Fluctuations electrical conductivity in a granular s-wave superconductor

The present study tries to evaluate the fluctuation electrical conductivity in a granular s-wave superconductor at the temperature near to the critical temperature. The evaluation is conducted under the condition of limited tunneling conductance between the grains and small impurity concentration. All the first order fluctuation corrections, involving the nonlocal scattered electron in a granular s-wave superconductor, are calculated in three dimensions and in the limit of clean. Using Green's function theory initially, the Cooperon (impurity vertex), λ(q,ε1,ε2), and the fluctuation propagator, Lk(q,Ωk), are calculated in the presence of impurities. Then, the three distinct contributions of Aslamazov-Larkin, Maki-Thompson, and Density of states are calculated by means of the Kubo formula. Analysis shows that the terms of Aslamazov-Larkin and anomalous Maki-Thompson have positive contributions to the conductivity in the clean limit, whereas the terms of Density of state and the regular Maki-Thompson have negative signs, leading to the reduction of total fluctuation conductivity.