RKKY interaction in monolayer and bilayer graphene: Exact results in terms of the Meijer-G functions

We present the results for the RKKY interaction in monolayer and bilayer graphene in terms of Meijer-G functions for the undoped, doped and biased cases. The results are obtained from the linear-response expression for susceptibility written in terms of the integral over Green's functions and using Dirac bands. The salient features of the large-distance behavior in each case will be discussed. For instance, for doped monolayer graphene, the interaction falls off as $R^{-2}$ and oscillates as the product of two terms, one being a $\{1+ \cos[(K-K').R]\}$-like interference term from both Dirac cones and the second term scaled by Fermi momentum $k_F$. For doped and unbiased bilayer graphene, the interaction decays as $R^{-2}\cos(k_FR)[e^{-k_FR}+\sin(k_FR)]\Phi_{K,K'}$ where $\Phi_{K,K'}$ is a similar Dirac-cone factor. For the gated bilayer graphene, $k_F$ must be replaced by another scaled momentum $k_U$, which depends on Fermi energy, gate voltage and the interlayer hopping energy, allowing the possibility of tuning of the interaction by gate voltage. References: M. Sherafati and S. Satpathy, PRB, 83, 165425 (2011); PRB, 84, 125416 (2011); AIP Conf. Proc. 1461, 24 (2012)