On the Pairing Instability in Rutile MO$_2$

The metal-dioxide family of compounds such as TiO$_2$, SnO$_2$, VO$_2$, NbO$_2$, and WO$_2$ have been of much recent interest for reasons as diverse as understanding novel correlated electron physics to designing new photo/electro-catalysis. All can be thought of as forming in a prototypical te\-tra\-gonal ``rutile'' structure, yet members with unpaired d$^1$ and d$^2$ electrons undergo a structural phase transition to a monoclinic, ``distorted rutile,'' structure. In some cases this metal pairing transition accompanies a metal to insulator transition, the precise role, however, is still not clear. Here we present a comparative first-principles study of the lattice instabilities in d$^1$ and d$^2$ MO$_2$ rutile, paying particular attention to the d$^2$ system WO$_2$, which remains metallic even in the distorted phase. We calculate the phonon dispersion in the tetragonal prototypical structure. Using unstable high symmetry modes as a guide, we calculate the energy surface around the high-symmetry structure and perform full structural relaxations in the distorted ground states. We elucidate the interplay between the electronic structure and the pairing transition and discuss the possibility of controlling it with strain.