Analytical expression for the RKKY interaction in doped graphene

We obtain an analytical expression for the Ruderman-Kittel-Kasuya-Yosida interaction $J$ in elctron- 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 the 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 $J\sim R^{-2}{k}_F\sin \left(2k_{F}R\right)\{1+\cos [(\vec{K}-\overrightarrow{K^{\prime }}).\vec{R}]\}$ 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.

Figure 1

RKKY interaction $J_{AA}$ obtained for the tight-binding bands (solid line), compared to the RKKY expression involving the Meijer G-function Eq. (15) (red dots) and its long-distance limit Eq. (19) (dashed line).

Figure 2

RKKY interaction for several cases. Black solid lines are the numerical results for the full tight-binding band structure, while the red/gray lines indicate the analytical results, Eqs. (15) and (25), for the linear bands.

Figure 3

The Meijer G-function as a function of $k_{F}R$ (black solid line) versus its asymptotic expansions [Eqs. (17) and (18)].

Figure 4

Beating pattern of the RKKY interaction for bond-centered moments separated along the zigzag direction.

Figure 5

Switching the exchange interaction between ferro and antiferro by changing the carrier density in graphene with gate voltage. The larger the distance between the moments, the earlier is the switching, which is controlled by $k_{F}R$.