Fully Ab Initio Finite-Size Corrections for Charged-Defect Supercell Calculations

In ab initio theory, defects are routinely modeled by supercells with periodic boundary conditions. Unfortunately, the supercell approximation introduces artificial interactions between charged defects. Despite numerous attempts, a general scheme to correct for these is not yet available. We propose a new and computationally efficient method that overcomes limitations of previous schemes and is based on a rigorous analysis of electrostatics in dielectric media. Its reliability and rapid convergence with respect to cell size is demonstrated for charged vacancies in diamond and GaAs.

Figure 1

Comparison of correction or extrapolation schemes (see text) for the formation energy of the $+2$ vacancy in diamond (plotted vs $L^{-1}$) in various supercells. A: bcc $1\times 1\times 1$ (32 atoms); B: simple cubic (sc) $2\times 2\times 2$ (64); C: fcc $4\times 4\times 4$ (128); D: sc $3\times 3\times 3$ (216); E: bcc $2\times 2\times 2$ (256); F: fcc $6\times 6\times 6$ (432); G: sc $4\times 4\times 4$ (512). The Fermi energy is set equal to the valence-band maximum. The horizontal dashed line indicates the converged result according to our new scheme.

Figure 2

Electrostatic potentials (see text, averaged along $x,y$) for ${\mathrm{V}}_{\mathrm{Ga}}^{3-}$ in a $3\times 3\times 3$ simple cubic GaAs supercell. The defect is located at $z=0\mathrm{bohr}$ with a periodic image at $z=31.38\mathrm{bohr}$.

Figure 3

Defect formation energies of the Ga vacancy in GaAs including supercell corrections as a function of the inverse number of atoms. The Fermi energy is set equal to the valence-band maximum and the Ga chemical potential to that of Ga metal.