Rashba spin precession in quantum-Hall edge channels

Quasi-one-dimensional edge channels are formed at the boundary of a two-dimensional electron system subject to a strong perpendicular magnetic field. We consider the effect of Rashba spin-orbit coupling, induced by structural inversion asymmetry, on their electronic and transport properties. Both our analytical and numerical results show that spin-split quantum-Hall edge channels exhibit properties analogous to that of Rashba-split quantum wires. Suppressed backscattering and a long spin lifetime make these edge channels an ideal system for observing voltage-controlled spin precession. Based on the latter, we propose a magnetless spin-dependent electron interferometer.

Figure 1

(Color online) Dispersion of the lowest Landau level, calculated within the longitudinal SO approximation, for a hard-wall confining potential of width $W$. The labels ↑, ↓ refer to eigenvalues of ${\sigma }_y$. The parameters used are $L_{\mathrm{so}}∕l_B=21.3$ and $W∕l_B=8.9$.

Figure 2

(Color online) Transverse probability-density profile for spin up (top) and spin down (bottom) right-moving edge states at a fixed energy $E=\hslash {\omega }_{\mathrm{c}}$. The transverse probability densities, shown here, correspond to the states marked by dots in Fig. 1. The spin labels ↑, ↓ refer to eigenvalues of ${\sigma }_y$. Parameters are the same as in Fig. 1.

Figure 3

(Color online) Spin-polarized conductances as a function of the inverse magnetic field $1∕\hslash {\omega }_{\mathrm{c}}$ for fixed Fermi energy $E_{\mathrm{F}}=0.03\mathrm{eV}$, which corresponds to three propagating transverse modes at zero magnetic field. The blue line represents $G_{\uparrow \uparrow }$, the red line $G_{\downarrow \uparrow }$, and the black line $G_{\uparrow \uparrow }+G_{\downarrow \uparrow }$. In plot (a) no Zeeman coupling is present, i.e., the gyromagnetic factor is $g=0$, whereas in plot (b) we have $g=-15$. The other parameters are $L∕{\lambda }_{\mathrm{F}}=40$, $W∕{\lambda }_{\mathrm{F}}=2$, and $L_{\mathrm{SO}}=L∕5$, ${\lambda }_{\mathrm{F}}$ denotes the Fermi wavelength.

Figure 4

(Color online) Conductance $G_{\uparrow \uparrow }$ of the spin-up edge channel at filling factor 1, computed both with transverse Rashba term (solid line) and without it (dashed line) and plotted here as a function of the Rashba SO coupling strength. The Zeeman coupling is set to zero. The other parameters are $E∕\hslash {\omega }_{\mathrm{c}}=1$, $L∕{\lambda }_{\mathrm{F}}=40$, and $W∕{\lambda }_{\mathrm{F}}=2$.

Figure 5

(Color online) $G_{\uparrow \uparrow }$ (a) and $G_{\downarrow \uparrow }$ (b) plotted as a function of SO coupling strength and the g-factor $g$, at filling factor 1. The red line represents the points for which $\mid {\nu }_{\mathrm{R}}∕{\nu }_{\mathrm{Z}}\mid =1$. (We computed ${\nu }_{\mathrm{R}}$ assuming $v_n^{\left(0\right)}={\omega }_{\mathrm{c}}{l}_B$, which is appropriate for a sharp edge potential.) The other parameters are $E_{\mathrm{F}}∕\hslash {\omega }_{\mathrm{c}}=1$, $L∕{\lambda }_{\mathrm{F}}=40$, and $W∕{\lambda }_{\mathrm{F}}=2$.

Figure 6

(Color online) Schematic view of the simplest interferometer setup. A right-moving edge channel (blue solid line) moving from lead a to lead b is split, via a first quantum point contact (QPC1), and recombines at a second quantum point contact (QPC2). The left-going edge channel (blue dashed line) formed at QPC2 is absorbed by the probe c in order to avoid additional interference.