Effective scheme to determine accurate defect formation energies and charge transition levels of point defects in semiconductors

We propose an effective method to accurately determine the defect formation energy ${E}_{\mathrm{f}}$ and charge transition level $\ensuremath{\varepsilon}$ of the point defects using exclusively cohesive energy ${E}_{\mathrm{coh}}$ and the fundamental band gap ${E}_{\mathrm{g}}$ of pristine host materials. We find that ${E}_{\mathrm{f}}$ of the point defects can be effectively separated into geometric and electronic contributions with a functional form: ${E}_{\mathrm{f}}=\ensuremath{\chi}{E}_{\mathrm{coh}}+\ensuremath{\lambda}{E}_{\mathrm{g}}$, where $\ensuremath{\chi}$ and $\ensuremath{\lambda}$ are dictated by the geometric and electronic factors of the point defects ($\ensuremath{\chi}$ and $\ensuremath{\lambda}$ are defect dependent). Such a linear combination of ${E}_{\mathrm{coh}}$ and ${E}_{\mathrm{g}}$ reproduces ${E}_{\mathrm{f}}$ with an accuracy better than 5% for electronic structure methods ranging from hybrid density-functional theory (DFT) to many-body random-phase approximation (RPA) and experiments. Accordingly, $\ensuremath{\varepsilon}$ is also determined by ${E}_{\mathrm{coh}}/{E}_{\mathrm{g}}$ and the defect geometric/electronic factors. The identified correlation is rather general for monovacancies and interstitials, which holds in a wide variety of semiconductors covering Si, Ge, phosphorenes, ZnO, GaAs, and InP, and enables one to obtain reliable values of ${E}_{\mathrm{f}}$ and $\ensuremath{\varepsilon}$ of the point defects for RPA and experiments based on semilocal DFT calculations.