I-U characteristics of diffusive SNS junctions with a multi-valley normal region

We study the I-U characteristics of SNS junctions with a multi-valley normal region. In doped semiconductors and semi-metals the relaxation of electrons to thermal equilibrium involves not only momentum relaxation within a given valley but also equilibration between different valleys. The associated inter-valley relaxation time τ_v may be significantly longer than the intra-band elastic relaxation time τ_el, yet much shorter than the inelastic relaxation time τ_in. This separation of time-scales has dramatic effects on the I-U characteristics of multi-valley SNS junctions. To study this we generalize the Larkin-Ovchinnikov kinetic equations to account for metals with multi-valley Fermi surfaces. Applying this formalism we find that the current response in voltage-bias is non-monotonic, there exists two maxima I_max,1, I_max,2 at voltages near (eτ_in)^-1 and (eτ_v)^-1, which may greatly exceed the temperature dependent critical current. In a current biased junction the I-U characteristic displays a jump in voltage of multiple orders of magnitude at a current I_jump at approximately the maxima in voltage bias, I_jump  ~ I_max,1.