Crossover from phase fluctuation to amplitude-dominated superconductivity: A model system

We have experimentally studied a model system that demonstrates the crossover from a superconductor that is dominated by phase fluctuations, to one in which the amplitude of the order parameter is the controlling influence on $T_c.$ This model system is comprised of two-dimensional granular Pb with an overlayer of Ag. The system displays many aspects of the phase diagram of the concentration dependence of $T_c$ in the high-$T_c$ superconductors, and this crossover has been applied to explain the phase diagram in that case. We point out the similarities and differences between the model system presented in this paper and the high-$T_c$ superconductors.