

Present studies on the puckered (p-) antimonene, also named $\alpha$ antimonene, are based on first-principles calculations, and no empirical potential has been proposed for the p-antimonene. We will thus parametrize a set of VFF model for the single-layer p-antimonene in this section. We will also derive the SW potential based on the VFF model for the single-layer p-antimonene.

Figure 183.

Structure for single-layer p-antimonene. (a) Top view illustrates the armchair direction is along the x-axis, while the zigzag direction is along the y-axis. The first Brillouin zone is shown in the inset. (b) Side view illustrates the puckered configuration. The pucker is perpendicular to the x-axis and is parallel with the y-axis. Sb atoms in the top/bottom group have different z-coordinates. (c) Atomic configuration.

The structure of the single-layer p-antimonene is shown in Figure 183 , which is similar as that of the black phosphorus as shown in Figure 178 . Structural parameters for p-antimonene are from the ab initio calculations [70]. The pucker of the p-antimonene is perpendicular to the x (armchair)-direction. The bases for the rectangular unit cell are $a_1=4.73$ Å and $a_2=4.36$ Å. There are four Sb atoms in the basic unit cell, and their relative coordinates are $(-u\mathrm{,0,}-v)$ , $(u\mathrm{,0,}v)$ , $(0.5-u\mathrm{,0.5,}v+w)$ , and $(0.5+u\mathrm{,0.5,}-v+w)$ with $u=0.044$ , $v=0.128$ , and $w=0.0338$ . The value of the dimensionless parameter $u$ is extracted from the geometrical parameters (bond lengths and bond angles) provided in [70]. The dimensionless parameters $v$ and $w$ are ratios based on the lattice constant in the out-of-plane z-direction, so an arbitrary value of $a_3=11.11$ Å is adopted in extracting the values of $v$ and $w$ . The value of $a_3$ has no effect on the actual position of each Sb atom. We note that the main purpose of the usage of $u$ , $v$ , and $w$ in representing atomic coordinates is to follow the same convention of black phosphorus. The resultant atomic coordinates are the same as that in [70].

Figure 184.

Eight atom types are introduced for the Sb atoms in the p-antimonene. (a) Top view and (b) side view.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
$r_{12}$	1.750	0.122	37.867	0.0	3.250
$r_{14}$	11.221	1.843	33.923	0.0	4.020

Table 367.

Two-body SW potential parameters for p-antimonene used by GULP [8] as expressed in Eq. (3).

The quantity ( $r_{ij}$ ) in the first line lists one representative term for the two-body SW potential between atoms i and j. Atom indexes are from Figure 183(c) .

As shown in Figure 183(b) , a specific feature in the puckered configuration of the p-antimonene is that Sb atoms in the top/bottom group are further divided into two subgroups with different z-coordinates. Specifically, in Figure 183(c) , there is a difference of $w{a}_3$ between the z-coordinate of atom 1 and the z-coordinates of atoms 2 and 3. Similarly, atom 4 is higher than atoms 5 and 6 for $w{a}_3$ along the z-direction. As a result of the nonzero value of $w$ , there are two different intergroup angles, i.e., ${\theta }_{134}{=88.3}^{\circ }$ and ${\theta }_{415}{=102.8}^{\circ }$ . We have $w=0$ for the ideal puckered configuration of the black phosphorus.

Figure 185.

Phonon dispersion for the single-layer p-antimonene. (a) The VFF model is fitted to the three acoustic branches in the long wave limit along the ΓX direction. The ab initio results (gray pentagons) are from [70]. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for the p-antimonene along ΓXSYΓ.

Table 366 shows five VFF terms for the single-layer p-antimonene; two of which are the bond stretching interactions shown by Eq. (1), while the other three terms are the angle bending interaction shown by Eq. (2). The force constant parameters are reasonably chosen to be the same for the two bond stretching terms denoted by $r_{12}$ and $r_{14}$ , as these two bonds have very close bond length value. The force constant parameters are the same for the two angle bending terms ${\theta }_{134}$ and ${\theta }_{415}$ , which have the same arm lengths. As a result, there are only three force constant parameters, i.e., $K_{12}=\mathrm{}{K}_{14}$ , $K_{123}$ , and $K_{134}=\mathrm{}{K}_{415}$ . These three force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the ΓX as shown in Figure 185(a) . The ab initio calculations for the phonon dispersion are from [70]. Similar phonon dispersion can also be found in other ab initio calculations [64, 75, 76]. We note that the lowest frequency branch around the Γ point from the VFF model is lower than the ab initio results. This branch is the flexural branch, which should be a quadratic dispersion. However, the ab initio calculations give a linear dispersion for the flexural branch due to the violation of the rigid rotational invariance in the first-principles package [20], so ab initio calculations typically overestimate the frequency of this branch. Figure 185(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

Figure 186.

Stress-strain relations for the p-antimonene of size $100\times 100$ Å. The p-antimonene is uniaxially stretched along the armchair or zigzag directions at temperatures 1 and 300 K.

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{123}$	7.435	95.380	0.122	0.122	0.0	3.250	0.0	3.250	0.0	4.545
${\theta }_{134}$	45.054	88.380	1.843	0.122	0.0	4.020	0.0	3.250	0.0	5.715
${\theta }_{415}$	47.338	102.800	1.843	0.122	0.0	4.020	0.0	3.250	0.0	6.105

Table 368.

Three-body SW potential parameters for p-antimonene used by GULP [8] as expressed in Eq. (4).

The first line ( ${\theta }_{ijk}$ ) presents one representative term for the three-body SW potential. The angle ${\theta }_{ijk}$ has the atom i as the apex. Atom indexes are from Figure 183(c) .

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Sb5─Sb4─Sb4	1.000	5.103	0.958	635.059	1.000	−0.094	38.498	0.056	4	0	0.0
Sb1─Sb2─Sb2	1.000	1.924	2.102	0.000	1.000	0.000	11.708	2.476	4	0	0.0
Sb5─Sb4─Sb6	1.000	0.000	0.000	431.139	1.000	0.030	0.000	0.000	4	0	0.0
Sb2─Sb3─Sb1	1.000	0.000	0.000	452.994	1.000	−0.222	0.000	0.000	4	0	0.0

Table 369.

SW potential parameters for p-antimonene used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Atom types in the first column are displayed in Figure 184 .

The parameters for the two-body SW potential used by GULP are shown in Table 367 . The parameters for the three-body SW potential used by GULP are shown in Table 368 . Parameters for the SW potential used by LAMMPS are listed in Table 369 . Eight atom types have been introduced for writing the SW potential script used by LAMMPS as shown in Figure 184 , which technically increases the cutoff for the bond stretching interaction between atom 1 and atom 2 in Figure 183(c) .

Figure 186 shows the stress-strain relations for the p-antimonene of size $100\times 100$ Å. The structure is uniaxially stretched in the armchair or zigzag directions at 1 and 300 K. The Young’s modulus is 18.3 and 65.2 N/m in the armchair and zigzag directions, respectively, at 1 K, which is obtained by linear fitting of the stress-strain relations in [0, 0.01]. The Poisson’s ratios from the VFF model and the SW potential are ${\nu }_{xy}=0.08$ and ${\nu }_{yx}=0.29$ . The third-order nonlinear elastic constant $D$ can be obtained by fitting the stress-strain relation to $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with E as the Young’s modulus. The values of $D$ are −22.1 and −354.1 N/m at 1 K along the armchair and zigzag directions, respectively. The ultimate stress is about 3.7 N/m at the critical strain of 0.37 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 6.4 N/m at the critical strain of 0.17 in the zigzag direction at the low temperature of 1 K.

Figure 187.

Phonon dispersion for the single-layer p-bismuthene. (a) The VFF model is fitted to the three acoustic branches in the long wave limit along the ΓX direction. The ab initio results (gray pentagons) are from [77]. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for the p-bismuthene along ΓXSYΓ.

93. p-Bismuthene

Present studies on the puckered (p-) bismuthene, which is also named $\alpha$ bismuthene, are based on first-principles calculations, and no empirical potential has been proposed for the p-bismuthene. We will thus parametrize a set of VFF model for the single-layer p-bismuthene in this section. We will also derive the SW potential based on the VFF model for the single-layer p-bismuthene.

The structure of the single-layer p-bismuthene is the same as p-antimonene as shown in Figure 183 . Structural parameters for p-bismuthene are from the ab initio calculations [77]. The pucker of the p-bismuthene is perpendicular to the x (armchair)-direction. The bases for the rectangular unit cell are $a_1=4.94$ Å and $a_2=4.55$ Å. There are four Bi atoms in the basic unit cell, and their relative coordinates are $(-u\mathrm{,0,}-v)$ , $(u\mathrm{,0,}v)$ , $(0.5-u\mathrm{,0.5,}v+w)$ , and $(0.5+u\mathrm{,0.5,}-v+w)$ with $u=0.0405$ , $v=0.130$ , and $w=0.0391$ . The value of the dimensionless parameter $u$ is extracted from the geometrical parameters provided in [77]. The dimensionless parameters $v$ and $w$ are ratios based on the lattice constant in the out-of-plane z-direction, so an arbitrary value of $a_3=11.81$ Å is adopted in extracting the values of $v$ and $w$ . The value of $a_3$ has no effect on the actual position of each Bi atom. We note that the main purpose of the usage of $u$ , $v$ , and $w$ in representing atomic coordinates is to follow the same convention of black phosphorus. The resultant atomic coordinates are the same as that in [77].

Figure 188.

Stress-strain relations for the p-bismuthene of size $100\times 100$ Å. The p-bismuthene is uniaxially stretched along the armchair or zigzag directions at temperatures 1 and 300 K.

VFF type	Bond stretching		Angle bending
Expression	$\frac{1}{2}{K}_{12}{\left(\text{Δ}{r}_{12}\right)}^2$	$\frac{1}{2}{K}_{14}{\left(\text{Δ}{r}_{14}\right)}^2$	$\frac{1}{2}{K}_{123}{\left(\text{Δ}{\theta }_{123}\right)}^2$	$\frac{1}{2}{K}_{134}{\left(\text{Δ}{\theta }_{134}\right)}^2$	$\frac{1}{2}{K}_{415}{\left(\text{Δ}{\theta }_{415}\right)}^2$
Parameter	7.675	7.675	2.267	8.347	8.347
$r_0$ or ${\theta }_0$	3.110	3.097	94.018	86.486	103.491

Table 370.

The VFF model for p-bismuthene.

The second line gives an explicit expression for each VFF term, where atom indexes are from Figure 183(c) . The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
$r_{12}$	1.777	0.109	46.775	0.0	3.401
$r_{14}$	12.322	1.872	45.998	0.0	4.301

Table 371.

Two-body SW potential parameters for p-bismuthene used by GULP [8] as expressed in Eq. (3).

The quantity ( $r_{ij}$ ) in the first line lists one representative term for the two-body SW potential between atoms i and j. Atom indexes are from Figure 183(c) .

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{123}$	2.408	94.018	0.109	0.109	0.0	3.401	0.0	3.401	0.0	4.745
${\theta }_{134}$	28.842	86.486	1.872	0.109	0.0	4.301	0.0	3.401	0.0	5.982
${\theta }_{415}$	30.388	103.491	1.872	0.109	0.0	4.301	0.0	3.401	0.0	6.473

Table 372.

Three-body SW potential parameters for p-bismuthene used by GULP [8] as expressed in Eq. (4).

The first line ( ${\theta }_{ijk}$ ) presents one representative term for the three-body SW potential. The angle ${\theta }_{ijk}$ has the atom i as the apex. Atom indexes are from Figure 183(c) .

As shown in Figure 183(b) , a specific feature in the puckered configuration of the p-bismuthene is that Bi atoms in the top/bottom group are further divided into two subgroups with different z-coordinates. Specifically, in Figure 183(c) , there is a difference of $w{a}_3$ between the z-coordinate of atom 1 and the z-coordinates of atoms 2 and 3. Similarly, atom 4 is higher than atoms 5 and 6 for $w{a}_3$ along the z-direction. As a result of the nonzero value of $w$ , there are two different intergroup angles, i.e., ${\theta }_{134}=86.486°$ and ${\theta }_{415}=103.491°$ . We have $w=0$ for the ideal puckered configuration of the black phosphorus.

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Bi1─Bi8─Bi8	1.000	2.737	1.571	112.813	1.000	−0.070	18.974	1.000	4	0	0.0
Bi1─Bi2─Bi2	1.000	2.808	1.532	0.000	1.000	0.000	19.577	0.888	4	0	0.0
Bi1─Bi8─Bi2	1.000	0.000	0.000	429.598	1.000	0.061	0.000	0.000	4	0	0.0
Bi2─Bi3─Bi1	1.000	0.000	0.000	452.618	1.000	−0.233	0.000	0.000	4	0	0.0

Table 373.

SW potential parameters for p-bismuthene used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Atom types in the first column are displayed in Figure 184 with elemental symbol Sb substituted by Bi.

Table 370 shows five VFF terms for the single-layer p-bismuthene; two of which are the bond stretching interactions shown by Eq. (1), while the other three terms are the angle bending interaction shown by Eq. (2). The force constant parameters are reasonably chosen to be the same for the two bond stretching terms denoted by $r_{12}$ and $r_{14}$ , as these two bonds have very close bond length value. The force constant parameters are the same for the two angle bending terms ${\theta }_{134}$ and ${\theta }_{415}$ , which have the same arm lengths. As a result, there are only three force constant parameters, i.e., $K_{12}=\mathrm{}{K}_{14}$ , $K_{123}$ , and $K_{134}=\mathrm{}{K}_{415}$ . These three force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the ΓX as shown in Figure 187(a) . The ab initio calculations for the phonon dispersion are from [77]. Similar phonon dispersion can also be found in other ab initio calculations [64]. Figure 187(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 371 . The parameters for the three-body SW potential used by GULP are shown in Table 372 . Parameters for the SW potential used by LAMMPS are listed in Table 373 . Eight atom types have been introduced for writing the SW potential script used by LAMMPS as shown in Figure 184 , which helps to increase the cutoff for the bond stretching interaction between atoms like 1 and 2 in Figure 183(c) .

Figure 188 shows the stress-strain relations for the p-bismuthene of size $100\times 100$ Å. The structure is uniaxially stretched in the armchair or zigzag directions at 1 and 300 K. The Young’s modulus is 10.2 and 26.2 N/m in the armchair and zigzag directions, respectively, at 1 K, which is obtained by linear fitting of the stress-strain relations in [0, 0.01]. The Poisson’s ratios from the VFF model and the SW potential are ${\nu }_{xy}=0.24$ and ${\nu }_{yx}=0.61$ . These values are very close to the ab initio calculations, e.g., ${\nu }_{xy}=0.261$ and ${\nu }_{yx}=0.648$ in [77]. The third-order nonlinear elastic constant $D$ can be obtained by fitting the stress-strain relation to $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with E as the Young’s modulus. The values of $D$ are −12.4 and −86.4 N/m at 1 K along the armchair and zigzag directions, respectively. The ultimate stress is about 2.6 N/m at the critical strain of 0.38 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 3.9 N/m at the critical strain of 0.29 in the zigzag direction at the low temperature of 1 K.

Figure 189.

Structure for single-layer p-MX, with M from group IV and X from group VI. (a) Top view. The armchair direction is along the x-axis, while the zigzag direction is along the y-axis. Red inset shows the first Brillouin zone. (b) Side view illustrates the puckered configuration. The pucker is perpendicular to the x-axis and is parallel with the y-axis. (c) Atomic configuration. Atom M (X) is represented by yellow smaller (gray larger) balls.

94. p-SiO

Present studies on the puckered (p-) SiO are based on first-principles calculations, and no empirical potential has been proposed for the p-SiO. We will thus parametrize the SW potential for the single-layer p-SiO in this section.

Figure 190.

Eight atom types are introduced for atoms in the p-MX, with M from group IV and X from group VI. (a) Top view and (b) side view.

The structure of the single-layer p-SiO is shown in Figure 189 , with M = Si and X = O. Structural parameters for p-SiO are from the ab initio calculations [78]. There are four atoms in the unit cell with relative coordinates as $(-u\mathrm{,0,}-v)$ , $(u\mathrm{,0,}v)$ , $(0.5-u\mathrm{,0.5,}v+w)$ , and $(0.5+u\mathrm{,0.5,}-v+w)$ with $u=0.1501$ , $v=0.0605$ , and $w=0.0800$ . The value of these dimensionless parameters are extracted from the geometrical parameters provided in [78], including lattice constants $a_1=4.701$ Å and $a_2=2.739$ Å, bond lengths $d_{12}=1.843$ Å and $d_{14}=2.859$ Å, and the angle ${\theta }_{145}{=96.0}^{\circ }$ . The dimensionless parameters $v$ and $w$ are ratios based on the lattice constant in the out-of-plane z-direction, which are arbitrarily chosen as $a_3=10.0$ Å. We note that the main purpose of the usage of $u$ , $v$ , and $w$ in representing atomic coordinates is to follow the same convention for all puckered structures in the present work. The resultant atomic coordinates are the same as that in [78].

As shown in Figure 189 , a specific feature in the puckered configuration of the p-SiO is that there is a small difference of $w{a}_3$ between the z-coordinate of atom 1 and the z-coordinates of atoms 2 and 3. Similarly, atom 4 is higher than atoms 5 and 6 for $w{a}_3$ along the z-direction. The sign of $w$ determines which types of atoms take the out-most positions, e.g., atoms 1, 5, and 6 are the out-most atoms if $w>0$ in Figure 189(c) , while atoms 2, 3, and 4 will take the out-most positions for $w<0$ . The p-SiO has a zigzag configuration as shown in Figure 191 , which is a specific case of the puckered structure shown in Figure 189 .

Figure 191.

Zigzag configuration of single-layer p-MX, with M from group IV and X = O. Atom M (O) is represented by purple larger (red smaller) balls.

Table 374 shows five VFF terms for the single-layer p-SiO; two of which are the bond stretching interactions shown by Eq. (1), while the other three terms are the angle bending interaction shown by Eq. (2). The force constant parameters are the same for the two angle bending terms ${\theta }_{134}$ and ${\theta }_{415}$ , which have the same arm lengths. All force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the ΓX as shown in Figure 192(a) . The ab initio calculations for the phonon dispersion are calculated from the SIESTA package [79]. The generalized gradients approximation is applied to account for the exchange-correlation function with Perdew, Burke, and Ernzerhof parameterization [80], and the double- $\zeta$ orbital basis set is adopted. Figure 192(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion.

Figure 192.

Phonon dispersion for the single-layer p-SiO. (a) The VFF model is fitted to the acoustic branches in the long wave limit along the ΓX direction. The ab initio calculations are calculated from SIESTA. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for the p-SiO along ΓXSYΓ.

The parameters for the two-body SW potential used by GULP are shown in Table 375 . The parameters for the three-body SW potential used by GULP are shown in Table 376 . Parameters for the SW potential used by LAMMPS are listed in Table 377 .

Figure 193 shows the stress-strain relations for the single-layer p-SiO of size $100\times 100$ Å. The structure is uniaxially stretched in the armchair or zigzag directions at 1 and 300 K. The structure of p-SiO is so soft along the armchair direction that the Young’s modulus is almost zero in the armchair direction. The Young’s modulus is 81.3 N/m in the zigzag direction at 1 K, which is obtained by linear fitting of the stress-strain relations in [0, 0.01]. The third-order nonlinear elastic constant $D$ can be obtained by fitting the stress-strain relation to $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with E as the Young’s modulus. The value of $D$ is −432.4 N/m at 1 K along the zigzag direction. The ultimate stress is about 5.3 N/m at the critical strain of 0.29 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 7.8 N/m at the critical strain of 0.23 in the zigzag direction at the low temperature of 1 K.

Figure 193.

Stress-strain relations for the single-layer p-SiO of size $100\times 100$ Å. The single-layer p-SiO is uniaxially stretched along the armchair or zigzag directions at temperatures 1 and 300 K.

VFF type	Bond stretching		Angle bending
Expression	$\frac{1}{2}{K}_{12}{\left(\text{Δ}{r}_{12}\right)}^2$	$\frac{1}{2}{K}_{14}{\left(\text{Δ}{r}_{14}\right)}^2$	$\frac{1}{2}{K}_{123}{\left(\text{Δ}{\theta }_{123}\right)}^2$	$\frac{1}{2}{K}_{134}{\left(\text{Δ}{\theta }_{134}\right)}^2$	$\frac{1}{2}{K}_{415}{\left(\text{Δ}{\theta }_{415}\right)}^2$
Parameter	12.191	12.191	4.817	3.123	3.123
$r_0$ or ${\theta }_0$	1.843	1.859	95.989	96.000	132.005

Table 374.

The VFF model for the single-layer p-SiO.

The second line gives an explicit expression for each VFF term, where atom indexes are from Figure 189(c) . The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
$r_{12}$	19.127	2.720	5.769	0.0	2.962
$r_{14}$	7.105	1.133	5.972	0.0	2.585

Table 375.

Two-body SW potential parameters for the single-layer p-SiO used by GULP [8] as expressed in Eq. (3).

The quantity ( $r_{ij}$ ) in the first line lists one representative term for the two-body SW potential between atoms i and j. Atom indexes are from Figure 189(c) .

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{123}$	314.008	95.989	2.720	2.720	0.0	2.962	0.0	2.962	0.0	3.720
${\theta }_{134}$	85.406	96.000	1.133	2.720	0.0	2.585	0.0	2.962	0.0	3.875
${\theta }_{415}$	152.982	132.005	1.133	2.720	0.0	2.585	0.0	2.962	0.0	4.194

Table 376.

Three-body SW potential parameters for the single-layer p-SiO used by GULP [8] as expressed in Eq. (4).

The first line ( ${\theta }_{ijk}$ ) presents one representative term for the three-body SW potential. The angle ${\theta }_{ijk}$ has the atom i as the apex. Atom indexes are from Figure 189(c) .

95. P-GeO

Present studies on the puckered (p-) GeO are based on first-principles calculations, and no empirical potential has been proposed for the p-GeO. We will thus parametrize the SW potential for the single-layer p-GeO in this section.

The structure of the single-layer p-GeO is shown in Figure 189 , with M=Ge and X=O. Structural parameters for p-GeO are from the ab initio calculations [78]. There are four atoms in the unit cell with relative coordinates such as $(-u\mathrm{,0,}-v)$ , $(u\mathrm{,0,}v)$ , $(0.5-u\mathrm{,0.5,}v+w)$ , and $(0.5+u\mathrm{,0.5,}-v+w)$ with $u=0.1622$ , $v=0.0616$ , and $w=0.0884$ . The values of these dimensionless parameters are extracted from the geometrical parameters provided in Ref. [78], including lattice constants $a_1=4.801$ Å and $a_2=3.055$ Å, bond lengths $d_{12}=1.956$ Å and $d_{14}=1.986$ Å, and the angle ${\theta }_{145}=93.3°$ . The dimensionless parameters $v$ and $w$ are ratios based on the lattice constant in the out-of-plane z-direction, which is arbitrarily chosen as $a_3=10.0$ Å. We note that the main purpose of the usage of $u$ , $v$ , and $w$ in representing atomic coordinates is to follow the same convention for all puckered structures in the present work. The resultant atomic coordinates are the same as that in Ref. [78].

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\text{θ}}_{\text{0}}$	$A_L$	$B_L$	$p$	$q$	Tol
Sn1─O2─O2	1.000	2.720	1.089	314.008	1.000	−0.104	19.127	0.105	4	0	0.0
Sn1─O1─O1	1.000	1.133	2.282	0.000	1.000	0.000	7.105	3.630	4	0	0.0
Sn1─O1─O2	1.000	0.000	0.000	85.406	1.000	−0.105	0.000	0.000	4	0	0.0
O1─Sn1─Sn2	1.000	0.000	0.000	152.982	1.000	−0.669	0.000	0.000	4	0	0.0

Table 377.

SW potential parameters for p-SiO used by LAMMPS [9], as expressed in Eqs. (9) and (10).

Atom types in the first column are displayed in Figure 190 , with M=Si and X=O.

Figure 194.

Phonon dispersion for the single-layer p-GeO. (a) The VFF model is fitted to the acoustic branches in the long wave limit along the ΓX direction. The ab initio calculations are calculated from SIESTA. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for the p-GeO along ΓXSYΓ.

As shown in Figure 189 , a specific feature in the puckered configuration of the p-GeO is that there is a small difference of $w{a}_3$ between the z-coordinate of atom 1 and the z-coordinates of atoms 2 and 3. Similarly, atom 4 is higher than atoms 5 and 6 for $w{a}_3$ along the z-direction. The sign of $w$ determines which types of atoms take the outmost positions, e.g., atoms 1, 5, and 6 are the outmost atoms if $w>0$ in Figure 189(c) , while atoms 2, 3, and 4 will take the outmost positions for $w<0$ . The p-GeO has a zigzag configuration as shown in Figure 191 , which is a specific case of the puckered structure shown in Figure 189 .

Figure 195.

Stress-strain relations for the single-layer p-GeO of size $100\times 100$ Å. The single-layer p-GeO is uniaxially stretched along the armchair or zigzag directions at temperatures 1 and 300 K.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{12}{\left(\text{Δ}{r}_{12}\right)}^2$	$\frac{1}{2}{K}_{14}{\left(\text{Δ}{r}_{14}\right)}^2$	$\frac{1}{2}{K}_{123}{\left(\text{Δ}{\theta }_{123}\right)}^2$	$\frac{1}{2}{K}_{134}{\left(\text{Δ}{\theta }_{134}\right)}^2$	$\frac{1}{2}{K}_{415}{\left(\text{Δ}{\theta }_{415}\right)}^2$
Parameter	12.191	12.191	4.817	3.123	3.123
$r_0$ or ${\theta }_0$	1.956	1.986	102.692	93.300	128.213

Table 378.

The VFF model for the single-layer p-GeO.

The second line gives an explicit expression for each VFF term, where atom indexes are from Figure 189(c) . The third line is the force constant parameters. Parameters are in the unit of $\text{eV}/{\mathrm{\AA }}^{\text{2}}$ for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

Table 378 shows five VFF terms for the single-layer p-GeO; two of which are the bond stretching interactions shown in Eq. (1), while the other three terms are the angle bending interaction shown in Eq. (2). The force constant parameters are the same for the two angle bending terms ${\theta }_{134}$ and ${\theta }_{415}$ , which have the same arm lengths. All force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the ΓX as shown in Figure 194(a) . The ab initio calculations for the phonon dispersion are calculated from the SIESTA package [79]. The generalized gradients approximation is applied to account for the exchange-correlation function with Perdew, Burke, and Ernzerhof parameterization [80], and the double- $\zeta$ orbital basis set is adopted. Figure 194(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
$r_{12}$	21.562	2.889	7.319	0.0	3.144
$r_{14}$	9.258	1.384	7.778	0.0	2.815

Table 379.

Two-body SW potential parameters for the single-layer p-GeO used by GULP [8], as expressed in Eq. (3).

The quantity ( $r_{ij}$ ) in the first line lists one representative term for the two-body SW potential between atoms i and j. Atom indexes are from Figure 189(c) .

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{123}$	326.824	102.692	2.889	2.889	0.0	3.144	0.0	3.144	0.0	3.928
${\theta }_{134}$	94.550	93.300	1.384	2.889	0.0	2.815	0.0	3.144	0.0	3.933
${\theta }_{415}$	152.646	128.213	1.384	2.889	0.0	2.815	0.0	3.144	0.0	4.284

Table 380.

Three-body SW potential parameters for the single-layer p-GeO used by GULP [8], as expressed in Eq. (4).

The first line ( ${\theta }_{ijk}$ ) presents one representative term for the three-body SW potential. The angle ${\theta }_{ijk}$ has the atom i as the apex. Atom indexes are from Figure 189(c) .

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Ge1─O2─O2	1.000	2.889	1.089	326.824	1.000	−0.220	21.562	0.105	4	0	0.0
Ge1─O1─O1	1.000	1.384	2.034	0.000	1.000	0.000	9.258	2.119	4	0	0.0
Ge1─O1─O2	1.000	0.000	0.000	94.550	1.000	−0.058	0.000	0.000	4	0	0.0
O1─Ge1─Ge2	1.000	0.000	0.000	152.646	1.000	−0.619	0.000	0.000	4	0	0.0

Table 381.

SW potential parameters for p-GeO used by LAMMPS [9], as expressed in Eqs. (9) and (10).

Atom types in the first column are displayed in Figure 190 , with M = Ge and X = O.

The parameters for the two-body SW potential used by GULP are shown in Table 379 . The parameters for the three-body SW potential used by GULP are shown in Table 380 . Parameters for the SW potential used by LAMMPS are listed in Table 381 .

Figure 195 shows the stress-strain relations for the single-layer p-GeO of size $100\times 100$ Å. The structure is uniaxially stretched in the armchair or zigzag directions at 1 and 300 K. The structure of p-GeO is so soft along the armchair direction that the Young’s modulus is almost zero in the armchair direction. The Young’s modulus is 14.5 and 78.9 N/m in the armchair and zigzag directions at 1 K, which are obtained by linear fitting of the stress-strain relations in [0, 0.01]. The Poisson’s ratios from the VFF model and the SW potential are ${\nu }_{xy}=0.09$ and ${\nu }_{yx}=0.65$ . The third-order nonlinear elastic constant $D$ can be obtained by fitting the stress-strain relation to $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with E as the Young’s modulus. The values of $D$ are 22.0 and −383.3 N/m at 1 K along the armchair and zigzag directions, respectively. The ultimate stress is about 4.8 N/m at the critical strain of 0.32 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 8.2 N/m at the critical strain of 0.23 in the zigzag direction at the low temperature of 1 K.

96. P-SnO

Present studies on the puckered (p-) SnO are based on first-principles calculations, and no empirical potential has been proposed for the p-SnO. We will thus parametrize the SW potential for the single-layer p-SnO in this section.

The structure of the single-layer p-SnO is shown in Figure 189 , with M = Sn and X = O. Structural parameters for p-SnO are from the ab initio calculations [78]. There are four atoms in the unit cell with relative coordinates such as $(-u\mathrm{,0,}-v)$ , $(u\mathrm{,0,}v)$ , $(0.5-u\mathrm{,0.5,}v+w)$ , and $(0.5+u\mathrm{,0.5,}-v+w)$ with $u=0.1485$ , $v=0.0818$ , and $w=0.0836$ . The values of these dimensionless parameters are extracted from the geometrical parameters provided in Ref. [78], including lattice constants $a_1=4.764$ Å and $a_2=3.400$ Å, bond lengths $d_{12}=2.127$ Å and $d_{14}=2.163$ Å, and the angle ${\theta }_{145}=90.0°$ . The dimensionless parameters $v$ and $w$ are ratios based on the lattice constant in the out-of-plane z-direction, which is arbitrarily chosen as $a_3=10.0$ Å. We note that the main purpose of the usage of $u$ , $v$ , and $w$ in representing atomic coordinates is to follow the same convention for all puckered structures in the present work. The resultant atomic coordinates are the same as that in Ref. [78].

Figure 196.

Phonon dispersion for the single-layer p-SnO. (a) The VFF model is fitted to the acoustic branches in the long wave limit along the ΓX direction. The ab initio calculations are calculated from SIESTA. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for the p-SnO along ΓXSYΓ.

As shown in Figure 189 , a specific feature in the puckered configuration of the p-SnO is that there is a small difference of $w{a}_3$ between the z-coordinate of atom 1 and the z-coordinates of atoms 2 and 3. Similarly, atom 4 is higher than atoms 5 and 6 for $w{a}_3$ along the z-direction. The sign of $w$ determines which types of atoms take the outmost positions, e.g., atoms 1, 5, and 6 are the outmost atoms if $w>0$ in Figure 189(c) , while atoms 2, 3, and 4 will take the outmost positions for $w<0$ . The p-SnO has a zigzag configuration as shown in Figure 191 , which is a specific case of the puckered structure shown in Figure 189 .

Figure 197.

Stress-strain relations for the single-layer p-SnO of size $100\times 100$ Å. The single-layer p-SnO is uniaxially stretched along the armchair or zigzag directions at temperatures 1 and 300 K.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{12}{\left(\text{Δ}{r}_{12}\right)}^2$	$\frac{1}{2}{K}_{14}{\left(\text{Δ}{r}_{14}\right)}^2$	$\frac{1}{2}{K}_{123}{\left(\text{Δ}{\theta }_{123}\right)}^2$	$\frac{1}{2}{K}_{134}{\left(\text{Δ}{\theta }_{134}\right)}^2$	$\frac{1}{2}{K}_{415}{\left(\text{Δ}{\theta }_{415}\right)}^2$
Parameter	9.208	9.208	2.835	3.023	3.023
$r_0$ or ${\theta }_0$	2.127	2.163	106.117	90.000	126.496

Table 382.

The VFF model for the single-layer p-SnO.

The second line gives an explicit expression for each VFF term, where atom indexes are from Figure 189(c) . The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

Table 382 shows five VFF terms for the single-layer p-SnO; two of which are the bond stretching interactions shown in Eq. (1), while the other three terms are the angle bending interaction shown in Eq. (2). The force constant parameters are the same for the two angle bending terms ${\theta }_{134}$ and ${\theta }_{415}$ , which have the same arm lengths. All force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the ΓX as shown in Figure 196(a) . The ab initio calculations for the phonon dispersion are calculated from the SIESTA package [79]. The generalized gradients approximation is applied to account for the exchange-correlation function with Perdew, Burke, and Ernzerhof parameterization [80], and the double- $\zeta$ orbital basis set is adopted. Figure 196(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
$r_{12}$	11.711	2.107	10.234	0.0	3.185
$r_{14}$	8.879	1.612	10.945	0.0	3.096

Table 383.

Two-body SW potential parameters for the single-layer p-SnO used by GULP [8], as expressed in Eq. (3).

The quantity ( $r_{ij}$ ) in the first line lists one representative term for the two-body SW potential between atoms i and j. Atom indexes are from Figure 189(c) .

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{123}$	82.293	106.117	2.107	2.107	0.0	3.185	0.0	3.185	0.0	4.082
${\theta }_{134}$	62.178	90.000	1.612	2.107	0.0	3.096	0.0	3.185	0.0	4.017
${\theta }_{415}$	96.214	126.496	1.612	2.107	0.0	3.096	0.0	3.185	0.0	4.426

Table 384.

Three-body SW potential parameters for the single-layer p-SnO used by GULP [8], as expressed in Eq. (4).

The first line ( ${\theta }_{ijk}$ ) presents one representative term for the three-body SW potential. The angle ${\theta }_{ijk}$ has the atom i as the apex. Atom indexes are from Figure 189(c) .

The parameters for the two-body SW potential used by GULP are shown in Table 383 . The parameters for the three-body SW potential used by GULP are shown in Table 384 . Parameters for the SW potential used by LAMMPS are listed in Table 385 .

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Sn1─O2─O2	1.000	2.107	1.512	82.293	1.000	−0.278	11.711	0.519	4	0	0.0
Sn1─O1─O1	1.000	1.612	1.921	0.000	1.000	0.000	8.879	1.623	4	0	0.0
Sn1─O1─O2	1.000	0.000	0.000	62.178	1.000	0.000	0.000	0.000	4	0	0.0
O1─Sn1─Sn2	1.000	0.000	0.000	96.214	1.000	−0.595	0.000	0.000	4	0	0.0

Table 385.

SW potential parameters for p-SnO used by LAMMPS [9], as expressed in Eqs. (9) and (10).

Atom types in the first column are displayed in Figure 190 , with M=Sn and X=O.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{12}{\left(\text{Δ}{r}_{12}\right)}^2$	$\frac{1}{2}{K}_{14}{\left(\text{Δ}{r}_{14}\right)}^2$	$\frac{1}{2}{K}_{123}{\left(\text{Δ}{\theta }_{123}\right)}^2$	$\frac{1}{2}{K}_{134}{\left(\text{Δ}{\theta }_{134}\right)}^2$	$\frac{1}{2}{K}_{415}{\left(\text{Δ}{\theta }_{415}\right)}^2$
Parameter	9.291	9.291	3.933	3.075	3.075
$r_0$ or ${\theta }_0$	1.757	1.849	105.384	118.100	104.288

Table 386.

The VFF model for the single-layer p-CS.

The second line gives an explicit expression for each VFF term, where atom indexes are from Figure 189(c) . The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

Figure 197 shows the stress-strain relations for the single-layer p-SnO of size $100\times 100$ Å. The structure is uniaxially stretched in the armchair or zigzag directions at 1 and 300 K. The structure of p-SnO is so soft along the armchair direction that the Young’s modulus is almost zero in the armchair direction. The Young’s modulus is 52.8 N/m in the zigzag direction at 1 K, which is obtained by linear fitting of the stress-strain relations in [0, 0.01]. The third-order nonlinear elastic constant $D$ can be obtained by fitting the stress-strain relation to $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with E as the Young’s modulus. The value of $D$ is −204.5 N/m at 1 K along the zigzag direction. The ultimate stress is about 3.8 N/m at the critical strain of 0.38 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 6.8 N/m at the critical strain of 0.26 in the zigzag direction at the low temperature of 1 K.

