Topological Edge States at a Tilt Boundary in Gated Multilayer Graphene

Despite much interest in engineering new topological surface (edge) states using structural defects, such topological surface states have not been observed yet. We show that recently imaged tilt boundaries in gated multilayer graphene should support topologically protected gapless edge states. We approach the problem from two perspectives: the microscopic perspective of a tight-binding model and an ab initio calculation on a bilayer, and the symmetry-protected topological (SPT) state perspective for a general multilayer. Hence, we establish the tilt-boundary edge states as the first concrete example of the edge states of symmetry-protected $\mathbb{Z}$-type SPT, protected by no-valley mixing, electron-number conservation, and time-reversal $T$ symmetries. Further, we discuss possible phase transitions between distinct SPTs upon symmetry changes. Combined with a recently imaged tilt-boundary network, our findings may explain the long-standing puzzle of subgap conductance in gated bilayer graphene. This proposal can be tested through future transport experiments on tilt boundaries. In particular, the tilt boundaries offer an opportunity for the in situ imaging of topological edge transport.

Popular Summary

Graphene, an atomically thin layer of carbon atoms, holds promise for electronics, but its lack of an electronic band gap limits its use in applications requiring a low-conductance “off” state. Multiple layers of graphene, on the other hand, develop a band gap in the presence of an electric field perpendicular to the layers. However, subgap conductance in gated multilayer graphene is unusually large and remains unexplained. In our paper, we show theoretically that multilayer graphene samples can host one-dimensional (1D) channels of transport along structural defects called tilt boundaries. We predict that these tilt boundaries not only have structural importance but also support a 1D channel of conduction that may influence subgap conduction.

Tilt boundaries occur at the border between regions in which one of the stacked layers is mismatched by one interatomic spacing, and they result from strain during growth. Our calculations show that the gapless 1D transport along the tilt boundaries is protected by the topological band structure of the gated multilayer graphene, similar to the protection of edge states of quantum Hall systems and those of two-dimensional topological insulators (quantum spin Hall systems).

This is a very unusual role for a structural defect; previously studied structural defects in monolayer systems mostly act as scatterers. Inspired by the important role symmetry plays in gated multilayer graphene, we have considered symmetry-change possibilities that can be driven externally by magnetic field or internally by interaction effects. This analysis allows us to place all possible gapped phases in few-layer graphene systems under the same umbrella. A particularly interesting prediction is for a band gap closing and reopening that might be relevant to recent low-temperature studies of gated bilayer graphene.

Our findings, combined with recent experimental observations of a network of such tilt boundaries, suggest that transport through novel topological kink states might explain the long-standing puzzle of subgap conductance. Furthermore, we have revealed the formal connection between gated multilayer graphene and various topological phases of interest.

Figure 1

A typical $AB\mathrm{\text{−}}BA$ tilt boundary under strain. The blue (red) filled circles mark the $a$ ($b$) sublattice sites. ${\gamma }_{ij}$ represent hopping matrices for a tight-binding model.

Figure 2

A tight-binding study in the presence of abrupt $AB\mathrm{\text{−}}BA$ tilt boundaries in gated bilayer graphene. (a) Schematic representation of the domain wall under strain. The grey (black) lines denote the upper (lower) layer, and solid blue (red) circles denote the $A$ ($B$) sublattice points. The tilt boundary along tangent vector $\overrightarrow{t}\parallel \widehat{y}$ is shaded. The Burger’s vector $\overrightarrow{b}=a\widehat{x}$ (green arrow) for interatomic spacing $a$ accounts for the difference between the solid green line and the dashed green line. (b) The resulting band structure. The edge states are marked with magenta lines. (c) Schematics of valley-momentum-locked edge states at a tilt boundary.

Figure 3

(a) The energy bands of $AB\mathrm{\text{−}}BA$ bilayer graphene connected to the two domain walls depicted in Fig. 2. The Fermi level is indicated with the dashed blue line, and the bands with the marked linear dispersion relation intersecting near the Fermi level are outlined by the purple (light) curves. (b) Top and side views of the charge distribution for a state near the $K_1$ point of (a). The yellow rectangles indicate the $AB\mathrm{\text{−}}BA$ domain-wall structure.