Quasiparticle branch mixing rates in superconducting aluminum

Abstract

The kinetic equation is used to compute the elastic and inelastic quasiparticle branch mixing rates for a superconducting film into which quasiparticles are injected via a tunnel barrier from a second superconducting film. Representative graphs are presented of the steady-state quasiparticle distribution, the quasiparticle charge imbalance ${Q}^{*}$ versus injection current, the charge relaxation rate ${\ensuremath{\tau}}_{{Q}^{*}}^{\ensuremath{-}1}$ vs $\frac{\ensuremath{\Delta}}{{k}_{B}{T}_{c}}$ for several values of elastic scattering rate, and the quasiparticle branch relaxation rate ${\ensuremath{\tau}}_{Q}^{\ensuremath{-}1}$ as a function of energy. The quasiparticle potential developed in the injection film is related to ${\ensuremath{\tau}}_{Q}^{\ensuremath{-}1}$ and thence to ${\ensuremath{\tau}}_{0}^{\ensuremath{-}1}$ a characteristic electron-phonon scattering time. Detailed measurements of ${\ensuremath{\tau}}_{Q}$ are reported for films of superconducting A1, some of which were doped with oxygen to give a range of transition temperatures from 1.2 to 2.1 K. From the dependence of ${\ensuremath{\tau}}_{{Q}^{*}}^{\ensuremath{-}1}$ on $\frac{\ensuremath{\Delta}}{{k}_{B}{T}_{c}}$, values are deduced for the gap anisotropy of the films. In the cleanest samples, ${\ensuremath{\tau}}_{0}=0.10\ifmmode\pm\else\textpm\fi{}0.02$ \ensuremath{\mu} sec, a value that is in good agreement with energy-gap relaxation and $2\ensuremath{\Delta}$- phonon (phonons of energy $\ensuremath{\gtrsim}2\ensuremath{\Delta}$) mean-free-path measurements, but a factor of about 4 smaller than that obtained from recombination time measurements and theoretical calculations. The value of ${\ensuremath{\tau}}_{0}^{\ensuremath{-}1}$ in the A1 films increases with the transition temperature ${T}_{c}$ as ${T}_{c}^{5}$ or ${T}_{c}^{6}$, instead of ${T}_{c}^{3}$ as predicted by simple theory. It is suggested that the rapid increase of ${\ensuremath{\tau}}_{0}^{\ensuremath{-}1}$ with ${T}_{c}$ may arise from either a strong dependence of ${\ensuremath{\alpha}}^{2}F(\ensuremath{\omega})$ on ${T}_{c}$ or from a small concentration of magnetic impurities.