First-principles finite-size correction schemes for point defects of Cu$_3$N

Point defects play a key role in determining semiconductor properties, such as electrical conductivity and photoluminescence, and often enable functional behavior. Accurate first-principles supercell simulations of point defects require reliable finite-size corrections. In this study, we combine PBE+U structural relaxations with HSE hybrid-functional calculations to determine defect formation energies and thermodynamic transition levels of Cu$_3$N. Finite-size trends are quantified using $\Gamma$-point calculations in supercells containing 256, 864, and 2048 atoms. We assess and extend the Makov-Payne and Lany-Zunger correction schemes by introducing additional $1/L^n$ terms, together with core-level potential alignment and defect-specific scaling models. Using Cu$_3$N as a case study, we show that charged vacancies with strongly localized defect states are accurately described by the Makov-Payne-type scaling ($1/L + 1/L^{3}$), whereas interstitial defects with shallow or weakly localized electronic character are better captured by a hydrogenic impurity model that accounts for defect-band dispersion. Residual trends for neutral or weakly localized defects are described by higher-order polynomial fits in $1/L^{3}$ and $1/L^{4}$. Hybrid-functional energetics corrected using PBE+U-based finite-size trends confirm the intrinsic $p$-type character of Cu$_3$N under the conditions considered and demonstrate that no single finite-size correction can be transferred across all defect types.