Multiscale analysis and numerical algorithm for the Schrödinger equations in heterogeneous media

摘要

In solid state physics, the most widely used techniques to calculate the electronic levels in nanostructures are the effective masses approximation (EMA) and its extension the multiband k · p method (see [9]). They have been particularly successful in the case of heterostructures (see, e.g. [4], [9], [11]). This paper discusses the multiscale analysis of the Schrödinger equation with rapidly oscillating coefficients. The new contributions obtained in this paper are the determination of the convergence rate for the approximate solutions, the definition of boundary layer solutions, and higher-order correctors. Consequently, a multiscale finite element method and some numerical results are presented. As one of the main results of this paper, we give a reasonable interpretation why the effective mass approximation is very accurate for calculating the band structures in semiconductor in the vicinity of Γ point, from the viewpoint of mathematics.