Critical fields of superconductors with magnetic impurities

TL;DR

This paper analyzes the critical magnetic fields in superconductors with magnetic impurities, revealing their ratios, effects of scattering, and implications for superconductor classification using weak-coupling theory.

Contribution

It provides a comprehensive evaluation of critical fields and the impact of magnetic and non-magnetic scattering on surface superconductivity within the weak-coupling framework.

Findings

The ratio Hc3/Hc2 ranges between 1.55 and 2.34 regardless of scattering.

Magnetic scattering shifts the maximum of the ratio to lower temperatures.

Surface superconductivity remains robust even in the gapless state.

Abstract

The upper critical field $H_{c2} $, the field $H_{c3}$ for nucleation of the surface superconductivity, and the thermodynamic $H_c $ are evaluated within the weak-coupling theory for the isotropic s-wave case and 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 ${\cal R}=H_{c3}/H_{c2}$ is within the window $1.55\lesssim {\cal R}\lesssim 2.34$, regardless of temperature, magnetic or non-magnetic scattering. While the non-magnetic impurities tend to flatten the ${\cal R}\left(T\right)$ variation, the magnetic scattering merely shifts the maximum of ${\cal R}\left(T\right)$ to lower temperatures. Surprisingly, while reducing the transition temperature, magnetic scattering has a milder impact on ${\cal R}$ than the non-magnetic scattering. The surface superconductivity is quite robust; in fact, the ratio ${\cal R}\approx 1.7$ 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 the 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.