Design and Analysis of Gallium Arsenide-Based Nanowire Using Coupled Non-Equilibrium Green Function for RF Hybrid Applications

Abstract

This research work uses sp³d⁵s* tight-binding models to design and analyze the structural properties of group IV and III-V oriented, rectangular Silicon (Si) and Gallium Arsenide (GaAs) Nanowires (NWs). The electrical characteristics of the NWs, which are shielded with Lanthanum Oxide (La₂O₃) material and the orientation with z [001] using the Non-Equilibrium Green Function (NEGF) method, have been analyzed. The electrical characteristics and the parameters for the multi-gate nanowires have been realized. A nanowire comprises a heavily doped n⁺ donor source and drains doping and n-donor doping at the channel. The specified nanowire has a gate length and channel length of 15 nm each, a source-drain device length L_SD = 35 nm, with La₂O₃ as 1 nm (gate dielectric oxide) each on the top and bottom of the core material (Si/GaAs). The Gate-All-Around (GAA) Si NW is superior with a high (I_ON/I_OFF ratio) of 1.06 × 10⁹, and a low leakage current, or OFF current (I_OFF), of 3.84 × 10⁻¹⁴ A. The measured values of the mid-channel conduction band energy (E_c) and charge carrier density (ρ) at V_G = V_D = 0.5 V are −0.309 eV and 6.24 × 10²³ C/cm³, respectively. The nanowires with hydrostatic strain have been determined by electrostatic integrity and increased mobility, making them a leading solution for upcoming technological nodes. The transverse dimensions of the rectangular nanowires with similar energy levels are realized and comparisons between Si and GaAs NWs have been performed.

1. Introduction

Over the last two decades in the semiconductor industry, the advanced structure of Metal Oxide Semiconductor Field Effect Transistors (MOSFETs), from planar to multi-gate design, has been proposed to achieve great electrostatic control over the channel. Various multi-gate structural designs named Double gate, Tri-gate, Pi, Omega, Top on-one side, and Gate-all-around (GAA) devices with nanotechnology approaches have been used for forthcoming applications. The GAA device architecture has high resistance properties. It exhibits static control on the gate over the conduction of the channel, which plays a major role in avoiding short-channel effects [1,2]. Natori et al. [3] have proposed GAA NW to resist (SCEs) by improving gate length and channel length scaling (L_G < 5 nm and L_ch < 15 nm). The primary benefit of GAA devices is that they have a higher I_ON/I_OFF ratio [4]. The reduction in OFF current (I_OFF) produces a high (I_ON/I_OFF) ratio. Batakala et al. [5] have demonstrated the comparison between Si and GaAs GAA MOSFETs. The gain and current-driving capacity of the GaAs material were efficient and, thus, based on this channel material, was selected following the application field. The reduction in leakage current had been achieved by considering major features, such as smaller threshold voltage (V_th ~0.3 V), channel length (L_ch ≥ 10 nm), shorter gate length, high dielectric constant oxide material, and the dopant concentration of the source, channel, and the drain, respectively [6].

Several studies have analyzed NWs with multi-gate arrangements. Wang et al. [7] reduced the computational complexity in SiNW by the scattering effects, and an approximation of the effective mass was carried out using Buttiker probes. Arun et al. [8] have proposed SiNW and exhibit the efficiency of the electrical characteristics by oxide thickness (t_ox) variation, dopant concentration, and thickness of the silicon (t_si) in the nanometer regime, respectively. Coquand et al. [9] have presented a study of channel performance and electrostatic control for Tri-gate nanowires by reducing gate length and channel width dimensions to the optimum limit. The optimization in the dimensions results in a low Subthreshold Slope (SS) and Drain-Induced Barrier lowering (DIBL). The effects of electron confinements on thin gate cross-sections surrounded by SiO₂ surface roughness have been discussed by Ramayya et al. [10]. The electron mobility was monotonously reduced due to the surface roughness scattering effect, which was dominant.

Cresti et al. [11] have addressed the surface roughness for an electron mobility reduction in GAA and DG nanowires. The surface roughness in the device had been chosen by transfer characteristic parameters, electron density, and low-field electron mobility. Jin et al. [12] have discussed the carrier scattering and backscattering mechanisms. The backscattering decreases the flow of current, electron density, and carrier velocity when the channel’s position varies. The carrier velocity works in the same operation in both carrier scattering and backscattering. The carrier velocity in carrier scattering may increase perhaps increase more in higher order than the backscattering mechanisms.

Numerous studies have been done on the transport characteristics, charge density distributions, and doping concentrations of SiNWs, which have been discussed both empirically and theoretically [13,14,15,16]. The carrier densities for various cross-sections (square, circular, elliptical, and rectangular) using Poisson Schrodinger (PS) and Drift Diffusion (DD) simulations had been investigated with variations in the gate length and channel position being discussed [17]. The source/drain connections were kept wider than the channel to decrease access resistance. The framework for single and double-gate nano MOSFETs using the NEGF coupled mode technique for NWs with six variants had been provided by Svizhenko et al. [18]. The coupled mode effects for nonuniformity of the transverse potential profile greatly impacted multi-gate nanowires. The transport calculations and the electrical properties, such as the transmission coefficient for Si and GaAs with different orientations using NEGF mechanisms, had been discussed elaborately by Luisier et al. [19,20,21].

