NANOSYSTEMS: PHYSICS, CHEMISTRY, MATHEMATICS, 2011, 2 (3), P. 7–28
WKB-BASED SCHEMES FOR TWO-BAND SCHRO¨ DINGER EQUATIONS IN THE HIGHLY OSCILLATORY REGIME
J. Geier – Institute for Analysis and Scientific Computing, Vienna University of Technology, Wien, Austria, jens.geier@tuwien.ac.at
A. Arnold – Institute for Analysis and Scientific Computing, Vienna University of Technology, Wien, Austria, anton.arnold@tuwien.ac.at
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 Schrödinger equations employed in quantum transport applications. We focus on the Kane-model and the two band – model. The accuracy of the presented method is illustrated on a numerical example.
Keywords: Schrödinger equation, Kane-model, two-band -model, highly oscillating wave functions, higher order WKB-approximation, asymptotically correct finite difference scheme
UDC 530.145
PACS 02.60.Lj, 02.60.Lh, 85.30.De