97. P-CS

Present studies on the puckered (p-) CS are based on first-principles calculations, and no empirical potential has been proposed for the p-CS. We will thus parametrize the SW potential for the single-layer p-CS in this section.

The structure of the single-layer p-CS is shown in Figure 189 , with M = C and X = S. Structural parameters for p-CS are from the ab initio calculations [78]. There are four atoms in the unit cell with relative coordinates such as $(-u\mathrm{,0,}-v)$ , $(u\mathrm{,0,}v)$ , $(0.5-u\mathrm{,0.5,}v+w)$ , and $(0.5+u\mathrm{,0.5,}-v+w)$ with $u=0.1302$ , $v=0.0733$ , and $w=\mathrm{}-0.0248$ . The values of these dimensionless parameters are extracted from the geometrical parameters provided in Ref. [78], including lattice constants $a_1=4.323$ Å and $a_2=2.795$ Å, bond lengths $d_{12}=1.757$ Å and $d_{14}=1.849$ Å, and the angle ${\theta }_{145}=118.1°$ . The dimensionless parameters $v$ and $w$ are ratios based on the lattice constant in the out-of-plane z-direction, which is arbitrarily chosen as $a_3=10.0$ Å. We note that the main purpose of the usage of $u$ , $v$ , and $w$ in representing atomic coordinates is to follow the same convention for all puckered structures in the present work. The resultant atomic coordinates are the same as that in Ref. [78].

Figure 198.

Phonon dispersion for the single-layer p-CS. (a) The VFF model is fitted to the acoustic branches in the long wave limit along the ΓX direction. The ab initio calculations are calculated from SIESTA. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for the p-CS along ΓXSYΓ.

As shown in Figure 189 , a specific feature in the puckered configuration of the p-CS is that there is a small difference of $w{a}_3$ between the z-coordinate of atom 1 and the z-coordinates of atoms 2 and 3. Similarly, atom 4 is higher than atoms 5 and 6 for $w{a}_3$ along the z-direction. The sign of $w$ determines which types of atoms take the outmost positions, e.g., atoms 1, 5, and 6 are the outmost atoms if $w>0$ in Figure 189(c) , while atoms 2, 3, and 4 will take the outmost positions for $w<0$ .

Figure 199.

Stress-strain relations for the single-layer p-CS of size $100\times 100$ Å. The single-layer p-CS is uniaxially stretched along the armchair or zigzag directions at temperatures 1 and 300 K. Inset shows the structure before (light) and after (dark) the structural transition around 0.16.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
$r_{12}$	8.898	1.894	4.765	0.0	2.669
$r_{14}$	5.791	1.220	5.844	0.0	2.600

Table 387.

Two-body SW potential parameters for the single-layer p-CS used by GULP [8], as expressed in Eq. (3).

The quantity ( $r_{ij}$ ) in the first line lists one representative term for the two-body SW potential between atoms i and j. Atom indexes are from Figure 189(c) .

Table 386 shows five VFF terms for the single-layer p-CS; two of which are the bond stretching interactions shown in Eq. (1), while the other three terms are the angle bending interaction shown in Eq. (2). The force constant parameters are the same for the two angle bending terms ${\theta }_{134}$ and ${\theta }_{415}$ , which have the same arm lengths. All force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the ΓX as shown in Figure 198(a) . The ab initio calculations for the phonon dispersion are calculated from the SIESTA package [79]. The generalized gradients approximation is applied to account for the exchange-correlation function with Perdew, Burke, and Ernzerhof parameterization [80], and the double- $\zeta$ orbital basis set is adopted. Figure 198(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion.

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{123}$	134.527	105.384	1.894	1.894	0.0	2.669	0.0	2.669	0.0	3.559
${\theta }_{134}$	79.992	118.100	1.220	1.894	0.0	2.600	0.0	2.669	0.0	4.046
${\theta }_{415}$	66.283	104.288	1.220	1.894	0.0	2.600	0.0	2.669	0.0	3.921

Table 388.

Three-body SW potential parameters for the single-layer p-CS used by GULP [8], as expressed in Eq. (4).

The first line ( ${\theta }_{ijk}$ ) presents one representative term for the three-body SW potential. The angle ${\theta }_{ijk}$ has the atom i as the apex. Atom indexes are from Figure 189(c) .

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
C1─S2─S2	1.000	1.894	1.410	134.527	1.000	−0.265	8.898	0.371	4	0	0.0
C1─S1─S1	1.000	1.220	2.131	0.000	1.000	0.000	5.791	2.637	4	0	0.0
C1─S1─S2	1.000	0.000	0.000	79.992	1.000	−0.471	0.000	0.000	4	0	0.0
S1─C1─C2	1.000	0.000	0.000	66.283	1.000	−0.247	0.000	0.000	4	0	0.0

Table 389.

SW potential parameters for p-CS used by LAMMPS [9], as expressed in Eqs. (9) and (10).

Atom types in the first column are displayed in Figure 190 , with M=C and X=S.

The parameters for the two-body SW potential used by GULP are shown in Table 387 . The parameters for the three-body SW potential used by GULP are shown in Table 388 . Parameters for the SW potential used by LAMMPS are listed in Table 389 .

Figure 199 shows the stress-strain relations for the single-layer p-CS of size $100\times 100$ Å. The structure is uniaxially stretched in the armchair or zigzag directions at 1 and 300 K. There is a structural transition around 0.16 at 1 K, where the C atom is twisted. The Young’s modulus is 16.2 and 70.5 N/m in the armchair and zigzag directions, respectively, at 1 K, which is obtained by linear fitting of the stress-strain relations in [0, 0.01]. The Poisson’s ratios from the VFF model and the SW potential are ${\nu }_{xy}=\mathrm{}-0.06$ and ${\nu }_{yx}=\mathrm{}-0.27$ . The third-order nonlinear elastic constant $D$ can be obtained by fitting the stress-strain relation to $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with E as the Young’s modulus. The values of $D$ are −27.3 and −447.2 N/m at 1 K along the armchair and zigzag directions, respectively. The ultimate stress is about 4.3 N/m at the critical strain of 0.38 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 5.2 N/m at the critical strain of 0.22 in the zigzag direction at the low temperature of 1 K.

Figure 200.

Phonon dispersion for the single-layer p-SiS. (a) The VFF model is fitted to the acoustic branches in the long wave limit along the ΓX direction. The ab initio calculations are calculated from SIESTA. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for the p-SiS along ΓXSYΓ.

98. P-SiS

Present studies on the puckered (p-) SiS are based on first-principles calculations, and no empirical potential has been proposed for the p-SiS. We will thus parametrize the SW potential for the single-layer p-SiS in this section.

Figure 201.

Stress-strain relations for the single-layer p-SiS of size $100\times 100$ Å. The single-layer p-SiS is uniaxially stretched along the armchair or zigzag directions at temperatures 1 and 300 K.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{12}{\left(\text{Δ}{r}_{12}\right)}^2$	$\frac{1}{2}{K}_{14}{\left(\text{Δ}{r}_{14}\right)}^2$	$\frac{1}{2}{K}_{123}{\left(\text{Δ}{\theta }_{123}\right)}^2$	$\frac{1}{2}{K}_{134}{\left(\text{Δ}{\theta }_{134}\right)}^2$	$\frac{1}{2}{K}_{415}{\left(\text{Δ}{\theta }_{415}\right)}^2$
Parameter	7.135	7.135	2.512	2.922	2.922
$r_0$ or ${\theta }_0$	2.300	2.344	93.554	96.500	111.710

Table 390.

The VFF model for the single-layer p-SiS.

The second line gives an explicit expression for each VFF term, where atom indexes are from Figure 189(c) . The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

The structure of the single-layer p-SiS is shown in Figure 189 , with M = Si and X = S. Structural parameters for p-SiS are from the ab initio calculations [78]. There are four atoms in the unit cell with relative coordinates such as $(-u\mathrm{,0,}-v)$ , $(u\mathrm{,0,}v)$ , $(0.5-u\mathrm{,0.5,}v+w)$ , and $(0.5+u\mathrm{,0.5,}-v+w)$ with $u=0.0884$ , $v=0.1093$ , and $w=0.0316$ . The values of these dimensionless parameters are extracted from the geometrical parameters provided in Ref. [78], including lattice constants $a_1=4.774$ Å and $a_2=3.352$ Å, bond lengths $d_{12}=2.300$ Å and $d_{14}=2.344$ Å, and the angle ${\theta }_{145}=96.5°$ . The dimensionless parameters $v$ and $w$ are ratios based on the lattice constant in the out-of-plane z-direction, which is arbitrarily chosen as $a_3=10.0$ Å. We note that the main purpose of the usage of $u$ , $v$ , and $w$ in representing atomic coordinates is to follow the same convention for all puckered structures in the present work. The resultant atomic coordinates are the same as that in Ref. [78].

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
$r_{12}$	3.878	0.797	13.992	0.0	2.977
$r_{14}$	6.051	1.301	15.094	0.0	3.217

Table 391.

Two-body SW potential parameters for the single-layer p-SiS used by GULP [8], as expressed in Eq. (3).

The quantity ( $r_{ij}$ ) in the first line lists one representative term for the two-body SW potential between atoms i and j. Atom indexes are from Figure 189(c) .

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{123}$	13.284	93.554	0.797	0.797	0.0	2.977	0.0	2.977	0.0	4.063
${\theta }_{134}$	21.310	96.500	1.301	0.797	0.0	3.217	0.0	2.977	0.0	4.232
${\theta }_{415}$	24.372	111.710	1.301	0.797	0.0	3.217	0.0	2.977	0.0	4.423

Table 392.

Three-body SW potential parameters for the single-layer p-SiS used by GULP [8], as expressed in Eq. (4).

The first line ( ${\theta }_{ijk}$ ) presents one representative term for the three-body SW potential. The angle ${\theta }_{ijk}$ has the atom i as the apex. Atom indexes are from Figure 189(c) .

As shown in Figure 189 , a specific feature in the puckered configuration of the p-SiS is that there is a small difference of $w{a}_3$ between the z-coordinate of atom 1 and the z-coordinates of atoms 2 and 3. Similarly, atom 4 is higher than atoms 5 and 6 for $w{a}_3$ along the z-direction. The sign of $w$ determines which types of atoms take the outmost positions, e.g., atoms 1, 5, and 6 are the outmost atoms if $w>0$ in Figure 189(c) , while atoms 2, 3, and 4 will take the outmost positions for $w<0$ .

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Si1─S2─S2	1.000	0.797	3.735	13.284	1.000	−0.062	3.878	34.661	4	0	0.0
Si1─S1─S1	1.000	1.301	2.474	0.000	1.000	0.000	6.051	5.276	4	0	0.0
Si1─S1─S2	1.000	0.000	0.000	21.310	1.000	−0.113	0.000	0.000	4	0	0.0
S1─Si1─Si2	1.000	0.000	0.000	24.372	1.000	−0.370	0.000	0.000	4	0	0.0

Table 393.

SW potential parameters for p-SiS used by LAMMPS [9], as expressed in Eqs. (9) and (10).

Atom types in the first column are displayed in Figure 190 , with M=Si and X=S.

Table 390 shows five VFF terms for the single-layer p-SiS; two of which are the bond stretching interactions shown in Eq. (1), while the other three terms are the angle bending interaction shown in Eq. (2). The force constant parameters are the same for the two angle bending terms ${\theta }_{134}$ and ${\theta }_{415}$ , which have the same arm lengths. All force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the ΓX as shown in Figure 200(a) . The ab initio calculations for the phonon dispersion are calculated from the SIESTA package [79]. The generalized gradients approximation is applied to account for the exchange-correlation function with Perdew, Burke, and Ernzerhof parameterization [80], and the double- $\zeta$ orbital basis set is adopted. Figure 200(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion.

The parameters for the two-body SW potential used by GULP are shown in Table 391 . The parameters for the three-body SW potential used by GULP are shown in Table 392 . Parameters for the SW potential used by LAMMPS are listed in Table 393 .

Figure 201 shows the stress-strain relations for the single-layer p-SiS of size $100\times 100$ Å. The structure is uniaxially stretched in the armchair or zigzag directions at 1 and 300 K. The Young’s modulus is 10.9 and 34.8 N/m in the armchair and zigzag directions, respectively, at 1 K, which is obtained by linear fitting of the stress-strain relations in [0, 0.01]. The Poisson’s ratios from the VFF model and the SW potential are ${\nu }_{xy}=0.04$ and ${\nu }_{yx}=0.12$ . The third-order nonlinear elastic constant $D$ can be obtained by fitting the stress-strain relation to $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with E as the Young’s modulus. The values of $D$ are −24.1 and −145.2 N/m at 1 K along the armchair and zigzag directions, respectively. The ultimate stress is about 2.3 N/m at the critical strain of 0.39 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 4.2 N/m at the critical strain of 0.25 in the zigzag direction at the low temperature of 1 K.

Figure 202.

Phonon dispersion for the single-layer p-GeS. (a) The VFF model is fitted to the acoustic branches in the long wave limit along the ΓX direction. The ab initio calculations are from Ref. 81. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for the p-GeS along ΓXSYΓ.

99. P-GeS

Present studies on the puckered (p-) GeS are based on first-principles calculations, and no empirical potential has been proposed for the p-GeS. We will thus parametrize the SW potential for the single-layer p-GeS in this section.

The structure of the single-layer p-GeS is shown in Figure 189 , with M = Ge and X = S. Structural parameters for p-GeS are from the ab initio calculations [78]. There are four atoms in the unit cell with relative coordinates such as $(-u\mathrm{,0,}-v)$ , $(u\mathrm{,0,}v)$ , $(0.5-u\mathrm{,0.5,}v+w)$ , and $(0.5+u\mathrm{,0.5,}-v+w)$ with $u=0.0673$ , $v=0.1173$ , and $w=0.0228$ . The values of these dimensionless parameters are extracted from the geometrical parameters provided in Ref. [78], including lattice constants $a_1=4.492$ Å and $a_2=3.642$ Å, bond lengths $d_{12}=2.462$ Å and $d_{14}=2.423$ Å, and the angle ${\theta }_{145}=94.4°$ . The dimensionless parameters $v$ and $w$ are ratios based on the lattice constant in the out-of-plane z-direction, which is arbitrarily chosen as $a_3=10.0$ Å. We note that the main purpose of the usage of $u$ , $v$ , and $w$ in representing atomic coordinates is to follow the same convention for all puckered structures in the present work. The resultant atomic coordinates are the same as that in Ref. [78].

Figure 203.

Stress-strain relations for the single-layer p-GeS of size $100\times 100$ Å. The single-layer p-GeS is uniaxially stretched along the armchair or zigzag directions at temperatures 1 and 300 K.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{12}{\left(\text{Δ}{r}_{12}\right)}^2$	$\frac{1}{2}{K}_{14}{\left(\text{Δ}{r}_{14}\right)}^2$	$\frac{1}{2}{K}_{123}{\left(\text{Δ}{\theta }_{123}\right)}^2$	$\frac{1}{2}{K}_{134}{\left(\text{Δ}{\theta }_{134}\right)}^2$	$\frac{1}{2}{K}_{415}{\left(\text{Δ}{\theta }_{415}\right)}^2$
Parameter	6.364	6.364	2.153	3.896	3.896
$r_0$ or ${\theta }_0$	2.462	2.423	95.402	94.400	104.837

Table 394.

The VFF model for the single-layer p-GeS.

The second line gives an explicit expression for each VFF term, where atom indexes are from Figure 189(c) . The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

As shown in Figure 189 , a specific feature in the puckered configuration of the p-GeS is that there is a small difference of $w{a}_3$ between the z-coordinate of atom 1 and the z-coordinates of atoms 2 and 3. Similarly, atom 4 is higher than atoms 5 and 6 for $w{a}_3$ along the z-direction. The sign of $w$ determines which types of atoms take the outmost positions, e.g., atoms 1, 5, and 6 are the outmost atoms if $w>0$ in Figure 189(c) , while atoms 2, 3, and 4 will take the outmost positions for $w<0$ .

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
$r_{12}$	2.096	0.351	18.371	0.0	2.926
$r_{14}$	6.694	1.571	17.234	0.0	3.398

Table 395.

Two-body SW potential parameters for the single-layer p-GeS used by GULP [8], as expressed in Eq. (3).

The quantity ( $r_{ij}$ ) in the first line lists one representative term for the two-body SW potential between atoms i and j. Atom indexes are from Figure 189(c) .

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{123}$	4.905	95.402	0.350	0.350	0.0	2.926	0.0	2.926	0.0	4.067
${\theta }_{134}$	20.842	94.400	1.571	0.350	0.0	3.398	0.0	2.926	0.0	4.292
${\theta }_{415}$	22.173	104.837	1.571	0.350	0.0	3.398	0.0	2.296	0.0	4.438

Table 396.

Three-body SW potential parameters for the single-layer p-GeS used by GULP [8], as expressed in Eq. (4).

The first line ( ${\theta }_{ijk}$ ) presents one representative term for the three-body SW potential. The angle ${\theta }_{ijk}$ has the atom i as the apex. Atom indexes are from Figure 189(c) .

Table 394 shows five VFF terms for the single-layer p-GeS; two of which are the bond stretching interactions shown in Eq. (1), while the other three terms are the angle bending interaction shown in Eq. (2). The force constant parameters are the same for the two angle bending terms ${\theta }_{134}$ and ${\theta }_{415}$ , which have the same arm lengths. All force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the ΓX as shown in Figure 202(a) . The ab initio calculations are from Ref. [81]. Figure 202(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion.

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Ge1─S4─S4	1.000	0.351	8.332	4.905	1.000	−0.094	2.096	1227.130	4	0	0.0
Ge1─S1─S1	1.000	1.571	2.163	0.000	1.000	0.000	6.694	2.830	4	0	0.0
Ge1─S1─S4	1.000	0.000	0.000	20.842	1.000	−0.077	0.000	0.000	4	0	0.0
S1─Ge1─Ge2	1.000	0.000	0.000	22.173	1.000	−0.256	0.000	0.000	4	0	0.0

Table 397.

SW potential parameters for p-GeS used by LAMMPS [9], as expressed in Eqs. (9) and (10).

Atom types in the first column are displayed in Figure 190 , with M=Ge and X=S.

The parameters for the two-body SW potential used by GULP are shown in Table 395 . The parameters for the three-body SW potential used by GULP are shown in Table 396 . Parameters for the SW potential used by LAMMPS are listed in Table 397 . Eight atom types have been introduced for writing the SW potential script used by LAMMPS as shown in Figure 190 with M = Ge and X = S, which helps to increase the cutoff for the bond stretching interaction between atom 1 and atom 2 in Figure 189(c) .

Figure 203 shows the stress-strain relations for the single-layer p-GeS of size $100\times 100$ Å. The structure is uniaxially stretched in the armchair or zigzag directions at 1 and 300 K. The Young’s modulus is 10.6 and 32.1 N/m in the armchair and zigzag directions, respectively, at 1 K, which is obtained by linear fitting of the stress-strain relations in [0, 0.01]. The Poisson’s ratios from the VFF model and the SW potential are ${\nu }_{xy}=0.10$ and ${\nu }_{yx}=0.29$ . The third-order nonlinear elastic constant $D$ can be obtained by fitting the stress-strain relation to $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with E as the Young’s modulus. The values of $D$ are −20.4 and −118.8 N/m at 1 K along the armchair and zigzag directions, respectively. The ultimate stress is about 2.4 N/m at the critical strain of 0.39 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 4.2 N/m at the critical strain of 0.24 in the zigzag direction at the low temperature of 1 K.

100. P-SnS

Present studies on the puckered (p-) SnS are based on first-principles calculations, and no empirical potential has been proposed for the p-SnS. We will thus parametrize the SW potential for the single-layer p-SnS in this section.

The structure of the single-layer p-SnS is shown in Figure 189 , with M = Sn and X = S. Structural parameters for p-SnS are from the ab initio calculations [78]. There are four atoms in the unit cell with relative coordinates such as $(-u\mathrm{,0,}-v)$ , $(u\mathrm{,0,}v)$ , $(0.5-u\mathrm{,0.5,}v+w)$ , and $(0.5+u\mathrm{,0.5,}-v+w)$ with $u=0.0426$ , $v=0.1284$ , and $w=0.0308$ . The values of these dimensionless parameters are extracted from the geometrical parameters provided in Ref. [78], including lattice constants $a_1=4.347$ Å and $a_2=4.047$ Å, bond lengths $d_{12}=2.728$ Å and $d_{14}=2.595$ Å, and the angle ${\theta }_{145}=89.0°$ . The dimensionless parameters $v$ and $w$ are ratios based on the lattice constant in the out-of-plane z-direction, which is arbitrarily chosen as $a_3=10.0$ Å. We note that the main purpose of the usage of $u$ , $v$ , and $w$ in representing atomic coordinates is to follow the same convention for all puckered structures in the present work. The resultant atomic coordinates are the same as that in Ref. [78].

Figure 204.

Phonon dispersion for the single-layer p-SnS. (a) The VFF model is fitted to the acoustic branches in the long wave limit along the ΓX direction. The ab initio calculations are from Ref. [81]. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for the p-SnS along ΓXSYΓ.

As shown in Figure 189 , a specific feature in the puckered configuration of the p-SnS is that there is a small difference of $w{a}_3$ between the z-coordinate of atom 1 and the z-coordinates of atoms 2 and 3. Similarly, atom 4 is higher than atoms 5 and 6 for $w{a}_3$ along the z-direction. The sign of $w$ determines which types of atoms take the outmost positions, e.g., atoms 1, 5, and 6 are the outmost atoms if $w>0$ in Figure 189(c) , while atoms 2, 3, and 4 will take the outmost positions for $w<0$ .

Figure 205.

Stress-strain relations for the single-layer p-SnS of size $100\times 100$ Å. The single-layer p-SnS is uniaxially stretched along the armchair or zigzag directions at temperatures 1 and 300 K.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{12}{\left(\text{Δ}{r}_{12}\right)}^2$	$\frac{1}{2}{K}_{14}{\left(\text{Δ}{r}_{14}\right)}^2$	$\frac{1}{2}{K}_{123}{\left(\text{Δ}{\theta }_{123}\right)}^2$	$\frac{1}{2}{K}_{134}{\left(\text{Δ}{\theta }_{134}\right)}^2$	$\frac{1}{2}{K}_{415}{\left(\text{Δ}{\theta }_{415}\right)}^2$
Parameter	4.163	4.163	1.776	5.841	5.841
$r_0$ or ${\theta }_0$	2.728	2.595	95.762	89.000	101.887

Table 398.

The VFF model for the single-layer p-SnS.

The second line gives an explicit expression for each VFF term, where atom indexes are from Figure 189(c) . The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of $\mathrm{\AA }$ ) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

Table 398 shows five VFF terms for the single-layer p-SnS; two of which are the bond stretching interactions shown in Eq. (1), while the other three terms are the angle bending interaction shown in Eq. (2). The force constant parameters are the same for the two angle bending terms ${\theta }_{134}$ and ${\theta }_{415}$ , which have the same arm lengths. All force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the ΓX as shown in Figure 204(a) . The ab initio calculations are from Ref. [81]. Figure 204(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
$r_{12}$	0.782	0.106	27.692	0.0	2.997
$r_{14}$	5.636	1.887	22.674	0.0	3.702

Table 399.

Two-body SW potential parameters for the single-layer p-SnS used by GULP [8], as expressed in Eq. (3).

The quantity ( $r_{ij}$ ) in the first line lists one representative term for the two-body SW potential between atoms i and j. Atom indexes are from Figure 189(c) .

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{123}$	1.968	95.762	0.106	0.106	0.0	2.997	0.0	2.997	0.0	4.197
${\theta }_{134}$	23.839	89.000	1.887	0.106	0.0	3.702	0.0	2.997	0.0	4.366
${\theta }_{415}$	24.888	101.887	1.887	0.106	0.0	3.702	0.0	2.997	0.0	4.566

Table 400.

Three-body SW potential parameters for the single-layer p-SnS used by GULP [8], as expressed in Eq. (4).

The first line ( ${\theta }_{ijk}$ ) presents one representative term for the three-body SW potential. The angle ${\theta }_{ijk}$ has the atom i as the apex. Atom indexes are from Figure 189(c) .

The parameters for the two-body SW potential used by GULP are shown in Table 399 . The parameters for the three-body SW potential used by GULP are shown in Table 400 . Parameters for the SW potential used by LAMMPS are listed in Table 401 . Eight atom types have been introduced for writing the SW potential script used by LAMMPS as shown in Figure 190 with M = Sn and X = S, which helps to increase the cutoff for the bond stretching interaction between atom 1 and atom 2 in Figure 189(c) .

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Sn1─S4─S4	1.000	0.106	28.347	1.971	1.000	−0.100	0.783	221784.222	4	0	0.0
Sn1─S1─S1	1.000	1.887	1.961	0.000	1.000	0.000	5.636	1.787	4	0	0.0
Sn1─S1─S4	1.000	0.000	0.000	23.839	1.000	0.017	0.000	0.000	4	0	0.0
S1─Sn1─Sn2	1.000	0.000	0.000	24.888	1.000	−0.206	0.000	0.000	4	0	0.0

Table 401.

SW potential parameters for p-SnS used by LAMMPS [9], as expressed in Eqs. (9) and (10).

Atom types in the first column are displayed in Figure 190 , with M=Sn and X=S.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{12}{\left(\text{Δ}{r}_{12}\right)}^2$	$\frac{1}{2}{K}_{14}{\left(\text{Δ}{r}_{14}\right)}^2$	$\frac{1}{2}{K}_{123}{\left(\text{Δ}{\theta }_{123}\right)}^2$	$\frac{1}{2}{K}_{134}{\left(\text{Δ}{\theta }_{134}\right)}^2$	$\frac{1}{2}{K}_{415}{\left(\text{Δ}{\theta }_{415}\right)}^2$
Parameter	10.120	10.120	4.505	3.910	3.910
$r_0$ or ${\theta }_0$	1.961	2.014	101.354	113.000	100.563

Table 402.

The VFF model for the single-layer p-CSe.

The second line gives an explicit expression for each VFF term, where atom indexes are from Figure 189(c) . The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of $\mathrm{\AA }$ ) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

Figure 205 shows the stress-strain relations for the single-layer p-SnS of size $100\times 100$ Å. The structure is uniaxially stretched in the armchair or zigzag directions at 1 and 300 K. The Young’s modulus is 9.6 and 24.5 N/m in the armchair and zigzag directions, respectively, at 1 K, which is obtained by linear fitting of the stress-strain relations in [0, 0.01]. The Poisson’s ratios from the VFF model and the SW potential are ${\nu }_{xy}=0.18$ and ${\nu }_{yx}=0.47$ . The third-order nonlinear elastic constant $D$ can be obtained by fitting the stress-strain relation to $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with E as the Young’s modulus. The values of $D$ are −14.7 and −80.3 N/m at 1 K along the armchair and zigzag directions, respectively. The ultimate stress is about 2.3 N/m at the critical strain of 0.36 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 3.1 N/m at the critical strain of 0.20 in the zigzag direction at the low temperature of 1 K.

101. P-CSe

Present studies on the puckered (p-) CSe are based on first-principles calculations, and no empirical potential has been proposed for the p-CSe. We will thus parametrize the SW potential for the single-layer p-CSe in this section.

The structure of the single-layer p-CSe is shown in Figure 189 , with M = C and X = Se. Structural parameters for p-CSe are from the ab initio calculations [78]. There are four atoms in the unit cell with relative coordinates such as $(-u\mathrm{,0,}-v)$ , $(u\mathrm{,0,}v)$ , $(0.5-u\mathrm{,0.5,}v+w)$ , and $(0.5+u\mathrm{,0.5,}-v+w)$ with $u=0.1079$ , $v=0.0894$ , and $w=\mathrm{}-0.0229$ . The values of these dimensionless parameters are extracted from the geometrical parameters provided in Ref. [78], including lattice constants $a_1=4.299$ Å and $a_2=3.034$ Å, bond lengths $d_{12}=1.961$ Å and $d_{14}=2.014$ Å, and the angle ${\theta }_{145}=113.0°$ . The dimensionless parameters $v$ and $w$ are ratios based on the lattice constant in the out-of-plane z-direction, which is arbitrarily chosen as $a_3=10.0$ Å. We note that the main purpose of the usage of $u$ , $v$ , and $w$ in representing atomic coordinates is to follow the same convention for all puckered structures in the present work. The resultant atomic coordinates are the same as that in Ref. [78].

Figure 206.

Phonon dispersion for the single-layer p-CSe. (a) The VFF model is fitted to the acoustic branches in the long wave limit along the ΓX direction. The ab initio calculations are calculated from SIESTA. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for the p-CSe along ΓXSYΓ.

As shown in Figure 189 , a specific feature in the puckered configuration of the p-CSe is that there is a small difference of $w{a}_3$ between the z-coordinate of atom 1 and the z-coordinates of atoms 2 and 3. Similarly, atom 4 is higher than atoms 5 and 6 for $w{a}_3$ along the z-direction. The sign of $w$ determines which types of atoms take the outmost positions, e.g., atoms 1, 5, and 6 are the outmost atoms if $w>0$ in Figure 189(c) , while atoms 2, 3, and 4 will take the outmost positions for $w<0$ .

Figure 207.

Stress-strain relations for the single-layer p-CSe of size $100\times 100$ Å. The single-layer p-CSe is uniaxially stretched along the armchair or zigzag directions at temperatures 1 and 300 K.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
$r_{12}$	6.141	1.114	7.394	0.0	2.700
$r_{14}$	7.411	1.316	8.226	0.0	2.828

Table 403.

Two-body SW potential parameters for the single-layer p-CSe used by GULP [8], as expressed in Eq. (3).

The quantity ( $r_{ij}$ ) in the first line lists one representative term for the two-body SW potential between atoms i and j. Atom indexes are from Figure 189(c) .

Table 402 shows five VFF terms for the single-layer p-CSe; two of which are the bond stretching interactions shown in Eq. (1), while the other three terms are the angle bending interaction shown in Eq. (2). The force constant parameters are the same for the two angle bending terms ${\theta }_{134}$ and ${\theta }_{415}$ , which have the same arm lengths. All force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the ΓX as shown in Figure 206(a) . The ab initio calculations for the phonon dispersion are calculated from the SIESTA package [79]. The generalized gradients approximation is applied to account for the exchange-correlation function with Perdew, Burke, and Ernzerhof parameterization [80], and the double- $\zeta$ orbital basis set is adopted. Figure 206(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion.

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{123}$	47.768	101.354	1.114	1.114	0.0	2.700	0.0	2.700	0.0	3.667
${\theta }_{134}$	52.464	113.000	1.316	1.114	0.0	2.828	0.0	2.700	0.0	4.157
${\theta }_{415}$	46.000	100.563	1.316	1.114	0.0	2.828	0.0	2.700	0.0	4.032

Table 404.

Three-body SW potential parameters for the single-layer p-CSe used by GULP [8], as expressed in Eq. (4).

The first line ( ${\theta }_{ijk}$ ) presents one representative term for the three-body SW potential. The angle ${\theta }_{ijk}$ has the atom i as the apex. Atom indexes are from Figure 189(c) .

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
C1─Se4─Se4	1.000	1.114	2.424	47.768	1.000	−0.197	6.141	4.802	4	0	0.0
C1─Se1─Se1	1.000	1.316	2.149	0.000	1.000	0.000	7.411	2.743	4	0	0.0
C1─Se1─Se4	1.000	0.000	0.000	52.464	1.000	−0.391	0.000	0.000	4	0	0.0
Se1─C1─C2	1.000	0.000	0.000	46.000	1.000	−0.183	0.000	0.000	4	0	0.0

Table 405.

SW potential parameters for p-CSe used by LAMMPS [9], as expressed in Eqs. (9) and (10).

Atom types in the first column are displayed in Figure 190 , with M=C and X=Se.

The parameters for the two-body SW potential used by GULP are shown in Table 403 . The parameters for the three-body SW potential used by GULP are shown in Table 404 . Parameters for the SW potential used by LAMMPS are listed in Table 405 . Eight atom types have been introduced for writing the SW potential script used by LAMMPS as shown in Figure 190 with M = C and X = Se, which helps to increase the cutoff for the bond stretching interaction between atom 1 and atom 2 in Figure 189(c) .

Figure 207 shows the stress-strain relations for the single-layer p-CSe of size $100\times 100$ Å. The structure is uniaxially stretched in the armchair or zigzag directions at 1 and 300 K. The Young’s modulus is 17.2 and 75.4 N/m in the armchair and zigzag directions, respectively, at 1 K, which is obtained by linear fitting of the stress-strain relations in [0, 0.01]. The Poisson’s ratios from the VFF model and the SW potential are ${\nu }_{xy}=\mathrm{}-0.02$ and ${\nu }_{yx}=\mathrm{}-0.11$ . The third-order nonlinear elastic constant $D$ can be obtained by fitting the stress-strain relation to $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with E as the Young’s modulus. The values of $D$ are −46.3 and −442.0 N/m at 1 K along the armchair and zigzag directions, respectively. The ultimate stress is about 3.0 N/m at the critical strain of 0.31 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 6.5 N/m at the critical strain of 0.20 in the zigzag direction at the low temperature of 1 K.

Figure 208.

Phonon dispersion for the single-layer p-SiSe. (a) The VFF model is fitted to the acoustic branches in the long wave limit along the ΓX direction. The ab initio calculations are calculated from SIESTA. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for the p-SiSe along ΓXSYΓ.

102. P-SiSe

Present studies on the puckered (p-) SiSe are based on first-principles calculations, and no empirical potential has been proposed for the p-SiSe. We will thus parametrize the SW potential for the single-layer p-SiSe in this section.

Figure 209.

Stress-strain relations for the single-layer p-SiSe of size $100\times 100$ Å. The single-layer p-SiSe is uniaxially stretched along the armchair or zigzag directions at temperatures 1 and 300 K.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{12}{\left(\text{Δ}{r}_{12}\right)}^2$	$\frac{1}{2}{K}_{14}{\left(\text{Δ}{r}_{14}\right)}^2$	$\frac{1}{2}{K}_{123}{\left(\text{Δ}{\theta }_{123}\right)}^2$	$\frac{1}{2}{K}_{134}{\left(\text{Δ}{\theta }_{134}\right)}^2$	$\frac{1}{2}{K}_{415}{\left(\text{Δ}{\theta }_{415}\right)}^2$
Parameter	7.192	7.192	3.222	5.890	5.890
$r_0$ or ${\theta }_0$	2.524	2.448	95.513	98.200	97.686

Table 406.

The VFF model for the single-layer p-SiSe.

The second line gives an explicit expression for each VFF term, where atom indexes are from Figure 189(c) . The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

The structure of the single-layer p-SiSe is shown in Figure 189 , with M = Si and X = Se. Structural parameters for p-SiSe are from the ab initio calculations [78]. There are four atoms in the unit cell with relative coordinates such as $(-u\mathrm{,0,}-v)$ , $(u\mathrm{,0,}v)$ , $(0.5-u\mathrm{,0.5,}v+w)$ , and $(0.5+u\mathrm{,0.5,}-v+w)$ with $u=0.0572$ , $v=0.1198$ , and $w=\mathrm{}-0.0011$ . The values of these dimensionless parameters are extracted from the geometrical parameters provided in Ref. [78], including lattice constants $a_1=4.400$ Å and $a_2=3.737$ Å, bond lengths $d_{12}=2.524$ Å and $d_{14}=2.448$ Å, and the angle ${\theta }_{145}=98.2°$ . The dimensionless parameters $v$ and $w$ are ratios based on the lattice constant in the out-of-plane z-direction, which is arbitrarily chosen as $a_3=10.0$ Å. We note that the main purpose of the usage of $u$ , $v$ , and $w$ in representing atomic coordinates is to follow the same convention for all puckered structures in the present work. The resultant atomic coordinates are the same as that in Ref. [78].

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
$r_{12}$	1.883	0.230	20.292	0.0	2.905
$r_{14}$	8.098	1.665	17.956	0.0	3.457

Table 407.

Two-body SW potential parameters for the single-layer p-SiSe used by GULP [8], as expressed in Eq. (3).

The quantity ( $r_{ij}$ ) in the first line lists one representative term for the two-body SW potential between atoms i and j. Atom indexes are from Figure 189(c) .

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{123}$	5.440	95.513	0.230	0.230	0.0	2.905	0.0	2.905	0.0	4.069
${\theta }_{134}$	28.616	98.200	1.665	0.230	0.0	3.457	0.0	2.905	0.0	4.379
${\theta }_{415}$	28.545	97.686	1.665	0.230	0.0	3.457	0.0	2.905	0.0	4.369

Table 408.

Three-body SW potential parameters for the single-layer p-SiSe used by GULP [8], as expressed in Eq. (4).

The first line ( ${\theta }_{ijk}$ ) presents one representative term for the three-body SW potential. The angle ${\theta }_{ijk}$ has the atom i as the apex. Atom indexes are from Figure 189(c) .

