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

A previous work [Shen, Ma, Xie, and Zhang, Phys. Rev. Lett. 92, 256603 (2004)] on two-dimensional quantum wells with Rashba type spin-orbit interaction under a strong perpendicular magnetic field is generalized to include the Dresselhaus coupling. The Rashba coupling and the Dresselhaus coupling interplay with the Zeeman effect in opposing ways. The former tends to produce a resonant spin Hall effect at certain magnetic fields while the latter suppresses it. Due to the resonant spin Hall effect, the spin Hall current is highly nonohmic at low temperatures. 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 become degenerate due to the Rashba coupling, and that the quantized charge Hall conductance changes by $2e^2∕h$ instead of $e^2∕h$ as the magnetic field changes through the resonant field.

Figure 1

(a) Energy levels in units of $\hslash \omega$ as a function of the dimensionless Rashba coupling ${\eta }_R$. The parameters are $\beta =0.9\times {10}^{-11}\mathrm{eV}\mathrm{m}$, $n_e=1.9\times {10}^{16}∕{\mathrm{m}}^2$, $m=0.05m_e$, and $g_s=4$, taken from Ref. 22 for the inversion heterostructures ${\mathrm{In}}_{0.53}{\mathrm{Ga}}_{0.47}\mathrm{As}∕{\mathrm{In}}_{0.52}{\mathrm{Al}}_{0.48}\mathrm{As}$. (b) Same as in (a), but for the Dresselhaus coupling ${\eta }_D$ (with the same strength of the Rashba coupling).

Figure 2

Average spin $S_z$ and spin Hall conductance as a function of $1∕B$ at $T=0$. The parameters are the same as in Fig. 1.

Figure 3

Spin Hall conductance vs $1∕B$ at several temperatures for Rashba coupling systems. The parameters are the same as those in Fig. 1. In the inset, temperature dependence of the height of the resonance peak and two side peaks is plotted.

Figure 4

Schematic illustration of the energy shift due to the electric field in the two degenerate levels near the resonant point.

Figure 5

Resonant spin current density as a function of the electric field at different temperatures. The spin current unit is $(-e∕4\pi )(N∕C)$. The filling factor at the resonance is $\nu =12.6$.