High-Order Harmonic Generation in 1D Single-Wall Carbon Nanotubes

Abstract

A comprehensive theoretical investigation of the process of high-order harmonic generation induced by intense few-cycle infrared laser pulses in one-dimensional single-wall carbon nanotubes is presented. The resulting emission spectra exhibit a non-perturbative plateau at high intensities. However, unlike more conventional systems such as atoms, molecules, or bulk solids, there is no simple scaling law governing the relationship between the cut-off frequency and the intensity. The interpretation of this distinctive behavior provides insights into the fundamental mechanism underlying high-order harmonic generation in these low-dimensional carbon allotropes. Employing a model for the emission dipole based on the saddle-point approximation, the study demonstrates that the initial step of harmonic emission is closely linked to the singular geometry of the band structure. This mechanism bears remarkable similarity to that observed in graphene but differs from the tunneling ionization/excitation process observed in gas systems and materials with finite band gaps. Notably, the pivotal role played by van Hove singularities in the generation of electron-hole pairs is demonstrated.

1. Introduction

Carbon nanotubes (CNTs) represent a compelling category of carbon-based nanostructures. They are comprised of hollow cylinders formed by the rolling of one-atom-thick sheets of carbon, exhibiting nanometer-scale diameters and aspect ratios that can reach up to ${10}^8$ [1], a magnitude notably greater than that of other one-dimensional materials. While the observation of such structures had been documented earlier [2, 3, 4], interest in studying CNTs surged following the seminal work by Iijima in 1991 [5]. Employing high-resolution transmission electron microscopy, Iijima characterized “graphite microtubules” comprised of concentric shells of carbon atoms, subsequently recognized as multi-wall carbon nanotubes (MWNTs). Later, tubes consisting of a single graphene layer were synthesized through arc discharge methods utilizing transition metal catalysts [6, 7]. These structures are known as single-wall carbon nanotubes (SWNTs) and have typical diameters around 1 nm. Depending on their geometry, SWNTs may exhibit either metallic or semiconducting behavior [8], whereas MWNTs typically behave akin to metallic materials [9]. Metallic nanotubes can sustain an axial current density of $4\times {10}^9\mathrm{A}/{\mathrm{cm}}^2$, three orders of magnitude higher than that of copper or aluminum [10]. Notably, all nanotubes demonstrate outstanding thermal conductivity along the axial direction, with measured values of approximately $3500\mathrm{W}{\mathrm{m}}^{-1}{\mathrm{K}}^{-1}$ [11]. However, they act as effective thermal insulators in the radial direction, exhibiting thermal conductivity values at room temperature comparable to soil ($1.52\mathrm{W}{\mathrm{m}}^{-1}{\mathrm{K}}^{-1}$) [12]. Owing to the presence of covalent ${\mathit{sp}}^2$ bonds between carbon atoms, CNTs possess exceptional strength and stiffness under stretching forces in the axial direction, with tensile strengths reaching up to 100 GPa [13]. Nevertheless, they tend to buckle under compression loads due to their hollow structure and high aspect ratio [14]. Furthermore, they display relatively low stiffness in the radial direction, characterized by a Young’s modulus on the order of several GPa [15], with experimental evidence suggesting that even van der Waals forces can deform two adjacent nanotubes [16]. These distinctive properties, among others, render CNTs highly valuable materials for technological applications across various scientific domains, encompassing nanotechnology, electronics, mechanics, optics, and composite materials [17].

High-order harmonic generation (HHG) is an extreme nonlinear optical phenomenon, where a target subjected to an intense laser pulse emits radiation in the form of high-frequency harmonics of the driving beam. This remarkable process has facilitated the expansion of coherent radiation into the extreme regions of the electromagnetic spectrum, addressing a fundamental challenge since the advent of the laser in 1960 [18]. The nonperturbative essence of HHG is marked by a distinctive feature in the harmonic spectrum: the emergence of a plateau structure followed by an abrupt cut-off [19]. This plateau can extend harmonic emission to thousands of orders, offering a diverse array of applications ranging from imaging and spectroscopy with sub-femtosecond resolution to the provision of coherent radiation sources in the XUV or even X-ray regimes [20, 21]. Experimental validation of harmonic generation was initially demonstrated by Franken et al. in 1961 [22], who observed the second harmonic of a ruby laser with a wavelength of 694 nm in a quartz crystal. However, it was not until the late 1980s that the availability of high-intensity laser sources enabled the observation of nonperturbative harmonics [23, 24]. The foundational principles of HHG were established during the 1990s, following rigorous theoretical and experimental endeavors that propelled the development of the field. Notably, in 1993, L’Huillier and Balcou observed up to the 135th harmonic in Ne utilizing pulses 1 ps long and intensities as high as ${10}^{15}\mathrm{W}/{\mathrm{cm}}^2$ [25], while Mackling et al. reported the observation of harmonics exceeding the 109th order for incident wavelengths in the near-infrared range using the same target [26]. Concurrently, in 1993, Schafer et al. introduced the renowned three-step model explaining HHG on a semiclassical basis [27], closely followed by Corkum, who published similar insights [28]. Subsequently, in 1994, Lewenstein et al. presented a comprehensive quantum theory delineating the underlying physics of HHG and offering quantitative predictions [29]. Since then, various research groups have reported the observation of progressively higher harmonic orders in gas systems [30, 31]. The repertoire of targets for high-order harmonic emission encompasses not only atomic or molecular systems but also solids and plasmas. Particularly, HHG from solid targets has incited significant interest due to their higher density, which engenders a qualitative enhancement in harmonic conversion efficiency, a consequence of the coherent nature of the process [32, 33, 34, 35, 36]. Currently, there has been a burgeoning interest in understanding the nonlinear optical response of lower-dimensional materials, such as 2D graphene [37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47], silicene [48], ${\mathrm{MoS}}_2$ [49], or hBN [50, 51, 52], and 1D silicon nanotubes [53] or CNTs [54, 55, 56]. For instance, the generation of up to the 9th harmonic in gapless graphene has been observed using a mid-infrared driving laser [57, 58]. Moreover, there is experimental evidence suggesting that the emission spectra of high-order harmonics in SWNTs can be tailored using electrolyte gating approaches, with up to the 11th harmonic being observed in semiconducting tubes [59]. These experimental endeavors underscore the feasibility of harnessing low-dimensional carbon allotropes as sources of high-order harmonic radiation.

Remarkably, the initial experiments concerning HHG in solid-state systems unearthed significant differences in the laws governing the spectral plateau and cut-off frequency compared to atomic or molecular systems [32]. In the latter, the cut-off frequency scales proportionally with the product of laser intensity and the square of the wavelength. Conversely, for semiconductors, the scaling exhibits linearity with the field amplitude. This observation underscores the paramount importance of comprehending the underlying mechanisms triggering HHG [36]. Within this framework, the distinctive energy band structure characteristic of lower-dimensional materials unveils novel paradigms.

This chapter delves into the investigation of the HHG process in single-walled carbon nanotubes (SWNTs) under the influence of intense ultrashort infrared pulses. Employing the tight-binding and zone folding approximations, Section 2 describes the structural properties of CNTs, elucidates the dynamical equations, and formulates the dipole expression governing harmonic emission. Subsequently, Section 3 scrutinizes the harmonic spectra, unveiling the hallmark non-perturbative features at fluences below the damage threshold. This section also includes an exhaustive discussion on the results obtained for different types of SWNT. Next, Section 4 introduces a model delineating the mechanism triggering harmonic emission, revealing its semblance to the mechanism reported for graphene [43] and highlighting the pivotal role of van Hove singularities [55]. Finally, Section 5 summarizes the main conclusions.

