Origin of donor compensation in monoclinic (Al$_x$Ga$_{1{\rm -}x})_2$O$_3$ alloys
(Al$_x$Ga$_{1{\rm -}x})_2$O$_3$ alloys are frequently used in heterostructures with monoclinic Ga$_2$O$_3$, resulting in a large conduction-band offset, which leads to charge carrier confinement, a property that is desirable for device applications. However, when (Al$_x$Ga$_{1{\rm -}x})_2$O$_3$ alloys are $n$-type doped with Si, the most efficient shallow donor, there is a significant reduction in the number of charge carriers when the Al content of the alloys is greater than 26%, rendering intentional doping ineffective. Here we show that this compensation is due to cation vacancies forming in response to donor doping. We use density functional theory with the HSE06 hybrid functional to study cation vacancies in monoclinic AlGaO$_3$ and monoclinic Al$_2$O$_3$. We find that vacancies prefer to occupy split-vacancy configurations, similar to vacancies in Ga$_2$O$_3$. Furthermore, by comparing the formation energy of the vacancy with the formation energy of Si donors, we show that vacancies are lower in energy than Si donors, independent of the Fermi level, as soon as the alloys contain more than 16% Al. Therefore, cation vacancies will compensate the donor doping, explaining experimental observations.