Character of electronic states in graphene antidot lattices: Flat bands and spatial localization

Graphene antidot lattices have recently been proposed as a new breed of graphene-based superlattice structures. We study electronic properties of triangular antidot lattices, with emphasis on the occurrence of dispersionless (flat) bands and the ensuing electron localization. Apart from strictly flat bands at zero energy (Fermi level), whose existence is closely related to the bipartite lattice structure, we also find quasiflat bands at low energies. We predict the real-space electron density profiles due to these localized states for a number of representative antidot lattices. We point out that the studied low-energy localized states compete with states induced by the superlattice-scale defects in this system, which have been proposed as hosts for electron-spin qubits. Furthermore, we suggest that local moments formed in these midgap zero-energy states may be at the origin of a surprising saturation of the electron dephasing length observed in recent weak localization measurements in graphene antidot lattices.

Figure 1

(Color online) Graphene antidot lattice: (a) Lattice structure described by the basis vectors ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$, with magnitude $\left|{\mathbf{a}}_1\right|=\left|{\mathbf{a}}_2\right|=La\sqrt{3}$; (b) hexagonal unit cell with a circular antidot; and (c) hexagonal unit cell with a triangular antidot. Vectors ${\mathit{\delta }}_1,{\mathit{\delta }}_2,{\mathit{\delta }}_3$ specify positions of the nearest neighbors of a carbon atom on sublattice $A$.

Figure 2

(Color online) Band structure for a {9,3} antidot lattice. Only bands above the Fermi level $\left(\mathcal{E}=0\right)$ are shown because of the particle-hole symmetry. $\Gamma$, $K$, $M$ stand for the high-symmetry points in the Brillouin zone.

Figure 3

(Color online) Band structure of an antidot lattice with triangular antidots. Only bands above the Fermi level $\left(\mathcal{E}=0\right)$ are shown because of the particle-hole symmetry. $\Gamma$, $K$, $M$ stand for the high-symmetry points in the Brillouin zone. The flat band at $\mathcal{E}=0$ is (a) sixfold degenerate $\left(D=6\right)$ and (b) ninefold degenerate $\left(D=9\right)$.

Figure 4

(Color online) Illustration of the spatial localization of the single-particle wave function corresponding to a flat band. (a) Tunneling-current distribution $I\left(\mathbf{r}\right)$, reflecting the population of only the $A$-sublattice sites. The sizes of the circles and the color brightness are proportional to $I\left(\mathbf{r}\right)$, which is scaled to $0\le I\le 1$. The electron amplitudes obey the sum rule shown in the inset. (b) Tunneling-current maxima along the $y$ direction, at positions indicated by the arrows in (a). Inset: tunneling-current distribution in the $x$ direction.

Figure 5

(Color online) Effect of an on-site impurity potential $V_e=-0.15t$ along the antidot edges: (a) band structure and the tunneling-current distributions corresponding to (b) the lowest-lying flat band and (c) the highest flat band.

Figure 6

(Color online) Effects of debonded C atoms in the antidot lattice {9,3.2}: (a) band structure and the tunneling-current distributions corresponding to (b) the low-lying quasiflat bands and (c) the energy bands within the range $0.125t\lesssim \mathcal{E}\lesssim 0.257t$.

Figure 7

(Color online) Effects of disorder in antidot lattices: (a) band structure of an antidot lattice with defects on sublattice $A$ (indicated by the arrow in the inset), (b) band structure in the presence of defects on both sublattices, and (c) unit cell with arrows indicating vacancies or C adatom defects. The dashed curves in (a) represent the band structure of an ideal lattice (without disorder). $\Gamma$, $K$, $M$ stand for the high-symmetry points in the Brillouin zone of the antidot lattice with the fourfold-enlarged unit cell.

Figure 8

(Color online) Tunneling-current distributions for a lattice with irregularly shaped antidots, shown for the four lowest electron bands: ${\mathcal{E}}_0=0$ (brown), ${\mathcal{E}}_1\left(\mathbf{k}\right)$ (blue), ${\mathcal{E}}_2\left(\mathbf{k}\right)$ (red), and ${\mathcal{E}}_3\left(\mathbf{k}\right)$ (violet). The tunneling-current magnitudes are represented by circles of different sizes. The sublattice polarization due to edge defects, along with the accompanying charge-density reconstruction, is shown in the zoomed part of the figure.