New Analytical Potential, Charge, and Current Distributions in JL MOSFETs: Differences between Accurate and Simplified Models

Abstract

The operation of metal-oxide-semiconductor-field-effect transistors (MOSFETs) cannot be conceived without the prior formation of inversion layers. Charge, potential, and current distributions in this layer are important quantities for all types of MOSFET configurations. The mathematical difficulty to solve analytically the nonlinear Poisson equation in Kingston-Neustadter (KN) model, where the charge density includes electrons and holes, led to Hauser and Littlejohn (HL) to consider a simplified but solvable model, of only electrons or only holes, and to numerous HL-like compact analytical models. Recently, a new and simple method that overcomes the mathematical difficulty to solve the nonlinear Poisson equation in the inversion layer of a MOS has been introduced, and the more accurate Kingston-Neustadter model was successfully solved. We summarize here the new method and given the analytical solutions for the KN and HL models, we compare the potential and charge distribution predictions and show that they may differ by orders of magnitude. We also briefly outline the analytical results for the inversion layer width, the effective ionized atoms concentration, and the drift-diffusion currents in single and double gate junctionless (JL) MOSFETs. Specific calculations of these quantities, as functions of impurity concentration, gate potential, and oxide layer width, will be shown and compared with the predictions of the most exemplary simplified model, the Hauser-Littlejohn model.

1. Introduction

The calculation of the electric potential in the inversion layer of a MOS requires the solution of the Poisson equation, whose complexity depends on how the charge density is modeled. The charge density of the inversion layer emerges as soon as the gate potential $V_G$ becomes larger than the threshold potential $V_{\mathit{Gt}}$ and the conduction band edge crosses the Fermi level. Among the first attempts to calculate the electric potential in the inversion layer of a MOS, stand out the Shockley theoretical work on semiconductors physics [1] and the Kingston and Neustadter [2] approach of 1955, where a charge density of electrons and holes is assumed, in general. Kingston and Neustadter were able to analytically perform the first integration of the nonlinear Poisson equation, but not the second integration. Since then, the analytical efforts to obtain the potential distribution for this accurate model were virtually abandoned and alternative methods, related to the calculation of MOS charges, capacitance, and currents, were devised to account for the analysis of the physical and phenomenological properties of the MOS and the field effect transistors. [3, 4, 5, 6, 7] Some years later, in 1968, Hauser and Littlejohn [8] considered a simplified but solvable model in which the charge density contains only electrons or only holes. This is also the characteristic of the numerous analytical compact models introduced since then [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23]. The literature on the operation of the MOS is overwhelming. A good reference for the standard approaches is Tisividis book [24].

Recently, the original problem of analytically solving the nonlinear Poisson equation in the inversion layer was successfully addressed [25]. In Section 2, we outline the inversion layer problem and the Kingston-Neustadter and Hauser-Littlejohn models. In Section 3, we introduce the bandstructure, relevant parameters, and basic relations to characterize a polarized metal oxide semiconductor. In Section 4, we present a summary of the new method that, based on displacement currents and fields arguments, made possible to replace the nonlinear Poisson equation of the inversion layer by a solvable second-order nonlinear differential equation, the solution of which solves also the original Poisson equation. In Section 5, we present the analytical formulas for the width of the inversion layer and the effective concentration of ionized impurities. In Section 6, we present the potential distributions for the KN and HL models, in a unified representation [25, 26]. We show that although these potentials are qualitatively similar, the predicted inversion layer widths are different. The same occurs with the charge distributions for the KN and HL models, studied in Section 7, as a function of the impurity concentration and the gate potential [26]. In Section 8, we present the analytical formula for the drift-diffusion current for a single-gate JL MOSFET and, in Section 9, the drift-diffusion current for a double-gate JL MOSFET [26].

2. The Kingston-Neustadter and the Hauser-Littlejohn models

In the original semiconductor theories of W. Shockley [1] it was assumed that in the most general case, there were electrons, holes as well as ionized donors and acceptors, and the charge density could be written as [1, 27]

$$\rho \left(y,z\right)=q\left(p\left(y,z\right)-n(y,z)-N_A^{-}+N_D^{+}\right),$$E1

where $q$ is the electric charge, $N_A^{-}$ and $N_D^{+}$ are the ionized acceptor and donor concentrations, and

$$n\left(y,z\right)=n_p{e}^{\mathit{e\varphi }\left(y,z\right)/k_{B}T},\text{and}p\left(y,z\right)=p_p{e}^{-\mathit{e\varphi }\left(y,z\right)/k_{B}T}.$$E2

where $k_B$ is the Boltzmann constant, $T$ is the temperature, $\varphi \left(y,z\right)$ is the local electric potential distribution of electrons and holes, and $n_p$ and $p_p$ are the minority and majority charge concentrations, which in terms of the intrinsic concentration $n_i$ and of the Fermi and thermal potentials, ${\varphi }_F$ and ${\varphi }_T=k_{B}T/e$, can be written as

$$n_p=n_i{e}^{-{\varphi }_F/{\varphi }_T},\text{and}{p}_p=n_i{e}^{{\varphi }_F/{\varphi }_T}$$E3

At high temperatures $N_A^{-}$ and $N_D^{+}$ approximately equal to $N_A$ and $N_D$, respectively. In a metal-oxide-semiconductor, the Kingston-Neustadter model assumption that the semiconductor has only one type of impurity, say type $p$, is accurate and there is no loss in generality when it is assumed that the charge of the ionized impurity concentration is $N_A$, which because of charge neutrality, deep in the bulk is $N_A=p_p-n_p$. Thus, in the KN model, the density can be written as [24]

$$\rho \left(y,z\right)=e\left[\left.p_p{e}^{-\varphi \left(y,z\right)/{\varphi }_T}-n_p{e}^{\varphi \left(y,z\right)/{\varphi }_T}-p_p+n_p\right)\right].$$E4

Using the mass action law $n_p=n_i^2/p_p$, the relation $n_i/p_p=e^{-{\varphi }_F/{\varphi }_T}$ and the fact that $p_p\simeq N_A$, the charge density of electrons and holes for a type $p$ semiconductor is written as [24]

$$\rho \left(z\right)={\mathit{eN}}_A\left[e^{-\varphi \left(y,z\right)/{\varphi }_T}-1-e^{-2{\varphi }_F/{\varphi }_T}\left(e^{\varphi \left(y,z\right)/{\varphi }_T}-1\right)\right].$$E5

Before a source-drain potential is applied, the 1D approximation is good enough and the density, in the KN model, can be written as

$$\rho \left(z\right)={\mathit{eN}}_A\left[e^{-\varphi \left(z\right)/{\varphi }_T}-1-e^{-2{\varphi }_F/{\varphi }_T}\left(e^{\varphi \left(z\right)/{\varphi }_T}-1\right)\right].$$E6

A common procedure to perform the first integration of the Poisson equation

$$\frac{d^2\varphi }{{\mathit{dz}}^2}=-\frac{\rho \left(z\right)}{{\epsilon }_s},$$E7

is to use the identity

$$\frac{d}{\mathit{dz}}{\left(\frac{\mathit{d\varphi }}{\mathit{dz}}\right)}^2=2\frac{\mathit{d\varphi }}{\mathit{dz}}\frac{d^2\varphi }{{\mathit{dz}}^2},$$E8

and transform the Poisson equation into

$$\frac{\mathit{d\varphi }}{\mathit{dz}}d\left(\frac{\mathit{d\varphi }}{\mathit{dz}}\right)=-\mathit{d\varphi }\frac{\rho \left(z\right)}{{\epsilon }_s}.$$E9

