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. In this study, we find that, for the standard geometry of a half-space sample in a magnetic field parallel to the surface, the ratio $\mathscr{R}$=H c3 /H c2 is within the window 1.55≲$\mathscr{R}$≲2.34 , regardless of temperature or the scattering type. While the nonmagnetic impurities tend to flatten the $\mathscr{R}$($\mathrm{T}$) variation, magnetic scattering merely shifts the maximum of $\mathscr{R}$($\mathrm{T}$) to lower temperatures. Surprisingly, while reducing the transition temperature, magnetic scattering has a milder impact on $\mathscr{R}$ than nonmagnetic scattering. The surface superconductivity is quite robust; in fact, the ratio $\mathscr{R}$≈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. Furthermore, we find a peculiar behavior of F c as a function of the pair-breaking scattering parameter at the low-$\mathrm{T}$ transition from gapped to gapless phases, which has recently been associated with the topological transition in the superconducting density of states.