Defect energetics in $\alpha$-${\mathrm{Al}}_2$${\mathrm{O}}_3$ and rutile Ti${\mathrm{O}}_2$

We report a theoretical survey of defect energetics in $\alpha$-${\mathrm{Al}}_2$ ${\mathrm{O}}_3$ and rutile Ti${\mathrm{O}}_2$ which we relate to structural and transport properties of these materials. The study of these crystals has required us to modify our computational methods based on the Mott-Littleton theory, which were previously confined to the treatment of cubic materials. We discuss the theoretical aspects of a new and quite general computational procedure, HADES III, which can be used for defect calculations on crystals of any symmetry. Our discussion pays particular attention to the effects on the calculated energetics of the use of Mott-Littleton methods adapted for anisotropic crystals. Other features, considered in detail, are the sensitivity of calculated defect energies to the choice of lattice potential and to the size of the atomistically simulated region surrounding the defect. We also compare our results for $\alpha$-${\mathrm{Al}}_2$ ${\mathrm{O}}_3$ and those of an earlier study of Dienes et al. Our calculations are then used to discuss the simplest features of the defect properties of pure and doped $\alpha$-${\mathrm{Al}}_2$ ${\mathrm{O}}_3$ and Ti${\mathrm{O}}_2$. The present results support the dominance of Schottky disorder in both crystals; cation Frenkel energies are high and anion Frenkel pairs may be of significance in $\alpha$-${\mathrm{Al}}_2$ ${\mathrm{O}}_3$. In addition we present a survey of doped alumina and of the effect of oxygen partial pressure on the defect structure of this material. Our results suggest that defect clustering will have a major influence on the properties of doped ${\mathrm{Al}}_2$ ${\mathrm{O}}_3$.