A first integration of this equation can easily be performed. Kingston and Neustadter obtained the electric field in the inversion layer as

$$\begin{array}{lll}\frac{1}{2}{\left(\frac{\mathit{d\varphi }\left(z\right)}{\mathit{dz}}\right)}^2& =& \frac{2{\mathit{eN}}_A}{{\epsilon }_s}{e}^{-{\varphi }_F/{\varphi }_T}\left({\varphi }_{T}cosh\frac{\varphi \left(z\right)-{\varphi }_F}{{\varphi }_T}+\varphi \left(z\right)sinh\frac{{\varphi }_F}{{\varphi }_T}-{\varphi }_{T}cosh\frac{{\varphi }_F}{{\varphi }_T}\right)\\ & \equiv & G\left(z,\varphi \left(z\right)\right).\end{array}$$E10

Here $e{\varphi }_F$ is the intrinsic Fermi energy. This first-order nonlinear differential equation could not be integrated once more. Since then, alternative, approximate, and numerical approaches have been introduced. Our purpose here is to show, in the next sections, that this problem has been overcome.

Some years later, in 1968, Hauser and Littlejohn considered a simplified but solvable charge density model where the difference p-n is replaced by only $p$ or $n$, and the impurity concentration is $N_A$ or $N_D$, depending on whether the space-charge region is p-type or n-type. If it is type p, the charge density in the inversion layer is written as

$${\rho }_H\left(z\right)=-{\mathit{eN}}_A{e}^{{\varphi }_H\left(z\right)/{\varphi }_T},$$E11

and the first integration, using also the identity (8), gives the electric field

$$\frac{1}{2}{\left(\frac{d{\varphi }_H\left(z\right)}{\mathit{dz}}\right)}^2=\frac{{\mathit{eN}}_A{\varphi }_T}{{\epsilon }_s}\left(e^{{\varphi }_H\left(z\right)/{\varphi }_T}-1\right)+\frac{1}{2}{\mathcal{E}}_b^2,$$E12

with ${\mathcal{E}}_b$ the electric field at $z=z_b$, the boundary between the inversion and depletion layer, where the HL model assumes that the potential vanishes, that is, ${\varphi }_H\left(z_b\right)=0$. This first-order differential equation can be integrated and the electrostatic potential becomes [8].

$${\varphi }_H\left(z\right)={\varphi }_{T}ln\left[-\frac{b}{a}{sech}^2\left(\frac{\left(z-z_b\right)\sqrt{b}}{2{\varphi }_T}+{tanh}^{-1}\sqrt{\frac{a}{b}+1}\right)\right],$$E13

with

$$a=\frac{2{\mathit{eN}}_0{\varphi }_T}{{\epsilon }_s}\text{and}b={\mathcal{E}}_b^2-a.$$E14

Notice that ${\varphi }_H^{\prime }\left(z_b\right)=2{\mathit{qN}}_0{V}_{\mathit{Gu}}/{\epsilon }_s\equiv -{\mathcal{E}}_b$. Here $V_{\mathit{Gu}}=V_{\mathit{Gt}}+{\mathcal{E}}_{\mathit{ox}}^t{t}_{\mathit{ox}}$, with $V_{\mathit{Gt}}$ the threshold gate potential, ${\mathcal{E}}_{\mathit{ox}}^t$ the electric field in the oxide layer at threshold, and $t_{\mathit{ox}}$ the oxide layer width, see Figure 1. In the following, we will refer to these models as the KN and HL models and one of our purposes is to compare their predictions. It is worth mentioning that besides these analytical attempts, specific analytical solutions for double-gate MOSFETs, using basically the HL model, were obtained by, among others, Y. Taur and by A. Ortiz-Conde et al. We will not address these works nor the numerous analytical compact models, nor the numerical approaches and simulation codes.

Figure 1.

A band structure of a metal-oxide-semiconductor with a type-$p$ semiconductor and a positive gate potential $V_G$ larger than the threshold potential. Several parameters that characterize the system are shown.

In the last 20 years, new geometries and multi-gate MOSFETs were introduced. The inversion layer phenomenon that occurs in the oxide-semiconductor interface of MOS in a single-gate (SG) MOSFET occurs also in each oxide-semiconductor interface of multi-gate MOSFETs. Hence, studying the inversion layer in a single MOS structure is essential. After obtaining the potential and charge distributions, as explicit functions of the relevant parameters, we will consider the analytical expression for the current distribution in a single gate JL MOSFET, and based on this theory, we will discuss the calculation of the analytical formulas for the potential distribution and current in a double gate JL MOSFET, as a simple extension of the SG MOSFET theory.

3. Parameters and basic relations in biased MOS structures

To fix the boundary conditions, the notation and some of the essential parameters used to characterize the MOS structure, we recall here some well-established relations and definitions. Most of the of these relations can be found in textbooks, and some good references are [1, 6, 24, 27, 28, 29, 30, 31]. We will assume that the potentials are measured from the flat-band condition and that the electric potential at $z_p$ (and beyond this point, i.e., in the bulk (see Figure 1)), is $\varphi \left(z_p\right)=$0. For gate potentials larger than the threshold potential, we will distinguish three characteristic layers between the $z=-t_{\mathit{ox}}$ and $z_p$. Between $-t_{\mathit{ox}}$ and $z=0$ the oxide-layer, where the electric potential is linear and given by:

$${\varphi }_{\mathit{ox}}\left(z\right)=V_G-\left(z+t_{\mathit{ox}}\right){\mathcal{E}}_{\mathit{ox}}\mathit{\text{for}}-t_{\mathit{ox}}<z<0,$$E15

and the capacitance per unit area is

$$C_{\mathit{ox}}=\frac{{\epsilon }_{\mathit{ox}}}{t_{\mathit{ox}}},$$E16

with ${\epsilon }_{\mathit{ox}}$ the electric permittivity in the oxide layer. From $z=0$ to $z_i$ we have the inversion layer, and from $z_i$ to $z_p$, the depletion layer. We will assume that the charge density of the ionized impurity atoms in the depletion layer is homogeneous and will be denoted as $N_d$. We will see below that $N_d\le N_A$, being $N_A$ the concentration of acceptor impurities. The precise concentration of the ionized impurity atoms will be determined by the continuity requirements. A closed formula will be obtained for this dynamic quantity.

The boundary conditions at the oxide-semiconductor interface, that is, at $z=0$, and at the edge of the depletion layer, say at $z_p$, are

$${\varphi }_{\mathit{ox}}\left(0\right)=V_G-{\mathcal{E}}_{\mathit{ox}}{t}_{\mathit{ox}},$$E17

and

$${\varphi }_d\left(z_p\right)=0,d{\varphi }_d\left(z_p\right)/\mathit{dz}=0,$$E18

respectively. Under these conditions, the electric potential distribution in the depletion layer is given by

$${\varphi }_d\left(z\right)=\frac{{\mathit{eN}}_d}{2{\epsilon }_s}{\left(z_p-z\right)}^2+\varphi \left(z_p\right)z_i<z<z_p,$$E19

with $z_p=z_i+w_p$, and $w_p=\sqrt{2{\epsilon }_s{V}_{\mathit{Gu}}/{\mathit{eN}}_A}$ the depletion layer width. The electric field, as a function of the electric potential, can be written as

$${\mathcal{E}}_d\left(z,{\varphi }_d\left(z\right)\right)=\sqrt{\frac{2{\mathit{eN}}_d{\varphi }_d\left(z\right)}{{\epsilon }_s}}{z}_i<z<z_p.$$E20

Here and in the following, we write ${\epsilon }_s$ instead of ${\epsilon }_r{\epsilon }_0$.

