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The induced-proximity effects of nearly commensurate lattice structure of agraphene layer on Ni(111) and Co(0001) substrates in the AC stackingconfiguration are addressed through an analytical tight-binding approach withinthe Slater-Koster method. A minimal Hamiltonian is constructed by consideringthe hybridizations of the magnetic $3d$-orbitals of Ni(Co) atoms with the$p_z$-orbitals of graphene, in addition to the atomic spin-orbit coupling andthe magnetization of the Ni(Co) atoms. A low-energy effective Hamiltonian forgraphene/Ni(Co) describing the perturbed $\pi$-bands in the vicinity of theDirac points is derived which enable us to get further insight on the physicalnature of the induced-effective couplings to the graphene layer. It is shownthat a magneto-spin-orbit type effect may emerge through two competingmechanisms simultaneously present, namely the proximity induced exchange andRashba spin-orbit interaction. Such effects results in giant exchangesplittings and robust Rashba spin-orbit coupling transferred to the graphenelayer in agreement with recent density functional theory calculations andexperimental observations. We further analyze the physical conditions for theappearance of intact Dirac cones in the minority spin bands as observed byrecent photoemission measurements with spin resolution.

Induced exchange and spin-orbit effects by proximity in graphene on Ni and Co