Using Density Functional Theory (DFT)-based techniques, the electronic characteristics of mechanical and crystalline properties of nanomaterials whose accuracies and efficiencies were reported when it is subjected to the temperature [22,23]. Kumarasinghe et al. [24] have investigated the electronic properties of pure and doped Si nanowires with dimensions up to 10 nm using large-scale Density Functional Theory (DFT) modeling. Three steps are used in modeling: (1) relaxation of the NW unit cell using the DFT method and extraction of the Hamiltonian and overlap matrices, (2) mode space transformation of the imported Hamiltonian matrices, and (3) non-orthogonal NEGF transport calculations using the mode space Hamiltonians, in a way that is self-consistent with the Poisson’s equation.

The Tight Binding (TB) methodology had been used by many researchers to address all the nanowire features that ultimately depend on the electronic structures. All varieties of nanowires and nanotubes (semiconducting, metallic, oxide, and others) were carried over in this method. The majority of research focuses on calculating the sub-band electronic structure of technological semiconductors and its relationship to factors such as shape, composition, and orientation, respectively [25].

Morioka et al. [26] have presented the electronic band structures of rectangular Si NWs using sp³d⁵s* tight-binding models. This method considers one excited s* orbital, p orbitals {p_x, p_y, and p_z}, and d orbitals {d_yz, d_zx, d_xy, d_3z²_-r², and d_x²_-y²}. The x, y, and z coordinate axes are set at [100], [010], and [001], respectively. The part of each atomic orbital typically depends on the nanowire’s width. The symmetrical character of the d₀₀₁ orbital has the same width dependence configuration of p₀₀₁, but the magnitude of orbital d₀₀₁ was found to be lesser, about nanowire based from p₀₀₁.

This work concentrates on designing a novel nanowire-based on GaAs material to use in RF hybrid applications. To improve the I_ON and I_OFF ratio, various methodologies have been utilized. The usage of high-ƙ dielectric material, such as La₂O₃, has shown various enhancements to create an optimal design to be used in RF application setups. This paper has been organized as follows. The basics of NEGF modeling have been discussed in Section 2. The proposed nanowire structure with six variants has been discussed in Section 3. The mathematical modeling of lanthanum oxide with the self-consistent methodology and the division of the 3D problem into 1D transport and 2D Schrodinger equations were investigated in Section 4. Section 5 discusses the lt and analysis of the work. Finally, Section 6 concludes the work and recommends the future aspects.

2. Basics of NEGF Modeling

The electronic properties of hydrogen-passivated compound semiconductor nanowires grown in different crystallographic orientations, specifically the band structures, band gaps, and effective electron masses, were discussed previously [27,28,29,30]. Horiguchi et al. [31] have discussed the Silicon nanowire bandgap dependency on the wire width using effective mass theory calculations and using the boundary conditions envelope between wire confinement potential and the barrier height confinement potential’s finiteness. Several authors have presented nanoscale modeling using green’s function, quantum transport modeling, density matrix calculation, and analyzing electronic devices in equilibrium conditions [32,33,34,35].

Seone et al. [36] have proposed the Gate-All-Around (GAA) Si nanowire MOSFET and the impact of current variability on the channel’s surface roughness was analyzed using 3-D real-space non-equilibrium Green’s function. Mazumder et al. [37] have proposed GAA GaAs TFET, which works under the tunneling phenomenon. The maximum I_ON/I_OFF ratio of TFET is achieved by adjusting the few electric gate insulator and GAA TFET channel architecture, which were been investigated to provide the best band-to-band tunneling and potential amplification. Montazeri et al. [38] have demonstrated the band structure for III–V compound semiconductor nanowires using k.p theory calculations. The calculation of the strain is used to determine the particular nanowire structure, and it had been employed using the elastic theory. The resulting calculated strain was called hydrostatic strain, which depends on the proportions of structural dimensions and is independent of the total size. Ren et al. [39] have modeled nanoscale MOSFETs and estimated the scattering and backscattering coefficients using the scattering theory. The critical length and carrier velocity at the source’s end and the channel’s start were identified using transport models. The ballistic, dual-gate nano transistors used for digital applications with a proper choice of the gate oxide thickness and scaling limit down to 10 nm were discussed [40,41]. Several studies have incorporated the operation of the nanowire in a ballistic regime using analytic models [42,43,44,45] and numerical simulations [46,47]. In the simulation study, the density of states, the electron density, and the conduction band energy (E_c) variations along the position of the channel were investigated [48].

3. Design of Proposed Novel GaAs Nanowire

The generic structure of the proposed nanowire has been designed with a rectangular cross-section with dimensions of 35 nm × 4.5 nm. The source and drain of the Si and GaAs-based nanowire material have a continuous n⁺ donor impurity concentration of (2 × 10²⁰ cm⁻³) and n donor doping of (1 × 10²⁰ cm⁻³) at the channel. The channel direction in this situation is longitudinal to the <001> z-axis, ‘x’ determines the channel width, and ‘y’ determines the current flow into the nanowires, as shown in Figure 1a. The electron movement in the longitudinal z direction is based on Kinetic Energy (E_z) and is called Transmission Probability T (E_z). The proposed nanowire dimensions are listed in

Table 1. Physical dimensions.

Device Parameters	Proposed Nanowire
Tox (upper) nm	1
Tox(lower) nm	1
Channel Length (Lch)	15 nm
Channel Width (Wch)	2.5 nm
Channel Height (Hch)	2.5 nm
Source Length (LS)	10 nm
Drain length (LD)	10 nm
S/D n + Donor doping (cm−3)	2 × 1020
n channel doping (cm−3)	1 × 1020
Shape	Rectangular
Transport	100
Confinement and Y direction	010

.