As is well known, for the band-edge bending to reach the threshold potential, as shown in Figure 1, the gate potential has to be greater than the threshold potential $V_{\mathit{Gt}}$, defined by

$$V_{\mathit{Gt}}=\left(E_g-E_{\mathit{Fs}}+{\mathcal{E}}_{\mathit{ox}}^t{t}_{\mathit{ox}}\right)/e.$$E21

Here, $E_g$ is the gap energy, $E_{\mathit{Fs}}$ the semiconductor Fermi energy, ${\mathcal{E}}_{\mathit{ox}}^t$ the electric field in the oxide layer at threshold, and $t_{\mathit{ox}}$ the oxide layer width. Once the inversion regime is established and $V_G$ is increased, the highly charged inversion layer emerges whose thickness, the inversion-layer width $z_i$, depends on the gate potential and on the impurity concentration. In terms of the parameters defined above, the condition for determining the inversion layer width is

$$V_{\mathit{Gt}}={\mathcal{E}}_{\mathit{ox}}^t{t}_{\mathit{ox}}+\varphi \left(z_i\right),$$E22

where

$$\varphi \left(z_i\right)=\frac{E_c-E_{\mathit{Fs}}}{e}\equiv V_{\mathit{Gu}},$$E23

is the surface potential threshold denoted here as $V_{\mathit{Gu}}$, with $u$ for umbral (from latin umbra—shadow, and liminaris—limit). After solving Poisson’s equation, we will present a closed formula for the inversion-layer width $z_i$.

The positive charge on the metallic side creates an electric field $\mathcal{E}\left(z\right)$ that penetrates on the semiconductor side, attracts electrons, and repels holes and, at the end, establishes an electric potential $\varphi \left(z\right)$ that redefines the valence and conduction band edges as $E_v-\mathit{e\varphi }\left(z\right)$ and $E_c-\mathit{e\varphi }\left(z\right)$, respectively. The inversion population process modifies the nature and the phenomenology of the physical system on the semiconductor side, whose precise description depends on whether the Poisson equation that governs the intertwined relation, between gate potential, charge distribution, and electric potential, can be fully or partially solved. In the next section, we summarize the main argument to transform the Poisson equation into a solvable nonlinear second-order differential equation.

4. Charge and field displacement in the inversion layer

When the gate potential $V_G$ becomes greater than the threshold potential $V_{\mathit{Gt}}$, the charge distribution ${\rho }_i$, regardless of how it is modeled, undergoes an inversion-population process (IPP). If the semiconductor is linear and the charge concentration ${\mathcal{N}}_i\left(z,\varphi ,T\right)$ in the growing inversion layer is highly localized, the balance between the time rate of the thermal energy $k_{B}T{\mathcal{N}}_i\left(z,\varphi ,T\right)$ and the time rate of the displacement field energy $u_E\left(\varphi ,T\right)=t\cdot \mathcal{D}$, can be written as

$$-{\varphi }_T{\int }_{{\rho }_i\left({\varphi }_t,T\right)}^{{\rho }_i\left(\varphi ,T\right)}d{\rho }_i\left(\varphi ,T\right)=\frac{1}{2}{\int }_{u_E\left({\varphi }_t,T\right)}^{u_E\left(\varphi ,T\right)}{\mathit{du}}_E.$$E24

where ${\rho }_i\left(\varphi ,T\right)=-e{\mathcal{N}}_i\left(z,\varphi ,T\right)$ is the charge distribution in the inversion layer, and ${\varphi }_t$ the electric potential at threshold. Assuming that $\mathcal{E}=-\mathit{d\varphi }\left(z\right)/\mathit{dz}\widehat{\mathbf{z}}$, we have

$$\frac{{\epsilon }_s}{2}{\left(\frac{\mathit{d\varphi }\left(z\right)}{\mathit{dz}}\right)}^2-\frac{{\epsilon }_s}{2}{\mathcal{E}}_t^2=-{\varphi }_T\left({\rho }_i\left(\varphi \right)-{\rho }_i\left({\varphi }_t\right)\right).$$E25

It was shown that when the charge density is, for example, the charge density of the KN model, the electric field obtained from this equation is equal to the electric field in Eq. (10). Taking into account the Poisson equation and the electric field in Eq. (10), we obtain the nonlinear second-order differential equation

$${\varphi }_T\frac{d^2\varphi \left(z\right)}{{\mathit{dz}}^2}=\frac{1}{2}{\left(\frac{\mathit{d\varphi }\left(z\right)}{\mathit{dz}}\right)}^2-\frac{1}{2}{\xi }^2,$$E26

where $\xi$ is a model and threshold-dependent parameter, given by

$${\xi }^2={\mathcal{E}}_t^2+2\frac{{\varphi }_T\rho \left({\varphi }_t\right)}{{\epsilon }_s}.$$E27

These are the main and most consequential results of the new method. In the next sections, we will apply this method for the HL and the KN models. The charge densities in these models are different; hence, the parameter $\xi$ is also different, while the differential equations are formally similar. Eq. (26) shows an important step toward the solution of the Poisson equation in the inversion layer. Eq. (26) is solvable and, as will be seen below, its solution is also a solution of the original Poisson equation. It has been shown in Ref. [25], that the analytical solution of (26) is

$$\varphi \left(z\right)={\varphi }_0-2{\varphi }_{T}ln\left(cosh\frac{\xi \left(z-z_0\right)}{2{\varphi }_T}-\frac{f_0}{\xi }sinh\frac{\xi \left(z-z_0\right)}{2{\varphi }_T}\right),$$E28

with ${\varphi }_0$ the potential and $f_0$ the electric field at $z=z_0$. Given this solution, and before we refer to the electric potential in the HL and KN models, it is convenient to consider the inversion layer width in general.

5. Inversion layer width and depletion layer concentration

When the continuity conditions are imposed at $z_i$, one obtains both the inversion layer width

$$z_i=\frac{2{\varphi }_T}{\xi }{cosh}^{-1}\left(\frac{-e^{\theta /2{\varphi }_T}{\xi }^2+\sqrt{{\mathcal{E}}_s^4-{\mathcal{E}}_s^2{\xi }^2\left(1-e^{\theta /{\varphi }_t}\right)}}{{\mathcal{E}}_s^2-{\xi }^2}\right),$$E29

where $\theta ={\varphi }_s-V_{\mathit{Gu}}$, and the concentration of the ionized impurity atoms in the inversion layer regime, given by

$$N_{\mathit{dl}}=\frac{{\epsilon }_s}{2\mathit{e\varphi }\left(z_i\right)}\left({\xi }^2+e^{-\left({\varphi }_s-\varphi \left(z_i\right)\right)/{\varphi }_t}\left({\mathcal{E}}_s^2-{\xi }^2\right)\right),$$E30

when $\theta \ge 0$, and by

$$N_{\mathit{ds}}=\frac{{\epsilon }_s{\mathcal{E}}_s^2}{2e{\varphi }_s},$$E31

when ${\varphi }_s<\varphi \left(z_i\right)$. Replacing the corresponding parameter $\xi$ one has the inversion layer width predicted in the corresponding model. See the next section.

