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

The charge-imbalance relaxation rate, 1F*τ_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 0.1Δ(T)kBTc1.4 for Er concentrations varying from 21 to 1660 at. ppm that produced estimated exchange scattering rates, τS^-1, from about 10⁹ sec^-1 to 5×{}10¹⁰ sec^-1. Measured values of 1F*τ_Q* were in good agreement with the Schmid-Sch\"on expression, ${1}{{F}*{{τ}}_{{Q}*}}=({{π}{Δ}}{4{k}B{T}c{{τ}}E})×{}{(1+2{{{τ}}E}{{{τ}}S})}1/2$, for ${{Δ}}{{k}B{T}c}{}0.8$, where ${{τ}}E^{{-}1}$ is the electron-phonon scattering rate estimated from the measured transition temperature. For larger values of ${{Δ}}{{k}B{T}c}$, the relaxation rate increased less rapidly with ${Δ}$. The appropriate Boltzmann equation was solved on a computer to obtain values for ${1}{{F}*{{τ}}_{{Q}*}}$ in the range $0.5<~{T}{{T}c}<~0.999999$. The computed values of ${1}{{F}*{{τ}}_{{Q}*}}$ agreed with several analytic expressions valid for ${{Δ}}{{k}B{T}c}{}1$, but not with the experimental data: The computed curves increased more rapidly than linearly with ${{Δ}}{{k}B{T}c}$ near ${T}c$, and the shape of the ${1}{{F}*{{τ}}_{{Q}*}}$ vs ${{Δ}}{{k}B{T}c}$ 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.