Long-Range Interaction between Adatoms in Graphene

We present a theory of electron-mediated interaction between adatoms in graphene. In the case of resonant scattering, relevant for hydrogenated graphene, a long-range $1/r$ interaction is found. This interaction can be viewed as a fermionic analog of the Casimir interaction, in which massless fermions play the role of photons. The interaction is an attraction or a repulsion depending on whether the adatoms reside on the same sublattice or on different sublattices, with attraction dominating for adatoms randomly distributed over both sublattices. The attractive nature of these forces creates an instability under which adatoms tend to aggregate.

Figure 1

Electron-mediated interaction between adatoms in graphene modeled by a hard-core potential: numerical results (black dots) and mean-field theory, Eq. (18) [red (medium gray) line]. The net interaction is a repulsion when adatoms are randomly placed on one of the sublattices, and an attraction when they are equally distributed over both sublattices. The $3/2$ power law (1) provides an accurate fit to the numerical results with the best-fit values ${\epsilon }_0=-0.75\mathrm{eV}$ (top curve) and ${\epsilon }_0=1.3\mathrm{eV}$ (bottom curve). System of size $48\times 82$ was used for simulation, each data point was averaged over 20 realizations of randomly generated adatom configurations. Inset: Attracting adatoms tend to aggregate. Phase diagram obtained from the free energy (19) is shown.

Figure 2

Electron-mediated interaction between adatoms depends on the type of sublattice: atoms on different sublattices, $A$ and $B$, attract (a), whereas atoms on the same sublattice repel (b) [see Eqs. (11, 13)]. The interaction is modulated by a prefactor which takes different values on the three sub-sublattices marked by 2, $2^{\prime }$ and $2^{\prime \prime }$: (a) $|\sin (\mathbf{K}\mathbf{r}+\varphi )|=|\sin \varphi |,|\sin (\varphi +\frac{2\pi }{3})|,|\sin (\varphi -\frac{2\pi }{3})|$; (b) ${cos}^2(\mathbf{K}\mathbf{r})=1$, $\frac{1}{4}$, $\frac{1}{4}$. The modulation results from interference between electronic states in valleys $K$ and $K^{\prime }$.

Figure 3

The interaction (10, 12) for zero and nonzero energy $\delta$ of the adatom resonance, Eq. (4). The interaction retains the $1/r$ form at distances $r\lesssim \hslash v_0/\delta$, decreasing more rapidly at larger $r$. When the system is doped away from neutrality, similar behavior is expected at distances shorter than the Fermi wavelength, $r\lesssim {\lambda }_F$.