The strain-stress relationships for coherent in-plane strain in heterostructures with monoclinic crystal systems: $\beta$-(Al$_x$Ga$_{1-x}$)$_2$O$_3$ on $(h0l)$ $\beta$-Ga$_2$O$_3$ as example

TL;DR

This paper derives the strain and stress relationships in monoclinic heterostructures, revealing multiple stable unit cells and stress directions, challenging previous assumptions about epitaxial growth in low-symmetry materials.

Contribution

It introduces a theoretical framework for understanding strain and stress in monoclinic epitaxial layers, identifying multiple stable configurations and stress directions.

Findings

Existence of three elastically stable unit cells for given templates.

Stress free directions can be oblique to the surface in low-symmetry materials.

Calculated band-to-band transitions for specific compositions.

Abstract

In this work we derive the state of strain or stress under symmetry conserving conditions in pseudomorphic lattices with monoclinic symmetry. We compare surface vectors across the template epitaxial layer interface and impose conditions of a stress free epitaxial layer. As a result, we demonstrate the existence, in theory, of exactly three possible unit cells which can establish onto a given template. We demonstrate this approach for a class of templates with $(h0l)$ planes and $\beta$-(Al$_x$Ga$_{1-x}$)$_2$O$_3$ on $(h0l)$ $\beta$-Ga$_2$O$_3$. We discuss the effects of composition $x$ and surface orientation onto the formation of three elastically stable unit cells, their strain and stress tensors, unit cell axes, unit cell volumes, lattice spacing, elastic potential energies, and stress free directions. The previous paradigm for epitaxial layer growth where the stress free direction is always perpendicular to the growing surface is not generally valid for low symmetry materials. In the example here, we find two possible competing domains with stress free direction oblique to the surface of the template for almost all planes $(h0l)$. We calculate the band-to-band transitions for $\beta$-(Al$_{0.1}$Ga$_{0.9}$)$_2$O$_3$ on $(h0l)$ $\beta$-Ga$_2$O$_3$ using the composition dependent deformation parameters and elastic coefficients reported prevoiously [Korlacki~\textit{et al.} Phys. Rev. Appl.~\textbf{18}, 064019 (2022)].