First-principles $\mathrm{DFT}+GW$ study of oxygen vacancies in rutile ${\text{TiO}}_2$

We perform first-principles calculations of the quasiparticle defect states, charge transition levels, and formation energies of oxygen vacancies in rutile titanium dioxide. The calculations are done within the recently developed combined $\mathrm{DFT}+GW$ formalism, including the necessary electrostatic corrections for the supercells with charged defects. We find the oxygen vacancy to be a negative $U$ defect, where $U$ is the defect electron addition energy. For Fermi level values below $\sim 2.8$ eV (relative to the valence-band maximum), we find the $+2$ charge state of the vacancy to be the most stable, while above $2.8$ eV we find that the neutral charge state is the most stable.

Figure 1

(a) Tetragonal primitive cell of ${\text{TiO}}_2$ in the rutile crystal structure. (b) The corresponding Brillouin zone.

Figure 2

Theoretical band structure of rutile ${\text{TiO}}_2$ calculated within DFT using the PBE exchange-correlation potential (red dotted lines) and using the $GW$ method (blue solid lines).

Figure 3

Charge densities of the localized states in the gap found in the HSE calculations with $+1$ charged supercells. Panel (a) shows the ground state, corresponding to a $+2$ charged oxygen vacancy and a polaron. Panel (b) shows the $+1$ charged oxygen vacancy. The isosurfaces show $10\%$ of the charge densities of the localized states.

Figure 4

Charge density of the localized defect state of the $+1$ charged oxygen vacancy found in the PBE calculation. The isosurface shows $10\%$ of the charge density. The atomic positions are the same as in Fig. 3.

Figure 5

Schematic illustration of the calculation of the charge transition levels ${\epsilon }^{2+/1+}$ and ${\epsilon }^{1+/0}$ within the $\mathrm{DFT}+GW$ formalism. Arrows indicate the actual paths used in our calculations.

Figure 6

Calculated formation energies of oxygen vacancies in rutile ${\text{TiO}}_2$ plotted as functions of Fermi level $E_{\mathrm{F}}$ in the (a) oxygen-rich and (b) titanium-rich growth conditions.