An envelope function formalism for lattice-matched heterostructures

TL;DR

This paper introduces a basis transformation in the envelope function method, enabling accurate electronic structure calculations in lattice-matched heterostructures without interface boundary conditions, by incorporating position-dependent matrix elements.

Contribution

It develops a basis transformation approach that extends the envelope function formalism to handle position-dependent matrix elements in heterostructures.

Findings

The formalism accurately models electronic structures in heterostructures.

It simplifies calculations by removing the need for interface boundary conditions.

The method is equivalent to standard formalism for two-band systems.

Abstract

The envelope function method traditionally employs a single basis set which, in practice, relates to a single material because the $k\cdot p$ matrix elements are generally only known in a particular basis. In this work, we defined a basis function transformation to alleviate this restriction. The transformation is completely described by the known inter-band momentum matrix elements. The resulting envelope function equation can solve the electronic structure in lattice matched heterostructures without resorting to boundary conditions at the interface between materials, while all unit-cell averaged observables can be calculated as with the standard envelope function formalism. In the case of two coupled bands, this heterostructure formalism is equivalent to the standard formalism while taking position dependent matrix elements.