Superheating Field of Extreme Type-II Superconductors Calculated from the Eliashberg Theory

Abstract

We calculate the superheating field $H_s$ of a type-II superconductor with a large Ginzburg-Landau parameter from the isotropic Eliashberg theory. It is shown that the temperature dependencies of $H_s(T)$ can be obtained from a thermodynamic approach for different scattering parameters $p=\hslash /2\pi {\tau }_N{k}_B{T}_c$ in a wide range of the nonmagnetic impurity scattering times ${\tau }_N$ ranging from the clean $(p\ll 1)$ to the dirty $(p\gg 1)$ limits. Specific calculations were done for Nb and Nb${}_3$Sn using their electron-phonon spectral functions extracted from tunneling measurements. We show that the ratio $H_s(T,p)/H_c(T)$ calculated from the Eliashberg theory exceeds that of the weak coupling BCS model, where $H_c$ is the thermodynamic magnetic field. We also show that, even though nonmagnetic impurities are pairbreakers in the current-carrying state, $H_s(0,p)$ for Nb${}_3$Sn at $T=0$ has a maximum at $p\approx 0.2$ at which the ratio $H_s/H_c$ is about $6\%$ higher than that in the BCS clean limit.