Is sodium a superconductor under high pressure?

Superconductivity has been predicted or measured for most alkali metals under high pressure, but the computed critical temperature (T_c) of sodium (Na) at the face-centered cubic (fcc) phase is vanishingly low. Here we report a thorough, first-principles investigation of superconductivity in Na under pressures up to 260 GPa, where the metal-to-insulator transition occurs. Linear-response calculations and density functional perturbation theory were employed to evaluate phonon distributions and the electron-phonon coupling for bcc, fcc, cI16, and tI19 Na. Our results indicate that the maximum electron-phonon coupling parameter, λ, is 0.5 for the cI16 phase, corresponding to a theoretical peak in the critical temperature at Tc 1.2 K. When pressure decreases or increases from 130 GPa, T_c drops quickly. This is mainly due to the lack of p-d hybridization in Na even at 260 GPa. Since current methods based on the Eliashberg and McMillian formalisms tend to overestimate the T_c (especially the peak values) of alkali metals, we conclude that under high pressure - before the metal-to-insulator transition at 260 GPa - superconductivity in Na is very weak, if it is measurable at all.