Asymptotic discontinuities in the RKKY interaction in the graphene Bernal bilayer

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 $R^{-5/2}$ 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 $\lambda =\sqrt{2}\pi \hslash v_F/t_{\perp }$ as $E_F\to E_g$ from below the gap edge, while it limits to $\lambda \to \infty$ if $E_F\to E_g$ from above the gap edge $(t_{\perp }$ 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.

Figure 1

Lattice structure and labeling convention for the Bernal (AB) stacked bilayer used in this work.

Figure 2

Band structure for the Bernal $\left(AB\right)$ stacked bilayer. (Left) Band structure for the Bernal $\left(AB\right)$ stacked bilayer with the bonding-antibonding gap edge at $E_F=\pm 0.4$ eV indicated by the horizontal dashed lines. The three $\mathbf{k}$-space vectors necessary for understanding the asymptotic RKKY interaction are also shown. (Right) Projection of the eigenfunction onto the bonding sublattice (i.e., the $A$ in the notation of Fig. 1).

Figure 3

Convention for the labeling of high-symmetry $K$ points used in this work, see also Table 1.

Figure 4

The full RKKY interaction and its asymptotic form for Fermi energies in the bonding-antibonding gap of the Bernal bilayer, plotted in the armchair direction. The interaction $J$ is measured in terms of the coupling constant $C$, Eq. (17), with negative values correspond to antiferromagnetic coupling and positive to ferromagnetic coupling. Each column of panels presents data for a different sublattice configuration: the first column [(a)–(d)] the case where both impurities are on the $A$ sublattice, the second column [(e)–(h)] to the case where the impurities occupy different sublattices, and the third column [(i)–(l)] to the case where both impurities are on the $B$ sublattice. Note that the asymptotic RKKY interaction for energies in the bonding-antibonding gap is independent of the layer position of the impurities, and hence in each panel contains only one set of asymptotic RKKY data.

Figure 5

The RKKY interaction at the bonding to antibonding gap edge: the full RKKY interaction and $O\left(R^{-2}\right)$ and $O\left(R^{-5/2}\right)$ asymptotic approximations to it, plotted along the armchair direction for impurities on the $A$ sublattice and for the Fermi energy placed at the bonding-antibonding gap edge $(E_F=t_{\perp }=0.4$ eV), with the upper inset panel presenting an extended range of impurity separations. For impurities on the $A$ sublattice, two Fermi surface spanning vectors drive the asymptotics: $2k_F^{\perp }$ and $k_F^{\perp }$ (the vector $k_F^{\perp }$ is illustrated in the lower inset panel). These lead, respectively, to $R^{-2}sin\left(2k_F^{\perp }\right)$ and $R^{-5/2}cos(k_F^{\perp }R-\pi /4)$ terms in the asymptotic expansion of the RKKY interaction, see Eq. (34) and Table 4. Evidently, the full RKKY interaction (full line) is in pronounced disagreement with the asymptotic RKKY interaction if only the $O\left(R^{-2}\right)$ term is considered (dotted line), but is well described when both $O\left(R^{-2}\right)$ and $O\left(R^{-5/2}\right)$ are included (dashed line). This $O\left(R^{-5/2}\right)$ term is important only on the $A$ sublattice [compare (d) and (l) of Fig. 4], and occurs due to the logarithmic divergence in the propagator on this sublattice when the doping level places the Fermi energy at the gap edge $E_F=t_{\perp }$.

Figure 6

Relative weights in the asymptotic RKKY interaction of the three Fermi surface spanning vectors that contribute for $E>t_{\perp }=0.4\mathrm{eV}$: $2k_F^{+},2k_F^{-}$, and $(k_F^{+}+k_F^{-})$. Shown are the ratios of the weights of the $2k_F^{-}$ and $(k_F^{+}+k_F^{-})$ spanning vectors to the weight of the $2k_F^{+}$ spanning vector, $W^{-}/W^{+}$ and $W^{+-}/W^{+}$, respectively (see Fig. 2 for geometry of these spanning vectors, and Eq. (35) for the asymptotic RKKY interaction expressed in terms of $W^{+},W^{+}$, and $W^{+-})$. For $E<t_{\perp }$, only $W^{+}$ is nonzero. The discontinuity in $W^{-}/W^{+}$ and $W^{+-}/W^{+}$ for impurities on the $A$ sublattice is related to the divergence of the real space propagator on this sublattice at $E_F=t_{\perp }=0.4\mathrm{eV}$.

Figure 7

RKKY impurity interaction $J$ as a function of Fermi energy for impurities on the $A$ sublattice (A-A), $B$ sublattice (B-B), and different sublattices (A-B), with the scale for $J$ given on the left-hand side. The vertical line indicates the energy of the bonding-antibonding gap edge. The impurity separation is $20a$ in all cases and the separation vector taken in the armchair direction $(a$ is the lattice parameter of graphene). The interaction is measured in terms of the coupling constant $C$, see Eq. (17); negative values correspond to an antiferromagnetic coupling, positive to a ferromagnetic coupling. A derivative discontinuity in the interaction for impurities on the $A$ sublattice occurs at $E_F=t_{\perp }=0.4$ eV, the edge of the bonding-antibonding gap. Also shown is the interaction for the case of intercalated (interstitial) impurities as a function of Fermi energy, with the scale for $J$ in this case given by the axis on the right-hand side, and the impurity separation again $20a$. For this impurity, the derivative discontinuity is considerably more pronounced.

