RKKY interactions in graphene: Dependence on disorder and Fermi energy

We report, how the indirect exchange interaction J(RKKY)(R) between magnetic moments at a distance R in graphene depends on nonmagmetic disorder strength W and gate voltage. First, a semiclassical method is used to rederive J(RKKY) in clean graphene, yielding the asymptotic decay 1/R2+alpha, where alpha = 1 is the power of the pseudogap at the Dirac point. Next, we perform numerical calculations with the Anderson tight-binding model on a honeycomb lattice. We observe that along the armchair direction J(RKKY) is more robust to nonmagnetic disorder than in other directions. This is explained semiclassically by the presence of more than one shortest path between two lattice sites in armchair directions, which is shown to reduce the disorder sensitivity compared to other directions. The distribution of J(RKKY) is calculated. We identify three different distribution shapes, repeated periodically along the zigzag direction, while only one kind, more narrow distribution, is observed along the armchair direction. We explain this by the different sensitivity to scattering phases. When increasing W, we find that the distribution crosses over to a logarithm-normal distribution. Its width is found to increase linearly with W. Moving away from the Dirac point, Friedel oscillations appear in addition to the one caused by the interference between two Dirac points. This results in a beating pattern. We study how this is effected by nonmagnetic disorder.