2. Model and methods

2.1 Structure of carbon nanotubes

Structurally, carbon nanotubes can be conceptualized as the consequence of rolling a strip of graphene [8]. Indeed, each SWNT is uniquely defined by a pair of integers ($n_1,n_2$), whereby the chiral vector ${\mathbf{C}}_h=n_1{\mathbf{a}}_1+n_2{\mathbf{a}}_2$ links two equivalent sites within the graphene lattice, being ${\mathbf{a}}_1=a\left(\sqrt{3}/2,1/2\right)$ and ${\mathbf{a}}_2=a\left(\sqrt{3}/2,-1/2\right)$ the lattice vectors, with $a=2.46$ Å denoting the lattice constant. The chiral vector, along with the translational period T—defined as the smallest lattice vector perpendicular to ${\mathbf{C}}_h$—governs the nanotube’s unit cell. This unit cell adopts a cylindrical geometry with a height $a_0=\mid \mathbf{T}\mid$ and diameter $d_t=\mid {\mathbf{C}}_h\mid /\pi$. Moreover, considering the diverse possible manners in which a graphene sheet can be rolled, it is useful to introduce the chiral angle $\theta$. This is the angle between ${\mathbf{C}}_h$ and ${\mathbf{a}}_1$ and ranges from 0° to 30° as a consequence of the hexagonal symmetry inherent to graphene’s honeycomb lattice. Notably, nanotubes of the ($n,n$) type exhibit a chiral angle of $\theta ={30}^{°}$, epitomizing an armchair pattern along the circumference, hence termed “armchair” ($\mathcal{A}$-tubes). Conversely, SWNTs of type ($n,0$), featuring a chiral angle of $\theta =0^{°}$, are denoted as “zigzag” ($\mathcal{Z}$) tubes. Zigzag tubes showcase a characteristic zigzag motif along the circumference, with carbon-carbon bonds aligned parallel to the nanotube axis. It merits attention that both armchair and zigzag tubes are achiral species, unlike chiral ($\mathcal{C}$) tubes, which are characterized by non-equal indices ($n_1,n_2\ne n_1\ne 0$) [60].

All species of nanotubes exhibit $a_0$-period translational symmetry along the tube axis ($z$-axis, conventionally defined), along with screw axes that denote pure rotational symmetries. Moreover, both chiral and achiral tubes manifest $\pi$-rotational symmetry around the U-axis, which is perpendicular to $z$. Additionally, $\mathcal{Z}$ and $\mathcal{A}$-tubes possess mirror reflection symmetries across the horizontal xy-plane (${\sigma }_h$) and the vertical yz-plane (${\sigma }_v$), attributes absent in $\mathcal{C}$ tubes [61]. These symmetry elements are encompassed within the line groups ${\mathbf{L}}_{\mathcal{ZA}}$ and ${\mathbf{L}}_{\mathcal{C}}$, derived from the product of the helical group and the dihedral point groups ${\mathbf{D}}_{\mathit{nh}}$ and ${\mathbf{D}}_n$, for achiral and chiral tubes, respectively, with $n$ representing the greatest common divisor of $n_1$ and $n_2$ [62]. Figure 1(a) illustrates the fundamental symmetry elements of the (8, 8) armchair configuration.

Figure 1.

(8, 8) armchair tube in the (a) real and (b) reciprocal spaces. Red and blue circles in (a) denote the two distinct sublattices of carbon atoms. The first BZ of the nanotube is composed of the green lines shown in (b). For comparison, this panel also shows the boundary of the first BZ of graphene (red hexagon) and the high symmetry points K, K′, and $\mathrm{\Gamma }$.

Each SWNT exemplifies a line group in the sense that the interrelation between carbon atoms and the tube’s symmetry operations is isomorphic: commencing with a solitary carbon atom and progressively applying the group elements, the entire tube is obtained. Consequently, the group symmetry engenders a set of quantum numbers endowed with complete physical significance. The translational periodicity of the tube ensures the conservation of quasi-momentum $k$. Furthermore, rotational symmetry around the z-axis conserves $m$, the projection of the orbital angular momentum along the tube axis. The presence of mirror reflection planes and $\pi$-rotational symmetry around the U-axis imposes additional conservation laws governing the parities of electronic states. Here, parity with respect to ${\sigma }_v$ is designated as $A$ for even states and $B$ for odd ones, while parity under $U$ or ${\sigma }_h$ transformations is denoted as $\pm$ for even and odd states, respectively. Consequently, the electron state $\mid k,m,A/B,\pm ⟩$ corresponds to a specific irreducible representation of the line group, and thus, its wave function transforms under symmetry operations akin to the basis of the corresponding representation.

Given the essentially one-dimensional nature of SWNTs, it proves advantageous to streamline their first Brillouin zone (BZ) to one dimension through the zone-folding technique. Accordingly, we consider the nanotube as a graphene layer with periodic boundary conditions along the circumferential direction, which can be succinctly expressed as:

$${\mathrm{\Psi }}_{\mathbf{k}}\left(\mathbf{r}+{\mathbf{C}}_h\right)=e^{i\mathbf{k}\cdot {\mathbf{C}}_h}{\mathrm{\Psi }}_{\mathbf{k}}\left(\mathbf{r}\right)={\mathrm{\Psi }}_{\mathbf{k}}\left(\mathbf{r}\right),$$E1

where ${\mathrm{\Psi }}_{\mathbf{k}}\left(\mathbf{r}\right)$ denotes the electron’s wave function in graphene. Although this approach disregards curvature effects, it is a satisfactory approximation since the mixing between the $\pi$ and $\sigma$ carbon orbitals is small and can be neglected near the Fermi level [63]. The condition outlined in Eq. (1) results in the quantization of permissible wave vectors along the circumferential direction as $\mathbf{k}\cdot {\mathbf{C}}_h=2\mathit{\pi m}$, where $m\in \mathrm{\mathbb{Z}}$. In this scheme, the wave vectors along the nanotube axis remain continuous, as no constraints are imposed on the $z$-direction. Consequently, the BZ of the nanotube encompasses a collection of lines with a length of $2\pi /a_0$ parallel to the tube axis. The number of lines $q=2N$, where $N$ represents the number of carbon atoms in the unit cell, corresponds to the order of the isogonal point group [64]. Each line is characterized by $m$, taking integer values within $\left(-q/2,q/2\right]$. According to the boundary condition imposed by Eq. (1), the $q$ lines are spaced apart by constant intervals of $2/d_t$. As an example, Figure 1(b) shows the allowed $\mathbf{k}$ vectors within the first BZ of the (8, 8) armchair configuration (depicted by green lines). This particular nanotube unit cell encompasses 32 carbon atoms, resulting in $q=16$, with $m$ ranging from 0 to $\pm 8$, and $d_t=10.9$ Å.

