Antiparallel spin Hall current in a bilayer with skew scattering

The spin Hall effect due to skew scattering is studied by solving the Boltzmann equation in a bilayer electron system with attractive impurity potentials in one layer and repulsive ones in the other. In such an impurity configuration, directions of the spin Hall current in two decoupled layers are antiparallel. An analytical formula for the magnitude of the antiparallel spin Hall current is derived in the crossover from the decoupled bilayer to the strongly coupled one with no spin Hall current with increasing ${\mathrm{\Delta }}_{\mathrm{SAS}}$, the energy separation between the symmetric and antisymmetric states of the motion perpendicular to the layer. When the impurity potential is short ranged and ${\mathrm{\Delta }}_{\mathrm{SAS}}\ll {\varepsilon }_{\mathrm{F}}$, with ${\varepsilon }_{\mathrm{F}}$ being the Fermi energy, the normalized antiparallel spin Hall conductivity is found to be ${[1+{\left(\omega {\tau }_p\right)}^2]}^{-1}$, with $\omega ={\mathrm{\Delta }}_{\mathrm{SAS}}/\hslash$ and ${\tau }_p$ being the momentum relaxation time at ${\varepsilon }_{\mathrm{F}}$. This formula is explained by extending the dynamics for the Hanle effect, which was originally developed for spin, to the pseudospin (layer) degree of freedom. The present finding suggests that the Hanle effect will be useful in understanding the pseudospin dynamics as well as the spin dynamics.

Figure 1

Coupled and decoupled double-quantum-well (DQW) structures. Wave functions for the motion perpendicular to the coupled symmetric DQW are the bonding (symmetric) $|G\rangle$ and antibonding (antisymmetric) $|E\rangle$ states, each of which is composed of wave functions in the left and right wells, $|L\rangle$ and $|R\rangle$. The energy separation ${\mathrm{\Delta }}_{\mathrm{SAS}}$ between $|G\rangle$ and $|E\rangle$ represents the strength of the coupling between $|L\rangle$ and $|R\rangle$. The DQW considered in this paper is assumed to have antiparallel extrinsic spin Hall current, ${\mathit{j}}^{\mathrm{sL}}=-{\mathit{j}}^{\mathrm{sR}}$, when it is decoupled.

Figure 2

Normalized antiparallel spin Hall conductivity ${\sigma }_{yx}^{\mathrm{s}3}/{\sigma }_0^{\mathrm{sH}}$. (a) The dependence on the pseudospin-precession frequency, $\omega ={\mathrm{\Delta }}_{\mathrm{SAS}}/\hslash$ [Eq. (69)], which shows that the momentum relaxation time ${\tau }_p$ determines the crossover position. (b) The dependence on the potential offset between the $L$ and $R$ layers $\hslash {\omega }_V$ [Eq. (67) at ${\tau }_l={\tau }_p$].