Charged Point Defects in the Flatland: Accurate Formation Energy Calculations in Two-Dimensional Materials

Impurities and defects frequently govern materials properties, with the most prominent example being the doping of bulk semiconductors where a minute amount of foreign atoms can be responsible for the operation of the electronic devices. Several computational schemes based on a supercell approach have been developed to get insights into types and equilibrium concentrations of point defects, which successfully work in bulk materials. Here, we show that many of these schemes cannot directly be applied to two-dimensional (2D) systems, as formation energies of charged point defects are dominated by large spurious electrostatic interactions between defects in inhomogeneous environments. We suggest two approaches that solve this problem and give accurate formation energies of charged defects in 2D systems in the dilute limit. Our methods, which are applicable to all kinds of charged defects in any 2D system, are benchmarked for impurities in technologically important $h$-BN and ${\mathrm{MoS}}_2$ 2D materials, and they are found to perform equally well for substitutional and adatom impurities.