Thermodynamically consistent stabilization of the drift-diffusion model for arbitrary band structures and carrier statistics

To this day, the drift-diffusion model remains the most widely applied semiconductor simulation tool. This is due to its unrivaled numerical robustness when it is discretized with the finite volume method and the Scharfetter–Gummel stabilization. Unfortunately, this stabilization is only valid for nondegenerate carrier statistics. Several extensions of the Scharfetter–Gummel scheme to degenerate semiconductors have been proposed; however, they either rely on additional approximations or lack the stability for a full-scale device simulation. In this paper, we address this issue and present a generalization of the Scharfetter–Gummel scheme using no further approximations. Our scheme works for arbitrary band structures and coarse grids and is guaranteed to be thermodynamically consistent. Similar to Scharfetter–Gummel, it leads to a diagonally dominant Jacobian (M-matrix) for the discrete continuity equation preserving its excellent stability properties. An implementation of the algorithm is available online via Zenodo under the MIT license. It has already been used in a 2D device simulation at 4K where it exhibited excellent stability at a negligible runtime penalty.