Resonant Intrinsic Spin Hall Effect in $p$-Type GaAs Quantum Well Structure

We study intrinsic spin Hall effect in $p$-type GaAs quantum well structure described by Luttinger Hamiltonian and a Rashba spin-orbit coupling arising from the structural inversion symmetry breaking. The Rashba term induces an energy level crossing in the lowest heavy hole subband, which gives rise to a resonant spin Hall conductance. The resonance may be used to identify the intrinsic spin Hall effect in experiments.

Figure 1

Dispersion of low lying subbands for $p$-type GaAs quantum well structure with the width $2L=83\AA$. (a) For Rashba coupling $\lambda =0$, (b) $\lambda ={\hslash }^2/mL$, and (c) $\lambda =3.0\times {\hslash }^2/mL$.

Figure 2

Spin Hall conductance of GaAs quantum well as a function of dimensionless Rashba coupling $\lambda /{\lambda }_0$, with ${\lambda }_0={\hslash }^2/mL$. The hole density $n_p=5.0\times {10}^{11}/{\mathrm{cm}}^2$ and thickness $2L=83\AA$. A finite lifetime $\tau =2.0\times {10}^{-11}\mathrm{s}$, equivalent to a mobility of ${10}^4{\mathrm{cm}}^2/\mathrm{s}\mathrm{V}$, is assumed.

Figure 3

The resonant Rashba coupling for the HH1 subbands as functions of hole density for various thickness $2L$ in GaAs quantum well.

Figure 4

Spin Hall conductance as a function of hole density for GaAs quantum well of half thickness $2L=83\AA$ and Rashba coupling $\lambda ={\hslash }^2/mL$. Dashed line is the spin Hall conductance at $\lambda =0$. The arrows A, B, and C indicate the doping concentrations with which the Fermi level reaches the $\Gamma$ point value of the subbands HH1, LH1, and HH2, respectively.