Interaction between the intrinsic edge state and the helical boundary state of topological insulator phase in bilayer graphene

• The TI helical state moves to nanoribbon edge in a gated ZENR-BG.
• The gapless modes of LD-BG at the two line defects are not equal to each other.
• The Kramers pairs are still valley polarized in a gated LD-BG.

Graphene has intrinsic edge states localized at zigzag edge or lattice defect. Helical boundary states can also be established in such a two-dimensional carbon material at the boundary of topological insulator (TI) phase realized by the extrinsic Rashba spin–orbital coupling (SOC) in gated bilayer graphene. We theoretically investigate the interaction between these two kinds of edge (boundary) states when they coexist in a bilayer graphene. We find that this interaction gives rise to some very interesting results. In a zigzag edged nanoribbon of bilayer graphene, it is possible that the TI helical state does not localize at the TI phase boundary. Instead it moves to the nanoribbon edge even though the SOC is absent therein. In a bulk lattice of bilayer graphene embedded with two line defects, the numbers of helical state subbands at the two line defects are not equal to each other. In such a case, the backscattering lacking is still forbidden since the Kramers pairs are valley polarized.