Supercell size scaling of density functional theory formation energies of charged defects

We address the calculation within density functional theory (DFT) of defect formation energies in alumina, a ceramic oxide often considered an archetype for a wide variety of other similar oxides. We examine the conditions under which calculated defect formation energies, especially those of charged defects, are independent of the principal approximations of the plane-wave DFT formalism, most significant of which is the finite-sized supercell in which the calculation must be performed. We introduce a variation on existing methods of extrapolation to infinite system size to reduce dependence of the result on finite-size errors in the electrostatic and elastic energies of a periodic supercell containing a defect. We also show how the results can be made relatively insensitive to the choice of exchange-correlation functional and pseudopotential by a suitable treatment of the chemical potentials of the atomic species. Our results for formation energies of charged defects are less sensitive than traditional approaches to supercell size and choices of exchange-correlation functional and pseudopotential, and differ notably from previous results.

Figure 1

(Color online) Supercell of ${\text{Al}}_2{\text{O}}_3$ containing $2\times 2\times 1$ copies of the hexagonal unit cell with 120 atoms. Left: side view along the $a$ axis. Right: top view down the $c$ axis.

Figure 2

(Color online) Correction of the formation energy of the aluminum vacancy $V_{\text{Al}}^{3-}$ by “average potential” shifts. Groups of bars represent different supercell sizes indicated on the axis. Within a group, different colored bars represent different choices of region over which to average the potential. The variation with cell size and choice of region over which to average the potential remains unacceptably large even at large supercell sizes. As the supercell size increases, the variation in the shift begins to converge, but still varies by 0.5 eV depending on the choice of region at $3\times 3\times 2$ hexagonal cells. As the size and shape of supercell are varied, even at the larger end of the size range, calculated formation energies still vary by 1.0 eV.

Figure 3

(Color online) Comparison of formation energies of the oxygen vacancy (a) without and (b) with the improved chemical potential scheme. With method (a), there is a spread of values of $\Delta E_f$ of over 1 eV between different XC functionals and PSPs. With method (b), the results all fall within about 0.2 eV of each other.

Figure 4

(Color online) The effect of the band-gap error on the formation energy of an oxygen vacancy. The lowermost line shows the unadjusted formation energy as a function of ${ϵ}_F$; the topmost line applies the full $m\times \Delta E_g$ adjustment and the middle line applies the adjustment of Eq. (9).

Figure 5

(Color online) 3D isosurface projected into the $ab$ plane of the defect state associated with the $V_{\text{O}}^0$ defect. Lobes associated with weight on 1NN oxygen $p$ states can be seen, but most of the weight is in a diffuse $s$ state on the vacancy site. The isosurface plotted corresponds to ${\left|\psi \left(\mathbf{r}\right)\right|}^2=0.1$.

Figure 6

(Color online) Finite-size errors in the formation energies of the principal intrinsic defects of alumina in their highest charge states. The results for larger supercells were obtained by embedding the relaxed structure of the smallest $2\times 2\times 1$ defect supercell in a region of perfect crystal, ignoring the effects of longer-ranged relaxations. (a) Aluminum vacancy, charge $-3$. (b) Aluminum interstitial, charge $+3$. (c) Oxygen vacancy, charge $+2$. (d) Oxygen interstitial, charge $-2$. This finite-size error reduction method works best for $V_{\text{Al}}^{3-}$ but leaves a residual uncertainty of no more than around 0.2 eV in all cases.

Figure 7

(Color online) Finite-size errors in the formation energies of the principal intrinsic defects of alumina in their highest charge states including the effects of long-ranged elastic relaxation. Note the compressed scales relative to the graphs of Fig. 6. (a) Aluminum vacancy, charge $-3$. (b) Aluminum interstitial, charge $+3$. (c) Oxygen vacancy, charge $+2$. (d) Oxygen interstitial, charge $-2$. While there is more variation than for the pure electrostatic case shown in Fig. 6, it is still possible to interpolate reliably to $v_M=0$ to within around 0.2 eV.

Figure 8

(Color online) Defect formation energies of the four main types of intrinsic point defect in alumina calculated in DFT with ultrasoft pseudopotentials and the LDA. The calculation of ${\mu }_{\text{O}}$ was carried out at $T=2371\text{K}$ and $p_{{\text{O}}_2}=0.2\text{atm}$.

Figure 9

(Color online) Defect formation energies of the split Al vacancy $V_{\text{Al},s}=2V_{\text{Al}}+{\text{Al}}_i$ and the $V_{\text{AlO}}$ defect alongside the $V_{\text{O}}$ and $V_{\text{Al}}$ single-atom vacancies. As with Fig. 8, results use ultrasoft pseudopotentials and the LDA. The calculation of ${\mu }_{\text{O}}$ was carried out at $T=2371\text{K}$ and $p_{{\text{O}}_2}=0.2\text{atm}$.