A Proof of Convergence of Gummel’s Algorithm for Realistic Device Geometries

Convergence is proven for a typical example from the class of highly successful decoupling algorithms for semiconductor simulation, collectively known as Gummel’s method, for one-, two- and three-dimensional models. Because a nonlinear equation is solved for the potential u at every step, the considered version corresponds closely to the algorithms used for numerical computation in practice. As opposed to most earlier publications, the dependence of the regularity of the solution on the device geometry and the nature of the boundary conditions for the system of mixed boundary value problems is considered. By a detailed analysis of the boundary conditions for a typical two- or three-dimensional device a theoretical regularity result by Murthy and Stampacchia can be applied. Hence it follows that for a physically realistic device geometry the solution is sufficiently regular for the algorithm to converge.