Charge-imbalance relaxation in the presence of a pair-breaking supercurrent in dirty, superconducting Al films

Abstract

The charge-imbalance relaxation rate, $\frac{1}{{F}^{*}{\ensuremath{\tau}}_{{Q}^{*}}}$, has been measured in dirty Al films in the presence of an applied supercurrent for $0.7\ensuremath{\lesssim}\frac{T}{{T}_{c}}\ensuremath{\lesssim}0.91$. The supercurrent ${I}_{s}$ induces an anisotropic energy gap, and hence allows elastic scattering to relax the charge imbalance. The effect of a supercurrent is equivalent to that of magnetic impurities, with a pair-breaking rate ${\ensuremath{\tau}}_{s}^{\ensuremath{-}1}=\frac{1}{2}D{(\frac{{p}_{s}}{\ensuremath{\hbar}})}^{2}$, where $D$ is the electron diffusion constant and ${p}_{s}$ is the momentum of a Cooper pair. The measured relaxation time, ${F}^{*}{\ensuremath{\tau}}_{{Q}^{*}}$, is found to depend on ${I}_{s}$ through the relation, ${F}^{*}{\ensuremath{\tau}}_{{Q}^{*}}({I}_{s})=\frac{{F}^{*}{\ensuremath{\tau}}_{{Q}^{*}}(0)}{{(1+b{{I}_{s}}^{2})}^{\frac{1}{2}}}$, where $b$ is a function of $\frac{T}{{T}_{c}}$. The measured temperature dependence of $b$ is consistent with previous measurements of $\frac{1}{{F}^{*}{\ensuremath{\tau}}_{{Q}^{*}}}$ in AlEr films, in which the Er is a pair-breaking magnetic impurity, and with the Schmid-Sch\"on theory. The measured magnitude of $b$ is in good agreement with the value expected from measured sample parameters. As with the data on AlEr, the measured value of $b$ is not consistent with results based on a numerical solution of the Boltzmann equation.