Superconducting properties of barium in three phases under high pressure from first principles

Electron-phonon coupling (EPC) in the three high-pressure phases of Ba is investigated using a pseudopotential plane-wave method based on density functional perturbation theory. The calculated values of superconducting critical temperature T_c of Ba-I and Ba-II under pressure are consistent well with the trends observed experimentally. Moreover, Ba-V is found to be superconducting with a maximum T_c exceeding 7.8 K at 45 GPa. With the increase of pressure, the values of T_c increase in Ba I and Ba-II but the value of T_c decreases in Ba-V. For Ba-I at pressures below 2 GPa, the increases of logarithmic average frequency ω_log and electron-phonon coupling parameters λ both contribute to the enhancement of T_c. For all the three phases at pressures above 2 GPa, T_c is found to be primarily determined by λ. Further investigation reveals that for all the three phases, the change in λ with pressure can be explained mainly by change in the phonon frequency. Thus for Ba-II and Ba-V, although they exhibit completely different superconducting behaviors, their superconductivities have the same origin; the pressure dependence of T_c is determined finally by the pressure dependence of phonon frequency.