Modeling phosphorus diffusion in three dimensions

A three-dimensional process simulator in which both the standard nonlinear diffusion model and a five-species kinetic model are implemented to model phosphorus diffusion in silicon is described. The reaction-diffusion equations are discretized using finite differences in space and the numerical method of lines. The resulting ordinary differential equations are solved using the system integrator LSODP. LSODP uses a backward difference formula for time integration and a Krylov projection method to solve the linear system in the inner Newton loop. Comparisons of the two diffusion models are run on several test problems. Results show that the Krylov approach works well for the mildly asymmetric system of the single species, but requires a large dimension (K_m≈20) subspace for the five-species model