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

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