A Singular Perturbation Approach for the Analysis of the Fundamental Semiconductor Equations.
Abstract
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 the authors 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 the 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 form 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. Therefore, the authors present a modification of the five point scheme which is uniformly convergent.
Document Details
- Document Type
- Technical Report
- Publication Date
- Feb 01, 1983
- Accession Number
- ADA127728
Entities
People
- Christian A. Ringhofer
- Peter A. Markowich
- Siegfried Selberherr
Organizations
- University of Wisconsin–Madison