Efficient Solution of the Schroedinger-Poisson Equations in Semiconductor Device Simulations

摘要

This paper reviews the numerical issues arising in the simulation of electronic states in highly confined semiconductor structures like quantum dots. For these systems, the main challenge lies in the efficient and accurate solution of the self-consistent one-band and multi-band Schroedinger-Poisson equations. After a brief introduction of the physical background, we first demonstrate that unphysical solutions of the Schroedinger equation due to the presence of material boundaries can be avoided by combining a suitable ordering of the differential operators with a robust discretization method like box discretization. Next, we discuss algorithms for the efficient solution of the resulting sparse matrix problems even on small computers. Finally, we introduce a predictor-corrector-type approach for the stabilizing the outer iteration loop that is needed to obtain a self-consistent solution of both Schroedinger's and Poisson's equation.