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 couples to one graphene sublattice and the $d_{xz},d_{yz}$ orbitals that couple directly to the second sublattice site. Such configuration is diagonal in pseudospin and spin space, yielding electron doping of the graphene and antiferromagnetic 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-neighbor sites, generating nondiagonal terms in the pseudospin 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 wave functions and their spin polarization.

Figure 1

Schematic picture of the configurations of a Co monolayer adsorbed on graphene. (Left) ATOP configuration: Co atoms are directly over the atoms of sublattice A and atoms of the sublattice B are in the hcp sites. (Right) HCP configuration: atoms of the sublattices A and B are at hcp sites. In both cases the magnetic order of cobalt, as well as the resulting magnetic order of the sublattices A and B, is indicated.

Figure 2

Positions of the Co first neighbor around A (left) and B (right), for the ATOP configuration. The orbitals that intervene in the overlaps of Co with A and B are drawn in each case.

Figure 3

Positions of the Co first neighbor around A (left) and B (right), for the HCP configuration. The orbitals that intervene in the overlaps of Co with A and B are drawn in each case.

Figure 4

Positions of the Co and B carbon atoms around a carbon atom A, in the HCP configuration of Co/graphene. $l=1,2,3$ represents the Co surrounding A, and $m=1,2,3$ represents the hops from A to B.

Figure 5

Band structure in the vicinity of the Dirac point for a monolayer of Co over graphene in the ATOP configuration (center left) and the HCP configuration (center right). The Fermi level (zero of energy) is indicated in both plots with a continuous line. In both cases the graphene layer is $n$ doped [17]. The eigenvectors corresponding to each band are indicated for both configurations. $k_{0x}$ is the adimensional wave vector in the $\widehat{x}$ direction. In this plot $p_y=0,p_x=\hslash k_x-\hslash K_{\xi }$, and $\xi =+1$. The behavior of the total magnetization of graphene ${\langle S_z\rangle }_T$, given by the contribution of all bands, as a function of the wave vector $k_x=k_{0x}(4\pi /3a)$, for the ATOP configuration (top left) and the HCP configuration (top right). The shaded regions (blue) indicate the zones in which the spin polarization is different from zero.