  The Josephson current is investigated in a superconducting graphene bilayer
where pristine graphene sheets can make in-plane or out-of-plane displacements
with respect to each other. The superconductivity can be of an intrinsic
nature, or due to a proximity effect. The results demonstrate that the
supercurrent responds qualitatively differently to relative displacement if the
superconductivity is due to either intralayer or interlayer spin-singlet
electron-electron pairing, thus providing a tool to distinguish between the two
mechanisms. Specifically, both the AA and AB stacking orders are studied with
antiferromagnetic spin alignment. For the AA stacking order with intralayer and
on-site pairing no current reversal is found. In contrast, the supercurrent may
switch its direction as a function of the in-plane displacement and
out-of-plane interlayer coupling for the cases of AA ordering with interlayer
pairing and AB ordering with either intralayer or interlayer pairing. In
addition to sign reversal, the Josephson signal displays many characteristic
fingerprints which derive directly from the pairing mechanism. Thus,
measurements of the Josephson current as a function of the graphene bilayer
displacement open up the means achieve deeper insights into the superconducting
pairing mechanism.
