Properties of (Ga$_{1-x}$In$_x$)$_2$O$_3$ over the whole $x$ range

Using density-functional ab initio theoretical techniques, we study (Ga$_{1-x}$In$_x$)$_2$O$_3$ in both its equilibrium structures (monoclinic $beta$ and bixbyite) and over the whole range of composition. We establish that the alloy exhibits a large and temperature-independent miscibility gap. On the low-$x$ side, the favored phase is isostructural with $beta$-Ga$_2$O$_3$; on the high-$x$ side, it is isostructural with bixbyite In$_2$O$_3$. The miscibility gap opens between approximately 15% and 55% In content for the bixbyite alloy grown epitaxially on In$_2$O$_3$, and 15% and 85% In content for the free-standing bixbyite alloy. The gap, volume and band offsets to the parent compound also exhibit anomalies as function of $x$. Specifically, the offsets in epitaxial conditions are predominantly type-B staggered, but have opposite signs in the two end-of-range phases.