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Magnetic field oscillations of the critical current in long ballistic graphene Josephson junctions

We study the Josephson current in long ballistic superconductor-monolayer graphene-superconductor junctions. As a first step, we have developed an efficient computational approach to calculate the Josephson current in tight-binding systems. This approach can be particularly useful in the long junction limit, which has hitherto attracted less theoretical interest but has recently become experimentally relevant. We use this computational approach to study the dependence of the critical current on the junction geometry, doping level, and an applied perpendicular magnetic field. In zero magnetic field we find a good qualitative agreement with the recent experiment of Ben Shalom et al. (Reference[12]) for the length dependence of the critical current. For highly doped samples our numerical calculations show a broad agreement with the results of the quasiclassical formalism. In this case the critical current exhibits Fraunhofer-like oscillations as a function of the magnetic field. However, for lower doping levels, where the cyclotron orbit becomes comparable to the characteristic geometrical length scales of the system, deviations from the results of the quasiclassical formalism appear. We argue that due to the exceptional tunability and long mean free path of graphene systems a new regime can be explored where geometrical and dynamical effects are equally important to understand the magnetic field dependence of the critical current.