RKKY interaction between adsorbed magnetic impurities in graphene: Symmetry and strain effects

The growing interest in carbon-based spintronics has stimulated a number of recent theoretical studies on the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction in graphene, with the aim of determining the most energetically favorable alignments between embedded magnetic moments. The RKKY interaction in undoped graphene decays faster than expected for conventional two-dimensional materials, and recent studies suggest that the adsorption configurations favored by many transition-metal impurities may lead to even shorter-ranged decays and possible sign-changing oscillations. Here, we show that these features emerge in a mathematically transparent manner when the symmetry of the configurations is included in the calculation. Furthermore, we show that by breaking the symmetry of the graphene lattice, via uniaxial strain, the decay rate, and hence the range, of the RKKY interaction can be significantly altered. Our results suggest that magnetic interactions between adsorbed impurities in graphene can be manipulated by careful strain engineering of such systems.

Figure 1

Schematic representation of the graphene lattice showing with the armchair ($A$) and zigzag ($Z$) directions and units of separation ($l_A$ and $l_Z$), the two-atom unit cell (shaded area) and lattice vectors ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$, and the bond lengths $R_1$, $R_2$, and $R_3$ between an atom on the lattice and its nearest neighbors. The filled and hollow symbols represent sites on different sublattices. The bottom panels show magnetic impurities ($X$) attached to graphene lattice atoms $x_i$ in the center-adsorbed (left) and bridge-adsorbed (right) configurations.

Figure 2

Numerical (symbols) and analytical (dashed lines) evaluations of the real (top panels) and imaginary (bottom panels) components of ${\Gamma }_{AB}$ for two center-adsorbed type impurities with separations of $30l_A$ in the left panels and $60l_Z$ in the right panels.

Figure 3

Numerical evaluation of the coupling between center-adsorbed impurities as a function of separation $D$ in the armchair (left) and zigzag (right) directions. The armchair (zigzag) results are multiplied by $D^7$ ($D^6$) to highlight the features discussed in the text. The red dashed line in the main panels highlights the boundary between AFM (above) and FM (below) couplings. In the armchair direction, an initial AFM interaction decays extremely rapidly as $D^{-10}$ before a sign change to FM and a decay of $D^{-7}$ at larger separations. Zigzag separations reveal that every third value of separation has an FM interaction approximately two orders of magnitude smaller than the AFM majority values. The insets in each case show log-log plots where dashed lines show the slopes corresponding to the relevant decay rates.

Figure 4

${\Gamma }_{AB}$ (top) and coupling (bottom) for bridge-adsorbed impurities separated in the armchair (left) and zigzag (right) directions. An excellent match is noted between numerical (symbols) and analytic (lines) results for ${\Gamma }_{AB}$ for separation of $35l_A$ and a separation of $60l_Z$ for both real (black) and imaginary (red) components. A monotonically decaying $D^{-3}$ FM interaction is seen in the armchair direction for the class of bridge adsorbates investigated, whereas a sign-changing oscillation is observed in the zigzag case. The phase of the oscillations is found to vary with the hopping parameter between the impurities and the carbon atoms, as shown in the inset.

Figure 5

Numerically evaluated indirect exchange interaction $J\left(\varepsilon \right)$ between two center-adsorbed impurities fixed distances apart in the (a) armchair and (b) zigzag directions as uniaxial strain is applied perpendicular to the separation direction. The results are normalized relative to the unstrained coupling $J\left(0\right)$. The inset in panel (a) shows a closeup of the region highlighted by a dotted rectangle in the main plot. (c) Log-log plots of coupling against zigzag direction separation for armchair direction strains of $\varepsilon =0.0$ (black, solid line), $0.05$ (red, dashed line), and $0.1$ (green, dashed-dotted line). The black dotted lines show linear regressions with slopes of $-6.8$, $-3.5$, and $-3.4$, respectively. (d) Decay exponent $\alpha$ as a function of strain for the cases shown in (c) and additional values.

Figure 6

The effect of strain on the indirect exchange interaction between bridge-adsorbed impurities separated in the armchair (left panels) and zigzag (right panels) direction. In all panels, red (green) plots correspond to an armchair (zigzag) direction strains. (a) and (b) show the separation dependence of numerically calculated interactions for unstrained (black, solid line) and 5% armchair (red, dashed line) or zigzag (green, dashed-dotted line) strains. The insets show log-log plots, confirming the persistence of the $D^{-3}$ decay rate. The bottom panels show the change in the coupling, relative to the unstrained coupling, as a function of strain for fixed separations of $80l_A$ [(c) and (d)] and $80l_Z$ [(e) and (f)]. For armchair strains [(c) and (e)] the large red dots represent numerical evaluations and the thin red lines the analytic predictions given in the text. Only numerical evaluations are shown for zigzag strain cases [(d) and (f)]. The dashed lines in (c) and (d) represent numerical evaluations for a second class of bridge atoms (see main text).