Non-perturbative valley Hall effect in graphenes

Valley Hall currents can be induced in graphene by breaking its version symmetry that opens a gap which is usually very small. A finite electric field can easily exceed the adiabatic limit on which the standard semiclassical theory for the valley Hall effect is based. Here we provide a non-perturbative formulation for valley Hall effect. We show that the non-perturbation effects of the electric field directly lead to an oscillating anomalous velocity in the coherent evolution of an electron, in contrast to the stationary one inherited to the adiabatic limit. By properly taking into account the decoherence effect, we obtain a steady-state valley Hall current that is in all orders of the electric field. It clearly shows nonlinearity in the Hall conductivity to high electric-field values and still retains the intrinsic role of the Berry curvature. Through the inspection of the non-perturbative effect of the electric field, it becomes obvious that the valley Hall conduction in gapped graphene is mediated by the interference between the electric field induced electron-hole excitations and the fermi sea.