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 $f=p/q$ flux quanta per plaquette (where p and q are mutually prime integers), we find that the mean-field transition temperature ${T}_{c}(f)\ensuremath{\approx}{T}_{c}{(0)q}^{\ensuremath{-}b}$ with $b=1/4.$ By contrast, a mean-field theory which neglects phase variation along the array predicts $b=1/2,$ 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.