The Landauer formula, as in reference [3], yields the following description of the drain current:

$$I_{DS}=\frac{e}{\pi h}{\displaystyle \sum _{n_v,n}{\displaystyle \underset{0}{\overset{\infty }{\int }}d{E}_z}}T(E_z)\times \left[f(E_{FL},E)-f(E_{FR,}E)\right]$$

(1)

$$f(E_F,E)={[1+\exp ((E-E_F)/k_{B}T)]}^{-1}$$

(2)

$$E=E_{n_v,n}(z_{\max })+E_z$$

(3)

where ‘n’ is the quantum number that matches the confinement in the wire cross-section; E_FL, and E_FR (=E_FL − eV_DS), where E_FL and E_FR are the fermi energy levels at the source and drain. Equation (2) is the Fermi-Dirac Distribution. E_nv, _n in Equation (3) fits into the particular valley n_v, where n_v = 1, 2, 3, represents the energy confinement level at the top barrier on the channel as (E_top =E_FL + k_BT). In Figure 1b, the potential energy distribution is along the z-axis, where the maximum energy at the uppermost oxide interface from the channel is represented as E_max. The Landauer equation can be simplified as follows:

$$I_{DS}=\frac{ek{}_{B}T}{\pi h}{\displaystyle \sum _{n_v,n}\ln \left(\frac{1+\exp ((E_{FL}-E_{nv,n})/k_{B}T)}{1+\exp ((E_{FR}-E_{nv,n})/k_{B}T)}\right)}$$

(4)

Equation (4) determines the current I_DS where confinement energy levels at the oxide interface top barrier z_max exist. Multiple gates or very thin film structures were necessary to control SCEs in III–V technologies, as suggested previously [3]. There are six variants shown in Figure 2. Each variant differs in the number of gates and their arrangement with natural length, as shown in

Table 2. Gate variant arrangement.

Gate Variant	No. of Gates ñ	Natural Length λn
Double Gate	2	1.149
GAA	4	0.8129
Omega Gate	3.4	0.8817
Pi Gate	3.14	0.9175
Top Gate	1	1.625
Tri-gate	3	0.939

.

The natural length λ_n can be calculated by:

$${\lambda }_n=\sqrt{\frac{{\epsilon }_{si}}{\tilde{n}{\epsilon }_{ox}}\left(1+\frac{{\epsilon }_{ox}{t}_{si}}{4{\epsilon }_{si}{t}_{ox}}\right)t_{si}{t}_{ox}}$$

(5)

where ‘ñ’ is referred to as the effective number of gates. The idea was to design devices with both doped and undoped channels that use mid-gap gate material and yields the highest gate efficiencies for sub-10 nm technology [27,28]. The device’s short-channel behavior has been enhanced by raising the equivalent gate number ‘ñ’ and by maintaining the size of the gate length (approximately) five to ten times greater than that of the natural length λ_n.

Scaling is possible with GAA devices because they are built with the gate in contact with the channel on all sides. The main benefit of GAA devices is that they have a higher I_ON/I_OFF ratio. Owing to the asymmetric characteristics of the electrostatic control, the tri-gate arrangement results in a lower gate-controlled charge and is 25% smaller when compared to the GAA SiNW for the specific W/H ratio because there are more channel sides placed towards the gate contact.

When the gate voltage V_G = 0 V, the potential in three-dimensional form has been distributed out over the length of the NW, as shown in Figure 3, which is represented in the order of Double gate, GAA, Omega, Pi, Top, and Tri-gate respectively. Three different effective masses (m_l, m_t, m_t), (m_t, m_l, m_t), and (m_t, m_t, m_l) have been considered for the x, y, and z directions. The (m_l) and (m_t) are the longitudinal and transverse effective masses whose value is equal to 0.98 m₀ and 0.19 m_0. The mass (m₀) is called free electron mass.

The sub-bands in the z longitudinal direction are more energetic than those in the valley pairs along x and y whose transport mass (m_t) is less than longitudinal mass (m_l). To simulate Δ valley for the electrons near the Z [001] direction for Si, the effective masses (m_x, m_y, and m_z) obtained for valleys 1, 2, and 3 are (0.19, 0.19, and 0.98), (0.38, 0.38, and 1.17), and (0.57, 0.57, and 1.36), respectively. Similarly, the effective masses (m_x, m_y, and m_z) for GaAs NW obtained for valleys 1, 2, and 3 are (0.067, 0.067, and 0.542), (0.134, 0.134, and 0.069), and (0.201, 0.201, and 0.676), respectively.

4. Mathematical Modeling of the Nanowire with La₂O₃

The proposed multi-gate device is a 3-D-dimensional nanowire with a source and drain doping concentration of 2 × 10²⁰ cm³. The source and drain are made of silicon or gallium arsenide that has been highly doped with n⁺ atoms. The device’s effective mass Hamiltonian has been denoted by the notation:

$$H^{3D}\psi (x,y,z)=E\psi (x,y,z)$$

(6)

$$H^{3D}=-\frac{h^2}{2m_x^{\ast }}\frac{{\partial }^2}{\partial x^2}-\frac{h^2}{2}\frac{\partial }{\partial y}\left(\frac{1}{m_y^{\ast }}\frac{\partial }{\partial y}\right)-\frac{h^2}{2}\frac{\partial }{\partial z}\left(\frac{1}{m_z^{\ast }}\frac{\partial }{\partial z}\right)+V(x,y,z)$$

(7)

where the conduction band edge profile is represented by V (x, y, z) [26], and m^∗_x, m^∗_y, and m^∗_z are the effective masses:

$$V(x,y,z)=E_{C1/2}(x,y)-\varphi (x,y,z)$$

(8)

where E_C1/2 (x, y) is the band gap of the nanowire core material (Si/GaAs), the point (x, y) links to the dioxide region, and (x, y, and z) corresponds to space potential. Due to the movement of electrons in the z-direction, the effective core mass in the transport direction is m*_z, and the effective oxide masses are represented by m*_x = m*_x (x,y) and m*_y = m*_y (x,y), respectively.