The Hamiltonian governing the electron dynamics within the periodic potential of the nanotube is expressed by the single-layer graphene (SLG) Hamiltonian $H_0\left(\mathbf{k}\right)$ restricted to the one-dimensional manifold of the nanotube, i.e., with $k$ satisfying Eq. (1). Thus, within the nearest-neighbors tight-binding approximation, the dispersion is given by $E_m^{\pm }\left(k\right)=\mp {\gamma }_0\mid f_m\left(k\right)\mid$, where ${\gamma }_0=-2.97$ eV denotes the tight-binding integral of graphene, and $f_m\left(k\right)$ represents the complex form factor:

$$f_m\left(k\right)=e^{-{\mathit{iak}}_1^m/\sqrt{3}}\left(1+2e^{i\sqrt{3}{\mathit{ak}}_1^m/2}cos\frac{{\mathit{ak}}_2^m}{2}\right),$$E2

where:

$$k_1^m=k_{\perp }^{m}cos{\theta }^{\prime }-ksin{\theta }^{\prime }$$E3

$$k_2^m=k_{\perp }^{m}sin{\theta }^{\prime }+kcos{\theta }^{\prime }$$E4

with ${\theta }^{\prime }=\pi /6-\theta$, and $k_{\perp }^m=2m/d_t$ representing the set of allowed wave vectors along the circumferential direction.

The Dirac points K and K′ degenerate at the Fermi level. Therefore, if K (or K′) represents an admissible wave vector for the SWNT, the corresponding bands exhibit degeneracy, rendering the nanotube metallic. This scenario is exemplified by the (8, 8) armchair tube depicted in Figure 1(b). Conversely, if K (or K’) is not a valid wave vector for the SWNT, the nanotube behaves as a small-gap semiconductor. Consequently, Eq. (1) furnishes the following general rule for predicting metallicity: a given ($n_1,n_2$) SWNT is metallic if $\left(n_1-n_2\right)\equiv 0$ (mod 3), and it is a small-gap semiconductor if $\left(n_1-n_2\right)\equiv 1,2$ (mod 3).

This rule is illustrated in the three examples of zigzag tubes presented in Figure 2. Panel (a) displays the metallic tube (12, 0) where $n\equiv 0$ (mod 3), intersecting the Dirac points at $m=‐8$ and 8. Conversely, panels (b) and (c) showcase the semiconducting tubes (13, 0) where $n\equiv 1$ (mod 3) and (14, 0) where $n\equiv 2$ (mod 3), respectively. As can be seen in the insets, the Dirac points are excluded from the BZ of both semiconducting species.

Figure 2.

First Brillouin zone of the zigzag tubes (12, 0), (13, 0), and (14, 0). The red hexagons in the three panels represent the boundary of the first BZ of graphene.

2.2 Dynamical equations and emission dipole

The symmetries inherent to single-walled carbon nanotubes establish the rules governing optical transitions between electronic states. When the incident beam’s wavelength significantly exceeds the translational period $a_0$, nearly direct transitions occur ($\mathrm{\Delta }k\approx 0$) due to the conservation of linear quasi-momentum. Within the dipole approximation, the interaction Hamiltonian is directly proportional to the momentum operator and transforms according to the polar vector representation of the nanotube’s symmetry group. As a result, an optical transition from state $\mid i⟩$ to state $\mid f⟩$ is permissible if the representations of the polar vector and the electronic states share a common component [61]. The $z$ component of a polar vector belongs to the non-degenerate representation ${}_0{A}_0^{-}$ and therefore exhibits even parity with respect to ${\sigma }_v$ and odd parity under ${\sigma }_h$ reflections and U-axis transformations [65]. Consequently, z-polarized light leaves the angular momentum $m$ and ${\sigma }_v$ parity quantum numbers unaffected but reverses the ${\sigma }_h$ parity of the interacting state. Therefore, for the transition $\mid i⟩\to \mid f⟩$ to be permissible, the states must share the same $k$, $m$, and ${\sigma }_v$-parity, while possessing opposite ${\sigma }_h$-parity quantum numbers [66]. Additionally, electronic intraband dynamics are subject to umklapp rules, governing the change of the quantum number $m$ when the k-trajectories exceed the first Brillouin zone [61]. Further elaboration on these umklapp rules will be provided in subsequent sections.

Let us consider a nanotube ($n_1,n_2$) irradiated by an intense laser pulse linearly polarized along its axis, $\mathbf{F}\left(t\right)=F\left(t\right){\mathbf{u}}_z$. The temporal evolution of the electron is governed by the time-dependent Hamiltonian $H\left(k,m,t\right)=H_0\left(k,m\right)+V_{\mathit{int}}\left(t\right)$, where $H_0\left(k,m\right)$ denotes the unperturbed Hamiltonian of graphene confined to the one-dimensional k-space manifold of the nanotube, and $V_{\mathit{int}}\left(t\right)=-q_{e}F\left(t\right)z$ represents the interaction potential due to the electric field in the dipole approximation. If $\mid {\varphi }_{k,m}^{\pm }⟩$ denote the eigenstates of the conduction and valence bands within the zone-folding approximation, the time-dependent wave function describing the electron in the periodic potential of the nanotube can be formulated as:

$$\psi \left(\mathbf{r},t\right)=\sum _m\int {\psi }_{k,m}\left(\mathbf{r},t\right)\mathit{dk}=\sum _m\int \left[C_m^{+}\left(k,t\right){\varphi }_{k,m}^{+}\left(\mathbf{r}\right)+C_m^{-}(k,t){\varphi }_{k,m}^{-}\left(\mathbf{r}\right)\right]\mathit{dk}.$$E5

Upon substitution of Eq. (5) into the time-dependent Schrödinger equation (TDSE), we obtain the following system of coupled two-level equations [43]:

$$i\mathrm{\hslash }\frac{d}{\mathit{dt}}{C}_m^{+}\left({\kappa }_t,t\right)=\left[E_m^{+}\left({\kappa }_t\right)-F\left(t\right)D_m\left({\kappa }_t\right)\right]C_m^{+}\left({\kappa }_t,t\right)-F\left(t\right)D_m\left({\kappa }_t\right)C_m^{-}\left({\kappa }_t,t\right),$$E6

$$i\mathrm{\hslash }\frac{d}{\mathit{dt}}{C}_m^{-}\left({\kappa }_t,t\right)=\left[E_m^{-}\left({\kappa }_t\right)-F\left(t\right)D_m\left({\kappa }_t\right)\right]C_m^{-}\left({\kappa }_t,t\right)-F\left(t\right)D_m\left({\kappa }_t\right)C_m^{+}\left({\kappa }_t,t\right),$$E7

where $\mathrm{\hslash }{\kappa }_t=\mathrm{\hslash }k-q_{e}A\left(t\right)/c$, with $q_e$ denoting the elementary charge, $c$ the speed of light, and $A\left(t\right)$ the vector potential. In Eqs. (6) and (7), $E_m^{\pm }$ represents the energy eigenvalues for band index $m$, and $D_m$ stands for the component of the transition matrix element of graphene parallel to the nanotube axis, constrained to the set of allowed wave vectors, which is given by:

$$D_m\left(k\right)=\frac{q_e}{2}\frac{\partial {\phi }_m}{\partial k},$$E8

where ${\phi }_m\left(k\right)$ represents the phase of the complex function $f_m\left(k\right)$ as defined by Eq. (2). To simplify the integration of Eqs. (6) and (7), we introduce the off-diagonal coefficients:

$$C_m^M\left({\kappa }_t,t\right)=C_m^{+}\left({\kappa }_t,t\right)-C_m^{-}\left({\kappa }_t,t\right),$$E9

$$C_m^P\left({\kappa }_t,t\right)=e^{-i{\phi }_m\left({\kappa }_t\right)}\left[C_m^{+}\left({\kappa }_t,t\right)+C_m^{-}({\kappa }_t,t)\right].$$E10

