First Principles Study of Formation Energies and Diffusion Mechanisms of Native Point Defects in Rutile TiO$_{2-x}$

We present results from first principles calculations for the formation energies and diffusion mechanisms of the oxygen vacancy (V$_{\rm O}$), titanium interstitial (Ti$_I$) and titanium antisite (Ti$_{\rm O}$) point defects in reduced rutile TiO$_{2-x}$. The calculations are performed in large supercells containing 240 to 270 atoms using the {\em ab initio} pseudopotential total energy method. As expected, the formation energy of Ti$_{\rm O}$ is found to be significantly higher than V$_{\rm O}$ and Ti$_I$, which have competitive formation energies as a function of the sample stoichiometry. The dominant charge states of V$_{\rm O}$ and Ti$_I$ are found to be $(2+)$ and $(4+)$, respectively, for a wide range of the electron chemical potential. The calculated diffusion barriers for Ti$_I^{4+}$ are lower than those of V$_{\rm O}^{2+}$, and do not present a significant anisotropy between the (100) open channels and the (110) direction. In fact, the barrier for the diffusion of Ti$_I^{4+}$ along the (100) open channels has an unexpectedly 0.1 eV higher barrier than along (110). The reasons for this observation and the implications of the present results are discussed in relation to bulk-defect assisted surface phenomena in rutile TiO$_2$.