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Multi-Band Effective Mass Approximations pp 87–126Cite as

Symmetries in Multiband Hamiltonians for Semiconductor Quantum Dots

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

Abstract

Our current understanding of the symmetries of multiband envelope function Hamiltonians for semiconductor quantum dots and their signatures in the energy level structure and wave function shapes is reviewed. We show how symmetry can be used to block-diagonalize the Hamiltonian matrix and consequently strongly reduce the computational effort. A detailed analysis of symmetries of several different model Hamiltonians reveals that the true symmetry of square-based pyramidal quantum dots is captured if either the interface effects are taken into account or additional higher energy bands are included in the multiband Hamiltonian. This indicates that multiband envelope function methods are fully capable of capturing the true atomistic symmetry of quantum dots in contrast to some widespread beliefs. In addition, we show that translational symmetry can be artificially introduced by the numerical method used, such as the plane wave method. Plane wave method introduces artificial quantum dot replica whose charges interact with charges in the real quantum dot and create an additional strain field in the real dot. This issue can be circumvented by the introduction of proper corrections in the procedure for calculation of Coulomb integrals and strain.

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Acknowledgements

Stanko Tomić* wishes to thank to the STFC e-Science, UK, for providing the computational resources, and the Royal Society, London, Research Grant “High Performance Computing in Modelling of Innovative High Efficiency Photovoltaic Devices”. Nenad Vukmirović was supported by European Community FP7 Marie Curie Career Integration Grant (ELECTROMAT), Serbian Ministry of Education, Science and Technological Development (project ON171017) and FP7 projects PRACE-2IP, PRACE-3IP, HP-SEE, and EGI-InSPIRE.

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  1. Joule Physics Laboratory, School of Computing, Science and Engineering, University of Salford, Manchester, M5 4WT, UK

    Stanko Tomić

  2. Scientific Computing Laboratory, Institute of Physics Belgrade, University of Belgrade, Pregrevica 118, 11080, Belgrade, Serbia

    Nenad Vukmirović

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  1. Stanko Tomić
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Correspondence to Stanko Tomić .

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  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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Tomić, S., Vukmirović, N. (2014). Symmetries in Multiband Hamiltonians for Semiconductor Quantum Dots. 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_3

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