Valley detection using a graphene gradual pn junction with spin-orbit coupling: an analytical conductance calculation

Graphene pn junction is the brick to build up variety of graphene nano-structures. The analytical formula of the conductance of graphene gradual pn junctions in the whole bipolar region has been absent up to now. In this paper, we analytically calculated that pn conductance with the spin-orbit coupling and stagger potential taken into account. Our analytical expression indicates that the energy gap causes the conductance to drop a constant value with respect to that without gap in a certain parameter region, and manifests that the curve of the conductance versus the stagger potential consists of two Gaussian peaks -- one valley contributes one peak. The latter feature allows one to detect the valley polarization without using double-interface resonant devices.