$γ$-phase Inclusions as Common Defects in Alloyed $β$-(Al$_x$Ga$_{1\text{-}x}$)$_2$O$_3$ and Doped $β$-Ga$_2$O$_3$ Films

$β$-Ga$_2$O$_3$ is a promising ultra-wide bandgap semiconductor whose properties can be further enhanced by alloying with Al. Here, using atomic-resolution scanning transmission electron microscopy (STEM), we find the thermodynamically-unstable $γ$-phase is a ubiquitous defect in both $β$-(Al$_x$Ga$_{1\text{-}x}$)$_2$O$_3$ films and doped $β$-Ga$_2$O$_3$ films grown by molecular beam epitaxy. For undoped $β$-(Al$_x$Ga$_{1\text{-}x}$)$_2$O$_3$ films we observe $γ$-phase inclusions between nucleating islands of the $β$-phase at lower growth temperatures (~400-600 $^{\circ}$C). In doped $β$-Ga$_2$O$_3$, a thin layer of the $γ$-phase is observed on the surfaces of films grown with a wide range of n-type dopants and dopant concentrations. The thickness of the $γ$-phase layer was most strongly correlated with the growth temperature, peaking at about 600 $^{\circ}$C. Ga interstitials are observed in $β$-phase, especially near the interface with the $γ$-phase. By imaging the same region of the surface of a Sn-doped $β$-(Al$_x$Ga$_{1\text{-}x}$)$_2$O$_3$ after ex-situ heating up to 400 $^{\circ}$C, a $γ$-phase region is observed to grow above the initial surface, accompanied by a decrease in Ga interstitials in the $β$-phase. This suggests that the diffusion of Ga interstitials towards the surface is likely the mechanism for growth of the surface $γ$-phase, and more generally that the more-open $γ$-phase may offer diffusion pathways to be a kinetically-favored and early-forming phase in the growth of Ga$_2$O$_3$.