Dynamic RKKY interaction in graphene

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, based on which the energetically favorable alignment between magnetic moments embedded in this material can be calculated. The general consensus is that the strength of the RKKY interaction in undoped graphene decays as $1/D^3$ or faster, where $D$ is the separation between magnetic moments. Such an unusually fast decay for a two-dimensional system suggests that the RKKY interaction may be too short ranged to be experimentally observed in graphene. Here we show in a mathematically transparent form that a far more long ranged interaction arises when the magnetic moments are taken out of their equilibrium positions and set in motion. We not only show that this dynamic version of the RKKY interaction in graphene decays far more slowly but also propose how it can be observed with currently available experimental methods.

Figure 1

Log-log plots of the numerically calculated spin susceptibility (in arbitrary units) probed at a distance $D$ from a single magnetic impurity. Top panels correspond to the case of static susceptibility whereas bottom panels are for finite frequencies. Left (right) panels are for undoped (doped) cases. (Red) dots are the numerically calculated values and (black) lines display the predicted power laws also highlighted on each of the main panels. Insets display the corresponding calculations for two impurities where exactly the same power law behavior is found.

Figure 2

(a) Diagonal element of the spin susceptibility ${\chi }_{0,0}$ (in arbitrary units) obtained for a single magnetic impurity located at site 0 plotted as a function of the excitation frequency $\omega$. A single peak of width $W_1$ is highlighted. (b) Fractional deviations of the linewidth [$(W-W_1)/W_1$] plotted as a function of the relevant separation (in units of the lattice parameter of graphene). The solid (blue) line corresponds to the case of two impurities a distance $D$ apart. The gray area delimits the region spanned by the two-impurity result if all possible directions are included. (c) The scattered (red) dots correspond to the linewidths $W$ of the susceptibility ${\chi }_{m,m}$ for different sites $m$ in the case of a disordered array of magnetic impurities randomly distributed on graphene, as shown schematically in the inset, plotted as a function of the corresponding nearest-neighbor separation, $D_{NN}$, i.e., the distance to the impurity nearest $m$ in the configuration.