Quasiparticle branch mixing rates in superconducting aluminum

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 ${\tau }_{Q^{*}}^{-1\mathrm{}}$ vs $\frac{\Delta }{k_B{T}_c}$ for several values of elastic scattering rate, and the quasiparticle branch relaxation rate ${\tau }_Q^{-1\mathrm{}}$ as a function of energy. The quasiparticle potential developed in the injection film is related to ${\tau }_Q^{-1\mathrm{}}$ and thence to ${\tau }_0^{-1\mathrm{}}$ a characteristic electron-phonon scattering time. Detailed measurements of ${\tau }_Q$ 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 ${\tau }_{Q^{*}}^{-1\mathrm{}}$ on $\frac{\Delta }{k_B{T}_c}$, values are deduced for the gap anisotropy of the films. In the cleanest samples, ${\tau }_0=0.10\mathrm{}\pm 0.02$ μ sec, a value that is in good agreement with energy-gap relaxation and $2\Delta$- phonon (phonons of energy $\gtrsim 2\Delta$) mean-free-path measurements, but a factor of about 4 smaller than that obtained from recombination time measurements and theoretical calculations. The value of ${\tau }_0^{-1\mathrm{}}$ 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 ${\tau }_0^{-1\mathrm{}}$ with $T_c$ may arise from either a strong dependence of ${\alpha }^{2}F\left(\omega \right)$ on $T_c$ or from a small concentration of magnetic impurities.