The strain-stress relationships for coherent in-plane strain in heterostructures with monoclinic crystal systems: β-(AlxGa1-x)2O3 on (h0l) β-Ga2O3 as example

We derive the relationship between the state of strain and stress in pseudomorphic epitaxial layers on a substrate with differing equilibrium lattice parameters under symmetry conserving conditions for the case of monoclinic crystal 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 up to three possible unit cells that can establish on a given template. We demonstrate this approach for a special case of symmetry-conserving strain for a class of templates with (h0l) surface orientations. We use β-(AlxGa1-x)2O3 on (h0l) β-Ga2O3 as an example. We discuss the effects of composition x and surface orientation on 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 only stress-free direction is perpendicular to the growing surface is not generally valid for low-symmetry materials. In the example here, we find that in addition to the stress-free growth direction for crystallographic planes with low Miller indices, two possible competing domains can exist with stress-free direction oblique to the surface of the template for almost all planes (h0l). We further demonstrate the effect of the orientation-dependent differently strained domains and calculate the strain-shifted anisotropic band-to-band transitions for β-(Al0.1Ga0.9)2O3 on (h0l) β-Ga2O3. We also provide examples for X-ray diffraction analyses which may help guiding the search for the occurrence of such differently strained unit cells in epitaxial layers grown by experiment, here by taking the (-201) surface orientation as case for demonstration.