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

Abstract

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 ...