  Superconducting hybrid junctions are revealing a variety of novel effects.
Some of them are due to the special layout of these devices, which often use a
coplanar configuration with relatively large barrier channels and the
possibility of hosting Pearl vortices. A Josephson junction with a quasi ideal
two-dimensional barrier has been realized by growing graphene on SiC with Al
electrodes. Chemical Vapor Deposition offers centimeter size monolayer areas
where it is possible to realize a comparative analysis of different devices
with nominally the same barrier. In samples with a graphene gap below 400 nm,
we have found evidence of Josephson coherence in presence of an incipient
Berezinskii-Kosterlitz-Thouless transition. When the magnetic field is cycled,
a remarkable hysteretic collapse and revival of the Josephson supercurrent
occurs. Similar hysteresis are found in granular systems and are usually
justified within the Bean Critical State model (CSM). We show that the CSM,
with appropriate account for the low dimensional geometry, can partly explain
the odd features measured in these junctions.
