Time-integration and iterative techniques for semiconductor diffusion modeling

Abstract

We investigate numerical integration, preconditioning, iterative solution and multigrid strategies for a class of réaction-diffusion systems used for modeling nonequilibrium phosphorus diffusion in silicon. These problems typically yield stiff systems of equations, and their efficient numerical simulation requires the use of stable integration strategies along with fast, robust algebraic system solvers. We compare the numerical performance of semi-implicit Runge-Kutta methods in conjunction with several standard nonsymmetric iterative solvers and multigrid methods. Our results demonstrate that block-diagonal preconditioning with node-based assembly of the discrete system dramatically improves the performance of iterative solvers. Numerical studies also reveal some interesting new aspects regarding the choice of integration schemes when using iterative methods to solve the linear systems. Unlike the case of direct solvers, where higher-order integration methods typically yield higher computational efficiency, the use of iterative solvers can significantly change or even reverse this trend.