First-principles calculations for charged defects at surfaces, interfaces, and two-dimensional materials in the presence of electric fields

We present a methodology to calculate the formation energy of a charged defect at a surface, an interface, or a two-dimensional material in the presence of a macroscopic electric field. We demonstrate that the proposed formalism corrects for electrostatic artifacts in standard repeated-slab calculations and allows us to extract reliably the formation energy in the isolated defect limit independently of vacuum thickness, slab thickness, or lateral supercell size. The formalism does not enter the self-consistency loop of a density functional theory (DFT) calculation, but requires as input only the electrostatic potential of the converged calculation. Thus, employing the proposed scheme does not require any changes in the DFT code, but only a postprocessing module that we provide for various codes.