Resonant spin Hall conductance in quantum Hall systems lacking bulk and structural inversion symmetry

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Following a previous work [Shen, Ma, Xie and Zhang, Phys. Rev. Lett. 92, 256603 (2004)] on the resonant spin Hall effect, we present detailed calculations of the spin Hall conductance in two-dimensional quantum wells in a strong perpendicular magnetic field. The Rashba coupling, generated by spin-orbit interaction in wells lacking bulk inversion symmetry, introduces a degeneracy of Zeeman-split Landau levels at certain magnetic fields. This degeneracy, if occuring at the Fermi energy, will induce a resonance in the spin Hall conductance below a characteristic temperature of order of the Zeeman energy. At very low temperatures, the spin Hall current is highly non-ohmic. The Dresselhaus coupling due to the lack of structure inversion symmetry partially or completely suppresses the spin Hall resonance. The condition for the resonant spin Hall conductance in the presence of both Rashba and Dresselhaus couplings is derived using a perturbation method. In the presence of disorder, we argue that the resonant spin Hall conductance occurs when the two Zeeman split extended states near the Fermi level becomes degenerate due to the Rashba coupling and that the the quantized charge Hall conductance changes by 2e²/h instead of e²/h as the magnetic field changes through the resonant field.