Valley degeneracy in biaxially strained aluminum arsenide quantum wells

TL;DR

This paper develops a comprehensive formalism for calculating electron subband energies and degeneracies in strained multi-valley quantum wells, with explicit results for AlAs grown along various orientations, including unconventional facets.

Contribution

It introduces a standardized rotation matrix and a generalized notation for analyzing electron behavior in strained multi-valley quantum wells, applicable to various orientations and miscut angles.

Findings

Determines critical well widths for valley degeneracy crossover in different orientations.

Defines a new primitive unit cell in momentum space for classifying inter-valley scattering.

Provides explicit calculations for AlAs quantum wells along multiple crystal facets.

Abstract

This paper details a complete formalism for calculating electron subband energy and degeneracy in strained multi-valley quantum wells grown along any orientation with explicit results for the AlAs quantum well case. A standardized rotation matrix is defined to transform from the conventional- cubic-cell basis to the quantum-well-transport basis whereby effective mass tensors, valley vectors, strain matrices, anisotropic strain ratios, and scattering vectors are all defined in their respective bases. The specific cases of (001)-, (110)-, and (111)-oriented aluminum arsenide (AlAs) quantum wells are examined, as is the unconventional (411) facet, which is of particular importance in AlAs literature. Calculations of electron confinement and strain in the (001), (110), and (411) facets determine the critical well width for crossover from double- to single-valley degeneracy in each system. The notation is generalized to include miscut angles, and can be adapted to other multi-valley systems. To help classify anisotropic inter-valley scattering events, a new primitive unit cell is defined in momentum space which allows one to distinguish purely in-plane inter-valley scattering events from those that requires an out-of-plane momentum scattering component.