Thermodynamics of two-dimensional josephson junctions

We derive the effective free energy of a two-dimensional Josephson junction in the presence of an external current and predict that the junction has a phase transition at a temperature T_J below the bulk transition temperature T_c. In the range T_J < T < T_c long-range phase correlation exists in the bulk, but is destroyed in the vicinity of the junction. The critical current I_c is reduced by thermal fluctuations, for a junction of size L, I_c∼ L^b(T) where b(T) < 0 for T_J < T < T_c (i.e. I_c vanishes at L→∞) while 0 < b(T) < 2 for T < T_J. Our results may account for the absence of an observable supercurrent at temperatures below T_c in YBa₂Cu₃O_x- and Bi₂Sr₂CaCu₂O₈-based junctions.