Fate of global superconductivity in arrays of long SNS junctions

Normal-metal films overlaid with arrays of superconducting islands undergo Berezinskii-Kosterlitz-Thouless (BKT) superconducting transitions at a temperature $T_{BKT}$. We present measurements of T$_{BKT}$ for arrays of mesoscopic Nb islands patterned on Au films for a range of island spacings $d$. We show that $T_{BKT} \sim 1/d^2$, and explain this dependence in terms of the quasiclassical prediction that the Thouless energy, rather than the superconducting gap, governs the inter-island coupling. We also find two deviations from the quasiclassical theory: (i) $T_{BKT}$ is sensitive to island height, because the islands are mesoscopic; and (ii) for widely spaced islands the transition appears to lead, not to a superconducting state, but to a finite-resistance "metallic" one.