The upper critical magnetic field, Hc2, the nucleation field of surface superconductivity, Hc3, and the thermodynamic critical magnetic field, Hc, are evaluated within the weak-coupling theory for the isotropic *s-*wave pairing with arbitrary transport and pair-breaking scattering [1,2]. We find that the maximum ratio, R=Hc3/Hc2, reached for a magnetic field parallel to the surface, remains within 1.55 < R(T)<2.34, regardless of temperature or the scattering type. While the nonmagnetic impurities tend to flatten the R(T) curve, magnetic scattering shifts the maximum of R(T) to lower temperatures. Surprisingly, magnetic scattering has a milder impact on R(T) than nonmagnetic scattering. The surface superconductivity is quite robust, the maximum ratio R = 1.7 is found even in the gapless state. Furthermore, we used Eilenberger’s energy function to evaluate the condensation energy Fc and the thermodynamic critical magnetic field Hc for arbitrary temperature and scattering parameters. By comparing Hc2 and Hc, we find that transport scattering promotes type-II behavior, but pair-breaking scattering pushes materials toward type-I superconductivity [3].

Finally, by conducting high precision and high-resolution calculations of Fcas a function of the pair-breaking scattering at very low temperatures we confirm the existence of a type-2.5 topological quantum phase transition from gapped to gapless state suggested recently from the discontinuity of the third derivative of Fc from the analytical Maki’s solutions for Fc at T=0 [4]. Here we show the temperature dependence of the third derivative and the temperature broadening of this quantum critical point to a cone-like structure as observed for other quantum critical phase transitions.

References

[1] V. G. Kogan and R. Prozorov, *“Critical Fields of Superconductors with Magnetic Impurities”,* Phys. Rev. B 106, 054505 (2022).

[2] V. G. Kogan and R. Prozorov, *“Changing the Type of Superconductivity by Magnetic and Potential Scattering”,* Phys. Rev. B 90, 180502 (2014).

[3] V. G. Kogan and R. Prozorov, *“Anisotropic Criteria for the Type of Superconductivity”,* Phys. Rev. B 90, 54516 (2014).

[4] Y. Yerin, A. A. Varlamov, and C. Petrillo, *“Topological Nature of the Transition between the Gap and the Gapless Superconducting States”,* EPL 138, (2022).

Keywords: upper critical field, third critical field, thermodynamic critical field, quantum topological phase transition