Fast Iterative Solution of Carrier Continuity Equations in 3D MOS /MESFET Simulations

We summarize our experience in solving the three-dimensional carrier continuity equations in our MOS/MES-FET simulator MINIMOS S. First we give a brief overview of the algebraic properties of the coefficient matrices. We show that the matrices are symmetrizable and can be solved by the symmetrized preconditioned CG algorithm. Since the symmetrization matrices are computationally infeasible due to their enormous dynamical range, we turn our focus to various iterative accelerated methods for nonsymmetric matrices. Of these methods the BICGS algorithm together with (unmodified) ILU preconditioning exhibits an optimum of reliability, convergence speed and memory consumption. A controllable level of fill-in during factorization can handle the badly conditioned systems which we frequently find in our simulations. 1 The Basic Partial Differential Equations In our MOS/MES-FET simulator MINIMOS 5 the static semiconductor equations are solved self-consistently on a three-dimensional rectangular domain using finite difference discretization. The static semiconductor equations (18] for the variables ( ,,P, n,p) consist of the Poisson equation div {f · grad,,P) = -p with the space charge p = q · (p n + Njj N;_ ), and of the carrier continuity equations divJ: = q · R divJ; = -q · R where the carrier transport is modelled by an extended drift--<liffusion approach 1: = q · μn · n · (grad,,P +~·grad ( n · k·i")) 1; = q · μP · p · (grad,,P } ·grad (P · k ·~,)) Let Fn ,p denote the effective driving forces for electrons and holes that depend on the local carrier temperatures Tn ,p [3][19] Fn ::: lgradtjJ ~ · grad ( k·:" · n) I Fp ::: lgrad,,P + ~ ·grad ('"~• · p) I