As shown in Figure 189 , a specific feature in the puckered configuration of the p-SiSe is that there is a small difference of $w{a}_3$ between the z-coordinate of atom 1 and the z-coordinates of atoms 2 and 3. Similarly, atom 4 is higher than atoms 5 and 6 for $w{a}_3$ along the z-direction. The sign of $w$ determines which types of atoms take the outmost positions, e.g., atoms 1, 5, and 6 are the outmost atoms if $w>0$ in Figure 189(c) , while atoms 2, 3, and 4 will take the outmost positions for $w<0$ .

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Si1─Se4─Se4	1.000	0.230	12.628	5.440	1.000	−0.096	1.883	7245.111	4	0	0.0
Si1─Se1─Se1	1.000	1.665	2.076	0.000	1.000	0.000	8.098	2.335	4	0	0.0
Si1─Se1─Se4	1.000	0.000	0.000	28.616	1.000	−0.143	0.000	0.000	4	0	0.0
Se1─Si1─Si2	1.000	0.000	0.000	28.545	1.000	−0.134	0.000	0.000	4	0	0.0

Table 409.

SW potential parameters for p-SiSe used by LAMMPS [9], as expressed in Eqs. (9) and (10).

Atom types in the first column are displayed in Figure 190 , with M=Si and X=Se.

Table 406 shows five VFF terms for the single-layer p-SiSe; two of which are the bond stretching interactions shown in Eq. (1), while the other three terms are the angle bending interaction shown in Eq. (2). The force constant parameters are the same for the two angle bending terms ${\theta }_{134}$ and ${\theta }_{415}$ , which have the same arm lengths. All force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the ΓX as shown in Figure 208(a) . The ab initio calculations for the phonon dispersion are calculated from the SIESTA package [79]. The generalized gradients approximation is applied to account for the exchange-correlation function with Perdew, Burke, and Ernzerhof parameterization [80], and the double- $\zeta$ orbital basis set is adopted. Figure 208(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion.

The parameters for the two-body SW potential used by GULP are shown in Table 407 . The parameters for the three-body SW potential used by GULP are shown in Table 408 . Parameters for the SW potential used by LAMMPS are listed in Table 409 . Eight atom types have been introduced for writing the SW potential script used by LAMMPS as shown in Figure 190 with M=Si and X=Se, which helps to increase the cutoff for the bond stretching interaction between atom 1 and atom 2 in Figure 189(c) .

Figure 209 shows the stress-strain relations for the single-layer p-SiSe of size $100\times 100$ Å. The structure is uniaxially stretched in the armchair or zigzag directions at 1 and 300 K. The Young’s modulus is 14.4 and 44.6 N/m in the armchair and zigzag directions, respectively, at 1 K, which is obtained by linear fitting of the stress-strain relations in [0, 0.01]. The Poisson’s ratios from the VFF model and the SW potential are ${\nu }_{xy}=0.09$ and ${\nu }_{yx}=0.30$ . The third-order nonlinear elastic constant $D$ can be obtained by fitting the stress-strain relation to $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with E as the Young’s modulus. The values of $D$ are −28.8 and −176.6 N/m at 1 K along the armchair and zigzag directions, respectively. The ultimate stress is about 3.1 N/m at the critical strain of 0.37 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 5.3 N/m at the critical strain of 0.21 in the zigzag direction at the low temperature of 1 K.

Figure 210.

Phonon dispersion for the single-layer p-GeSe. (a) The VFF model is fitted to the acoustic branches in the long wave limit along the ΓX direction. The ab initio calculations are from Ref. [81]. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for the p-GeSe along ΓXSYΓ.

103. P-GeSe

Present studies on the puckered (p-) GeSe are based on first-principles calculations, and no empirical potential has been proposed for the p-GeSe. We will thus parametrize the SW potential for the single-layer p-GeSe in this section.

The structure of the single-layer p-GeSe is shown in Figure 189 , with M = Ge and X = Se. Structural parameters for p-GeSe are from the ab initio calculations [78]. There are four atoms in the unit cell with relative coordinates such as $(-u\mathrm{,0,}-v)$ , $(u\mathrm{,0,}v)$ , $(0.5-u\mathrm{,0.5,}v+w)$ , and $(0.5+u\mathrm{,0.5,}-v+w)$ with $u=0.0439$ , $v=0.1258$ , and $w=\mathrm{}-0.0080$ . The values of these dimensionless parameters are extracted from the geometrical parameters provided in Ref. [78], including lattice constants $a_1=4.302$ Å and $a_2=3.965$ Å, bond lengths $d_{12}=2.661$ Å and $d_{14}=2.544$ Å, and the angle ${\theta }_{145}=97.4°$ . The dimensionless parameters $v$ and $w$ are ratios based on the lattice constant in the out-of-plane z-direction, which is arbitrarily chosen as $a_3=10.0$ Å. We note that the main purpose of the usage of $u$ , $v$ , and $w$ in representing atomic coordinates is to follow the same convention for all puckered structures in the present work. The resultant atomic coordinates are the same as that in Ref. [78].

Figure 211.

Stress-strain relations for the single-layer p-GeSe of size $100\times 100$ Å. The single-layer p-GeSe is uniaxially stretched along the armchair or zigzag directions at temperatures 1 and 300 K.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_{12}{\left(\text{Δ}{r}_{12}\right)}^2$	$\frac{1}{2}{K}_{14}{\left(\text{Δ}{r}_{14}\right)}^2$	$\frac{1}{2}{K}_{123}{\left(\text{Δ}{\theta }_{123}\right)}^2$	$\frac{1}{2}{K}_{134}{\left(\text{Δ}{\theta }_{134}\right)}^2$	$\frac{1}{2}{K}_{415}{\left(\text{Δ}{\theta }_{415}\right)}^2$
Parameter	5.063	5.063	2.249	5.927	5.927
$r_0$ or ${\theta }_0$	2.661	2.544	96.322	97.000	93.964

Table 410.

The VFF model for the single-layer p-GeSe.

The second line gives an explicit expression for each VFF term, where atom indexes are from Figure 189(c) . The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

As shown in Figure 189 , a specific feature in the puckered configuration of the p-GeSe is that there is a small difference of $w{a}_3$ between the z-coordinate of atom 1 and the z-coordinates of atoms 2 and 3. Similarly, atom 4 is higher than atoms 5 and 6 for $w{a}_3$ along the z-direction. The sign of $w$ determines which types of atoms take the outmost positions, e.g., atoms 1, 5, and 6 are the outmost atoms if $w>0$ in Figure 189(c) , while atoms 2, 3, and 4 will take the outmost positions for $w<0$ .

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
$r_{12}$	0.962	0.115	25.070	0.0	2.938
$r_{14}$	6.572	1.846	20.943	0.0	3.628

Table 411.

Two-body SW potential parameters for the single-layer p-GeSe used by GULP [8], as expressed in Eq. (3).

The quantity ( $r_{ij}$ ) in the first line lists one representative term for the two-body SW potential between atoms i and j. Atom indexes are from Figure 189(c) .

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{123}$	2.614	96.322	0.115	0.115	0.0	2.938	0.0	2.938	0.0	4.133
${\theta }_{134}$	25.087	97.400	1.846	0.115	0.0	3.628	0.0	2.938	0.0	4.455
${\theta }_{415}$	24.789	93.964	1.846	0.155	0.0	3.628	0.0	2.938	0.0	4.404

Table 412.

Three-body SW potential parameters for the single-layer p-GeSe used by GULP [8], as expressed in Eq. (4).

The first line ( ${\theta }_{ijk}$ ) presents one representative term for the three-body SW potential. The angle ${\theta }_{ijk}$ has the atom i as the apex. Atom indexes are from Figure 189(c) .

Table 410 shows five VFF terms for the single-layer p-GeSe; two of which are the bond stretching interactions shown in Eq. (1), while the other three terms are the angle bending interaction shown in Eq. (2). The force constant parameters are the same for the two angle bending terms ${\theta }_{134}$ and ${\theta }_{415}$ , which have the same arm lengths. All force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the ΓX as shown in Figure 210(a) . The ab initio calculations are from Ref. [81]. Figure 210(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion.

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Ge1─Se4─Se4	1.000	0.115	25.561	2.614	1.000	−0.110	0.962	143723.555	4	0	0.0
Ge1─Se1─Se1	1.000	1.846	1.965	0.000	1.000	0.000	6.572	1.804	4	0	0.0
Ge1─Se1─Se4	1.000	0.000	0.000	25.087	1.000	−0.129	0.000	0.000	4	0	0.0
Se1─Ge1─Ge2	1.000	0.000	0.000	24.789	1.000	−0.069	0.000	0.000	4	0	0.0

Table 413.

SW potential parameters for p-GeSe used by LAMMPS [9], as expressed in Eqs. (9) and (10).

Atom types in the first column are displayed in Figure 190 , with M=Ge and X=Se.

The parameters for the two-body SW potential used by GULP are shown in Table 411 . The parameters for the three-body SW potential used by GULP are shown in Table 412 . Parameters for the SW potential used by LAMMPS are listed in Table 413 . Eight atom types have been introduced for writing the SW potential script used by LAMMPS as shown in Figure 190 with M = Ge and X = Se, which helps to increase the cutoff for the bond stretching interaction between atom 1 and atom 2 in Figure 189(c) .

Figure 211 shows the stress-strain relations for the single-layer p-GeSe of size $100\times 100$ Å. The structure is uniaxially stretched in the armchair or zigzag directions at 1 and 300 K. The Young’s modulus is 11.1 and 32.0 N/m in the armchair and zigzag directions, respectively, at 1 K, which is obtained by linear fitting of the stress-strain relations in [0, 0.01]. The Poisson’s ratios from the VFF model and the SW potential are ${\nu }_{xy}=0.14$ and ${\nu }_{yx}=0.42$ . The third-order nonlinear elastic constant $D$ can be obtained by fitting the stress-strain relation to $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with E as the Young’s modulus. The values of $D$ are −19.3 and −114.7 N/m at 1 K along the armchair and zigzag directions, respectively. The ultimate stress is about 2.6 N/m at the critical strain of 0.36 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 3.9 N/m at the critical strain of 0.20 in the zigzag direction at the low temperature of 1 K.

104. P-SnSe

Present studies on the puckered (p-) SnSe are based on first-principles calculations, and no empirical potential has been proposed for the p-SnSe. We will thus parametrize the SW potential for the single-layer p-SnSe in this section.

The structure of the single-layer p-SnSe is shown in Figure 189 , with M = Sn and X = Se. Structural parameters for p-SnSe are from the ab initio calculations [78]. There are four atoms in the unit cell with relative coordinates such as $(-u\mathrm{,0,}-v)$ , $(u\mathrm{,0,}v)$ , $(0.5-u\mathrm{,0.5,}v+w)$ , and $(0.5+u\mathrm{,0.5,}-v+w)$ with $u=0.0313$ , $v=0.1358$ , and $w=0.0074$ . The values of these dimensionless parameters are extracted from the geometrical parameters provided in Ref. [78], including lattice constants $a_1=4.453$ Å and $a_2=4.260$ Å, bond lengths $d_{12}=2.887$ Å and $d_{14}=2.730$ Å, and the angle ${\theta }_{145}=92.5°$ . The dimensionless parameters $v$ and $w$ are ratios based on the lattice constant in the out-of-plane z-direction, which is arbitrarily chosen as $a_3=10.0$ Å. We note that the main purpose of the usage of $u$ , $v$ , and $w$ in representing atomic coordinates is to follow the same convention for all puckered structures in the present work. The resultant atomic coordinates are the same as that in Ref. [78].

Figure 212.

Phonon dispersion for the single-layer p-SnSe. (a) The VFF model is fitted to the acoustic branches in the long wave limit along the ΓX direction. The ab initio calculations are from Ref. [82]. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for the p-SnSe along ΓXSYΓ.

As shown in Figure 189 , a specific feature in the puckered configuration of the p-SnSe is that there is a small difference of $w{a}_3$ between the z-coordinate of atom 1 and the z-coordinates of atoms 2 and 3. Similarly, atom 4 is higher than atoms 5 and 6 for $w{a}_3$ along the z-direction. The sign of $w$ determines which types of atoms take the outmost positions, e.g., atoms 1, 5, and 6 are the outmost atoms if $w>0$ in Figure 189(c) , while atoms 2, 3, and 4 will take the outmost positions for $w<0$ .

Figure 213.

Stress-strain relations for the single-layer p-SnSe of size $100\times 100$ Å. The single-layer p-SnSe is uniaxially stretched along the armchair or zigzag directions at temperatures 1 and 300 K.

VFF type	Bond stretching		Angle bending
Expression	$\frac{1}{2}{K}_{12}{\left(\text{Δ}{r}_{12}\right)}^2$	$\frac{1}{2}{K}_{14}{\left(\text{Δ}{r}_{14}\right)}^2$	$\frac{1}{2}{K}_{123}{\left(\text{Δ}{\theta }_{123}\right)}^2$	$\frac{1}{2}{K}_{134}{\left(\text{Δ}{\theta }_{134}\right)}^2$	$\frac{1}{2}{K}_{415}{\left(\text{Δ}{\theta }_{415}\right)}^2$
Parameter	3.872	3.872	3.157	7.674	7.674
$r_0$ or ${\theta }_0$	2.887	2.730	95.087	92.500	95.411

Table 414.

The VFF model for the single-layer p-SnSe.

The second line gives an explicit expression for each VFF term, where atom indexes are from Figure 189(c) . The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

Table 414 shows five VFF terms for the single-layer p-SnSe; two of which are the bond stretching interactions shown in Eq. (1), while the other three terms are the angle bending interaction shown in Eq. (2). The force constant parameters are the same for the two angle bending terms ${\theta }_{134}$ and ${\theta }_{415}$ , which have the same arm lengths. All force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the ΓX as shown in Figure 212(a) . The ab initio calculations are from Ref. [82]. Figure 212(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
$r_{12}$	0.565	0.056	34.734	0.0	3.088
$r_{14}$	5.811	1.989	27.773	0.0	3.895

Table 415.

Two-body SW potential parameters for the single-layer p-SnSe used by GULP [8], as expressed in Eq. (3).

The quantity $\left(r_{ij}\right)$ in the first line lists one representative term for the two-body SW potential between atoms i and j. Atom indexes are from Figure 189(c) .

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{123}$	2.777	95.087	0.056	0.056	0.0	3.088	0.0	3.088	0.0	4.357
${\theta }_{134}$	27.996	92.500	1.989	0.056	0.0	3.895	0.0	3.088	0.0	4.530
${\theta }_{415}$	28.193	95.411	1.989	0.056	0.0	3.895	0.0	3.088	0.0	4.576

Table 416.

Three-body SW potential parameters for the single-layer p-SnSe used by GULP [8], as expressed in Eq. (4).

The first line ( ${\theta }_{ijk}$ ) presents one representative term for the three-body SW potential. The angle ${\theta }_{ijk}$ has the atom i as the apex. Atom indexes are from Figure 189(c) .

The parameters for the two-body SW potential used by GULP are shown in Table 415 . The parameters for the three-body SW potential used by GULP are shown in Table 416 . Parameters for the SW potential used by LAMMPS are listed in Table 417 . Eight atom types have been introduced for writing the SW potential script used by LAMMPS as shown in Figure 190 with M = Sn and X = Se, which helps to increase the cutoff for the bond stretching interaction between atom 1 and atom 2 in Figure 189(c) .

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Sn1─Se4─Se4	1.000	0.056	55.166	2.777	1.000	−0.089	0.565	3537820.961	4	0	0.0
Sn1─Se1─Se1	1.000	1.989	1.959	0.000	1.000	0.000	5.811	1.776	4	0	0.0
Sn1─Se1─Se4	1.000	0.000	0.000	27.996	1.000	−0.044	0.000	0.000	4	0	0.0
Se1─Sn1─Sn2	1.000	0.000	0.000	28.193	1.000	−0.094	0.000	0.000	4	0	0.0

Table 417.

SW potential parameters for p-SnSe used by LAMMPS [9], as expressed in Eqs. (9) and (10).

Atom types in the first column are displayed in Figure 190 , with M = Sn and X = Se.

VFF type	Bond stretching		Angle bending
Expression	$\frac{1}{2}{K}_{12}{\left(\text{Δ}{r}_{12}\right)}^2$	$\frac{1}{2}{K}_{14}{\left(\text{Δ}{r}_{14}\right)}^2$	$\frac{1}{2}{K}_{123}{\left(\text{Δ}{\theta }_{123}\right)}^2$	$\frac{1}{2}{K}_{134}{\left(\text{Δ}{\theta }_{134}\right)}^2$	$\frac{1}{2}{K}_{415}{\left(\text{Δ}{\theta }_{415}\right)}^2$
Parameter	13.235	13.235	4.210	3.901	3.901
$r_0$ or ${\theta }_0$	2.164	2.181	103.122	110.000	90.854

Table 418.

The VFF model for the single-layer p-CTe.

The second line gives an explicit expression for each VFF term, where atom indexes are from Figure 189(c) . The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

Figure 213 shows the stress-strain relations for the single-layer p-SnSe of size $100\times 100$ Å. The structure is uniaxially stretched in the armchair or zigzag directions at 1 and 300 K. The Young’s modulus is 11.4 and 34.1 Nm⁻¹ in the armchair and zigzag directions, respectively, at 1 K, which is obtained by linear fitting of the stress-strain relations in [0, 0.01]. The Poisson’s ratios from the VFF model and the SW potential are ${\nu }_{xy}=0.11$ and ${\nu }_{yx}=0.33$ . The third-order nonlinear elastic constant $D$ can be obtained by fitting the stress-strain relation to $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with E as the Young’s modulus. The values of $D$ are −22.0 and −128.8 Nm⁻¹ at 1 K along the armchair and zigzag directions, respectively. The ultimate stress is about 2.3 Nm⁻¹ at the critical strain of 0.32 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 3.3 Nm⁻¹ at the critical strain of 0.15 in the zigzag direction at the low temperature of 1 K.

105. P-CTe

Present studies on the puckered (p-) CTe are based on first-principles calculations, and no empirical potential has been proposed for the p-CTe. We will thus parametrize the SW potential for the single-layer p-CTe in this section.

The structure of the single-layer p-CTe is shown in Figure 189 , with M = C and X = Te. Structural parameters for p-CTe are from the ab initio calculations [78]. There are four atoms in the unit cell with relative coordinates such as $(-u\mathrm{,0,}-v)$ , $(u\mathrm{,0,}v)$ , $(0.5-u\mathrm{,0.5,}v+w)$ , and $(0.5+u\mathrm{,0.5,}-v+w)$ with $u=0.0837$ , $v=0.1041$ , and $w=\mathrm{}-0.0371$ . The values of these dimensionless parameters are extracted from the geometrical parameters provided in Ref. [78], including lattice constants $a_1=3.889$ Å and $a_2=3.390$ Å, bond lengths $d_{12}=2.164$ Å and $d_{14}=2.181$ Å, and the angle ${\theta }_{145}=110.0°$ . The dimensionless parameters $v$ and $w$ are ratios based on the lattice constant in the out-of-plane z-direction, which is arbitrarily chosen as $a_3=10.0$ Å. We note that the main purpose of the usage of $u$ , $v$ , and $w$ in representing atomic coordinates is to follow the same convention for all puckered structures in the present work. The resultant atomic coordinates are the same as that in Ref. [78].

Figure 214.

Phonon dispersion for the single-layer p-CTe. (a) The VFF model is fitted to the acoustic branches in the long wave limit along the ΓX direction. The ab initio calculations are calculated from SIESTA. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for the p-CTe along ΓXSYΓ.

As shown in Figure 189 , a specific feature in the puckered configuration of the p-CTe is that there is a small difference of $w{a}_3$ between the z-coordinate of atom 1 and the z-coordinates of atoms 2 and 3. Similarly, atom 4 is higher than atoms 5 and 6 for $w{a}_3$ along the z-direction. The sign of $w$ determines which types of atoms take the outmost positions, e.g., atoms 1, 5, and 6 are the outmost atoms if $w>0$ in Figure 189(c) , while atoms 2, 3, and 4 will take the outmost positions for $w<0$ .

Figure 215.

Stress-strain relations for the single-layer p-CTe of size $100\times 100$ Å. The single-layer p-CTe is uniaxially stretched along the armchair or zigzag directions at temperatures 1 and 300 K.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
$r_{12}$	4.165	0.424	10.965	0.0	2.643
$r_{14}$	12.519	1.569	11.313	0.0	3.106

Table 419.

Two-body SW potential parameters for the single-layer p-CTe used by GULP [8], as expressed in Eq. (3).

The quantity $\left(r_{ij}\right)$ in the first line lists one representative term for the two-body SW potential between atoms i and j. Atom indexes are from Figure 189(c) .

Table 418 shows five VFF terms for the single-layer p-CTe; two of which are the bond stretching interactions shown in Eq. (1), while the other three terms are the angle bending interaction shown in Eq. (2). The force constant parameters are the same for the two angle bending terms ${\theta }_{134}$ and ${\theta }_{415}$ , which have the same arm lengths. All force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the ΓX as shown in Figure 214(a) . The ab initio calculations for the phonon dispersion are calculated from the SIESTA package [79]. The generalized gradients approximation is applied to account for the exchange-correlation function with Perdew et al. parameterization [80], and the double- $\zeta$ orbital basis set is adopted. Figure 214(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion.

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{123}$	13.040	103.122	0.424	0.424	0.0	2.643	0.0	2.643	0.0	3.639
${\theta }_{134}$	29.206	110.000	1.569	0.424	0.0	3.106	0.0	2.643	0.0	4.280
${\theta }_{415}$	25.795	90.854	1.569	0.424	0.0	3.106	0.0	2.643	0.0	4.043

Table 420.

Three-body SW potential parameters for the single-layer p-CTe used by GULP [8], as expressed in Eq. (4).

The first line $\left({\theta }_{ijk}\right)$ presents one representative term for the three-body SW potential. The angle ${\theta }_{ijk}$ has the atom i as the apex. Atom indexes are from Figure 189(c) .

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
C1─Te4─Te4	1.000	0.424	6.232	13.040	1.000	−0.227	4.165	338.925	4	0	0.0
C1─Te1─Te1	1.000	1.569	1.979	0.000	1.000	0.000	12.519	1.866	4	0	0.0
C1─Te1─Te4	1.000	0.000	0.000	29.206	1.000	−0.342	0.000	0.000	4	0	0.0
Te1─C1─C2	1.000	0.000	0.000	25.795	1.000	−0.015	0.000	0.000	4	0	0.0

Table 421.

SW potential parameters for p-CTe used by LAMMPS [9], as expressed in Eqs. (9) and (10).

Atom types in the first column are displayed in Figure 190 , with M = C and X = Te.

The parameters for the two-body SW potential used by GULP are shown in Table 419 . The parameters for the three-body SW potential used by GULP are shown in Table 420 . Parameters for the SW potential used by LAMMPS are listed in Table 421 . Eight atom types have been introduced for writing the SW potential script used by LAMMPS as shown in Figure 190 with M = C and X = Te, which helps to increase the cutoff for the bond stretching interaction between atom 1 and 2 in Figure 189(c) .

Figure 215 shows the stress-strain relations for the single-layer p-CTe of size $100\times 100$ Å. The structure is uniaxially stretched in the armchair or zigzag directions at 1 and 300 K. The Young’s modulus is 10.8 and 89.1 Nm⁻¹ in the armchair and zigzag directions, respectively, at 1 K, which is obtained by linear fitting of the stress-strain relations in [0, 0.01]. The Poisson’s ratios from the VFF model and the SW potential are ${\nu }_{xy}=0.02$ and ${\nu }_{yx}=0.20$ . The third-order nonlinear elastic constant $D$ can be obtained by fitting the stress-strain relation to $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with E as the Young’s modulus. The values of $D$ are −15.3 and −419.6 Nm⁻¹ at 1 K along the armchair and zigzag directions, respectively. The ultimate stress is about 3.0 Nm⁻¹ at the critical strain of 0.43 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 9.4 Nm⁻¹ at the critical strain of 0.21 in the zigzag direction at the low temperature of 1 K.

Figure 216.

Phonon dispersion for the single-layer p-SiTe. (a) The VFF model is fitted to the acoustic branches in the long wave limit along the ΓX direction. The ab initio calculations are from Ref. [83]. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for the p-SiTe along ΓXSYΓ.

106. P-SiTe

Present studies on the puckered (p-) SiTe are based on first-principles calculations, and no empirical potential has been proposed for the p-SiTe. We will thus parametrize the SW potential for the single-layer p-SiTe in this section.

Figure 217.

Stress-strain relations for the single-layer p-SiTe of size $100\times 100$ Å. The single-layer p-SiTe is uniaxially stretched along the armchair or zigzag directions at temperatures 1 and 300 K.

VFF type	Bond stretching		Angle bending
Expression	$\frac{1}{2}{K}_{12}{\left(\text{Δ}{r}_{12}\right)}^2$	$\frac{1}{2}{K}_{14}{\left(\text{Δ}{r}_{14}\right)}^2$	$\frac{1}{2}{K}_{123}{\left(\text{Δ}{\theta }_{123}\right)}^2$	$\frac{1}{2}{K}_{134}{\left(\text{Δ}{\theta }_{134}\right)}^2$	$\frac{1}{2}{K}_{415}{\left(\text{Δ}{\theta }_{415}\right)}^2$
Parameter	9.479	2.892	3.145	10.111	10.111
$r_0$ or ${\theta }_0$	2.641	2.772	102.142	100.200	92.760

Table 422.

The VFF model for the single-layer p-SiTe.

The second line gives an explicit expression for each VFF term, where atom indexes are from Figure 189(c) . The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

The structure of the single-layer p-SiTe is shown in Figure 189 , with M = Si and X = Te. Structural parameters for p-SiTe are from the ab initio calculations [78]. There are four atoms in the unit cell with relative coordinates such as $(-u\mathrm{,0,}-v)$ , $(u\mathrm{,0,}v)$ , $(0.5-u\mathrm{,0.5,}v+w)$ , and $(0.5+u\mathrm{,0.5,}-v+w)$ with $u=0.0581$ , $v=0.1363$ , and $w=\mathrm{}-0.0173$ . The values of these dimensionless parameters are extracted from the geometrical parameters provided in Ref. [78], including lattice constants $a_1=4.300$ Å and $a_2=4.109$ Å, bond lengths $d_{12}=2.641$ Å and $d_{14}=2.772$ Å, and the angle ${\theta }_{145}=100.200°$ . The dimensionless parameters $v$ and $w$ are ratios based on the lattice constant in the out-of-plane z-direction, which is arbitrarily chosen as $a_3=10.0$ Å. We note that the main purpose of the usage of $u$ , $v$ , and $w$ in representing atomic coordinates is to follow the same convention for all puckered structures in the present work. The resultant atomic coordinates are the same as that in Ref. [78].

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
$r_{12}$	2.394	0.194	24.324	0.0	2.999
$r_{14}$	3.818	1.722	29.522	0.0	3.864

Table 423.

Two-body SW potential parameters for the single-layer p-SiTe used by GULP [8], as expressed in Eq. (3).

The quantity $\left(r_{ij}\right)$ in the first line lists one representative term for the two-body SW potential between atoms i and j. Atom indexes are from Figure 189(c) .

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{123}$	4.868	102.142	0.194	0.194	0.0	2.999	0.0	2.999	0.0	4.204
${\theta }_{134}$	43.419	100.200	1.722	0.194	0.0	3.864	0.0	2.999	0.0	4.577
${\theta }_{415}$	42.155	92.760	1.722	0.194	0.0	3.864	0.0	2.999	0.0	4.289

Table 424.

Three-body SW potential parameters for the single-layer p-SiTe used by GULP [8], as expressed in Eq. (4).

The first line $\left({\theta }_{ijk}\right)$ presents one representative term for the three-body SW potential. The angle ${\theta }_{ijk}$ has the atom i as the apex. Atom indexes are from Figure 189(c) .

As shown in Figure 189 , a specific feature in the puckered configuration of the p-SiTe is that there is a small difference of $w{a}_3$ between the z-coordinate of atom 1 and the z-coordinates of atoms 2 and 3. Similarly, atom 4 is higher than atoms 5 and 6 for $w{a}_3$ along the z-direction. The sign of $w$ determines which types of atoms take the outmost positions, e.g., atoms 1, 5, and 6 are the outmost atoms if $w>0$ in Figure 189(c) , while atoms 2, 3, and 4 will take the outmost positions for $w<0$ .

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Si1─Te4─Te4	1.000	0.194	15.442	4.868	1.000	−0.210	2.394	17093.960	4	0	0.0
Si1─Te1─Te1	1.000	1.722	2.245	0.000	1.000	0.000	3.818	3.360	4	0	0.0
Si1─Te1─Te4	1.000	0.000	0.000	43.419	1.000	−0.177	0.000	0.000	4	0	0.0
Te1─Si1─Si2	1.000	0.000	0.000	42.155	1.000	−0.048	0.000	0.000	4	0	0.0

Table 425.

SW potential parameters for p-SiTe used by LAMMPS [9], as expressed in Eqs. (9) and (10).

Atom types in the first column are displayed in Figure 190 , with M = Si and X = Te.

Table 422 shows five VFF terms for the single-layer p-SiTe; two of which are the bond stretching interactions shown in Eq. (1), while the other three terms are the angle bending interaction shown in Eq. (2). The force constant parameters are the same for the two angle bending terms ${\theta }_{134}$ and ${\theta }_{415}$ , which have the same arm lengths. All force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the ΓX as shown in Figure 216(a) . The ab initio calculations are from Ref. [83]. Figure 216(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion.

The parameters for the two-body SW potential used by GULP are shown in Table 423 . The parameters for the three-body SW potential used by GULP are shown in Table 424 . Parameters for the SW potential used by LAMMPS are listed in Table 425 . Eight atom types have been introduced for writing the SW potential script used by LAMMPS as shown in Figure 190 with M = Si and X = Te, which helps to increase the cutoff for the bond stretching interaction between atom 1 and 2 in Figure 189(c) .

Figure 217 shows the stress-strain relations for the single-layer p-SiTe of size $100\times 100$ Å. The structure is uniaxially stretched in the armchair or zigzag directions at 1 and 300 K. The Young’s modulus is 14.0 and 53.6 Nm⁻¹ in the armchair and zigzag directions, respectively, at 1 K, which is obtained by linear fitting of the stress-strain relations in [0, 0.01]. The Poisson’s ratios from the VFF model and the SW potential are ${\nu }_{xy}=0.12$ and ${\nu }_{yx}=0.47$ . The third-order nonlinear elastic constant $D$ can be obtained by fitting the stress-strain relation to $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with E as the Young’s modulus. The values of $D$ are −32.9 and −183.2 Nm⁻¹ at 1 K along the armchair and zigzag directions, respectively. The ultimate stress is about 2.2 Nm⁻¹ at the critical strain of 0.24 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 7.0 Nm⁻¹ at the critical strain of 0.22 in the zigzag direction at the low temperature of 1 K.

107. P-GeTe

Present studies on the puckered (p-) GeTe are based on first-principles calculations, and no empirical potential has been proposed for the p-GeTe. We will thus parametrize the SW potential for the single-layer p-GeTe in this section.

Figure 218.

Phonon dispersion for the single-layer p-GeTe. (a) The VFF model is fitted to the acoustic branches in the long wave limit along the ΓX direction. The ab initio calculations are calculated from SIESTA. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for the p-GeTe along ΓXSYΓ.

The structure of the single-layer p-GeTe is shown in Figure 189 , with M = Ge and X = Te. Structural parameters for p-GeTe are from the ab initio calculations [78]. There are four atoms in the unit cell with relative coordinates such as $(-u\mathrm{,0,}-v)$ , $(u\mathrm{,0,}v)$ , $(0.5-u\mathrm{,0.5,}v+w)$ , and $(0.5+u\mathrm{,0.5,}-v+w)$ with $u=0.0538$ , $v=0.1422$ , and $w=\mathrm{}-0.0216$ . The values of these dimensionless parameters are extracted from the geometrical parameters provided in Ref. [78], including lattice constants $a_1=4.376$ Å and $a_2=4.238$ Å, bond lengths $d_{12}=2.736$ Å and $d_{14}=2.883$ Å, and the angle ${\theta }_{145}=100.4°$ . The dimensionless parameters $v$ and $w$ are ratios based on the lattice constant in the out-of-plane z-direction, which is arbitrarily chosen as $a_3=10.0$ Å. We note that the main purpose of the usage of $u$ , $v$ , and $w$ in representing atomic coordinates is to follow the same convention for all puckered structures in the present work. The resultant atomic coordinates are the same as that in Ref. [78].

Figure 219.

Stress-strain relations for the single-layer p-GeTe of size $100\times 100$ Å. The single-layer p-GeTe is uniaxially stretched along the armchair or zigzag directions at temperatures 1 and 300 K.

VFF type	Bond stretching		Angle bending
Expression	$\frac{1}{2}{K}_{12}{\left(\text{Δ}{r}_{12}\right)}^2$	$\frac{1}{2}{K}_{14}{\left(\text{Δ}{r}_{14}\right)}^2$	$\frac{1}{2}{K}_{123}{\left(\text{Δ}{\theta }_{123}\right)}^2$	$\frac{1}{2}{K}_{134}{\left(\text{Δ}{\theta }_{134}\right)}^2$	$\frac{1}{2}{K}_{415}{\left(\text{Δ}{\theta }_{415}\right)}^2$
Parameter	7.074	7.074	2.611	5.876	5.876
$r_0$ or ${\theta }_0$	2.736	2.883	101.517	100.400	91.402

Table 426.

The VFF model for the single-layer p-GeTe.

The second line gives an explicit expression for each VFF term, where atom indexes are from Figure 189(c) . The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

As shown in Figure 189 , a specific feature in the puckered configuration of the p-GeTe is that there is a small difference of $w{a}_3$ between the z-coordinate of atom 1 and the z-coordinates of atoms 2 and 3. Similarly, atom 4 is higher than atoms 5 and 6 for $w{a}_3$ along the z-direction. The sign of $w$ determines which types of atoms take the outmost positions, e.g., atoms 1, 5, and 6 are the outmost atoms if $w>0$ in Figure 189(c) , while atoms 2, 3, and 4 will take the outmost positions for $w<0$ .

Table 426 shows five VFF terms for the single-layer p-GeTe; two of which are the bond stretching interactions shown in Eq. (1), while the other three terms are the angle bending interaction shown by Eq. (2). The force constant parameters are the same for the two angle bending terms ${\theta }_{134}$ and ${\theta }_{415}$ , which have the same arm lengths. All force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the ΓX as shown in Figure 218(a) . The ab initio calculations for the phonon dispersion are calculated from the SIESTA package [79]. The generalized gradients approximation is applied to account for the exchange-correlation function with Perdew et al. parameterization [80], and the double- $\zeta$ orbital basis set is adopted. Figure 218(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
$r_{12}$	1.708	0.165	28.018	0.0	3.072
$r_{14}$	9.854	1.745	34.542	0.0	4.005

Table 427.

Two-body SW potential parameters for the single-layer p-GeTe used by GULP [8], as expressed in Eq. (3).

The quantity $\left(r_{ij}\right)$ in the first line lists one representative term for the two-body SW potential between atoms i and j. Atom indexes are from Figure 189(c) .

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{123}$	3.626	101.517	0.165	0.165	0.0	3.072	0.0	3.072	0.0	4.307
${\theta }_{134}$	23.509	100.400	1.745	0.165	0.0	4.005	0.0	3.072	0.0	4.659
${\theta }_{415}$	22.756	91.402	1.745	0.165	0.0	4.005	0.0	3.072	0.0	4.434

Table 428.

Three-body SW potential parameters for the single-layer p-GeTe used by GULP [8], as expressed in Eq. (4).

The first line $\left({\theta }_{ijk}\right)$ presents one representative term for the three-body SW potential. The angle ${\theta }_{ijk}$ has the atom i as the apex. Atom indexes are from Figure 189(c) .

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Ge1─Te4─Te4	1.000	0.165	18.665	3.626	1.000	−0.200	1.708	38204.858	4	0	0.0
Ge1─Te1─Te1	1.000	1.745	2.295	0.000	1.000	0.000	9.854	3.725	4	0	0.0
Ge1─Te1─Te4	1.000	0.000	0.000	23.509	1.000	−0.181	0.000	0.000	4	0	0.0
Te1─Ge1─Ge2	1.000	0.000	0.000	22.756	1.000	−0.024	0.000	0.000	4	0	0.0

Table 429.

SW potential parameters for p-GeTe used by LAMMPS [9], as expressed in Eqs. (9) and (10).

Atom types in the first column are displayed in Figure 190 , with M = Ge and X = Te.

The parameters for the two-body SW potential used by GULP are shown in Table 427 . The parameters for the three-body SW potential used by GULP are shown in Table 428 . Parameters for the SW potential used by LAMMPS are listed in Table 429 . Eight atom types have been introduced for writing the SW potential script used by LAMMPS as shown in Figure 190 with M = Ge and X = Te, which helps to increase the cutoff for the bond stretching interaction between atom 1 and 2 in Figure 189(c) .

