Topological confinement states in ABA trilayer graphene with antiparallel electric field

In AB-stacked bilayer graphene, the introduction of an out-of-plane electric field can break the inversion symmetry to open up a nontrivial bulk gap hosting the quantum valley Hall state. However, in ABA-stacked trilayer graphene, the out-of-plane electric field cannot open up a nontrivial valley band gap. By applying a pair of antiparallel electric fields, we theoretically propose three kinds of schemes to open up bulk gaps that harbor the quantum valley Hall effect in the ABA-stacked trilayer graphene. By further considering the small-angle twisted trilayer graphene, we can obtain the topological confinement states along the naturally formed domain walls between ABA- and BAB-stacked regions. It is noteworthy that the synergic effect between the antiparallel electric fields and the magnetic field can open up a nontrivial band gap possessing the quantum valley Hall effect and quantum Hall effect simultaneously. Our work not only theoretically proposes how to realize the quantum valley Hall effect in ABA-trilayer graphene, but also provides an ideal platform to explore the hybrid topological phases.