Spurious Solutions in the Multiband Effective Mass Theory Applied to Low Dimensional Nanostructures

B. Lassen, R. V. N. Melnik, Morten Willatzen

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Abstract

In this paper we analyze a long standing problem of the appearance of spurious, non-physical solutions arising in the application of the effective mass theory to low dimensional nanostructures. The theory results in a system of coupled eigenvalue PDEs that is usually supplemented by interface boundary conditions that can be derived from a variational formulation of the problem. We analyze such a system for the envelope functions and show that a failure to restrict their Fourier expansion coefficients to small k components would lead to the appearance of non-physical solutions. We survey the existing methodologies to eliminate this difficulty and propose a simple and effective solution. This solution is demonstrated on an example of a two-band model for both bulk materials and low-dimensional nanostructures. Finally, based on the above requirement of small k, we derive a model for nanostructures with cylindrical symmetry and apply the developed model to the analysis of quantum dots using an eight-band model.
Original languageEnglish
JournalCommunications in Computational Physics
Volume6
Issue number4
Pages (from-to)699-729
ISSN1815-2406
DOIs
Publication statusPublished - 2009
Externally publishedYes

Keywords

  • Effective envelope theory
  • Quantum confinement
  • Abrupt interfaces
  • Multiband models
  • k space
  • Fourier coefficients
  • Highly oscillatory integrals
  • Variational formulation
  • Coupled systems of PDEs
  • Multiple scales
  • Continuum and atomistic models
  • Eigenvalue problem
  • Interface boundary conditions
  • Band gap
  • Spurious solutions
  • Low dimensional semiconductor nanostructures

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