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$_3$.

中文翻译：

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钨青铜钠超导转变温度——基于第一性原理计算的理论研究——

使用第一性原理计算，我们检查了钠钨青铜中超导性的转变温度 $T_{\rm c}$ (Na$_x$WO$_3$，其中 $x$ 等于或小于 unity )。尽管 $T_{\rm c}$ 相对较低 $T_{\rm c}( <\sim 3 {\rm K})$，但有趣的是，它对 $x$ 的特征指数依赖性已在 $ \sim 0.2 < x < \sim 0.4$。基于包含等离子体效应的 $T_{\rm c}$ 的 McMillan 方程，我们成功地再现了 $T_{\rm c}$ 的绝对值及其 $x$ 依赖性。我们还发现等离子体效应对于估计 $T_{\rm c}$ 以及声子至关重要。由于计算出的 $T_{\rm c}$ 可能不会超过 $\sim 20$ K，即使对于 $x <\sim 0.1$，低 $T_{\rm c}$ 的超导性可以通过通常的声子机制来解释，包括等离子体效应。另一方面，最近的实验在 $x\sim 0.05$ 的 Na$_x$WO$_3$ 系统表面发现了高达约 90 K 的高 $T_{\rm c}$用我们的结果来解释。这种差异表明需要另一种机制来阐明 Na$_x$WO$_3$ 的高 $T_{\rm c}$ 超导性的性质。