Bounds on the range of density-functional-theory point-defect levels in semiconductors and insulators

Defects in semiconductors and insulators are characterized by their levels, which are defined as the values of the Fermi level at which the charge state of the defect changes. Kohn–Sham density functional theory calculations for charged defects have been widely and successfully used to predict defect levels. Due to their lower computational cost and demonstrated ability to predict levels spanning the measured band gap, semilocal exchange–correlation functionals are widely used in these calculations. However, there is a potential pitfall in using semilocal functionals: although they often predict accurate energies for adding or removing electrons from states that are localized near the defect, the famous band gap error results in overly small energies for adding or removing electrons from extended band edge states. As a result, electrons (or holes) that should occupy localized states may become partially or fully delocalized. In order to help detect and analyze such cases, we introduce bounds on the defect levels that can be obtained using a given functional, supercell, and Brillouin zone sampling. Since these bounds correspond to the charge transition levels of the corresponding defect-free supercell, comparison with the bounds reveals when a calculated level is behaving in a bulk-like rather than defect-like manner. We find that the bounds depend significantly on supercell size due to band-filling effects that arise from the finite charge density created when one electron is added to or removed from a finite-sized supercell, and this size dependence helps explain the success of defect level calculations using semilocal functionals.