Competition between spin-orbit interaction and exchange coupling within a honeycomb lattice ribbon

Spin density patterns of a pinned magnetic impurity that is embedded in a honeycomb lattice with zigzag edges are investigated by employing a mean-field assisted Landauer-Keldysh formalism. Both the intrinsic spin-orbit coupling and the extrinsic localized magnetic moments are considered, and the effects of the pinning directions and the species of the sublattice on the electron spins are analyzed. A local time-reversal symmetry breaking cannot destroy the equilibrium edge-state spin accumulation that is induced by intrinsic spin-orbit coupling when the pinning field lies in the plane of the ribbon and the embedding position is the A-site at the edge. The induced local spin can be either parallel or antiparallel to the localized impurity moment, depending on the location of the pinned impurity, because itinerant electrons are found only at the B-site of the edge, but not at the A-site.