Theory of superconductivity in reentrant superconductors: Tunneling in paramagnetic phase

We compute the temperature- ($T-$) dependent order parameter $\Delta \mathrm{}\left(T\right)\mathrm{}$, condensation energy $H_c^2\mathrm{}\left(T\right)\mathrm{}$, and frequency-dependent density of states $N_s\left(\omega \right)$, for a reentrant superconductor in the paramagnetic phase. Our parameters are chosen to be appropriate to Er${\mathrm{Rh}}_4$ ${\mathrm{B}}_4$. To characterize microscopically the superconductivity, we numerically solve the analog of the Eliashberg equations using a simple diffusion model for the spin-fluctuation propagator. We find that, although the coupling between the local spins and the superconducting electrons is small by some measures, it nevertheless leads to results which strongly deviate from those observed in both the BCS and Abrikosov-Gorkov theory. Pair breaking increases overall as the temperature is lowered; this effect is partially compensated by the accompanying softening of the spin-fluctuation modes and the presence of the crystal-field splitting. All of these factors contribute to make the transition out of the superconducting state first order. Our results for $N_s\left(\omega \right)$ compare reasonably well with recent point-contact tunneling experiments. The calculated form for $\Delta \mathrm{}\left(T\right)\mathrm{}$ is consistent with that derived from experimental measurements of a Fraunhoffer diffraction pattern using a Josephson-junction configuration.