6. Potential distributions

As mentioned above, the difference between one model and another lies in the charge density and therefore in the parameter $\xi$. Now we will be more specific.

6.1 Potential distribution in Hauser-Littlejohn model

In the solvable Hauser-Littlejohn model, assuming a type $p$ semiconductor, the charge density and the parameter $\xi$ are

$${\rho }_H\left(z\right)=-{\mathit{eN}}_0{e}^{e{\varphi }_H\left(z\right)/k_{B}T}\text{and}{\xi }_H^2={\mathcal{E}}_b^2-\frac{2{\mathit{eN}}_0{\varphi }_T}{{\epsilon }_s}.$$E32

For the boundary conditions considered by Hauser and Littlejohn, that is, for $z_0=z_b$, and at this point ${\varphi }_0={\varphi }_b=0$ and $f_0=-{\mathcal{E}}_b$, the potential distribution becomes

$${\varphi }_H\left(z\right)=-2{\varphi }_{T}ln\left(cosh\frac{{\xi }_H\left(z-z_b\right)}{2{\varphi }_T}+\frac{{\mathcal{E}}_b}{{\xi }_H}sinh\frac{{\xi }_H\left(z-z_b\right)}{2{\varphi }_T}\right),$$E33

with ${\mathcal{E}}_b=\sqrt{2{\mathit{eN}}_d{V}_{\mathit{Gu}}/{\epsilon }_s}$. It is not difficult to show that the HL solution in (13), obtained by direct integration, reduces rigorously to ${\varphi }_H\left(z\right)$ (see also Appendix of Ref. [25]). In this model, $z_b=z_i$ defines the boundary between the inversion and depletion layer, and Hauser and Littlejohn assume, that the electric potential at this point vanishes, and we can write also the potential distribution for the HL model in terms of the surface parameters at the oxide-semiconductor interface, that is, at $z_0=0$, where ${\varphi }_0={\varphi }_s$ and $f_0=-{\mathcal{E}}_{\mathit{Hs}}$. In this case, the potential distribution is written as

$${\varphi }_L\left(z\right)={\varphi }_s-2{\varphi }_{T}ln\left(cosh\frac{{\xi }_{H}z}{2{\varphi }_T}+\frac{{\mathcal{E}}_{\mathit{Hs}}}{{\xi }_H}sinh\frac{{\xi }_{H}z}{2{\varphi }_T}\right).$$E34

Here the surface electric field ${\mathcal{E}}_{\mathit{Hs}}$ is

$${\mathcal{E}}_{\mathit{Hs}}\left(z\right)={\left(-{\mathit{eN}}_A{\varphi }_T{e}^{e\left({\varphi }_s-V_{\mathit{Gu}}\right)/{\varphi }_T}+{\xi }_H^2\right)}^{1/2}.$$E35

Notice that the assumption ${\varphi }_H\left(z_b\right)$ = 0, implies that ${\varphi }_H\left(z\right)={\varphi }_L\left(z\right)-V_{\mathit{Gu}}$. In terms of ${\varphi }_L\left(z\right)$, the charge density in the Hauser-Littlejohn model is

$${\rho }_H\left(z\right)=-{\mathit{eN}}_A{e}^{e\left({\varphi }_L\left(z\right)-V_{\mathit{Gu}}\right)/k_{B}T}.$$E36

In Figure 2 we plot the potential energy $-e{\varphi }_L\left(z\right)$ as function of $z$, for different values of the impurity concentration, and for different surface potentials, as well. In the next subsection, by comparing with the more accurate KN model, it will be seen that the HL model predicts much smaller inversion layer widths, and grows faster near the oxide-semiconductor interface. The potential distribution ${\varphi }_H\left(z\right)$ is well known and has been widely used as an approximate result for double-gate devices [15, 16].

Figure 2.

The potential energy in the HL model as a function of $z$ for different values of the impurity concentration $N_A$ and different surface potentials ${\varphi }_s$. The small circles indicate the limit (at $z_i$) between the inversion and the depletion layers. The dashed black lines define the threshold energies $-{\mathit{eV}}_{\mathit{Gu}}$, in eVs.

6.2 Potential distribution in the Kingston-Neustadter model

In the important Kingston-Neustadter model, the charge density can be written as

$${\rho }_K\left(z\right)=2{\mathit{eN}}_A{e}^{-{\varphi }_F/{\varphi }_T}\left[sinh\frac{{\varphi }_F-{\varphi }_K\left(z\right)}{{\varphi }_T}-sinh\frac{{\varphi }_F}{{\varphi }_T}\right].$$E37

The parameter $\xi$ is

$${\xi }_K^2=\frac{8{\mathit{eN}}_A}{{\epsilon }_s}{e}^{-{\varphi }_F/{\varphi }_T}\left({\varphi }_F-{\varphi }_T\right)sinh\frac{{\varphi }_F}{{\varphi }_T},$$E38

and the potential distribution, for ${\varphi }_0={\varphi }_s$, $f_0=-{\mathcal{E}}_s$ and $z_0=0$, becomes

$${\varphi }_K\left(z\right)={\varphi }_s-2{\varphi }_{T}ln\left(cosh\frac{{\xi }_{K}z}{2{\varphi }_T}+\frac{{\mathcal{E}}_s}{{\xi }_K}sinh\frac{{\xi }_{K}z}{2{\varphi }_T}\right).$$E39

Given this potential distribution, an alternative representation for the electric field in the inversion layer is

$${\mathcal{E}}_K\left(z\right)={\xi }_K\frac{sinh\frac{{\xi }_{K}z}{2{\varphi }_T}+\frac{{\mathcal{E}}_s}{{\xi }_K}cosh\frac{{\xi }_{K}z}{2{\varphi }_T}}{cosh\frac{{\xi }_{K}z}{2{\varphi }_T}+\frac{{\mathcal{E}}_s}{{\xi }_K}sinh\frac{{\xi }_{K}z}{2{\varphi }_T}}.$$E40

Here the surface electric field is

$${\mathcal{E}}_s={\left[\frac{2{\mathit{eN}}_A}{{\epsilon }_s}{e}^{-{\varphi }_F/{\varphi }_T}\left({\varphi }_{T}cosh\frac{{\varphi }_s-{\varphi }_F}{{\varphi }_T}+{\varphi }_{s}sinh\frac{{\varphi }_F}{{\varphi }_T}-{\varphi }_{T}cosh\frac{{\varphi }_F}{{\varphi }_T}\right)\right]}^{1/2}.$$E41

It was shown in Ref. [25]), by substitution and numerical evaluation, that the function ${\varphi }_K\left(z\right)$ in (39) solves also the original Poisson equation and the electric field in (40), matches with the electric field $\sqrt{2G\left(z,\varphi \left(z\right)\right)}$ in (10), that was obtained by Kingston and Neustadter after the first integration of the original Poisson equation J. Reiter has shown also [32], by numerically solving the differential equation, that the difference between the numerical and analytical solutions is less than 0.5%, in the entire parametric space.

In Figure 3 we plot the potential energy $-e{\varphi }_K\left(z\right)$ as functions of $z$, for different values of the impurity concentrations $N_A$, as well as for different values of the surface potential ${\varphi }_s$. In this figure, we indicate the Fermi levels (dashed black lines) and the borders between the inversion and depletion layers. These points, where the potential energies cross the Fermi levels, define the inversion layer widths $z_i$. The behavior of ${\varphi }_K\left(z\right)$ is qualitatively similar to that of ${\varphi }_L\left(z\right)$ in Figure 2. It grows also rapidly for gate potentials $V_G$ larger than the threshold potential $V_{\mathit{Gt}}$, or surface potentials ${\psi }_s$ larger than the umbral potential $V_{\mathit{Gu}}$.

