Multiband Hamiltonians of the Luttinger-Kohn Theory and Ellipticity Requirements

TL;DR

This paper analyzes the ellipticity conditions of multiband Hamiltonians derived from the Luttinger-Kohn theory, revealing that many models violate these conditions, which affects their physical accuracy in nanostructure modeling.

Contribution

It provides a detailed analysis of ellipticity violations in $k \, \cdot \, p$ models, especially the 6x6 Hamiltonians, highlighting their impact on physical properties.

Findings

Many derived models fail to meet ellipticity requirements.

Ellipticity violations significantly affect physical property predictions.

The analysis emphasizes the need for ellipticity considerations in model development.

Abstract

Modern applications require a robust and theoretically strong tool for the realistic modeling of electronic states in low dimensional nanostructures. The $k \cdot p$ theory has fruitfully served this role for the long time since its creation. During last two decades several problems have been detected in connection with the application of the $k \cdot p$ approach to such nanostructures. These problems are closely related to the violation of the ellipticity conditions for the underlying system, the fact that until recently has been largely overlooked. We demonstrate that in many cases the models derived by a formal application of the Luttinger-Kohn theory fail to satisfy the ellipticity requirements. The detailed analysis, presented here on an example of the $6 \times 6$ Hamiltonians, shows that this failure has a strong impact on the physically important properties conventionally studied with these models.