Strong-Coupling Effects in the Pressure Dependence of Superconductivity

We discuss the effect of hydrostatic pressure on the superconducting properties of the strong-coupling metals Pb and Hg. We solve the Éliashberg gap equations at zero and at finite pressures to obtain the gap ${\Delta }_0$ in the excitation spectrum at $T=0\mathrm{}$ 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 ${\Delta }_0$ and $T_c$ do not scale in the same way under pressure. The gap is relatively more affected, so that the ratio $\frac{2{\Delta }_0}{k_B{T}_c}$ 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 $\frac{2{\Delta }_0}{k_B{T}_c}$. In Hg, no experimental results exist at the moment, but we predict similar strong-coupling effects.