Dependence of the Electrical Conductivity and Thermoelectric Power of Pure and Aluminum-Doped Rutile on Equilibrium Oxygen Pressure and Temperature

Abstract

The electrical conductivity $\ensuremath{\sigma}$ and thermoelectric power $Q$ have been measured as a function of equilibrium oxygen pressure in crystals of pure and Al-doped rutile. The pressure was varied over the range ${10}^{\ensuremath{-}4}$ mm Hg to \ensuremath{\sim}${10}^{3}$ mm Hg with the temperature held constant at several fixed values in the vicinity of 1000\ifmmode^\circ\else\textdegree\fi{}K. The electrical conductivity of the pure material varied with pressure according to the relation $\ensuremath{\sigma}\ensuremath{\propto}{p}^{\ensuremath{-}\frac{1}{x}}$ with $x\ensuremath{\approx}6$ for $p>10$ mm Hg and $x\ensuremath{\approx}5$ for $p<10$ mm Hg. For the Al-doped crystal, $\ensuremath{\sigma}$ reached a minimum for $p={p}_{0}$ mm Hg. At low pressures $\ensuremath{\sigma}$ followed a relation of the form $\ensuremath{\sigma}\ensuremath{\propto}{p}^{\ensuremath{-}\frac{1}{x}}$ with $x\ensuremath{\sim}5$, whereas at high pressures a relation of the form $\ensuremath{\sigma}\ensuremath{\propto}{p}^{+\frac{1}{x}}$ with $x\ensuremath{\sim}5$ was obeyed. The thermoelectric power exhibited a reversal near ${p}_{0}$, being negative at lower pressures and positive for $p>{p}_{0}$. For pure rutile a defect model involving anion vacancies and estimated values of equilibrium constants leads to a pressure dependence of the conductivity: $\ensuremath{\sigma}\ensuremath{\propto}{p}^{\ensuremath{-}\frac{1}{6}}$, whereas for a model involving titanium interstitials the pressure dependence is $\ensuremath{\sigma}\ensuremath{\propto}{p}^{\ensuremath{-}\frac{1}{5}}$. On the basis of the present measurements it seems that for pure rutile the anion vacancy mechanism obtains at "high" pressures with the titanium interstitials mechanism predominating at "low" pressures. For Al-doped rutile, both the anion vacancy model and the titanium interstitials model are shown to give rise to approximately the observed pressure dependence of conductivity. The anion vacancy model equations are solved in detail (assuming certain values of the equilibrium constants). The titanium interstitials model equations are not solved explicitly for this case but it is shown that for a certain combination of reaction constants the observed pressure dependence may be approximated. Assuming a two-band formalism to be valid---and therefore, a formalism into which the results derived from any defect model must fit---we have analyzed the data and shown that reasonable agreement with observation results. Some indications that hole mobilities at 1000\ifmmode^\circ\else\textdegree\fi{}K are considerably larger (\ensuremath{\sim}180 ${\mathrm{cm}}^{2}$/V sec) than those of the electrons are discussed. Assuming high-temperature polar scattering to obtain, a hole effective mass ${10}^{\ensuremath{-}2}{m}_{0}$ (where ${m}_{0}$ is the free-electron mass) is deduced as a consequence of the mobility calculations.