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 $R_{\square }^c$ for the normal-state sheet resistance. $R_{\square }^c$ 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 $R_{\square }^c=5.7\mathrm{}$ kΩ. For the fractional correction from the next term in the expansion, the Bethe approximation gives $\frac{-3\mathrm{}}{5_Z^{}{}_{}{}^2}$, or -4% for $Z=4\mathrm{}$.