  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.
