The Schrödinger–Poisson self-consistency in layered quantum semiconductor structures

Abstract

We develop a self-consistent solution of the Schrödinger and Poisson equations in semiconductor heterostructures with arbitrary doping profiles and layer geometries. An algorithm for this non-linear problem is presented in a multiband k⋅P framework for the electronic band structure using the finite element method. The discretized functional integrals associated with the Schrödinger and Poisson equations are used in a variational approach. The finite element formulation allows us to evaluate functional derivatives needed to linearize Poisson’s equation in a natural manner. Illustrative examples are presented using a number of heterostructures including single quantum wells, an asymmetric double quantum well, p-i-n-i superlattices, and trilayer superlattices.