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,
\[\left(\frac{\epsilon_{i+1,j}-\epsilon_{i,j}}{\Delta_i}+\frac{\epsilon_{i,j+1}-\epsilon_{i,j}}{\Delta_j}\right)\left(\frac{\phi_{i+1,j}-\phi_{i,j}}{\Delta_i}+\frac{\phi_{i,j+1}-\phi_{i,j}}{\Delta_j}\right)+\epsilon_{i,j}\left(\frac{\phi_{i+1,j}-2\phi_{i,j}+\phi_{i-1,j}}{\Delta_i^2}+\frac{\phi_{i,j+1}-2\phi_{i,j}+\phi_{i,j-1}}{\Delta_j^2}\right)\\= -q(p-n+N_D-N_A)\\ \frac{\epsilon_{n+1}}{\Delta^2}\phi_{n+1}-\frac{\epsilon_{n+1}+\epsilon_{n}}{\Delta^2}\phi_n +\frac{\epsilon_{n}}{\Delta^2}\phi_{n-1}= -q(p-n+N_D-N_A) \] 标签：phi,right,frac,Poisson,epsilon,nabla,差分,一维,Delta From： https://www.cnblogs.com/ghzhan/p/17691987.html