Effect of lattice instability on superconductivity in sodium tungsten bronze

The effect of lattice instability on the electronic properties of the tungsten bronzes, $M_x\mathrm{W}{\mathrm{O}}_3\mathrm{}(0<x<\mathrm{}1\mathrm{})\mathrm{}$, is considered. A model of the free energy which describes the various phases as local minima in configuration space is shown to provide a basis for understanding the structures observed and transformations between them when $M$ is an alkali metal. For the case of ${\mathrm{Na}}_x\mathrm{W}{\mathrm{O}}_3$, the effect on the superconducting transition temperature of a phonon which is assumed to soften as a function of $x$ is explicitly calculated. Tunneling between the local free-energy minima is assisted by this soft phonon. Good agreement is obtained with recent experimental observations of a dramatic increase in $T_c$ as $x$ decreases and approaches the critical value for transition between the superconducting and semiconducting tetragonal phases. Noteworthy features of this work are that the structural transformation does not correspond to a simple condensation of the soft phonon, and also that the phonon softening and configurational tunneling are considered simultaneously.