Anomalous transport in random superconducting composite systems

Abstract

We have studied anomalous diffusion in the random superconducting network (RSN) with a wide distribution of conductivity. We consider a composite medium of superconducting and normal conducting regions in which the normal conducting component obeys a transfer-rate distribution of the form W−(1+α) (0<α<1). In the static (dc) case below the percolation threshold pc, one finds that the dc conductivity varies as (pc−p)−s′ in the vicinity of pc with s′=1/α. Above the percolation threshold and in the RSN limit, the superconducting component is considered to possess a large but finite transfer rate Ws. In this limit, the dc conductivity follows the p=pc behavior for small Ws, crossing over to the behavior of ordinary percolation at a crossover value of the superconducting transfer rate Ws,co, which is found to vary as (p−pc)−(1+α)/α. The results are in good accord with scaling relations. Right at the percolation threshold, the frequency-dependent conductivity is calculated in the RSN limit. The real part of the conductivity (σR) at low frequencies initially follows the dc behavior, crossing over to the behavior σR∼ω1−α at high frequencies. The crossover frequencies are estimated for various relevant regions. The imaginary part of the conductivity (σI) has even more complex behaviors. At high frequencies, σI varies as ω1−α, being the same as the real part. The results are in accord with the scaling relations generalized to finite frequencies. The model is also numerically solved in the effective medium approximation to compare with the analytic results. Good agreements are found.