Dielectric Relaxations in Reduced Rutile (TiO2−x) at Low Temperatures

The complex dielectric constant of reduced rutile, $\mathrm{Ti}{\mathrm{O}}_{2-x}$, has been measured at frequencies between ${10}^2$ and ${10}^5$ Hz at temperatures between 1.2 and 50°K. Three distinct relaxation processes have been observed: process I at temperatures around 4.0°K with activation energy $Q=1.0\mathrm{}\pm 0.5\times {10}^{-3\mathrm{}}$ eV and pre-exponential factor ${\tau }^{*}={10}^{-6\mathrm{}}$ sec, process II at temperatures around 15°K with $Q=2.0\mathrm{}\pm 0.5\times {10}^{-2\mathrm{}}$ eV and ${\tau }^{*}={10}^{-7\mathrm{}}-{10}^{-8\mathrm{}}$ sec, and process III at temperatures around 11°K with $Q=8.0\mathrm{}\pm 0.5\times {10}^{-3\mathrm{}}$ eV and ${\tau }^{*}\approx {10}^{-7\mathrm{}}$ sec. All three relaxation processes are ascribed to polaron hopping between normal cation lattice sites around complex ionic defect cores. By increasing the concentration of trivalent impurity, the most likely ionic core responsible for process I is found to be an oxygen vacancy associated with one trivalent substitutional impurity, while the core most likely responsible for process II is an interstitial ${\mathrm{Ti}}^{+3\mathrm{}}$ ion in association with two trivalent substitutional impurity ions. An ion core consisting of a single pentavalent substitutional impurity accounts well for process III.