An Isaac Newton Institute Workshop

Effective Computational Methods for Highly Oscillatory Problems: The Interplay between Mathematical Theory and Applications

Spurious Solutions in Applications of Effective Envelope Theory: Roots of the Problem and Methodologies for its Remedy

Authors: Roderick Melnik (Wilfrid Laurier University and University of Waterloo, Canada), Benny Lassen (University of Southern Denmark), Morten Willatzen (University of Southern Denmark)

Abstract

The appearance of spurious solutions is a long standing problem in the application of the envelope function approximation. The subject of this talk is the exact envelope function theory developed by M.G. Burt and further extended by B.A. Foreman. Using this theory, multiband models can be derived for nanoscale heterostructures, similar to the “effective mass” models developed by J.M. Luttinger and W. Kohn that opened the door to modern treatments of important classes of bulk semiconductor materials and nanostructures. In applying this approach, we have to undergo several levels of approximations of the Hamiltonian. In this talk we analyze such approximations critically and show how spurious non-physical solutions in the resulting coupled systems of partial differential equations can be avoided. The discussion is motivated by our research into low-dimensional semiconductor nanostructures, in particular quantum dots.

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