Intrinsic valley Hall effect in graphene

If electrons are incident from an armchair graphene ribbon into the bulk graphene region, the electronic diffraction occurs. Because of the different triangular wrapping of the energy dispersion between valleys K and K′, the electrons of valley K tend to be diffracted to one side and those of valley K′ to the other side. When the current is injected from the armchair ribbon of a four-terminal graphene device, the major portion of the incident current of valley K flows through one side arm and the minor portion through the other side arm. The ratio between them is derived to be 1+4E/3 in the low energy limit, where E is the energy in units of hopping parameter. The major arm for valley K is the minor arm for valley K′. This results in the rise of the valley Hall effect, which is an intrinsic property of graphene stemming from the different electronic structure of the two valleys. The valley Hall conductance is calculated to be (2E/3)G0 with G0 being the conductance supported by the injection ribbon.