RKKY coupling in graphene

We study the carrier-mediated exchange interaction, the so-called Ruderman-Kittel-Kasuya-Yoshida (RKKY) coupling, between two magnetic impurity moments in graphene using exact diagonalization on the honeycomb lattice. By using the tight-binding nearest-neighbor band structure of graphene we also avoid the use of a momentum cutoff which plagues perturbative results in the Dirac continuum model formulation. We extract both the short and long impurity-impurity distance behavior and show on a qualitative agreement with earlier perturbative results in the long-distance limit but also report on a few new findings. In the bulk the RKKY coupling is proportional to 1/vertical bar R vertical bar(3) and displays [1+cos(2k(D).R)] -type oscillations. A-A sublattice coupling is always ferromagnetic whereas A-B subattice coupling is always antiferromagnetic and three times as large. We also study the effect of edges in zigzag graphene nanoribbons (ZGNRs). We find that for impurities on the edge the RKKY coupling decays exponentially because of the localized zero-energy edge states and we also conclude that a nonperturbative treatment is essential for these edge impurities. For impurities inside a ZGNR the bulk characteristics are quickly regained.