# Compact Trap-Assisted-Tunneling Model for Line Tunneling Field-Effect-Transistor Devices
## 2 Model Presentation
### Compact Implementation  
For compact implementation, \(G_{\text{TAT}}\) given by (9) is approximated using a Gaussian distribution function as follows:  
\[G_{TAT}=G_{TAT\_max}exp\left(-\left(\frac{x-x_{max}}{x_{width}}\right)^{2}\right) \tag{16}\]  
where \(x_{\text{width}}\) is the width of the Gaussian distribution. \(G_{\text{TAT\_max}}\) is the maximum \(G_{\text{TAT}}\) along the channel region, and \(x_{\text{max}}\) is the \(x\)-point where \(G_{\text{TAT}}\) is maximum. \(G_{\text{TAT\_max}}\) always occurs where \(n=p\). This can be seen in Figure 3 which shows \(n\), \(p\) (left axis), and \(G_{\text{TAT}}\) (right axis) at different \(V_{\text{gs}}\)s for \(V_{\text{ds}}\)=0.5 V, for an LTFET with \(T_{\text{j}}\) = 10 nm. It can be seen from Figure 3 that at the point where \(n=p\), \(G_{\text{TAT}}\) is maximum. The \(x\) co-ordinate where this occurs is labelled as \(\text{x}_{\text{max}}\). \(x_{\text{max}}\) can be found by equating \(n=p\), as follows: and solving for \(x\), that is. \[n_{i}exp\left(\frac{mx_{max}+\varphi_{s}-\varphi_{quasi}}{v_{th}}\right)=n_{i }exp\left(-\frac{mx_{max}+\varphi_{s}}{v_{th}}\right) \tag{17}\]  
\[x_{max}=\frac{\varphi_{quasi}-\left(2\varphi_{s}\right)}{2m} \tag{18}\]  
\(G_{\text{TAT\_max}}\) is found by using \(x_{\text{max}}\) in (9), as follows:  
\[G_{TAT\_max}=\frac{n(x_{max})p(x_{max})-n_{i}^{2}}{\frac{\tau_{\text{op}}}{1+ \Gamma_{p}}[n(x_{max})+n_{1}]+\frac{\tau_{0n}}{1+\Gamma_{n}}[p(x_{max})+p_{1}]} \tag{19}\]where \(n(x_{\rm max})\) and \(p(x_{\rm max})\) can be found using \(x=x_{\rm max}\) in (11). It should be mentioned that \(\Gamma\) is independent of \(x\); constant electric field in the channel makes \(\Gamma\) calculated from (12) essentially independent of \(x\). In order to find \(x_{\rm width}\), the standard deviation (\(\sigma\)) of the Gaussian distribution is utilized. \(\sigma\) is obtained using standard equations [21]. Utilizing the Empirical Rule in Statistics and Probability [21] which states that 99.7% of values in a normal distribution lie within 3 standard deviations of the mean value [21], the width of Gaussian distribution was approximated at 3 standard deviations of the maximum value of the function, that is, the width was estimated to be the \(x\) co-ordinate (xwidth) that satisfied this condition, 3\(\sigma\)_of_\(G_{TAT}\), symbolized by \(G_{TAT\_3\sigma}\). Substituting this condition in (9), along with (11), assuming mid-gap trap level so that \(n_{1}=p_{1}=n_{i}\), (9) can be given by  
\[G_{TAT\_3\sigma}\ =\ \frac{n_{i}^{2}exp\left(\frac{-\varphi_{\rm quasi}}{v_{ \rm th}}\right)-n_{i}^{2}}{\frac{\tau_{0p}}{1+\Gamma_{p}}(n+n_{i})+\frac{\tau_ {0p}}{1+\Gamma_{n}}(p+n_{i})} \tag{20}\]  
It can be seen from Figures 3 and 4 that at any bias, the width of the Gaussian distribution \(n>>p\). Furthermore, assuming equivalent \(\tau_{0n}=\tau_{0p}=\tau_{0}\), and \(\Gamma_{n}\)-\(\Gamma_{p}\) which is true for the materials with \(m_{\rm e}\)-\(m_{\rm p}\) including silicon, and advanced III-V InGaAs-based materials [13], (20) can be expressed as  
\[G_{TAT\_3\sigma}\ =\ \frac{n_{i}^{2}exp\left(\frac{-\varphi_{\rm quasi}}{v_{ \rm th}}\right)-n_{i}^{2}}{\frac{\tau_{0}}{1+\Gamma_{n}}\left[n_{i}\left(exp \left(\frac{m_{\rm width}+\varphi_{\rm s}-\varphi_{\rm quasi}}{v_{\rm th}} \right)+2\right)\right]} \tag{21}\]  
Figure 3: (**a–d**) \(n\), \(p\) (left-axis), and \(G_{\rm TAT}\) (right-axis) at \(V_{\rm gs}=0.4\), 0.8, 1.2, and 1.6 V, respectively, all at \(V_{\rm ds}=0.5\) V. The location where \(n=p\), and of \(G_{\rm TAT\_Max}\), that is at \(x=x_{\rm max}\) are identified by arrows.