Mean-field theory for arrays of Josephson-coupled wires

Abstract

We describe a mean-field theory for phase transitions in a Josephson junction array consisting of two sets of parallel wire networks, arranged at right angles and coupled together by Josephson interactions. In contrast to earlier treatments, we include the variation of the superconducting phase along the individual wires; such variation is always present if the wires have finite thickness and are sufficiently long. The mean-field result is obtained by treating the individual wires exactly and the coupling between them within the mean-field approximation. For a perpendicular applied magnetic field of strength f ϭp/q flux quanta per plaquette ͑where p and q are mutually prime integers͒, we find that the mean-field transition temperature T c ( f )ϷT c (0)q Ϫb with bϭ1/4. By contrast, a mean-field theory which neglects phase variation along the array predicts bϭ1/2, and gives a T c which diverges in the thermodynamic limit. The model with phase variations agrees somewhat better with experiment on large arrays than does the approximation which neglects phase variations.

Critical currents of Josephson-coupled wire arrays

“…4 This same effect causes the critical current of the array to saturate at a finite value which depends on the Josephson penetration depth, even as N → ∞. 5,6 The exactness of mean-field theory is believed to have an analog in the dynamics of these arrays. Chandra et al 7,8 have studied a version of local dynamics equivalent, in the large-N limit, to the resistively-shunted junction (RSJ) model with Gaussian thermal noise and no applied current (they use a different numerical approach from that followed here).…”