Nonlocal effect on the fluctuation conductivity of granular superconductors (Q627640)

The paper is devoted to the development of the theory of the superconducting fluctuations near the transition into the superconducting state with an analysis of fluctuation corrections for electrical conductivity where the pairing leads to three contributions named the Aslamazov-Larkin (AL), the Maki-Thompson (MT), and density of states (DOS) terms. The fluctuation corrections of the first order are considered in the case of the small impurity concentrations (which are low enough that the localization length to be larger than the grain size) on the base of a simplified model where metal grains form a regular lattice and taking into account non-local electron scattering in the clean superconductor. Restricting to the region of temperatures near the critical temperature and zero field, it is assumed that the main contribution to the macroscopic resistivity of the granular system comes from the intergranular tunneling. First, the Hamiltonian describing the system is written, and the Kubo formula is used to derive an expression for the electrical conductivity in a granular superconductor. In the framework of the Matsubara imaginary time formalism, the electromagnetic response operator defined on Matsubara frequencies is presented as the correlator of two one-electron Green's functions averaged over impurity positions and accounting for interactions. All diagrams which contribute to the conductivity of the granular metal are described. Scattering of the electrons inside the grains by impurities is included in the Born approximation giving rise to a scattering mean free time. Then there are calculated the impurity vortex (or Cooperon) and fluctuation propagator in the presence of dilute impurities. The Cooperon is calculated in momentum representation by using the explicit expression for the polarization operator. The propagator of superconducting fluctuations is defined by using above-mentioned diagrams as a function of the total energy in the Cooper channel at arbitrary mean free path and magnetic fields. The non-local fluctuation propagator is defined via the polarization operator determined as a loop of two single particle Green's functions in the particle-particle channel. Then the total zero-field fluctuation conductivity is defined by the corrections to the conductivity due to AL, DOS and MT contributions. These corrections are considered in the dirty and clean cases. It is shown that in the clean case the large negative DOS contribution is compensated by the positive anomalous MT one and the total fluctuation correction in clean case is reduced to the AL term, only.