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Multi-Band Effective Mass Approximations pp 247–272Cite as

Transient Simulation of k⋅p-Schrödinger Systems Using Discrete Transparent Boundary Conditions

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

Abstract

This chapter deals with the derivation and analysis of discrete transparent boundary conditions (TBCs) for transient systems of Schrödinger-type equations in one space dimension. These systems occur i.e. in the physics of layered semiconductor devices as the so called k⋅p-Schrödinger equations, which are a well established tool for band structure calculations.The new TBCs are constructed directly for the chosen finite difference scheme, in order to ensure the stability of the underlying scheme and to completely avoid any numerical reflections. The discrete TBCs are constructed using the solution of the exterior problem with Laplace and \(\mathcal{Z}\)-transformation, respectively.These discrete TBCs can easily be obtained by an inverse \(\mathcal{Z}\)-transformation based on FFT, but these exact discrete TBCs are non-local in time and thus very costly. Hence, as a remedy, we present approximate discrete TBCs, that allow a fast calculation of the boundary terms using a sum-of-exponentials approach.

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Author information

Authors and Affiliations

  1. Institut für Mathematik, Technische Universität Berlin, Strasse des 17. Juni 136, 10623, Berlin, Germany

    Andrea Zisowsky

  2. Institut für Analysis und Scientific Computing, Technische Universität Wien, Wiedner Hauptstr. 8, 1040, Wien, Austria

    Anton Arnold

  3. Lehrstuhl für Angewandte Mathematik und Numerische Analysis, Fachbereich C Mathematik und Naturwissenschaften, Bergische Universität Wuppertal, Gaußstr. 20, 42119, Wuppertal, Germany

    Matthias Ehrhardt

  4. Forschungsgruppe “Partielle Differentialgleichungen”, Weierstrass-Institut für Angewandte Analysis und Stochastik, Mohrenstrasse 39, 10117, Berlin, Germany

    Thomas Koprucki

Authors
  1. Andrea Zisowsky
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  2. Anton Arnold
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  3. Matthias Ehrhardt
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  4. Thomas Koprucki
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Corresponding author

Correspondence to Andrea Zisowsky .

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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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Zisowsky, A., Arnold, A., Ehrhardt, M., Koprucki, T. (2014). Transient Simulation of k⋅p-Schrödinger Systems Using Discrete Transparent Boundary Conditions. 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_7

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