I'm currently trying to fully understand the van Roosbroeck system in 1D (and hopefully more later). So far, I've understood the Scharfetter-Gummel discretization, as well as the Gummel iteration. However, I'm somewhat lost on the precise sequence of operations in the Gummel iteration. It appears to me as this:
1.a. Solve linear Poisson equation given an initial guess for carrier densities.
OR
1.b. Solve linearized Poisson equation given an initial guess for both carrier densities and potential.
I'm not quite sure which one to choose and why.
2. Solve for carrier densities given the potential just found.
3. Repeat 1 and 2 until convergence.
However I'm currently confused at:
1. I do not currently know the matrix operator to get the carrier densities from the potential.
2. The Scharfetter-Gummel scheme appears to find current density at 1 less point than the potential. Does this mean that the current-density-from-potential/carrier-density matrix is not square but an NxN+1 matrix?
I've looked at the Nanohub Drift-Diffusion tutorial PDFs by Prof. Vasileska, but I'm always looking for any other obscure but useful resources that people may have found along the way.