Figure 219 shows the stress-strain relations for the single-layer p-GeTe of size $100\times 100$ Å. The structure is uniaxially stretched in the armchair or zigzag directions at 1 and 300 K. The Young’s modulus is 8.1 and 41.6 N/m in the armchair and zigzag directions, respectively, at 1 K, which is obtained by linear fitting of the stress-strain relations in [0, 0.01]. The Poisson’s ratios from the VFF model and the SW potential are ${\nu }_{xy}=0.09$ and ${\nu }_{yx}=0.49$ . The third-order nonlinear elastic constant $D$ can be obtained by fitting the stress-strain relation to $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with E as the Young’s modulus. The values of $D$ are −10.5 and −143.7 N/m at 1 K along the armchair and zigzag directions, respectively. The ultimate stress is about 2.6 N/m at the critical strain of 0.53 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 5.3 N/m at the critical strain of 0.21 in the zigzag direction at the low temperature of 1 K.

108. P-SnTe

Present studies on the puckered (p-) SnTe are based on first-principles calculations, and no empirical potential has been proposed for the p-SnTe. We will thus parametrize the SW potential for the single-layer p-SnTe in this section.

The structure of the single-layer p-SnTe is shown in Figure 189 , with M = Sn and X = Te. Structural parameters for p-SnTe are from the ab initio calculations [78]. There are four atoms in the unit cell with relative coordinates such as $(-u\mathrm{,0,}-v)$ , $(u\mathrm{,0,}v)$ , $(0.5-u\mathrm{,0.5,}v+w)$ , and $(0.5+u\mathrm{,0.5,}-v+w)$ with $u=0.0478$ , $v=0.1567$ , and $w=\mathrm{}-0.0050$ . The values of these dimensionless parameters are extracted from the geometrical parameters provided in Ref. [78], including lattice constants $a_1=4.581$ Å and $a_2=4.542$ Å, bond lengths $d_{12}=2.931$ Å and $d_{14}=3.164$ Å, and the angle ${\theta }_{145}=96.0°$ . The dimensionless parameters $v$ and $w$ are ratios based on the lattice constant in the out-of-plane z-direction, which is arbitrarily chosen as $a_3=10.0$ Å. We note that the main purpose of the usage of $u$ , $v$ , and $w$ in representing atomic coordinates is to follow the same convention for all puckered structures in the present work. The resultant atomic coordinates are the same as that in Ref. [78].

Figure 220.

Phonon dispersion for the single-layer p-SnTe. (a) The VFF model is fitted to the acoustic branches in the long wave limit along the ΓX direction. The ab initio calculations are calculated from SIESTA. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for the p-SnTe along ΓXSYΓ.

As shown in Figure 189 , a specific feature in the puckered configuration of the p-SnTe is that there is a small difference of $w{a}_3$ between the z-coordinate of atom 1 and the z-coordinates of atoms 2 and 3. Similarly, atom 4 is higher than atoms 5 and 6 for $w{a}_3$ along the z-direction. The sign of $w$ determines which types of atoms take the outmost positions, e.g., atoms 1, 5, and 6 are the outmost atoms if $w>0$ in Figure 189(c) , while atoms 2, 3, and 4 will take the outmost positions for $w<0$ .

Figure 221.

Stress-strain relations for the single-layer p-SnTe of size $100\times 100$ Å. The single-layer p-SnTe is uniaxially stretched along the armchair or zigzag directions at temperatures 1 and 300 K.

VFF type	Bond stretching		Angle bending
Expression	$\frac{1}{2}{K}_{12}{\left(\text{Δ}{r}_{12}\right)}^2$	$\frac{1}{2}{K}_{14}{\left(\text{Δ}{r}_{14}\right)}^2$	$\frac{1}{2}{K}_{123}{\left(\text{Δ}{\theta }_{123}\right)}^2$	$\frac{1}{2}{K}_{134}{\left(\text{Δ}{\theta }_{134}\right)}^2$	$\frac{1}{2}{K}_{415}{\left(\text{Δ}{\theta }_{415}\right)}^2$
Parameter	7.074	7.074	2.611	2.611	5.876
$r_0$ or ${\theta }_0$	2.931	3.164	101.578	96.000	94.045

Table 430.

The VFF model for the single-layer p-SnTe.

The second line gives an explicit expression for each VFF term, where atom indexes are from Figure 189(c) . The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

Table 430 shows five VFF terms for the single-layer p-SnTe; two of which are the bond stretching interactions shown in Eq. (1), while the other three terms are the angle bending interaction shown in Eq. (2). The force constant parameters are the same for the two angle bending terms ${\theta }_{134}$ and ${\theta }_{415}$ , which have the same arm lengths. All force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the ΓX as shown in Figure 220(a) . The ab initio calculations for the phonon dispersion are calculated from the SIESTA package [79]. The generalized gradients approximation is applied to account for the exchange-correlation function with Perdew et al. parameterization [80], and the double- $\zeta$ orbital basis set is adopted. Figure 220(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
$r_{12}$	1.662	0.131	36.901	0.0	3.241
$r_{14}$	11.054	1.777	50.109	0.0	4.349

Table 431.

Two-body SW potential parameters for the single-layer p-SnTe used by GULP [8], as expressed in Eq. (3).

The quantity ( $r_{ij}$ ) in the first line lists one representative term for the two-body SW potential between atoms i and j. Atom indexes are from Figure 189(c) .

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{123}$	3.170	101.578	0.131	0.131	0.0	3.241	0.0	3.241	0.0	4.562
${\theta }_{134}$	20.298	96.000	1.777	0.131	0.0	4.349	0.0	3.241	0.0	4.766
${\theta }_{415}$	20.176	94.045	1.777	0.131	0.0	4.349	0.0	3.241	0.0	4.595

Table 432.

Three-body SW potential parameters for the single-layer p-SnTe used by GULP [8], as expressed in Eq. (4).

The first line ( ${\theta }_{ijk}$ ) presents one representative term for the three-body SW potential. The angle ${\theta }_{ijk}$ has the atom i as the apex. Atom indexes are from Figure 189(c) .

The parameters for the two-body SW potential used by GULP are shown in Table 431 . The parameters for the three-body SW potential used by GULP are shown in Table 432 . Parameters for the SW potential used by LAMMPS are listed in Table 433 . Eight atom types have been introduced for writing the SW potential script used by LAMMPS as shown in Figure 190 with M = Sn and X = Te, which helps to increase the cutoff for the bond stretching interaction between atom 1 and 2 in Figure 189(c) .

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Sn1─Te4─Te4	1.000	0.131	24.712	3.170	1.000	−0.201	1.662	124727.735	4	0	0.0
Sn1─Te1─Te1	1.000	1.777	2.448	0.000	1.000	0.000	11.054	5.028	4	0	0.0
Sn1─Te1─Te4	1.000	0.000	0.000	20.298	1.000	−0.105	0.000	0.000	4	0	0.0
Te1─Sn1─Sn2	1.000	0.000	0.000	20.176	1.000	−0.071	0.000	0.000	4	0	0.0

Table 433.

SW potential parameters for p-SnTe used by LAMMPS [9], as expressed in Eqs. (9) and (10).

Atom types in the first column are displayed in Figure 190 , with M = Sn and X = Te.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_r{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\theta }{\left(\text{Δ}\theta \right)}^2$
Parameter	18.387	3.465
$r_0$ or ${\theta }_0$	2.279	116.218

Table 434.

The VFF model for silicene.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction.

Figure 221 shows the stress-strain relations for the single-layer p-SnTe of size $100\times 100$ Å. The structure is uniaxially stretched in the armchair or zigzag directions at 1 and 300 K. The Young’s modulus is 6.6 and 38.5 Nm⁻¹ in the armchair and zigzag directions, respectively, at 1 K, which is obtained by linear fitting of the stress-strain relations in [0, 0.01]. The Poisson’s ratios from the VFF model and the SW potential are ${\nu }_{xy}=0.10$ and ${\nu }_{yx}=0.57$ . The third-order nonlinear elastic constant $D$ can be obtained by fitting the stress-strain relation to $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with E as the Young’s modulus. The values of $D$ are −7.2 and −114.5 N/m at 1 K along the armchair and zigzag directions, respectively. The ultimate stress is about 2.4 Nm⁻¹ at the critical strain of 0.61 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 5.3 N/m at the critical strain of 0.21 in the zigzag direction at the low temperature of 1 K.

109. Silicene

There have been several empirical potentials available for the silicene. A many-body potential based on the Lennard-Jones and Axilrod-Teller functions was used to describe the interaction within the single-layer silicene [85]. The modified embedded-atom potential [86] was used by Pei et al. to simulate the thermal transport in the single-layer silicene in 2013 [87]. The environment-dependent interatomic potential [88] was also used to simulate the silicene [89]. In particular, the original set of SW parameters [4] for the silicon was found to be not suitable for the planar silicene, so two sets of optimized parameters for the SW potential were proposed to simulate the thermal conductivity in the single-layer silicene in 2014 [90]. We will develop a new SW potential to describe the interaction within the silicene in this section, with specific focus on the mechanical properties of the silicene.

Figure 222.

Structure for the buckled M, with M from group V. (a) Top view. The armchair direction is along the horizontal direction, while the zigzag direction is along the vertical direction. The unit cell is displayed by the blue rhombus. Inset shows the first Brillouin zone. (b) Side view illustrates the buckled configuration of height $h$ .

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Si─Si	19.343	1.668	16.186	0.0	3.075

Table 435.

Two-body SW potential parameters for silicene used by GULP [8], as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
Si─Si─Si	142.310	116.218	1.668	1.668	0.0	3.075	0.0	3.075	3.075	4.181

Table 436.

Three-body SW potential parameters for silicene used by GULP [8], as expressed in Eq. (4).

Figure 223.

Phonon dispersion for the silicene. (a) The VFF model is fitted to the three acoustic branches in the long wave limit along the ΓM direction. The ab initio results (gray pentagons) are from Ref. [84]. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for the silicene along ΓMKΓ.

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Si─Si─Si	1.000	1.668	1.844	142.310	1.000	-0.442	19.343	2.091	4	0	0.0

Table 437.

SW potential parameters for silicene used by LAMMPS [9], as expressed in Eqs. (9) and (10).

The structure of the silicene is shown in Figure 222 , with structural parameters from the ab initio calculations [84]. The silicene has a buckled configuration as shown in Figure 222(b) , where the buckle is along the zigzag direction. The height of the buckle is $h=0.45$ Å and the lattice constant is 3.87 Å, which results in a bond length of 2.279 Å.

Figure 224.

Stress-strain relations for the silicene of size $100\times 100$ Å. The silicene is uniaxially stretched along the armchair or zigzag directions at temperatures 1 and 300 K. Left inset shows the buckled configuration for the silicene at the uniaxial strain 0.07 at 1 K along the armchair direction. Right inset: the buckled configuration becomes planar for the silicene at the uniaxial strain of 0.08 at 1 K along the armchair direction.

Table 434 shows the VFF model for the silicene. The force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the ΓM as shown in Figure 223(a) . The ab initio calculations for the phonon dispersion are from Ref. [84]. Similar phonon dispersion can also be found in other ab initio calculations [84, 91–102]. Figure 223(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

We note that the present SW potential is fitted perfectly to the three acoustic phonon branches, so it can give a nice description for the elastic deformation of the silicene. As a trade off, the optical phonons are overestimated by the present SW potential. Hence, the present SW potential is in particular suitable for the simulation of mechanical or thermal processes, which are dominated by acoustic phonons, while the present SW potential may cause a systematic error for the optical absorption process, which mainly involves the optical phonons. One can introduce the long-range interactions (e.g., the second-nearest-neighboring interaction) to give a good description for both acoustic and optical phonon branches, see one such example for borophene in Ref. [14]. It is because the long-range interaction mainly contributes to the acoustic phonon branches, while it makes only neglectable contribution to the optical phonon branches. As another solution, the SW potential can give reasonable descriptions for the optical phonon branches by reducing its accuracy in describing acoustic phonon branches as done in Ref. [90].

The parameters for the two-body SW potential used by GULP are shown in Table 435 . The parameters for the three-body SW potential used by GULP are shown in Table 436 . Parameters for the SW potential used by LAMMPS are listed in Table 437 .

Figure 224 shows the stress-strain relations for the silicene of size $100\times 100$ Å. The structure is uniaxially stretched in the armchair or zigzag directions at 1 and 300 K. The Young’s modulus is 63.3 Nm⁻¹ in both armchair and zigzag directions at 1 K, which are obtained by linear fitting of the stress-strain relations in [0, 0.01]. The Young’s modulus is isotropic for the silicene. The value of the Young’s modulus agrees with the value of 63.8 Nm⁻¹ from the ab initio calculations [103]. The Poisson’s ratios from the VFF model and the SW potential are ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.15$ , which are smaller but comparable with the ab initio results of 0.325 [103]. The third-order nonlinear elastic constant $D$ can be obtained by fitting the stress-strain relation to $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with E as the Young’s modulus. The values of $D$ are −212.5 and −267.5 Nm⁻¹ at 1 K along the armchair and zigzag directions, respectively. The ultimate stress is about 8.6 Nm⁻¹ at the critical strain of 0.23 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 7.8 Nm⁻¹ at the critical strain of 0.23 in the zigzag direction at the low temperature of 1 K.

The stress-strain curves shown in Figure 224 disclose a structural transition at the strain around 0.076 for the silicene at the low temperature of 1 K. The buckled configuration of the silicene is flattened during this structural transition, which can be seen from these two insets in Figure 224 . This structural transition was also observed in the ab initio calculations [95], where the critical strain for the structural transition is 0.2. At temperatures above 300 K, this structural transition is blurred by stronger thermal vibrations; i.e., the buckled configuration of the silicene can be strongly disturbed by the thermal vibration at higher temperatures.

Figure 225.

Phonon dispersion for the germanene. (a) The VFF model is fitted to the three acoustic branches in the long wave limit along the ΓM direction. The ab initio results (gray pentagons) are from Ref. 84. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for the germanene along ΓMKΓ.

110. Germanene

In a recent work, the Tersoff potential was applied to simulate the thermal conductivity of the germanene nanoribbon [104]. We will provide the SW potential to describe the interaction within the germanene in this section.

The structure of the germanene is shown in Figure 222 , with structural parameters from the ab initio calculations [84]. The germanene has a buckled configuration as shown in Figure 222(b) , where the buckle is along the zigzag direction. The height of the buckle is $h=0.69$ Å and the lattice constant is 4.06 Å, which results in a bond length of 2.443 Å.

Table 438 shows the VFF model for the germanene. The force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the ΓM as shown in Figure 225(a) . The ab initio calculations for the phonon dispersion are from Ref. [84]. Similar phonon dispersion can also be found in other ab initio calculations [84, 92, 93, 98, 101, 102, 105]. Figure 225(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

Figure 226.

Stress-strain relations for the germanene of size $100\times 100$ Å. The germanene is uniaxially stretched along the armchair or zigzag directions at temperatures 1 and 300 K. Left bottom inset shows the buckled configuration for the germanene at the uniaxial strain 0.14 at 1 K along the armchair direction. Right top inset: the buckled configuration becomes planar for the germanene at the uniaxial strain of 0.16 at 1 K along the armchair direction.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_r{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\theta }{\left(\text{Δ}\theta \right)}^2$
Parameter	18.387	3.465
$r_0$ or ${\theta }_0$	2.443	112.358

Table 438.

The VFF model for germanene.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Ge─Ge	19.570	1.607	21.372	0.0	3.252

Table 439.

Two-body SW potential parameters for germanene used by GULP [8], as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
Ge─Ge─Ge	107.735	112.358	1.607	1.607	0.0	3.252	0.0	3.252	3.252	4.4

Table 440.

Three-body SW potential parameters for germanene used by GULP [8], as expressed in Eq. (4).

The parameters for the two-body SW potential used by GULP are shown in Table 439 . The parameters for the three-body SW potential used by GULP are shown in Table 440 . Parameters for the SW potential used by LAMMPS are listed in Table 441 .

Figure 226 shows the stress-strain relations for the germanene of size $100\times 100$ Å. The structure is uniaxially stretched in the armchair or zigzag directions at 1 and 300 K. The Young’s modulus is 53.2 Nm⁻¹ in both armchair and zigzag directions at 1 K, which are obtained by linear fitting of the stress-strain relations in [0, 0.01]. The Young’s modulus is isotropic for the germanene. The Poisson’s ratios from the VFF model and the SW potential are ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.19$ . The third-order nonlinear elastic constant $D$ can be obtained by fitting the stress-strain relation to $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with E as the Young’s modulus. The values of $D$ are −229.2 and −278.2 Nm⁻¹ at 1 K along the armchair and zigzag directions, respectively. The ultimate stress is about 7.5 Nm⁻¹ at the critical strain of 0.26 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 7.3 Nm⁻¹ at the critical strain of 0.27 in the zigzag direction at the low temperature of 1 K.

The stress-strain curves shown in Figure 226 disclose a structural transition for the germanene at the low temperature of 1 K. The critical strains for the structural transition are 0.15 and 0.16 along the armchair and zigzag directions, respectively. The buckled configuration of the germanene is flattened during this structural transition, which can be seen from these two insets in Figure 226 . At temperatures above 300 K, this structural transition is blurred by stronger thermal vibrations; i.e., the buckled configuration of the germanene can be strongly disturbed by the thermal vibration at higher temperatures.

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Ge─Ge─Ge	1.000	1.607	2.024	107.735	1.000	-0.380	19.570	3.205	4	0	0.0

Table 441.

SW potential parameters for germanene used by LAMMPS [9], as expressed in Eqs. (9) and (10).

Figure 227.

Phonon dispersion for the stanene. (a) The VFF model is fitted to the three acoustic branches in the long wave limit along the ΓM direction. The ab initio results (gray pentagons) are from Ref. [105]. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for the stanene along ΓMKΓ.

111. Stanene

There are several available empirical potentials for the description of the interaction within the stanene. The modified embedded atom method potential was applied to simulate mechanical properties for the stanene [106]. A VFF model was fitted for the stanene in 2015 [107]. The Tersoff potential was parameterized to describe the interaction for stanene [108]. In this section, we will develop the SW potential for the stanene.

Figure 228.

Stress-strain relations for the stanene of size $100\times 100$ Å. The stanene is uniaxially stretched along the armchair or zigzag directions at temperatures 1 and 300 K. Left inset: the buckled configuration for the stanene at the uniaxial strain 0.14 at 1 K along the armchair direction. Right inset: the buckled configuration becomes planar for the stanene at the uniaxial strain of 0.16 at 1 K along the armchair direction.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_r{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\theta }{\left(\text{Δ}\theta \right)}^2$
Parameter	10.489	1.372
$r_0$ or ${\theta }_0$	2.836	111.224

Table 442.

The VFF model for stanene.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction.

The structure of the stanene is shown in Figure 222 , with structural parameters from the ab initio calculations [105]. The stanene has a buckled configuration as shown in Figure 222(b) , where the buckle is along the zigzag direction. The height of the buckle is $h=0.86$ Å and the lattice constant is 4.68 Å, which results in a bond length of 2.836 Å.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Sn─Sn	19.542	2.227	42.047	0.0	3.758

Table 443.

Two-body SW potential parameters for stanene used by GULP [8], as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
Sn─Sn─Sn	98.863	111.224	2.227	2.227	0.0	3.758	0.0	3.758	3.758	5.076

Table 444.

Three-body SW potential parameters for stanene used by GULP [8], as expressed in Eq. (4).

Table 442 shows the VFF model for the stanene. The force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the ΓM as shown in Figure 227(a) . The ab initio calculations for the phonon dispersion are from Ref. [105] with the spin-orbit coupling effect. Similar phonon dispersion can also be found in other ab initio calculations [101, 102, 105, 109–111]. Figure 227(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 443 . The parameters for the three-body SW potential used by GULP are shown in Table 444 . Parameters for the SW potential used by LAMMPS are listed in Table 445 .

Figure 228 shows the stress-strain relations for the stanene of size $100\times 100$ Å. The structure is uniaxially stretched in the armchair or zigzag directions at 1 and 300 K. The Young’s modulus is 17.0 Nm⁻¹ in both armchair and zigzag directions at 1 K, which are obtained by linear fitting of the stress-strain relations in [0, 0.01]. The Young’s modulus is isotropic for the stanene. The Poisson’s ratios from the VFF model and the SW potential are ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.29$ . The third-order nonlinear elastic constant $D$ can be obtained by fitting the stress-strain relation to $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with E as the Young’s modulus. The values of $D$ are −37.2 Nm⁻¹ and −69.4 Nm⁻¹ at 1 K along the armchair and zigzag directions, respectively. The ultimate stress is about 3.5 Nm⁻¹ at the critical strain of 0.32 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 3.6 Nm⁻¹ at the critical strain of 0.29 in the zigzag direction at the low temperature of 1 K.

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Sn─Sn─Sn	1.000	2.227	1.687	98.863	1.000	-0.362	19.542	1.709	4	0	0.0

Table 445.

SW potential parameters for stanene used by LAMMPS [9], as expressed in Eqs. (9) and (10).

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_r{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\theta }{\left(\text{Δ}\theta \right)}^2$
Parameter	2.128	1.175
$r_0$ or ${\theta }_0$	2.890	94.372

Table 446.

The VFF model for indiene.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction.

The stress-strain curves shown in Figure 228 disclose a structural transition for the stanene at the low temperature of 1 K. The critical strain for the structural transition is about 0.15 along the armchair and zigzag directions. The buckled configuration of the stanene is flattened during this structural transition, which can be seen from these two insets in Figure 228 . At temperatures above 300 K, this structural transition is blurred by stronger thermal vibrations, i.e., the buckled configuration of the stanene can be strongly disturbed by the thermal vibration at higher temperatures.

112. Indiene

In this section, we will develop the SW potential for the indiene, i.e., the single layer of indium atoms. The structure of the indiene is shown in Figure 222 , with structural parameters from the ab initio calculations [112]. The indiene has a buckled configuration as shown in Figure 222(b) , where the buckle is along the zigzag direction. The lattice constant is 4.24 Å and the bond length is 2.89 Å, which results in the buckling height of $h=1.536$ Å.

Figure 229.

Phonon dispersion for the indiene. (a) The VFF model is fitted to the three acoustic branches in the long wave limit along the ΓM direction. The ab initio results (gray pentagons) are from Ref. [112]. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for the indiene along ΓMKΓ.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
In─In	1.537	0.946	41.855	0.0	3.565

Table 447.

Two-body SW potential parameters for indiene used by GULP [8], as expressed in Eq. (3).

Table 446 shows the VFF model for the indiene. The force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the ΓM as shown in Figure 229(a) . The ab initio calculations for the phonon dispersion are from Ref. [112]. We note that the lowest frequency branch around the Γ point from the VFF model is lower than the ab initio results. This branch is the flexural branch, which should be a quadratic dispersion. However, the ab initio calculations give a linear dispersion for the flexural branch due to the violation of the rigid rotational invariance in the first-principles package [20], so ab initio calculations typically overestimate the frequency of this branch. Figure 229(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

Figure 230.

Stress-strain relations for the indiene of size $100\times 100$ Å. The indiene is uniaxially stretched along the armchair or zigzag directions at temperatures 1 K.

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
In─In─In	9.745	94.372	0.946	0.946	0.0	3.565	0.0	4.565	3.565	4.686

Table 448.

Three-body SW potential parameters for indiene used by GULP [8], as expressed in Eq. (4).

The parameters for the two-body SW potential used by GULP are shown in Table 447 . The parameters for the three-body SW potential used by GULP are shown in Table 448 . Parameters for the SW potential used by LAMMPS are listed in Table 449 .

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
In─In─In	1.000	0.946	3.768	9.745	1.000	-0.076	1.537	52.262	4	0	0.0

Table 449.

SW potential parameters for indiene used by LAMMPS [9], as expressed in Eqs. (9) and (10).

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_r{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\theta }{\left(\text{Δ}\theta \right)}^2$
Parameter	15.372	5.138
$r_0$ or ${\theta }_0$	2.270	94.209

Table 450.

The VFF model for blue phosphorus.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
P─P	5.706	0.491	13.276	0.0	2.798

Table 451.

Two-body SW potential parameters for blue phosphorus used by GULP [8], as expressed in Eq. (3).

Figure 230 shows the stress-strain relations for the indiene of size $100\times 100$ Å. The structure is uniaxially stretched in the armchair or zigzag directions at 1 K. The Young’s modulus is 8.4 Nm⁻¹ in both armchair and zigzag directions at 1 K, which are obtained by linear fitting of the stress-strain relations in [0, 0.01]. The Young’s modulus of the indiene is very small; i.e., the indiene is very soft. As a result, we find that the structure becomes unstable at room temperature. The Poisson’s ratios from the VFF model and the SW potential are ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.18$ . The third-order nonlinear elastic constant $D$ can be obtained by fitting the stress-strain relation to $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with E as the Young’s modulus. The values of $D$ are −42.0 and −50.2 Nm⁻¹ at 1 K along the armchair and zigzag directions, respectively. The ultimate stress is about 0.77 Nm⁻¹ at the critical strain of 0.16 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 0.73 Nm⁻¹ at the critical strain of 0.19 in the zigzag direction at the low temperature of 1 K.

Figure 231.

Phonon dispersion for the single-layer blue phosphorus. (a) The VFF model is fitted to the three acoustic branches in the long wave limit along the ΓM direction. The ab initio results (gray pentagons) are from Ref. [55]. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for the blue phosphorus along ΓMKΓ.

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
P─P─P	16.605	94.209	0.491	0.491	0.0	2.798	0.0	2.798	2.798	3.677

Table 452.

Three-body SW potential parameters for blue phosphorus used by GULP [8], as expressed in Eq. (4).

113. Blue phosphorus

The blue phosphorus is also named $\beta$ -phosphorus. Present studies on the blue phosphorus are based on first-principles calculations, and no empirical potential has been proposed for the blue phosphorus. We will thus parametrize a set of VFF model for the single-layer blue phosphorus in this section. We will also derive the SW potential based on the VFF model for the single-layer blue phosphorus.

Figure 232.

Stress-strain relations for the blue phosphorus of size $100\times 100$ Å. The blue phosphorus is uniaxially stretched along the armchair or zigzag directions at temperatures 1 and 300 K.

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
P─P─P	1.000	0.491	5.699	16.605	1.000	−0.073	5.706	228.424	4	0	0.0

Table 453.

SW potential parameters for blue phosphorus used by LAMMPS [9], as expressed in Eqs. (9) and (10).

The structure of the single-layer blue phosphorus is shown in Figure 222 , with structural parameters from the ab initio calculations [55]. The blue phosphorus has a buckled configuration as shown in Figure 222(b) , where the buckle is along the zigzag direction. The height of the buckle is $h=1.211$ Å. The lattice constant is 3.326 Å, and the bond length is 2.270 Å.

Table 450 shows the VFF model for the single-layer blue phosphorus. The force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the ΓM as shown in Figure 231(a) . The ab initio calculations for the phonon dispersion are from Ref. [55]. Similar phonon dispersion can also be found in other ab initio calculations [61, 64, 84]. Figure 231(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 451 . The parameters for the three-body SW potential used by GULP are shown in Table 452 . Parameters for the SW potential used by LAMMPS are listed in Table 453 .

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_r{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\theta }{\left(\text{Δ}\theta \right)}^2$
Parameter	15.372	5.138
$r_0$ or ${\theta }_0$	2.510	91.964

Table 454.

The VFF model for b-arsenene.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
As-As	6.418	0.482	19.846	0.0	3.060

Table 455.

Two-body SW potential parameters for b-arsenene used by GULP [8], as expressed in Eq. (3).

Figure 232 shows the stress-strain relations for the single-layer blue phosphorus of size $100\times 100$ Å. The structure is uniaxially stretched in the armchair or zigzag directions at 1 and 300 K. The Young’s modulus is 60.5 and 60.6 Nm⁻¹ in the armchair and zigzag directions, respectively, at 1 K, which is obtained by linear fitting of the stress-strain relations in [0, 0.01]. The Young’s modulus is isotropic for the blue phosphorus. The Poisson’s ratios from the VFF model and the SW potential are ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.18$ . The third-order nonlinear elastic constant $D$ can be obtained by fitting the stress-strain relation to $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with E as the Young’s modulus. The values of $D$ are −195.3 and −237.0 Nm⁻¹ at 1 K along the armchair and zigzag directions, respectively. The ultimate stress is about 8.0 Nm⁻¹ at the critical strain of 0.22 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 7.6 Nm⁻¹ at the critical strain of 0.25 in the zigzag direction at the low temperature of 1 K.

Figure 233.

Phonon dispersion for the single-layer b-arsenene. (a) The VFF model is fitted to the three acoustic branches in the long wave limit along the ΓM direction. The ab initio results (gray pentagons) are from Ref. [70]. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for the b-arsenene along ΓMKΓ.

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
As-As-As	14.845	91.964	0.482	0.482	0.0	3.060	0.0	3.060	3.060	4.004

Table 456.

Three-body SW potential parameters for b-arsenene used by GULP [8], as expressed in Eq. (4).

114. B-Arsenene

Present studies on the buckled (b-) arsenene, which is also named $\beta$ arsenene, are based on first-principles calculations, and no empirical potential has been proposed for the b-arsenene. We will thus parametrize a set of VFF model for the single-layer b-arsenene in this section. We will also derive the SW potential based on the VFF model for the single-layer b-arsenene.

Figure 234.

Stress-strain relations for the b-arsenene of size 100×100 Å. The b-arsenene is uniaxially stretched along the armchair or zigzag directions at temperatures 1 and 300 K.

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
As─As─As	1.000	0.482	6.349	14.845	1.000	-0.034	6.418	367.693	4	0	0.0

Table 457.

SW potential parameters for b-arsenene used by LAMMPS [9], as expressed in Eqs. (9) and (10).

The structure of the single-layer b-arsenene is shown in Figure 222 , with structural parameters from the ab initio calculations [70]. The b-arsenene has a buckled configuration as shown in Figure 222(b) , where the buckle is along the zigzag direction. The height of the buckle is $h=1.40$ Å. The lattice constant is 3.61 Å, and the bond length is 2.51 Å.

Table 454 shows the VFF model for the single-layer b-arsenene. The force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the ΓM as shown in Figure 233(a) . The ab initio calculations for the phonon dispersion are from Ref. [70]. Similar phonon dispersion can also be found in other ab initio calculations [64, 72, 73]. We note that the lowest frequency branch around the Γ point from the VFF model is lower than the ab initio results. This branch is the flexural branch, which should be a quadratic dispersion. However, the ab initio calculations give a linear dispersion for the flexural branch due to the violation of the rigid rotational invariance in the first-principles package [20], so ab initio calculations typically overestimate the frequency of this branch. Figure 233(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_r{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\theta }{\left(\text{Δ}\theta \right)}^2$
Parameter	15.372	5.138
$r_0$ or ${\theta }_0$	2.890	90.927

Table 458.

The VFF model for b-antimonene.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction.

The parameters for the two-body SW potential used by GULP are shown in Table 455 . The parameters for the three-body SW potential used by GULP are shown in Table 456 . Parameters for the SW potential used by LAMMPS are listed in Table 457 .

Figure 234 shows the stress-strain relations for the single-layer b-arsenene of size 100×100 Å. The structure is uniaxially stretched in the armchair or zigzag directions at 1 and 300 K. The Young’s modulus is 50.8 and 49.9 Nm⁻¹ in the armchair and zigzag directions, respectively, at 1 K, which is obtained by linear fitting of the stress-strain relations in [0, 0.01]. The Young’s modulus is isotropic for the b-arsenene. The Poisson’s ratios from the VFF model and the SW potential are ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.21$ . The third-order nonlinear elastic constant D can be obtained by fitting the stress-strain relation to $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with E as the Young’s modulus. The values of D are −127.6 and -153.6 Nm⁻¹ at 1 K along the armchair and zigzag directions, respectively. The ultimate stress is about 7.6 Nm⁻¹ at the critical strain of 0.24 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 7.2 Nm⁻¹ at the critical strain of 0.28 in the zigzag direction at the low temperature of 1 K.

Figure 235.

Phonon dispersion for the single-layer b-antimonene. (a) The VFF model is fitted to the three acoustic branches in the long wave limit along the ΓM direction. The ab initio results (gray pentagons) are from Ref. 70. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for the b-antimonene along ΓMKΓ.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Sb─Sb	8.173	0.523	34.879	0.0	3.505

Table 459.

Two-body SW potential parameters for b-antimonene used by GULP [8], as expressed in Eq. (3).

115. b-Antimonene

The buckled (b-) antimonene is a Sb allotrope, which is also named $\beta$ antimonene. Present studies on the b-antimonene are based on first-principles calculations, and no empirical potential has been proposed for the b-antimonene. We will, thus, parametrize a set of VFF model for the single-layer b-antimonene in this section. We will also derive the SW potential based on the VFF model for the single-layer b-antimonene.

Figure 236.

Stress-strain relations for the b-antimonene of size $100\times 100$ Å. The b-antimonene is uniaxially stretched along the armchair or zigzag directions at temperatures 1 and 300 K.

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
Sb─Sb─Sb	14.100	90.927	0.523	0.523	0.0	3.505	0.0	3.503	3.505	4.577

Table 460.

Three-body SW potential parameters for b-antimonene used by GULP [8], as expressed in Eq. (4).

The structure of the single-layer b-antimonene is shown in Figure 222 , with structural parameters from the ab initio calculations [70]. The b-antimonene has a buckled configuration as shown in Figure 222(b) , where the buckle is along the zigzag direction. The height of the buckle is $h=1.65$ Å. The lattice constant is 4.12 Å, and the bond length is 2.89 Å.

Table 458 shows the VFF model for the single-layer b-antimonene. The force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the ΓM as shown in Figure 235(a) . The ab initio calculations for the phonon dispersion are from Ref. 70. Similar phonon dispersion can also be found in other ab initio calculations [64, 72, 73]. Figure 235(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Sb─Sb─Sb	1.000	0.523	6.702	14.100	1.000	−0.016	8.173	466.184	4	0	0.0

Table 461.

SW potential parameters for b-antimonene used by LAMMPS [9], as expressed in Eqs. (9) and (10).

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_r{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\theta }{\left(\text{Δ}\theta \right)}^2$
Parameter	11.529	3.853
$r_0$ or ${\theta }_0$	3.045	90.901

Table 462.

The VFF model for b-bismuthene.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction.

The parameters for the two-body SW potential used by GULP are shown in Table 459 . The parameters for the three-body SW potential used by GULP are shown in Table 460 . Parameters for the SW potential used by LAMMPS are listed in Table 461 .

Figure 236 shows the stress-strain relations for the single-layer b-antimonene of size $100\times 100$ Å. The structure is uniaxially stretched in the armchair or zigzag directions at 1 and 300 K. The Young’s modulus is 39.6 N/m in both armchair and zigzag directions at 1 K, which are obtained by linear fitting of the stress-strain relations in [0, 0.01]. The Young’s modulus is isotropic for the b-antimonene. The Poisson’s ratios from the VFF model and the SW potential are ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.24$ . The third-order nonlinear elastic constant $D$ can be obtained by fitting the stress-strain relation to $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with E as the Young’s modulus. The values of $D$ are −62.6 and −91.5 N/m at 1 K along the armchair and zigzag directions, respectively. The ultimate stress is about 7.1 N/m at the critical strain of 0.28 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 6.7 N/m at the critical strain of 0.31 in the zigzag direction at the low temperature of 1 K.

Figure 237.

Phonon dispersion for the single-layer b-bismuthene. (a) The VFF model is fitted to the three acoustic branches in the long wave limit along the ΓM direction. The ab initio results (gray pentagons) are from [64]. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for the b-bismuthene along ΓMKΓ.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Bi─Bi	6.805	0.552	42.985	0.0	3.693

Table 463.

Two-body SW potential parameters for b-bismuthene used by GULP [8], as expressed in Eq. (3).

116. b-Bismuthene

The buckled (b-) bismuthene is a Bi allotrope, which is also named $beta$ bismuthene. Most studies on the b-bismuthene are based on first-principles calculations, while a modified Morse potential was proposed for the b-bismuthene in 2013 [113]. We will parametrize a set of VFF model for the single-layer b-bismuthene in this section. We will also derive the SW potential based on the VFF model for the single-layer b-bismuthene.

Figure 238.

Stress-strain relations for the b-bismuthene of size $100\times 100$ Å. The b-bismuthene is uniaxially stretched along the armchair or zigzag directions at temperatures 1 and 300 K.

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
Bi─Bi─Bi	10.574	90.901	0.552	0.552	0.0	3.693	0.0	3.693	3.693	4.821

Table 464.

Three-body SW potential parameters for b-bismuthene used by GULP [8], as expressed in Eq. (4).

The structure of the single-layer b-bismuthene is shown in Figure 222 , with structural parameters from the ab initio calculations [70]. The b-bismuthene has a buckled configuration as shown in Figure 222(b) , where the buckle is along the zigzag direction. The height of the buckle is $h=1.73$ Å. The lattice constant is 4.34 Å, and the bond length is 3.045 Å.