Figure 3.

The potential energy in the KN model as a function of $z$ for different values of the impurity concentration $N_A$ and different surface potentials ${\varphi }_s$. The small circles indicate the limit (at $z_i$) between the inversion and the depletion layers. The dashed black lines define the threshold energies $-{\mathit{eV}}_{\mathit{Gu}}$ (or Fermi levels).

It is now possible to compare the electric potentials and the inversion layer widths predicted by the Kingston-Neustadter and the Hauser-Littlejohn models. In Figure 4, we plot the potential energies, predicted in both models, as functions of $z$, and for different values of the impurity concentration $N_A$ and different values of the surface potential ${\varphi }_s$. The potential distributions are qualitatively similar but quantitatively different. In the HL model, the inversion layer widths are smaller than those predicted in the KN model, by almost a factor of 2, the electric potentials are also smaller but near the oxide-semiconductor interface, the electric potentials in the HL model grow faster than in the KN model. In general, the potential and the layer widths are underestimated in the HL model. Because of the smaller inversion layer width and the rapid grow of ${\varphi }_H\left(z\right)$, the charge distribution in the HL model is much more localized and closer to the 2D assumption.

Figure 4.

The electric potential energy in the KN (continuous) and HL (dashed) model as a function of $z$ for different values of the impurity concentration $N_A$ and different surface potentials ${\varphi }_s$. The small circles indicate the limit (at $z_i$) between the inversion and the depletion layers.

In Figure 5 we plot the inversion layer widths in the KN model as functions of the gate potential $V_G={\varphi }_s+\kappa t_{\mathit{ox}}{\mathcal{E}}_s$, for different values of the impurity concentration. For these plots, the silicon parameter $\kappa ={\epsilon }_s/{\epsilon }_{\mathit{ox}}=3$ and oxide-layer width $t_{\mathit{ox}}=2\mathrm{nm}$ were considered.

Figure 5.

The inversion-layer width $z_i$ in the KN model as function of $V_G$, for different impurity concentrations. In the upper axes, the threshold potentials is written as $V_{\mathit{Gt}}=V_{\mathit{Gu}}+{\mathrm{\Delta }}_{\mathit{ox}}$, with ${\mathrm{\Delta }}_{\mathit{ox}}=t_{\mathit{ox}}{\mathrm{E}}_{\mathit{ox}}^t=\kappa t_{\mathit{ox}}{\mathrm{E}}_s^t$, the potential drop in the oxide layer.

Given the explicit functions ${\varphi }_L$, ${\varphi }_K$, ${\xi }_H$ and ${\xi }_K$, we will in the next section evaluate the charge density distributions.

7. Charge density distributions

An important consequence of the inversion population phenomenon is the rapid increase of the charge density in the inversion layer. Replacing the electric potentials ${\varphi }_L\left(z\right)$ and ${\varphi }_K\left(z\right)$ in the corresponding density, Eqs. (36) and (37) we have, for the HL model, the charge density

$${\rho }_H\left(z\right)=-{\mathit{eN}}_{\mathit{Hd}}{e}^{\left({\varphi }_s-V_{\mathit{Gu}}\right)/{\varphi }_T}{\left(cosh\frac{{\xi }_{H}z}{2{\varphi }_T}+\frac{{\mathcal{E}}_{\mathit{Hs}}}{{\xi }_H}sinh\frac{{\xi }_{H}z}{2{\varphi }_T}\right)}^2,$$E42

and, for the Kingston-Littlejohn model, the charge density

$$\begin{array}{lll}{\rho }_K\left(z\right)& =& {\mathit{eN}}_{\mathit{Kd}}\{e^{-{\varphi }_s/{\varphi }_T}{\left(cosh\frac{{\xi }_{H}z}{2{\varphi }_T}+\frac{{\mathcal{E}}_s}{{\xi }_H}sinh\frac{{\xi }_{H}z}{2{\varphi }_T}\right)}^2-1\\ & & -\left.e^{-2{\varphi }_F/{\varphi }_T}\left[1-e^{{\varphi }_s/{\varphi }_T}{\left(cosh\frac{{\xi }_{H}z}{2{\varphi }_T}+\frac{{\mathcal{E}}_s}{{\xi }_H}sinh\frac{{\xi }_{H}z}{2{\varphi }_T}\right)}^{-2}\right]\right\}.\end{array}$$E43

To visualize the similarities and differences between these charge densities, we plot them side by side in Figure 6, and the surface densities in Figure 7. The densities in Figure 6 are plotted as functions of the distance $z$ to the oxide-semiconductor interface, for different values of the impurity concentration and for different values of the surface potential ${\varphi }_s$. The qualitative behavior is similar, with rapid growth in the inversion layer and almost constant densities in the depletion layers. In these layer, we take into account the effective impurity ionization $N_d$. The effect of this concentration is observed in Figure 6a where the charge density varies continuously at $z_i$ and vanishes at the border of the depletion layer at $z_p=z_i+w_p$. The densities in the HL model behave as shown in Figure 6b with a practically constant concentration in the depletion layers. As observed before, the HL predicts thinner inversion layers. Near the inversion layer edge, around $z_i$, the densities are quantitatively similar, but as one approaches the oxide interface, the differences grow. To visualize this difference we plot in Figure 7, the surface densities

Figure 6.

Charge density in the KN (left) and HL (right) models as a function of $z$, and for different values of $N_A$. The small circles indicate the limit (at $z_i$) between the inversion and the depletion layers.

Figure 7.

The charge density at $z=0$ in the KN (left) and the HL (right) model, as functions of the surface potential ${\varphi }_s$, for two values of the impurity concentration $N_A={10}^{23}{m}^{-3}$ and $N_A={10}^{24}{m}^{-3}$.

$${\rho }_{\mathit{Hs}}\left(z\right)=-{\mathit{eN}}_{\mathit{Hd}}{e}^{\left({\varphi }_s-V_{\mathit{Gu}}\right)/{\varphi }_T},$$E44

and

$${\rho }_{\mathit{Ks}}\left(z\right)=-2{\mathit{eN}}_{\mathit{Kd}}{e}^{-{\varphi }_F/{\varphi }_T}\left(sinh\frac{{\varphi }_s-{\varphi }_F}{{\varphi }_T}+sinh\frac{{\varphi }_F}{{\varphi }_T}\right)$$E45

From Figure 7, it is clear that the difference between the charge densities in the HL and KN models grows with ${\varphi }_s$ by orders of magnitude. We also see that due to the exponential function in (44), the density at the surface, in the HL model, is practically insensitive to the impurity concentration, while in the KN model, the surface density in (45) is sensitive to the impurity concentration, as could be expected.

8. Drift current in a JL MOSFET channel

When the biased MOS system is part of a single or multiple gate MOSFETs, the charge and potential distributions obtained before play an important role in the nature and magnitude of the drift-diffusion currents moving in the MOSFET channel(s), parallel to the oxide-semiconductor interface(s). To simplify the analysis, we will have the so-called junctionless MOSFETs. As is well known and will be seen here, the charge carriers in the inversion layer play the most important role.

