Phase-selective growth of $\kappa$- vs $\beta$-Ga$_2$O$_3$ and (In$_x$Ga$_{1-x}$)$_2$O$_3$ by In-mediated metal exchange catalysis in plasma-assisted molecular beam epitaxy

Its piezo- and potentially ferroelectric properties make the metastable kappa polymorph of Ga$_2$O$_3$ an interesting material for multiple applications, while In-incorporation into any polymorphs of Ga$_2$O$_3$ allows to lower their bandgap. In this work, we provide a guideline to achieve single phase $\kappa$-, $\beta$-Ga$_2$O$_3$ as well as their (In$_x$Ga$_{1-x}$)$_2$O$_3$ alloys up to x = 0.14 and x = 0.17 respectively, using In-mediated metal exchange catalysis in plasma assisted molecular beam epitaxy (MEXCAT-MBE). The polymorph transition from $\kappa$ to $\beta$ is also addressed, highlighting the fundamental role played by the thermal stability of the $\kappa$-Ga$_2$O$_3$. Additionally, we also demonstrate the possibility to grow ($\bar{2}$01) $\beta$-Ga$_2$O$_3$ on top of $\alpha$-Al$_2$O$_3$ (0001) at temperatures at least 100 {\deg}C above those achievable with conventional non-catalyzed MBE, opening the road for increased crystal quality in heteroepitaxy. The role of the substrate, as well as strain and structural defects in the growth of $\kappa$-Ga$_2$O$_3$ is also investigated by growing simultaneously on three different materials: (i) $\alpha$-Al$_2$O$_3$ (0001), (ii) 20 nm of ($\bar{2}$01) $\beta$-Ga$_2$O$_3$ on $\alpha$-Al$_2$O$_3$ (0001) and (iii) ($\bar{2}$01) $\beta$-Ga$_2$O$_3$ single crystal.