Superconductivity of $\beta$-Gallium.

Elemental gallium can exist in several phases under ambient conditions. The stable $\alpha$ phase has a superconducting transition temperature, $T_c$, of 0.9~K. By contrast, the $T_c$ of the metastable $\beta$ phase is around 6~K. To understand the significant improvement in $T_c$ in the $\beta$ phase, we first calculate the electronic structure, phonon dispersion, and the electron-phonon coupling of gallium in the $\alpha$ and $\beta$ phase. Next, we solve the Eliashberg equations to obtain the superconducting gaps and the transition temperatures. Using these results, we relate the increased $T_c$ in the $\beta$ phase to structural differences between the phases that affect the electronic and phonon properties. The structure motif of the $\alpha$ phase is Ga$_2$ dimers, which form strong covalent bonds leading to bonding and antibonding states that suppress the density of states at the Fermi level. The $\beta$-Ga structure consists of arrays of Ga chains that favor strong coupling between the lattice vibrations and the electronic states near the Fermi level. The increased density of states and strong coupling to the phonons for the $\beta$-Ga chains compared to the $\alpha$ Ga$_2$ dimers enhance superconductivity in the $\beta$-Ga phase.