Summary: NANOSYSTEMS: PHYSICS, CHEMISTRY, MATHEMATICS, 2011, 1 (1), P. 1­22
UDC nan
WKB-based schemes for two-band Schr¨odinger
equations in the highly oscillatory regime
Jens Geier1
, Anton Arnold1
1
Institute for Analysis and Scientific Computing, Vienna University of Technology,
Wiedner Hauptstr. 8, A-1040 Wien, Austria.
jens.geier@tuwien.ac.at, anton.arnold@tuwien.ac.at
PACS 02.60.Lj, 02.60.Lh, 85.30.De
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 Schr¨odinger equations employed in quantum trans-
port applications. We focus on the Kane­model and the two band k·p­model. The accuracy of the presented
method is illustrated on a numerical example.
Keywords: Schr¨odinger equation, Kane­model, two-band k · p­model, highly oscillating wave functions,

Source: Arnold, Anton - Institut für Analysis und Scientific Computing, Technische Universität Wien

Collections: Mathematics