These coefficients allow us to rewrite the equations and overcome numerical instabilities arising from the divergence of $D_m$ near the Dirac points [55]:

$$\begin{array}{lll}i\mathrm{\hslash }\frac{d}{\mathit{dt}}{C}_m^M\left({\kappa }_t,t\right)& =& \frac{E_m^{+}\left({\kappa }_t\right)+E_m^{-}\left({\kappa }_t\right)}{2}{C}_m^M\left({\kappa }_t,t\right)\\ & & +\frac{E_m^{+}\left({\kappa }_t\right)-E_m^{-}\left({\kappa }_t\right)}{2}{e}^{i{\phi }_m\left({\kappa }_t\right)}{C}_m^P\left({\kappa }_t,t\right),\end{array}$$E11

$$\begin{array}{lll}i\mathrm{\hslash }\frac{d}{\mathit{dt}}{C}_m^P\left({\kappa }_t,t\right)& =& \frac{E_m^{+}\left({\kappa }_t\right)+E_m^{-}\left({\kappa }_t\right)}{2}{C}_m^P\left({\kappa }_t,t\right)\\ & & +\frac{E_m^{+}\left({\kappa }_t\right)-E_m^{-}\left({\kappa }_t\right)}{2}{e}^{-i{\phi }_m\left({\kappa }_t\right)}{C}_m^M\left({\kappa }_t,t\right).\end{array}$$E12

The wave function $\psi \left(\mathbf{r},t\right)$ of the Bloch electron during its interaction with an external electric field is described by Eq. (5) after solving the dynamical equations (6) and (7) for the probability coefficients $C_m^{\pm }\left(k,t\right)$. Due to the combination of the Coulomb interaction with the nanotube network and the force exerted by the driving field, the electron undergoes oscillatory motion along the axial direction. Consequently, it emits radiation in the form of electromagnetic waves. In the dipole approximation, the power radiated $P\left(t\right)$ follows Larmor’s semiclassical formula:

$$P\left(t\right)=\frac{2}{3c^3}{\left|a\left(t\right)\right|}^2$$E13

where $a\left(t\right)$ is the acceleration of the electric dipole, which corresponds to the second time derivative of the expectation value

$$d\left(t\right)=⟨\psi \left(t\right)|q_e\widehat{z}|\psi \left(t\right)⟩=i\frac{q_e}{2}\sum _m\int \left[C_m^{M\ast }\frac{\partial C_m^M}{\partial {\kappa }_t}+C_m^{P\ast }\frac{\partial C_m^P}{\partial {\kappa }_t}\right]\mathit{dk}.$$E14

The m-intraband component of the emission spectra can be computed using [55]:

$$a_m^{\mathit{\text{intra}}}\left(t\right)=\frac{{q_e}^2}{{\mathrm{\hslash }}^2}F\left(t\right)\int \left[{\left|C_m^{+}\right|}^2\frac{{\partial }^2{E}_m^{+}}{\partial {\kappa }_t^2}+{\left|C_m^{-}\right|}^2\frac{{\partial }^2{E}_m^{-}}{\partial \kappa t^2}\right]\mathit{dk}.$$E15

Finally, the total emission spectra and its intraband contribution are obtained by taking the Fourier transform of Eq. (13).

3. High-order harmonic emission spectra

Let us then turn our attention to the investigation of high-order harmonic emission spectra resulting from mid-infrared few-cycle driving pulses at varying intensities targeting SWNTs of diverse types and dimensions. These driving pulses are characterized by a sinusoidal envelope $F\left(t\right)=F_0{sin}^2\left(\mathit{\pi t}/8T\right)sin\left({\omega }_{0}t\right)$ for $0\le t\le 8T$, where $F_0$ denotes the field amplitude, $T$ represents the period, and ${\omega }_0=2\pi /T$ denotes the field frequency. Local field corrections to the driving field amplitude are neglected, as the incident beam, directed perpendicular to the nanotube, traverses walls of atomic-size thickness. We delve into the discussion of the spectra computed using these driving pulses in the subsequent sections.

3.1 Response from armchair nanotubes

All $\left(n,n\right)$ type tubes exhibit the $\pi$-band structure shown in Figure 3(a), with the valence and conduction bands crossing at $a_{0}k=2\pi /3$, indicating that they are always metallic. Optical transitions $A_0^{+}\to B_0^{+}$ and $B_n^{+}\to A_n^{+}$ induced by z-polarized drivers are forbidden in any $\mathcal{A}$-type tube due to the ${\sigma }_h$-parity selection rule. The rest of the bands $\mid m\mid =1,\dots ,n-1$ are doublets where optical transitions are allowed for any $a_{0}k\in \left(0,\pi \right]$, but forbidden at $a_{0}k=0$. The intraband dynamics is also subject to the $\mathit{\text{umklapp}}$ rule $m^{\prime }=m\pm n$ (mod $2n$), causing $m$ to shift to $m^{\prime }$ when the first Brillouin zone is exceeded through $a_{0}k=\pi$ [67]. The density of states (DOS) has a small non-null value nearby the Fermi level. The van Hove singularities at the zero-slope points of the band’s diagram can be easily identified.

Figure 3.

(a) Band structure of (8, 8) $\mathcal{A}$-type tube. The energy is given in units of the frequency of a $3\mathrm{\mu m}$ laser ($\mathrm{\hslash }{\omega }_0=0.41$ eV). The horizontal axis represents the wave vector times the translational period $a_0=2.46$ Å. The pink-filled area in the background shows the DOS. (b) Harmonic yield from (8, 8) armchair driven by a 3 μm, 28 fs (2.9 cycles) FWHM pulse at $5\times {10}^{10}\mathrm{W}/{\mathrm{cm}}^2$ peak intensity. The solid (dashed) line represents the total (intraband) harmonic spectrum. (c) Contribution from different $m$ bands to the total harmonic yield.

Figure 3(b) displays the calculated harmonic yield from a (8, 8) $\mathcal{A}$-tube driven by a $3\mathrm{\mu m}$ wavelength laser beam with peak intensity of $5\times {10}^{10}\mathrm{W}/{\mathrm{cm}}^2$. The harmonic emission is linearly polarized along the tube axis. The emergence of a spectral plateau extending up to a cut-off frequency corresponding to the 9th harmonic is clearly observed. Due to the centrosymmetric structure of the system, only odd-order harmonics are present in the spectrum. In Figure 3(c), the contributions to the emission spectra corresponding to different values of $m$ are shown. Notably, the higher-order harmonics are mainly contributed by bands $\mid m\mid =7$, which include the first van Hove singularity. Remarkably, the band gap at the first van Hove singularity nearly resonates with a $6{\omega }_0$ transition, enhancing the interband component of the harmonic emission around this resonance. The other points of the Brillouin zone contribute mainly to the third harmonic with intraband radiation.

