Topological Crystalline Insulator Phase in Graphene Multilayers

While the experimental progress on three dimensional topological insulators is rapid, the development of their 2D counterparts has been comparatively slow, despite their technological promise. The main reason is materials challenges of the to date only realizations of 2D topological insulators, in semiconductor quantum wells. Here we identify a 2D topological insulator in a material which does not face similar challenges and which is by now most widely available and well charaterized: graphene. For certain commensurate interlayer twists, graphene multilayers are insulators with sizable band gaps. We show that they are moreover in a topological phase protected by crystal symmetry. As its fundamental signature, this topological state supports one-dimensional boundary modes. They form low-dissipation quantum wires that can be defined purely electrostatically.

Figure 1

SE-even graphene bilayer with an interlayer twist angle of 38.213°. The structure has $C_2$ symmetry on $S$. For clarity the top layer atoms are depicted smaller than those in the lower layer.

Figure 2

Spectrum of a 60 unit cell wide ribbon implementing the lattice model described in the main text at an interlayer rotation angle of 38.213°. Here, the edges are perpendicular to the axis $S$ in Fig. 1, so as to preserve $\mathcal{M}$ symmetry. Two degenerate pairs of edge modes are demonstrated, one pair per edge. In the zoom into the low-energy region shown in the inset the two pairs of modes are depicted with an (artificial) energy offset. Momentum is measured in units of $1/a$, the inverse lattice constant of graphene.

Figure 3

Spectrum of a ribbon of the same width as in Fig. 2, but in the perpendicular crystallographic direction. Periodic boundary conditions are applied and two flux tubes parallel to $S$, each inducing a sign change in the interlayer coupling $\gamma$, are inserted. The flux tubes (gray cylinders in the lattice cartoon picture of the lower panel) break $\mathcal{M}$ symmetry on the lattice scale, yet four topological boundary modes per flux tube are observed (momentum is measured relative to the $K$ points; states from both $K$ points are shown). Inset (right): Zoom into the low-energy region with an (artificial) energy offset for different modes. Inset (left): The same spectrum for a ribbon with additional pseudospin off-diagonal interlayer hopping ($A$-to-$B$ and $B$-to-$A$), as in bilayer graphene. The same boundary modes are observed.

Figure 4

Spectra as in Fig. 3 for a ribbon with flux tubes at angles 23.4° (a) and 8.7° (b) with $S$. Because of the rotational invariance of Eq. (1) zero modes are again observed, although the flux tubes break $\mathcal{M}$ on the lattice scale.

Figure 5

Spectrum of the same ribbon as shown in Fig. 3, but with the flux tubes replaced by two line potentials, generated by pairs of oppositely charged wires (brown and green cylinders in the lattice cartoon of the lower panel).