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We present a first-principles study of defect formation and electronicstructure of erbium (Er)-doped yttria (Y$_2$O$_3$). This is an emergingmaterial for spin-photon interfaces in quantum information science due to thenarrow linewidth optical emission from Er dopants at standard telecommunicationwavelengths and their potential for quantum memories and transducers. Wecalculate formation energies of neutral, negatively, and positively charged Erdopants and find the charge neutral configuration to be the most stable,consistent with experiment. Of the two substitutional sites of Er for Y, the$C_2$ (more relevant for quantum memories) and $C_{3i}$ (more relevant forquantum transduction), we identify the former as possessing the lowestformation energy. The electronic properties are calculated using thePerdew-Burke-Ernzerhof (PBE) functional along with the Hubbard $U$ parameterand 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 densityfunctional theory (DFT) approach underestimates the band gap of the host andlacks a first-principles justification for $U$. To overcome these issues, weperformed 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 modificationsin the band gap and the $4f$ splittings depending on the choice of tuningparameters. We also computed the many-particle electronic excitation energiesand compared them with experimental values from photoluminescence.

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