Table 462 shows the VFF model for the single-layer b-bismuthene. The force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the ΓM as shown in Figure 237(a) . The ab initio calculations for the phonon dispersion are from [64]. Similar phonon dispersion can also be found in other ab initio calculations [77]. We note that the lowest-frequency branch around the Γ point from the VFF model is lower than the ab initio results. This branch is the flexural branch, which should be a quadratic dispersion. However, the ab initio calculations give a linear dispersion for the flexural branch due to the violation of the rigid rotational invariance in the first-principles package [20], so ab initio calculations typically overestimate the frequency of this branch. Figure 237(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Bi─Bi─Bi	1.000	0.552	6.690	10.574	1.000	−0.016	6.805	462.978	4	0	0.0

Table 465.

SW potential parameters for b-bismuthene used by LAMMPS [9], as expressed in Eqs. (9) and (10).

The parameters for the two-body SW potential used by GULP are shown in Table 463 . The parameters for the three-body SW potential used by GULP are shown in Table 464 . Parameters for the SW potential used by LAMMPS are listed in Table 465 .

Figure 238 shows the stress-strain relations for the single-layer b-bismuthene of size $100\times 100$ Å. The structure is uniaxially stretched in the armchair or zigzag directions at 1 and 300 K. The Young’s modulus is 27.0 N/m in both armchair and zigzag directions at 1 K, which are obtained by linear fitting of the stress-strain relations in [0, 0.01]. The Young’s modulus is isotropic for the b-bismuthene. The value of the Young’s modulus is close to the value of 23.9 N/m from the ab initio calculations [77]. The Poisson’s ratios from the VFF model and the SW potential are ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.25$ , which are comparable with the ab initio results of 0.327 [77]. The third-order nonlinear elastic constant $D$ can be obtained by fitting the stress-strain relation to $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with E as the Young’s modulus. The values of $D$ are −34.3 and −54.5 N/m at 1 K along the armchair and zigzag directions, respectively. The ultimate stress is about 5.2 N/m at the critical strain of 0.29 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 4.9 N/m at the critical strain of 0.33 in the zigzag direction at the low temperature of 1 K.

117. b-CO

Present studies on the buckled (b-) CO are based on first-principles calculations, and no empirical potential has been proposed for the b-CO. We will thus parametrize a set of SW potential for the single-layer b-CO in this section.

Figure 239.

Structure for single-layer buckled MX (b-MX), with M from group IV and X from group VI, or both M and X from group IV, or M from group III and X from group V. (a) Top view. The armchair direction is along the horizontal direction, while the zigzag direction is along the vertical direction. The unit cell is displayed by the blue rhombus. Inset shows the first Brillouin zone. (b) Side view illustrates the buckled configuration of height $h$ .

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_r{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\theta }{\left(\text{Δ}\theta \right)}^2$
Parameter	16.063	5.221
$r_0$ or ${\theta }_0$	1.636	97.181

Table 466.

The VFF model for b-CO.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction.

The structure of the single-layer b-CO is shown in Figure 239 . The structural parameters are from the ab initio calculations [78]. The b-CO has a buckled configuration as shown in Figure 239(b) , where the buckle is along the zigzag direction. This structure can be determined by two independent geometrical parameters, including the lattice constant 2.454 Å and the bond length 1.636 Å.

Figure 240.

Phonon dispersion for the single-layer b-CO. (a) The VFF model is fitted to the two in-plane acoustic branches in the long wave limit along the ΓM direction. The ab initio results (gray pentagons) are calculated from SIESTA. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for the b-CO along ΓMKΓ.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
C─O	7.656	1.054	3.582	0.0	2.293

Table 467.

Two-body SW potential parameters for b-CO used by GULP [8], as expressed in Eq. (3).

Table 466 shows the VFF model for the single-layer b-CO. The force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the ΓM as shown in Figure 240(a) . The ab initio calculations for the phonon dispersion are calculated from the SIESTA package [79]. The generalized gradients approximation is applied to account for the exchange-correlation function with Perdew, Burke, and Ernzerhof parameterization [80], and the double- $\zeta$ orbital basis set is adopted. Figure 240(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

Figure 241.

Stress-strain relations for the b-CO of size $100\times 100$ Å. The b-CO is uniaxially stretched along the armchair or zigzag directions at temperatures 1 and 300 K.

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
C-O-O	65.778	97.181	1.054	1.054	0.0	2.293	0.0	2.293	0.0	3.352
O-C-C	65.778	97.181	1.054	1.054	0.0	2.293	0.0	2.293	0.0	3.352

Table 468.

Three-body SW potential parameters for b-CO used by GULP [8], as expressed in Eq. (4).

The parameters for the two-body SW potential used by GULP are shown in Table 467 . The parameters for the three-body SW potential used by GULP are shown in Table 468 . Parameters for the SW potential used by LAMMPS are listed in Table 469 .

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
C─O─O	1.000	1.054	2.175	65.778	1.000	−0.125	7.812	2.900	4	0	0.0
O─C─C	1.000	1.054	2.175	65.778	1.000	−0.125	7.812	2.900	4	0	0.0

Table 469.

SW potential parameters for b-CO used by LAMMPS [9], as expressed in Eqs. (9) and (10).

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer b-CO under uniaxial tension at 1.0 and 300.0 K. Figure 241 shows the stress-strain curve for the tension of a single-layer b-CO of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer b-CO is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer b-CO. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 99.1 and 98.8 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.08$ .

There is no available value for nonlinear quantities in the single-layer b-CO. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −513.8 and −542.0 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 9.4 N/m at the ultimate strain of 0.18 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 9.2 N/m at the ultimate strain of 0.21 in the zigzag direction at the low temperature of 1 K.

118. b-CS

Present studies on the buckled (b-) CS are based on first-principles calculations, and no empirical potential has been proposed for the b-CS. We will thus parametrize a set of SW potential for the single-layer b-CS in this section.

The structure of the single-layer b-CS is shown in Figure 239 . The structural parameters are from the ab initio calculations [78]. The b-CS has a buckled configuration as shown in Figure 239(b) , where the buckle is along the zigzag direction. This structure can be determined by two independent geometrical parameters, including the lattice constant 2.836 Å and the bond length 1.880 Å.

Figure 242.

Phonon dispersion for the single-layer b-CS. (a) The VFF model is fitted to the two in-plane acoustic branches in the long wave limit along the ΓM direction. The ab initio results (gray pentagons) are calculated from SIESTA. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for the b-CS along ΓMKΓ.

Table 470 shows the VFF model for the single-layer b-CS. The force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the ΓM as shown in Figure 242(a) . The ab initio calculations for the phonon dispersion are calculated from the SIESTA package [79]. The generalized gradients approximation is applied to account for the exchange-correlation function with Perdew, Burke, and Ernzerhof parameterization [80], and the double- $\zeta$ orbital basis set is adopted. Figure 242(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 471 . The parameters for the three-body SW potential used by GULP are shown in Table 472 . Parameters for the SW potential used by LAMMPS are listed in Table 473 .

Figure 243.

Stress-strain relations for the b-CS of size $100\times 100$ Å. The b-CS is uniaxially stretched along the armchair or zigzag directions at temperatures 1 and 300 K.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_r{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\theta }{\left(\text{Δ}\theta \right)}^2$
Parameter	9.390	4.722
$r_0$ or ${\theta }_0$	1.880	97.921

Table 470.

The VFF model for b-CS.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer b-CS under uniaxial tension at 1.0 and 300.0 K. Figure 243 shows the stress-strain curve for the tension of a single-layer b-CS of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer b-CS is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer b-CS. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 63.5 and 63.6 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.05$ .

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
C─S	6.014	1.233	6.246	0.0	2.641

Table 471.

Two-body SW potential parameters for b-CS used by GULP [8], as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
C-S-S	61.413	97.921	1.233	1.233	0.0	2.641	0.0	2.641	0.0	3.874
S-C-C	61.413	97.921	1.233	1.233	0.0	2.641	0.0	2.641	0.0	3.874

Table 472.

Three-body SW potential parameters for b-CS used by GULP [8], as expressed in Eq. (4).

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
C─S─S	1.000	1.233	2.142	61.413	1.000	−0.138	6.014	2.703	4	0	0.0
S─C─C	1.000	1.233	2.142	61.413	1.000	−0.138	6.014	2.703	4	0	0.0

Table 473.

SW potential parameters for b-CS used by LAMMPS [9], as expressed in Eqs. (9) and (10).

There is no available value for nonlinear quantities in the single-layer b-CS. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −352.5 and −372.0 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 5.7 N/m at the ultimate strain of 0.17 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 5.5 N/m at the ultimate strain of 0.20 in the zigzag direction at the low temperature of 1 K.

Figure 244.

Phonon dispersion for the single-layer b-CSe. (a) The VFF model is fitted to the two in-plane acoustic branches in the long wave limit along the ΓM direction. The ab initio results (gray pentagons) are calculated from SIESTA. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for the b-CSe along ΓMKΓ.

119. b-CSe

Present studies on the buckled (b-) CSe are based on first-principles calculations, and no empirical potential has been proposed for the b-CSe. We will thus parametrize a set of SW potential for the single-layer b-CSe in this section.

The structure of the single-layer b-CSe is shown in Figure 239 . The structural parameters are from the ab initio calculations [78]. The b-CSe has a buckled configuration as shown in Figure 239(b) , where the buckle is along the zigzag direction. This structure can be determined by two independent geometrical parameters, including the lattice constant 3.063 Å and the bond length 2.055 Å.

Figure 245.

Stress-strain relations for the b-CSe of size $100\times 100$ Å. The b-CSe is uniaxially stretched along the armchair or zigzag directions at temperatures 1 and 300 K.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_r{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\theta }{\left(\text{Δ}\theta \right)}^2$
Parameter	10.425	5.031
$r_0$ or ${\theta }_0$	2.055	96.362

Table 474.

The VFF model for b-CSe.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction.

Table 474 shows the VFF model for the single-layer b-CSe. The force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the ΓM as shown in Figure 244(a) . The ab initio calculations for the phonon dispersion are calculated from the SIESTA package [79]. The generalized gradients approximation is applied to account for the exchange-correlation function with Perdew, Burke, and Ernzerhof parameterization [80], and the double- $\zeta$ orbital basis set is adopted. Figure 244(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
C─Se	7.691	1.298	8.917	0.0	2.872

Table 475.

Two-body SW potential parameters for b-CSe used by GULP [8], as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
C-Se-Se	61.215	96.362	1.298	1.298	0.0	2.872	0.0	2.872	0.0	4.184
Se-C-C	61.215	96.362	1.298	1.298	0.0	2.872	0.0	2.872	0.0	4.184

Table 476.

Three-body SW potential parameters for b-CSe used by GULP [8], as expressed in Eq. (4).

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
C─Se─Se	1.000	1.298	2.212	61.215	1.000	−0.111	7.691	3.137	4	0	0.0
Se─C─C	1.000	1.298	2.212	61.215	1.000	−0.111	7.691	3.137	4	0	0.0

Table 477.

SW potential parameters for b-CSe used by LAMMPS [9], as expressed in Eqs. (9) and (10).

The parameters for the two-body SW potential used by GULP are shown in Table 475 . The parameters for the three-body SW potential used by GULP are shown in Table 476 . Parameters for the SW potential used by LAMMPS are listed in Table 477 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer b-CSe under uniaxial tension at 1.0 and 300.0 K. Figure 245 shows the stress-strain curve for the tension of a single-layer b-CSe of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer b-CSe is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer b-CSe. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 61.6 and 61.4 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.09$ .

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_r{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\theta }{\left(\text{Δ}\theta \right)}^2$
Parameter	9.367	4.311
$r_0$ or ${\theta }_0$	2.231	97.239

Table 478.

The VFF model for b-CTe.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
C─Te	8.314	1.440	12.387	0.0	3.127

Table 479.

Two-body SW potential parameters for b-CTe used by GULP [8], as expressed in Eq. (3).

There is no available value for nonlinear quantities in the single-layer b-CSe. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −306.6 and −324.9 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 6.1 N/m at the ultimate strain of 0.19 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 5.9 N/m at the ultimate strain of 0.22 in the zigzag direction at the low temperature of 1 K.

120. b-CTe

Present studies on the buckled (b-) CTe are based on first-principles calculations, and no empirical potential has been proposed for the b-CTe. We will thus parametrize a set of SW potential for the single-layer b-CTe in this section.

The structure of the single-layer b-CTe is shown in Figure 239 . The structural parameters are from the ab initio calculations [78]. The b-CTe has a buckled configuration as shown in Figure 239(b) , where the buckle is along the zigzag direction. This structure can be determined by two independent geometrical parameters, including the lattice constant 3.348 Å and the bond length 2.231 Å.

Figure 246.

Phonon dispersion for the single-layer b-CTe. (a) The VFF model is fitted to the two in-plane acoustic branches in the long wave limit along the ΓM direction. The ab initio results (gray pentagons) are calculated from SIESTA. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for the b-CTe along ΓMKΓ.

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
C-Te-Te	54.451	97.239	1.440	1.440	0.0	3.127	0.0	3.127	0.0	4.573
Te-C-C	54.451	97.239	1.440	1.440	0.0	3.127	0.0	3.127	0.0	4.573

Table 480.

Three-body SW potential parameters for b-CTe used by GULP [8], as expressed in Eq. (4).

Figure 247.

Stress-strain relations for the b-CTe of size $100\times 100$ Å. The b-CTe is uniaxially stretched along the armchair or zigzag directions at temperatures 1 and 300 K.

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
C─Te─Te	1.000	1.440	2.172	54.451	1.000	−0.126	8.314	2.883	4	0	0.0
Te─C─C	1.000	1.440	2.172	54.451	1.000	−0.126	8.314	2.883	4	0	0.0

Table 481.

SW potential parameters for b-CTe used by LAMMPS [9], as expressed in Eqs. (9) and (10).

Table 478 shows the VFF model for the single-layer b-CTe. The force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the ΓM as shown in Figure 246(a) . The ab initio calculations for the phonon dispersion are calculated from the SIESTA package [79]. The generalized gradients approximation is applied to account for the exchange-correlation function with Perdew, Burke, and Ernzerhof parameterization [80], and the double- $\zeta$ orbital basis set is adopted. Figure 246(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 479 . The parameters for the three-body SW potential used by GULP are shown in Table 480 . Parameters for the SW potential used by LAMMPS are listed in Table 481 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer b-CTe under uniaxial tension at 1.0 and 300.0 K. Figure 247 shows the stress-strain curve for the tension of a single-layer b-CTe of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer b-CTe is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer b-CTe. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 48.8 and 48.3 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.12$ .

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_r{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\theta }{\left(\text{Δ}\theta \right)}^2$
Parameter	9.315	3.300
$r_0$ or ${\theta }_0$	1.884	96.676

Table 482.

The VFF model for b-SiO.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction.

There is no available value for nonlinear quantities in the single-layer b-CTe. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −306.6 and −324.9 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 5.2 N/m at the ultimate strain of 0.20 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 5.0 N/m at the ultimate strain of 0.23 in the zigzag direction at the low temperature of 1 K.

Figure 248.

Phonon dispersion for the single-layer b-SiO. (a) The VFF model is fitted to the two in-plane acoustic branches in the long wave limit along the ΓM direction. The ab initio results (gray pentagons) are calculated from SIESTA. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for the b-SiO along ΓMKΓ.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Si─O	5.819	1.200	6.299	0.0	2.636

Table 483.

Two-body SW potential parameters for b-SiO used by GULP [8], as expressed in Eq. (3).

121. b-SiO

Present studies on the buckled (b-) SiO are based on first-principles calculations, and no empirical potential has been proposed for the b-SiO. We will thus parametrize a set of SW potential for the single-layer b-SiO in this section.

Figure 249.

Stress-strain relations for the b-SiO of size $100\times 100$ Å. The b-SiO is uniaxially stretched along the armchair or zigzag directions at temperatures 1 and 300 K.

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
Si-O-O	40.695	96.676	1.200	1.200	0.0	2.636	0.0	2.636	0.0	3.845
O-Si-Si	40.695	96.676	1.200	1.200	0.0	2.636	0.0	2.636	0.0	3.845

Table 484.

Three-body SW potential parameters for b-SiO used by GULP [8], as expressed in Eq. (4).

The structure of the single-layer b-SiO is shown in Figure 239 . The structural parameters are from the ab initio calculations [78]. The b-SiO has a buckled configuration as shown in Figure 239(b) , where the buckle is along the zigzag direction. This structure can be determined by two independent geometrical parameters, including the lattice constant 2.815 Å and the bond length 1.884 Å.

Table 482 shows the VFF model for the single-layer b-SiO. The force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the ΓM as shown in Figure 248(a) . The ab initio calculations for the phonon dispersion are calculated from the SIESTA package [79]. The generalized gradients approximation is applied to account for the exchange-correlation function with Perdew, Burke, and Ernzerhof parameterization [80], and the double- $\zeta$ orbital basis set is adopted. Figure 248(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Si─O─O	1.000	1.200	2.197	40.695	1.000	−0.116	5.819	3.043	4	0	0.0
O─Si─Si	1.000	1.200	2.197	40.695	1.000	−0.116	5.819	3.043	4	0	0.0

Table 485.

SW potential parameters for b-SiO used by LAMMPS [9], as expressed in Eqs. (9) and (10).

The parameters for the two-body SW potential used by GULP are shown in Table 483 . The parameters for the three-body SW potential used by GULP are shown in Table 484 . Parameters for the SW potential used by LAMMPS are listed in Table 485 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer b-SiO under uniaxial tension at 1.0 and 300.0 K. Figure 249 shows the stress-strain curve for the tension of a single-layer b-SiO of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer b-SiO is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer b-SiO. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 51.3 and 50.3 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.11$ .

There is no available value for nonlinear quantities in the single-layer b-SiO. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −247.8 and −253.6 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 5.2 N/m at the ultimate strain of 0.20 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 5.1 N/m at the ultimate strain of 0.23 in the zigzag direction at the low temperature of 1 K.

Figure 250.

Phonon dispersion for the single-layer b-SiS. (a) The VFF model is fitted to the two in-plane acoustic branches in the long wave limit along the ΓM direction. The ab initio results (gray pentagons) are calculated from SIESTA. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for the b-SiS along ΓMKΓ.

122. b-SiS

Present studies on the buckled (b-) SiS are based on first-principles calculations, and no empirical potential has been proposed for the b-SiS. We will thus parametrize a set of SW potential for the single-layer b-SiS in this section.

The structure of the single-layer b-SiS is shown in Figure 239 . The structural parameters are from the ab initio calculations [78]. The b-SiS has a buckled configuration as shown in Figure 239(b) , where the buckle is along the zigzag direction. This structure can be determined by two independent geometrical parameters, including the lattice constant 3.299 Å and the bond length 2.321 Å.

Figure 251.

Stress-strain relations for the b-SiS of size $100\times 100$ Å. The b-SiS is uniaxially stretched along the armchair or zigzag directions at temperatures 1 and 300 K.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_r{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\theta }{\left(\text{Δ}\theta \right)}^2$
Parameter	8.441	4.802
$r_0$ or ${\theta }_0$	2.321	90.581

Table 486.

The VFF model for b-SiS..

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of $\mathrm{\AA }$ ) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction.

Table 486 shows the VFF model for the single-layer b-SiS. The force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the ΓM as shown in Figure 250(a) . The ab initio calculations for the phonon dispersion are calculated from the SIESTA package [79]. The generalized gradients approximation is applied to account for the exchange-correlation function with Perdew, Burke, and Ernzerhof parameterization [80], and the double- $\zeta$ orbital basis set is adopted. Figure 250(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Si─S	6.897	1.264	14.510	0.0	3.177

Table 487.

Two-body SW potential parameters for b-SiS used by GULP [8], as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
Si-S-S	45.954	90.581	1.264	1.264	0.0	3.177	0.0	3.177	0.0	4.506
S-Si-Si	45.954	90.581	1.264	1.264	0.0	3.177	0.0	3.177	0.0	4.506

Table 488.

Three-body SW potential parameters for b-SiS used by GULP [8], as expressed in Eq. (4).

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Si─S─S	1.000	1.264	2.514	45.954	1.000	−0.010	6.897	5.687	4	0	0.0
S─Si─Si	1.000	1.264	2.514	45.954	1.000	−0.010	6.897	5.687	4	0	0.0

Table 489.

SW potential parameters for b-SiS used by LAMMPS [9], as expressed in Eqs. (9) and (10).

The parameters for the two-body SW potential used by GULP are shown in Table 487 . The parameters for the three-body SW potential used by GULP are shown in Table 488 . Parameters for the SW potential used by LAMMPS are listed in Table 489 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer b-SiS under uniaxial tension at 1.0 and 300.0 K. Figure 251 shows the stress-strain curve for the tension of a single-layer b-SiS of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer b-SiS is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer b-SiS. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 45.5 and 45.8 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.13$ .

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_r{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\theta }{\left(\text{Δ}\theta \right)}^2$
Parameter	8.441	4.802
$r_0$ or ${\theta }_0$	2.477	90.590

Table 490.

The VFF model for b-SiSe.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Si─Se	7.857	1.349	18.822	0.0	3.391

Table 491.

Two-body SW potential parameters for b-SiSe used by GULP [8], as expressed in Eq. (3).

There is no available value for nonlinear quantities in the single-layer b-SiS. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −196.4 and −217.9 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 5.1 N/m at the ultimate strain of 0.21 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 4.9 N/m at the ultimate strain of 0.24 in the zigzag direction at the low temperature of 1 K.

123. b-SiSe

Present studies on the buckled (b-) SiSe are based on first-principles calculations, and no empirical potential has been proposed for the b-SiSe. We will thus parametrize a set of SW potential for the single-layer b-SiSe in this section.

The structure of the single-layer b-SiSe is shown in Figure 239 . The structural parameters are from the ab initio calculations [78]. The b-SiSe has a buckled configuration as shown in Figure 239(b) , where the buckle is along the zigzag direction. This structure can be determined by two independent geometrical parameters, including the lattice constant 3.521 Å and the bond length 2.477 Å.

Figure 252.

Phonon dispersion for the single-layer b-SiSe. (a) The VFF model is fitted to the two in-plane acoustic branches in the long wave limit along the ΓM direction. The ab initio results (gray pentagons) are calculated from SIESTA. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for the b-SiSe along ΓMKΓ.

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
Si-Se-Se	45.968	90.590	1.349	1.349	0.0	3.391	0.0	3.391	0.0	4.810
Se-Si-Si	45.968	90.590	1.349	1.349	0.0	3.391	0.0	3.391	0.0	4.810

Table 492.

Three-body SW potential parameters for b-SiSe used by GULP [8], as expressed in Eq. (4).

Figure 253.

Stress-strain relations for the b-SiSe of size $100\times 100$ Å. The b-SiSe is uniaxially stretched along the armchair or zigzag directions at temperatures 1 and 300 K.

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Si─Se─Se	1.000	1.349	2.514	45.968	1.000	−0.010	7.857	5.6683	4	0	0.0
Se─Si─Si	1.000	1.349	2.514	45.968	1.000	−0.010	7.857	5.6683	4	0	0.0

Table 493.

SW potential parameters for b-SiSe used by LAMMPS [9], as expressed in Eqs. (9) and (10).

Table 490 shows the VFF model for the single-layer b-SiSe. The force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the ΓM as shown in Figure 252(a) . The ab initio calculations for the phonon dispersion are calculated from the SIESTA package [79]. The generalized gradients approximation is applied to account for the exchange-correlation function with Perdew, Burke, and Ernzerhof parameterization [80], and the double- $\zeta$ orbital basis set is adopted. Figure 252(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 491 . The parameters for the three-body SW potential used by GULP are shown in Table 492 . Parameters for the SW potential used by LAMMPS are listed in Table 493 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer b-SiSe under uniaxial tension at 1.0 and 300.0 K. Figure 253 shows the stress-strain curve for the tension of a single-layer b-SiSe of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer b-SiSe is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer b-SiSe. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 41.8 and 41.9 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.15$ .

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_r{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\theta }{\left(\text{Δ}\theta \right)}^2$
Parameter	8.418	4.349
$r_0$ or ${\theta }_0$	2.690	90.779

Table 494.

The VFF model for b-SiTe.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction.

There is no available value for nonlinear quantities in the single-layer b-SiSe. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −169.9 and −188.0 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 4.9 N/m at the ultimate strain of 0.22 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 4.7 N/m at the ultimate strain of 0.25 in the zigzag direction at the low temperature of 1 K.

Figure 254.

Phonon dispersion for the single-layer b-SiTe. (a) The VFF model is fitted to the three acoustic branches in the long wave limit along the ΓM direction. The ab initio results (gray pentagons) are from Ref. 83. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for the b-SiTe along ΓMKΓ.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Si─Te	9.285	1.473	26.181	0.0	3.685

Table 495.

Two-body SW potential parameters for b-SiTe used by GULP [8], as expressed in Eq. (3).

124. b-SiTe

Present studies on the buckled (b-) SiTe are based on first-principles calculations, and no empirical potential has been proposed for the b-SiTe. We will thus parametrize a set of SW potential for the single-layer b-SiTe in this section.

Figure 255.

Stress-strain relations for the b-SiTe of size $100\times 100$ Å. The b-SiTe is uniaxially stretched along the armchair or zigzag directions at temperatures 1 and 300 K.

	K (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
Si-Te-Te	41.952	90.779	1.473	1.473	0.0	3.685	0.0	3.685	0.0	5.232
Te-Si-Si	41.952	90.779	1.473	1.473	0.0	3.685	0.0	3.685	0.0	5.232

Table 496.

Three-body SW potential parameters for b-SiTe used by GULP [8], as expressed in Eq. (4).

The structure of the single-layer b-SiTe is shown in Figure 239 . The structural parameters are from the ab initio calculations [83]. The b-SiTe has a buckled configuration as shown in Figure 239(b) , where the buckle is along the zigzag direction. This structure can be determined by two independent geometrical parameters, e.g., the lattice constant 3.83 Å and the bond length 2.689 Å. The resultant height of the buckle is $h=1.53$ Å.

Table 494 shows the VFF model for the single-layer b-SiTe. The force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the ΓM as shown in Figure 254(a) . The ab initio calculations for the phonon dispersion are from [83]. Figure 254(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Si─Te─Te	1.000	0.725	4.497	27.653	1.000	−0.014	4.428	113.714	4	0	0.0
Te─Si─Si	1.000	0.725	4.497	27.653	1.000	−0.014	4.428	113.714	4	0	0.0

Table 497.

SW potential parameters for b-SiTe used by LAMMPS [9], as expressed in Eqs. (9) and (10).

The parameters for the two-body SW potential used by GULP are shown in Table 495 . The parameters for the three-body SW potential used by GULP are shown in Table 496 . Parameters for the SW potential used by LAMMPS are listed in Table 497 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer b-SiTe under uniaxial tension at 1.0 and 300.0 K. Figure 255 shows the stress-strain curve for the tension of a single-layer b-SiTe of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer b-SiTe is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer b-SiTe. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 34.3 and 34.6 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. These values agree with the ab initio result at 0 K temperature, e.g., 34.1 N/m in [83]. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.18$ , which agrees with the ab initio result [83] of 0.18.

There is no available value for nonlinear quantities in the single-layer b-SiTe. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −119.3 and −137.2 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 4.6 N/m at the ultimate strain of 0.24 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 4.4 N/m at the ultimate strain of 0.27 in the zigzag direction at the low temperature of 1 K.

125. b-GeO

Present studies on the buckled (b-) GeO are based on first-principles calculations, and no empirical potential has been proposed for the b-GeO. We will thus parametrize a set of SW potential for the single-layer b-GeO in this section.

Figure 256.

Phonon dispersion for the single-layer b-GeO. (a) The VFF model is fitted to the two in-plane acoustic branches in the long wave limit along the ΓM direction. The ab initio results (gray pentagons) are calculated from SIESTA. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for the b-GeO along ΓMKΓ.

The structure of the single-layer b-GeO is shown in Figure 239 . The structural parameters are from the ab initio calculations [78]. The b-GeO has a buckled configuration as shown in Figure 239(b) , where the buckle is along the zigzag direction. This structure can be determined by two independent geometrical parameters, including the lattice constant 3.124 Å and the bond length 2.032 Å.

Table 498 shows the VFF model for the single-layer b-GeO. The force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the ΓM as shown in Figure 256(a) . The ab initio calculations for the phonon dispersion are calculated from the SIESTA package [79]. The generalized gradients approximation is applied to account for the exchange-correlation function with Perdew, Burke, and Ernzerhof parameterization [80], and the double- $\zeta$ orbital basis set is adopted. Figure 256(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

Figure 257.

Stress-strain relations for the b-GeO of size $100\times 100$ Å. The b-GeO is uniaxially stretched along the armchair or zigzag directions at temperatures 1 and 300 K.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_r{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\theta }{\left(\text{Δ}\theta \right)}^2$
Parameter	9.315	3.300
$r_0$ or ${\theta }_0$	2.032	100.475

Table 498.

The VFF model for b-GeO.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Ge─O	7.390	1.413	8.524	0.0	2.879

Table 499.

Two-body SW potential parameters for b-GeO used by GULP [8], as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
Ge-O-O	47.962	100.475	1.413	1.413	0.0	2.879	0.0	2.879	0.0	4.267
O-Ge-Ge	47.962	100.475	1.413	1.413	0.0	2.879	0.0	2.879	0.0	4.267

Table 500.

Three-body SW potential parameters for b-GeO used by GULP [8], as expressed in Eq. (4).

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Ge─O─O	1.000	1.413	2.037	47.962	1.000	−0.182	7.390	2.136	4	0	0.0
O─Ge─Ge	1.000	1.413	2.037	47.962	1.000	−0.182	7.390	2.136	4	0	0.0

Table 501.

SW potential parameters for b-GeO used by LAMMPS [9], as expressed in Eqs. (9) and (10).

The parameters for the two-body SW potential used by GULP are shown in Table 499 . The parameters for the three-body SW potential used by GULP are shown in Table 500 . Parameters for the SW potential used by LAMMPS are listed in Table 501 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer b-GeO under uniaxial tension at 1.0 and 300.0 K. Figure 257 shows the stress-strain curve for the tension of a single-layer b-GeO of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer b-GeO is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer b-GeO. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 47.5 and 46.8 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.11$ .

There is no available value for nonlinear quantities in the single-layer b-GeO. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −224.6 and −232.8 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 4.9 N/m at the ultimate strain of 0.20 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 4.8 N/m at the ultimate strain of 0.23 in the zigzag direction at the low temperature of 1 K.

Figure 258.

Phonon dispersion for the single-layer b-GeS. (a) The VFF model is fitted to the two in-plane acoustic branches in the long wave limit along the ΓM direction. The ab initio results (gray pentagons) are calculated from SIESTA. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for the b-GeS along ΓMKΓ.

126. b-GeS

Present studies on the buckled (b-) GeS are based on first-principles calculations, and no empirical potential has been proposed for the b-GeS. We will thus parametrize a set of SW potential for the single-layer b-GeS in this section.

The structure of the single-layer b-GeS is shown in Figure 239 . The structural parameters are from the ab initio calculations [78]. The b-GeS has a buckled configuration as shown in Figure 239(b) , where the buckle is along the zigzag direction. This structure can be determined by two independent geometrical parameters, including the lattice constant 3.485 Å and the bond length 2.428 Å.

Figure 259.

Stress-strain relations for the b-GeS of size $100\times 100$ Å. The b-GeS is uniaxially stretched along the armchair or zigzag directions at temperatures 1 and 300 K.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_r{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\theta }{\left(\text{Δ}\theta \right)}^2$
Parameter	8.322	3.516
$r_0$ or ${\theta }_0$	2.428	91.725

Table 502.

The VFF model for b-GeS.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Ge─S	7.657	1.363	17.377	0.0	3.338

Table 503.

Two-body SW potential parameters for b-GeS used by GULP [8], as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
Ge-S-S	35.249	91.725	1.363	1.363	0.0	3.338	0.0	3.338	0.0	4.761
S-Ge-Ge	35.249	91.725	1.363	1.363	0.0	3.338	0.0	3.338	0.0	4.761

Table 504.

Three-body SW potential parameters for b-GeS used by GULP [8], as expressed in Eq. (4).

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Ge─S─S	1.000	1.363	2.448	35.249	1.000	−0.030	7.657	5.030	4	0	0.0
S─Ge─Ge	1.000	1.363	2.448	35.249	1.000	−0.030	7.657	5.030	4	0	0.0

Table 505.

SW potential parameters for b-GeS used by LAMMPS [9], as expressed in Eqs. (9) and (10).

Table 502 shows the VFF model for the single-layer b-GeS. The force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the ΓM as shown in Figure 258(a) . The ab initio calculations for the phonon dispersion are calculated from the SIESTA package [79]. The generalized gradients approximation is applied to account for the exchange-correlation function with Perdew, Burke, and Ernzerhof parameterization [80], and the double- $\zeta$ orbital basis set is adopted. Figure 258(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 503 . The parameters for the three-body SW potential used by GULP are shown in Table 504 . Parameters for the SW potential used by LAMMPS are listed in Table 505 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer b-GeS under uniaxial tension at 1.0 and 300.0 K. Figure 259 shows the stress-strain curve for the tension of a single-layer b-GeS of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer b-GeS is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer b-GeS. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 34.9 and 34.1 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.18$ .

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_r{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\theta }{\left(\text{Δ}\theta \right)}^2$
Parameter	8.322	3.516
$r_0$ or ${\theta }_0$	2.568	91.406

Table 506.

The VFF model for b-GeSe.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Ge─S	8.498	1.430	21.745	0.0	3.526

Table 507.

Two-body SW potential parameters for b-GeSe used by GULP [8], as expressed in Eq. (3).

There is no available value for nonlinear quantities in the single-layer b-GeS. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity D can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with E as the Young’s modulus. The values of D from the present SW potential are −128.1 and −135.0 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 4.5 N/m at the ultimate strain of 0.23 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 4.3 N/m at the ultimate strain of 0.27 in the zigzag direction at the low temperature of 1 K.

Figure 260.

Phonon dispersion for the single-layer b-GeSe. (a) The VFF model is fitted to the two in-plane acoustic branches in the long wave limit along the ΓM direction. The ab initio results (gray pentagons) are calculated from SIESTA. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for the b-GeSe along ΓMKΓ.

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
Ge─Se─Se	34.791	91.406	1.430	1.430	0.0	3.526	0.0	3.526	0.0	5.021
Se─Ge─Ge	34.791	91.406	1.430	1.430	0.0	3.526	0.0	3.526	0.0	5.021

Table 508.

Three-body SW potential parameters for b-GeSe used by GULP [8], as expressed in Eq. (4).

127. B-GeSe

Present studies on the buckled (b-) GeSe are based on first-principles calculations, and no empirical potential has been proposed for the b-GeSe. We will thus parametrize a set of SW potential for the single-layer b-GeSe in this section.

Figure 261.

Stress-strain relations for the b-GeSe of size $100\times 100$ Å. The b-GeSe is uniaxially stretched along the armchair or zigzag directions at temperatures 1 and 300 K.

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Ge─Se─Se	1.000	1.430	2.466	34.791	1.000	−0.025	8.498	5.205	4	0	0.0
Se─Ge─Ge	1.000	1.430	2.466	34.791	1.000	−0.025	8.498	5.205	4	0	0.0

Table 509.

SW potential parameters for b-GeSe used by LAMMPS [9], as expressed in Eqs. (9) and (10).

The structure of the single-layer b-GeSe is shown in Figure 239 . The structural parameters are from the ab initio calculations [78]. The b-GeSe has a buckled configuration as shown in Figure 239(b) , where the buckle is along the zigzag direction. This structure can be determined by two independent geometrical parameters, including the lattice constant 3.676 Å and the bond length 2.568 Å.

Table 506 shows the VFF model for the single-layer b-GeSe. The force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the ΓM as shown in Figure 260(a) . The ab initio calculations for the phonon dispersion are calculated from the SIESTA package [79]. The generalized gradient approximation is applied to account for the exchange-correlation function with Perdew, Burke, and Ernzerhof parameterization [80], and the double- $\zeta$ orbital basis set is adopted. Figure 260(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 507 . The parameters for the three-body SW potential used by GULP are shown in Table 508 . Parameters for the SW potential used by LAMMPS are listed in Table 509 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer b-GeSe under uniaxial tension at 1 and 300 K. Figure 261 shows the stress-strain curve for the tension of a single-layer b-GeSe of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer b-GeSe is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer b-GeSe. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 31.6 and 31.5 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}={\nu }_{yx}=0.19$ .

There is no available value for nonlinear quantities in the single-layer b-GeSe. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −105.2 and −118.3 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 4.4 N/m at the ultimate strain of 0.24 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 4.2 N/m at the ultimate strain of 0.28 in the zigzag direction at the low temperature of 1 K.

128. B-GeTe

Present studies on the buckled (b-) GeTe are based on first-principles calculations, and no empirical potential has been proposed for the b-GeTe. We will thus parametrize a set of SW potential for the single-layer b-GeTe in this section.

The structure of the single-layer b-GeTe is shown in Figure 239 . The structural parameters are from the ab initio calculations [78]. The b-GeTe has a buckled configuration as shown in Figure 239(b) , where the buckle is along the zigzag direction. This structure can be determined by two independent geometrical parameters, including the lattice constant 3.939 Å and the bond length 2.768 Å.

