Influence of dephasing on the quantum Hall effect and the spin Hall effect

We study the influence of the phase relaxation process on Hall resistance and spin Hall current of a mesoscopic two-dimensional four-terminal Hall cross bar with or without Rashba spin-orbit interaction (SOI) in a perpendicular uniform magnetic field. We find that the plateaus of the Hall resistance with even number of edge states can survive for very strong phase relaxation when the system size is much longer than the phase coherence length. On the other hand, the odd integer Hall resistance plateaus arising from the SOI are easily destroyed by the weak phase relaxation during the competition between the magnetic field and the SOI which delocalize the edge states. In addition, we have also studied the transverse spin Hall current and found that it exhibits resonant behavior whenever the Fermi level crosses the Landau band of the system. The phase relaxation process weakens the resonant spin Hall current and enhances the nonresonant spin Hall current.