These results can be further explained by examining the impact of the dipole matrix element on the solutions of the dynamical equations. Indeed, according to Eqs. (6) and (7), the coupling between the valence and conduction bands during the interaction with the laser field is governed by the matrix element $D_m$, which, for $\mathcal{A}$-type tubes, is given by [68]:

$$D_m\left(k\right)=-\frac{a_0{q}_e}{2}\frac{sin\left(\frac{a_{0}k}{2}\right)sin\left(\frac{\mathit{m\pi }}{n}\right)}{1+4cos\left(\frac{a_{0}k}{2}\right)\left[cos\left(\frac{\mathit{m\pi }}{n}\right)+cos\left(\frac{a_{0}k}{2}\right)\right]}.$$E16

Figure 4 illustrates $D_m$ in the (8, 8)-armchair tube for various band indices $m$. Notably, $D_m$ is null for $m=0$, 8 across any $a_{0}k\in \left(-\pi ,\pi \right]$, thus decoupling the system of equations. Initial conditions assume that the states below the Fermi level are fully occupied, $C_m^{-}\left(k,0\right)=1$, while the conduction band is empty, $C_m^{+}\left(k,0\right)=0$. Since the states $B_n^{+}$ ($A_0^{+}$) and $A_n^{+}$ ($B_0^{+}$) are decoupled, their population contributes to the spectra solely through intraband transitions. Remarkably, this conclusion aligns with the ${\sigma }_h$ symmetry-induced selection rule.

Figure 4.

Matrix element $D_m\left(k\right)$ corresponding to the (8, 8) armchair tube for different $m$ values. The vertical dotted lines at $a_{0}k=\pm 2\pi /3$ indicate the band’s crossing at the Dirac points for $m=8$.

The coupling exhibits symmetry with respect to $a_{0}k=0$ for bands with equal $\mid m\mid$ and displays peaks of magnitude increasing with $\mid m\mid$ at wave vectors corresponding to the first four van Hove singularities in the DOS. The absolute maximum of $D_m$ is attained at the first van Hove singularity, corresponding to bands located nearest to the Dirac points, $\mid m\mid =n-1$. This result suggests that the interaction with the driving laser is much more efficient for these states and, consequently, that the interband emission spectra are mainly due to transitions within states $\mid m\mid =n-1$, as shown in Figure 3(c). It is noteworthy that the increased band curvature near the Dirac points entails sharper van Hove singularities, further enhancing the efficiency of interband transitions.

3.2 Emission spectra from zigzag nanotubes

While armchair tubes always exhibit metallic character, the characteristics of zigzag nanotubes can vary, encompassing both metallic and semiconducting behavior depending on the chiral index ($n,0$). The anisotropic dispersion of graphene bands at the vicinity of the Dirac points results in three different types of $\mathcal{Z}$-tubes. In metallic tubes where $n\equiv 0$ (mod 3), the valence and conduction bands $\mid m_F\mid =2n/3$ degenerate at the Fermi level. Additionally, the non-degenerate bands closest to them, $m=m_F\pm 1$, exhibit varying dispersion despite being symmetrically located with respect to K, as illustrated in Figures 2(a) and 5(a). Conversely, for semiconducting tubes with $n\equiv 1$ or $n\equiv 2$ (mod 3), the same dispersion asymmetry manifests in distinct variations of transition energies, as shown in Figures 2(b) and 5(b), and Figures 2(c) and 5(c), respectively.

Figure 5.

$\pi$-band structures and DOS of (a) (12, 0), (b) (13, 0), and (c) (14, 0) zigzag SWNTs. The energy and horizontal axis are given in the same units as in Figure 3. It is noteworthy that for all $\mathcal{Z}$-tubes, $a_0=4.26$ Å.

All $\mathcal{Z}$-type tubes exhibit band diagrams akin to those depicted in Figure 5. Notably, the zero-slope points consistently occur at $a_{0}k=0$, aligning with the van Hove singularities. A noteworthy characteristic of zigzag tubes is the absence of symmetry-related constraints on interband transitions induced by z-polarized beams. This absence stems from the fact that valence and conduction bands form either multiplets or singlets with the same ${\sigma }_v$ parity and opposite ${\sigma }_h$ parity. Moreover, the $\mathit{\text{umklapp}}$ rule governing intraband transitions that extend beyond the first Brillouin zone mirrors that observed in $\mathcal{A}$-type tubes: $m^{\prime }=m\pm n$ (mod $2n$) [67]. Close to the Fermi level, the DOS remains null for semiconducting species, while metallic tubes exhibit a small but nonzero DOS, mirroring the behavior observed in $\mathcal{A}$-type nanotubes.

For $\mathcal{Z}$-tubes, the matrix element that couples the valence and conduction bands in Eqs. (6) and (7) is given by [68]:

$$D_m\left(k\right)=-\frac{{\mathit{aq}}_e}{2}\frac{sin{\mathrm{\Theta }}_m^{k}sin{\mathrm{\Xi }}_m^k}{1+4cos{\mathrm{\Theta }}_m^k\left[cos{\mathrm{\Xi }}_m^k+cos{\mathrm{\Theta }}_m^k\right]}$$E17

where

$${\mathrm{\Theta }}_m^k=\frac{\mathit{\pi m}}{2n}+\frac{\sqrt{3}\mathit{ak}}{4}$$E18

$${\mathrm{\Xi }}_m^k=\frac{3\mathit{\pi m}}{2n}-\frac{\sqrt{3}\mathit{ak}}{4}$$E19

with $a$ being the lattice constant of graphene. Similar to $\mathcal{A}$-type species, in zigzag nanotubes $D_m$ is symmetric with respect to $a_{0}k=0$ for bands with equal $\mid m\mid$. In the (12, 0) metallic tube, depicted in Figure 6(a), $D_m$ diverges at $a_{0}k=0$ for $\mid m\mid =m_F$ and reaches maxima of decreasing magnitude for the rest of $m$ values. Note that the maximal values of $D_m$ are always located in the proximity of $a_{0}k=0$. Semiconducting species exhibit a similar behavior, albeit the matrix element remains finite in all bands, as demonstrated in Figures 7(a) and 8(a). Given that $D_m$ is generally nonzero in $\mathcal{Z}$-tubes, the dipole emission consists of both intraband and interband contributions for all $m$. Interband transitions predominantly occur in the vicinity of $a_{0}k=0$, where both $D_m$ and the DOS reach maximal values.

Figure 6.

(a) Matrix element $D_m\left(k\right)$ corresponding to the (12, 0) zigzag tube for different $m$ values. The inset shows the relative location of the different bands with respect to K. (b) Harmonic yield from the (12, 0) $\mathcal{Z}$-tube. The parameters of the driver are the same as those in Figure 3. (c) Contribution to the harmonic yield from the different values of $\mid m\mid$.

Figure 7.

(a) Matrix element $D_m\left(k\right)$ corresponding to the (13, 0) semiconducting zigzag tube for different $m$ values. The inset illustrates the relative location of the different bands with respect to the K point. (b) Harmonic yield from the (13, 0) $\mathcal{Z}$-tube. The contributions to the total harmonic yield from the different values of $\mid m\mid$ are depicted in panel (c). The parameters of the driver are the same as those in Figure 3.

Figure 8.

(a) Matrix element $D_m\left(k\right)$ corresponding to the semiconducting (14, 0) $\mathcal{Z}$-type tube. The inset illustrates the relative location of the different bands with respect to the K point. (b) Harmonic yield from the (14, 0) $\mathcal{Z}$-tube. (c) Contributions to the total harmonic yield from the different values of $m$. The parameters of the driver are the same as those in Figure 3.