Figure 262.

Phonon dispersion for the single-layer b-GeTe. (a) The VFF model is fitted to the two in-plane acoustic branches in the long wave limit along the ΓM direction. The ab initio results (gray pentagons) are calculated from SIESTA. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for the b-GeTe along ΓMKΓ.

Table 510 shows the VFF model for the single-layer b-GeTe. The force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the ΓM as shown in Figure 262(a) . The ab initio calculations for the phonon dispersion are calculated from the SIESTA package [79]. The generalized gradient approximation is applied to account for the exchange-correlation function with Perdew, Burke, and Ernzerhof parameterization [80], and the double- $\zeta$ orbital basis set is adopted. Figure 262(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

Figure 263.

Stress-strain relations for the b-GeTe of size $100\times 100$ Å. The b-GeTe is uniaxially stretched along the armchair or zigzag directions at temperatures 1 and 300 K.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_r{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\theta }{\left(\text{Δ}\theta \right)}^2$
Parameter	8.322	3.516
$r_0$ or ${\theta }_0$	2.768	90.718

Table 510.

The VFF model for b-GeTe.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of ev/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of $\mathrm{\AA }$ ) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction.

The parameters for the two-body SW potential used by GULP are shown in Table 511 . The parameters for the three-body SW potential used by GULP are shown in Table 512 . Parameters for the SW potential used by LAMMPS are listed in Table 513 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer b-GeTe under uniaxial tension at 1 and 300 K. Figure 263 shows the stress-strain curve for the tension of a single-layer b-GeTe of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer b-GeTe is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer b-GeTe. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 27.7 and 28.0 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}={\nu }_{yx}=0.21$ .

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Ge─Te	9.704	1.513	29.352	0.0	3.791

Table 511.

Two-body SW potential parameters for b-GeTe used by GULP [8], as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
Ge─Te─Te	33.832	90.718	1.513	1.513	0.0	3.791	0.0	3.791	0.0	5.381
Te─Ge─Ge	33.832	90.718	1.513	1.513	0.0	3.791	0.0	3.791	0.0	5.381

Table 512.

Three-body SW potential parameters for b-GeTe used by GULP [8], as expressed in Eq. (4).

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Ge─Te─Te	1.000	1.513	2.506	33.832	1.000	−0.013	9.704	5.605	4	0	0.0
Te─Ge─Ge	1.000	1.513	2.506	33.832	1.000	−0.013	9.704	5.605	4	0	0.0

Table 513.

SW potential parameters for b-GeTe used by LAMMPS [9], as expressed in Eqs. (9) and (10).

There is no available value for nonlinear quantities in the single-layer b-GeTe. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −80.4 and −95.9 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 4.3 N/m at the ultimate strain of 0.26 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 4.1 N m⁻¹ at the ultimate strain of 0.30 in the zigzag direction at the low temperature of 1 K.

Figure 264.

Phonon dispersion for the single-layer b-SnO. (a) The VFF model is fitted to the two in-plane acoustic branches in the long wave limit along the ΓM direction. The ab initio results (gray pentagons) are calculated from SIESTA. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for the b-SnO along ΓMKΓ.

129. B-SnO

Present studies on the buckled (b-) SnO are based on first-principles calculations, and no empirical potential has been proposed for the b-SnO. We will thus parametrize a set of SW potential for the single-layer b-SnO in this section.

The structure of the single-layer b-SnO is shown in Figure 239 . The structural parameters are from the ab initio calculations [78]. The b-SnO has a buckled configuration as shown in Figure 239(b) , where the buckle is along the zigzag direction. This structure can be determined by two independent geometrical parameters, including the lattice constant 3.442 Å and the bond length 2.204 Å.

Figure 265.

Stress-strain relations for the b-SnO of size $100\times 100$ Å. The b-SnO is uniaxially stretched along the armchair or zigzag directions at temperatures 1 and 300 K.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_r{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\theta }{\left(\text{Δ}\theta \right)}^2$
Parameter	9.315	3.300
$r_0$ or ${\theta }_0$	2.204	102.677

Table 514.

The VFF model for b-SnO.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of ev/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of $\mathrm{\AA }$ ) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction.

Table 514 shows the VFF model for the single-layer b-SnO. The force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the ΓM as shown in Figure 264(a) . The ab initio calculations for the phonon dispersion are calculated from the SIESTA package [79]. The generalized gradient approximation is applied to account for the exchange-correlation function with Perdew, Burke, and Ernzerhof parameterization [80], and the double- $\zeta$ orbital basis set is adopted. Figure 264(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Sn─O	9.133	1.609	11.798	0.0	3.146

Table 515.

Two-body SW potential parameters for b-SnO used by GULP [8], as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
Sn─O─O	52.875	102.677	1.609	1.609	0.0	3.146	0.0	3.146	0.0	4.702
O─Sn─Sn	52.875	102.677	1.609	1.609	0.0	3.146	0.0	3.146	0.0	4.702

Table 516.

Three-body SW potential parameters for b-SnO used by GULP [8], as expressed in Eq. (4).

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Sn─O─O	1.000	1.609	1.955	52.875	1.000	−0.219	9.133	1.760	4	0	0.0
O─Sn─Sn	1.000	1.609	1.955	52.875	1.000	−0.219	9.133	1.760	4	0	0.0

Table 517.

SW potential parameters for b-SnO used by LAMMPS [9], as expressed in Eqs. (9) and (10).

The parameters for the two-body SW potential used by GULP are shown in Table 515 . The parameters for the three-body SW potential used by GULP are shown in Table 516 . Parameters for the SW potential used by LAMMPS are listed in Table 517 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer b-SnO under uniaxial tension at 1 and 300 K. Figure 265 shows the stress-strain curve for the tension of a single-layer b-SnO of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer b-SnO is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer b-SnO. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 43.7 and 43.8 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}={\nu }_{yx}=0.12$ .

There is no available value for nonlinear quantities in the single-layer b-SnO. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −199.9 and −215.1 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 4.7 N/m at the ultimate strain of 0.21 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 4.5 N/m at the ultimate strain of 0.24 in the zigzag direction at the low temperature of 1 K.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_r{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\theta }{\left(\text{Δ}\theta \right)}^2$
Parameter	6.909	2.710
$r_0$ or ${\theta }_0$	2.616	91.793

Table 518.

The VFF model for b-SnS.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of ev/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of $\mathrm{\AA }$ ) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Sn─S	7.392	1.472	23.416	0.0	3.597

Table 519.

Two-body SW potential parameters for b-SnS used by GULP [8], as expressed in Eq. (3).

130. B-SnS

Present studies on the buckled (b-) SnS are based on first-principles calculations, and no empirical potential has been proposed for the b-SnS. We will thus parametrize a set of SW potential for the single-layer b-SnS in this section.

The structure of the single-layer b-SnS is shown in Figure 239 . The structural parameters are from the ab initio calculations [78]. The b-SnS has a buckled configuration as shown in Figure 239(b) , where the buckle is along the zigzag direction. This structure can be determined by two independent geometrical parameters, including the lattice constant 3.757 Å and the bond length 2.616 Å.

Figure 266.

Phonon dispersion for the single-layer b-SnS. (a) The VFF model is fitted to the two in-plane acoustic branches in the long wave limit along the ΓM direction. The ab initio results (gray pentagons) are calculated from SIESTA. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for the b-SnS along ΓMKΓ.

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
Sn─S─S	27.243	91.793	1.472	1.472	0.0	3.597	0.0	3.597	0.0	5.132
S─Sn─Sn	27.243	91.793	1.472	1.472	0.0	3.597	0.0	3.597	0.0	5.132

Table 520.

Three-body SW potential parameters for b-SnS used by GULP [8], as expressed in Eq. (4).

Table 518 shows the VFF model for the single-layer b-SnS. The force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the ΓM as shown in Figure 266(a) . The ab initio calculations for the phonon dispersion are calculated from the SIESTA package [79]. The generalized gradient approximation is applied to account for the exchange-correlation function with Perdew, Burke, and Ernzerhof parameterization [80], and the double- $\zeta$ orbital basis set is adopted. Figure 266(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

Figure 267.

Stress-strain relations for the b-SnS of size $100\times 100$ Å. The b-SnS is uniaxially stretched along the armchair or zigzag directions at temperatures 1 and 300 K.

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Sn─S─S	1.000	1.472	2.444	27.243	1.000	−0.031	7.392	4.994	4	0	0.0
S─Sn─Sn	1.000	1.472	2.444	27.243	1.000	−0.031	7.392	4.994	4	0	0.0

Table 521.

SW potential parameters for b-SnS used by LAMMPS [9], as expressed in Eqs. (9) and (10).

The parameters for the two-body SW potential used by GULP are shown in Table 519 . The parameters for the three-body SW potential used by GULP are shown in Table 520 . Parameters for the SW potential used by LAMMPS are listed in Table 521 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer b-SnS under uniaxial tension at 1 and 300 K. Figure 267 shows the stress-strain curve for the tension of a single-layer b-SnS of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer b-SnS is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer b-SnS. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 23.8 and 24.4 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}={\nu }_{yx}=0.20$ .

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_r{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\theta }{\left(\text{Δ}\theta \right)}^2$
Parameter	6.909	2.710
$r_0$ or ${\theta }_0$	2.747	90.923

Table 522.

The VFF model for b-SnSe.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of ev/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of $\mathrm{\AA }$ ) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction.

There is no available value for nonlinear quantities in the single-layer b-SnS. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −71.8 and −88.3 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 3.5 N/m at the ultimate strain of 0.25 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 3.4 N/m at the ultimate strain of 0.29 in the zigzag direction at the low temperature of 1 K.

131. B-SnSe

Present studies on the buckled (b-) SnSe are based on first-principles calculations, and no empirical potential has been proposed for the b-SnSe. We will thus parametrize a set of SW potential for the single-layer b-SnSe in this section.

Figure 268.

Phonon dispersion for the single-layer b-SnSe. (a) The VFF model is fitted to the two in-plane acoustic branches in the long wave limit along the ΓM direction. The ab initio results (gray pentagons) are calculated from SIESTA. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for the b-SnSe along ΓMKΓ.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Si─Se	7.976	1.510	28.471	0.0	3.765

Table 523.

Two-body SW potential parameters for b-SnSe used by GULP [8], as expressed in Eq. (3).

The structure of the single-layer b-SnSe is shown in Figure 239 . The structural parameters are from the ab initio calculations [78]. The b-SnSe has a buckled configuration as shown in Figure 239(b) , where the buckle is along the zigzag direction. This structure can be determined by two independent geometrical parameters, including the lattice constant 3.916 Å and the bond length 2.747 Å.

Figure 269.

Stress-strain relations for the b-SnSe of size $100\times 100$ Å. The b-SnSe is uniaxially stretched along the armchair or zigzag directions at temperatures 1 and 300 K.

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
Sn─Se─Se	26.294	90.923	1.510	1.510	0.0	3.765	0.0	3.765	0.0	5.349
Se─Sn─Sn	26.294	90.923	1.510	1.510	0.0	3.765	0.0	3.765	0.0	5.349

Table 524.

Three-body SW potential parameters for b-SnSe used by GULP [8], as expressed in Eq. (4).

Table 522 shows the VFF model for the single-layer b-SnSe. The force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the ΓM as shown in Figure 268(a) . The ab initio calculations for the phonon dispersion are calculated from the SIESTA package [79]. The generalized gradient approximation is applied to account for the exchange-correlation function with Perdew, Burke, and Ernzerhof parameterization [80], and the double- $\zeta$ orbital basis set is adopted. Figure 268(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Sn─Se─Se	1.000	1.510	2.494	26.294	1.000	−0.016	7.976	5.482	4	0	0.0
Se─Sn─Sn	1.000	1.510	2.494	26.294	1.000	−0.016	7.976	5.482	4	0	0.0

Table 525.

SW potential parameters for b-SnSe used by LAMMPS [9], as expressed in Eqs. (9) and (10).

The parameters for the two-body SW potential used by GULP are shown in Table 523 . The parameters for the three-body SW potential used by GULP are shown in Table 524 . Parameters for the SW potential used by LAMMPS are listed in Table 525 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer b-SnSe under uniaxial tension at 1 and 300 K. Figure 269 shows the stress-strain curve for the tension of a single-layer b-SnSe of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer b-SnSe is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer b-SnSe. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 22.0 and 22.2 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}={\nu }_{yx}=0.22$ .

There is no available value for nonlinear quantities in the single-layer b-SnSe. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −61.6 and −73.69 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 3.5 N/m at the ultimate strain of 0.27 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 3.3 N/m at the ultimate strain of 0.31 in the zigzag direction at the low temperature of 1 K.

132. B-SnTe

Present studies on the buckled (b-) SnTe are based on first-principles calculations, and no empirical potential has been proposed for the b-SnTe. We will thus parametrize a set of SW potential for the single-layer b-SnTe in this section.

The structure of the single-layer b-SnTe is shown in Figure 239 . The structural parameters are from the ab initio calculations [78]. The b-SnTe has a buckled configuration as shown in Figure 239(b) , where the buckle is along the zigzag direction. This structure can be determined by two independent geometrical parameters, including the lattice constant 4.151 Å and the bond length 2.947 Å.

Figure 270.

Phonon dispersion for the single-layer b-SnTe. (a) The VFF model is fitted to the two in-plane acoustic branches in the long wave limit along the ΓM direction. The ab initio results (gray pentagons) are calculated from SIESTA. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for the b-SnTe along ΓMKΓ.

Table 526 shows the VFF model for the single-layer b-SnTe. The force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the ΓM as shown in Figure 270(a) . The ab initio calculations for the phonon dispersion are calculated from the SIESTA package [79]. The generalized gradient approximation is applied to account for the exchange-correlation function with Perdew, Burke, and Ernzerhof parameterization [80], and the double- $\zeta$ orbital basis set is adopted. Figure 270(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

Figure 271.

Stress-strain relations for the b-SnTe of size $100\times 100$ Å. The b-SnTe is uniaxially stretched along the armchair or zigzag directions at temperatures 1 and 300 K.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_r{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\theta }{\left(\text{Δ}\theta \right)}^2$
Parameter	6.909	2.710
$r_0$ or ${\theta }_0$	2.947	89.542

Table 526.

The VFF model for b-SnTe.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of ev/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of $\mathrm{\AA }$ ) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction.

The parameters for the two-body SW potential used by GULP are shown in Table 527 . The parameters for the three-body SW potential used by GULP are shown in Table 528 . Parameters for the SW potential used by LAMMPS are listed in Table 529 .

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Sn─Te	8.864	1.559	37.713	0.0	4.019

Table 527.

Two-body SW potential parameters for b-SnTe used by GULP [8], as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
Sn─Te─Te	24.867	89.542	1.559	1.559	0.0	4.019	0.0	4.019	0.0	5.670
Te─Sn─Sn	24.867	89.542	1.559	1.559	0.0	4.019	0.0	4.019	0.0	5.670

Table 528.

Three-body SW potential parameters for b-SnTe used by GULP [8], as expressed in Eq. (4).

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Sn─Te─Te	1.000	1.559	2.577	24.867	1.000	0.008	8.864	6.378	4	0	0.0
Te─Sn─Sn	1.000	1.559	2.577	24.867	1.000	0.008	8.864	6.378	4	0	0.0

Table 529.

SW potential parameters for b-SnTe used by LAMMPS [9], as expressed in Eqs. (9) and (10).

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer b-SnTe under uniaxial tension at 1 and 300 K. Figure 271 shows the stress-strain curve for the tension of a single-layer b-SnTe of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer b-SnTe is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer b-SnTe. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 19.6 and 19.1 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}={\nu }_{yx}=0.23$ .

Figure 272.

Phonon dispersion for the single-layer b-SnGe. (a) The VFF model is fitted to the three acoustic branches in the long wave limit along the ΓM direction. The ab initio results (gray pentagons) are from [114]. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for the b-SnGe along ΓMKΓ.

There is no available value for nonlinear quantities in the single-layer b-SnTe. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −48.7 and −54.3 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 3.4 N/m at the ultimate strain of 0.29 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 3.3 N/m at the ultimate strain of 0.33 in the zigzag direction at the low temperature of 1 K.

133. B-SnGe

Present studies on the buckled SnGe (b-SnGe) are based on first-principles calculations, and no empirical potential has been proposed for the b-SnGe. We will thus parametrize a set of SW potential for the single-layer b-SnGe in this section.

The structure of the single-layer b-SnGe is shown in Figure 239 . The structural parameters are from the ab initio calculations [114]. The b-SnGe has a buckled configuration as shown in Figure 239(b) , where the buckle is along the zigzag direction. This structure can be determined by two independent geometrical parameters, e.g., the lattice constant 4.27 Å and the bond length 2.57 Å. The resultant height of the buckle is $h=0.73$ Å.

Figure 273.

Stress-strain relations for the b-SnGe of size $100\times 100$ Å. The b-SnGe is uniaxially stretched along the armchair or zigzag directions at temperatures 1 and 300 K.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_r{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\theta }{\left(\text{Δ}\theta \right)}^2$
Parameter	8.390	3.112
$r_0$ or ${\theta }_0$	2.570	112.350

Table 530.

The VFF model for b-SnGe.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of ev/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of $\mathrm{\AA }$ ) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction.

Table 530 shows the VFF model for the single-layer b-SnGe. The force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the ΓM as shown in Figure 272(a) . The ab initio calculations for the phonon dispersion are from [114]. Figure 272(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Sn─Ge	13.674	2.267	21.812	0.0	3.777

Table 531.

Two-body SW potential parameters for b-SnGe used by GULP [8], as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
Sn─Ge─Ge	77.881	112.350	2.267	2.267	0.0	3.777	0.0	3.777	0.0	5.833
Ge─Sn─Sn	77.881	112.350	2.267	2.267	0.0	3.777	0.0	3.777	0.0	5.833

Table 532.

Three-body SW potential parameters for b-SnGe used by GULP [8], as expressed in Eq. (4).

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Sn─Ge─Ge	1.000	2.267	1.666	77.881	1.000	−0.380	13.674	0.826	4	0	0.0
Ge─Sn─Sn	1.000	2.267	1.666	77.881	1.000	−0.380	13.674	0.826	4	0	0.0

Table 533.

SW potential parameters for b-SnGe used by LAMMPS [9], as expressed in Eqs. (9) and (10).

The parameters for the two-body SW potential used by GULP are shown in Table 531 . The parameters for the three-body SW potential used by GULP are shown in Table 532 . Parameters for the SW potential used by LAMMPS are listed in Table 533 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer b-SnGe under uniaxial tension at 1 and 300 K. Figure 273 shows the stress-strain curve for the tension of a single-layer b-SnGe of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer b-SnGe is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer b-SnGe. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 36.8 N/m along both armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}={\nu }_{yx}=0.11$ .

Figure 274.

Phonon dispersion for the single-layer b-SiGe. (a) The VFF model is fitted to the three acoustic branches in the long wave limit along the ΓM direction. The ab initio results (gray pentagons) are from [114]. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for the b-SiGe along ΓMKΓ.

There is no available value for nonlinear quantities in the single-layer b-SnGe. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −171.6 and −197.0 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 4.9 N/m at the ultimate strain of 0.27 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 5.0 N m⁻¹ at the ultimate strain of 0.31 in the zigzag direction at the low temperature of 1 K.

Figure 275.

Stress-strain relations for the b-SiGe of size $100\times 100$ Å. The b-SiGe is uniaxially stretched along the armchair or zigzag directions at temperatures 1 and 300 K.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_r{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\theta }{\left(\text{Δ}\theta \right)}^2$
Parameter	16.390	3.112
$r_0$ or ${\theta }_0$	2.310	114.702

Table 534.

The VFF model for b-SiGe.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of ev/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of $\mathrm{\AA }$ ) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction.

134. B-SiGe

Present studies on the buckled SiGe (b-SiGe) are based on first-principles calculations, and no empirical potential has been proposed for the b-SiGe. We will thus parametrize a set of SW potential for the single-layer b-SiGe in this section.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Si─Ge	22.576	2.122	14.237	0.0	3.417

Table 535.

Two-body SW potential parameters for b-SiGe used by GULP [8], as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
Si─Ge─Ge	87.197	114.702	2.122	2.122	0.0	3.417	0.0	3.417	0.0	5.314
Ge─Si─Si	87.197	114.702	2.122	2.122	0.0	3.417	0.0	3.417	0.0	5.314

Table 536.

Three-body SW potential parameters for b-SiGe used by GULP [8], as expressed in Eq. (4).

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Si─Ge─Ge	1.000	2.122	1.610	87.197	1.000	−0.418	22.576	0.702	4	0	0.0
Ge─Si─Si	1.000	2.122	1.610	87.197	1.000	−0.418	22.576	0.702	4	0	0.0

Table 537.

SW potential parameters for b-SiGe used by LAMMPS [9], as expressed in Eqs. (9) and (10).

The structure of the single-layer b-SiGe is shown in Figure 239 . The structural parameters are from the ab initio calculations [114]. The b-SiGe has a buckled configuration as shown in Figure 239(b) , where the buckle is along the zigzag direction. This structure can be determined by two independent geometrical parameters, e.g., the lattice constant 3.89 Å and the bond length 2.31 Å. The resultant height of the buckle is $h=0.55$ Å.

Table 534 shows the VFF model for the single-layer b-SiGe. The force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the ΓM as shown in Figure 274(a) . The ab initio calculations for the phonon dispersion are from [114]. Figure 274(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 535 . The parameters for the three-body SW potential used by GULP are shown in Table 536 . Parameters for the SW potential used by LAMMPS are listed in Table 537 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer b-SiGe under uniaxial tension at 1 and 300 K. Figure 275 shows the stress-strain curve for the tension of a single-layer b-SiGe of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer b-SiGe is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer b-SiGe. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 54.6 and 54.3 N/m along the armchair and zigzag directions, respectively. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}={\nu }_{yx}=0.16$ .

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_r{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\theta }{\left(\text{Δ}\theta \right)}^2$
Parameter	10.315	2.880
$r_0$ or ${\theta }_0$	2.520	113.298

Table 538.

The VFF model for b-SnSi.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of ev/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of $\mathrm{\AA }$ ) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction.

There is no available value for nonlinear quantities in the single-layer b-SiGe. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −186.7 and −233.5 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 11.0 N/m at the ultimate strain of 0.31 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 11.2 N/m at the ultimate strain of 0.31 in the zigzag direction at the low temperature of 1 K.

135. B-SnSi

Present studies on the buckled SnSi (b-SnSi) are based on first-principles calculations, and no empirical potential has been proposed for the b-SnSi. We will thus parametrize a set of SW potential for the single-layer b-SnSi in this section.

The structure of the single-layer b-SnSi is shown in Figure 239 . The structural parameters are from the ab initio calculations [114]. The b-SnSi has a buckled configuration as shown in Figure 239(b) , where the buckle is along the zigzag direction. This structure can be determined by two independent geometrical parameters, e.g., the lattice constant 4.21 Å and the bond length 2.52 Å. The resultant height of the buckle is $h=0.67$ Å.

Figure 276.

Phonon dispersion for the single-layer b-SnSi. (a) The VFF model is fitted to the three acoustic branches in the long wave limit along the ΓM direction. The ab initio results (gray pentagons) are from [114]. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for the b-SnSi along ΓMKΓ.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Sn─Si	16.463	2.260	20.164	0.0	3.713

Table 539.

Two-body SW potential parameters for b-SnSi used by GULP [8], as expressed in Eq. (3).

Table 538 shows the VFF model for the single-layer b-SnSi. The force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the ΓM as shown in Figure 276(a) . The ab initio calculations for the phonon dispersion are from [114]. Figure 276(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

Figure 277.

Stress-strain relations for the b-SnSi of size $100\times 100$ Å. The b-SnSi is uniaxially stretched along the armchair or zigzag directions at temperatures 1 and 300 K.

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
Sn─Si─Si	75.415	113.298	2.260	2.260	0.0	3.713	0.0	3.713	0.0	5.751
Si─Sn─Sn	75.415	113.298	2.260	2.260	0.0	3.713	0.0	3.713	0.0	5.751

Table 540.

Three-body SW potential parameters for b-SnSi used by GULP [8], as expressed in Eq. (4).

The parameters for the two-body SW potential used by GULP are shown in Table 539 . The parameters for the three-body SW potential used by GULP are shown in Table 540 . Parameters for the SW potential used by LAMMPS are listed in Table 541 .

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Sn─Si─Si	1.000	2.260	1.643	75.415	1.000	−0.396	16.463	0.773	4	0	0.0
Si─Sn─Sn	1.000	2.260	1.643	75.415	1.000	−0.396	16.463	0.773	4	0	0.0

Table 541.

SW potential parameters for b-SnSi used by LAMMPS [9], as expressed in Eqs. (9) and (10).

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer b-SnSi under uniaxial tension at 1 and 300 K. Figure 277 shows the stress-strain curve for the tension of a single-layer b-SnSi of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer b-SnSi is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer b-SnSi. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 39.0 and 38.4 N/m along the armchair and zigzag directions, respectively. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}={\nu }_{yx}=0.14$ .

There is no available value for nonlinear quantities in the single-layer b-SnSi. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −150.5 and −174.8 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 6.5 N/m at the ultimate strain of 0.28 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 6.6 N/m at the ultimate strain of 0.32 in the zigzag direction at the low temperature of 1 K.

136. B-InP

Present studies on the buckled InP (b-InP) are based on first-principles calculations, and no empirical potential has been proposed for the b-InP. We will thus parametrize a set of SW potential for the single-layer b-InP in this section.

The structure of the single-layer b-InP is shown in Figure 239 . The structural parameters are from the ab initio calculations [114]. The b-InP has a buckled configuration as shown in Figure 239(b) , where the buckle is along the zigzag direction. This structure can be determined by two independent geometrical parameters, e.g., the lattice constant 4.17 Å and the bond length 2.46 Å. The resultant height of the buckle is $h=0.51$ Å.

Table 542 shows the VFF model for the single-layer b-InP. The force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the ΓM as shown in Figure 278(a) . The ab initio calculations for the phonon dispersion are from [114]. Figure 278(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

Figure 278.

Phonon dispersion for the single-layer b-InP. (a) The VFF model is fitted to the three acoustic branches in the long wave limit along the ΓM direction. The ab initio results (gray pentagons) are from [114]. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for the b-InP along ΓMKΓ.

The parameters for the two-body SW potential used by GULP are shown in Table 543 . The parameters for the three-body SW potential used by GULP are shown in Table 544 . Parameters for the SW potential used by LAMMPS are listed in Table 545 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer b-InP under uniaxial tension at 1 and 300 K. Figure 279 shows the stress-strain curve for the tension of a single-layer b-InP of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer b-InP is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer b-InP. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 39.3 and 38.3 N/m along the armchair and zigzag directions, respectively. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}={\nu }_{yx}=0.17$ .

Figure 279.

Stress-strain relations for the b-InP of size $100\times 100$ Å. The b-InP is uniaxially stretched along the armchair or zigzag directions at temperatures 1 and 300 K.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_r{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\theta }{\left(\text{Δ}\theta \right)}^2$
Parameter	12.903	2.384
$r_0$ or ${\theta }_0$	2.460	115.895

Table 542.

The VFF model for b-InP.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of ev/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of $\mathrm{\AA }$ ) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
In─P	20.610	2.306	18.311	0.0	3.651

Table 543.

Two-body SW potential parameters for b-InP used by GULP [8], as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
In─P─P	70.782	115.895	2.306	2.306	0.0	3.651	0.0	3.651	0.0	5.696
P─In─In	70.782	115.895	2.306	2.306	0.0	3.651	0.0	3.651	0.0	5.696

Table 544.

Three-body SW potential parameters for b-InP used by GULP [8], as expressed in Eq. (4).

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
In─P─P	1.000	2.306	1.583	70.782	1.000	−0.437	20.610	0.648	4	0	0.0
P─In─In	1.000	2.306	1.583	70.782	1.000	−0.437	20.610	0.648	4	0	0.0

Table 545.

SW potential parameters for b-InP used by LAMMPS [9], as expressed in Eqs. (9) and (10).

There is no available value for nonlinear quantities in the single-layer b-InP. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −119.3 and −132.0 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 9.2 N/m at the ultimate strain of 0.35 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 9.1 N m⁻¹ at the ultimate strain of 0.31 in the zigzag direction at the low temperature of 1 K.

137. B-InAs

Present studies on the buckled InAs (b-InAs) are based on first-principles calculations, and no empirical potential has been proposed for the b-InAs. We will thus parametrize a set of SW potential for the single-layer b-InAs in this section.

The structure of the single-layer b-InAs is shown in Figure 239 . The structural parameters are from the ab initio calculations [114]. The b-InAs has a buckled configuration as shown in Figure 239(b) , where the buckle is along the zigzag direction. This structure can be determined by two independent geometrical parameters, e.g., the lattice constant 4.28 Å and the bond length 2.55 Å. The resultant height of the buckle is $h=0.62$ Å.

Figure 280.

Phonon dispersion for the single-layer b-InAs. (a) The VFF model is fitted to the three acoustic branches in the long wave limit along the ΓM direction. The ab initio results (gray pentagons) are from [114]. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for the b-InAs along ΓMKΓ.

Table 546 shows the VFF model for the single-layer b-InAs. The force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the ΓM as shown in Figure 280(a) . The ab initio calculations for the phonon dispersion are from [114]. Figure 280(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 547 . The parameters for the three-body SW potential used by GULP are shown in Table 548 . Parameters for the SW potential used by LAMMPS are listed in Table 549 .

Figure 281.

Stress-strain relations for the b-InAs of size $100\times 100$ Å. The b-InAs is uniaxially stretched along the armchair or zigzag directions at temperatures 1 and 300 K.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_r{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\theta }{\left(\text{Δ}\theta \right)}^2$
Parameter	10.903	2.384
$r_0$ or ${\theta }_0$	2.550	114.115

Table 546.

The VFF model for b-InAs.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of ev/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of $\mathrm{\AA }$ ) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer b-InAs under uniaxial tension at 1 and 300 K. Figure 281 shows the stress-strain curve for the tension of a single-layer b-InAs of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer b-InAs is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer b-InAs. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 33.9 and 34.2 N/m along the armchair and zigzag directions, respectively. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}={\nu }_{yx}=0.17$ .

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
In─As	18.099	2.320	21.141	0.0	3.766

Table 547.

Two-body SW potential parameters for b-InAs used by GULP [8], as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
In─As─As	64.931	114.115	2.320	2.320	0.0	3.766	0.0	3.766	0.0	5.847
As─In─In	64.931	114.115	2.320	2.320	0.0	3.766	0.0	3.766	0.0	5.847

Table 548.

Three-body SW potential parameters for b-InAs used by GULP [8], as expressed in Eq. (4).

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
In─As─As	1.000	2.320	1.624	64.931	1.000	−0.409	18.099	0.730	4	0	0.0
As─In─In	1.000	2.320	1.624	64.931	1.000	−0.409	18.099	0.730	4	0	0.0

Table 549.

SW potential parameters for b-InAs used by LAMMPS [9], as expressed in Eqs. (9) and (10).

There is no available value for nonlinear quantities in the single-layer b-InAs. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −85.0 and −130.2 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 7.3 N/m at the ultimate strain of 0.32 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 7.5 N/m at the ultimate strain of 0.32 in the zigzag direction at the low temperature of 1 K.

Figure 282.

Phonon dispersion for the single-layer b-InSb. (a) The VFF model is fitted to the three acoustic branches in the long wave limit along the ΓM direction. The ab initio results (gray pentagons) are from [114]. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for the b-InSb along ΓMKΓ.

138. B-InSb

Present studies on the buckled InSb (b-InSb) are based on first-principles calculations, and no empirical potential has been proposed for the b-InSb. We will thus parametrize a set of SW potential for the single-layer b-InSb in this section.

The structure of the single-layer b-InSb is shown in Figure 239 . The structural parameters are from the ab initio calculations [114]. The b-InSb has a buckled configuration as shown in Figure 239(b) , where the buckle is along the zigzag direction. This structure can be determined by two independent geometrical parameters, e.g., the lattice constant 4.57 Å and the bond length 2.74 Å. The resultant height of the buckle is $h=0.73$ Å.

Figure 283.

Stress-strain relations for the b-InSb of size $100\times 100$ Å. The b-InSb is uniaxially stretched along the armchair or zigzag directions at temperatures 1 and 300 K.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_r{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\theta }{\left(\text{Δ}\theta \right)}^2$
Parameter	8.903	2.384
$r_0$ or ${\theta }_0$	2.740	113.012

Table 550.

The VFF model for b-InSb.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of ev/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of $\mathrm{\AA }$ ) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction.

Table 550 shows the VFF model for the single-layer b-InSb. The force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the ΓM as shown in Figure 282(a) . The ab initio calculations for the phonon dispersion are from [114]. Figure 282(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
In-Sb	16.706	2.445	28.182	0.0	4.034

Table 551.

Two-body SW potential parameters for b-InSb used by GULP [8], as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
In─Sb─Sb	61.578	113.012	2.445	2.445	0.0	4.034	0.0	4.034	0.0	6.243
Sb─In─In	61.578	113.012	2.445	2.445	0.0	4.034	0.0	4.034	0.0	6.243

Table 552.

Three-body SW potential parameters for b-InSb used by GULP [8], as expressed in Eq. (4).

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
In─Sb─Sb	1.000	2.445	1.650	61.578	1.000	−0.391	16.706	0.788	4	0	0.0
Sb─In─In	1.000	2.445	1.650	61.578	1.000	−0.391	16.706	0.788	4	0	0.0

Table 553.

SW potential parameters for b-InSb used by LAMMPS [9], as expressed in Eqs. (9) and (10).

The parameters for the two-body SW potential used by GULP are shown in Table 551 . The parameters for the three-body SW potential used by GULP are shown in Table 552 . Parameters for the SW potential used by LAMMPS are listed in Table 553 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer b-InSb under uniaxial tension at 1 and 300 K. Figure 283 shows the stress-strain curve for the tension of a single-layer b-InSb of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer b-InSb is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer b-InSb. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 28.6 and 28.9 N/m along the armchair and zigzag directions, respectively. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}={\nu }_{yx}=0.17$ .

There is no available value for nonlinear quantities in the single-layer b-InSb. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −85.4 and −121.0 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 5.8 N/m at the ultimate strain of 0.31 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 6.0 N m⁻¹ at the ultimate strain of 0.33 in the zigzag direction at the low temperature of 1 K.

Figure 284.

Phonon dispersion for the single-layer b-GaAs. (a) The VFF model is fitted to the three acoustic branches in the long wave limit along the ΓM direction. The ab initio results (gray pentagons) are from [114]. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for the b-GaAs along ΓMKΓ.

139. b-GaAs

Present studies on the buckled GaAs (b-GaAs) are based on first-principles calculations, and no empirical potential has been proposed for the b-GaAs. We will thus parametrize a set of SW potential for the single-layer b-GaAs in this section.

Figure 285.

Stress-strain relations for the b-GaAs of size $100\times 100$ Å. The b-GaAs is uniaxially stretched along the armchair or zigzag directions at temperatures 1 and 300 K.

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_r{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\theta }{\left(\text{Δ}\theta \right)}^2$
Parameter	12.903	3.284
$r_0$ or ${\theta }_0$	2.360	114.513

Table 554.

The VFF model for b-GaAs.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of ${\text{eV/}\mathrm{\AA }}^{\text{2}}$ for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of $\mathrm{\AA }$ ) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction.

The structure of the single-layer b-GaAs is shown in Figure 239 . The structural parameters are from the ab initio calculations [114]. The b-GaAs has a buckled configuration as shown in Figure 239(b) , where the buckle is along the zigzag direction. This structure can be determined by two independent geometrical parameters, e.g., the lattice constant 3.97 Å and the bond length 2.36 Å. The resultant height of the buckle is $h=0.55$ Å.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Ga─As	18.485	2.161	15.510	0.0	3.489

Table 555.

Two-body SW potential parameters for b-GaAs used by GULP [8], as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
Ga─As─As	91.177	114.513	2.161	2.161	0.0	3.489	0.0	3.489	0.0	5.423
As─Ga─Ga	91.177	114.513	2.161	2.161	0.0	3.489	0.0	3.489	0.0	5.423

Table 556.

Three-body SW potential parameters for b-GaAs used by GULP [8], as expressed in Eq. (4).

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Ga─As─As	1.000	2.161	1.614	91.177	1.000	−0.415	18.485	0.711	4	0	0.0
As─Ga─Ga	1.000	2.161	1.614	91.177	1.000	−0.415	18.485	0.711	4	0	0.0

Table 557.

SW potential parameters for b-GaAs used by LAMMPS [9], as expressed in Eqs. (9) and (10).

Table 554 shows the VFF model for the single-layer b-GaAs. The force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the ΓM as shown in Figure 284(a) . The ab initio calculations for the phonon dispersion are from Ref. [114]. Figure 284(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 555 . The parameters for the three-body SW potential used by GULP are shown in Table 556 . Parameters for the SW potential used by LAMMPS are listed in Table 557 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer b-GaAs under uniaxial tension at 1 and 300 K. Figure 285 shows the stress-strain curve for the tension of a single-layer b-GaAs of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer b-GaAs is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer b-GaAs. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 50.5 and 50.9 N/m along the armchair and zigzag directions, respectively. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.13$ .

