Orbital upper critical field of type-II superconductors with pair breaking

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.

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