WKB-based schemes for two-band Schrödinger equations in the highly oscillatory regime

Abstract

An efficient and accurate numerical method is presented for the solution of highly oscillatory differential equations in one spatial dimension. While standard methods would require a very fine grid to resolve the oscillations, the presented approach uses first an analytic WKB-type transformation, which filters out the dominant oscillations. The resulting ODE-system is much smoother and can hence be discretized on a much coarser grid, with significantly reduced numerical costs. Here we are concerned with stationary two-band Schrodinger equations employed in quantum transport applications. 

We focus on the Kane–model and the two band 𝑘 ⋅ 𝑝–model. The accuracy of the presented method is illustrated on a numerical example.