A well-established formula for the calculation of the source-drain current is Pao-Sah’s formula [5]

$$I_D=\frac{{\mathit{eD}}_n{W}_x}{L}{\int }_0^{z_p}n\left(y,z\right)\mathit{dz}{\int }_0^L\mathit{dy}\frac{\mathit{d\eta }\left(y\right)}{\mathit{dy}}.$$E46

Here $L$ is the channel length that we will assume alongside the $y$ direction, $W_x$ the channel depth, $D_n$ the diffusion coefficient, and $z_p=z_i+w_p$ the inversion plus depletion layers widths. The semiconductor layer width $W_z$ may be greater or smaller than $z_p$. When the drift and diffusion currents are taken into account, the distribution $n\left(y,z\right)$ becomes

$$n\left(y,z\right)=n_A{e}^{\left(\varphi \left(z\right)-\eta \left(y\right)-{\varphi }_{\mathit{Fs}}\right)/{\varphi }_T},$$E47

where $\varphi \left(z\right)$ is the electrical potential due to the gate potential, $\eta \left(y\right)$ is the quasi-Fermi level modified by the drain-source potential $U_D$ along the channel, and ${\varphi }_{\mathit{Fs}}=\left(E_c-E_{\mathit{Fs}}\right)/e$ is the equilibrium Fermi potential in the bulk, that we will assume to be measured from the conduction band edge. Usually $\mathit{dz}$ is replaced by $-\mathit{d\varphi }/\mathcal{E}\left(z,\varphi \right)$, where $\mathcal{E}\left(z,\varphi \right)=\sqrt{2G\left(z,\varphi \right)}$. We do not need to make this change because we do have the electric potential as an explicit function of Z. We keep the integration upon $z$, but we will change the integration upon $y$ by the integration upon $\eta$. We also split the integration in $z$ in two parts, one for the inversion region and one for the depletion layer. In this case,

$$I_D=\frac{{\mathit{eD}}_n{w}_x{n}_i}{L}\left[{\int }_0^{z_i}{e}^{{\varphi }_i\left(z\right)/{\varphi }_T}\mathit{dz}+{\int }_{z_i}^{z_i+w_p}{e}^{{\varphi }_d\left(z\right)/{\varphi }_T}\mathit{dz}\right]{\int }_0^{U_D}{e}^{\left(-\eta \left(y\right)-{\varphi }_{\mathit{Fs}}\right)/{\varphi }_T}\mathit{d\eta },$$E48

with the function ${\varphi }_i$ given by the electric potential defined in (39), and the function ${\varphi }_d$ given above for the electric potentials in the depletion layer, Eqs. (30) and (31). Similarly, the current in the HL model is straightforwardly calculated when the electric potential defined in (34) is replaced for ${\varphi }_i$. In this section we will present results only for the KN model and we will drop the subscript K. The integrations are direct and we obtain, in the KN model, the current

$$\begin{array}{lll}{I}_D& =& \frac{{\mathit{eD}}_n{w}_x{n}_i}{L}\left[e^{\left({\varphi }_s-{\varphi }_{\mathit{Fs}}\right)/{\varphi }_T}{\left.\frac{2{\varphi }_{T}sinh\frac{\mathit{\eta z}}{2{\varphi }_T}}{\eta \left(cosh\frac{\mathit{\eta z}}{2{\varphi }_T}+\frac{{\mathcal{E}}_s}{\eta }sinh\frac{\mathit{\eta z}}{2{\varphi }_T}\right)}\right|}_0^{z_i}\right.\\ & & \left.+\frac{\sqrt{{\mathit{\pi \epsilon }}_s{\varphi }_T}}{\sqrt{{\mathit{eN}}_d}}{e}^{-{\varphi }_F/2{\varphi }_T}\mathrm{Erf}\left[\frac{\sqrt{{\mathit{eN}}_d}{w}_p}{2\sqrt{{\epsilon }_s{\varphi }_T}}\right]\right]\left(1-e^{-U_D/{\varphi }_T}\right),\end{array}$$E49

which reduces to

$$\begin{array}{lll}{I}_D& =& \frac{{\mathit{eD}}_n{w}_x{n}_i}{L}\left[\frac{2{\varphi }_T}{\xi }{e}^{\left({\varphi }_s-{\varphi }_{\mathit{Fs}}\right)/{\varphi }_T}{e}^{\left(V_{\mathit{Gu}}-{\varphi }_s\right)/2{\varphi }_T}sinh\frac{\xi z_i}{2{\varphi }_T}\right.\\ & & \left.+\frac{\sqrt{{\mathit{\pi \epsilon }}_s{\varphi }_T}}{\sqrt{{\mathit{eN}}_d}}{e}^{-{\varphi }_F/2{\varphi }_T}\mathrm{Erf}\left[\frac{\sqrt{{\mathit{eN}}_d}{w}_p}{2\sqrt{{\epsilon }_s{\varphi }_T}}\right]\right]\left(1-e^{-U_D/{\varphi }_T}\right).\end{array}$$E50

Since $V_{\mathit{Gu}}={\varphi }_{\mathit{Fs}}$, this becomes

$$\begin{array}{lll}{I}_D& =& \frac{{\mathit{eD}}_n{w}_x{n}_i}{L}\left[\frac{2{\varphi }_T}{\xi }{e}^{\left({\varphi }_s-V_{\mathit{Gu}}\right)/2{\varphi }_T}sinh\frac{\xi z_i}{2{\varphi }_T}\right.\\ & & +\left.\frac{\sqrt{{\mathit{\pi \epsilon }}_s{\varphi }_T}}{\sqrt{{\mathit{eN}}_d}}{e}^{-{\varphi }_F/2{\varphi }_T}\mathrm{Erf}\left[\frac{\sqrt{{\mathit{eN}}_d}{w}_p}{2\sqrt{{\epsilon }_s{\varphi }_T}}\right]\right]\left(1-e^{-U_D/{\varphi }_T}\right)\equiv I_{\mathit{Di}}+I_{\mathit{Dd}}.\end{array}$$E51

As mentioned before, the concentration $N_d$ should be replaced by $N_{\mathit{ds}}$, when ${\varphi }_s<V_{\mathit{Gu}}$, and by $N_{\mathit{dl}}$ when ${\varphi }_s>V_{\mathit{Gu}}$. We shall present now some graphs to visualize the behavior of these currents when both the gate and source-drain potentials are applied. For all graphs shown in this section, we will assume the mobility $\mu =400$cm²/Vs in the depletion and inversion layers. These quantities can of course be changed. The impurity concentrations $N_A$ will be indicated in the graphs.

We will plot now the depletion and inversion layer currents and the total current, separately, as functions of the source-drain potential $U_D$ and of the surface potential ${\varphi }_s$. Since $V_G={\varphi }_s+\kappa t_{\mathit{ox}}{\mathcal{E}}_s$, plotting as a function of ${\varphi }_s$ is almost as plotting as function of $V_G$, not exactly as will be seen below. In all cases, of this section, we will assume that the mobility $\mu =400$cm${}^2/$Vm is the same in the depletion and inversion layers, $L=0.4\mu \mathrm{m}$ and $W_x=1.0\mu \mathrm{m}$. In Figure 8 we plot the depletion, inversion, and total currents as functions of $U_D$ and of ${\varphi }_s$. In ($a$) the depletion current below and above the threshold, in ($b$) the inversión layer current and in ($c$) the sum of both currents. Above the threshold, the current $I_{\mathit{Dd}}$ grows with $U_D$ but is independent of $V_G$. Both currents $I_{\mathit{Dd}}$ and $I_{\mathit{Di}}$ grow monotonously with $U_D$ and tend to a saturation value at higher values. For this graph, we assumed also that $W_x=1\mu \mathrm{m}$ and $L=W_x/2.5$.

