Slope of Hc2 close to Tc versus the size of the Cooper pairs: The role of disorder in Dynes superconductors

The size of the Cooper pair ${\ensuremath{\xi}}_{\mathrm{pair}}$ is one of the basic characteristics of a superconductor, but it is not possible to measure it directly. It might be argued that ${\ensuremath{\xi}}_{\mathrm{pair}}$ can be determined from the value ${\ensuremath{\xi}}_{\mathrm{slope}}$ extracted from the measurement of the slope of ${H}_{c2}$ close to ${T}_{c}$. Taking into account both pair-conserving and pair-breaking scattering on impurities within the recently developed theory of Dynes superconductors, we perform an explicit calculation of ${\ensuremath{\xi}}_{\mathrm{pair}}$ and ${\ensuremath{\xi}}_{\mathrm{slope}}$. We show that the two quantities agree only in clean superconductors. In particular, when the pair-breaking disorder approaches the quantum critical point, ${\ensuremath{\xi}}_{\mathrm{pair}}$ stays finite, whereas ${\ensuremath{\xi}}_{\mathrm{slope}}$ diverges.