Analytical Expression for the RKKY Interaction in Doped Graphene

TL;DR

This paper derives an analytical expression for the RKKY interaction in doped graphene, revealing its oscillatory behavior and distance dependence, and corrects previous inaccuracies in the literature.

Contribution

It provides the first analytical formula for the RKKY interaction in doped graphene using Meijer G-functions, clarifying its oscillatory nature and distance scaling.

Findings

The RKKY interaction oscillates with distance and Fermi wavevector.

The interaction decays as R^{-2} for doped graphene, similar to 2D electron gases.

At zero doping, the decay reverts to R^{-3}.

Abstract

We obtain an analytical expression for the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction $J$ in electron or hole doped graphene for linear Dirac bands. The results agree very well with the numerical calculations for the full tight-binding band structure in the regime where the linear band structure is valid. The analytical result, expressed in terms of the Meijer G-function, consists of a product of two oscillatory terms, one coming from the interference between the two Dirac cones and the second coming from the finite size of the Fermi surface. For large distances, the Meijer G-function behaves as a sinusoidal term, leading to the result $J \sim R^{-2} k_F \sin (2 k_F R) {1 + \cos[(K-K').R]}$ for moments located on the same sublattice. The $R^{-2}$ dependence, which is the same for the standard two-dimensional electron gas, is universal irrespective of the sublattice location and the distance direction of the two moments except when $k_F =0$ (undoped case), where it reverts to the $R^{-3}$ dependence. These results correct several inconsistencies found in the literature.