Iteration scheme for the solution of the two-dimensional Schrödinger-Poisson equations in quantum structures

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Abstract
A fast and robust iterative method for obtaining self-consistent solutions to the coupled system of Schrödinger’s and Poisson’s equations is presented. Using quantum mechanical perturbation theory, a simple expression describing the dependence of the quantum electron density on the electrostatic potential is derived. This expression is then used to implement an iteration scheme, based on a predictor-corrector type approach, for the solution of the coupled system of differential equations. We find that this iteration approach simplifies the software implementation of the nonlinear problem, and provides excellent convergence speed and stability. We demonstrate the approach by presenting an example for the calculation of the two-dimensional bound electron states within the cross section of a GaAs-AlGaAs based quantum wire. For this example, the convergence is six times faster by applying our predictor-corrector approach compared to a corresponding underrelaxation algorithm.

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