Understanding and correcting the spurious interactions in charged supercells

The supercell technique is widely spread for the simulation of charged point defects. Charged defects in a supercell are unfortunately subjected to spurious image interactions, which are usually handled by introducing two correcting terms: a Madelung-type correction that accounts for the electrostatic interactions of repeated charges in a compensating background and a potential alignment term that refers the charged supercell to the electron reservoir. We demonstrate that the Madelung correction already brings a large potential shift that slowly converges as $1/L$ with increasing supercell sizes. We hence define a potential alignment devoid of any double counting. We finally propose a simple evaluation for the nearest-neighbor interaction that removes the remaining spurious hybridization of the defect wave functions between images. The application of these three corrections together drastically speeds up the convergence with respect to supercell size for all defects that are not too shallow.

Figure 1

Convergence as a function of the supercell size of the formation energy (top) of a silicon interstitial Si${}_{\mathrm{Tet}}^{2+}$ and (middle) of a silicon vacancy V${}_{\mathrm{Si}}^{2+}$. The raw energies are represented with circles. The Madelung corrected energies are represented with squares. The horizontal lines represent the converged values, and the thin dashed lines are tentative extrapolations with the usual function ${\gamma }_1{N}^{-1/3}+{\gamma }_2{N}^{-1}$. (bottom) A cut of the difference in electronic densities between the defective and host cells, $n^{\mathrm{defect}}\left(\mathbf{r}\right)-n^{\mathrm{bulk}}\left(\mathbf{r}\right)$, along the (110) direction, passing through the bond centers for a 1000-atom cubic supercell.

Figure 2

Electrostatic potentials created by a single point charge in an infinite sample (solid red line), an array of point charges with compensating background (dashed black line), and a single point charge in an infinite sample shifted by the Madelung potential $-v_M$ (dot-dashed blue line). The parameters (lattice constant, dielectric constant) have been chosen to mimic a cubic 512-atom supercell of NaCl. The green diamonds represent the deviation of the Na 2$s$ core levels with respect to the bulk Na 2$s$ levels, as obtained from a real calculation of a 512-atom supercell containing a vacancy V${}_{\mathrm{Cl}}^{+}$. The horizontal lines show the asymptotic values of the single point charge potentials.

Figure 3

Schematics illustrating the role of the potential alignment $\Delta V$. (top) The system we intend to simulate: a single charged defect ${\mathrm{D}}^{\mathrm{q}}$ in a single supercell (dark blue), embedded in the infinite bulk (light pink). (bottom) The system we actually calculate with the supercell approach: an array of replicated defects ${\mathrm{D}}^{\mathrm{q}}$. (middle) The corresponding running average potentials, with a solid red line for the truly isolated defect and a dashed blue line for the supercell approach.

Figure 4

Convergence as a function of the supercell size of the formation energy of a sodium vacancy V${}_{\mathrm{Na}}^{-}$. The raw energies are represented with circles. The Madelung corrected energies are represented with squares. The data with potential alignment following Eq. (9) together with the Madelung correction are represented by diamonds. The data with potential alignment following Ref. 9 together with the Makov-Payne monopole and quadrupole corrections are represented by open triangles. The horizontal line represents our converged value, and the thin dashed line is a tentative extrapolation with the usual function ${\gamma }_1{N}^{-1/3}+{\gamma }_2{N}^{-1}$.

Figure 5

Convergence as a function of the supercell size of the formation energy of a chlorine vacancy V${}_{\mathrm{Cl}}^{+}$. The raw energies are represented with circles. The Madelung corrected energies are represented with squares. The data with potential alignment following Eq. (9) together with the Madelung correction are diamonds. The triangles represent the final data including the removal of the neighbor interaction according to Eq. (10). The horizontal line represents the converged value, and the thin dashed lines are tentative extrapolations with the usual function ${\gamma }_1{N}^{-1/3}+{\gamma }_2{N}^{-1}$.

Figure 6

Convergence as a function of the supercell size of the formation energy of (top) a silicon interstitial Si${}_{\mathrm{Tet}}^{2+}$ and (bottom) a silicon vacancy V${}_{\mathrm{Si}}^{2+}$. The raw energies are represented with circles. The Madelung corrected energies are represented with squares. The data with potential alignment following Eq. (9) together with the Madelung correction are diamonds. The triangles represent the final data, which include the removal of the neighbor interaction according to Eq. (10). The horizontal lines represent the converged value, and the thin dashed lines are tentative extrapolations with the usual function ${\gamma }_1{N}^{-1/3}+{\gamma }_2{N}^{-1}$.