Ferromagnetic order induced on graphene by Ni/Co proximity effects

We build a tight-binding Hamiltonian describing Co/Ni over graphene, contemplating ATOP (a Co/Ni atom on top of each Carbon atom of one graphene sublattice) and HCP (one Co/Ni atom per Graphene plaquette) configurations. For the ATOP configuration the orbitals involved, for the Co/Ni, are the $d_{z^2-r^2}$ which most strongly couple to one graphene sublattice and the $d_{xz}$, $d_{yz}$ orbitals that couple directly to the second sublattice site. Such configuration is diagonal in pseudo-spin and spin space, yielding electron doping of the graphene and antiferro-magnetic ordering in the primitive cell in agreement with DFT calculations. The second, HCP configuration is symmetric in the graphene sublattices and only involves coupling to the $d_{xz}$, $d_{yz}$ orbitals. The register of the lattices in this case allows for a new coupling between nearest neighbour sites, generating non-diagonal terms in the pseudo-spin space and novel spin-kinetic couplings mimicking a spin-orbit coupling generated by a magnetic coupling. The resulting proximity effect in this case yields ferromagnetic order in the graphene substrate. We derive the band structure in the vicinity of the K points for both configurations, the Bloch wavefunctions and their spin polarization.