Electrical control of the RKKY interaction in bilayer graphene

The Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction between impurity spins is calculated for bilayer graphene in the presence of a layer symmetry-breaking external electric field. We find that for intercalated impurities (i.e., impurity atoms between the two constituent layers of the bilayer) the interaction is extraordinarily sensitive to such a field. In particular, (i) the form of the RKKY interaction may be tuned between oscillatory, ferromagnetic, and antiferromagnetic simply by varying the external field, and (ii) the strength of the RKKY interaction may be increased by an order of magnitude by application of an external field. This sensitivity arises directly from the “Mexican hat” form that the low-energy spectrum takes in an applied field. These finding suggest that heterostructures of intercalated magnetic atoms in bilayer graphene may represent a possible system for electrical control over magnetic structure.

Figure 1

Two intercalated impurities in AB-stacked bilayer graphene. Each impurity spin couples via a contact interaction to the ten surrounding carbon atoms of the bilayer (light shaded/green atoms).

Figure 2

Hexagonal Brillouin zone of bilayer graphene with the high-symmetry $K$-points labeled.

Figure 3

The total density of states of biased bilayer graphene $\rho \left(E\right)={\sum }_i{\rho }_i\left(E\right)$ (right) and a schematic illustration of the low-energy spectrum in vicinity of the $K$ point (middle). The two low-energy bands are separated by a band gap $E_g=V{t}_{\perp }/\sqrt{V^2+t_{\perp }^2}$ and the distance between the two minima (maxima) is given by $\mathrm{\Delta }k=V\sqrt{(V^2+2t_{\perp }^2)/(V^2+t_{\perp }^2)}/\left(2\hslash v_F\right)$. The DOS is defined piecewise in the energy intervals on the left. The function $u\left(E\right)$ must be inserted in its $\eta \to 0$ limit as defined in Eq. (17).

Figure 4

Overview of the RKKY interaction for a fixed Fermi energy of 100 meV. The left panel presents a density plot of the spin coupling type—ferromagnetic or antiferromagnetic—as a function of impurity separation and interlayer bias. The low-energy band structure for three representative values of this interlayer bias (0.1, 0.2, and 0.3 eV) is displayed in the associated panels indicated by arrows; the dashed lines on the density plots correspond to these bias values. The right panel shows the magnitude of the RKKY interaction plotted over the same variables, with the interaction clearly massively enhanced for values of the interlayer bias that place the Fermi energy in the Mexican hat region. The connection vector between the impurities is assumed to be in the armchair direction, and thus the intervalley scattering factors $f$ [see Eqs. (23, 24, 25)] are all unity. The temperature is set to 10 K.

Figure 5

The RKKY interaction for a fixed Fermi energy of 100 meV and a selection of bias potentials representative of the behavior seen in Fig. 4. The form of the RKKY interaction evolves from oscillatory at low bias, through a massively enhanced antiferromagnetic coupling, to a ferromagnetic form. The RKKY interaction is evaluated at 10 K and the connection vector of the impurities $\mathbf{R}$ is taken to be in the armchair direction, and thus the intervalley scattering factors [Eqs. (23, 24, 25)] are all unity.

Figure 6

Effect of finite temperature on the RKKY interaction. The Fermi energy is pinned to a value of 100 meV and the interlayer bias is fixed at 0.26 eV. While the RKKY interaction changes form from antiferromagnetic to ferromagnetic, it does not significantly reduce in magnitude. The connection vector of the impurities $\mathbf{R}$ is taken to be in the armchair direction and thus the intervalley scattering factors [Eqs. (23, 24, 25)] are all unity.

Figure 7

RKKY interaction at fixed particle density. The particle density at $E_F=100$ meV and zero bias is maintained for all finite values of the interlayer bias. The inset displays the large $R$ behavior. The connection vector of the impurities $\mathbf{R}$ is taken to be in the armchair direction and thus the intervalley scattering factors [Eqs. (23, 24, 25)] are all unity.

Figure 8

The RKKY interaction at the Dirac point for a range of bias potentials, $T=10$ K. The connection vector of the impurities $\mathbf{R}$ is taken to be in the armchair direction, and thus the intervalley scattering factors [Eqs. (23, 24, 25)] are all unity.