Figure 6(b) illustrates the harmonic yield from the metallic (12, 0) $\mathcal{Z}$-tube at a driving peak intensity of $5\times {10}^{10}\mathrm{W}/{\mathrm{cm}}^2$ and $3\mathrm{\mu m}$ wavelength. The spectrum notably exhibits the emergence of the 6-photon resonance, with the lower-order harmonics almost suppressed and the higher ones well resolved. However, the intraband component of the emission spectra displays a noisy structure extending to arbitrary high frequencies, stemming from the divergence of the transition matrix element at the Dirac points. This unphysical spectrum is counteracted by the interband emission, ensuring that the total spectrum remains finite.

The contributions from different values of $\mid m\mid$ are shown in Figure 6(c). Notably, the total yield is predominantly attributed to the contribution from $\mid m\mid =9$, corresponding to states clustered at the first van Hove singularity, where $D_m$ reaches its maximal finite value. Conversely, the other bands, where $D_m$ is weaker, primarily contribute to the lower harmonics. Consequently, the high-order emission spectra in metallic $\mathcal{Z}$-type nanotubes are primarily governed by transitions between states $\mid m\mid =\mid m_F\mid +1$ at wave vectors close to the first van Hove singularity, mirroring the behavior observed in armchair tubes.

The emission spectra from semiconducting tubes belonging to the two different types, $n\equiv 1$ and $n\equiv 2$ (mod 3), are depicted in Figures 7(b) and 8(b), respectively. Employing the same peak intensity and wavelength for the driving field as in the previously discussed cases, both spectra exhibit similarities, albeit with slightly lower efficiency observed for the tube with a smaller radius (13, 0). Subtle differences between the spectra arise from the distinct band dispersion at the two first van Hove singularities, positioned at opposite sides of the K point.

Notably, the intraband component does not contribute to the higher harmonic orders in either case, mirroring the behavior observed in metallic $\mathcal{A}$ and $\mathcal{Z}$ nanotubes. Furthermore, the spectra of these semiconducting tubes exhibit signals near the 2nd and 4th harmonics, resonating with the band gap at the two first van Hove singularities, as illustrated in Figures 5(b) and (c).

The contributions from the different bands are shown in Figures 7(c) and 8(c). In both cases, the total yield effectively results from the coherent addition of two contributions, corresponding to bands at the first and second van Hove singularities, where $D_m$ reaches maximal values. This characteristic is distinct from armchair or metallic zigzag tubes, where only the contribution from the band crossing the first van Hove singularity is pertinent to HHG. Consequently, the resonances at the first and second van Hove singularities (near the 2nd and 4th harmonics, respectively) are clearly discernible in both figures, contributing to the complex structure of the spectral region below the 5th harmonic.

3.3 Impact of nanotube size on the spectral response

The harmonic yield from $\mathcal{A}$-type tubes of various diameters is depicted in Figure 9(a). Notably, while the fundamental characteristics of the spectra remain consistent across different diameters, there is an observable increase in the efficiency of harmonics around resonance (5th and 7th) with increasing tube diameter. This trend can be attributed to several factors. Firstly, the slope of the bands closest to $a_{0}k=\pm 2\pi /3$ grows as the tube diameter increases, providing a larger number of electronic states available at the maximal values of $D_m$. Consequently, as the diameter increases, the BZ points of bands $\mid m\mid =n-1$ approach the Dirac points, causing $D_m$ to approach singularity, thereby enhancing the efficiency of the harmonics. Moreover, with an increase in diameter, the resonance near the fifth harmonic undergoes a red-shift due to the reduced gap energy of the $\mid m\mid =n-1$ bands. Interestingly, despite these variations, the cut-off frequency of the spectra remains relatively unchanged, irrespective of the nanotube’s size. This observation underscores the robustness of the harmonic generation process across different nanotube diameters.

Figure 9.

Harmonic yield from metallic and semiconducting tubes of different diameters. (a) Harmonic yield from (8, 8), (9, 9), (10, 10), (11, 11), and (12, 12) armchair tubes. The diameters of these tubes are $d_t=1.09$, 1.22, 1.36, 1.50, and 1.63 nm, respectively. The electric field parameters are consistent with those in Figure 3. (b) Metallic zigzag tubes with diameters 1.18, 1.41, and 1.65 nm, respectively. (c) Semiconducting type $n\equiv 1$ zigzag tubes with $d_t=1.24$, 1.49, and 1.73 nm. (c) Type $n\equiv 2$ zigzag tubes with diameters 1.33, 1.57, and 1.81 nm.

The remaining panels in Figure 9 provide additional insights into the dependence of HHG on the nanotube’s diameter for $\mathcal{Z}$-tubes. In metallic tubes, depicted in Figure 9(b), the spectral efficiency increases with the tube diameter. Lower-order harmonics are better resolved and exhibit a red shift as the diameter increases, attributed to the reduced resonance with the gap energy of the $m=m_F\pm 1$ bands. These observations are consistent with the trends observed in panel (a) for $\mathcal{A}$-tubes, suggesting that in metallic tubes, the interband component of the emission dipole becomes more prominent as the diameter increases. This phenomenon arises from the proximity of the first van Hove singularity to the Dirac points, resulting in a smaller band gap and a stronger matrix element.

For semiconducting tubes, as shown in Figure 9(b) and (c), there is no clear correlation between the nanotube diameter and either the spectral efficiency or the resolution of harmonic peaks. The fundamental characteristics of the spectra are preserved, and a consistent spectral complexity is observed at low frequencies across different diameters. Notably, as the diameter increases, the differences in the band structure among tubes become less significant near the Fermi level, akin to observations in metallic nanotubes. This finding elucidates the absence of a monotonous increase in spectral efficiency with diameter in semiconducting tubes.

3.4 Effect of chirality on the harmonic response

The impact of chirality on the harmonic response of carbon nanotubes is a subject of significant interest. Chirality, determined by the chiral angle, dictates the electronic properties of nanotubes, influencing their conductivity and band structure. Consequently, understanding how chirality affects high-order harmonic generation (HHG) provides valuable insights into the nanotube’s optical response and potential applications in nonlinear optics and ultrafast photonics. To investigate this, we analyze the harmonic emission spectra of carbon nanotubes with varying chiral angles. Unlike zigzag species, chiral tubes lack mirror reflection planes, leading to singlet electronic bands with opposite (±) U-parity within the $a_{0}k\in \left(0,\pi \right)$ domain. This absence of symmetry-related constraints on optical transitions induced by z-polarized light sets the stage for exploring the effects of chirality on HHG.

Figure 10(a)–(c) illustrates the first Brillouin zone (BZ), the band structure, and the transition matrix element corresponding to the metallic (8, 2) $\mathcal{C}$-type tube. Metallic chiral SWNTs with $n_1-n_2\equiv 0$ (mod 3) exhibit band degeneracy at the Fermi level at k-values dependent on the structural properties of the tube [64]. The matrix element $D_m$ diverges at these points, where the DOS presents a small non-zero value, similar to $\mathcal{Z}$-tubes. The van Hove singularities occur at wave vectors where $D_m$ reaches finite maximal values. Similarly, semiconducting $\mathcal{C}$-type tubes exhibit the same behavior as their achiral counterparts, with $D_m$ maximal in the proximity of the van Hove singularities. Therefore, the conclusions drawn above for the $\mathcal{Z}$-species are also applicable to $\mathcal{C}$-type tubes.

Figure 10.

