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Discrete Transparent Boundary Conditions for Multi-Band Effective Mass Approximations

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Multi-Band Effective Mass Approximations

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 94))

Abstract

This chapter is concerned with the derivation and numerical testing of discrete transparent boundary conditions (DTBCs) for stationary multi-band effective mass approximations (MEMAs). We analyze the continuous problem and introduce transparent boundary conditions (TBCs). The discretization of the differential equations is done with the help of finite difference schemes.A fully discrete approach is used in order to develop DTBCs that are completely reflection-free. The analytical and discrete dispersion relations are analyzed in depth and the limitations of the numerical computations are shown. We extend the results of earlier works on DTBCs for the scalar Schrödinger equation by considering alternative finite difference schemes.The introduced schemes and their corresponding DTBCs are tested numerically on an example with a single barrier potential. The d-band k⋅p-model is introduced as most general MEMA. We derive DTBCs for the d-band k⋅p-model and test our results on a quantum well nanostructure.

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Author information

Authors and Affiliations

  1. Technische Universität Berlin, DFG Research Center MATHEON, Strasse des 17. Juni 136, 10623, Berlin, Germany

    Dirk Klindworth

  2. Bergische Universität Wuppertal, Fachbereich C – Mathematik und Naturwissenschaften, Lehrstuhl für Angewandte Mathematik und Numerische Analysis, Gaußstrasse 20, 42119, Wuppertal, Germany

    Matthias Ehrhardt

  3. Forschungsgruppe “Partielle Differentialgleichungen”, Weierstraß–Institut für Angewandte Analysis und Stochastik, Mohrenstrasse 39, 10117, Berlin, Germany

    Thomas Koprucki

Authors
  1. Dirk Klindworth
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  2. Matthias Ehrhardt
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  3. Thomas Koprucki
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Corresponding author

Correspondence to Dirk Klindworth .

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Editors and Affiliations

  1. & Numerische Mathematik, Bergische Universität Wuppertal Lehrstuhl für Angewandte Mathematik, Wuppertal, Germany

    Matthias Ehrhardt

  2. Forschungsgruppe “Partielle Differentialgleichungen", Weierstraß-Institut für Angewandte Analysis und Stochastik, Berlin, Germany

    Thomas Koprucki

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Klindworth, D., Ehrhardt, M., Koprucki, T. (2014). Discrete Transparent Boundary Conditions for Multi-Band Effective Mass Approximations. In: Ehrhardt, M., Koprucki, T. (eds) Multi-Band Effective Mass Approximations. Lecture Notes in Computational Science and Engineering, vol 94. Springer, Cham. https://doi.org/10.1007/978-3-319-01427-2_8

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