Transverse electric field-induced quantum valley Hall effects in zigzag-edge graphene nanoribbons

We have investigated gapless edge states in zigzag-edge graphene nanoribbons under a transverse electric field across the opposite edges by using a tight-binding model and the density functional theory calculations. The tight-binding model predicted that a quantum valley Hall effect occurs at the vacuum-nanoribbon interface under a transverse electric field and, in the presence of edge potentials with opposite signs on opposite edges, an additional quantum valley Hall effect occurs under a much lower field. Dangling bonds inevitable at the edges of real nanoribbons, functional groups terminating the edge dangling bonds, and spin polarizations at the edges result in the edge potentials. The density functional theory calculations confirmed that asymmetric edge terminations, such as one having hydrogen at an edge and fluorine at the other edge, lead to the quantum valley Hall effect even in the absence of a transverse electric field. The electric field-induced half-metallicity in the antiferromagnetic phase, which has been intensively investigated in the last decade, was revealed to originate from a half-metallic quantum valley Hall effect.