The wavefunction of the three-dimensional Hamiltonian (x, y, and z) in the longitudinal z direction is given as:

$$\psi (x,y,z)={\displaystyle \sum _m{\phi }_b}(z){\psi }_b(x,y;z)$$

(9)

The bth mode eigen function ψ_b (x, y; z) represented in two-dimensional (2D) Schro dinger equation is given as:

$$H^{2D}\psi (x,y;z)=E_b(x){\psi }_b(x,y;z)$$

(10)

where

$$H^{2D}=-\frac{h^2}{2}\frac{\partial }{\partial x}\left(\frac{1}{m_x^{\ast }(x,y)}\frac{\partial }{\partial y}\right)-\frac{h^2}{2}\frac{\partial }{\partial y}\left(\frac{1}{m_y^{\ast }(x,y)}\frac{\partial }{\partial y}\right)+V(x,y;z)$$

(11)

$$H^{2D}=-\frac{h^2}{2}\frac{\partial }{\partial x}\left(\frac{1}{m_x^{\ast }(x)}\frac{\partial }{\partial x}\right)-\frac{h^2}{2}\frac{\partial }{\partial y}\left(\frac{1}{m_y^{\ast }(y)}\frac{\partial }{\partial y}\right)+V(x,y;z)$$

(12)

Under boundary conditions, the wave functions at the margins of the two-dimensional (2D) cross-section plane is known as uncoupled mode space method, which eliminates the coupling among several modes (or sub-bands), and φ_b (z) satisfies as follows:

$$\left\{-\frac{h^2}{2m_z^{\ast }}\frac{{\partial }^2}{\partial z^2}+E_b(z)\right\}{\phi }_b(z)=E{\phi }_b(z)$$

(13)

The Schrodinger Equation (13) with open boundary conditions describes the 1-D transport problem, and further, the NEGF technique [31] has been used to solve it. The primary notation for the sub-band b using 1-D Green’s function (G_b) is as follows:

$$G_b={[E-H_b^{1D}-{\displaystyle \sum S.b}-{\displaystyle \sum D.b}]}^{-1}$$

(14)

where

$$H^{1D}=-\frac{h^2}{2m_z^{\ast }}\frac{{\partial }^2}{\partial z^2}+E_b(z)$$

(15)

Σ_S,b and Σ_D,b are the S/D self-energies of sub-band b, respectively. The 1D charge density n_k^1D(z) in the bth sub-band is then obtained via:

$$n_b^{1D}(x)=\frac{1}{2\pi {\Delta }_x}{\displaystyle \int dE{f}_s}G{\Gamma }_{S.b}{G}_b^{†}+f_D{G}_k{\Gamma }_{D.b}{G}_b^{†}$$

(16)

where Δ_x is the lattice spacing, and Γ_S,b and Γ_D,b are defined by:

$${\Gamma }_{S.b}=i\left({\sum }_{S.b}-{\sum }_{S,b}^{†}\right),$$

(17)

$${\Gamma }_{D.b}=i\left({\sum }_{D.b}-{\sum }_{D,b}^{†}\right),$$

(18)

The Fermi Distribution functions and the Fermi Energies at the source and drain are given as follows:

$$f_{SD}(E)=\frac{1}{1+e^{\left(E-E_F^{SD}\right)/k_{B}T}}$$

(19)

The 3-D quantum charge density has been employed in Poisson’s equation after one-dimensional (1D) charge densities of each sub-band are resolve as follows:

$$n^{3D}(x,y,z)={{\displaystyle \sum _k{n}_b^{1D}(x)\left|{\psi }_b(x,y;z\right|}}^2$$

(20)

$${\nabla }^2\varphi (x,y,z)=-\frac{q}{\in }\left(N_D(x,y,z)-n^{3D}(x,y,z)\right)$$

(21)

Equation (21) determines the potential and doping profile (N_D) of (x, y, and z). The current in Equation (22) is calculated using the Landauer–Buttiker formula, once self-consistency and charge distributions are attained:

$$I_D=2\frac{q}{h}{\displaystyle \sum _b{\displaystyle \int dE{T}_b}}(E)(f_s(E)-f_D(E)),$$

(22)

where the Transmission Probability T_b (E) for sub-band ‘b’ is given by:

$$T_b(E)=Tr({\Gamma }_{S.b}{G}_b{\Gamma }_{D.b}{G}_b^{†})$$

(23)

4.1. Numerical Approaches

The two-dimensional (2-D) Schrodinger equations and the 1-D NEGF equation numerical solutions have been presented. The mass discontinuity across the lanthanum oxide (La₂O₃) contact has been included in the 2-D Schrodinger equation using the following k-space approach.

