Unified model of the insulator-metal transition in ${\mathrm{Ti}}_2$${\mathrm{O}}_3$ and the high-temperature transitions in ${\mathrm{V}}_2$${\mathrm{O}}_3$

A unified band-crossing model of the insulator-metal transition in ${\mathrm{Ti}}_2$ ${\mathrm{O}}_3$ and the high-temperature tranistion in ${\mathrm{V}}_2$ ${\mathrm{O}}_3$ is presented. The model is based on a Hartree-Fock free-energy calculation including electron-electron Coulomb energy, elastic energy, and band-electron entropy, and taking into account the partial overlap of $e\left(\pi \right)$ bonding and antibonding bands. Below the transition temperature, the $a_{ 1}$ bonding band is taken to lie below the $e\left(\pi \right)$ bonding band in ${\mathrm{Ti}}_2$ ${\mathrm{O}}_3$, and above the Fermi level in the $e\left(\pi \right)$ bonding band in ${\mathrm{V}}_2$ ${\mathrm{O}}_3$. This difference in ordering in the model calculation is found to be determined largely by the difference in electron-electron Coulomb energy for ${\mathrm{Ti}}_2$ ${\mathrm{O}}_3$ and ${\mathrm{V}}_2$ ${\mathrm{O}}_3$, due to the difference in number of electrons per cation. The transitions in ${\mathrm{Ti}}_2$ ${\mathrm{O}}_3$ and ${\mathrm{V}}_2$ ${\mathrm{O}}_3$ are ascribed to the crossing of the $a_{ 1}$ bonding band and the $e\left(\pi \right)$ bonding band as a function of temperature, due mainly to competition between electron-electron energy and band-electron entropy. The model calculation is applied to a description of the behavior of lattice parameters, specific heat, and magnetic susceptibility through the transition temperature in ${\mathrm{Ti}}_2$ ${\mathrm{O}}_3$ and ${\mathrm{V}}_2$ ${\mathrm{O}}_3$. The effect of the substitution of impurities on the nature of the transitions is also discussed. The calculated properties of ${\mathrm{Ti}}_2$ ${\mathrm{O}}_3$ are found to be at best only in qualitative agreement with experiment, and in the case of the magnetic susceptibility, in serious disagreement. It is suggested that the disagreement may be in part due to the formation of singlet excitons in