Grid quality and its influence on accuracy and convergence in device simulation

Abstract

Convergence problems can significantly limit the practical value of numerical device simulation, especially where the ability to obtain stable results for a wide range of process conditions is crucial. Solving the semiconductor device equations is difficult for several reasons: solutions exhibit extremely rapid spatial variations in thin boundary layers at p-n junctions and inversion/depletion layers, equations are strongly nonlinear, a unique steady-state solution may not exist for a given set of bias conditions, loss of accuracy is possible when evaluating physical model equations. At the same time, the solution can be quite sensitive to the boundary conditions, making a certain level of accuracy necessary to ensure convergence of the nonlinear iteration. As a result, convergence problems are known to exist. Remedies for these problems are, if at all, usually found heuristically, rarely understanding the reasons for the problem or why a particular approach works better than others. In this paper, an analysis of the situation is presented and an example demonstrates how improving grid quality can help solve convergence problems. A tradeoff between reducing discretization error and improving convergence and stability is demonstrated. The approach is justified on the grounds of basic finite element theory and demonstrated using a practical application (power MOS breakdown simulation).