Study of tunneling process effects on the fluctuation conductivity of a granular s-wave superconductor in nanometer-scale

We investigate thermal transport in a granular s-wave dirty superconductor at nanometer-scale in 3d in the limit of large tunneling conductance, JT ≫ 1 at near and far from the critical temperature. Calculations is carried out by using Green's function, we obtain the impurity vertex (Cooperon) and fluctuations propagator in the presence of impurities. Then, by concerning Kubo formula we evaluate the three distinct contributions of the Aslamazov-Larkin (AL), Maki-Thompson (MT) and density of states (DOS). The distinctive contributions to fluctuations conductivity depend differently on the tunneling because of their different natures. Therefore, we will show that in the limit of ε ≪ JTη/Tc, where η is the mean level spacing in a grain and ϵ=lnTTc≃(T−Tc)/Tc is the reduced temperature, the tunneling is effective also there is a crossover to the specific behavior of a homogeneous system, as T → Tc, from the point of view of the fluctuating Cooper pairs. In the limit of ε ≫ JTη/Tc, the tunneling is not effective, and the system behaves as an ensemble of real zero-dimensional grains.