Band-gap trend of corundum oxides α−M2O3 ( M=Co

In recent years, ${d}^{6}$ transition-metal corundum oxides $\ensuremath{\alpha}\phantom{\rule{4.pt}{0ex}}\text{\ensuremath{-}}\phantom{\rule{4.pt}{0ex}}{M}_{2}{\mathrm{O}}_{3}$ ($M=\mathrm{Co}$, Rh, Ir) have garnered significant interest due to their notable $p$-type conductivity; however, their electronic properties remain controversial. In this study, we employ first-principles calculations within different functional levels to systematically investigate the geometry and electronic structure of $\phantom{\rule{4pt}{0ex}}\ensuremath{\alpha}\phantom{\rule{4.pt}{0ex}}\text{\ensuremath{-}}\phantom{\rule{4.pt}{0ex}}{M}_{2}{\mathrm{O}}_{3}$. Our findings reveal that these oxides have a relatively small difference between the indirect and direct band gap, contradicting previous studies. Additionally, we demonstrate that the band gaps of these oxides are closely associated with the ligand field splitting of the cation M d orbitals and the experimentally observed nonmonotonic trend of the direct band-gap variation can be explained by the orbital-dependent Coulomb and exchange interactions. This study enhances our understanding of how the involvement of $d$ orbitals impacts the band gap of transition-metal oxides.