(a) First Brillouin zone of the (8, 2) $\mathcal{C}$-type tube. (b) Band structure of the chiral tube (8, 2). The energy is given in units of the tight binding integral ${\gamma }_0$. The degenerate $\mid m\mid =6$ bands are highlighted in red, while the bands at the first $\mid m\mid =7$ and second $\mid m\mid =5$ van Hove singularities are depicted in blue and green, respectively. (c) Transition matrix element $D_m\left(k\right)$ corresponding to these highlighted bands. (d) Comparison of harmonic yields from (7, 7) armchair, (10, 3) chiral, and (12, 0) zigzag tubes. The achiral species are metallic, while the chiral one is semiconducting. The diameters of the three tubes are similar: 0.95 nm, 0.93 nm, and 0.94 nm, respectively. (e) Harmonic yields from $\mathcal{A}$ (8, 8), $\mathcal{C}$ (12, 3), and $\mathcal{Z}$ (14, 0) tubes. In this case, the zigzag tube is semiconducting, while the other two are metallic. The diameters of the three tubes are 1.09 nm, 1.08 nm, and 1.10 nm, respectively. In both panels, the driving pulse is the same as for the rest of the cases analyzed in this chapter.

Figure 10(d) presents a comparison of the harmonic yields from $\mathcal{A}$, $\mathcal{Z}$, and $\mathcal{C}$-type SWNTs with diameters of approximately 0.95 nm, driven at a peak intensity of $5\times {10}^{10}\mathrm{W}/{\mathrm{cm}}^2$ and a wavelength of $3\mathrm{\mu m}$. The yield from the semiconducting chiral species (10, 3) with a chirality angle of $\theta ={13}^{°}$ is clearly distinct from the spectra of the other two metallic tubes (7, 7) and (12, 0), which are very similar. All spectra exhibit similar cut-off frequencies, higher harmonic yields for the semiconducting species, and enhancement of the low harmonic spectrum in the semiconducting tube due to resonance. Additionally, Figure 10(e) displays the harmonic yield from three tubes of larger diameter (∼ 1.10 nm) generated by the same laser pulse. Once again, the semiconducting tube exhibits the most efficient spectra. In this case, the chiral tube (12, 3) with a chirality angle of $\theta ={11}^{°}$ is metallic, and its harmonic yield resembles that produced by the metallic (8, 8) armchair. Finally, in Figure 11, we compare the harmonic yield from two semiconducting nanotubes of different chirality with diameters of approximately 0.85 nm. The spectral structure and overall efficiency of all harmonic orders are similar in both cases, although they exhibit some differences in the 5th order and in the perturbative spectral tail. Nevertheless, these differences are much less significant than those observed between semiconducting and metallic tubes of the same diameter.

Figure 11.

Harmonic yield from (8, 4) chiral with a chirality angle of $\theta ={19}^{°}$ and (11, 0) zigzag nanotubes. Both nanotubes are semiconducting species with diameters of 0.83 and 0.86 nm, respectively. The parameters of the driving pulse are the same as in Figure 10.

In conclusion, for nanotubes of similar diameter, semiconducting species exhibit more efficient HHG compared to their metallic counterparts. In fact, tubes with similar diameter and the same conducting character produce similar spectra, regardless of their chirality. Therefore, the harmonic spectra are highly sensitive to changes in the chiral angle $\theta$ only when it alters the tube’s conductivity from semiconducting to metallic or vice versa. Otherwise, variations in $\theta$ have minimal impact on the harmonic response.

3.5 Dependence of the spectral response on the driving intensity

All previous spectral analyses were conducted using a consistent mid-infrared 8-cycle pulse with a peak intensity of $5\times {10}^{10}\mathrm{W}/{\mathrm{cm}}^2$ and a wavelength of $3\mathrm{\mu m}$. However, to explore the dependency of HHG on intensity, we present in Figure 12(a) the spectral yield from the (16, 0) $\mathcal{Z}$-tube exposed to a driving pulse with the same duration and wavelength but with a peak intensity of $5\times {10}^{12}\mathrm{W}/{\mathrm{cm}}^2$. Notably, the spectral plateau extends further into the extreme ultraviolet range, although the complex structure observed at lower intensities becomes more pronounced.

Figure 12.

(a) Harmonic yield from the (16, 0) $\mathcal{Z}$-tube driven by a $3\mu \mathrm{m}$ wavelength at a peak intensity of $5\times {10}^{12}\mathrm{W}/{\mathrm{cm}}^2$. (b) Scaling of the cut-off with intensity from the (16, 0) zigzag tube. The blue diamonds represent the result of numerical integration of the TDSE, while the red circles are obtained from the semiclassical SPAM considering the electron-hole pair created at the first van Hove singularity, attained at $a_{0}k=0$, $\mid m\mid =11$. The green-filled area represents intensities above the damage threshold. The inset illustrates the band structure at $\mid m\mid =5$ and 11, along with the maximum excursion of the electron-hole pair in the BZ corresponding to intensity points $A$, $B$, and $C$. According to the Umklapp rule, band $m=11$ shifts to $m^{\prime }=5$ if the intraband oscillation exceeds the first Brillouin zone through $a_{0}k=\pi$.

The scaling of the cut-off frequency with the intensity of the driving field is presented in Figure 12(b). We highlight with a green background the intensities at which nanotubes are expected to experience damage, considering a damage fluence threshold of $150\mathrm{mJ}/{\mathrm{cm}}^2$ [38]. The behavior of the cut-off frequency with intensity resembles that observed in graphene [43]. Notably, the cut-off frequency reaches a saturation point at higher intensities. This saturation corresponds to a photon energy of approximately 12.7 eV, which matches the gap achieved by the electron-hole pairs generated at the first van Hove singularity when the oscillation of the quasimomentum ${\kappa }_t$ is at its maximum. This saturation point corresponds to the interband transition $E_5^{-}\to E_5^{+}$, as depicted in the inset of Figure 12(b). Similar to graphene, there is no simple law (e.g., linear or quadratic) governing the dependence of the cut-off frequency on the field amplitude. This behavior allows to unveil the mechanism underlying high-harmonic generation (HHG) in these materials.

4. The generation mechanism

The relationship between the cut-off frequency and the intensity of the driving beam, as depicted in Figure 12(b), mirrors previous findings in graphene [43] and, subsequently, in armchair SWNTs [55]. In these previous works, it was demonstrated that the mechanism triggering HHG in graphene could be extrapolated to $\mathcal{A}$-type SWNTs by interchanging the roles of Dirac points and van Hove singularities. Remarkably, the semiclassical model based on the saddle-point approximation (SPAM) described in [55] imposes no assumptions regarding chirality or metallicity, thus rendering it universally applicable to SWNTs.

According to this semiclassical model, the q-th order emission dipole can be mathematically represented as:

$$d_m^{\updownarrow }\left(q{\omega }_0\right)=\frac{i{\mathcal{D}}_0^m}{q_e}{\int }_{-\infty }^{\infty }{\int }_k{e}^{\frac{i}{\mathrm{\hslash }}\left[S_m\left(k,t,t_H\right)+q\mathrm{\hslash }{\omega }_{0}t\right]}{D}_m\left({\kappa }_t\right)\mathit{dk}\mathit{dt}.$$E20

In this equation, $S_m\left(k,t,t_H\right)$ represents the action, ${\kappa }_t$ denotes the kinetic quasimomentum, ${\mathcal{D}}_0^m$ is a constant, and $t_H$ represents the time at which the electron crosses the first van Hove singularity, corresponding to the creation of the electron-hole pair. Eq. (20) essentially adapts the expression derived for graphene [43] to the one-dimensional manifold of SWNTs, reversing the roles between the Dirac points and the first van Hove singularity.

