Formation energies of charged defects in 2D materials - a new perspective

Formation energies of defects in semiconductors play a major role in a wide range of properties. Supercell schemes have been widely used for calculations. In the conventional formulation, a divergence arises from periodic Coulomb interactions and, in the “jellium” scheme, is removed by setting the average electrostatic potential to zero. A posteriori corrections are used to determine the infinite-supercell limit. For 2D materials, where unscreened Coulomb tails are present in the vacuum regions, additional complications arise. In this work, we present an alternative formulation, derived from statistical mechanics, which dictates that supercells are naturally neutral: “charged defects” are merely ionized, by trading carriers with the energy bands (charge neutrality of the crystal is an essential ingredient of the statistical mechanics of electrons in semiconductors). We show that the jellium approach can be derived from the statistical-mechanics-backed theory by invoking ad hoc approximations. We report density-functional-theory calculations showing that the differences between the two methods are especially large in 2D materials, e.g., h-BN, where they can be of order 1 eV. Convergence rates are excellent.