Is Sodium a Superconductor Under Pressure?

Superconductivity has been discovered in compressed Li with a critical temperature ($T_{\rm c}$) of 14 K. The other alkali metals are, theoretically, predicted to become superconductors under pressure. Sodium (Na) is the notable exception. Previous {\it ab initio} calculations considered superconductivity only in the BCC and FCC structures of alkali metals; however, Na goes through complicated, structural phase transitions at higher pressures until it becomes an insulator around 260 GPa. We have performed first-principles linear response calculations for four metallic phases (BCC, FCC, cI16 and tI19) of Na to compute lattice dynamics and the electron-phonon spectral function. The electron-phonon coupling parameter as well as $T_{\rm c}$ were then determined as functions of pressure. Our results suggest that the critical temperature for Na rises with increasing pressure to a maximum $T_{\rm c}$ of 1.2 K in the cI16 phase, then it decreases rapidly to zero K at higher pressures.