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_r{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\theta }{\left(\text{Δ}\theta \right)}^2$
Parameter	16.050	3.022
$r_0$ or ${\theta }_0$	2.250	117.152

Table 558.

The VFF model for b-GaP.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of ${\text{eV/}\mathrm{\AA }}^{\text{2}}$ for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of $\mathrm{\AA }$ ) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Ga─P	21.948	2.152	12.814	0.0	3.350

Table 559.

Two-body SW potential parameters for b-GaP used by GULP [8], as expressed in Eq. (3).

There is no available value for nonlinear quantities in the single-layer b-GaAs. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −199.5 and −258.6 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 8.3 N/m at the ultimate strain of 0.27 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 8.3 N/m at the ultimate strain of 0.30 in the zigzag direction at the low temperature of 1 K.

140. b-GaP

Present studies on the buckled GaP (b-GaP) are based on first-principles calculations, and no empirical potential has been proposed for the b-GaP. We will thus parametrize a set of SW potential for the single-layer b-GaP in this section.

The structure of the single-layer b-GaP is shown in Figure 239 . The structural parameters are from the ab initio calculations [114]. The b-GaP has a buckled configuration as shown in Figure 239(b) , where the buckle is along the zigzag direction. This structure can be determined by two independent geometrical parameters, e.g., the lattice constant 3.84 Å and the bond length 2.25 Å. The resultant height of the buckle is $h=0.40$ Å.

Figure 286.

Phonon dispersion for the single-layer b-GaP. (a) The VFF model is fitted to the three acoustic branches in the long wave limit along the ΓM direction. The ab initio results (gray pentagons) are from Ref. [114]. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for the b-GaP along ΓMKΓ.

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
Ga─P─P	95.438	117.152	2.152	2.152	0.0	3.350	0.0	3.350	0.0	5.246
P─Ga─Ga	95.438	117.152	2.152	2.152	0.0	3.350	0.0	3.350	0.0	5.246

Table 560.

Three-body SW potential parameters for b-GaP used by GULP [8], as expressed in Eq. (4).

Figure 287.

Stress-strain relations for the b-GaP of size $100\times 100$ Å. The b-GaP is uniaxially stretched along the armchair or zigzag directions at temperatures 1 and 300 K.

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Ga─P─P	1.000	2.152	1.557	95.438	1.000	−0.456	21.948	0.597	4	0	0.0
P─Ga─Ga	1.000	2.152	1.557	95.438	1.000	−0.456	21.948	0.597	4	0	0.0

Table 561.

SW potential parameters for b-GaP used by LAMMPS [9], as expressed in Eqs. (9) and (10).

Table 558 shows the VFF model for the single-layer b-GaP. The force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the ΓM as shown in Figure 286(a) . The ab initio calculations for the phonon dispersion are from Ref. [114]. Figure 286(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 559 . The parameters for the three-body SW potential used by GULP are shown in Table 560 . Parameters for the SW potential used by LAMMPS are listed in Table 561 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer b-GaP under uniaxial tension at 1 and 300 K. Figure 287 shows the stress-strain curve for the tension of a single-layer b-GaP of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer b-GaP is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer b-GaP. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 57.2 and 57.4 N/m along the armchair and zigzag directions, respectively. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.14$ .

VFF type	Bond stretching	Angle bending
Expression	$\frac{1}{2}{K}_r{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\theta }{\left(\text{Δ}\theta \right)}^2$
Parameter	12.050	3.022
$r_0$ or ${\theta }_0$	2.570	114.791

Table 562.

The VFF model for b-AlSb.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of ${\text{eV/}\mathrm{\AA }}^{\text{2}}$ for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of $\mathrm{\AA }$ ) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Al─Sb	20.580	2.365	21.812	0.0	3.803

Table 563.

Two-body SW potential parameters for b-AlSb used by GULP [8], as expressed in Eq. (3).

There is no available value for nonlinear quantities in the single-layer b-GaP. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −186.4 and −261.6 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 11.3 N/m at the ultimate strain of 0.29 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 10.9 N/m at the ultimate strain of 0.29 in the zigzag direction at the low temperature of 1 K.

Figure 288.

Phonon dispersion for the single-layer b-AlSb. (a) The VFF model is fitted to the three acoustic branches in the long wave limit along the ΓM direction. The ab initio results (gray pentagons) are from Ref. [114]. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for the b-AlSb along ΓMKΓ.

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
Al─Sb─Sb	85.046	114.791	2.365	2.365	0.0	3.803	0.0	3.803	0.0	5.915
Sb─Al─Al	85.046	114.791	2.365	2.365	0.0	3.803	0.0	3.803	0.0	5.915

Table 564.

Three-body SW potential parameters for b-AlSb used by GULP [8], as expressed in Eq. (4).

141. b-AlSb

Present studies on the buckled AlSb (b-AlSb) are based on first-principles calculations, and no empirical potential has been proposed for the b-AlSb. We will thus parametrize a set of SW potential for the single-layer b-AlSb in this section.

Figure 289.

Stress-strain relations for the b-AlSb of size $100\times 100$ Å. The b-AlSb is uniaxially stretched along the armchair or zigzag directions at temperatures 1 and 300 K.

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Al─Sb─Sb	1.000	2.365	1.608	85.046	1.000	−0.419	20.580	0.697	4	0	0.0
Sb─Al─Al	1.000	2.365	1.608	85.046	1.000	−0.419	20.580	0.697	4	0	0.0

Table 565.

SW potential parameters for b-AlSb used by LAMMPS [9], as expressed in Eqs. (9) and (10).

The structure of the single-layer b-AlSb is shown in Figure 239 . The structural parameters are from the ab initio calculations [114]. The b-AlSb has a buckled configuration as shown in Figure 239(b) , where the buckle is along the zigzag direction. This structure can be determined by two independent geometrical parameters, e.g., the lattice constant 4.33 Å and the bond length 2.57 Å. The resultant height of the buckle is $h=0.60$ Å.

Table 562 shows the VFF model for the single-layer b-AlSb. The force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the ΓM as shown in Figure 288(a) . The ab initio calculations for the phonon dispersion are from Ref. [114]. Figure 288(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 563 . The parameters for the three-body SW potential used by GULP are shown in Table 564 . Parameters for the SW potential used by LAMMPS are listed in Table 565 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer b-AlSb under uniaxial tension at 1 and 300 K. Figure 289 shows the stress-strain curve for the tension of a single-layer b-AlSb of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer b-AlSb is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer b-AlSb. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 41.7 and 42.0 N/m along the armchair and zigzag directions, respectively. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.15$ .

There is no available value for nonlinear quantities in the single-layer b-AlSb. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −142.4 and −190.8 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 8.1 N/m at the ultimate strain of 0.29 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 8.1 N/m at the ultimate strain of 0.31 in the zigzag direction at the low temperature of 1 K.

142. BO

Present studies on BO are based on first-principles calculations, and no empirical potential has been proposed for BO. We will thus parametrize a set of SW potential for the single-layer BO in this section.

The structure of the single-layer BO is shown in Figure 290 with M = B and X = O. The structural parameters are from the ab initio calculations [115]. BO has a bi-buckled configuration as shown in Figure 290(b) , where the buckle is along the zigzag direction. Two buckling layers are symmetrically integrated through the interior B─B bonds, forming a bi-buckled configuration. This structure can be determined by three independent geometrical parameters, e.g., the lattice constant 2.44 Å, the bond length $d_{\text{B}-\text{O}}=1.52$ Å, and the bond length $d_{\text{B}-\text{B}}=1.77$ Å.

VFF type	Bond stretching		Angle bending
Expression	$\frac{1}{2}{K}_{\text{B}-\text{O}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{B}-\text{B}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{BOO}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{BBO}}{\left(\text{Δ}\theta \right)}^2$
Parameter	23.030	15.512	5.577	6.209
$r_0$ or ${\theta }_0$	1.520	1.770	106.764	112.059

Table 566.

The VFF model for BO.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of ${\text{eV/}\mathrm{\AA }}^{\text{2}}$ for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of $\mathrm{\AA }$ ) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
B1-O1	11.725	1.207	2.669	0.0	2.197
B1-B2	6.749	0.875	4.908	0.0	2.392

Table 567.

Two-body SW potential parameters for BO used by GULP [8], as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
B1─O1─O1	107.486	106.764	1.207	1.207	0.0	2.197	0.0	2.197	0.0	3.333
B1─B2─O1	87.662	112.059	0.875	1.207	0.0	2.392	0.0	2.197	0.0	3.198

Table 568.

Three-body SW potential parameters for BO used by GULP [8], as expressed in Eq. (4).

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
B1─O1─O1	1.000	1.207	1.820	107.486	1.000	−0.288	11.725	1.256	4	0	0.0
B1─B2─B2	1.000	0.875	2.734	0.000	1.000	0.000	6.749	8.377	4	0	0.0
B1─B2─O1	1.000	0.000	0.000	87.662	1.000	−0.376	0.000	0.000	4	0	0.0

Table 569.

SW potential parameters for BO used by LAMMPS [9], as expressed in Eqs. (9) and (10).

Figure 290.

Structure of bi-buckled MX crystal, with M from group III and X from group VI. (a) Top view. The unit cell is highlighted by a blue parallelogram. Inset shows the first Brillouin zone of the reciprocal lattice space. (b) Side view displays the bi-buckled configuration. (c) Perspective view. M atoms are represented by larger gray balls. X atoms are represented by smaller red balls.

Table 566 shows the VFF model for the single-layer BO. The force constant parameters are determined by fitting to the six low-frequency branches in the phonon dispersion along the ΓM as shown in Figure 291(a) . The ab initio calculations for the phonon dispersion are from Ref. [115]. Figure 291(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 567 . The parameters for the three-body SW potential used by GULP are shown in Table 568 . Parameters for the SW potential used by LAMMPS are listed in Table 569 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer BO under uniaxial tension at 1 and 300 K. Figure 292 shows the stress-strain curve for the tension of a single-layer BO of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer BO is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer BO. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 299.6 and 297.7 N/m along the armchair and zigzag directions, respectively. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.11$ .

Figure 291.

Phonon dispersion for the single-layer BO. (a) The VFF model is fitted to the six low-frequency branches along the ΓM direction. The ab initio results (gray pentagons) are from Ref. [115]. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for the BO along ΓMKΓ.

Figure 292.

Stress-strain relations for BO of size $100\times 100$ Å. BO is uniaxially stretched along the armchair or zigzag directions at temperatures 1 and 300 K.

There is no available value for nonlinear quantities in the single-layer BO. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −1554.7 and −1585.2 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 28.9 N/m at the ultimate strain of 0.19 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 28.2 N/m at the ultimate strain of 0.24 in the zigzag direction at the low temperature of 1 K.

143. AlO

Present studies on AlO are based on first-principles calculations, and no empirical potential has been proposed for AlO. We will thus parametrize a set of SW potential for the single-layer AlO in this section.

The structure of the single-layer AlO is shown in Figure 290 with M = Al and X = O. The structural parameters are from the ab initio calculations [115]. AlO has a bi-buckled configuration as shown in Figure 290(b) , where the buckle is along the zigzag direction. Two buckling layers are symmetrically integrated through the interior Al-Al bonds, forming a bi-buckled configuration. This structure can be determined by three independent geometrical parameters, e.g., the lattice constant 2.96 Å, the bond length $d_{\text{Al}-\text{O}}=1.83$ Å, and the bond length $d_{\text{Al}-\text{Al}}=2.62$ Å.

VFF type	Bond stretching		Angle bending
Expression	$\frac{1}{2}{K}_{\text{Al}-\text{O}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{Al}-\text{Al}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{AlOO}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{AlAlO}}{\left(\text{Δ}\theta \right)}^2$
Parameter	18.189	6.410	3.182	1.318
$r_0$ or ${\theta }_0$	1.830	2.620	107.947	110.956

Table 570.

The VFF model for AlO.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of ${\text{eV/}\mathrm{\AA }}^{\text{2}}$ for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of $\mathrm{\AA }$ ) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Al1─O1	13.758	1.488	5.608	0.0	2.655
Al1─Al2	3.609	0.678	23.560	0.0	3.287

Table 571.

Two-body SW potential parameters for AlO used by GULP [8], as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
Al1─O1─O1	64.759	107.947	1.488	1.488	0.0	2.655	0.0	2.655	0.0	4.043
Al1─O1─Al2	12.688	110.956	1.488	0.678	0.0	2.655	0.0	3.287	0.0	4.213

Table 572.

Three-body SW potential parameters for AlO used by GULP [8], as expressed in Eq. (4).

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Al1─O1─O1	1.000	1.488	1.785	64.759	1.000	−0.308	13.758	1.145	4	0	0.0
Al1─Al2─Al2	1.000	0.678	4.846	0.000	1.000	0.000	3.609	111.363	4	0	0.0
Al1─Al2─O1	1.000	0.000	0.000	12.688	1.000	−0.358	0.000	0.000	4	0	0.0

Table 573.

SW potential parameters for AlO used by LAMMPS [9], as expressed in Eqs. (9) and (10).

Figure 293.

Phonon dispersion for the single-layer AlO. (a) The VFF model is fitted to the six low-frequency branches along the ΓM direction. The ab initio results (gray pentagons) are from Ref. [115]. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for AlO along ΓMKΓ.

Table 570 shows the VFF model for the single-layer AlO. The force constant parameters are determined by fitting to the six low-frequency branches in the phonon dispersion along the ΓM as shown in Figure 293(a) . The ab initio calculations for the phonon dispersion are from Ref. [115]. Figure 293(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 571 . The parameters for the three-body SW potential used by GULP are shown in Table 572 . Parameters for the SW potential used by LAMMPS are listed in Table 573 .

Figure 294.

Stress-strain relations for the AlO of size $100\times 100$ Å. The AlO is uniaxially stretched along the armchair or zigzag directions at temperatures 1 and 300 K.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer AlO under uniaxial tension at 1 and 300 K. Figure 294 shows the stress-strain curve for the tension of a single-layer AlO of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer AlO is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer AlO. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 149.3 and 148.2 N/m along the armchair and zigzag directions, respectively. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.19$ .

There is no available value for nonlinear quantities in the single-layer AlO. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −563.9 and −565.6 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 19.6 N/m at the ultimate strain of 0.26 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 20.4 N/m at the ultimate strain of 0.34 in the zigzag direction at the low temperature of 1 K.

Figure 295.

Phonon dispersion for the single-layer GaO. (a) The VFF model is fitted to the six low-frequency branches along the ΓM direction. The ab initio results (gray pentagons) are from Ref. [115]. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for the GaO along ΓMKΓ.

144. GaO

Present studies on the GaO are based on first-principles calculations, and no empirical potential has been proposed for the GaO. We will thus parametrize a set of SW potential for the single-layer GaO in this section.

The structure of the single-layer GaO is shown in Figure 290 with M = Ga and X = O. The structural parameters are from the ab initio calculations [115]. The GaO has a bi-buckled configuration as shown in Figure 290(b) , where the buckle is along the zigzag direction. Two buckling layers are symmetrically integrated through the interior Ga-Ga bonds, forming a bi-buckled configuration. This structure can be determined by three independent geometrical parameters, e.g., the lattice constant 3.12 Å, the bond length $d_{\text{Ga}-\text{O}}=1.94$ Å, and the bond length $d_{\text{Ga}-\text{Ga}}=2.51$ Å.

VFF type	Bond stretching		Angle bending
Expression	$\frac{1}{2}{K}_{\text{Ga}-\text{O}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{Ga}-\text{Ga}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{GaOO}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{GaGaO}}{\left(\text{Δ}\theta \right)}^2$
Parameter	18.189	6.410	3.182	1.628
$r_0$ or ${\theta }_0$	1.940	2.510	107.051	111.794

Table 574.

The VFF model for GaO.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of ${\text{eV/}\mathrm{\AA }}^{\text{2}}$ for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of $\mathrm{\AA }$ ) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Ga1─O1	15.178	1.550	7.082	0.0	2.807
Ga1─Ga2	4.225	0.890	19.846	0.0	3.257

Table 575.

Two-body SW potential parameters for GaO used by GULP [8], as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
Ga1─O1─O1	62.149	107.051	1.550	1.550	0.0	2.807	0.0	2.807	0.0	4.262
Ga1─O1─Ga2	18.443	111.794	1.550	0.890	0.0	2.807	0.0	3.257	0.0	4.269

Table 576.

Three-body SW potential parameters for GaO used by GULP [8], as expressed in Eq. (4).

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Ga1─O1─O1	1.000	1.550	1.811	62.149	1.000	−0.293	15.178	1.227	4	0	0.0
Ga1─Ga2─Ga2	1.000	0.890	3.661	0.000	1.000	0.000	4.225	31.685	4	0	0.0
Ga1─Ga2─O1	1.000	0.000	0.000	18.443	1.000	−0.371	0.000	0.000	4	0	0.0

Table 577.

SW potential parameters for GaO used by LAMMPS [9], as expressed in Eqs. (9) and (10).

Figure 296.

Stress-strain relations for the GaO of size $100\times 100$ Å. GaO is uniaxially stretched along the armchair or zigzag directions at temperatures 1 and 300 K.

Table 574 shows the VFF model for the single-layer GaO. The force constant parameters are determined by fitting to the six low-frequency branches in the phonon dispersion along the ΓM as shown in Figure 295(a) . The ab initio calculations for the phonon dispersion are from Ref. [115]. Figure 295(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 575 . The parameters for the three-body SW potential used by GULP are shown in Table 576 . Parameters for the SW potential used by LAMMPS are listed in Table 577 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer GaO under uniaxial tension at 1 and 300 K. Figure 296 shows the stress-strain curve for the tension of a single-layer GaO of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer GaO is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer GaO. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 137.2 and 136.6 N/m along the armchair and zigzag directions, respectively. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.22$ .

VFF type	Bond stretching		Angle bending
Expression	$\frac{1}{2}{K}_{\text{In}-\text{O}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{In}-\text{In}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{InOO}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{InInO}}{\left(\text{Δ}\theta \right)}^2$
Parameter	16.916	4.250	2.171	1.138
$r_0$ or ${\theta }_0$	2.160	2.860	107.328	111.538

Table 578.

The VFF model for InO.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of ${\text{eV/}\mathrm{\AA }}^{\text{2}}$ for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of $\mathrm{\AA }$ ) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
In1-O1	17.600	1.735	10.884	0.0	3.128
In1-In2	3.440	0.945	33.453	0.0	3.682

Table 579.

Two-body SW potential parameters for InO used by GULP [8], as expressed in Eq. (3).

There is no available value for nonlinear quantities in the single-layer GaO. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −467.5 and −529.6 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 19.6 N/m at the ultimate strain of 0.28 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 20.8 N/m at the ultimate strain of 0.35 in the zigzag direction at the low temperature of 1 K.

145. InO

Present studies on InO are based on first-principles calculations, and no empirical potential has been proposed for InO. We will thus parametrize a set of SW potential for the single-layer InO in this section.

Figure 297.

Phonon dispersion for the single-layer InO. (a) The VFF model is fitted to the six low-frequency branches along the ΓM direction. The ab initio results (gray pentagons) are from Ref. [115]. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for InO along ΓMKΓ.

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
In1─O1─O1	42.946	107.328	1.735	1.735	0.0	3.128	0.0	3.128	0.0	4.754
In1─O1─In2	12.470	111.538	1.735	0.945	0.0	3.128	0.0	3.682	0.0	4.800

Table 580.

Three-body SW potential parameters for InO used by GULP [8], as expressed in Eq. (4).

The structure of the single-layer InO is shown in Figure 290 with M = In and X = O. The structural parameters are from the ab initio calculations [115]. InO has a bi-buckled configuration as shown in Figure 290(b) , where the buckle is along the zigzag direction. Two buckling layers are symmetrically integrated through the interior In-In bonds, forming a bi-buckled configuration. This structure can be determined by three independent geometrical parameters, e.g., the lattice constant 3.48 Å, the bond length $d_{\text{In}-\text{O}}=2.16$ Å, and the bond length $d_{\text{In}-\text{In}}=2.86$ Å.

Figure 298.

Stress-strain relations for InO of size $100\times 100$ Å. InO is uniaxially stretched along the armchair or zigzag directions at temperatures 1 and 300 K.

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
In1─O1─O1	1.000	1.735	1.803	42.946	1.000	−0.298	17.600	1.201	4	0	0.0
In1─In2─In2	1.000	0.945	3.895	0.000	1.000	0.000	3.440	41.864	4	0	0.0
In1─In2─O1	1.000	0.000	0.000	12.470	1.000	−0.367	0.000	0.000	4	0	0.0

Table 581.

SW potential parameters for InO used by LAMMPS [9], as expressed in Eqs. (9) and (10).

Table 578 shows the VFF model for the single-layer InO. The force constant parameters are determined by fitting to the six low-frequency branches in the phonon dispersion along the ΓM as shown in Figure 297(a) . The ab initio calculations for the phonon dispersion are from Ref. [115]. Figure 297(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 579 . The parameters for the three-body SW potential used by GULP are shown in Table 580 . Parameters for the SW potential used by LAMMPS are listed in Table 581 .

VFF type	Bond stretching		Angle bending
Expression	$\frac{1}{2}{K}_{\text{B}-\text{S}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{B}-\text{B}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{BSS}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{BBS}}{\left(\text{Δ}\theta \right)}^2$
Parameter	17.138	16.385	5.144	3.861
$r_0$ or ${\theta }_0$	1.940	1.720	102.691	115.613

Table 582.

The VFF model for BS.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of ${\text{eV/}\mathrm{\AA }}^{\text{2}}$ for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of $\mathrm{\AA }$ ) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer InO under uniaxial tension at 1 and 300 K. Figure 298 shows the stress-strain curve for the tension of a single-layer InO of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer InO is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer InO. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 85.7 N/m along the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.29$ .

There is no available value for nonlinear quantities in the single-layer InO. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −157.3 and −210.9 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 19.9 N/m at the ultimate strain of 0.38 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 23.6 N/m at the ultimate strain of 0.39 in the zigzag direction at the low temperature of 1 K.

Figure 299.

Phonon dispersion for the single-layer BS. (a) The VFF model is fitted to the six low-frequency branches along the ΓM direction. The ab initio results (gray pentagons) are from Ref. [115]. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for the BS along ΓMKΓ.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
B1─S1	13.021	1.417	7.082	0.0	2.769
B1─B2	14.613	1.809	4.376	0.0	2.602

Table 583.

Two-body SW potential parameters for BS used by GULP [8], as expressed in Eq. (3).

146. BS

Present studies on BS are based on first-principles calculations, and no empirical potential has been proposed for the BS. We will thus parametrize a set of SW potential for the single-layer BS in this section.

Figure 300.

Stress-strain relations for the BS of size $100\times 100$ Å. The BS is uniaxially stretched along the armchair or zigzag directions at temperatures 1 and 300 K.

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
B1─S1─S1	82.459	102.691	1.417	1.417	0.0	2.769	0.0	2.769	0.0	4.139
B1─B2─S1	102.002	115.613	1.809	1.417	0.0	2.602	0.0	2.769	0.0	3.717

Table 584.

Three-body SW potential parameters for BS used by GULP [8], as expressed in Eq. (4).

The structure of the single-layer BS is shown in Figure 290 with M = B and X = S. The structural parameters are from the ab initio calculations [115]. BS has a bi-buckled configuration as shown in Figure 290(b) , where the buckle is along the zigzag direction. Two buckling layers are symmetrically integrated through the interior B-B bonds, forming a bi-buckled configuration. This structure can be determined by three independent geometrical parameters, e.g., the lattice constant 3.03 Å, the bond length $d_{\text{B}-\text{S}}=1.94$ Å, and the bond length $d_{\text{B}-\text{B}}=1.72$ Å.

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
B1─S1─S1	1.000	1.417	1.955	82.459	1.000	−0.220	13.021	1.758	4	0	0.0
B1─B2─B2	1.000	1.809	1.438	0.000	1.000	0.000	14.613	0.408	4	0	0.0
B1─B2─S1	1.000	0.000	0.000	102.002	1.000	−0.432	0.000	0.000	4	0	0.0

Table 585.

SW potential parameters for BS used by LAMMPS [9], as expressed in Eqs. (9) and (10).

Table 582 shows the VFF model for the single-layer BS. The force constant parameters are determined by fitting to the six low-frequency branches in the phonon dispersion along the ΓM as shown in Figure 299(a) . The ab initio calculations for the phonon dispersion are from Ref. [115]. Figure 299(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 583 . The parameters for the three-body SW potential used by GULP are shown in Table 584 . Parameters for the SW potential used by LAMMPS are listed in Table 585 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer BS under uniaxial tension at 1 and 300 K. Figure 300 shows the stress-strain curve for the tension of a single-layer BS of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer BS is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer BS. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 179.4 and 178.5 N/m along the armchair and zigzag directions, respectively. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.16$ .

There is no available value for nonlinear quantities in the single-layer BS. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −793.2 and −823.2 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 20.0 N/m at the ultimate strain of 0.21 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 19.5 N/m at the ultimate strain of 0.28 in the zigzag direction at the low temperature of 1 K.

Figure 301.

Phonon dispersion for the single-layer AlS. (a) The VFF model is fitted to the six low-frequency branches along the ΓM direction. The ab initio results (gray pentagons) are from Ref. [115]. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for AlS along ΓMKΓ.

147. AlS

Present studies on AlS are based on first-principles calculations, and no empirical potential has been proposed for AlS. We will thus parametrize a set of SW potential for the single-layer AlS in this section.

The structure of the single-layer AlS is shown in Figure 290 with M = Al and X = S. The structural parameters are from the ab initio calculations [115]. AlS has a bi-buckled configuration as shown in Figure 290(b) , where the buckle is along the zigzag direction. Two buckling layers are symmetrically integrated through the interior Al─Al bonds, forming a bi-buckled configuration. This structure can be determined by three independent geometrical parameters, e.g., the lattice constant 3.57 Å, the bond length $d_{\text{Al}-\text{S}}=2.32$ Å, and the bond length $d_{\text{Al}-\text{Al}}=2.59$ Å.

VFF type	Bond stretching		Angle bending
Expression	$\frac{1}{2}{K}_{\text{Al}-\text{S}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{Al}-\text{Al}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{AlSS}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{AlAlS}}{\left(\text{Δ}\theta \right)}^2$
Parameter	11.065	4.912	3.210	1.900
$r_0$ or ${\theta }_0$	2.320	2.590	100.600	117.324

Table 586.

The VFF model for AlS.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of ${\text{eV/}\mathrm{\AA }}^{\text{2}}$ for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of $\mathrm{\AA }$ ) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Al1─S1	11.476	1.618	14.485	0.0	3.289
Al1─Al2	4.575	1.280	22.499	0.0	3.500

Table 587.

Two-body SW potential parameters for AlS used by GULP [8], as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
Al1─S1─S1	46.910	100.600	1.618	1.618	0.0	3.289	0.0	3.289	0.0	4.877
Al1─S1─Al2	26.090	117.324	1.280	1.618	0.0	3.500	0.0	3.289	0.0	4.853

Table 588.

Three-body SW potential parameters for AlS used by GULP [8], as expressed in Eq. (4).

Figure 302.

Stress-strain relations for AlS of size $100\times 100$ Å. AlS is uniaxially stretched along the armchair or zigzag directions at temperatures 1 and 300 K.

Table 586 shows the VFF model for the single-layer AlS. The force constant parameters are determined by fitting to the six low-frequency branches in the phonon dispersion along the ΓM as shown in Figure 301(a) . The ab initio calculations for the phonon dispersion are from Ref. [115]. Figure 301(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 587 . The parameters for the three-body SW potential used by GULP are shown in Table 588 . Parameters for the SW potential used by LAMMPS are listed in Table 589 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer AlS under uniaxial tension at 1 and 300 K. Figure 302 shows the stress-strain curve for the tension of a single-layer AlS of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer AlS is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer AlS. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 85.2 and 84.6 N/m along the armchair and zigzag directions, respectively. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.22$ .

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Al1─S1─S1	1.000	1.618	2.032	46.910	1.000	−0.184	11.476	2.112	4	0	0.0
Al1─Al2─Al2	1.000	1.280	2.735	0.000	1.000	0.000	4.575	8.388	4	0	0.0
Al1─Al2─S1	1.000	0.000	0.000	26.090	1.000	−0.459	0.000	0.000	4	0	0.0

Table 589.

SW potential parameters for AlS used by LAMMPS [9], as expressed in Eqs. (9) and (10).

VFF type	Bond stretching		Angle bending
Expression	$\frac{1}{2}{K}_{\text{Ga}-\text{S}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{Ga}-\text{Ga}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{GaSS}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{GaGaS}}{\left(\text{Δ}\theta \right)}^2$
Parameter	10.014	6.133	2.925	1.900
$r_0$ or ${\theta }_0$	2.360	2.470	100.921	117.065

Table 590.

The VFF model for GaS.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of ${\text{eV/}\mathrm{\AA }}^{\text{2}}$ for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of $\mathrm{\AA }$ ) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction.

There is no available value for nonlinear quantities in the single-layer AlS. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −289.7 and −302.4 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 11.9 N/m at the ultimate strain of 0.25 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 11.9 N/m at the ultimate strain of 0.36 in the zigzag direction at the low temperature of 1 K.

148. GaS

Present studies on GaS are based on first-principles calculations, and no empirical potential has been proposed for the GaS. We will thus parametrize a set of SW potential for the single-layer GaS in this section.

Figure 303.

Phonon dispersion for the single-layer GaS. (a) The VFF model is fitted to the six low-frequency branches along the ΓM direction. The ab initio results (gray pentagons) are from Ref. [115]. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for GaS along ΓMKΓ.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Ga1─S1	10.825	1.658	15.510	0.0	3.349
Ga1─Ga2	6.316	1.506	18.610	0.0	3.434

Table 591.

Two-body SW potential parameters for GaS used by GULP [8], as expressed in Eq. (3).

The structure of the single-layer GaS is shown in Figure 290 with M = Ga and X = S. The structural parameters are from the ab initio calculations [115]. GaS has a bi-buckled configuration as shown in Figure 290(b) , where the buckle is along the zigzag direction. Two buckling layers are symmetrically integrated through the interior Ga─Ga bonds, forming a bi-buckled configuration. This structure can be determined by three independent geometrical parameters, e.g., the lattice constant 3.64 Å, the bond length $d_{\text{Ga}-\text{S}}=2.36$ Å, and the bond length $d_{\text{Ga}-\text{Ga}}=2.47$ Å.

Figure 304.

Stress-strain relations for GaS of size $100\times 100$ Å. GaS is uniaxially stretched along the armchair or zigzag directions at temperatures 1 and 300 K.

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
Ga1─S1─S1	43.355	100.921	1.658	1.658	0.0	3.349	0.0	3.349	0.0	4.972
Ga1─S1─Ga2	30.536	117.065	1.658	1.506	0.0	3.349	0.0	3.434	0.0	4.809

Table 592.

Three-body SW potential parameters for GaS used by GULP [8], as expressed in Eq. (4).

Table 590 shows the VFF model for the single-layer GaS. The force constant parameters are determined by fitting to the six low-frequency branches in the phonon dispersion along the ΓM as shown in Figure 303(a) . The ab initio calculations for the phonon dispersion are from Ref. [115]. Figure 303(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 591 . The parameters for the three-body SW potential used by GULP are shown in Table 592 . Parameters for the SW potential used by LAMMPS are listed in Table 593 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer GaS under uniaxial tension at 1 and 300 K. Figure 304 shows the stress-strain curve for the tension of a single-layer GaS of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer GaS is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer GaS. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 76.2 N/m along the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.23$ .

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Ga1─S1─S1	1.000	1.658	2.020	43.355	1.000	−0.189	10.825	2.052	4	0	0.0
Ga1─Ga2─Ga2	1.000	1.506	2.280	0.000	1.000	0.000	6.316	3.615	4	0	0.0
Ga1─Ga2─S1	1.000	0.000	0.000	30.536	1.000	−0.455	0.000	0.000	4	0	0.0

Table 593.

SW potential parameters for GaS used by LAMMPS [9], as expressed in Eqs. (9) and (10).

There is no available value for nonlinear quantities in the single-layer GaS. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −254.5 and −269.8 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 10.8 N/m at the ultimate strain of 0.26 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 11.0 N/m at the ultimate strain of 0.36 in the zigzag direction at the low temperature of 1 K.

Figure 305.

Phonon dispersion for the single-layer InS. (a) The VFF model is fitted to the six low-frequency branches along the ΓM direction. The ab initio results (gray pentagons) are from Ref. [115]. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for InS along ΓMKΓ.

VFF type	Bond stretching		Angle bending
Expression	$\frac{1}{2}{K}_{\text{In}-\text{S}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{In}-\text{In}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{InSS}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{InInS}}{\left(\text{Δ}\theta \right)}^2$
Parameter	10.014	4.533	2.179	1.412
$r_0$ or ${\theta }_0$	2.560	2.820	100.624	117.305

Table 594.

The VFF model for InS.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of ${\text{eV/}\mathrm{\AA }}^{\text{2}}$ for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of $\mathrm{\AA }$ ) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction.

149. InS

Present studies on the InS are based on first-principles calculations, and no empirical potential has been proposed for the InS. We will thus parametrize a set of SW potential for the single-layer InS in this section.

Figure 306.

Stress-strain relations for InS of size $100\times 100$ Å. InS is uniaxially stretched along the armchair or zigzag directions at temperatures 1 and 300 K.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
In1─S1	12.652	1.787	21.475	0.0	3.629
In1─In2	5.202	1.454	31.620	0.0	3.833

Table 595.

Two-body SW potential parameters for InS used by GULP [8], as expressed in Eq. (3).

The structure of the single-layer InS is shown in Figure 290 with M = In and X = S. The structural parameters are from the ab initio calculations [115]. InS has a bi-buckled configuration as shown in Figure 290(b) , where the buckle is along the zigzag direction. Two buckling layers are symmetrically integrated through the interior In─In bonds, forming a bi-buckled configuration. This structure can be determined by three independent geometrical parameters, e.g., the lattice constant 3.94 Å, the bond length $d_{\text{In}-\text{S}}=2.56$ Å, and the bond length $d_{\text{In}-\text{In}}=2.82$ Å.

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
In1─S1─S1	31.876	100.624	1.787	1.787	0.0	3.629	0.0	3.629	0.0	5.382
In1─S1─In2	19.900	117.305	1.787	1.454	0.0	3.629	0.0	3.833	0.0	5.325

Table 596.

Three-body SW potential parameters for InS used by GULP [8], as expressed in Eq. (4).

Table 594 shows the VFF model for the single-layer InS. The force constant parameters are determined by fitting to the six low-frequency branches in the phonon dispersion along the ΓM as shown in Figure 305(a) . The ab initio calculations for the phonon dispersion are from Ref. [115]. Figure 305(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 595 . The parameters for the three-body SW potential used by GULP are shown in Table 596 . Parameters for the SW potential used by LAMMPS are listed in Table 597 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer InS under uniaxial tension at 1 and 300 K. Figure 306 shows the stress-strain curve for the tension of a single-layer InS of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer InS is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer InS. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 52.9 and 53.2 N/m along the armchair and zigzag directions, respectively. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.29$ .

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
In1─S1─S1	1.000	1.787	2.031	31.876	1.000	−0.184	12.652	2.108	4	0	0.0
In1─In2─In2	1.000	1.454	2.635	0.000	1.000	0.000	5.202	7.067	4	0	0.0
In1─In2─S1	1.000	0.000	0.000	19.990	1.000	−0.459	0.000	0.000	4	0	0.0

Table 597.

SW potential parameters for InS used by LAMMPS [9], as expressed in Eqs. (9) and (10).

There is no available value for nonlinear quantities in the single-layer InS. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −115.9 and −141.1 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 10.2 N/m at the ultimate strain of 0.32 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 11.8 N/m at the ultimate strain of 0.42 in the zigzag direction at the low temperature of 1 K.

150. BSe

Present studies on the BSe are based on first-principles calculations, and no empirical potential has been proposed for the BSe. We will thus parametrize a set of SW potential for the single-layer BSe in this section.

The structure of the single-layer BSe is shown in Figure 290 with M=B and X=Se. The structural parameters are from the ab initio calculations [115]. The BSe has a bi-buckled configuration as shown in Figure 290(b) , where the buckle is along the zigzag direction. Two buckling layers are symmetrically integrated through the interior B─B bonds, forming a bi-buckled configuration. This structure can be determined by three independent geometrical parameters, e.g., the lattice constant 3.25 Å, the bond length $d_{\text{B}-\text{Se}}=2.10$ Å, and the bond length $d_{\text{B}-\text{B}}=1.71$ Å.

Figure 307.

Phonon dispersion for the single-layer BSe. (a) The VFF model is fitted to the six low-frequency branches along the ΓM direction. The ab initio results (gray pentagons) are from Ref. [115]. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for BSe along ΓMKΓ.

Figure 308.

