Monte Carlo simulation of a model small-particle superconductor: Effects of disorder

We describe Monte Carlo simulations of a model three-dimensional inhomogeneous superconductor in which small superconducting grains are coupled together by Josephson tunneling. Disorder is included, but Coulomb effects arising from finite grain capacitances are omitted. An ordered simple cubic array of grains is found to exhibit a phase-ordering transition to a state of long-range phase coherence and zero resistivity. The phase-ordering transition becomes conspicuously well separated from the single-grain transition when the intergrain normal-state resistance is about $\frac{\hslash }{e^2}\sim 4000 \Omega$. The specific heat also changes from bulklike to single-particle-like behavior at this resistance. The specific-heat anomaly arising from the phase-ordering transition is found to be very weak. Disorder is studied in a model incorporting random intergrain coupling, via site dilution, and a model with random single-grain transition temperatures. The site-diluted model shows few qualitative differences from the ordered lattice, except that long-range phase coherence disappears as expected below the percolation threshold. If the volume fraction of superconducting grains is a function of temperature, then the phase-ordering temperature is found sometimes to occur above the peak in the specific heat, in agreement with experiments on granular A1.