4.1.1. K-Space Solutions of Two-Dimensional Schrodinger Equations

Let’s first rewrite Equation (25) as follows:

$$\psi (x,y)={\psi }_b(x,y;z)$$

(24)

where A_k’s are expansion coefficients and |K〉 is a basic set. The eigenvalue problem is solved by substituting Equation (24) in Equation (10) and multiplying 〈L\ by the equation sides:

$${\displaystyle \sum _K{H}_{LK}^{2D}}{A}_K=E_b{A}_L$$

(25)

where H_LK^2DA_{K =} 〈L\H^2D|K〉. In the standard k-space solution [19]:

$$|K\rangle =\sqrt{\frac{2}{L_x}}\sqrt{\frac{2}{L_y}}\sin (k_{i}x)\sin (k_{j}y)$$

(26)

Here, L_x and L_y are the cross-side section’s lengths in the x and y directions, respectively.

$$k_i=\frac{\pi i}{L_x}(i=1,\dots N_x)$$

(27)

$$k_j=\frac{\pi j}{L_y}(j=1,\dots N_y)$$

(28)

The corresponding grid numbers in the x and y directions are N_x and N_y. It must be noted that the K index is derived with the indices i and j by the formula K = (N_x (i − 1) + j) in Equation (26).

A rectangular cross-section with core/oxide interfaces at (x₁ and x₂) and (y₁ and y₂), respectively (see Figure 4). Equation (29) defines the effective asymmetrical masses at the core/oxide interfaces for the Hamiltonian using H_LK^2D and it is given as follows:

$$H_{LK}^{2D}=H_{LK}^0+H_{LK}^{(x)}+H_{LK}^{(y)}$$

(29)

where

$$H_{LK}^0=\frac{4}{L_x{L}_y}{\displaystyle \underset{0}{\overset{L_x}{\int }}{\displaystyle \underset{0}{\overset{L_y}{\int }}dxdy\sin (k_u}}x)\sin (k_{v}y)\left(\frac{h^2{k}_i^2}{2m_x^{\ast }(x,y)}+\frac{h^2{k}_i^2}{2m_y^{\ast }(x,y)}+V(x,y)\right)\times \sin (k_{i}x)\sin (k_{j}y)$$

(30)

$$H_{LK}^{(x)}=\frac{2h^2{k}_i}{L_x{L}_y}(\sin (k_u{x}_1)\cos (k_i{x}_1)-(\sin (k_u{x}_2)\cos (k_i{x}_2))\left(\frac{1}{m_{core,x}^{\ast }}-\frac{1}{m_{ox}^{\ast }}\right)\times {\displaystyle \underset{y_1}{\overset{y_2}{\int }}dy}\sin (k_{j}y)\sin (k_{v}y)$$

(31)

$$H_{LK}^{(y)}=\frac{2h^2{k}_j}{L_x{L}_y}(\sin (k_u{y}_1)\cos (k_j{y}_1)-(\sin (k_v{y}_2)\cos (k_j{y}_2))\left(\frac{1}{m_{core,y}^{\ast }}-\frac{1}{m_{ox}^{\ast }}\right)\times {\displaystyle \underset{x_1}{\overset{x_2}{\int }}dx}\sin (k_{i}x)\sin (k_{u}x)$$

(32)

where $m_{core,x}^{\ast }$ and $m_{core,y}^{\ast }$ are the effective core masses in the x and y directions. The u and v are the indices that are mapped with L in a parallel fashion to the index K, respectively.

4.1.2. Product Space Solutions of 2-D Schrodinger Equations

From Figure 4, the rectangular cross-section with effective mass is represented as follows:

$$m_x^{*}(x,y)=\{\begin{array}{l}{m}_x^{*}(x){ \mathrm{if} \mathrm{y}}_1\le y\le y_2\\ m_{ox}^{*}{ \mathrm{if} \mathrm{y} < \mathrm{y}}_1 or{ \mathrm{y} > \mathrm{y}}_2\end{array}$$

(33)

where

$$m_x^{*}(x)=\{\begin{array}{l}{m}_{core,x}^{*}{ \mathrm{if} \mathrm{x}}_1\le x\le x_2\\ m_{ox}^{*}{ \mathrm{if} \mathrm{x} < \mathrm{x}}_1 or{ \mathrm{x} > \mathrm{x}}_2\end{array}$$

(34)

For a good approximation, it is written as follows:

$$m_x^{*}(x,y)=m_x^{*}(x) \mathrm{for} 0\le y\le L_y$$

(35)

The oxide region on either side of the core has been considered as (y₁ < y < y₂) and the amplitude in the top and bottom of the oxide regions are considered as (y < y₁ and y > y₂), respectively. The band gap of the oxide materials is substantially wider than that of the core material. Equations (36) and (37) define the effective masses in the y direction:

$$m_y^{*}(x,y)=m_y^{*}(y) \mathrm{for} 0\le x\le L_x$$

(36)

where

$$m_y^{*}(y)=\{\begin{array}{l}{m}_{core,x}^{*}{ \mathrm{if} \mathrm{y}}_1\le y\le y_2\\ m_{ox}^{*}{ \mathrm{if} \mathrm{y} < \mathrm{y}}_1 or{ \mathrm{y} > \mathrm{y}}_2\end{array}$$

(37)

