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 Hc 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 R = H-c3/H-c2 is within the window 1.55 less than or similar to R less than or similar to 2.34, regardless of temperature or the scattering type. While the nonmagnetic impurities tend to flatten the R(T) variation, magnetic scattering merely shifts the maximum of R(T) to lower temperatures. Surprisingly, while reducing the transition temperature, magnetic scattering has a milder impact on R than nonmagnetic scattering. The surface superconductivity is quite robust; in fact, the ratio R approximate to 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.

Automated summary

This study evaluates the upper critical field Hc2, the field Hc3 for nucleation of surface superconductivity, and the thermodynamic field Hc within the weak-coupling theory for an isotropic s-wave case with arbitrary transport and pair-breaking scattering. The results show that, regardless of temperature or scattering type, the ratio R = Hc3/Hc2 for a half-space sample in a magnetic field parallel to the surface falls within the range 1.55 ≤ R ≤ 2.34. Nonmagnetic impurities flatten the R(T) variation, while magnetic scattering only shifts the maximum of R(T) to lower temperatures.