Valley-Hall kink and edge states in multilayer graphene

We report on a theoretical study of one-dimensional (1D) states localized at few-layer graphene system ribbon edges and at interfaces between few-layer graphene systems with different valley-Hall conductivities. These 1D states are topologically protected when valley mixing is neglected. We address the influence on their properties of stacking arrangement, interface structure, and external electric field perpendicular to the layers. We find that 1D states are generally absent at multilayer ribbon armchair direction edges, but present irrespective of crystallographic orientation at any internal valley-Hall interface of an ABC-stacked multilayer.

Figure 1

Schematic illustration of the relationship between valley-Hall effects and 1D conduction channels at interfaces expected on the basis of continuum model considerations. The number of 1D modes per valley is an integer $n$ evaluated from differences between valley Chern numbers $n=|{\tilde{N}}_3^l-{\tilde{N}}_3^r|$. When the layer number $N$ is odd, the integral of the Berry curvatures is half-odd integer, a property that is related to the half-quantized Hall effect of Dirac systems. We find that the number of 1D channels is then usually reduced to the integer part of the integrated Berry-curvature difference. The sense of the arrows directed perpendicular to the page indicates the sign of the Chern number associated with each valley. Upper panel: illustration of ABC trilayer graphene with opposite electric-field signs in the left and right regions with three 1D modes per valley. Lower panel: junction formed between pentalayer and trilayer regions under a uniform bias potential. In this case, the valley-Hall conductivities have the same sign but different magnitude on the opposite sides of the interface and the number of 1D channels is expected to be proportional to the difference of the individual Berry-curvature integrals. The continuum model picture illustrated here can be invalidated by atomic-scale physics at the interface, particularly when the number of layers on opposite sides of the interface is different.

Figure 2

Band structures of ABC- and ABA-stacked, zigzag- and armchair-terminated trilayer and tetralayer ribbons in the presence of a uniform electric field. (a) ABC-stacked layers with zigzag edges, (b) ABA-stacked layers with zigzag edges, (c) ABC-stacked layers with armchair edges, and (d) ABA-stacked layers with armchair edges. In each figure, the left and right panels represent trilayer and tetralayer ribbons, respectively. For ABC stacking the electric field opens a bulk band gap containing edge states in the zigzag case but not in the armchair case. The number of edge-state branches is independent of the ribbon width, whereas the number of bulk-state branches is proportional to ribbon width. In the case of ABA stacking, ribbons are metallic in the trilayer, whereas a small gap opens in the tetralayer geometry.

Figure 3

Upper Panel: band structure of bilayer, trilayer, and tetralayer graphene zigzag ribbons with an electric-field sign change at the ribbon center. We can clearly observe two, three, and four 1D valley-Hall kink states with a common sign of velocity located around the valley point. The branches, which correspond to wave functions localized at the ribbon center, are plotted in red. The other branches that cross the bulk gap are doubly-degenerate edge states. Lower Panel: band structure of bilayer, trilayer, and tetralayer graphene armchair ribbons with an electric-field sign change at the ribbon center. As in the zigzag case, we can clearly identify two, three, and four three 1D kink-state branches. In armchair edges, we do not find 1D edge-state channels.