Charge-imbalance relaxation in the presence of a pair-breaking interaction in superconducting AlEr films

Abstract

The charge-imbalance relaxation rate, $\frac{1}{{F}^{*}{\ensuremath{\tau}}_{{Q}^{*}}}$, has been measured in dirty superconducting AlEr films in which Er is a pair-breaking magnetic impurity that induces charge relaxation through elastic exchange scattering. Measurements were made in the range $0.1\ensuremath{\lesssim}\frac{\ensuremath{\Delta}(T)}{{k}_{B}{T}_{c}}\ensuremath{\lesssim}1.4$ for Er concentrations varying from 21 to 1660 at. ppm that produced estimated exchange scattering rates, ${\ensuremath{\tau}}_{S}^{\ensuremath{-}1}$, from about ${10}^{9}$ ${\mathrm{sec}}^{\ensuremath{-}1}$ to 5\ifmmode\times\else\texttimes\fi{}${10}^{10}$ ${\mathrm{sec}}^{\ensuremath{-}1}$. Measured values of $\frac{1}{{F}^{*}{\ensuremath{\tau}}_{{Q}^{*}}}$ were in good agreement with the Schmid-Sch\"on expression, $\frac{1}{{F}^{*}{\ensuremath{\tau}}_{{Q}^{*}}}=(\frac{\ensuremath{\pi}\ensuremath{\Delta}}{4{k}_{B}{T}_{c}{\ensuremath{\tau}}_{E}})\ifmmode\times\else\texttimes\fi{}{(1+2\frac{{\ensuremath{\tau}}_{E}}{{\ensuremath{\tau}}_{S}})}^{\frac{1}{2}}$, for $\frac{\ensuremath{\Delta}}{{k}_{B}{T}_{c}}\ensuremath{\lesssim}0.8$, where ${\ensuremath{\tau}}_{E}^{\ensuremath{-}1}$ is the electron-phonon scattering rate estimated from the measured transition temperature. For larger values of $\frac{\ensuremath{\Delta}}{{k}_{B}{T}_{c}}$, the relaxation rate increased less rapidly with $\ensuremath{\Delta}$. The appropriate Boltzmann equation was solved on a computer to obtain values for $\frac{1}{{F}^{*}{\ensuremath{\tau}}_{{Q}^{*}}}$ in the range $0.5<~\frac{T}{{T}_{c}}<~0.999999$. The computed values of $\frac{1}{{F}^{*}{\ensuremath{\tau}}_{{Q}^{*}}}$ agreed with several analytic expressions valid for $\frac{\ensuremath{\Delta}}{{k}_{B}{T}_{c}}\ensuremath{\ll}1$, but not with the experimental data: The computed curves increased more rapidly than linearly with $\frac{\ensuremath{\Delta}}{{k}_{B}{T}_{c}}$ near ${T}_{c}$, and the shape of the $\frac{1}{{F}^{*}{\ensuremath{\tau}}_{{Q}^{*}}}$ vs $\frac{\ensuremath{\Delta}}{{k}_{B}{T}_{c}}$ curves was qualitatively different. This discrepancy suggests that either the generally accepted expression for exchange charge relaxation is incorrect, or that the Boltzmann equation is inappropriate for these calculations.