The adjacent side of the oxide effective mass regions is inappropriate for the above reasons. Therefore, it is written as under good approximation:

$$H^{2D}=-\frac{h^2}{2}\frac{\partial }{\partial x}\left(\frac{1}{m_x^{\ast }(x)}\frac{\partial }{\partial x}\right)-\frac{h^2}{2}\frac{\partial }{\partial y}\left(\frac{1}{m_y^{\ast }(y)}\frac{\partial }{\partial y}\right)+V(x,y)$$

(38)

The following equation is the product-space solution:

$$|K\rangle ={\chi }_i(x){\xi }_j(y),$$

(39)

Equation (40) determines the 1D Schrodinger equation in the x direction, where χ_i is the ith Eigen function:

$$\left\{-\frac{h^2}{2}\frac{\partial }{\partial x}\left(\frac{1}{m_x^{\ast }(x)}\frac{\partial }{\partial x}\right)-\overline{V}(x)\right\}{\chi }_i(x)={\in }_i{\chi }_i(x)$$

(40)

where

$$\overline{V}(x)=\frac{1}{y_2-y_1}{\displaystyle \underset{y_1}{\overset{y_2}{\int }}dz V(x,y)}$$

(41)

ζ_j(y) is the jth eigen function and V(x,y) is the confinement potential for the following 1-D Schrodinger equation in the y-direction:

$$\left\{-\frac{h^2}{2}\frac{\partial }{\partial y}\left(\frac{1}{m_y^{\ast }(y)}\frac{\partial }{\partial y}\right)-\overline{V}(y)\right\}{\xi }_i(y)={\in }_i{\zeta }_j(y)$$

(42)

where

$$\overline{V}(x)=\frac{1}{y_2-y_1}{\displaystyle \underset{y_1}{\overset{y_2}{\int }}dz V(x,y)}$$

(43)

Substituting Equation (39) in Equation (38) and obtaining Equations (40)–(43).

$$\begin{array}{ll}{H}^{2D}|K\rangle \hfill & =-{\xi }_j(y)\frac{h^2}{2}\frac{\partial }{\partial x}\left(\frac{1}{m_x^{\ast }(x)}\frac{\partial {\chi }_i(x)}{\partial x}\right)-{\chi }_i(x)\frac{h^2}{2}\frac{\partial }{\partial y}\left(\frac{1}{m_y^{\ast }(y)}\frac{\partial {\xi }_j(y)}{\partial y}\right)+V(x,y){\chi }_i(x){\xi }_j(y)\hfill \\ \hfill & =({\in }_i+{\in }_j+V(x,y)-\overline{V}(x)-\overline{V}(y))|K\rangle \hfill \end{array}$$

(44)

By multiplying 〈L\ in Equation (44), we obtain:

$$H_{LK}^{2D}=\hspace{0.17em}{\in }_L{\delta }_{LK}+\langle L\backslash (V(x,y)-\overline{V}(x)-\overline{V}(y))|K\rangle$$

(45)

where

$${\in }_L={\in }_i+{\in }_j$$

(46)

After resolving the Schrodinger equations, the eigenvalue problem and the product-space solution have been found. Equations (40) and (42) illustrate one dimensional (1-D) version of the k-space solution approach, which was introduced in the previous section and has been employed in the modeling.

5. Analysis of the GaAs-Based Nanowire

There are six possible structures that have been considered in the simulated NWs. Two distinct materials (Si and GaAs), the Double Gate (DG), Gate-All-Around (GAA), Omega, Pi, Top, and Tri-gate variants have been discussed. Figure 2 depicts the rectangular structure with the physical dimensions of all six variants. The crystallographic orientation z <001> direction has coincided with the channel transport direction. The design parameters of the nanowire, listed in Table 1, have been considered for modeling. The conduction band margins of the NWs for different dielectrics had been addressed previously [49]. Higher gate dielectric constant materials have lower conduction band edges. The SiO₂ has a greater conduction band edge than La₂O₃ when used as a gate dielectric oxide. Thus, lanthanum oxide (La₂O₃) has been chosen as a better choice for a gate dielectric oxide and it is one of the best reasons to provide conduction at lower energies. The Si and GaAs rectangular nanowire simulations have been designed with the same wire length (L_wire = 35 nm). Figure 5 shows the comparison between the first and last state energy. The first and sixteenth energy levels of conduction band electrons in a rectangular wire differ by 22% at the left contact of the fermi level E_FL = −5 eV.

Based on the full-band model (sp³d⁵s*) model, Figure 6 illustrates transmission coefficients for the 2.5 nm wire in the conduction band. The conduction band reaches high transmission when the thickness of the wire get decreases, as shown previously [50,51].

The ballistic current has been calculated by a comparison of the transmission and energy. The transmission steps depend on the channel and have high transmission regions at an energy E = 2.6 eV. The energy differences are nearly parallel; the higher transmission obtained for both Si and GaAs nanowires are 2.8892 eV and 3.5768 at 2.6 eV. Hence, the GaAs is 1.23 times greater than Si NW. Maximum transmission can be achieved with an increase in wire dimension. Higher transmission had been achieved using different orientations with an increase in gate bias, as shown previously [20]. The transmission spectrum has been fixed with zero gate bias (V_G) and a drain voltage (V_D) of 0.6 V. When the gate voltage increases, higher transmission is achieved due to the lowering of the barrier. To normalize the current density in ballistic conditions, the effective width (W_eff) is assumed to be four times the channel width (W_ch), as shown previously [11,52].

Figure 7 shows the normalized current density spectra (i_z/i_z, avg) calculated by (T × (f_L − f_R)), where T is the transmission and (f_L − f_R) are the left and right fermi level contacts. The normalized current density distribution is uniform in the GaAs NW, and this uniformity occurs when the Wagner number (Wa > 5), as shown previously [53].

