Modulation of Kekulé adatom ordering due to strain in graphene

Intervalley scattering of carriers in graphene at “top” adatoms may give rise to a hidden Kekulé ordering pattern in the adatom positions. This ordering is the result of a rapid modulation in the electron-mediated interaction between adatoms at the wave vector $\mathit{K}-{\mathit{K}}^{\prime }$, which has been shown experimentally and theoretically to dominate their spatial distribution. Here we show that the adatom interaction is extremely sensitive to strain in the supporting graphene, which leads to a characteristic spatial modulation of the Kekulé order as a function of adatom distance. Our results suggest that the spatial distributions of adatoms could provide a way to measure the type and magnitude of strain in graphene and the associated pseudogauge field with high accuracy.

Figure 1

Sketch of the Kekulé index ${\nu }_{\mathit{r}}=0,1,2$ (red, green, blue) of sites on each (A/B) sublattice in graphene. Hidden Kekulé ordering of top adatoms (in yellow) corresponds to adsorption on sites with equal ${\nu }_{\mathit{r}}$ (here ${\nu }_{\mathit{r}}=0$) as a result of adatom interaction mediated by carriers in graphene that undergo intervalley scattering. Sublattice correlation of adatoms may also arise from the same interaction.

Figure 2

Interaction potential between two adatoms (with ${\epsilon }_0=-0.15t$) on unstrained graphene, weakly attached ($t^{\prime }=0.7t$) to (a) the same and (b) different subblattices. The interaction potential results from the scattering of graphene carriers at the adatom sites. The potential is attractive for equal sublattices and repulsive otherwise. By color-coding the potential according to the Kekulé character ${\nu }_{\mathit{r}}$ of the separation vector $\mathit{r}={\mathit{r}}_2-{\mathit{r}}_1$, we see that at equilibrium the two adatoms rest on the same sublattice at sites with equal Kekulé character $[{\nu }_{\mathit{r}}=0\left(\mathrm{red}\right)\Rightarrow {\nu }_{{\mathit{r}}_1}={\nu }_{{\mathit{r}}_2}]$. Regardless of the radial dependence of the interaction potential [multiplied here by $\left(\right|\mathit{r}|/a_0{)}^3$ for visibility], its Kekulé components (colored insets) satisfy the simple forms in Eqs. (4) and (5), up to small corrections from higher angular harmonics.

Figure 3

Interaction potential between two adatoms as in Fig. 2, with a 1% uniform strain present in graphene, either (a) and (d) uniaxial ${\epsilon }_{xx}$, (b) and (e) uniaxial ${\epsilon }_{yy}$, or (c) and (f) a uniform shear ${\epsilon }_{xy}$. Equal-sublattice configurations are still preferred, but the Kekulé character acquires a modulation with distance that reflects the change in the intervalley separation $\mathit{K}-{\mathit{K}}^{\prime }$ by a pseudogauge field $2\mathit{A}\sim ({\epsilon }_{xx}-{\epsilon }_{yy},-2{\epsilon }_{xy})$.

Figure 4

Interaction potential (a) and (b) $U_{AA}$ and (c) and (d) $U_{AB}$ along the $x$ and $y$ directions at fixed ${\epsilon }_0=-0.15t$ as $t^{\prime }/t$ is varied. The atomic-scale Kekulé oscillations arise only in cuts along the $x$ direction [(a) and (c)]. In (a) and (c), dots denote positions with ${\nu }_{\mathit{r}}=0$ (same Kekuké character of adatoms). $U_{AB}$ and $U_{AA}$ have opposite signs, but the sign is inverted in an attractive-repulsive crossover that appears around $|t^{\prime }|\approx 1.5|t|$ [31].

Figure 5

Spatial map of the $U_{AA}\left(\mathit{r}\right)$ and $U_{AB}\left(\mathit{r}\right)$ adatom interaction in the strong-coupling limit ($t^{\prime }=5t$, ${\epsilon }_0=-0.15t$). (a) and (b) The case without strain and (c) and (d) the case with a 1% uniaxial strain along the $x$ direction.

Figure 6

Interaction potential for same-sublattice adatoms along the $x$ axis in the presence of an on-site electron-electron repulsion $U$. (a) The interaction potential for values of $U$ below $U_c$. (b) The evolution of the interaction potential in the ferromagnetic regime $U>U_c$.

Figure 7

$U_{AA}\left(r\right)$ cut at $y=0$ analogous to Fig. 2 in the main text as the damping factor of electrons in graphene is reduced from $0.04t$ to $0.005t$. The value chosen in the main text is $\delta =0.03t$. Note that as coherence increases, the potential becomes stronger and decays as $1/r^3$ (dashed line) at large distances.