Spin-resolved Hall effect driven by spin-orbit coupling

Spin and electric Hall currents are calculated numerically in a two-dimensional mesoscopic system with Rashba and Dresselhaus spin-orbit coupling by means of the Landauer-B\"uttiker formalism. It is found that both electric and spin Hall currents circulate when two spin-orbit couplings coexist, while the electric Hall conductance vanishes if either one is absent. The electric and spin Hall conductances are suppressed in strong disorder, but survive in weak disorder. Physically it can be understood that the spin transverse ``force'' generated by spin-orbit coupling is responsible for the formation of the spin Hall current and the lack of transverse reflection symmetry is the origin of the electric Hall current.