Intrinsic $\mathrm{Li}\mathrm{Nb}{\mathrm{O}}_3$ point defects from hybrid density functional calculations

The formation energies and charge transition levels of the most relevant $\mathrm{Li}\mathrm{Nb}{\mathrm{O}}_3$ intrinsic point defects, i.e., Nb antisites and Li as well Nb vacancies, are studied from first principles. Thereby isolated defects are modeled in the framework of the density functional theory with local and hybrid exchange-correlation functionals. The inclusion of nonlocal exchange opens the $\mathrm{Li}\mathrm{Nb}{\mathrm{O}}_3$ fundamental band gap by nearly 2 eV and modifies considerably the relative stability of the investigated defects with respect to values calculated with local functionals. On the other hand, supercell symmetry and finite-size errors in calculations using periodic boundary conditions are found to have a major influence on the outcome of the simulations. It is found that, in particular, the Nb vacancy causes a long-range strain field and requires very large supercells for its adequate modeling. Compared to previous theoretical results we find an enhanced stability of the Nb vacancy with respect to the other defects. The V${}_{\mathrm{Nb}}^{-5}$ is predicted to be stable for Fermi level positions in the upper part of the band gap.

Figure 1

Ball-and-stick models for defect-free LN (a) and material with ${\mathrm{Nb}}_{\mathrm{Li}}$ (b), ${\mathrm{V}}_{\mathrm{Li}}$ (c), and ${\mathrm{V}}_{\mathrm{Nb}}$ (d) point defects. O, small circles; Li, large dark circles; Nb atoms, large light circles.

Figure 2

Stability range of the chemical potentials (in eV) of the $\mathrm{Li}\mathrm{Nb}{\mathrm{O}}_3$ constituents. The region enclosed between points B, D, and F satisfies Eq. (3), while the region enclosed between points A, C, E, and G satisfies Eq. (4). The shaded region is the intersection between them and represents the thermodynamically allowed range of the chemical potentials.

Figure 3

Electronic band structure of ferroelectric $\mathrm{Li}\mathrm{Nb}{\mathrm{O}}_3$ calculated with DFT-PBE (dashed lines) and hybrid DFT (straight lines). Application of the hybrid potential leads to a band-gap opening of 1.84 eV.

Figure 4

Defect formation energy difference between HSE and PBE (at Fermi energy $E_F=0$) of ${\mathrm{Nb}}_{\mathrm{Li}}$, ${\mathrm{V}}_{\mathrm{Li}}$, and ${\mathrm{V}}_{\mathrm{Nb}}$ in LN as a function of the charge state.

Figure 5

Defect formation energies of ${\mathrm{Nb}}_{\mathrm{Li}}^{+}4$, ${\mathrm{V}}_{\mathrm{Li}}^{-}1$, and ${\mathrm{V}}_{\mathrm{Nb}}^{-}1$ calculated in hexagonal cells. Lines represent the linear fit of the calculated points. The effect of the Madelung correction (in red) is indicated.

Figure 6

Defect formation energies within PBE, HSE, and PBE + XC of ${\mathrm{Nb}}_{\mathrm{Li}}$, ${\mathrm{V}}_{\mathrm{Li}}$, and ${\mathrm{V}}_{\mathrm{Nb}}$ in LN as a function of the Fermi energy.

Figure 7

HSE total density of states (arbitrary units) for stable charge states of ${\mathrm{Nb}}_{\mathrm{Li}}$, ${\mathrm{V}}_{\mathrm{Li}}$, and ${\mathrm{V}}_{\mathrm{Nb}}$ as well as defect-free LN.

Figure 8

HSE electron density difference for stable charge states of ${\mathrm{Nb}}_{\mathrm{Li}}$, ${\mathrm{V}}_{\mathrm{Li}}$, and ${\mathrm{V}}_{\mathrm{Nb}}$ along a (2$\overline{11}$0) plane. Atomic color coding as in Fig. 1. Blue and red regions represent electron depletion and accumulation, respectively.