Near-edge optical properties of $\beta $-Ga$_{\mathrm{2}}$O$_{\mathrm{3}}$ determined through first-principles calculations

$\beta -$Ga$_{\mathrm{2}}$O$_{\mathrm{3}}$ is a wide band-gap material of interest for many applications, including high-power electronics and optoelectronics. The electronic and optical properties are especially interesting due to its wide band gap, reported in the literature between 4.4-5.0 eV. We use first-principles calculations including density functional theory (DFT) and many-body perturbation theory (GW) to investigate the discrepancy in the reported values of the fundamental band gap and whether the nature is direct or indirect. We find that the band gap is indirect but only 29 meV lower in energy than the direct gap. By comparing the imaginary part of the dielectric function to the calculated optical matrix elements for $\Gamma $---$\Gamma $ electronic transitions, we verify the directional-dependence of the absorption onsets for the material. This anisotropy can explain the broad range of reported band gaps. The calculated radiative recombination coefficients demonstrate that despite being an indirect-gap material, intrinsic deep-UV light emission is possible with $\beta $-Ga$_{\mathrm{2}}$O$_{\mathrm{3}}$ at high excitation.