A hybrid density functional study on energetics of native defects in anatase $\rm TiO_2$

The energetics and electronic structures of native defects in anatase $\rm TiO_2$ have been studied by means of hybrid density-functional calculations. Here, we show that oxygen vacancy ($V_{\rm O}$) and titanium interstitial ($\rm Ti_\textit{i}$) are both shallow donors and are likely to form with a substantial concentration in an oxygen poor condition. While titanium vacancies ($V_{\rm Ti}$) is a deep acceptor. The charge neutrality showed that Fermi level is pinned at the conduction band minimum in good agreement with the common observations of n-type conductivity in a reduced $\rm TiO_2$. Self-trap hole ($\rm O_O$) states are localized at oxygen anions. On the other hand the self-trapped electron ($\rm Ti_{Ti}$) cannot be produced in the bulk. Although, $\rm Ti_\textit{i}$ is the strongest candidate for the unintentional $n$-type conductivity owning to its low formation energy, we show that the post-growth of $V_{\rm O}$ in anatase is also possible under annealing temperature and pressure.