Limiting magnetic fields of superconductors with arbitrary potential and pair-breaking scattering*

The upper critical magnetic field, H_c2(T), the nucleation field of surface superconductivity, H_c3(T), and the thermodynamic critical magnetic field, Hc(T), were evaluated within the weak-coupling Eilenberger formalism with arbitrary TRANSPORT and pair-breaking scatterings. We find that the maximum of H_c3(T)/H_c2(T) is surprisingly robust and is confined to the interval between 1.5 and 2.4, even in the gapless state. Similar calculations of the nucleation field were conducted for ultra-thin films in a parallel magnetic field. There we find a substantial enhancement of the transition temperature Tc by the field.  We also find regimes of field-induced re-entrant superconductivity. Finally, we discuss a singularity in the free energy third derivative corresponding to a type-2½ topological quantum phase transition from a gapped to a gapless superconducting state.