Phase transitions in a disordered granular superconductor near percolation

The properties of a disordered granular superconductor consisting of superconducting grains of size comparable to the zero-temperature bulk superconducting coherence length embedded in a nonsuperconducting host are studied by means of a randomly diluted Josephson tunnel junction model near a percolation threshold pc. A replicated (n{→}0) continuum Landau-Ginzburg field theory describing the macroscopic properties of this material is derived from first principles. The mean-field phase diagram as a function of temperature T, applied magnetic field H, and grain concentration p exhibits a Meissner phase, an Abrikosov vortex lattice, and a spin-glass phase all arising from the low-temperature phase coherence of the condensate wave function among the grains. For H=0, in the superconducting phase, the macroscopic superfluid density ρₛ{~}(p-pc{)}ᵗas p→{p}c$ where t=3 in mean-field theory. The spin-glass phase arises from frustration among loops of the percolating network in the presence of an applied magnetic field. Here ${{ρ}}ₛ=0, leading to complete flux penetration on average but with a frozen-in distribution of randomly oriented tunneling supercurrents leading to power-law decaying ~{x}^{{{-}}(d{{-}}2)}$ local fluctuations in the B field. In the low-temperature limit, vortices are shown to consist of spin-glass cores in a superconducting background.