WIAS Preprint No. 1133, (2006)

Electronic states in semiconductor nanostructures and upscaling to semi-classical models


  • Koprucki, Thomas
    ORCID: 0000-0001-6235-9412
  • Kaiser, Hans-Christoph
  • Fuhrmann, Jürgen
    ORCID: 0000-0003-4432-2434

2010 Mathematics Subject Classification

  • 82D37 34L40


  • Semiconductor nanostructures, kp method, electronic states, band structure, semiclassical models, upscaling, quantum wells, semiconductor lasers


In semiconductor devices one basically distinguishes three spatial scales: The atomistic scale of the bulk semiconductor materials (sub-Angstroem), the scale of the interaction zone at the interface between two semiconductor materials together with the scale of the resulting size quantization (nanometer) and the scale of the device itself (micrometer). The paper focuses on the two scale transitions inherent in the hierarchy of scales in the device. We start with the description of the band structure of the bulk material by kp Hamiltonians on the atomistic scale. We describe how the envelope function approximation allows to construct kp Schroedinger operators describing the electronic states at the nanoscale which are closely related to the kp Hamiltonians. Special emphasis is placed on the possible existence of spurious modes in the kp Schroedinger model on the nanoscale which are inherited from anomalous band bending on the atomistic scale. We review results of the mathematical analysis of these multi-band kp Schroedinger operators. Besides of the confirmation of the main facts about the band structure usually taken for granted, key results are conditions on the coefficients of the kp Schroedinger operator for the nanostructure, which exclude spurious modes and an estimate of the size of the band gap. Using these results, we give an overview of properties of the electronic band structure of strained quantum wells. Further, the assumption of flat-band conditions across the nanostructure allows for upscaling of quantum calculations to state equations for semi-classical models. We demonstrate this approach for parameters such as the quantum corrected band-edges, the effective density of states, the optical response, and the optical peak gain. Further, we apply the kp Schroedinger theory to low gap quantum wells, a case where a proper rescaling of the optical matrix element is necessary to avoid spurious modes. Finally, we discuss the application of the kp Schroedinger models to biased quantum wells, the operation mode of electro-optic modulators.

Appeared in

  • Analysis, Modeling and Simulation of Multiscale Problems, A. Mielke, ed., Springer, Heidelberg, 2006, pp. 365--394

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