The one-dimensional electron density (N_1D) along the channel has been plotted against Si and GaAs NW. The comparison has been noticed specifically at the midchannel ‘z’. The NEGF calculations are made to compute electron density and the electrostatic potential at the interface. At zero gate bias, there is no creation of a potential barrier and electrons to penetrate the channel. The electron density (N_1D ~ 1 × 10²⁰) cm⁻³ has been obtained in the OFF state when V_G = 0 V at the source and decreases more at the midchannel. The electron density increases at the midchannel due to three reasons: (a) Surface Roughness, (b) Higher Gate bias voltage (V_G > 0.3 V), or (c) when channel doping is greater than source-drain doping. Here, the middle of the channel has a low electron concentration, which maintains a higher concentration at the drain. The electron densities are uniform throughout the height of the channel, and the GaAs have attained a higher electron concentration at the midchannel than Si NW, as shown in Figure 8.

The conduction band energy depends upon the function of both y and z, which is a function of width and length, whereas the sub-band energy minima depends on length [33]. Using the relationship with the carrier velocity, it is concluded that the frequency of electron transmission and channel length are inversely proportional with each other. Thus, the saturation current (I_ON) increases when the channel length gets reduced. The carriers can travel more easily through shorter gate lengths and channel lengths in comparison to a longer channel, as shown previously [45]. The conduction band edge profiles for the GAA variant by fixing V_D = 0.5 V and V_G have been varied between 0 and 1, as shown in Figure 9. The device gets off at low gate voltages. When gate voltage increases, the potential barrier gets lower, and the energy attained by the electrons will move faster from source to drain and gets lowered with an increase in drain bias.

The conduction band energy decreases at the midchannel when gate voltage V_G increases from 0 to 1 V. Each band energy differs with a voltage of 0.1 V. Due to a higher impurity concentration than that of the channel, a sudden peak charge density (ρ) of 6.75 × 10²⁶ Coul.m⁻³ was produced at the source and drain when V_G = 0 V. With the increase in gate voltage V_G = 0.5 V, the charge density (ρ) over the length of the channel decreases from 6.48 × 10²⁶ Coul.m⁻³ to 6.24 × 10²⁶ Coul.m⁻³ from the source to the midchannel and then increases when it reaches near the drain terminal, as shown in Figure 10. The charge density increases at the midchannel when gate voltage increases from 0 to 1 V with voltage difference of 0.1 V.

Table 3. Comparison results of Si and GaAs Tri-gate NW.

Parameter	Si NW	GaAs NW
IDmax(A)	6.08 × 10−8	8.20 × 10−8
Transmission	2.889	3.576
Normalized Current Density	1	1.06 × 10−6
Electron Density, N1D(#/cm3)	1.76 × 1020	1.98 × 1020
ION (A)	5.6 × 10−13	8.21 × 10−7
IOFF (A)	3.75 × 10−23	2.62 × 10−15
ION/IOFF	1.49 × 1010	3.13 × 108
Subthreshold Slope SS (mV/decade) at VG = 0.1 V to 0.6 V	8 mV to 107 mV	8 mV to 99 mV

shows the comparison modeling results of Si and GaAs Trigate NW. Though the geometrical dimensions are identical for Si and GaAs NW, the accumulation of electrons in GaAs NW is 11% more when it varies with gate voltage when compared to Si NW. Hence, it is evident from the results that the increase in electron density of GaAs NW results in a decrease in current density, which shows that the current density depends upon the property and nature of the material and is independent of electron density. The inversion charge shifts away from the interface in the charge on the quantum modulation effect on Si and GaAs NW, which has been demonstrated to have no impact on the Subthreshold Slope (SS), as shown previously [54]. The log-scale (I_D-V_GS) transfer curve with V_GS at the subthreshold region has been used to calculate the Subthreshold Slope (SS), which is defined as −[d(log₁₀I_D)/dV_G].

The transfer characteristic curve for Si NW is shown in Figure 11. The GAA device has the highest ON current of 4.09 × 10⁻⁵ A. Our simulation results of 15 nm Si NW are compared with GaAs NW. Figure 12 shows the transfer characteristics curve for Si and GaAs Tri-gate NW. Here, the I_ON current of the GaAs Tri-gate nanowire is 10⁻⁷ A (V_S = 0 V and V_D = 0.6 V), and with the silicon nanowire is 10⁻¹³ A (V_S = 0 V and V_D = 0.6 V). This analysis shows that the gallium arsenide nanowire, due to its larger I_ON current, has more advantages over other types of devices. The results of I_ON, I_OFF, and I_ON/I_OFF are identical values in the simulation results of the Double Gate and Omega variants, as shown clearly in

Table 4. Comparison results of Si NW variants.

