## 3 Numerical Solution Scheme
\(\frac{\partial\eta_{c}}{\partial n}\) can be found by derivative of Equation (2) by the carrier density:  
\[\left(\frac{\partial\eta_{c}}{\partial n}\right)=\frac{1}{2}\left(\frac{2\pi \hbar^{2}}{m_{\epsilon}^{*}k_{B}T_{e}}\right)^{\frac{3}{2}}\frac{1}{F_{\frac{1 }{2}}(\eta_{c})}; \tag{25}\]  
\(\frac{\partial\eta_{c}}{\partial T_{e}}\) can be found by taking the derivative of equation Equation (2) by the electronic temperature:  
\[\left(\frac{\partial n_{c}}{\partial T_{e}}\right)=-\frac{3}{\sqrt{2}}n\times( T_{e})^{-\frac{5}{2}}\left(\frac{\pi\hbar^{2}}{m_{\epsilon}^{*}k_{B}}\right)^{ \frac{3}{2}}\frac{1}{F_{\frac{1}{2}}(\eta_{c})}. \tag{26}\]  
The boundary conditions can be rewritten in the finite-difference form as follows,  
\[\frac{n_{1}^{k+1}-n_{1}^{k}}{\Delta t}+\frac{2J_{1}^{k}}{\Delta z}=(S_{n})_{1} ^{k}-\gamma(n_{1}^{k})^{3}+\delta_{1}^{k}n_{1}^{k}; \tag{27}\]  
\[(C_{a})_{1}^{k}\frac{(T_{a})_{1}^{k+1}-(T_{a})_{1}^{k}}{\Delta t}=\frac{2}{( \Delta z)^{2}}(k_{a})_{1+\frac{1}{2}}^{k}(T_{2}^{k}-T_{1}^{k})+\frac{(C_{e-h}) _{1}^{k}}{(\tau_{e-p})_{1}^{k}}\left[T_{1}^{k}-(T_{a})_{1}^{k}\right]; \tag{28}\]  
\[\frac{T_{1}^{k+1}-T_{1}^{k}}{\Delta t}=(1-\psi)f_{1}^{k}+\psi f_{1}^{k+1} \tag{29}\]  
Initially we set them all except the first three to be equal to the corresponding old values (at time step \(k\)):  
\[\left(A_{i}^{k+1}\right)^{(0)}=A_{i}^{k}. \tag{36}\]  
Here, \((0)\) means the \(0^{th}\) step of the corrector. With this assumption, Equation (19) becomes  
\[a_{i}T_{i-1}^{k+1}+b_{i}T_{i}^{k+1}+c_{i}T_{i+1}^{k+1}=r_{i}, \tag{37}\]  
which can be represented as a tridiagonal system of equations,  
\[ \begin{bmatrix}b_{1}&c_{1}&...&&0\\ a_{2}&b_{2}&c_{2}&&\\ &a_{3}&b_{3}&&...\\...&&&...&c_{N-1}\\ 0&0&...&a_{N}&b_{N}\end{bmatrix} \times \begin{bmatrix}T_{1}^{k+1}\\ T_{2}^{k+1}\\...\\ T_{N-1}^{k+1}\\ T_{N}^{k+1}\end{bmatrix} = \begin{bmatrix}r_{1}\\ r_{2}\\...\\ r_{N-1}\\ r_{N}\end{bmatrix} , \tag{38}\]  
where  
\[a_{i}=-\psi\frac{\Delta t}{\Delta z(C_{e-h})_{i}^{k+1}}\left(k_{B}[H_{0}^{1}( \eta_{e})+H_{0}^{1}(\eta_{h})]_{i-\frac{1}{2}}^{k+1}\times l_{i-1}^{k+1}+(k_ {e}+k_{h})_{i-\frac{1}{2}}^{k+1}/\Delta z\right), \tag{39}\]  
\[ \begin{split} b_{i}=1&-\psi\frac{\Delta t}{(C_{e-h})_{i}^{ k+1}}\times\Bigg{(}-\frac{k_{B}}{\Delta z}\left[H_{0}^{1}(\eta_{e})+H_{0}^{1}( \eta_{h})\right]_{i+\frac{1}{2}}^{k+1}\times l_{i}^{k+1}\\ &\quad+\frac{k_{B}}{\Delta z}\left[H_{0}^{1}(\eta_{e})+H_{0}^{1}( \eta_{h})\right]_{i-\frac{1}{2}}^{k+1}\times l_{i-1}^{k+1}\\ &\quad-\left[(k_{e}+k_{h})_{i-\frac{1}{2}}^{k+1}+(k_{e}+k_{h})_{i +\frac{1}{2}}^{k+1}\right]/(\Delta z)^{2}-(C_{e-h})_{i}^{k+1}/(\tau_{e-p})_{i }^{k+1}\\ &\quad-\frac{3}{2}(\frac{\partial n}{\partial t})_{i}^{k+1}k_{B} \left(H_{\frac{1}{2}}^{\frac{3}{2}}(\eta_{e})+H_{\frac{1}{2}}^{\frac{3}{2}}( \eta_{h})\right)_{i}^{k+1}-\frac{3}{2}k_{B}n_{i}^{k+1}\left(\frac{\partial n}{ \partial t}\right)_{i}^{k+1}\times\\ &\Bigg{\{}\left(\frac{\partial\eta_{e}}{\partial n}\right)_{i}^{ k+1}\left[1-\left(H_{\frac{1}{2}}^{\frac{3}{2}}(\eta_{e})\right)_{i}^{k+1}\left(H_{ \frac{1}{2}}^{-\frac{1}{2}}(\eta_{e})\right)_{i}^{k+1}\right]\\ &\quad+\left.\left(\frac{\partial\eta_{h}}{\partial n}\right)_{i}^{ k+1}\left[1-\left(H_{\frac{1}{2}}^{\frac{3}{2}}(\eta_{h})\right)_{i}^{k+1} \left(H_{\frac{1}{2}}^{-\frac{1}{2}}(\eta_{h})\right)_{i}^{k+1}\right]\right\} \Bigg{)},\end{split}  \tag{40}\]  
\[ \begin{split} c_{i}=\psi\frac{\Delta t}{\Delta z(C_{e-h})_{i}^{k+1 }}\left(k_{B}[H_{0}^{1}(\eta_{e})+H_{0}^{1}(\eta_{h})]_{i+\frac{1}{2}}^{k+1} \times l_{i}^{k+1}-(k_{e}+k_{h})_{i+\frac{1}{2}}^{k+1}/\Delta z\right),\end{split}  \tag{41}\]  
\[