# Compact Trap-Assisted-Tunneling Model for Line Tunneling Field-Effect-Transistor Devices
## 2 Model Presentation
### Schenk Model  
The Schenk model [7; 10] works by modifying the lifetime parameter (\(\tau_{0}\)) in the original Shockley-Read-Hall (SRH) recombination-generation model through the \(\Gamma\) parameter as follows:  
\[\tau_{n(p)}=\frac{\tau_{0n(p)}}{1+\Gamma_{n(p)}(E)} \tag{8}\]  
where \(n(p)\) in (8) represent electrons and holes, respectively. The \(\Gamma\) accounts for the \(E\)-driven, and phonon-assisted TAT process, and is calculated by the Schenk model. If there is any TAT present, \(\Gamma\) is usually significantly higher than 1 [5]. Otherwise, it is zero. The lifetime given by (8) features in the SRH model expression, which is used to give \(G_{\text{TAT}}\) if \(\Gamma>0\), as follows:  
\[G_{TAT}=\frac{np-n_{i}^{2}}{\frac{\tau_{0p}}{\Gamma+\Gamma_{p}}(n+n_{1})+\frac {\tau_{0p}}{1+\Gamma_{n}}(p+p_{1})} \tag{9}\]  
where \(n_{i}\) is an intrinsic carrier concentration, \(n_{1}=n_{i}exp(E_{T}/kT)\), \(p_{1}=n_{i}exp(E_{T}/kT)\), and \(k\) and \(T\) are the Boltzmann constant and temperature, respectively. \(n\) and \(p\) in (9) are given by  
\[n=n_{i}exp\bigg{(}\frac{\varphi(x)-\varphi_{\text{equasi}}}{v_{th}}\bigg{)}, \ p=n_{i}exp\bigg{(}-\frac{\varphi(x)}{v_{th}}\bigg{)} \tag{10}\]  
where \(v_{\text{th}}\) and \(\varphi_{\text{equasi}}\) present thermal voltage and electron quasi Fermi level, respectively. Substituting for \(\varphi_{\text{channel}}\) from (6) in (10), \(n\) and \(p\) are given by  
\[n(x)=n_{i}exp\bigg{(}\frac{mx+\varphi_{s}-\varphi_{\text{quasi}}}{v_{th}} \bigg{)},\ p(x)=n_{i}.exp\bigg{(}-\frac{mx+\varphi_{s}}{v_{th}}\bigg{)}. \tag{11}\]  
\(\Gamma\) is calculated by the following equation. A detailed derivation of this equation is given in [7]. \[ \begin{split}\Gamma_{n(p)}=&\left(1+\frac{\left( \hbar\Theta_{n(p)}\right)^{\frac{3}{2}}\sqrt{E_{n(p)}-E_{n(p)}}}{E_{n(p)}\hbar \omega_{0}}\right)^{-\frac{1}{2}}\frac{\left(\hbar\Theta_{n(p)}\right)^{\frac{ 3}{2}}\left(E_{n(p)}-E_{n(p)}\right)^{\frac{1}{4}}}{2\sqrt{E_{n(p)}-E_{n(p)}} }\bigg{(}\frac{\hbar\Theta_{n(p)}}{kT}\bigg{)}^{\frac{3}{2}}.\\ exp\bigg{(}-\frac{E_{n(p)}-E_{n(p)}}{\hbar\omega_{0}}+\frac{ \hbar\omega_{0}-kT}{2\hbar\omega_{0}}+\frac{2E_{n(p)}+kT}{2\hbar\omega_{0}} \ln\frac{E_{n(p)}}{\epsilon_{R}}-\frac{E_{n(p)}}{\hbar\omega_{0}}\ln\frac{E_{ n(p)}}{\epsilon_{R}}+\frac{E_{n(p)}-E_{n(p)}}{kT}-\frac{4}{3}\bigg{(}\frac{E_{n(p)}-E_{n(p)} }{\hbar\Theta_{n(p)}}\bigg{)}^{\frac{3}{2}}\bigg{)}.\end{split}  \tag{12}\] \begin{table}
\begin{tabular}{c l l} \hline \hline
**Symbol** & \multicolumn{1}{c}{**Description**} & \multicolumn{1}{c}{**Value/Unit**} \\ \hline \(\varphi_{\text{source}}\) & Potential at source edge & V \\ \hline \(N_{a}\), \(N_{d}\) & Source, channel doping & cm\({}^{-3}\) \\ \hline \(x_{\text{source}}\), \(x_{\text{channel}}\) & \(x\) co-ordinate in source, channel & cm \\ \hline \(L_{\text{dep}}\) & Depletion length in source & cm \\ \hline \(\varepsilon_{sl}\), \(\varepsilon_{ox}\) & Silicon, oxide permittivity & \(11.9_{c0}\), \(25_{c0}\) \\ \hline \(V_{\text{fb}}\), & Flat band voltage & V \\ \hline \(q\) & Electron charge & \(1.6\times 10^{-19}\) C \\ \hline \(\varphi_{\text{s}}\varphi_{\text{i}}\varphi_{\text{channel}}\) & Surface potential at \(x_{\text{channel}}=0\), junction potential at \(x=T_{\text{j}}\), and potential as a function of \(x_{\text{channel}}\), respectively & V \\ \hline \(\varphi_{\text{dep}}\) & Depletion potential in source & V \\ \hline \(C_{\text{ox}}\) & Gate capacitance & F/cm\({}^{2}\) \\ \hline \(V_{\text{bis}}\) & Channel/source junction built-in potential & V \\ \hline \hline \end{tabular}
\end{table} 
Table 2: List of symbols specifically used in (1)–(7). Here, \(\hbar\) is the reduced Planck's constant, \(\hbar\omega_{0}\) is the effective phonon energy (\(E_{\text{phon}}\)), \(\varepsilon_{\text{R}}=SE_{\text{phon}}\) is the lattice relaxation energy [17] which is known to influence the subthreshold slope (SS), where \(S\) is the Huang-Rhys factor [18] which is a measure of electron-phonon coupling.