Parameter	DG	GAA	OMEGA	PI	TOP	TRI
ION (A)	4.04 × 10−5	4.09 × 10−5	4.04 × 10−5	3.46 × 10−5	1.99 × 10−5	3.38 × 10−7
IOFF (A)	4.35 × 10−14	3.84 × 10−14	4.35 × 10−14	3.62 × 10−13	5.03 × 10−8	5.64 × 10−13
ION/IOFF	0.92 × 109	1.06 × 109	0.92 × 109	0.95 × 108	0.95 × 103	0.6 × 106
Conduction band Ec in eV at 15 nm	−0.310	−0.309	−0.310	−0.330	−0.424	−0.334
Charge density ρ (Coul.m−3) at 15 nm	6.42 × 1023	6.24 × 1023	6.42 × 1023	1.01 × 1024	4.6 × 1024	1.01 × 1024
Normalized current density	1.03 × 1029	1.08 × 1029	1.03 × 1029	1.06 × 1029	1.07 × 1029	1.08 × 1029
Subthreshold slope (mV)	106	176	106	107	115	108
No. of nodes	9	9	10	10	10	9
No. of iterations	21	21	21	16	16	15
NEGF simulation time in secs	0.277	0.159	0.278	0.250	0.173	0.543
Schrodinger simulation time in secs	0.323	0.333	0.365	0.369	0.348	0.879
Poisson simulation time in secs	0.666	0.582	0.690	0.758	0.711	4.061

comparison results.

It was observed previously [55] that when the gate length increases to 35 nm, the I_ON/I_OFF ratio remains high in GAA NWs compared to all other gates. The Omega and the Double gate NWs modeling results remain the same and high when compared to the Ω -gate MoS₂FET [56,57], which are shown in Table 4. The GAA has a smaller leakage current than any other gate and a higher conduction band energy of 8% and 37% (at 15 nm) than the Tri-gate and Pi gate. Thus, the GAA Si NW also shows a good Subthreshold Slope (SS) of 176 mV/dec, which is 39% greater than the Tri-gate NW. Therefore, the GAA device has been chosen as a better electrostatic control device.

Our proposed method has been compared with a previous study [11], where channel length (L_ch = 15 nm) and oxide thickness (t_ox = 1 nm) are the same. Hence, it has been observed that the increase in the gate length (L_G ≥ 15 nm) and silicon dioxide material (SiO₂) material leads to an increase in leakage current. Thus, in our proposed method, the OFF current (I_OFF) has been reduced by various parameters, such as (1) oxide material with high dielectric constant, (2) gate length scaling, and (3) low threshold voltage.

Table 5. Comparison of various Si nanowires with conventional nanowires.

Nanowire	Oxide (nm)	Lch (nm)	LG (nm)	VGS, VDS (V)	ION (A)	IOFF (A)	ION/IOFF
[4]	HFO2	5	20	(0.6,0.6)	4.5 × 10−5	0.1 × 10−6	4500
[6]	SiO2	15	5	(1,0.5)	1 × 10−5	8.8 × 10−16	0.11 × 1011
[8]	SiO2	11	10	(1,1)	0.87 × 10−3	3.4 × 10−12	0.25 × 109
[12]	SiO2	15	15	(0.3,0.5)	1.22 × 10−5	3 × 10−11	0.40 × 106
[36]	SiO2	6	6	(0.6, 0.001)	1 × 10−6	1 × 10−13	1 × 107
[49]	La2O3	10	10	(1,0.5)	5.5 × 103	0.84 × 10−8	6.54 × 1011
[51]	SiO2	5	6	(0.6,0.6)	9.8 × 10−7	3.4 × 10−13	2.88 × 106
This work	La2O3	15	2	(1, 0.6)	4.09 × 10−5	3.84 × 10−14	1.06 × 109
This work	La2O3	15	2	(0.6, 0.6)	2.11 × 10−6	3.84 × 10−14	0.55 × 108

summarizes the Si NW GAA variant with existing SiNW for an S/D doping concentration of 2 × 10²⁰ cm⁻³. According to modeling results, the GAA NW has achieved a high I_ON/I_OFF ratio (1.06 × 10⁹) when the width-to-height (W/H) ratio dimensions are equal to 1. For fixed gate and drain voltages, the small (W/H) ratio changes in the geometrical dimensions result in a low I_ON/I_OFF ratio, a high electron density, Subthreshold Slope (SS), and Drain Induced Barrier Lowering (DIBL). The smooth and rough surface in the channel also differs between ON and OFF currents. However, the I_ON/I_OFF ratios are the same in both cases, the smooth surface produces a high ON current (1 × 10⁻⁶ A) as in Ref. [32].

6. Conclusions and Future Recommendations

The electrical characteristics of the NWs shielded with Lanthanum Oxide (La₂O₃) material and the orientation with [001] z using the Non-Equilibrium Green Function (NEGF) method were analyzed. Using the NEGF technique, the performance of Silicon and Gallium Arsenide Nanowires with multi-gate structural arrangements, the electrical characteristics, and their parameters are computed. The comparison between all the nanowire variants was simulated. The semi-empirical tight-binding technique (sp³d⁵s*) was used to determine the transmission coefficient of Silicon and Gallium Arsenide nanowires for [001] orientations. The transverse dimensions of rectangular nanowires with similar energy levels have been examined, and the comparisons between Silicon and Gallium Arsenide NWs were investigated. The III–V compound semiconductor, such as GaAs NW, shows an attractive simulation in a few parameter results, such as transmission and electron density, compared to Silicon NW. Considering the issue of leakage current reduction, Silicon NWs are more suitable than Gallium Arsenide NWs.

In future work, the comparison between the same wire (Si or GaAs) with different orientations and the same orientations for different materials (Si and GaAs) should be investigated. The problems solved by Gallium Arsenide have focused on III–V compounds along with Silicon or IV–IV compounds, which could be used for applications such as energy storage, flexible electronics, and biomedical devices. Additionally, the development of new synthesis techniques may lead to the production of nanowires with novel compositions and improved properties.