Density-functional calculations of defect formation energies using the supercell method: Brillouin-zone sampling

Abstract

Using the DFT supercell method, the BZ sampling error in the formation energy and atomic structure are investigated for vacancy and interstitial defects in diamond and silicon. We find that the $\mathbf{k}$-point sampling errors in the total energy vary considerably depending on the charge state and defect type without systematic cancellation, even for the same size of supercell. The error in the total energy increases with decreasing electronic perturbation of the defect system relative to the perfect bulk; this effect originates in the localization of electronic states due to the symmetry reduction induced by the presence of a defect. The error in the total energy is directly transferred to the formation energy, and consequently changes the thermodynamic stability of charge states and shifts the ionization levels. In addition, in force calculations and atomic structure determinations, the $\mathbf{k}$-point sampling error is observed to increase as the charge becomes more negative. The $\ensuremath{\Gamma}$-point sampling results in erroneously large relaxation of the four atoms surrounding a vacancy in diamond. We suggest that stronger repulsions between electrons occupying degenerate defect levels at $\ensuremath{\Gamma}$-point compared to those occupying split energy levels at other $\mathbf{k}$ points induces larger atomic movements.