Universal criterion for the onset of superconductivity in granular films

In a granular film, modeled by a Josephson-coupled lattice, virtual tunneling of quasiparticles results in a critical value R_ᶜ for the normal-state sheet resistance. R_ᶜ is a universal upper bound for zero-temperature superconductivity and is computed in descending powers of z, the number of grain neighbors. Mean-field theory for a square lattice gives the leading term R_ᶜ=5.7 k{Ω}. For the fractional correction from the next term in the expansion, the Bethe approximation gives -35Z^_², or -4% for $Z=4$.