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RKKY Interactions in Graphene: Dependence on Disorder and Gate Voltage

We report the dependence of Ruderman-Kittel-Kasuya-Yoshida\,(RKKY) interaction on nonmagmetic disorder and gate voltage in grapheme. First the semiclassical method is employed to reserve the expression for RKKY interaction in clean graphene. Due to the pseudogap at Dirac point, the RKKY coupling in undoped grapheme is found to be proportional to $1/R^3$. Next, we investigate how the RKKY interaction depends on nonmagnetic disorder strength and gate voltage by studying numerically the Anderson tight-binding model on a honeycomb lattice. We observe that the RKKY interaction along the armchair direction is more robust to nonmagnetic disorder than in other directions. This effect can be explained semiclassically: The presence of multiple shortest paths between two lattice sites in the armchair directions is found to be responsible for the reduceddisorder sensitivity. We also present the distribution of the RKKY interaction for the zigzag and armchair directions. We identify three different shapes of the distributions which are repeated periodically along the zigzag direction, while only one kind, and more narrow distribution, is observed along the armchair direction. Moreover, we find that the distribution of amplitudes of the RKKY interaction crosses over from a non-Gaussian shape with very long tails to a completely log-normal distribution when increasing the nonmagnetic disorder strength. The width of the log-normal distribution is found to linearly increase with the strength of disorder, in agreement with analytical predictions. At finite gate voltage near the Dirac point, Friedel oscillation appears in addition to the oscillation from the interference between two Dirac points. This results in a beating pattern. We study how these beating patterns are effected by the nonmagnetic disorder in doped graphene.