Electronic Band Structures and Excitonic Properties of Delafossites: A $\textit{GW}$-BSE study

We report the band structures and excitonic properties of delafossites $CuMO_2$ (M = Al, Ga, In, Sc, Y, Cr) calculated using the state-of-the-art $\textit{GW}$-BSE approach. We find that all the delafossites are indirect band gap semiconductors with large exciton binding energies, varying from 0.24 to 0.44 eV. The lowest and strongest exciton, mainly contributed from either Cu 3$\textit{d}$ $\rightarrow$ Cu 3$\textit{p}$ (Al, Ga, In) or Cu 3$\textit{d}$ $\rightarrow$ M 3$\textit{d}$ (M = Sc, Y, Cr) transitions, is always located at $L$ point of the rhombohedral Brillouin zone. Taking the electron-hole effect into account, our theoretical band gaps exhibit nice agreement with experiments.