Josephson critical current of long SNS junctions in the presence of a magnetic field

We evaluate the Josephson critical current of a long and wide two-dimensional superconductor-normal metal-superconductor (SNS) junction, taking into account the effect of electron reflection off the side edges of the junction. Considering clean junctions, we find that the effect of edges alters the usual Fraunhofer-like dependence of the Josephson critical current~$I_c$ on the magnetic flux~$\Phi$. At relatively weak fields, $B\lesssim \Phi_0/W^2$, the edge effect lifts zeros of the $I_c(\Phi)$ dependence and gradually shifts the maxima of that function by $\Phi_0/2$. (Here $W$ is the width of the junction and $\Phi_0$ the magnetic flux quantum.) At higher fields, $B\gtrsim\Phi_0/W^2$, the edge effect leads to an accelerated decay of the critical current $I_c(\Phi)$ with increasing $\Phi$. Our results are robust with respect to the roughness of realistic boundaries. Finally, we discuss the role of mesoscopic fluctuations of $I_c(\Phi)$ originating from the scattering off the edges, and compare our findings to recent experiments.