As $D_m$ varies smoothly in reciprocal space, the principal contributions to the integral in Eq. (20) originate from the stationary points of the exponential function’s phase [29]. This leads to the following conditions:

$${\int }_{t_H}^t{v}_m^{+}\left({\kappa }_{\tau }\right)\mathit{d\tau }={\int }_{t_H}^t{v}_m^{-}\left({\kappa }_{\tau }\right)\mathit{d\tau },$$E21

$$E_m^{+}\left({\kappa }_t\right)-E_m^{-}\left({\kappa }_t\right)=q\mathrm{\hslash }{\omega }_0.$$E22

Here, $v_m^{\pm }\left({\kappa }_t\right)$ represents the velocities of the valence and conduction band electrons, respectively. Consequently, the emission of the q-th order harmonic occurs at time $t$ if the semiclassical trajectories of the electron-hole pair coincide at that moment. This implies the process outlined in Figure 13: (1) initiation of electron quivering in the valence band at point $k$ of band $m$; (2) electron reaching the first van Hove singularity at time $t_H$, promoting to the conduction band, and generating a hole in the valence band; (3) oscillation of the electron-hole pair according to the band’s geometry; and (4) eventual recombination at time $t$, emitting a photon resonant with the band gap. Notably, the first step of this mechanism is associated with non-adiabatic excitation and can occur at any point during the interaction, irrespective of the field maxima.

Figure 13.

Mechanism for high-order harmonic generation in SWNTs.

It is crucial to emphasize that both conditions given by Eqs. (21) and (22) must be simultaneously satisfied for photon emission to occur. This conclusion offers insight into the observed saturation of the cut-off frequency depicted in Figure 12(b): the maximum achievable energy of the emitted photon is constrained by the maximum band gap attainable by the electron-hole pair during their excursion, regardless the intensity of the driving beam.

To illustrate these concepts further, consider Figure 14. Panels (a) and (b) depict the classical trajectories of two electrons, originating from different wave vectors $k_A$ and $k_B$ at the onset of interaction, reaching the first van Hove singularity of a semiconducting (16, 0) zigzag SWNT at times $t_{H_A}$ and $t_{H_B}$, respectively. The scenarios presented correspond to a driving laser pulse with a peak intensity of $5\times {10}^{11}\mathrm{W}/{\mathrm{cm}}^2$ and a wavelength of $3\mathrm{\mu m}$. Trajectories shown in panel (a) intersect at time $t$, fulfilling Eq. (21) and leading to photon emission. Conversely, trajectories in panel (b) do not intersect at $t$, failing to meet the condition on semiclassical trajectories.

Figure 14.

(a) and (b) Electron and hole trajectories corresponding to points A and B illustrated in (c). The trajectory of the electron (black solid line) and hole (dashed line) are represented as a function of time. The figure corresponds to a driving laser pulse of 3 μm with peak intensity of $5\times {10}^{11}\mathrm{W}/{\mathrm{cm}}^2$ targeting a (16, 0) zigzag tube. (c) Energy map of the maximum gap energy of the quivering electron for the classical trajectories computed with the SPAM, being $t_H$ the time of the creation of the electron-hole pair and $t$ the time of the potential photon emission. The red area represents the points where the electron-hole trajectories intersect in direct space at time $t$ and, therefore, where there is photon emission.

Figure 14(c) provides a map of the energy gap as a function of the initial and final times $t_H$ and $t$. Points A and B correspond to the scenarios depicted in panels (a) and (b), respectively. The regions shaded in red represent solutions to the system of Eqs. (21) and (22), indicating electron-hole pairs created at $t_H$ that recombine in the same crystal cell at time $t$. These regions correspond to instances when the harmonic photon can be effectively emitted.

According to the map, point A reprsents an electron-hole pair created at $t_H\simeq 0.4\mathrm{T}$, recombining at $t\simeq 0.9\mathrm{T}$, and emitting a photon with energy $\simeq 17\mathrm{\hslash }{\omega }_0$. While other points in the map may potentially emit higher frequency harmonics at time $t$, photon emission is precluded as the electron-hole pairs fail to recombine. Thus, the maximum photon energy is determined by the energy maxima of the map confined to the red regions. Hence, the harmonic cut-off predicted by Figure 14(c) aligns closely with the results derived from the integration of the TDSE in Figure 12(b). Specifically, the prediction suggests the harmonic cut-off around $\simeq 21\mathrm{\hslash }{\omega }_0$, which is slightly lower than the maximum available gap $\simeq 23\mathrm{\hslash }{\omega }_0$. Notably, the agreement between the red circles representing the SPAM prediction and the results of the numerical calculations (blue diamonds) underscores the reliability of the SPAM approach in accurately predicting the harmonic cut-off frequency.

5. Conclusions

This chapter presents a comprehensive investigation into the phenomenon of high-order harmonic generation (HHG) induced by intense few-cycle pulses in single-wall carbon nanotubes. The observed spectral yield exhibits a plateau, indicative of the non-perturbative nature of HHG. Semiconducting SWNTs are found to be more efficient sources of high-order harmonics compared to metallic ones of the same diameter. The spectral yield in semiconducting tubes results from the coherent addition of contributions from states at the first and second van Hove singularities, while in metallic tubes, only states at the first van Hove singularity contribute. Lower-order harmonics arise from both intra- and inter-band dynamics, whereas higher-order harmonics predominantly stem from inter-band transitions.

Interestingly, spectral similarities are observed among SWNTs of the same diameter but with similar conduction characteristics, suggesting that metallicity rather than chirality influences the spectral yield. The efficiency of spectral yield in metallic tubes increases with diameter up to a certain value, while no significant correlation is observed between the diameter and spectral efficiency or resolution in semiconducting tubes. As the intensity of the driving beam increases, the spectral plateau extends toward the extreme ultraviolet regime, with a nontrivial dependence on the cut-off scaling.

Furthermore, the saturation of the cut-off photon energy, irrespective of the driving beam intensity, is attributed to the maximum band gap attainable by electron-hole pairs generated at the first van Hove singularity during their motion through the Brillouin zone (BZ). This saturation underscores the non-adiabatic nature of the first step in the generation mechanism of high-order harmonics, which is distinct from gas or bulk solid systems where the generation is linked to the maximum amplitude of the field.

Overall, the study unveils the fundamental mechanism underlying high-order harmonic generation in these low-dimensional carbon structures, akin to the process observed in graphene. The findings have significant implications for understanding and controlling HHG in SWNTs and related materials, paving the way for tailored applications in nanophotonics and ultrafast optics.

Acknowledgments

We acknowledge economic support from the Spanish Ministerio de Ciencia, Innovación y Universidades and the Agencia Estatal de Investigacion (10.13039/501100011033) (PID2019-106910GB-100 and PID2022-142340NB-I00). This project has also received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (Grant Agreement No. 851201).

Conflict of interest

The authors declare no conflict of interest.

Abbreviations

BZ	Brillouin zone
CNT	carbon nanotube
DOS	density of states
FWHM	full width at half maximum
HHG	high-order harmonic generation
MWNT	multi-wall carbon nanotube
SPAM	saddle-point approximation model
SLG	single-layer graphene
SWNT	single-wall carbon nanotube
TDSE	time-dependent Schrödinger equation
XUV	extreme ultraviolet