Counterpropagating topological interface states in graphene patchwork structures with regular arrays of nanoholes

Topological properties of graphene with nanoholes are theoretically investigated in detail. It is revealed that two types of hole arrangements, honeycomb array and triangular array, give rise to two topologically distinct states. We give a concise two-step strategy for choosing array structures, where two types of Brillouin zone folding, corner-to-corner or corner-to-center, play crucial roles. The topological nontriviality is confirmed by interface states at a domain wall between the honeycomb and triangular hole arrays, which show a cross-shaped band structure resembling helical edge states in quantum spin Hall states. We also diagnose the topology of the states from the view point of the topological crystalline insulator and show that the parity index against the $C_2$ rotation is a key quantity. These findings indicate that nanotechnology-based geometrical design and manipulation are promising in exploration of topological phases of matter.

Figure 1

(a) Brillouin zone for the $2\times 2$ superstructure compared with that of graphene as a typical corner-to-corner folding and a schematic picture for the corresponding band dispersion folded by the hole arrays matching this condition. The Dirac cones are preserved in this case. (b) Brillouin zone for the $\sqrt{3}\times \sqrt{3}$ superstructure as a typical corner-to-center folding, and the corresponding band structures with a gap induced by the superstructure. (c), (d) Lattice structure with $1\times 1$ (pristine graphene) and $4\times 4$ superstructures of arrays of nanoholes formed by removing six carbon atoms. (e), (f) Honeycomb and triangular hole arrays by an extra $\sqrt{3}\times \sqrt{3}$ ordering on top of the $4\times 4$ superstructure.

Figure 2

Bulk band dispersions for the sequence of the corner-to-corner folding induced by six-carbon hexagonal holes. (a) Pristine graphene. (b) $4\times 4$ structure. (c) $11\times 11$ structure ($11=3\times 4-1$).

Figure 3

Bulk band structures for the $4\sqrt{3}\times 4\sqrt{3}$ superstructure of (a) the honeycomb hole array and (b) the triangular hole array. The left panels are derived from the tight-binding model, while the right panels are from DFT calculations, where the parities and the irreducible representations for the valence top and the conduction bottom are respectively denoted.

Figure 4

(a) Schematic picture for the graphene patchwork with an interface between two regions with $4\sqrt{3}\times 4\sqrt{3}$ honeycomb and triangular hole arrays. (b) Band structure of the graphene patchwork with the brown dispersions representing the topological interface states. For the numerical calculations, we work on a stripelike structure of infinite length in the $x$ direction including two regions (honeycomb and triangular hole arrays) with the same width $16\times 4\times {\left({\tilde{\mathit{a}}}_2\right)}_y$ alternating in the $y$ direction.

Figure 5

(a) Local current distribution and wave function for the pseudospin-up interface state. The size of the arrows in the left panel represents the magnitude of the current, while the size and color of the disk on each of the lattice sites in the right panel represent the modulus and the phase of the wave function, respectively. Here, (a)–(c) are obtained at $k_{\parallel }=0.04\pi /\left|3{\mathit{a}}_1\right|$ for which the pseudospin-up state can be readily isolated from the down state by their energies [see Fig. 4], whereas (d) and (e) [and also Eqs. (4, 5, 6, 7)] are for $k_{\parallel }=0$. (b), (c) Closeup views of the regions enclosed by the (b) orange and (c) green dashed lines in panel (a). (d), (e) Bulk $|p_{+}+i{d}_{+}\rangle$ state in the regions with the (d) honeycomb and (e) triangular hole arrays, where $|p_{+}\rangle$ and $|d_{+}\rangle$ states are given by $|p_{+}\rangle \propto |p_x\rangle +i|p_y\rangle$ and $|d_{+}\rangle \propto |d_{x^2-y^2}\rangle -i|d_{xy}\rangle$, respectively.

Figure 6

(a) $5\times 5$ superstructure where the circles and double circles are the guides for the two extra $\sqrt{3}\times \sqrt{3}$ superstructures. (b) Band structure for the interface between $5\sqrt{3}\times 5\sqrt{3}$ triangle and honeycomb hole arrays. (c) 24-carbon hexagonal hole and (e) 42-carbon hexagonal hole. (d) Band structure for the patchwork of $8\sqrt{3}\times 8\sqrt{3}$ with 24-carbon hexagonal holes. (f) Band structure for the patchwork of $8\sqrt{3}\times 8\sqrt{3}$ with 42-carbon hexagonal holes. The setup for the calculation is the same as that in Fig. 4, except for that the width of each region is $16\times 5\times {\left({\tilde{\mathit{a}}}_2\right)}_y$ for panel (b) and $12\times 8\times {\left({\tilde{\mathit{a}}}_2\right)}_y$ for panels (d) and (f).

Figure 7

Schematic description of the superstructure to gap out the Dirac cones. (a), (c) Gapped cases with the opposite mass terms. (b) Critical point with the gapless Dirac cones. This conceptually works as an intermediate state between panels (a) and (c), but in practice there is no need to fabricate this structure.

Figure 8

(a)–(f) Bulk band structures for the $n\sqrt{3}\times n\sqrt{3}$ superstructures ($n=4$, 5, and 6). The $n=4$ case is the reproduction of Fig. 4 for comparing with others. The upper (lower) panels are for the triangular (honeycomb) hole array. (g) Schematic picture for the $6\sqrt{3}\times 6\sqrt{3}$ superstructure. (h) Band structure for the patchwork between $6\sqrt{3}\times 6\sqrt{3}$ triangular and honeycomb hole arrays. The setup for the calculation is the same as that in Fig. 4.

Figure 9

Bulk wave functions at the $\mathrm{\Gamma }$ point corresponding to the $E_{1g}$ and $E_{2u}$ states near the Fermi energy, where the $E_{1g}$ and $E_{2u}$ states are resolved into {$p_x$, $p_y$, $d_{x^2-y^2}$, $d_{xy}$}. The upper panels are for the honeycomb hole arrays, while the lower panels are for the triangular hole arrays. The $|p_{+}+i{d}_{+}\rangle$ states in Figs. 5 and 5 are also reproduced here for easy comparison. The wave functions are presented in the same way as in Fig. 5.