A unified compact model for electrostatics of III–V GAA transistors with different geometries

Abstract

In this work, a physics-based unified compact model for III-V GAA FET electrostatics is proposed. The model considers arbitrary cross-sectional geometry of GAA FETs viz. rectangular, circular and elliptical. A comprehensive model for cuboid GAA FETs is developed first using the constant charge density approximation. The model is then combined with the earlier developed model for cylindrical GAA FETs to have a unified representation. The efficacy of the model is validated by comparing it with simulation data from a 2D coupled Poisson-Schrödinger solver. The proposed model is found to be, (a) accurate for GAA FETs with different geometries, dimensions and channel materials and (b) computationally efficient.

Appendices

Appendix 1

According to the CCDA, the $|\varPsi _{i,j}(x,y)|^2$ can be approximated, to satisfy the condition of normalization as,

$$\begin{aligned} \int _{-W/2}^{W/2}\int _{-H/2}^{H/2}|\varPsi _{i,j}(x,y)|^2{\text {d}}x{\text {d}}y=1 \end{aligned}$$

(24)

Equation (24) is written with the assumption that the entire wavefunction lies within the semiconductor channel. Neglecting the wavefunction penetration in the oxide for the normalization of the wavefunction has little impact on the physics that are more sensitive to the calculation of the quantized energy levels. The finite height of the semiconductor-dielectric barrier is considered in the model via the values of $E_{i,j}$. Since with CCDA, n(x,y) is assumed to be constant within the semiconductor channel, $|\varPsi _{i,j}(x,y)|^2$ will also constant.

Evaluating (24) $|\varPsi _{i,j}(x,y)|^2$ can be written as,

$$\begin{aligned} |\varPsi _{i,j}(x,y)|^2 = \frac{1}{WH} \end{aligned}$$

(25)

Appendix 2

Equation (7) with CCDA can be written as,

$$\begin{aligned} \frac{\partial ^2\varPhi (x,y)}{\partial x^2}+\frac{\partial ^2\varPhi (x,y)}{\partial y^2} = -\frac{Q_s}{\varepsilon _s WH} \end{aligned}$$

(26)

Since the source term $-Q_s/\varepsilon _s WH$ is a constant, $\varPhi (x,y)$ can be assumed to be of the form $\varPhi (x,y) = Ax^2 + By^2 + Cx + Dy + E$. Substituting in (26),

$$\begin{aligned} A+B = -\frac{Q_s}{2\varepsilon _s WH} \end{aligned}$$

(27)

The boundary conditions for $\varPhi (x,y)$ in a III-V GAA FETs are

$$\begin{aligned}&\frac{\partial \varPhi (x,y)}{\partial x} \bigg |_{x=0} = 0 \end{aligned}$$

(28)

$$\begin{aligned}&\frac{\partial \varPhi (x,y)}{\partial y} \bigg |_{y=0} = 0 \end{aligned}$$

(29)

$$\begin{aligned}&\varPhi (x,y) \bigg |_{x=0,y=0} = \varPhi _c \end{aligned}$$

(30)

Also,

$$\begin{aligned}&\frac{1}{H}\int_{-H/2}^{H/2}\varPhi (-W/2,y){\text {d}}y = \frac{1}{W}\int_{-W/2}^{W/2}\varPhi (x,-H/2){\text {d}}x \end{aligned}$$

(31)

Using the above conditions and analytical solution of $\varPhi (x,y)$ can be written as,

$$\begin{aligned} \varPhi (x,y) = -\frac{Q_s}{2\varepsilon _sW^2+H^2}\left[ \frac{H}{W}x^2+\frac{W}{H}y^2\right] + \varPhi _c \end{aligned}$$

(32)

Appendix 3

The perturbation term can be written as,

$$\begin{aligned} \varDelta Ee_{i,j}\,=\,<\varPsi ^*_{i,j}|\widetilde{\varPhi }|\varPsi _{i,j}>\end{aligned}$$

(33)

where the perturbing potential is given by, $\widetilde{\varPhi }(x,y) = \varPhi (x,y) - \varPhi _c$. Here $\varPhi _c$ is used as the reference to calculate $\hbox {Eg}_{\mathrm{i,j}}$. Using (11) $\widetilde{\varPhi }$(x,y) can be written as,

$$\begin{aligned} \widetilde{\varPhi }(x,y) = -\frac{Q_s}{2\varepsilon _sW^2+H^2}\left[ \frac{H}{W}x^2+\frac{W}{H}y^2\right] \end{aligned}$$

(34)

Using (34), (33) can be written as,

$$\begin{aligned}&\varDelta Ee_{i,j} = \\\nonumber &-q\int_{-W/2}^{W/2}\int_{-H/2}^{H/2} |\varPsi _{i,j}(x,y)|^2\left( -\frac{Q_s}{2\varepsilon _sW^2+H^2} \left[ \frac{H}{W}x^2+\frac{W}{H}y^2\right] \right) {\text {d}}x{\text {d}}y \end{aligned}$$

(35)

Substituting $|\varPsi _{i,j}(x,y)|^2 = 1/WH$ and integrating (35), the perturbation term can be derived as,

$$\begin{aligned} \varDelta Ee_{i,j} = \frac{qQ_sWH}{12\varepsilon _s(W^2+H^2)} \end{aligned}$$

(36)