RKKY Interaction in Graphene from Lattice Green's Function

TL;DR

This paper presents a numerical and analytical study of the RKKY interaction between magnetic impurities in graphene, revealing a 1/R^3 decay, sublattice-dependent magnetic coupling, and directional oscillations, extending understanding beyond previous models.

Contribution

The authors develop a lattice Green's function method to compute the RKKY interaction in graphene for large distances and derive corrected analytical long-distance behavior including phase factors.

Findings

RKKY interaction decays as 1/R^3 in graphene.

Interaction is ferromagnetic on the same sublattice, antiferromagnetic on opposite.

Oscillations depend on direction and include additional phase factors.

Abstract

We study the exchange interaction $J$ between two magnetic impurities in graphene (the 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. % avoids the use of a cutoff function often invoked in the literature to curtail the diverging contributions from the linear bands and yields results that are valid for all 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 graphene are that unlike the $J \propto (2k_FR)^{-2} \sin (2k_FR) $ behavior of an ordinary 2D metal in the long-distance limit, $J$ in graphene falls off as $1/R^3$, shows the $1 + \cos ((K-K').R)$-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.