Ruderman-Kittel-Kasuya-Yosida interaction at finite temperature: Graphene and bilayer graphene

We investigate the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction between magnetic impurities in both single layer and Bernal stacked bilayer graphene, finding a number of striking anomalies in the temperature dependence of this interaction. In undoped single layer graphene the strength of the RKKY interaction for substitutional impurities anomalously increases upon increasing temperature, an effect that persists up to and beyond room temperature. For impurities intercalated in the Bernal stacked bilayer and a doping that places the chemical potential near the antibonding band edge, a qualitative change of the RKKY interaction with temperature occurs: a low-temperature oscillatory interaction develops into a high-temperature antiferromagnetic coupling, accompanied by an overall increase of the interaction strength. The origin of the temperature anomalies can be traced back to specific features of the density of states: the vanishing density of states at the apex of the Dirac cone in single layer graphene, and the “kink” in the density of states at the antibonding band edge in the case of the Bernal bilayer.

Figure 1

Temperature dependence of the impurity interaction for two spin moments on the $A$ sublattice of pristine graphene ($\mu =0\mathrm{eV}$). The full (black) line is the exact RKKY result of Eq. (10) while the broken (red) line is the asymptotic result, valid for large $R$, given by Eq. (21). The interaction $\mathcal{J}$ is measured in terms of the coupling constant $C$ and multiplied by a factor of ${10}^5$; positive values correspond to ferromagnetic coupling and negative to antiferromagnetic coupling. The impurities are separated by $50a$ (with $a$ the graphene lattice constant), and the impurity vector is taken to be in the armchair direction. Other possible sublattice combinations yield similar results. The inset displays the turning point in the temperature dependence as a function of the separation.

Figure 2

Shown is the RKKY interaction for $A$ sublattice impurities in doped graphene ($\mu =0.1$ eV) as a function of impurity separation and for temperatures $T=10\mathrm{K}, T=200\mathrm{K}$, and $T=500\mathrm{K}$. In all cases the impurity separation vector is in the armchair direction. The asymptotic result [see Eq. (21)] is also shown as the broken line. The interaction $\mathcal{J}$ is measured in terms of the coupling constant $C$ and multiplied by a factor ${10}^5$; positive values correspond to ferromagnetic coupling and negative to antiferromagnetic coupling.

Figure 3

Temperature dependence of RKKY interaction for plaquette impurities separated by $R=20a$ (with $a$ the lattice constant of graphene). The interaction $\mathcal{J}$ is measured in terms of the coupling constant $C$; positive values correspond to ferromagnetic coupling and negative to antiferromagnetic coupling. In each case the nature of the coupling of the plaquette impurity to the graphene layer is indicated by the label ${\mathcal{J}}_{(i,j)}$: this represents the RKKY interaction between two plaquette impurities in which the first (second) impurity couples via spin flips between $i\mathrm{th}$ ($j\mathrm{th}$) nearest-neighbor carbon atoms or via on-site spin flips $i,j=0$. These processes are illustrated in the right-hand panel of the figure for spin flips between (a) first, (b) second, (c) third nearest neighbors, and (d) on-site spin flips. Evidently, the nature of the local coupling has a pronounced effect on the temperature dependence.

Figure 4

Left-hand panel: band structure of the $AB$ bilayer in which the width of the band line represents the contribution of that band to the asymptotic RKKY interaction between intercalated impurities in the bilayer. Evidently, the antibonding band edge is associated with a switching of the Fermi surface that drives the interaction from the low-energy chiral band to the high-energy antibonding band. The right-hand panel illustrates the impact this has on the RKKY interaction via the quantities ${\sum }_{R>R_c}J\left(\mathbf{R}\right)$ and ${\sum }_{R>R_c}\left|J\left(\mathbf{R}\right)\right|$. The former will, by cancellation, be close to zero for an oscillatory interaction and attain its maximum value for a monotonic interaction while the latter quantity measures the strength of the RKKY interaction. Clearly, as the Fermi energy crosses the antibonding band edge the “switching” of the Fermi circle driving the RKKY interaction results in a rapid change of the RKKY both in both form and in magnitude.

Figure 5

Anomalous temperature behavior at the antibonding band edge: The RKKY interaction for intercalated impurities shown as a function of separation for six different doping levels of the bilayer, and in each case for the temperatures 10, 100, 200, and 300 K. The interaction $\mathcal{J}$ is measured in terms of the coupling constant $C$; positive values corresponding to ferromagnetic coupling and negative to antiferromagnetic coupling. In all panels the impurity separation vector is taken in the armchair direction. At doping levels that place the chemical potential far from the antibonding band edge ($E_{\mathrm{g}}=0.4\mathrm{eV}$) the RKKY interaction shows the expected temperature damping, i.e., the form of the interaction is preserved while the amplitude decreases with increasing temperature. However, at the antibonding band edge (c), a strikingly different temperature dependence may be observed in that (i) the form of the RKKY interaction itself changes with temperature, from a low-temperature oscillatory behavior to a high-temperature antiferromagnetic behavior, and (ii) the magnitude of the interaction increases with temperature.

Figure 6

Temperature dependence of the sums ${\sum }_{R>R_c}\mathcal{J}\left(\mathbf{R}\right)$ (lower panel) and ${\sum }_{R>R_c}\left|\mathcal{J}\left(\mathbf{R}\right)\right|$ as a function of temperature for chemical potentials of 0.36, 0.37, 0.38, and 0.39 eV. The first of these two sums will be close to zero in the case of an oscillatory interaction, and attain its largest value and be equal in magnitude to the second sum for a monotonic interaction. This second sum evidently simply measures the strength of the RKKY interaction. As may be seen in the lower panel of this figure, there is a crossover from a low-temperature oscillatory form to a monotonic (AFM) form of the interaction with increasing temperature. At this crossover the magnitude of the interaction also increases significantly (see upper panel). For chemical potentials placed further from the antibonding band edge this crossover occurs at higher temperatures, before being lost altogether.

Figure 7

Overview of the temperature-dependent RKKY interaction for the intercalated impurity geometry. The right-hand column shows the RKKY interaction (scaled by $R^2/\mu$ for clarity), while the left-hand column displays the regions of ferromagnetic and antiferromagnetic interaction, indicated by the dark (black) shaded and light shaded (red) areas respectively. While the $T=0$ discontinuity at $0.4\mathrm{eV}$ in the interaction may still be observed at 10 K, it is progressively smoothed out with increasing temperature as the FM and AFM bands merge across the $T=0$ discontinuity. The vertical lines are at $R=20a$ and the interaction on these lines is displayed in Fig. 8; the horizontal lines indicate the gap energy of $E_{\mathrm{g}}=0.4\mathrm{eV}$.

Figure 8

Temperature smearing of the RKKY antibonding band-edge singularity: Shown is the RKKY interaction as a function of doping for intercalated magnetic impurities for temperatures of 10, 100, 200, and 300 K. The interaction $\mathcal{J}$ is measured in terms of the coupling constant $C$; positive values corresponding to ferromagnetic coupling and negative to antiferromagnetic coupling. In the low-temperature regime (10 K) a distinct derivative discontinuity is seen at the antibonding band edge of 0.4 eV, which is, however, rapidly smoothed with increasing temperature.