Josephson junction in a thin film

The phase difference φ(y) for a vortex at a line Josephson junction in a thin film attenuates at large distances as a power law, unlike the case of a bulk junction where it approaches exponentially the constant values at infinities. The field of a Josephson vortex is a superposition of fields of standard Pearl vortices distributed along the junction with the line density φ′(y)/2π. We study the integral equation for φ(y) and show that the phase is sensitive to the ratio l/Λ, where l=λ_J²/λ_L, Λ = 2λ_L²/d, λ_L, and λ_J are the London and Josephson penetration depths, and d is the film thickness. For l≪Λ, the vortex "core" of the size l is nearly temperature independent, while the phase "tail" scales as √lΛ/y²=λ_J√2λ _L/d/y²; i.e., it diverges as T →T_c. For l≫Λ, both the core and the tail have nearly the same characteristic length √lΛ.