Role of electrical field in quantum Hall effect of graphene

The ballistic motion of carriers of graphene in an orthogonal electromagnetic field is investigated to explain quantum Hall effect of graphene under experimental conditions. With the electrical field, all electronic eigen-states have the same expectation value of the velocity operator, or classically, all carriers move in cycloid-like curves with the same average velocity. This velocity is the origin of the Hall conductance and its magnitude is just appropriate so that the quantized Hall conductance is exactly independent of the external field. Electrical field changes each Landau level into a bundle of energies. Hall conductance plateaus occur in small fields as bundle gaps exist and are destroyed in intermediate fields as bundles overlap. As the electrical field tends to the critical point, all bundles have the same width, and bundle gaps increase to infinity rapidly. As a result, saturation of the Hall conductance may be observed. Electrical field thus demonstrates nonlinear effects on the Hall conductance.