RKKY interactions in graphene: Dependence on disorder and Fermi energy

We report, how the indirect exchange interaction $J_{\mathrm{RKKY}}\left(R\right)$ between magnetic moments at a distance $R$ in graphene depends on nonmagmetic disorder strength $W$ and gate voltage. First, a semiclassical method is used to rederive $J_{\mathrm{RKKY}}$ in clean graphene, yielding the asymptotic decay $1/R^{2+\alpha }$, where $\alpha =1$ is the power of the pseudogap at the Dirac point. Next, we perform numerical calculations with the Anderson tight-binding model on a honeycomb lattice. We observe that along the armchair direction $J_{\mathrm{RKKY}}$ is more robust to nonmagnetic disorder than in other directions. This is explained semiclassically by the presence of more than one shortest path between two lattice sites in armchair directions, which is shown to reduce the disorder sensitivity compared to other directions. The distribution of $J_{\mathrm{RKKY}}$ is calculated. We identify three different distribution shapes, repeated periodically along the zigzag direction, while only one kind, more narrow distribution, is observed along the armchair direction. We explain this by the different sensitivity to scattering phases. When increasing $W$, we find that the distribution crosses over to a logarithm-normal distribution. Its width is found to increase linearly with $W$. Moving away from the Dirac point, Friedel oscillations appear in addition to the one caused by the interference between two Dirac points. This results in a beating pattern. We study how this is effected by nonmagnetic disorder.

Figure 1

(a) Propagating paths of an electron between positions 0 and $\mathit{R}$. (b) Graphene lattice, where the two sublattices are denoted as A and B. Two representative directions (zigzag and armchair) are indicated as dashed gray lines. ${\theta }_{\mathit{R}}$ is the angle between $\mathit{R}$ and the unit vector $\widehat{\mathit{x}}$.

Figure 2

$J_{\mathrm{RKKY}}$ multiplied by $R^3$, along (a) ${\theta }_R=0$ (zigzag-AA), (b) ${\theta }_R=\pi /12.5$, (c) ${\theta }_R=\pi /10$, and (d) ${\theta }_R=\pi /3$ (armchair-AA) direction in the diffusive regime, averaged over 1600 disorder configurations. ${\theta }_R$ is defined in Fig. 1b ($5\times {10}^5$ sites, $M=5\times {10}^3$).

Figure 3

Schematic diagram of the shortest path in (a) the zigzag-AA and (b) armchair-AA directions. ${\delta }_{1\left(2\right)}$ denote the phase shifts due to nonmagnetic disorder.

Figure 4

The distribution of $\sqrt{J_{\mathrm{RKKY}}^2}$ multiplied by $R^3$, for (a) the first, (b) second, (c) third of the triplet of points along zigzag-AA direction, see Fig. 4f, and (d) for the armchair direction. For comparison, these distributions are plotted together in (e) ($W=t$, $2\times {10}^4$ sites $M={10}^3$).

Figure 5

(a) Averaged density of states in a double logarithmic plot. Inset: linear plot ($M=7\times {10}^3$). (b) Geometrical averaged $J_{\mathrm{RKKY}}$ along zigzag-AA direction ($M=5\times {10}^3$). $3\times {10}^7$ sites are used. Black lines in (a) are fittings to $\rho \left(E\right)=\gamma \left|E\right|$ with $\gamma =0.95,1.07,1.3$ for $W=0,0.5t,t$, respectively.

Figure 6

(a) Absolute value of $J_{\mathrm{RKKY}}$ in a disordered sample as a function of distance $R$. (b) The distribution of the logarithm of $|J_{\mathrm{RKKY}}|$ for $R=50\sqrt{3}a$ and different $W$. The inset of (b) shows the distribution of the amplitude $J_{\mathrm{RKKY}}$ itself for $W=4t$. $1.8\times {10}^5$ sites and $M=3\times {10}^3$ are used. Black dashed lines are logarithm-normal distribution functions with $\sigma =2.4,3.2,4.35$ and $\alpha =32.5,50,72.5$ for $W=4t,6t,8t$, respectively (see text).

Figure 7

Plot of the width of the distribution of $|J_{\mathrm{RKKY}}|$ as a function of $W$ in units of $t$. Red line: linear fitting, yielding $\sigma =W/2+0.34$.

Figure 8

(a) $J_{\mathrm{RKKY}}$ multiplied by $R^2$ in doped graphene at the chemical potential $\mu =0.1t$ along the zigzag-AB direction. Density plots of $J_{\mathrm{RKKY}}$ for (b) $\mu =0.2t$ and (c) $\mu =t$ for all directions. In (a), the results of calculations with KPM and the lattice Green's function method are shown as solid blue and dashed red lines, respectively. ( $7.2\times {10}^5$ sites, $M=5\times {10}^3$.)

Figure 9

Average $J_{\mathrm{RKKY}}$ multiplied by $R^2$, along (a) zigzag-AA, (b) armchair-AA directions at $\mu =0.2t$ in the diffusive regime. $N=1600$. Lattice size and $M$ as in Fig. 2.

Figure 10

(a) $J_{\mathrm{RKKY}}$ as a function of $M$ ($5\times {10}^5$ sites). (b) The smallest $M$ that yields a convergent result as a function of $R$.

Figure 11

$J_{\mathrm{RKKY}}$ as a function of system size $L$. $50\sqrt{3}a$ is used as the longest distance $R_F$ ($M=5\times {10}^3$).