Diamagnetism of percolative granular superconductors and diluted Josephson arrays

We discuss diamagnetic fluctuations in percolative granular superconductors modeled by randomly diluted Josephson arrays in which junctions are present with probability p and absent with probability 1-p. Results are obtained using mean-field theory and scaling arguments, which provide a qualitative picture for two- and three-dimensional granular materials. Asymptotically exact critical exponents are obtained in an {ε} expansion about d=6-{ε} dimensions. In zero external field, the singular part of the average diamagnetic susceptibility of finite clusters at zero temperature is proportional to {}p-pc{{}}^{t{{-}}2{ν}}$, where t is the exponent controlling the growth of the macroscopic conductivity in diluted resistor networks and {ν} is the percolation correlation length exponent. In nonzero external magnetic field, the disordered Josephson network exhibits spin-glassy behavior, and the singular part of the mean-square fluctuation in the magnetic moment of finite clusters is proportional to {}p-pc{{}}^{{t}₂{{-}}2{ν}}$, where ${t}₂=(d-2)ν+{{φ}}₂$ and ${{φ}}₂$ is a crossover exponent associated with the square of the temperature. We evaluate ${{φ}}₂$ to first order in {ε} and find it to be identical to an exponent describing corrections to scaling in the probability distribution for the resistance of a diluted resistor network at the percolation threshold, i.e., φ₂=1+0(ε²).