Asymptotic discontinuities in the RKKY interaction in the graphene Bernal bilayer

Abstract

We derive the asymptotics of the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction between magnetic impurities in the graphene Bernal bilayer using a four-band model of the bilayer spectrum. We find three distinct regimes depending on the position of the Fermi energy in the bilayer spectrum: in the bonding-antibonding gap, at the gap edge, and outside the gap. In particular, for impurities on the bilayer bonding sublattice (the ``$A$'' sublattice) and Fermi energies close to the bonding-antibonding gap edge ${E}_{g}$, we identify a (integrable) logarithmic divergence of the integrand of the RKKY exchange integral. This divergence drives a number of novel RKKY effects for impurities on the $A$ sublattice: (i) an asymptotic ${R}^{\ensuremath{-}5/2}$ term at the gap edge and (ii) a derivative discontinuity in RKKY interaction as a function of the Fermi energy at the gap edge. In the case of intercalated impurities (impurities between the two layers of the bilayer), we find a remarkable discontinuity in the period of the RKKY oscillation at the gap edge. The period of the oscillation limits to $\ensuremath{\lambda}=\sqrt{2}\ensuremath{\pi}\ensuremath{\hbar}{v}_{F}/{t}_{\ensuremath{\perp}}$ as ${E}_{F}\ensuremath{\rightarrow}{E}_{g}$ from below the gap edge, while it limits to $\ensuremath{\lambda}\ensuremath{\rightarrow}\ensuremath{\infty}$ if ${E}_{F}\ensuremath{\rightarrow}{E}_{g}$ from above the gap edge $({t}_{\ensuremath{\perp}}$ is the interlayer coupling, ${v}_{F}$ the Fermi velocity of graphene). The origin of this discontinuity we attribute to (i) the $A$ sublattice divergence and (ii) interference effects driven by the intrinsic valley degree of freedom of graphene. On this basis, we predict that the magnetic response of intercalated bilayer graphene will show a profound sensitivity to doping for Fermi energies near the bonding-antibonding gap edge.