Strong-Coupling Effects in the Pressure Dependence of Superconductivity

Abstract

We discuss the effect of hydrostatic pressure on the superconducting properties of the strong-coupling metals Pb and Hg. We solve the \'Eliashberg gap equations at zero and at finite pressures to obtain the gap ${\ensuremath{\Delta}}_{0}$ in the excitation spectrum at $T=0$ and the critical temperature ${T}_{c}$. The calculations are effected for kernels at zero pressure and repeated with kernels appropriate to a 5% volume decrease. At nonzero pressure the phonon parts of the kernels are rescaled for the upward shift in the phonon frequencies and adjusted for changes in the electron-ion pseudopotential form factor. It is found that ${\ensuremath{\Delta}}_{0}$ and ${T}_{c}$ do not scale in the same way under pressure. The gap is relatively more affected, so that the ratio $\frac{2{\ensuremath{\Delta}}_{0}}{{k}_{B}{T}_{c}}$ tends towards the BCS weak-coupling limit of 3.52 with decreasing volume and, hence, decreasing electron-phonon interaction. In Pb, our results are in good agreement with experiment for both the change in ${T}_{c}$ and in the ratio $\frac{2{\ensuremath{\Delta}}_{0}}{{k}_{B}{T}_{c}}$. In Hg, no experimental results exist at the moment, but we predict similar strong-coupling effects.