Quasiparticle lifetimes and tunneling times in a superconductor-insulator-superconductor tunnel junction with spatially inhomogeneous electrodes

The low-energy quasiparticle scattering and recombination lifetimes for a proximity sandwich of two superconductors S and S’ with different bulk energy gaps, are calculated as a function of the spatial coordinate and temperature. The spatial dependence of the order parameter and density of states are calculated on the basis of a microscopic model of the proximity effect, based on the Usadel equations, for dirty superconductors in thermal equilibrium. A zero boundary resistance between S and S’ and a Boltzmann-like energy distribution of the excess quasiparticles are assumed. In the case of a small diffusion time constant an effective quasiparticle relaxation rate into and excitation rate out of the reduced gap region in the SS’ sandwich are obtained as a function of (finite, but low) temperature and strength of the proximity effect, determined by the parameter ${\gamma }_{\mathit{m}}$, by averaging over the energies and positions of the quasiparticles. In the same way effective tunneling times for electrons and holes tunneling out of the trap in the SS’ sandwich to the other electrode of an SS’IS’ ’S junction are determined as a function of temperature, voltage, and ${\gamma }_{\mathit{m}}$.