Transition Temperature of Superconductivity in Sodium Tungsten Bronze -Theoretical Study Based on First-principles Calculations-

Using first-principles calculations, we examine the transition temperature $T_{\rm c}$ of superconductivity in sodium tungsten bronze ( Na$_x$WO$_3$, where $x$ is equal to or less than unity ). Although $T_{\rm c}$ is relatively low $T_{\rm c}( <\sim 3 {\rm K})$, it is interesting that its characteristic exponential dependence on $x$ has been experimentally observed at $\sim 0.2 < x < \sim 0.4$. On the basis of the McMillan equation for $T_{\rm c}$ including the effect of plasmons, we succeed in reproducing the absolute values of $T_{\rm c}$ and its $x$ dependence. We also find that the plasmon effect is crucial for the estimation of $T_{\rm c}$ as well as phonons. Since the calculated $T_{\rm c}$ may not exceed $\sim 20$ K even for $x <\sim 0.1$, the superconductivity at a low $T_{\rm c}$ can be interpreted by the usual phonon mechanism, including the plasmon effect. On the other hand, a high $T_{\rm c}$ up to about 90 K, which is found on the surface of a Na$_x$WO$_3$ system at $x\sim 0.05$ by recent experiments, cannot be explained by our results. This discrepancy suggests that another mechanism is required to clarify the nature of the high-$T_{\rm c}$ superconductivity of Na$_x$WO$_3$.