Slope of $H_{c2}$ close to $T_c$ versus the size of the Cooper pairs: The role of disorder in Dynes superconductors

TL;DR

This paper investigates the relationship between the Cooper pair size and the slope of the upper critical field near the critical temperature in Dynes superconductors, highlighting the effects of disorder and impurity scattering.

Contribution

It provides a theoretical analysis showing that the Cooper pair size and the slope-derived length scale only agree in clean superconductors, with disorder causing divergence.

Findings

$\xi_{ m pair}$ remains finite at the quantum critical point.

$\xi_{ m slope}$ diverges near the quantum critical point.

Disorder affects the relationship between pair size and $H_{c2}$ slope.

Abstract

The size of the Cooper pair $\xi_{\rm pair}$ is one of the basic characteristics of a superconductor, but it is not possible to measure it directly. It might be argued that $\xi_{\rm pair}$ can be determined from the value $\xi_{\rm slope}$ 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 $\xi_{\rm pair}$ and $\xi_{\rm slope}$. We show that the two quantities agree only in clean superconductors. In particular, when the pair-breaking disorder approaches the quantum critical point, $\xi_{\rm pair}$ stays finite, whereas $\xi_{\rm slope}$ diverges.