# Compact Trap-Assisted-Tunneling Model for Line Tunneling Field-Effect-Transistor Devices
## 2 Model Presentation
### Potential Model  
Figure 2a shows the TAT generation rate (\(G_{\text{TAT}}\)) contour plot at \(V_{\text{gs}}=0\) V and \(V_{\text{ds}}=0.5\) V for an LTFET with \(T_{\text{j}}=10\) nm. Figure 2b shows the \(G_{\text{TAT}}\) extracted from the contour plot along the cutline indicated in Figure 2a, along with \(G_{\text{TAT}}\) at several \(V_{\text{gs}}\) biases ranging from 0 to 2 V. The region shown along the cutline is the reference region of the model, where all parameters including potential and \(G_{\text{TAT}}\) are calculated and assumed to be constant in the \(y\) and \(z\) directions. It can be seen that the \(G_{\text{TAT}}\) is significant only in the channel region, and for that reason, this model calculates \(G_{\text{TAT}}\) only in the channel region.  
\begin{table}
\begin{tabular}{c c c} \hline
**Mechanism** & **Region Dominated** & **Threshold Voltage** \\ \hline
1D BTBT & Typically subthreshold region. & \\ \hline TAT & Typically subthreshold and intermediate bias regions [5], and in extreme cases [13] the high bias region as well. & TAT \(<\) 2D BTBT \(<\) 1D BTBT \\ \hline \end{tabular}
\end{table}
Table 1: Current mechanisms in line tunneling TFETs and their threshold voltages.  
Figure 2: \(Cont.\)  
The potential model is based on solving the Poisson equation twice in the source and \(R_{\text{overlap}}\) regions. As a result, the potential and electric field (\(E\)) equations in the respective regions are given below [16]. Table 2 mentions the meaning of each symbol used in the following equations:  
\[\frac{\partial\varphi_{source}}{\partial x_{source}}=\frac{qN_{a}}{\varepsilon_{ si}}\Big{(}x_{source}-L_{dep}\Big{)} \tag{1}\]  
\[\frac{\partial\varphi_{channel}}{\partial x_{channel}}=-\frac{qN_{d}}{ \varepsilon_{si}}x_{channel}-\frac{\varepsilon_{ox}}{\varepsilon_{si}\alpha_{ ox}}\Big{(}V_{gs}-V_{fb}-\varphi_{s}\Big{)} \tag{2}\]  
\[\varphi_{source}(x_{source})=\frac{qN_{a}}{2\varepsilon_{si}}\Big{(}x_{source} -L_{dep}\Big{)}^{2}+\varphi_{dep} \tag{3}\]  
\[\varphi_{channel}(x_{channel})=-\frac{qN_{d}}{2\varepsilon_{si}}x_{channel}^ {2}-\frac{\varepsilon_{ox}}{\varepsilon_{si}\alpha_{ox}}\Big{(}V_{gs}-V_{fb}- \varphi_{s}\Big{)}x_{channel}+\varphi_{s} \tag{4}\]  
\[\varphi_{s}=V_{gs}-V_{fb}+\frac{q(N_{a}+N_{d})}{C_{ox}}T_{j}+\frac{q\varepsilon _{si}N_{a}}{C_{ox}^{2}} \tag{5}\]  
\[-\gamma\sqrt{V_{gs}-V_{fb}+\frac{q(N_{a}+N_{d})}{2\varepsilon_{si}}T_{j}^{2} +\frac{q(N_{a}+N_{d})}{C_{ox}}T_{j}+\frac{q\varepsilon_{si}N_{a}}{2C_{ox}^{2} }-\varphi_{dep}} \tag{6}\]  
where \(\gamma=\sqrt{2_{si}qN_{d}}/C_{ox}\). The potential profile within the channel is assumed to be linear.  
\[\varphi_{channel}(x_{channel})=mx_{channel}+\varphi_{s} \tag{7}\]  
where \(m\) is the slope of the linear potential profile in the channel, \(m=(\varphi_{s}-\varphi_{j})/T_{j}\), or the electric field. Equation (7) is not derived from (2). It is a convenient assumption for compact modeling purposes. It can be seen in Figure 2c that potential follows a linear potential profile within the channel. Due to the strong band-bending, this assumption quasi-breaks down at higher \(V_{gs}\) bias. This consequence is explained later in Section 3. Because of the linear potential profile, \(E\) is constant throughout the channel and \(\varphi_{j}\) can be derived as  
\[\varphi_{j}=\varphi_{s}+\frac{qT_{j}^{2}(N_{d}+2N_{a})}{2\varepsilon_{si}}- \sqrt{\frac{\Big{(}2 qN_{a}T_{j}^{2}\Big{)}(\varphi_{s}+V_{bis})}{\varepsilon _{si}}}+\frac{q^{2}T_{j}^{4}\Big{(}N_{d}N_{a}+N_{a}^{2}\Big{)}}{\varepsilon_{ si}^{2}} \tag{8}\]