Halide donors in monoclinic- and corundum-phase Ga$_2$O$_3$ and Al$_2$O$_3$
We present a systematic first-principles investigation of halide impurities (F and Cl) in Ga$_2$O$_3$ and Al$_2$O$_3$, considering both monoclinic and corundum phases. Our study of the structural properties, formation energies, and charge-state transition levels establishes the relative stability of different atomic configurations and charge states. We find that F and Cl on oxygen sites act as shallow donors in Ga$_2$O$_3$ in both the monoclinic and corundum phases. However, their behavior differs substantially as the band gap increases with greater Al compositions. Fluorine is prone to $DX$-center formation with increased Al composition, leading to self-compensation at 38% Al concentration in monoclinic (Al$_x$Ga$_{1-x}$)$_2$O$_3$ and 70% Al concentration in corundum (Al$_x$Ga$_{1-x}$)$_2$O$_3$. Chlorine is more resistant to $DX$-center formation: in monoclinic (Al$_x$Ga$_{1-x}$)$_2$O$_3$, Cl$_\mathrm{O}$ on the lowest-energy oxygen site shows an onset of $DX$ behavior at 50% alloy composition, while in corundum (Al$_x$Ga$_{1-x}$)$_2$O$_3$ this onset for Cl$_\mathrm{O}$ occurs only at Al concentrations as high as 84%. We also study F and Cl interstitials, finding that they act as compensating centers but also exhibit migration barriers that are low enough for them to be removed by post-growth annealing. Surprisingly, in both monoclinic and corundum Al$_2$O$_3$, Cl$_\mathrm{O}$ exhibits a relatively shallow transition level located at $\sim$0.48 eV below the conduction-band minimum, much shallower than F$_\mathrm{O}$ and other donor candidates. These remarkable results identify Cl as an unusually promising donor candidate in (Al$_x$Ga$_{1-x}$)$_2$O$_3$ alloys and even pure Al$_2$O$_3$, although high formation energies and compensation will render observation of true $n$-type conductivity in Al$_2$O$_3$ difficult.