We study via Monte Carlo simulation the effective superfluid density $n_s$ and the real part of the integrated fluctuation conductivity, ${\gamma }_2$, of a model granular superconductor in which the individual superconducting grains are coupled via Josephson tunneling. The phase-ordering transition temperature $T_c$ is determined as the temperature at which $n_s$ goes to zero. Above an intergrain normal-state resistance $R\sim R_0=\frac{\hslash }{e^2}$, $T_c$ falls significantly below the single-grain transition temperature $T_{c0\mathrm{}}$, in agreement with our previous Monte Carlo results, and $n_s$ deviates substantially from typical bulk behavior. At temperature $T=0\mathrm{}$, we show analytically that $n_s$ in site-diluted samples is proportional to the effective conductance of the sample in its normal state. It follows that the zero-temperature penetration depth ${\lambda }_p\mathrm{}\left(0\right)\mathrm{}$ of the granular superconductor varies as the square root of the normal-state resistivity. Near percolation, ${\lambda }_p\mathrm{}\left(0\right)\mathrm{}\propto {(p-p_c)}^{-\frac{t}{2}}$, where $t$ is the percolation exponent describing effective conductivity in composites of normal metal and insulator. A sum rule is derived for ${\gamma }_2$, relating it to the Josephson coupling energy. ${\gamma }_2$ is found to have two characteristic contributions. One is due to thermodynamic fluctuations and appears near $T_c$ in ordered and weakly diluted lattices of superconducting grains. The other arises from "impurity modes" associated with sites near vacancies in site-diluted lattices. This contribution persists at all temperatures near or below $T_c$, and dominates over the first contribution above a site dilution of about 10%. The possibility of observing these effects experimentally is discussed.

• Received 25 April 1983

DOI:https://doi.org/10.1103/PhysRevB.28.5053