Theory of scanning gate microscopy imaging of the supercurrent distribution in a planar Josephson junction

We theoretically investigate the mapping of the supercurrent distribution in\na planar superconductor-normal-superconductor junction in the presence of a\nperpendicular magnetic field via the scanning gate microscopy technique. We\nfind that the distribution of counter-propagating supercurrents aligned in\nJosephson vortices can be mapped by the change of the critical current induced\nby the tip of the scanning probe, if the flux in the junction is set close to\nmaxima of the Fraunhofer pattern. Instead, when the magnetic field drives the\njunction to a supercurrent minimum in the Fraunhofer pattern, the\nsuperconducting phase adapts, and the tip always increases the supercurrent.\nThe perpendicular magnetic field leads to the formation of Josephson vortices,\nwhose extension for highly transparent junctions depends on the current\ncirculation direction. We show that this leads to an asymmetric supercurrent\ndistribution in the junction and that this can be revealed by scanning gate\nmicroscopy. We explain our findings on the basis of numerical calculations for\nboth short- and long-junction limits and provide a phenomenological model for\nthe observed phenomena.\n