Distinguishing erbium dopants in Y$_2$O$_3$ by site symmetry: \textit{ ab initio} theory of two spin-photon interfaces

We present a first-principles study of defect formation and electronic structure of erbium (Er)-doped yttria (Y$_2$O$_3$). This is an emerging material for spin-photon interfaces in quantum information science due to the narrow linewidth optical emission from Er dopants at standard telecommunication wavelengths and their potential for quantum memories. We calculate formation energies of neutral, negatively, and positively charged Er dopants and find the configuration to be the most stable, consistent with experiment. Of the two substitutional sites of Er for Y, the $C_2$ and $C_{3i}$, we identify the former (with lower site symmetry) as possessing the lowest formation energy. The electronic properties are calculated using the Perdew-Burke-Ernzerhof (PBE) functional along with the Hubbard $U$ parameter {\color{black} and spin-orbit coupling (SOC)}, which yields a $\sim$ 6 $\mu_B$ orbital and a $\sim$ 3 $\mu_B$ spin magnetic moment, and 11 electrons in the Er $4f$ shell, confirming the formation of charge-neutral Er$^{3+}$. This standard density functional theory (DFT) approach underestimates the band gap of the host and lacks a first-principles justification for $U$. To overcome these issues we performed screened hybrid functional (HSE) calculations, including a negative $U$ for the $4f$ orbitals, with mixing ($\alpha$) and screening ($w$) parameters. These produced robust electronic features with slight modifications in the band gap and the $4f$ splittings depending on the choice of tuning parameters. We also computed the many-particle electronic excitation energies and compared them with experimental values from photoluminescence.