Figure 8

The full RKKY interaction and its asymptotic form plotted for the strongly doped Bernal bilayer with the impurity separation vector in the armchair direction. (a) and (b) correspond to the case where both impurities are on the $A$ sublattice, (c) and (d) to the case where the impurities occupy different sublattice, and (e) and (f) to the case where both impurities occupy the $B$ sublattice. The interaction $J$ is measured in terms of the coupling constant to the Dirac gas, Eq. (17); positive values corresponding to ferromagnetic coupling and negative to antiferromagnetic coupling. Note that the figure legend apply to the data shown in all panels.

Figure 9

Interaction for intercalated impurities in the $AB$ bilayer with the impurity separation vector taken in the armchair direction. (a) to (f) show the full RKKY interaction, $J$, and the asymptotic RKKY interaction, $J_{\text{asympt}}$, for specific Fermi energies, while (g) shows a density plot of $J{R}^2/E_F$ plotted vs Fermi energy $E_F$ and impurity separation $R$. Note that the interaction is measured in terms of the coupling constant $C$, see Eq. (17), and hence negative values of $J$ correspond to an antiferromagnetic interaction and positive to a ferromagnetic interaction. At $E_F=t_{\perp }=0.4$ eV, the interaction changes sharply (though continuously) from an oscillatory to antiferromagnetic character; the Fermi surface spanning vector driving the asymptotics however changes discontinuously, compare (c) and (d) and see also Fig. 11. The dashed vertical line in (g) indicates the line section of the interaction plotted in Fig. 7, while the dot dashed lines in (a) to (f) show the full RKKY interaction for a different coupling scheme for the intercalated impurity, see text for details.

Figure 10

Relative weights of the Fermi surface spanning vectors in the RKKY interaction energy for intercalated (interstitial) and plaquet impurities in the Bernal bilayer. The impurity separation vector is taken in the armchair direction in all cases. For $E>t_{\perp }=0.4$ eV, the two distinct Fermi circles allow for three Fermi surface spanning vectors: $2k_F^{+},2k_F^{-}$, and $(k_F^{+}+k_F^{-})$. Shown are the ratios of the weights of the $2k_F^{-}$ and $(k_F^{+}+k_F^{-})$ spanning vectors to the weight of the $2k_F^{+}$ spanning vector, $W^{-}/W^{+}$ and $W^{+-}/W^{+}$, respectively (see inset picture for geometry of these spanning vectors, and Eq. (35) for the asymptotic RKKY interaction expressed in terms of $W^{+},W^{+}$, and $W^{+-})$. Above the bonding-antibonding gap edge $\left(E_F=t_{\perp }\right)$ the intercalated impurity interaction is dominated by the $2k_F^{-}$ spanning vector.

Figure 11

Asymptotic discontinuity in the RKKY interaction of intercalated impurities: half periods, i.e., node spacing, of the intercalated impurity RKKY interaction in the armchair direction (full circles), along with the period corresponding to the $k_F^{+}$ Fermi surface spanning vector $(E_F<t_{\perp }=0.4$ eV) and the $k_F^{-}$ spanning vector $(E_F>t_{\perp }=0.4$ eV). Inset picture indicates these spanning vectors in relation to the Bernal bilayer band structure.

Figure 12

Plaquet impurity interaction for the $AB$ bilayer. (a) to (f) show the full RKKY interaction, $J$, and the asymptotic RKKY interaction, $J_{\text{asympt}}$, for specific Fermi energies, while (g) shows a density plot of $J{R}^2/E_F$ plotted vs Fermi energy $E_F$ and impurity separation $R$. At $E_F=t_{\perp }=0.4$ eV, the interaction energy changes sharply (though continuously). The dashed vertical line indicates the line section of the interaction plotted in Fig. 7.

Figure 13

Intercalated impurity interaction for a series of Fermi energies plotted as a function of interlayer separation, for a coupling scheme in which the impurity couples to the Dirac gas via second nearest-neighbor spin flips. Note the dramatic change in the RKKY period as the Fermi energy crosses the bonding to antibonding gap edge of $E_{\mathrm{gap}}=0.4$ eV.

Figure 14

Intercalated impurity interaction for a series of Fermi energies plotted as a function of interlayer separation, for a coupling scheme in which the impurity couples to the Dirac gas via nearest-neighbor spin flips. For this coupling scheme, in contrast to the result of Fig. 13, there is no marked change in the RKKY period at the gap edge of $E_{\mathrm{gap}}=0.4$ eV.

Figure 15

Exponent $n={log}_{10}[J\left(R\right)/J\left(R_0\right)]/{log}_{10}(R/R_0)$ of the RKKY interaction of plaquet impurities, $J\left(R\right)\propto R^{-n}$, for undoped graphene in various coupling schemes. Here, $R_0$ is the largest separation that can be calculated within floating point precision $(\approx 35a)$.

Figure 16

The RKKY interaction of plaquet impurities in doped single-layer graphene in various coupling schemes for the impurity. In the fully coherent scheme, all spin-flip processes have equal weight; in the nearest-neighbor coherent scheme only intersite spin flips between nearest-neighbor carbon atoms are allowed. Note that for the fully coherent coupling scheme the interaction is multiplied by a scale factor of ${10}^5$ for ease of viewing.