Although there are numerous experimental studies on defects in  rutile TiO₂, the formation mechanisms of these defects such as Schottky and Frenkel defects are not completely understood. In this work first-principles, electronic structure, plane-wave pseudopotential calculations are performed to investigate the structure and energies associated with defects in TiO₂. The computational approaches are density functional theory (DFT) with the generalized gradient and ultrasoft pseudopotential approximations in a supercell model. To date, we have compared the configuration of Schottky (V_Ti^-4 + 2V_O⁺²), Frenkel (Ti_i⁺⁴ + V_Ti^-4) and Anti-Frenkel (V_O⁺² + O_i^-2) defects. 

      Our DFT calculation results predict that Frenkel defects are energetically favorable in TiO₂ compared to Schottky defects. The lowest Frenkel DFE is about 1.98 eV, which is lower than the lowest Schottky DFE (3.01 eV) and is also lower than former empirical calculation results (4.21-7.50eV). This finding agrees with space charge measurement results although the results of other calculations show that the Schottky DFE is much lower than the Frenkel DFE (see, e.g., Ikeda, Chiang et al 1993, Dawson, Bristowe et al. 1997). The calculations also indicate that both Frenkel and Schottky defects prefer to cluster together rather than spread out across the unit cell.

      The DFT calculations coupled with thermodynamics calculations are used to investigate individual defects in various charge states. The atoms in a bulk-like, fully three-dimensional 2×2×3 unit cell are relaxed from their initial positions to lower the energy of the system. Our calculations suggest that the defect structure of rutile TiO₂ mainly depends on the defect charge state, temperature and oxygen partial pressure.

i) Influence of P_O2

      It is found that in bulk TiO₂ oxygen vacancies and Ti vacancies are more stable when they exist in fully charged states at room temperature. The DFEs of vacancies are found to be  weakly sensitive to the charge state when they are nearly fully charged.  The situation for interstitials are more complex due to the conflict between geometrical and electrochemical factors. The calculations do predict that O interstitials prefer the neutral charge state while the Ti interstitial prefers the +3 charge state.

      The most stable defect in rutile depends on the Fermi level, temperature and defect charge state. When T = 300 K, V_O is more stable than the Ti interstitial over a wide range of Fermi levels. When T > 1400 K, the Ti interstitial is more stable than V_O. This result gives a reasonable explanation for the controversial finding of dominant point defects in rutile TiO₂ at low O₂ partial pressures. At low temperatures (T < 800 K), the intrinsic point defects are hard to form in pure bulk TiO₂ (their DFEs > 0). However, at high temperatures (T > 1250 K), the Ti interstitials with +3 charge automatically form in the pure bulk rutile TiO₂ (their DFEs < 0).