  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$_3$.
