The orbital upper critical field $H_{ c2} $ is evaluated for isotropic materials with arbitrary transport and pair-breaking scattering rates. It is shown that unlike transport scattering which enhances $H_{ c2} $, the pair breaking suppresses the upper critical field and reduces the dimensionless ratio $h^*(0)=H_{ c2}(0)/T_c(dH_{ c2}/dT)_{ T_c} $ from the Helfand-Werthamer value of $\approx 0.7$ to 0.5 for a strong pair-breaking. $h^*(T)$ is evaluated for arbitrary transport and pair-breaking scattering. A phenomenological model for the pair-breaking suppression by magnetic fields is introduced. It shows qualitative features such as a positive curvature of $H_{ c2}(T) $ and the low temperature upturn usually associated with multi-band superconductivity.
Categories: cond-mat.supr-con