We study the exchange interaction $J$ between two magnetic impurities in undoped graphene (the Ruderman-Kittel-Kasuya-Yosida [RKKY] interaction) by directly computing the lattice Green’s function for the tight-binding band structure for the honeycomb lattice. The method allows us to compute $J$ numerically for much larger distances than can be handled by finite-lattice calculations as well as for small distances. In addition, we rederive the analytical long-distance behavior of $J$ for linearly dispersive bands and find corrections to the oscillatory factor that were previously missed in the literature. The main features of the RKKY interaction in half-filled graphene are that unlike the $J\propto (2k_{F}R){}^{-2}\sin (2k_{F}R)$ behavior of an ordinary two-dimensional metal in the long-distance limit, $J$ in graphene falls off as $1/R^3$, shows the $1+\cos [(K-K^{\prime })·R]$-type oscillations with additional phase factors depending on the direction, and exhibits a ferromagnetic interaction for moments on the same sublattice and an antiferromagnetic interaction for moments on the opposite sublattices as required by particle-hole symmetry. The computed $J$ with the full band structure agrees with our analytical results in the long-distance limit, including the oscillatory factors with the additional phases.

• Received 27 August 2010

DOI:https://doi.org/10.1103/PhysRevB.83.165425