Finite-size corrections for defect-involving vertical transitions in supercell calculations

A correction method for vertical transition levels (VTLs) involving defect states calculated with a supercell technique is formulated and its effectiveness is systematically verified with ten defects in prototypical materials: cubic-BN, GaN, MgO, and 3C-SiC. Without any corrections, the absolute errors are around 1 eV with moderate size supercells in most cases. In contrast, when our correction method is adopted, the absolute errors are reduced and become less than 0.12 eV in all the cases. Our correction scheme is general and will have the potential for wide application as it is adaptive for evaluating various quantities at fixed geometry, as represented by those relevant to the generalized Koopmans' theorem.

Figure 1

(a) Relative VTLs from $+1$ to neutral charge transition of the oxygen vacancy ($V_{\mathrm{O}}$) in MgO without corrections, with the eFNV corrections using the static [eFNV] or high-frequency dielectric tensor $\left[\mathrm{eFNV}\right({\varepsilon }_{\infty }$)] [10], and with corrections using the present method. (b) Relative formation energies of $V_{\mathrm{O}}^0$ at the $V_{\mathrm{O}}^{+1}$ atomic configurations in MgO without corrections. The horizontal axis is the inverse of the cube root of the number of atoms in the supercells. The uncorrected values are fitted with a function of $a{N}^{-1}+b{N}^{-1/3}+c$, and energy zeros are set at the respective extrapolated ones.

Figure 2

Relative VTLs without corrections, with point-charge (PC) corrections, and with PC plus alignment corrections of (a) ${\mathit{V}}_{\mathrm{B}}(-2/-3)$, (b) $V_{\mathrm{B}}(-3/-2)$, (c) $V_{\mathrm{N}}(+2/+3)$, and (d) $V_{\mathrm{N}}(+3/+2)$ in c-BN; (e) $V_{\mathrm{Ga}}(-2/-3)$ and (f) $V_{\mathrm{N}}(+2/+3)$ in GaN; (g) $V_{\mathrm{Mg}}(-1/-2)$, (h) $V_{\mathrm{O}}(+1/+2)$, and (i) $V_{\mathrm{O}}(+2/+1)$ in MgO; and (j) $V_{\mathrm{C}}(+1/+2)$ in 3C-SiC as a function of the inverse of the cube root of the number of atoms in the supercells. The energy zeros are set at the values with the PC plus alignment corrections at the largest supercells.

Figure 3

Relative VTLs without corrections, with PC corrections, and with PC plus alignment corrections estimated with the second-smallest supercells (128 atoms for GaN and 216 atoms for c-BN, MgO, and 3C-SiC). The energy zeros are the same as that in Fig. 2. The shaded area within $\pm 0.12$ eV highlights the errors of the PC plus alignment corrections.