Issue in obtaining solution of Poisson eq. for self-consistent calculation in NEGF

Hi,

I’ve been working on an exercise matlab code posted by Prof. S. Datta. ( https://nanohub.org/resources/19564 ) In that example, the code performs 1D simple self-consistent calculation btw NEGF and Poisson eq. It uses Newton-Raphson method to get a solution of Poisson eq. as below and it converges smoothly after few iterations. (refer Appendix B in linked thesis for detail of Newton-Raphson method: https://engineering.purdue.edu/gekcogrp/publications/theses/PhD_11_2011_Sunhee_Lee_PhD_Thesis_main.pdf)

dN=n-Nd+((1/beta)*D2*U); dU=(-beta)*(inv(D2-(beta*diag(D))))*dN;U=U+dU;

However, if I get a solution from direct inversion as below, it doesn’t converges at all.

N=n-Nd; U =(-beta)*inv(D2)*N;

I found that in a simple case like PN junction, direct inversion works well. Why it is not working in this case? In what case I need to work with more numerically stable method such as Newton-Raphson method?

I will be appreciated if you can help me.

Thanks. Youngseok Kim

### 1 Responses

1. When you are using direct inversion U =(-beta)*inv(D2)*N to solve poisson equation, the calculated U cannot be given directly as U for next iteration. Add a fraction of this U say Unew=Uprev+0.1*U to arrive at convergence. This fractional value depends on your parameters.