Anisotropic charge screening and supercell size convergence of defect formation energies

One of the main sources of error associated with the calculation of defect formation energies using plane-wave density functional theory (DFT) is finite size error resulting from the use of relatively small simulation cells and periodic boundary conditions. Most widely used methods for correcting this error, such as that of Makov and Payne, assume that the dielectric response of the material is isotropic and can be described using a scalar dielectric constant $\epsilon$. However, this is strictly only valid for cubic crystals, and cannot work in highly anisotropic cases. Here we introduce a variation of the technique of extrapolation based on the Madelung potential that allows the calculation of well-converged dilute limit defect formation energies in noncubic systems with highly anisotropic dielectric properties. As an example of the implementation of this technique we study a selection of defects in the ceramic oxide Li${}_2$TiO${}_3$ which is currently being considered as a lithium battery material and a breeder material for fusion reactors.

Figure 1

Crystal structure of $\beta -$Li${}_2$TiO${}_3$. Yellow, green, and red spheres represent titanium, lithium, and oxygen ions, respectively. The black outline represents a single unit cell.

Figure 2

Formation energy of the V${}_{\mathrm{Ti}}^{-4}$ defect for a range of supercells with differing $v_M$. The wide variation in the points demonstrates that is not possible to fit a single straight line of the form $E_f\left(v_M\right)=E_f^{\infty }+b{v}_M$ to the data.

Figure 3

Formation energy of V${}_{\mathrm{Ti}}^{-4}$ as a function of $v_M^{\mathrm{scr}}$, where ${\overline{\epsilon }}^{\mathrm{DFPT}}$ has been used in the calculation of $v_M^{\mathrm{scr}}$. The black dashed line represents a fit of the form given in Eq. (5) to the raw data and the red dashed line is fitted to the corrected data.

Figure 4

Densities of states for perfect Li${}_2$TiO${}_3$ and the V${}_{\mathrm{Ti}}^{-4}$, Li${}_{\mathrm{Ti}}^{-3}$, and O${}_i^{-2}$ defects. The long dashed green lines, intermediate dashed blue lines, and short dashed red lines correspond to $s$-, $p$-, and $d$-derived states, respectively, with the sum plotted using the solid black line. The valence band maximum is indicated with a dashed gray vertical line and the conduction band minimum with a dotted gray vertical line.

Figure 5

Defect formation energy as a function of $v_M^{\mathrm{scr}}$ for the (a) V${}_{\mathrm{Ti}}^{-4}$, (b) Li${}_{\mathrm{Ti}}^{-3}$, and (c) O${}_i^{-2}$ defects, where nonzero elements of ${\overline{\epsilon }}^{\mathrm{eff}}$ are fitted to the data. The wide variation in the data shown in Fig. 2 has disappeared, so one can interpolate to $v_M^{\mathrm{scr}}=0$ and extract defect formation energies in the dilute limit. Note that simulations in the largest $3\times 2\times 2$, $2\times 3\times 2$, and $2\times 2\times 3$ supercells were only performed for the V${}_{\mathrm{Ti}}^{-4}$ defect.

Figure 6

(a) Atomic arrangement surrounding an as yet unoccupied interstitial site in Li${}_2$TiO${}_3$ and (b) same atoms after introduction of an O${}_i^{-2}$ defect. Clearly visible in (a) is the distorted rocksalt structure of Li${}_2$TiO${}_3$ and the distortion arising arising due to the defect is illustrated in (b). It is this distortion that causes the significant change in ${\overline{\epsilon }}^{\mathrm{eff}}$ for this defect. All atom positions have been extracted from simulations in the $2\times 2\times 2$ supercells.