Figure 8.

Depletion, inversion, and total current as functions of the source-drain potential $U_D$ and of the surface potential ${\varphi }_s$. In ($a$) the depletion current $I_{\mathit{Dd}}$ below and above the threshold, in ($b$) the inversión layer current $I_{\mathit{Di}}$ and in ($c$) the total current $I_D=I_{\mathit{Dd}}+I_{\mathit{Di}}$. The depletion current $I_{\mathit{Dd}}$ as a function of $U_D$ grow monotonously toward a saturation value, as a function of ${\varphi }_s$ the depletion current grows steadily until the gate potential reaches the threshold value, where $I_{\mathit{Dd}}$ experiences a sharp change. Above this threshold, the depletion current becomes independent of the gate potential. At variance with $I_{\mathit{Dd}}$, the inversión layer current $I_{\mathit{Di}}$, as a function of the ${\varphi }_s$ (i.e. of the gate potential), starts growing at the threshold potential, and for a fixed value of the gate potential, the inversion layer current also grows with $U_D$ and tends to a saturation value at higher values. For this graph $\mu =400{\mathit{cm}}^2/\mathit{Vs}$,$W_x=1\mathit{\mu m}$,$L=W_x/2.5$.

Due to the nonlinear relationship between the gate and the surface potential and the nonlinear relationship between the surface electric field ${\mathcal{E}}_s$ and ${\varphi }_s$, the growths of the gate current inversion layer as a function of ${\varphi }_s$ and as a function of $V_G$ should look different. In fact, this difference can be seen in Figure 9 ($a$) and ($b$), where the same inversion current is plotted, in one case as a function of ${\varphi }_s$, and as a function of $V_G$ in the other case. In Figure 9 ($a$) the current $I_{\mathit{Di}}$ grows exponentially as a function of ${\varphi }_s$ while the currents $I_{\mathit{Di}}$ in Figure 9 ($b$) grow almost linearly as function of $V_G$. Since the depletion current remains almost constant above the threshold behavior the linear behavior of $I_{\mathit{Di}}$ occurs also in the total current $I_D$.

Figure 9.

The inversion-layer current $I_{\mathit{Di}}$ as function of the surface potential ${\varphi }_s$ (panel ($a$)) and the gate potential $V_G$ (panel ($b$)). In ($a$) the current is plotted as a function of (equidistant values) of${\varphi }_s$, and in ($b$), the same current but as a function of (equidistant values) of $V_G$. In the last case, it is evident that at larger values of $V_G$ the current becomes a linear function of $V_G$.

In Figure 10, we plot the total drift-diffusion current $I_D$ as a function of the gate potential $V_G$, for different values of the impurity concentration. As mentioned before the inversion layer current and the total current behave, at larger values of the gate potential, as linear functions of this variable, with lower slopes at higher impurity concentrations. It can be seen also in Figure 10, that the threshold potential for higher impurity concentration is larger. These threshold potentials and the so-called operational threshold potentials, determined at the intersection of the linear current with the abscissa, are indicated with arrows. The operational threshold potentials are larger than the gate potential thresholds. In the particular case considered here, the differences are of the order of 0.3 V. Given the device parameters, the operational potentials can also be predicted. To complete the comparison of the KN and HL model predictions, we plot in Figure 11 the drift-diffusion currents in the KN and HL models assuming the same device parameters. The predict currents are also different. The currents in the simplified HL model are smaller than the currents in the most accurate KN model. For these graphs, we assumed that the impurity concentration is $N_A={10}^{23}{\mathrm{m}}^{-3}$.

Figure 10.

The total source-drain current $I_D=I_{\mathit{Dd}}+I_{\mathit{Di}}$ in silicon MOSFETs as functions of the gate potential, for three different values of the impurity concentration. Here, we assume that the mobility, in the depletion and inversion layers, is $\mu =400{\mathit{cm}}^2/\mathit{Vm}$, $U_D=\mathit{0.001}V$, and channel length $L=0.5\mathit{\mu m}$. The gate potential thresholds $V_{\mathit{Gt}}$ are given on the graph. These thresholds and the so-called operational threshold potentials, defined generally at the intersection of the abscissa with the extension of the linear current (see dashed lines), are indicated with arrows.

Figure 11.

The drift-diffusion currents in the KN and HL models as functions of the gate potential. The HL model predicts smaller currents than the KN model.

The theory presented in this section, and before, can be applied almost straightforwardly to more complex devices, among them, to the double-gate (DG) MOSFET studied in the next section, and to multi-gate MOSFETs.

9. The double gate JL MOSFET

Explicit calculations for other MOSFET configurations can be performed almost as a trivial extension of the results presented in the last sections. We will refer only and briefly to the double-gate MOSFET. Consider a double-gate device like the one sketched in Figure 12, where the oxide layers are at a distance $t_{\mathit{Si}}$ and the front and back gate voltages $V_{\mathit{Gf}}$ and $V_{\mathit{Gb}}$ are assumed to have opposite polarizations. In this case, holes accumulate near the back oxide-semiconductor interface and electrons near the front oxide-semiconductor. Before the source-drain potential $U_D$ is applied, we can write the electric potential in the inversion layers in the 1D approximation as

Figure 12.

A double gate structure as a generalization of the single gate MOS of Figure 1. We assume here that the back gate potential is negative and attracts holes. The potential distribution ${\varphi }_{\mathit{DG}}\left(z\right)$ (red curve) is schematically shown.

$${\varphi }_{\mathit{iDG}}\left(z\right)={\varphi }_{\mathit{\text{if}}}\left(z\right)+{\varphi }_{\mathit{ib}}\left(z\right).$$E52

Assuming that the charge densities are described by the Kingston-Neustadter-like distributions, the potential distributions in the front and back inversion layers, respectively, can be written as

$${\varphi }_{\mathit{\text{if}}}\left(z\right)={\varphi }_{\mathit{sf}}-2{\varphi }_{T}ln\left(cosh\frac{{\xi }_{K}z}{2{\varphi }_T}+\frac{{\mathcal{E}}_{\mathit{Kfs}}}{{\xi }_K}sinh\frac{{\xi }_{K}z}{2{\varphi }_T}\right),$$E53

and

$${\varphi }_{\mathit{ib}}\left(z\right)=-{\varphi }_{\mathit{sb}}+2{\varphi }_{T}ln\left(cosh\frac{{\xi }_K{z}^{\prime }}{2{\varphi }_T}+\frac{{\mathcal{E}}_{\mathit{Kbs}}}{{\xi }_K}sinh\frac{{\xi }_K{z}^{\prime }}{2{\varphi }_T}\right).$$E54

Here, $z^{\prime }=t_{\mathit{Si}}-z$, ${\varphi }_{\mathit{sf}}$ and ${\varphi }_{\mathit{sb}}$ are the front and back surface potentials, and ${\mathcal{E}}_{\mathit{Kfs}}$ and ${\mathcal{E}}_{\mathit{Kbs}}$ are the front and back surface electric fields. In the symmetric case with $t_{\mathit{oxf}}=t_{\mathit{osb}}$ and $V_{\mathit{Gf}}=-V_{\mathit{Gb}}$, the potential function ${\varphi }_{\mathit{DG}}\left(z\right)$ is asymmetric, as sketched in Figure 12 (red curve), and the potential energy $q{\varphi }_{\mathit{iDG}}\left(z\right)$ is symmetric, as shown in Figure 13 ($a$), where a silicon layer $t_{\mathit{Si}}\simeq 40\mathrm{nm}$ is assumed, which for an impurity concentration $N_A=1\times {10}^{18}{\mathrm{cm}}^{-3}$ is $\simeq \left(1/2\right)\left(w_{\mathit{pd}}+w_{\mathit{nd}}+z_{\mathit{\text{if}}}+z_{\mathit{ib}}\right)$, where $w_{\mathit{pd}}$ and $w_{\mathit{nd}}$ are the depletion layers widths and $z_{\mathit{\text{if}}}$ and $z_{\mathit{ib}}$ the inversion layer widths.

