Comparison of Iteration Schemes for the Solution of theMultidimensional Schr?dinger-Poisson Equations

摘要

We present a fast and robust iterative method for obtaining self-consistent solutions to thecoupled system of Schr?dinger's and Poisson's equations in quantum structures. A simpleexpression describing the dependence of the quantum electron density on the electrostaticpotential is used to implement a predictor – corrector type iteration scheme for the solutionof the coupled system of differential equations. This approach simplifies the softwareimplementation of the nonlinear problem, and provides excellent convergence speed andstability. We demonstrate the algorithm by presenting an example for the calculation ofthetwo-dimensional bound electron states within the cross-section of a GaAs-AlGaAs basedquantum wire. For this example, six times fewer iterations are needed when our predictor – correctorapproach is applied, compared to a corresponding underrelaxation algorithm.