Critical fields of superconductors with magnetic impurities

The upper critical field ${H}_c2$, the field ${H}_c3$ for nucleation of the surface superconductivity, and the thermodynamic field ${H}_c$ are evaluated within the weak-coupling theory for the isotropic s-wave case with arbitrary transport, and pair-breaking scattering. We find that, for the standard geometry of a half-space sample in a magnetic field parallel to the surface, the ratio $\mathcal{R}={H}_c3/{H}_c2$ is within the window $1.55\ensuremath{\lesssim}\mathcal{R}\ensuremath{\lesssim}2.34$, regardless of temperature or the scattering type. While the nonmagnetic impurities tend to flatten the $\mathcal{R}(T)$ variation, magnetic scattering merely shifts the maximum of $\mathcal{R}(T)$ to lower temperatures. Surprisingly, while reducing the transition temperature, magnetic scattering has a milder impact on $\mathcal{R}$ than nonmagnetic scattering. The surface superconductivity is quite robust; in fact, the ratio $\mathcal{R}\ensuremath{\approx}1.7$ even in the gapless state. We used Eilenberger's energy functional to evaluate the condensation energy ${F}_c$ and the thermodynamic critical field ${H}_c$ for any temperature and scattering parameters. By comparing ${H}_c2$ and ${H}_c$, we find that, unlike transport scattering, the pair-breaking pushes materials toward type-I behavior. We find a peculiar behavior of ${F}_c$ as a function of the pair-breaking scattering parameter at the low-$T$ transition from gapped to gapless phases, which has recently been associated with the topological transition in the superconducting density of states.