Poisson 方程有限差分(一维+二维)

Poisson equation can be writtern as follows:

\[\nabla\cdot[\epsilon(r)\nabla\phi(r)] = -q(p-n+N_D-N_A)\\ \nabla\epsilon(r)\cdot\nabla\phi(r) + \epsilon(r)\nabla^2\phi(r) = -q(p-n+N_D-N_A) \]

Since Poisson equation is based on the finite difference method (FDM), discretization is required for implementation. In 1D example, the discretization with grid size can be written as,

\[\left(\frac{\epsilon_{n+1}-\epsilon_{n}}{\Delta}\right)\left(\frac{\phi_{n+1}-\phi_{n}}{\Delta}\right)+\epsilon_n\frac{\phi_{n+1}-2\phi_{n}+\phi_{n-1}}{\Delta^2}= -q(p-n+N_D-N_A)\\ \frac{\epsilon_{n+1}}{\Delta^2}\phi_{n+1}-\frac{\epsilon_{n+1}+\epsilon_{n}}{\Delta^2}\phi_n +\frac{\epsilon_{n}}{\Delta^2}\phi_{n-1}= -q(p-n+N_D-N_A) \]

For 2D- examples,

\[\left(\frac{\epsilon_{i+1,j}-\epsilon_{i,j}}{\Delta_i}\bm{i}+\frac{\epsilon_{i,j+1}-\epsilon_{i,j}}{\Delta_j}\bm{j}\right)\left(\frac{\phi_{i+1,j}-\phi_{i,j}}{\Delta_i}\bm{i}+\frac{\phi_{i,j+1}-\phi_{i,j}}{\Delta_j}\bm{j}\right)+\epsilon_{i,j}\left(\frac{\phi_{i+1,j}-2\phi_{i,j}+\phi_{i-1,j}}{\Delta_i^2}+\frac{\phi_{i,j+1}-2\phi_{i,j}+\phi_{i,j-1}}{\Delta_j^2}\right)\\= -q(p-n+N_D-N_A) \]

Then

\[\frac{\epsilon_{i+1,j}\phi_{i+1,j}-(\epsilon_{i+1,j}+\epsilon_{i,j})\phi_{i,j}+\epsilon_{i,j}\phi_{i-1,j}}{\Delta_i^2}+\frac{\epsilon_{i,j+1}\phi_{i,j+1}-(\epsilon_{i,j+1}+\epsilon_{i,j})\phi_{i,j}+\epsilon_{i,j}\phi_{i,j-1}}{\Delta_j^2} = -q(p-n+N_D-N_A)\\ J\bm{\phi} = -q(p-n+N_D-N_A)\\ J\bm{\phi} = -\rho \]

\[-J\Delta\phi - \frac{\partial\rho}{\partial\phi}\Delta\phi = -(J+\frac{\partial\rho}{\partial\phi})\Delta\phi=R\\ R = J\phi + \rho \]

\[n = n_{xz}e^{\frac{q\phi-q\phi_{old}}{k_{B}T}},\quad p = n_ie^{-\frac{q\phi}{k_BT}}\\ \rho = q(p-n+N_D-N_A)\\ \frac{\partial\rho}{\partial\phi} = -\frac{q^2}{k_BT}(n+p) \]