Figure 13.

Electric potential of drift-diffusion current in a symmetric double-gate JL MOSFET. In ($a$) the electric potential for a DG MOS where the silicon thickness $t_{\mathit{Si}}$ is smaller than the expected depletion plus inversion layers width $w_p+w_n+z_{\mathit{\text{if}}}+z_{\mathit{ib}}$. Some parameters are assumed equal to those in Ref. [23], i.e., $L=11\mathit{\mu m}$, $W_x=10\mu$m. However, nowhere in Ref. [23] are the impurity concentration and mobilities given. In this reference, the silicon layer width $t_{\mathit{Si}}$ is assumed equal to 8 nm, slightly larger than the sum of the inversion layer widths for $N_A=1\times {10}^{18}{\mathit{cm}}^{-3}$. Here, we assume that $t_{\mathit{Si}}\simeq 40\mathit{nm}$.

Given the potential distribution ${\varphi }_{\mathit{iDG}}\left(z\right)$, we can, using Pao-Sah’s [5] formula, obtain the analytical expression for the drift-diffusion current in the DG JL MOSFET channel as a function of the relevant parameters. In the symmetric case, the total current in the DG JL MOSFET channels can be written as

$$\begin{array}{l}{I}_{\mathit{DG}}=\frac{{\mathit{eD}}_n{W}_x{n}_i}{L}\left[{\int }_0^{z_{\mathit{\text{if}}}}{e}^{{\varphi }_{\mathit{\text{if}}}\left(z\right)/{\varphi }_T}\mathit{dz}+{\int }_{z_{\mathit{\text{if}}}}^{t_{\mathit{Si}}/2}{e}^{{\varphi }_{\mathit{pd}}\left(z\right)/{\varphi }_T}\mathit{dz}+{\int }_{t_{\mathit{Si}}/2}^{t_{\mathit{Si}}-z_{\mathit{ib}}}{e}^{-{\varphi }_{\mathit{nd}}\left(z\right)/{\varphi }_T}\mathit{dz}\right.\\ \left.+{\int }_{t_{\mathit{Si}}-z_{\mathit{ib}}}^{t_{\mathit{Si}}}{e}^{-{\varphi }_{\mathit{ib}}\left(z\right)/{\varphi }_T}\mathit{dz}\right]\times {\int }_0^{U_D}{e}^{\left(-\eta \left(y\right)-{\varphi }_{\mathit{Fs}}\right)/{\varphi }_T}\mathit{d\eta }.\end{array}$$E55

All integrals in this equation can be performed analytically, and for $z_{\mathit{\text{if}}}+z_{\mathit{ib}}<t_{\mathit{Si}}<w_p+w_n+z_{\mathit{\text{if}}}+z_{\mathit{ib}}$, and $z_{\mathit{\text{if}}}=z_{\mathit{ib}}$, the current becomes

$$\begin{array}{lll}{I}_{\mathit{DG}}& =& \frac{{\mathit{eD}}_n{w}_x{n}_i}{L}\left(1-e^{-U_D/{\varphi }_T}\right)\left[\frac{4{\varphi }_T}{\xi }{e}^{\left({\varphi }_s-V_{\mathit{Gu}}\right)/2{\varphi }_T}sinh\frac{\xi z_i}{2{\varphi }_T}\right.\\ & & \left.+\frac{\sqrt{{\mathit{\pi \epsilon }}_s{\varphi }_T}}{\sqrt{{\mathit{eN}}_d}}{e}^{-{\varphi }_F/2{\varphi }_T}\left(\mathrm{Erf}\left[\frac{\sqrt{{\mathit{eN}}_d}{w}_p}{2\sqrt{{\epsilon }_s{\varphi }_T}}\right]+\mathrm{Erf}\left[\frac{\sqrt{{\mathit{eN}}_d}\left(t_{\mathit{Si}}-w_p-2z_{\mathit{\text{if}}}\right)}{2\sqrt{{\epsilon }_s{\varphi }_T}}\right]\right)\right]\\ & =& I_{\mathit{DGi}}+I_{\mathit{DGd}}.\end{array}$$E56

It can easily be seen that the current in the DG JL MOSFET is practically twice the charge and current in the single gate MOSFET channel. [25] In fact, it is exactly twice when $t_{\mathit{Si}}=2w_p+2z_{\mathit{\text{if}}}$.

In Figure 13($b$) we plot the drift-diffusion current of Eq. (56) as a function of the gate potential, for three different values of the source-drain voltage $U_D$. For this graph we assume that the impurity concentration is $N_A=1\times {10}^{18}{\mathrm{cm}}^{-3}$, the channel length is $L=11\mu$m, the gate width is $W_x=10\mu$m, and the mobility is assumed equal to 40cm²/Vs. Some of these parameters are the same as those used in Figure 7 of Ref. [23], where a simplified charge distribution is assumed; however, we cannot compare with that reference because some parameters, such as the impurities concentration and mobilities, are not given.

It is clear that the analytical expression for the current in Eq. (51) can also be used for asymmetric DG JL MOSFETs. The general formulas for the inversion layer width, surface electric field, and effective concentration of ionized impurity atoms, introduced for the single gate MOS in Section 5, are also valid with the appropriate changes for the double gate MOSFET. The ability to evaluate these quantities, particularly the widths of the inversion layers, is, for a given impurity concentration, an important input in the design of MOSFETs and in the definition of some device parameters, such as the silicon layer width $t_{\mathit{Si}}$.

10. Conclusions

We presented here an outline of the new theoretical approach for the analytical calculation of potential distributions and drift-diffusion currents in MOS and JL MOSFETs. We presented a summary of the novel method that makes it possible to solve analytically the nonlinear Poisson equation in the inversion layer of MOSs and MOSFETs. We apply this method to the more precise Kingston-Neustadter charge density model, unsolved until now, where the presence of electrons and holes in the inversion layer is assumed, and also to the most representative of the simplified compact analytical models, the Hauser-Littlejohn model, which assumes that only electrons or only holes are present. The predictions of both models for the potential and charge distributions, as well as for the drift-diffusion current, were compared and we have shown that they generally differ. In the HL model, the widths of the inversion layer are much smaller, and the charge distributions are almost 2D, with a higher surface density and practically insensitive to impurity concentration. The drift-diffusion currents predicted by the HL model are also different and smaller than the currents predicted by the KN model. Independent of these comparisons with the standard approaches, we have presented here several analytical formulas for potential and charge distributions and new analytical formulas for the drift-diffusion current in single and double-gate MOSFETs.

Acknowledgments

I acknowledge valuable comments by Herbert P. Simanjuntak, Jürgen Reiter, and A. Robledo-Martinez.

Conflict of interest

The author declares no conflict of interest.