A nonlinear multigrid method for one-dimensional semiconductor device simulation: results for the diode

摘要

This paper studies a multigrid method for the solution of the semiconductor device simulation problem. Although the real impact of multigrid will always be in two or more dimensions, here the possibility of the method is investigated for the one-dimensional case. The essential difficulty for multigrid for the semiconductor problem is the possible adverse effect of very coarse grids on the convergence rate of the method, and the difficult computation of a sufficiently close initial approximation for the nonlinear iterative solver on the coarse grids.For the solution on the coarsest grid, continuation is applied together with Newton iteration. The latter is stabilized by the possible insertion of collective symmetric Gauss—Seidel relaxation sweeps for smoothing the iterands.The multigrid method makes use of a box method and a dedicated nonlinear prolongation that is adapted to the Scarfetter—Gummel discretisation.A standard diode-model problem, both with forward and with reversed bias, is used to show that true multigrid convergence can be obtained indeed, even with very coarse meshes on the coarse grids. However, V-cycles are not always sufficient and W-cycles may be required.