A Multigrid Preconditioner for the Semiconductor Equations

A multigrid preconditioned conjugate gradient algorithm is introduced into a semiconductor device modeling code DANCIR. This code simulates a wide variety of semiconductor devices by numerically solving the drift-diffusion equations. The most time-consuming aspect of the simulation is the solution of three linear systems within each iteration of the Gummel method. The original version of DANCIR uses a conjugate gradient iteration preconditioned by an incomplete Cholesky factorization. In this paper, we consider the replacement of the Cholesky preconditioner by a multigrid preconditioner. To adapt the multigrid method to the drift-diffusion equations, interpolation, projection, and coarse grid discretization operators need to be developed. These operators must take into account a number of physical aspects that are present in typical devices: wide-scale variation in the partial differential equation (PDE) coefficients, small-scale phenomena such as contact points, and an oxide layer. Additionally, suitable...