Edge spin current and spin polarization in quantum Hall regime

We study the edge spin current and spin polarization, and their repsonses to an external electric field in a two-dimensional electron gas with a Rashba spin-orbit coupling in the quantum Hall regime. The edge state carries a large spin current, which drops sharply away from the boundary. The spin Hall current of the edge states has a resonance at a critical magnetic field, accompanied by a spin rotation in the bulk. The spin Hall current is shown to be proportional to the spin polarization, which provides an explicit way to extract the spin current in experiment.

Figure 1

(Color online) Energy spectra in the unit of $\hslash \omega$ vs the effective Rashba coupling $\eta =\lambda m{l}_b∕{\hslash }^2$ at the edges $y_0=\pm (L∕2-4l_b)$. The parameters are $\lambda =0.96\times {10}^{-4}\hslash \mathrm{c}$, $n_e=1.9\times {10}^{12}∕{\mathrm{cm}}^2$, $g_s=4$, and $m=0.05m_e$, suitable for ${\mathrm{In}}_{053}{\mathrm{Ga}}_{0.47}\mathrm{As}∕{\mathrm{In}}_{0.52}{\mathrm{Al}}_{0.48}$ As in Ref. 12. The inset shows the level anticrossing of the edge states.

Figure 2

(Color online) The charge Hall current in unit of $-e{l}_b\omega$, the spin Hall current in unit of $\hslash l_b\omega ∕2$, $S^z$ and $S^y$ in unit of $\hslash ∕2$ in the twenty lowest energy states near the boundary $y_0=L∕2-\zeta l_b$. The charge and spin currents as well as $S^y$ are antisymmetric, while $S^z$ is symmetric with respect to $y_0$. The magnetic field $B=6.1\mathrm{T}$, for which the thirteenth and fourteenth levels are degnerated far away from the two edges.[8] The arrowed line with numbers indicate the order of the energy eigenstates from the lowest to higher ones.

Figure 3

(Color online) The spin polarization and spin current of the two levels that are almost degenerated subjected to an fixed external electric field, $E=0.01\mathrm{V}∕\mathrm{m}$. The left is for $y_0=L∕2-\zeta l_b$ and the right is for $y_0=-L∕2-\zeta l_b$. (a) The solid line is for the two levels with an energy gap $\Delta E$ in the bulk less than the electric field energy $eE{l}_b$; (b) the dashed line for the two levels with $\Delta E\simeq eE{l}_b$; (c) the dashed-dotted line for the two levels of $\Delta E⪢eE{l}_b$. (The energy gap $\Delta E$ is controllable by the magnetic field near the resonant point.)

Figure 4

(Color online) The spin Hall conductance (pillar shape) and charge Hall conductance (staircase shape) as a function of $1∕B$ for a fixed chemical potential, $\mu =14.539\mathrm{meV}$. The inset is spin Hall conductance for a fixed density of charge carriers, $n_e=1.9\times {10}^{12}{\mathrm{cm}}^{-2}$.