Comparison of two methods for circumventing the Coulomb divergence in supercell calculations for charged point defects

Density-functional-theory calculations were performed for the unrelaxed $+2$ Si vacancy and $+2$ self-interstitial utilizing periodic boundary conditions and two different methods—the uniform background charge method and the local moment counter charge method—for circumventing the divergence of the Coulomb potential. Formation energies in nominal 64-, 216-, 512-, 1000-, and 1728-atom supercells were converged with respect to Brillouin zone sampling and then extrapolated to an infinite sized supercell by fitting to a polynomial in odd powers of $1∕L$ where $L$ is the cubed root of the supercell volume. The extrapolated values from the two methods agreed very well ($2\mathrm{meV}$ difference for the vacancy and $13\mathrm{meV}$ difference for the interstitial) as expected from inspection of their respective energy expressions. The extrapolated values and fitting parameters were then employed to evaluate analytic correction formulas that have been proposed to remove spurious electrostatic contributions from defect formation energies. The results indicate that existing formulas are not capable of removing these contributions and that further development is needed in this area.