Effective sine-Gordon model for the static properties of narrow window junctions

We show that the maximum tunneling current for a long thin window Josephson junction in the presence of an external magnetic field is given to a good approximation by the solution of an effective sine-Gordon model with appropriate rescalings. This model is obtained via a reduction of the two-dimensional system to a one-dimensional integro-differential equation (nonlocal sine-Gordon). The passive region introduces an effective Josephson characteristic length. Using numerical simulations we show that for a narrow junction the cases of inline and overlap current feed are well described by the effective model. We discuss the influence of the width of the junction and find that the model gives a good approximation for the behavior of large area windows in the inline geometry, while it deviates for the overlap.