Strain and composition dependencies of the near bandgap optical transitions in monoclinic (Al$_x$Ga$_{1-x}$)$_2$O$_3$ alloys with coherent biaxial in-plane strain on (010) Ga$_2$O$_3$

The bowing of the energy of the three lowest band-to-band transitions in $\beta$-(Al$_{x}$Ga$_{1-x}$)$_2$O$_3$ alloys was resolved using a combined density functional theory (DFT) and generalized spectroscopic ellipsometry (GSE) approach. The DFT calculations of the electronic band structure of both, $\beta$-Ga$_2$O$_3$ and $\theta$-Al$_2$O$_3$, allow extracting of the linear portion of the energy shift in the alloys, and provide a method for quantifying the role of coherent strain present in the $\beta$-(Al$_{x}$Ga$_{1-x}$)$_2$O$_3$ thin films on (010) $\beta$-Ga$_2$O$_3$ substrates. The energies of band-to-band transitions were obtained using the spectroscopic ellipsometry eigenpolarization model approach [A. Mock et al., Phys. Rev. B 95, 165202 (2017)]. After subtracting the effects of strain which also induces additional bowing and after subtraction of the linear portion of the energy shift due to alloying, the bowing parameters associated with the three lowest band-to-band transitions in monoclinic $\beta$-(Al$_{x}$Ga$_{1-x}$)$_2$O$_3$ are found.