Josephson current in a graphene SG/ferromagnetic barrier/SG junction

The Josephson current passing through a SG1/FB/SG2 graphene junction, where SG and FB are those parts of a graphene layer which are induced into the superconducting state and into the ferromagnetic state, respectively, and where the small thickness of the FB layer L is studied. The ferromagnetic barrier strength is taken to be given by χH ∼ HL/hvF, where H is the strength of the exchange energy and vF ∼ 106 m/s is the Fermi velocity of quasiparticles. The eigenstates of the relativistic quasiparticles in the graphene are taken to be the solutions of the Dirac Bogoliubov-de Gennes equations. It is found that the energy levels of the Andreev bound states for the Weyl-Dirac particles in the SG1/FB/SG2 junction are independent of the direction of the spins and that they depend on the strength of ferromagnetic barrier potential. The critical supercurrent is seen to vary in an oscillatory (periodic) manner as χH is varied. The oscillatory behavior of the critical supercurrent carried by the Cooper pairs formed by massless the Weyl-Dirac particles is different from the behavior of the supercurrent carried by the Cooper pairs formed by non-relativistic particles in a conventional SC/FI/SC (FI being a ferromagnetic insulator) junction. In those types of junctions, the supercurrent does not exhibit a similar oscillatory dependence