Long-range Josephson coupling through ferromagnetic graphene

We study the Josephson effect in graphene-based ballistic superconductor-ferromagnet-superconductor (SFS) junctions. We find an oscillatory Josephson coupling I(c)R(N) of F graphene whose amplitude is nonvanishing for a half-metallic graphene, increases for the exchange fields h above the Fermi energy E(F) and shows only a slow damping at strong exchange fields h >> E(F). We interpret this long-range Josephson coupling as the result of the exchange mediated Andreev-Klein process at FS interfaces which enhances the induced antiparallel-spin superconducting correlations in F graphene by increasing h above EF. We further demonstrate the existence of regular temperature-induced transitions between 0 and pi couplings in the plane of T and h where the phase boundaries have distinct shapes at the two regimes of h below and above E(F).