Valley-dependent odd quantum Hall states induced by side potential in graphene

Abstract

The valley degree of freedom endows graphene with many novel physical properties. In this paper, we propose a new method to construct valley-related quantum Hall states using side potentials. We found that due to the effect of side potential, zigzag graphene nanoribbons have rich and practical edge states related to valleys. If the side potential is combined with the out-of-plane exchange field and the Rashba spin–orbit coupling, the zigzag graphene nanoribbons have the characteristics of valley selective anomalous quantum Hall effect. If we further consider the staggered sublattice potential, as long as the strength of the staggered sublattice potential is weaker than the strength of the out-of-plane exchange field, this valley selection feature is robust to the staggered sublattice potential. However, once the strength of the staggered sublattice potential is stronger than that of the out-of-plane exchange field, the valley selection will disappear. Still if a specific side potential is applied, the system will have the characteristics of a typical valley Hall effect. Here we want to emphasize that in the absence of side potential, although the edge state of the system will have the characteristics of the valley hall effect, the energy band of such edge state is mixed with the bulk band; if a specific side potential is applied, these edge states can be brought into the bulk band gap. In order to verify the effectiveness of our proposed method, we further use this method to construct the quantum valley-spin Hall effect in the toy Kane–Mele model.