Computational design of optimal heterostructures for $\beta$-Ga$_2$O$_3$

Ga$_2$O$_3$ is a wide-bandgap material of interest for a wide variety of devices, many of these requiring heterostructures, for instance to achieve carrier confinement. A common method to create such heterostructures is to alloy with In$_2$O$_3$ or Al$_2$O$_3$. However, the lattice constants of these materials are significantly different from those of Ga$_2$O$_3$, leading to large amounts of strain in the resulting heterostructure. If the thickness of the heterostructure is increased, this can lead to cracking. By considering alloys of In$_2$O$_3$ and Al$_2$O$_3$, the lattice constants can be tailored to those of Ga$_2$O$_3$, while still keeping a sizable conduction-band offset. We use density functional theory with hybrid functionals to investigate the structural and electronic properties of In$_2$O$_3$ and Al$_2$O$_3$ alloys in the bixbyite, corundum, and monoclinic structures. We find that the lattice constants increase with In incorporation. Bandgaps decrease nonlinearly with increasing In concentration. We find the (In$_{\rm 0.25}$Al$_{\rm 0.75}$)$_{\rm 2}$O$_{\rm 3}$ monoclinic structure to be of particular interest, as it closely matches the Ga$_2$O$_3$ lattice constants while providing an indirect/direct bandgap of 5.94/5.70 eV and a conduction-band offset of 1 eV compared to Ga$_2$O$_3$.