Stress-strain relations for the BSe of size $100\times 100$ Å. The BSe is uniaxially stretched along the armchair or zigzag directions at temperatures 1 and 300 K.

VFF type	Bond stretching		Angle bending
Expression	$\frac{1}{2}{K}_{\text{B}-\text{Se}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{B}-\text{B}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{BSeSe}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{BBSe}}{\left(\text{Δ}\theta \right)}^2$
Parameter	17.138	15.227	5.144	3.113
$r_0$ or ${\theta }_0$	2.100	1.710	101.394	116.681

Table 598.

The VFF model for BSe.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction.

Table 598 shows the VFF model for the single-layer BSe. The force constant parameters are determined by fitting to the six low-frequency branches in the phonon dispersion along the ΓM as shown in Figure 307(a) . The ab initio calculations for the phonon dispersion are from [115]. Figure 307(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 599 . The parameters for the three-body SW potential used by GULP are shown in Table 600 . Parameters for the SW potential used by LAMMPS are listed in Table 601 .

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
B1─Se1	14.825	1.491	9.724	0.0	2.985
B1─B2	17.700	2.252	4.275	0.0	2.691

Table 599.

Two-body SW potential parameters for BSe used by GULP [8], as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
B1─S1─Se1	77.850	101.394	1.491	1.491	0.0	2.985	0.0	2.985	0.0	4.440
B1─B2─Se1	104.372	116.681	2.252	1.491	0.0	2.691	0.0	2.985	0.0	3.923

Table 600.

Three-body SW potential parameters for BSe used by GULP [8], as expressed in Eq. (4).

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer BSe under uniaxial tension at 1.0 and 300.0 K. Figure 308 shows the stress-strain curve for the tension of a single-layer BSe of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer BSe is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer BSe. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 157.3 and 156.4 N/m along the armchair and zigzag directions, respectively. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.19$ .

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
B1─Se1─Se1	1.000	1.491	2.002	77.850	1.000	−0.198	14.825	1.968	4	0	0.0
B1─B2─B2	1.000	2.252	1.195	0.000	1.000	0.000	17.700	0.166	4	0	0.0
B1─B2─Se1	1.000	0.000	0.000	104.372	1.000	−0.449	0.000	0.000	4	0	0.0

Table 601.

SW potential parameters for BSe used by LAMMPS [9], as expressed in Eqs. (9) and (10).

There is no available value for nonlinear quantities in the single-layer BSe. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −627.0 and −655.6 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 19.2 N/m at the ultimate strain of 0.23 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 18.7 N/m at the ultimate strain of 0.31 in the zigzag direction at the low temperature of 1 K.

Figure 309.

Phonon dispersion for the single-layer AlSe. (a) The VFF model is fitted to the six low-frequency branches along the ΓM direction. The ab initio results (gray pentagons) are from [115]. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for the AlSe along ΓMKΓ.

VFF type	Bond stretching		Angle bending
Expression	$\frac{1}{2}{K}_{Al-Se}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{Al-Al}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{AlSeSe}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{AlAlSe}{\left(\text{Δ}\theta \right)}^2$
Parameter	9.831	4.487	2.916	1.659
$r_0$ or ${\theta }_0$	2.470	2.570	99.846	117.926

Table 602.

The VFF model for AlSe.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Al1─Se1	11.362	1.694	18.610	0.0	3.493
Al1─Al2	4.974	1.558	21.812	0.0	3.570

Table 603.

Two-body SW potential parameters for AlSe used by GULP [8], as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
Al1─Se1─Se1	41.235	99.846	1.694	1.694	0.0	3.493	0.0	3.493	0.0	5.164
Al1─Se1─Al2	26.418	117.926	1.558	1.694	0.0	3.570	0.0	3.493	0.0	5.029

Table 604.

Three-body SW potential parameters for AlSe used by GULP [8], as expressed in Eq. (4).

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Al1─Se1─Se1	1.000	1.694	2.062	41.235	1.000	−0.171	11.362	2.260	4	0	0.0
Al1─Al2─Al2	1.000	1.558	2.292	0.000	1.000	0.000	4.974	3.704	4	0	0.0
Al1─Al2─Se1	1.000	0.000	0.000	26.418	1.000	−0.468	0.000	0.000	4	0	0.0

Table 605.

SW potential parameters for AlSe used by LAMMPS [9], as expressed in Eqs. (9) and (10).

Figure 310.

Stress-strain relations for the AlSe of size $100\times 100$ Å. The AlSe is uniaxially stretched along the armchair or zigzag directions at temperatures 1 and 300 K.

151. AlSe

Present studies on the AlSe are based on first-principles calculations, and no empirical potential has been proposed for the AlSe. We will thus parametrize a set of SW potential for the single-layer AlSe in this section.

The structure of the single-layer AlSe is shown in Figure 290 with M = Al and X = Se. The structural parameters are from the ab initio calculations [115]. The AlSe has a bi-buckled configuration as shown in Figure 290(b) , where the buckle is along the zigzag direction. Two buckling layers are symmetrically integrated through the interior Al-Al bonds, forming a bi-buckled configuration. This structure can be determined by three independent geometrical parameters, e.g., the lattice constant 3.78 Å, the bond length $d_{\text{Al}-\text{Se}}=2.47$ Å, and the bond length $d_{\text{Al}-\text{Al}}=2.57$ Å.

Table 602 shows the VFF model for the single-layer AlSe. The force constant parameters are determined by fitting to the six low-frequency branches in the phonon dispersion along the ΓM as shown in Figure 309(a) . The ab initio calculations for the phonon dispersion are from [115]. Figure 309(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 603 . The parameters for the three-body SW potential used by GULP are shown in Table 604 . Parameters for the SW potential used by LAMMPS are listed in Table 605 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer AlSe under uniaxial tension at 1.0 and 300.0 K. Figure 310 shows the stress-strain curve for the tension of a single-layer AlSe of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer AlSe is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer AlSe. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 69.4 and 69.2 N/m along the armchair and zigzag directions, respectively. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.24$ .

There is no available value for nonlinear quantities in the single-layer AlSe. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −217.3 and −231.9 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 10.4 N/m at the ultimate strain of 0.27 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 10.5 N/m at the ultimate strain of 0.38 in the zigzag direction at the low temperature of 1 K.

152. GaSe

Present studies on the GaSe are based on first-principles calculations, and no empirical potential has been proposed for the GaSe. We will thus parametrize a set of SW potential for the single-layer GaSe in this section.

The structure of the single-layer GaSe is shown in Figure 290 with M=Ga and X=Se. The structural parameters are from the ab initio calculations [115]. The GaSe has a bi-buckled configuration as shown in Figure 290(b) , where the buckle is along the zigzag direction. Two buckling layers are symmetrically integrated through the interior Ga-Ga bonds, forming a bi-buckled configuration. This structure can be determined by three independent geometrical parameters, e.g., the lattice constant 3.82 Å, the bond length $d_{\text{Ga}-\text{Se}}=2.50$ Å, and the bond length $d_{\text{Ga}-\text{Ga}}=2.46$ Å.

VFF type	Bond stretching		Angle bending
Expression	$\frac{1}{2}{K}_{\text{Ga}-\text{Se}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{Ga}-\text{Ga}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{GaSeSe}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{GaGaSe}}{\left(\text{Δ}\theta \right)}^2$
Parameter	10.014	5.400	2.925	1.701
$r_0$ or ${\theta }_0$	2.500	2.460	99.636	118.092

Table 606.

The VFF model for GaSe.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Ga1─Se1	11.798	1.706	19.531	0.0	3.533
Ga1─Ga2	6.479	1.765	18.311	0.0	3.502

Table 607.

Two-body SW potential parameters for GaSe used by GULP [8], as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
Ga1─Se1─Se1	40.978	99.636	1.706	1.706	0.0	3.533	0.0	3.533	0.0	5.218
Ga1─Se1─Ga2	31.031	118.092	1.765	1.706	0.0	3.502	0.0	3.533	0.0	4.985

Table 608.

Three-body SW potential parameters for GaSe used by GULP [8], as expressed in Eq. (4).

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Ga1─Se1─Se1	1.000	1.706	2.070	40.978	1.000	−0.167	11.798	2.305	4	0	0.0
Ga1─Ga2─Ga2	1.000	1.765	1.984	0.000	1.000	0.000	6.479	1.888	4	0	0.0
Ga1─Ga2─Se1	1.000	0.000	0.000	31.031	1.000	−0.471	0.000	0.000	4	0	0.0

Table 609.

SW potential parameters for GaSe used by LAMMPS [9], as expressed in Eqs. (9) and (10).

Figure 311.

Phonon dispersion for the single-layer GaSe. (a) The VFF model is fitted to the six low-frequency branches along the ΓM direction. The ab initio results (gray pentagons) are from [115]. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for the GaSe along ΓMKΓ.

Table 606 shows the VFF model for the single-layer GaSe. The force constant parameters are determined by fitting to the six low-frequency branches in the phonon dispersion along the ΓM as shown in Figure 311(a) . The ab initio calculations for the phonon dispersion are from [115]. Figure 311(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 607 . The parameters for the three-body SW potential used by GULP are shown in Table 608 . Parameters for the SW potential used by LAMMPS are listed in Table 609 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer GaSe under uniaxial tension at 1.0 and 300.0 K. Figure 312 shows the stress-strain curve for the tension of a single-layer GaSe of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer GaSe is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer GaSe. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 68.3 and 67.9 N/m along the armchair and zigzag directions, respectively. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.25$ .

Figure 312.

Stress-strain relations for the GaSe of size $100\times 100$ Å. The GaSe is uniaxially stretched along the armchair or zigzag directions at temperatures 1 and 300 K.

There is no available value for nonlinear quantities in the single-layer GaSe. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −206.1 and −219.8 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 10.5 N/m at the ultimate strain of 0.27 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 10.8 N/m at the ultimate strain of 0.39 in the zigzag direction at the low temperature of 1 K.

VFF type	Bond stretching		Angle bending
Expression	$\frac{1}{2}{K}_{\text{In}-\text{Se}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{In}-\text{In}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{InSeSe}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{InInSe}}{\left(\text{Δ}\theta \right)}^2$
Parameter	9.812	4.185	2.090	1.227
$r_0$ or ${\theta }_0$	2.690	2.810	99.296	118.361

Table 610.

The VFF model for InSe.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
In1─Se1	13.281	1.822	26.181	0.0	3.797
In1─In2	5.414	1.661	31.174	0.0	2.890

Table 611.

Two-body SW potential parameters for InSe used by GULP [8], as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
In1─Se1─Se1	28.853	99.296	1.822	1.822	0.0	3.797	0.0	3.797	0.0	5.601
In1─Se1─In2	19.120	118.361	1.822	1.661	0.0	3.797	0.0	3.890	0.0	5.489

Table 612.

Three-body SW potential parameters for InSe used by GULP [8], as expressed in Eq. (4).

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
In1─Se1─Se1	1.000	1.822	2.084	28.853	1.000	−0.162	13.281	2.377	4	0	0.0
In1─In2─In2	1.000	1.661	2.342	0.000	1.000	0.000	5.414	4.094	4	0	0.0
In1─In2─Se1	1.000	0.000	0.000	19.120	1.000	−0.475	0.000	0.000	4	0	0.0

Table 613.

SW potential parameters for InSe used by LAMMPS [9], as expressed in Eqs. (9) and (10).

Figure 313.

Phonon dispersion for the single-layer InSe. (a) The VFF model is fitted to the six low-frequency branches along the ΓM direction. The ab initio results (gray pentagons) are from [115]. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for the InSe along ΓMKΓ.

153. InSe

Present studies on the InSe are based on first-principles calculations, and no empirical potential has been proposed for the InSe. We will thus parametrize a set of SW potential for the single-layer InSe in this section.

The structure of the single-layer InSe is shown in Figure 290 with M=In and X=Se. The structural parameters are from the ab initio calculations [115]. The InSe has a bi-buckled configuration as shown in Figure 290(b) , where the buckle is along the zigzag direction. Two buckling layers are symmetrically integrated through the interior In-In bonds, forming a bi-buckled configuration. This structure can be determined by three independent geometrical parameters, e.g., the lattice constant 4.10 Å, the bond length $d_{\text{In}-\text{Se}}=2.69$ Å, and the bond length $d_{\text{In}-\text{In}}=2.81$ Å.

Figure 314.

Stress-strain relations for the InSe of size $100\times 100$ Å. The InSe is uniaxially stretched along the armchair or zigzag directions at temperatures 1 and 300 K.

Table 610 shows the VFF model for the single-layer InSe. The force constant parameters are determined by fitting to the six low-frequency branches in the phonon dispersion along the ΓM as shown in Figure 313(a) . The ab initio calculations for the phonon dispersion are from [115]. Figure 313(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 611 . The parameters for the three-body SW potential used by GULP are shown in Table 612 . Parameters for the SW potential used by LAMMPS are listed in Table 613 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer InSe under uniaxial tension at 1.0 and 300.0 K. Figure 314 shows the stress-strain curve for the tension of a single-layer InSe of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer InSe is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer InSe. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 45.7 and 45.8 N/m along the armchair and zigzag directions, respectively. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.30$ .

VFF type	Bond stretching		Angle bending
Expression	$\frac{1}{2}{K}_{\text{B}-\text{Te}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{B}-\text{B}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{BTeTe}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{BBTe}}{\left(\text{Δ}\theta \right)}^2$
Parameter	13.287	14.502	5.466	2.515
$r_0$ or ${\theta }_0$	2.310	1.710	100.809	117.156

Table 614.

The VFF model for BTe.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
B1─Te1	13.727	1.619	14.237	0.0	3.277
B1─B2	24.140	2.933	4.275	0.0	2.830

Table 615.

Two-body SW potential parameters for BTe used by GULP [8], as expressed in Eq. (3).

There is no available value for nonlinear quantities in the single-layer InSe. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −81.6 and −103.5 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 9.8 N/m at the ultimate strain of 0.35 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 11.9 N/m at the ultimate strain of 0.44 in the zigzag direction at the low temperature of 1 K.

154. BTe

Present studies on the BTe are based on first-principles calculations, and no empirical potential has been proposed for the BTe. We will thus parametrize a set of SW potential for the single-layer BTe in this section.

Figure 315.

Phonon dispersion for the single-layer BTe. (a) The VFF model is fitted to the six low-frequency branches along the ΓM direction. The ab initio results (gray pentagons) are from [115]. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for the BTe along ΓMKΓ.

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
B1─Te1─Te1	80.622	100.809	1.619	1.619	0.0	3.277	0.0	3.277	0.0	4.863
B1─B2─Te1	116.3301	117.156	2.933	1.619	0.0	2.830	0.0	3.277	0.0	4.199

Table 616.

Three-body SW potential parameters for BTe used by GULP [8], as expressed in Eq. (4).

The structure of the single-layer BTe is shown in Figure 290 with M=B and X=Te. The structural parameters are from the ab initio calculations [115]. The BTe has a bi-buckled configuration as shown in Figure 290(b) , where the buckle is along the zigzag direction. Two buckling layers are symmetrically integrated through the interior B-B bonds, forming a bi-buckled configuration. This structure can be determined by three independent geometrical parameters, e.g., the lattice constant 3.56 Å, the bond length $d_{\text{B}-\text{Te}}=2.31$ Å, and the bond length $d_{\text{B}-\text{B}}=1.71$ Å.

Figure 316.

Stress-strain relations for the BTe of size $100\times 100$ Å. The BTe is uniaxially stretched along the armchair or zigzag directions at temperatures 1 and 300 K.

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
B1─Te1─Te1	1.000	1.619	2.024	80.622	1.000	−0.188	13.727	2.073	4	0	0.0
B1─B2─B2	1.000	2.933	0.965	0.000	1.000	0.000	24.140	0.058	4	0	0.0
B1─B2─Te1	1.000	0.000	0.000	116.301	1.000	−0.456	0.000	0.000	4	0	0.0

Table 617.

SW potential parameters for BTe used by LAMMPS [9], as expressed in Eqs. (9) and (10).

Table 614 shows the VFF model for the single-layer BTe. The force constant parameters are determined by fitting to the six low-frequency branches in the phonon dispersion along the ΓM as shown in Figure 315(a) . The ab initio calculations for the phonon dispersion are from [115]. Figure 315(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 615 . The parameters for the three-body SW potential used by GULP are shown in Table 616 . Parameters for the SW potential used by LAMMPS are listed in Table 617 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer BTe under uniaxial tension at 1.0 and 300.0 K. Figure 316 shows the stress-strain curve for the tension of a single-layer BTe of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer BTe is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer BTe. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 130.6 and 129.7 N/m along the armchair and zigzag directions, respectively. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.16$ .

VFF type	Bond stretching		Angle bending
Expression	$\frac{1}{2}{K}_{\text{Al}-\text{Te}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{Al}-\text{Al}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{AlTeTe}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{AlAlTe}}{\left(\text{Δ}\theta \right)}^2$
Parameter	8.077	3.859	2.820	1.518
$r_0$ or ${\theta }_0$	2.700	2.580	99.124	118.495

Table 618.

The VFF model for AlTe.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of $\mathrm{\AA }$ ) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction.

There is no available value for nonlinear quantities in the single-layer BTe. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −560.4 and −588.7 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 14.9 N/m at the ultimate strain of 0.22 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 14.4 N/m at the ultimate strain of 0.27 in the zigzag direction at the low temperature of 1 K.

Figure 317.

Phonon dispersion for the single-layer AlTe. (a) The VFF model is fitted to the six low-frequency branches along the ΓM direction. The ab initio results (gray pentagons) are from [115]. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for the AlTe along ΓMKΓ.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Al1─Te1	10.971	1.821	26.572	0.0	3.809
Al1─Al2	5.523	2.002	22.154	0.0	3.716

Table 619.

Two-body SW potential parameters for AlTe used by GULP [8], as expressed in Eq. (3).

155. AlTe

Present studies on the AlTe are based on first-principles calculations, and no empirical potential has been proposed for the AlTe. We will thus parametrize a set of SW potential for the single-layer AlTe in this section.

Figure 318.

Stress-strain relations for the AlTe of size $100\times 100$ Å. The AlTe is uniaxially stretched along the armchair or zigzag directions at temperatures 1 and 300 K.

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
Al1─Te1─Te1	38.638	99.124	1.821	1.821	0.0	3.809	0.0	3.809	0.0	5.614
Al1─Te1─Al2	29.574	118.495	2.002	1.821	0.0	3.716	0.0	3.809	0.0	5.330

Table 620.

Three-body SW potential parameters for AlTe used by GULP [8], as expressed in Eq. (4).

The structure of the single-layer AlTe is shown in Figure 290 with M=Al and X=Te. The structural parameters are from the ab initio calculations [115]. The AlTe has a bi-buckled configuration as shown in Figure 290(b) , where the buckle is along the zigzag direction. Two buckling layers are symmetrically integrated through the interior Al-Al bonds, forming a bi-buckled configuration. This structure can be determined by three independent geometrical parameters, e.g., the lattice constant 4.11 Å, the bond length $d_{\text{Al}-\text{Te}}=2.70$ Å, and the bond length $d_{\text{Al}-\text{Al}}=2.58$ Å.

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Al1─Te1─Te1	1.000	1.821	2.091	38.638	1.000	−0.159	10.971	2.415	4	0	0.0
Al1─Al2─Al2	1.000	2.002	1.856	0.000	1.000	0.000	5.523	1.379	4	0	0.0
Al1─Al2─Te1	1.000	0.000	0.000	29.574	1.000	−0.477	0.000	0.000	4	0	0.0

Table 621.

SW potential parameters for AlTe used by LAMMPS [9], as expressed in Eqs. (9) and (10).

Table 618 shows the VFF model for the single-layer AlTe. The force constant parameters are determined by fitting to the six low-frequency branches in the phonon dispersion along the ΓM as shown in Figure 317(a) . The ab initio calculations for the phonon dispersion are from [115]. Figure 317(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 619 . The parameters for the three-body SW potential used by GULP are shown in Table 620 . Parameters for the SW potential used by LAMMPS are listed in Table 621 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer AlTe under uniaxial tension at 1.0 and 300.0 K. Figure 318 shows the stress-strain curve for the tension of a single-layer AlTe of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer AlTe is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer AlTe. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 55.8 and 54.9 N/m along the armchair and zigzag directions, respectively. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.24$ .

There is no available value for nonlinear quantities in the single-layer AlTe. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −171.4 and −179.0 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 8.4 N/m at the ultimate strain of 0.27 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 8.6 N/m at the ultimate strain of 0.39 in the zigzag direction at the low temperature of 1 K.

Figure 319.

Phonon dispersion for the single-layer GaTe. (a) The VFF model is fitted to the six low-frequency branches along the ΓM direction. The ab initio results (gray pentagons) are from [115]. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for the GaTe along ΓMKΓ.

156. GaTe

Present studies on the GaTe are based on first-principles calculations, and no empirical potential has been proposed for the GaTe. We will thus parametrize a set of SW potential for the single-layer GaTe in this section.

The structure of the single-layer GaTe is shown in Figure 290 with M=Ga and X=Te. The structural parameters are from the ab initio calculations [115]. The GaTe has a bi-buckled configuration as shown in Figure 290(b) , where the buckle is along the zigzag direction. Two buckling layers are symmetrically integrated through the interior Ga-Ga bonds, forming a bi-buckled configuration. This structure can be determined by three independent geometrical parameters, e.g., the lattice constant 4.13 Å, the bond length $d_{\text{Ga}-\text{Te}}=2.70$ Å, and the bond length $d_{\text{Ga}-\text{Ga}}=2.46$ Å.

VFF type	Bond stretching		Angle bending
Expression	$\frac{1}{2}{K}_{\text{Ga}-\text{Te}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{Ga}-\text{Ga}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{GaTeTe}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{GaGaTe}}{\left(\text{Δ}\theta \right)}^2$
Parameter	7.382	4.366	2.841	1.519
$r_0$ or ${\theta }_0$	2.700	2.460	99.781	117.978

Table 622.

The VFF model for GaTe.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
Ga1─Te1	10.179	1.849	26.572	0.0	3.817
Ga1─Ga2	6.750	2.239	18.311	0.0	3.634

Table 623.

Two-body SW potential parameters for GaTe used by GULP [8], as expressed in Eq. (3).

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
Ga1─Te1─Te1	40.060	99.781	1.849	1.849	0.0	3.817	0.0	3.817	0.0	5.642
Ga1─Te1─Ga2	34.354	117.978	2.239	1.849	0.0	3.634	0.0	3.817	0.0	5.238

Table 624.

Three-body SW potential parameters for GaTe used by GULP [8], as expressed in Eq. (4).

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
Ga1─Te1─Te1	1.000	1.849	2.065	40.060	1.000	−0.170	10.179	2.274	4	0	0.0
Ga1─Ga2─Ga2	1.000	2.239	1.623	0.000	1.000	0.000	6.750	0.728	4	0	0.0
Ga1─Ga2─Te1	1.000	0.000	0.000	34.354	1.000	−0.469	0.000	0.000	4	0	0.0

Table 625.

SW potential parameters for GaTe used by LAMMPS [9], as expressed in Eqs. (9) and (10).

Figure 320.

Stress-strain relations for the GaTe of size $100\times 100$ Å. The GaTe is uniaxially stretched along the armchair or zigzag directions at temperatures 1 and 300 K.

Table 622 shows the VFF model for the single-layer GaTe. The force constant parameters are determined by fitting to the six low-frequency branches in the phonon dispersion along the ΓM as shown in Figure 319(a) . The ab initio calculations for the phonon dispersion are from [115]. Figure 319(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 623 . The parameters for the three-body SW potential used by GULP are shown in Table 624 . Parameters for the SW potential used by LAMMPS are listed in Table 625 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer GaTe under uniaxial tension at 1.0 and 300.0 K. Figure 320 shows the stress-strain curve for the tension of a single-layer GaTe of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer GaTe is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer GaTe. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 55.2 and 55.3 N/m along the armchair and zigzag directions, respectively. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.23$ .

VFF type	Bond stretching		Angle bending
Expression	$\frac{1}{2}{K}_{\text{In}-\text{Te}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{In}-\text{In}}{\left(\text{Δ}r\right)}^2$	$\frac{1}{2}{K}_{\text{InTeTe}}{\left(\text{Δ}\theta \right)}^2$	$\frac{1}{2}{K}_{\text{InInTe}}{\left(\text{Δ}\theta \right)}^2$
Parameter	5.592	3.928	2.419	1.227
$r_0$ or ${\theta }_0$	2.890	2.810	99.148	118.477

Table 626.

The VFF model for InTe.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
In1─Te1	8.707	1.950	34.879	0.0	4.077
In1─In2	6.312	2.068	31.174	0.0	4.015

Table 627.

Two-body SW potential parameters for InTe used by GULP [8], as expressed in Eq. (3).

There is no available value for nonlinear quantities in the single-layer GaTe. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −183.2 and −195.6 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 7.8 N/m at the ultimate strain of 0.26 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 7.8 N/m at the ultimate strain of 0.37 in the zigzag direction at the low temperature of 1 K.

157. InTe

Present studies on the InTe are based on first-principles calculations, and no empirical potential has been proposed for the InTe. We will thus parametrize a set of SW potential for the single-layer InTe in this section.

Figure 321.

Phonon dispersion for the single-layer InTe. (a) The VFF model is fitted to the six low-frequency branches along the ΓM direction. The ab initio results (gray pentagons) are from [115]. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for the InTe along ΓMKΓ.

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
In1─Te1─Te1	33.178	99.148	1.950	1.950	0.0	4.077	0.0	4.077	0.0	6.011
In1─Te1─In2	22.833	118.477	2.068	1.950	0.0	4.015	0.0	4.077	0.0	5.741

Table 628.

Three-body SW potential parameters for InTe used by GULP [8], as expressed in Eq. (4).

The structure of the single-layer InTe is shown in Figure 290 with M=In and X=Te. The structural parameters are from the ab initio calculations [115]. The InTe has a bi-buckled configuration as shown in Figure 290(b) , where the buckle is along the zigzag direction. Two buckling layers are symmetrically integrated through the interior In-In bonds, forming a bi-buckled configuration. This structure can be determined by three independent geometrical parameters, e.g., the lattice constant 4.40 Å, the bond length $d_{\text{In}-\text{Te}}=2.89$ Å, and the bond length $d_{\text{In}-\text{In}}=2.81$ Å.

Figure 322.

Stress-strain relations for the InTe of size $100\times 100$ Å. The InTe is uniaxially stretched along the armchair or zigzag directions at temperatures 1 and 300 K.

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
In1─Te1─Te1	1.000	1.950	2.090	33.178	1.000	−0.159	8.707	2.410	4	0	0.0
In1─In2─In2	1.000	2.068	1.942	0.000	1.000	0.000	6.312	1.704	4	0	0.0
In1─In2─Te1	1.000	0.000	0.000	22.833	1.000	−0.477	0.000	0.000	4	0	0.0

Table 629.

SW potential parameters for InTe used by LAMMPS [9], as expressed in Eqs. (9) and (10).

Table 626 shows the VFF model for the single-layer InTe. The force constant parameters are determined by fitting to the six low-frequency branches in the phonon dispersion along the ΓM as shown in Figure 321(a) . The ab initio calculations for the phonon dispersion are from [115]. Figure 321(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 627 . The parameters for the three-body SW potential used by GULP are shown in Table 628 . Parameters for the SW potential used by LAMMPS are listed in Table 629 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer InTe under uniaxial tension at 1.0 and 300.0 K. Figure 322 shows the stress-strain curve for the tension of a single-layer InTe of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer InTe is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer InTe. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 40.6 and 40.9 N/m along the armchair and zigzag directions, respectively. The Poisson’s ratio from the VFF model and the SW potential is ${\nu }_{xy}=\mathrm{}{\nu }_{yx}=0.23$ .

There is no available value for nonlinear quantities in the single-layer InTe. We have thus used the nonlinear parameter $B=0.5d^4$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third-order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are −130.4 and −142.2 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 5.9 N/m at the ultimate strain of 0.26 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 5.9 N/m at the ultimate strain of 0.38 in the zigzag direction at the low temperature of 1 K.

158. Borophene

Most existing theoretical studies on the monolayer of boron atoms (borophene) are based on the first-principles calculations. The ReaxFF force field model was developed for the borophene recently [117]. The present authors have provided the VFF model and the SW potential to describe the atomic interaction within the borophene [14], which includes the second-nearest-neighboring interactions. In the present work, we present a more efficient SW potential with only the first-nearest-neighboring interactions.

The structure of the borophene is shown in Figure 323 , with structural parameters from the ab initio calculations [116]. Borophene has a puckered configuration as shown in Figure 323(b) , where the pucker is perpendicular to the x-direction. The height of the pucker is $h=0.911$ Å, which is the distance between the top chain and the bottom chain along the out-of-plane z-direction. The two lattice bases are $a_1=2.866$ Å and $a_2=1.614$ Å for the in-plane rectangular unit cell. There are two inequivalent boron atoms in the unit cell. Boron atoms are categorized into the top chain and the bottom chain. The top chain includes atoms like 1, 4, and 7. The bottom chain includes atoms like 2, 3, 5, and 6.

Figure 323.

Structure for borophene. (a) Top view. Atoms are categorized into top chains and bottom chains. The top chain includes atoms like 1, 4, and 7. The bottom chain includes atoms like 2, 3, 5, and 6. The unit cell is shown by blue rectangle. The first Brillouin zone is shown by red rectangle on the left. (b) Perspective view illustrates the puckered configuration, with $h$ as the distance between the top and bottom chains along the out-of-plane z-direction. The pucker is perpendicular to the x-axis and is parallel with the y-axis.

Table 630 shows four VFF terms for the borophene; two of which are the bond stretching interaction shown by Eq. (1), while the other two terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the ΓX as shown in Figure 324(a) . The ab initio calculations for the phonon dispersion are from [116]. Similar phonon dispersion can also be found in other ab initio calculations [118]. Figure 324(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

Figure 324.

Phonon dispersion for the borophene. (a) The VFF model is fitted to the three acoustic branches in the long wave limit along the ΓY direction. The ab initio results (gray pentagons) are from [116]. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for the borophene along ΓYSXΓ.

The parameters for the two-body SW potential used by GULP are shown in Table 631 . The parameters for the three-body SW potential used by GULP are shown in Table 632 . Parameters for the SW potential used by LAMMPS are listed in Table 633 . We note that 12 atom types have been introduced for the simulation of borophene using LAMMPS, because the angles around atom 1 in Figure 323(a) are not distinguishable in LAMMPS. We thus need to differentiate these angles by assigning these six neighboring atoms (2, 3, 4, 5, 6, 7) with different atom types. Figure 325 shows that 12 atom types are necessary for the purpose of differentiating all 6 neighbors around 1 B atom.

Figure 325.

Twelve atom types are introduced for the boron atoms in the borophene. Atoms B₁, B₃, B₅, B₇, B₉, and B₁₁ are from the bottom group. Atoms B₂, B₄, B₆, B₈, B₁₀, and B₁₂ are from the top group.

Figure 326.

Stress-strain relations for the borophene of size $100\times 100$ Å. The borophene is uniaxially stretched along the x or y directions at temperatures 1 and 300 K.

Figure 326 shows the stress-strain relations for the borophene of size $100\times 100$ Å. The structure is uniaxially stretched in the x or y directions at 1 and 300 K. The Young’s modulus is 162.7 and 385.0 N/m in the x and y directions, respectively, at 1 K, which is obtained by linear fitting of the stress-strain relations in [0, 0.01]. These values are in good agreement with the ab initio results at 0 K temperature, e.g., 170 and 398 N/m in [119], or 166 and 389 N/m in [116], or 163 and 399 N/m in [120]. Previous ab initio calculations obtained negative Poisson’s ratio for the uniaxial stretching of the borophene in the x and y directions, e.g., −0.02 and −0.04 in Refs 119 and 116. The Poisson’s ratio from the present SW potential is −0.03 and −0.07 along the x and y directions, respectively, which is quite comparable with the ab initio results.

VFF type	Bond stretching		Angle bending
Expression	$\frac{1}{2}{K}_{14}{\left(\text{Δ}{r}_{14}\right)}^2$	$\frac{1}{2}{K}_{12}{\left(\text{Δ}{r}_{12}\right)}^2$	$\frac{1}{2}{K}_{134}{\left(\text{Δ}{\theta }_{134}\right)}^2$	$\frac{1}{2}{K}_{135}{\left(\text{Δ}{\theta }_{135}\right)}^2$
Parameter	20.673	6.025	3.523	4.651
$r_0$ or ${\theta }_0$	1.614	1.880	64.581	99.318

Table 630.

The VFF model for borophene.

The second line gives an explicit expression for each VFF term, where atom indexes are from Figure 323(a) . The third line is the force constant parameters. Parameters are in the unit of eV/Ų for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle ${\theta }_{ijk}$ has atom i as the apex.

	A (eV)	ρ (Å)	B (Å^4)	$r_{\text{min}}$ (Å)	$r_{\text{max}}$ (Å)
$r_{14}$	8.974	0.971	3.393	0.0	2.240
$r_{12}$	2.098	0.618	6.246	0.0	2.419

Table 631.

Two-body SW potential parameters for borophene used by GULP [8], as expressed in Eq. (3).

The quantity ( $r_{ij}$ ) in the first line lists one representative term for the two-body SW potential between atoms i and j. Atom indexes are from Figure 323(a) .

	K (eV)	${\theta }_0$ (°)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\text{min}12}$ (Å)	$r_{\text{max}12}$ (Å)	$r_{\text{min}13}$ (Å)	$r_{\text{max}13}$ (Å)	$r_{\text{min}23}$ (Å)	$r_{\text{max}23}$ (Å)
${\theta }_{134}$	32.074	64.581	0.971	0.618	0.0	2.240	0.0	2.419	0.0	2.419
${\theta }_{135}$	23.668	99.318	0.618	0.618	0.0	2.419	0.0	2.410	2.240	3.047

Table 632.

Three-body SW potential parameters for borophene used by GULP [8], as expressed in Eq. (4).

The first line ( ${\theta }_{ijk}$ ) presents one representative term for the three-body SW potential. The angle ${\theta }_{ijk}$ has the atom i as the apex. Atom indexes are from Figure 323(a) .

	$\mathbf{ϵ}$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\mathbf{cos}{\theta }_0$	$A_L$	$B_L$	$p$	$q$	Tol
B1─B5─B5	1.000	0.971	2.307	0.000	1.000	0.000	8.974	3.817	4	0	0.0
B1─B2─B2	1.000	0.618	3.914	0.000	1.000	0.000	2.098	42.820	4	0	0.0
B1─B2─B5	1.000	0.000	0.000	32.074	1.000	0.429	0.000	0.000	4	0	0.0
B1─B2─B4	1.000	0.000	0.000	23.668	1.000	−0.162	0.000	0.000	4	0	0.0

Table 633.

SW potential parameters for borophene used by LAMMPS [9], as expressed in Eqs. (9) and (10).

Atom types in the first column are displayed in Figure 325 .

The third-order nonlinear constant ( $D$ ) can be obtained by fitting the stress-strain relation to the function $\sigma =\mathrm{}Eϵ+\frac{1}{2}D{ϵ}^2$ , with $E$ as the Young’s modulus. The obtained values of $D$ are −1100.1 and −2173.6 N/m in the x and y directions, respectively. The ultimate stress is about 12.3 N/m at the critical strain of 0.17 in the x-direction at the low temperature of 1 K, which agree quite well with the ab initio results at 0 K [116, 118, 120]. The ultimate stress is about 32.9 N/m at the critical strain of 0.16 in the y-direction at the low temperature of 1 K, which are quite comparable with ab initio results at 0 K [116, 118, 120].

159. Conclusion remarks

As a final remark, we note some major advantages and deficiencies for the SW potential parameters provided in the present work. On the one hand, the key feature of the SW potential is its high efficiency, which is maintained by using minimum potential parameters in the present work, so the interaction range is limited to the first-nearest-neighboring atoms. As a result, the present SW potential parameters are of high computational efficiency. On the other hand, since the interaction is limited to short-range, the optical branches in the phonon spectrum are typically overestimated by the present SW potential. It is because we have ignored the long-range interactions, which contribute mostly to the acoustic phonon branches while have neglectable contribution to the optical phonon branches. The short-range interaction has thus been strengthened to give an accurate description for the acoustic phonon branches and the elastic properties, which leads to the overestimation of the optical phonon branches as a trade-off. Hence, there will be systematic overestimation for simulating optical processes using the present SW parameters.

We also note that the mathematical form of the SW potential is not suitable for the atomic-thick planar structures, such as graphene and b-BN, because the SW potential is not able to resist the bending motion of these real planar crystals [121, 122].

In conclusion, we have provided the SW potential parameters for 156 layered crystals. The supplemental resources for all simulations in the present work are available online in [1], including a Fortran code to generate crystals’ structures, files for molecular dynamics simulations using LAMMPS, files for phonon calculation with the SW potential using GULP, and files for phonon calculations with the valence force field model using GULP.

Acknowledgments

The work is supported by the Recruitment Program of Global Youth Experts of China, the National Natural Science Foundation of China (NSFC) under Grant No. 11504225, and the start-up funding from Shanghai University.