## 3 Numerical Solution Scheme
\begin{split} r_{i}=T_{i}^{k}+&(1-\psi)\frac{\Delta t }{(C_{e-h})_{i}^{k}}f_{i}^{k}+\psi\frac{\Delta t}{(C_{e-h})_{i}^{k+1}}\times \\ &\Bigg{(}S_{i}^{k+1}-(E_{g})_{i+\frac{1}{2}}^{k+1}\times\frac{l_{ i}^{k+1}}{\Delta z}+(E_{g})_{i-\frac{1}{2}}^{k+1}\times\frac{l_{i-1}^{k+1}}{\Delta z} \\ &+\frac{(C_{e-h})_{i}^{k+1}}{(\tau_{e-p})_{i}^{k+1}}(T_{a})_{i}^ {k+1}-(\frac{\partial n}{\partial t})_{i}^{k+1}(E_{g})_{i}^{k+1}\\ &-n_{i}^{k+1}\left\{\left(\frac{\partial E_{g}}{\partial n}\right) _{i}^{k+1}\left(\frac{\partial n}{\partial t}\right)_{i}^{k+1}+\left(\frac{ \partial E_{g}}{\partial T_{a}}\right)_{i}^{k+1}\left(\frac{\partial T_{a}}{ \partial t}\right)_{i}^{k+1}\right\}\Bigg{)}\end{split}  \tag{42}\]for \(i=2,...,N-1\), and the boundary conditions are  
\[ \begin{split} b_{1}=1-\psi&\frac{\Delta t}{(C_{e-h})_{1 }^{k+1}}\times\Bigg{(}-\frac{k_{B}}{\Delta z}\left[H_{0}^{1}(\eta_{e})+H_{0}^{ 1}(\eta_{h})\right]_{1}^{k+1}\times J_{1}^{k+1}\\ &-\frac{(k_{e}+k_{h})_{\frac{3}{2}}^{k+1}}{(\Delta z)^{2}}-(C_{e- h})_{1}^{k+1}/(\tau_{e-p})_{1}^{k+1}\\ &-\frac{3}{2}k_{B}(\frac{\partial n}{\partial t})_{1}^{k+1}\left( H_{\frac{1}{2}}^{\frac{3}{2}}(\eta_{e})+H_{\frac{1}{2}}^{\frac{3}{2}}(\eta_{h}) \right)_{1}^{k+1}-\frac{3}{2}k_{B}n_{1}^{k+1}\left(\frac{\partial n}{\partial t }\right)_{1}^{k+1}\times\\ &\Bigg{\{}\left(\frac{\partial\eta_{e}}{\partial n}\right)_{1}^{k +1}\left[1-\left(H_{\frac{1}{2}}^{\frac{3}{2}}(\eta_{e})\right)_{1}^{k+1} \left(H_{\frac{1}{2}}^{-\frac{1}{2}}(\eta_{e})\right)_{1}^{k+1}\right]\\ &+\left.\left(\frac{\partial\eta_{h}}{\partial n}\right)_{1}^{k+1 }\left[1-\left(H_{\frac{1}{2}}^{\frac{3}{2}}(\eta_{h})\right)_{1}^{k+1}\left( H_{\frac{1}{2}}^{-\frac{1}{2}}(\eta_{h})\right)_{1}^{k+1}\right]\right\}\Bigg{)},\end{split}  \tag{43}\]  
\[ \begin{split} c_{1}=-\psi&\frac{2\Delta t}{\Delta z (C_{e-h})_{1}^{k+1}}\left(-k_{B}[H_{0}^{1}(\eta_{e})+H_{0}^{1}(\eta_{h})]_{ \frac{3}{2}}^{k+1}\times J_{1}^{k+1}+(k_{e}+k_{h})_{\frac{3}{2}}^{k+1}/\Delta z \right),\end{split}  \tag{44}\]  
\[ \begin{split} r_{1}=& T_{1}^{k}+(1-\psi)\Delta tf_{1 }^{k}+\psi\frac{\Delta t}{(C_{e-h})_{1}^{k+1}}\times\\ &\Bigg{(}S_{1}^{k+1}-2(E_{g})_{\frac{3}{2}}^{k+1}\times\frac{J_{ 1}^{k+1}}{\Delta z}+\frac{(C_{e-h})_{1}^{k+1}}{(\tau_{e-p})_{i}1^{k+1}}(T_{a}) _{1}^{k+1}-\left(\frac{\partial n}{\partial t}\right)_{1}^{k+1}(E_{g})_{1}^{k +1}\\ &-n_{1}^{k+1}\left\{\left(\frac{\partial E_{g}}{\partial n}\right) _{1}^{k+1}\left(\frac{\partial n}{\partial t}\right)_{1}^{k+1}+\left(\frac{ \partial E_{g}}{\partial T_{a}}\right)_{1}^{k+1}\left(\frac{\partial T_{a}}{ \partial t}\right)_{1}^{k+1}\right\}\Bigg{)}\end{split}  \tag{45}\]  
and  
\[ \begin{split} a_{N}=&-\psi\frac{\Delta t}{\Delta z (C_{e-h})_{N}^{k+1}}\times\\ &\Big{(}2k_{B}[H_{0}^{1}(\eta_{e})+H_{0}^{1}(\eta_{h})]_{N-\frac{ 1}{2}}^{k+1}\times J_{N-1}^{k+1}+(k_{e}+k_{h})_{N-\frac{1}{2}}^{k+1}/\Delta z \Big{)},\end{split}  \tag{46}\]  
\[ \begin{split} b_{N}=1-&\psi&\frac{\Delta t }{(C_{e-h})_{N}^{k+1}}\times\Bigg{(}-2\frac{k_{B}}{\Delta z}\left[H_{0}^{1}( \eta_{e})+H_{0}^{1}(\eta_{h})\right]_{N-\frac{1}{2}}^{k+1}\times J_{N-1}^{k+1} \\ &-\frac{2(k_{e}+k_{h})_{N-\frac{1}{2}}^{k+1}}{(\Delta z)^{2}}-(C_ {e-h})_{N}^{k+1}/(\tau_{e-p})_{N}^{k+1}\\ &-\frac{3}{2}k_{B}\left(\frac{\partial n}{\partial t}\right)_{N}^{ k+1}\left(H_{\frac{1}{2}}^{\frac{3}{2}}(\eta_{e})+H_{\frac{1}{2}}^{\frac{3}{2}}( \eta_{h})\right)_{N}^{k+1}-\frac{3}{2}k_{B}n_{N}^{k+1}\left(\frac{\partial n}{ \partial t}\right)_{N}^{k+1}\times\\ &\Bigg{\{}\left(\frac{\partial\eta_{e}}{\partial n}\right)_{N}^{k+1 }\left[1-\left(H_{\frac{1}{2}}^{\frac{3}{2}}(\eta_{e})\right)_{N}^{k+1}\left( H_{\frac{1}{2}}^{-\frac{1}{2}}(\eta_{e})\right)_{N}^{k+1}\right]\] \[+\left.\left(\frac{\partial\eta_{h}}{\partial n}\right)_{N}^{k+1 }\left[1-\left(H_{\frac{1}{2}}^{\frac{3}{2}}(\eta_{h})\right)_{N}^{k+1}\left( H_{\frac{1}{2}}^{-\frac{1}{2}}(\eta_{h})\right)_{N}^{k+1}\right]\right\} \Bigg{)},\end{split}