A singular perturbation approach for the analysis of the fundamental semiconductor equations

This paper is concerned with a singular perturbation analysis of the two-dimensional steady-state semiconductor equations and of the usual finite difference scheme consisting of the five point discretization of Poisson´s equation and of the Scharfetter--Gummel discretization of the continuity equations. By appropriate scaling we transform the semiconductor equations into a singularly perturbed elliptic system with nonsmooth data. The singular perturbation parameter is defined as the minimal Debeye-length of the device under consideration. Singular perturbation theory allows to distinguish between regions of strong and of weak variation of solutions, so called layers and smooth regions, and to describe solutions qualitatively in these regions. This information is used to analyze the stability and convergence of the discretization scheme. Particular emphasis is put on the construction of efficient grids. It is shown that the Scharfetter-Gummel method is uniformly convergent, i.e., the global error contribution coming from the continuity equations is small when the maximal mesh size is small, independent of the gradient of the solution. Layer jumps are automatically resolved. The five point scheme however is not uniformly convergent. Large gradients of solutions require a graded mesh if solutions inside the layers are to be resolved accurately. This can lead to an intolerably large number of gridpoints. Therefore, we present a modification of the five point scheme which is uniformly convergent.