Unidirectional spin-orbit interaction and spin-helix state in a (110)-oriented GaAs/(Al,Ga)As quantum well

The Dresselhaus spin-orbit interaction is quantitatively investigated in a (110)-oriented GaAs quantum well by means of time- and spatially resolved Kerr rotation. The experimental results directly demonstrate a unidirectional out-of-plane spin-orbit interaction that linearly depends on the electron momentum along the $\left[1\overline{1}0\right]$ direction and vanishes for the electron momentum along the [001] direction. Spatially resolved measurements of the diffusion-driven spin precession dynamics provide evidence of the formation of a persistent spin-helix state in this system.

Figure 1

(a) Schematic description of the oblique experimental geometry and the SOI vs the electron momentum on the Fermi surface. (b) Schematic description of the electron spin dynamics and the two spin components $(S_{\parallel }$ and $S_{\perp })$ detected by the polar magneto-optical Kerr effect. (c), (d) TRKR measurements upon introduction of a positive current (black circles), a negative current (red circles) along the $\left[1\overline{1}0\right]$ direction, and no current (blue circles). The solid and dashed green lines are fitted curves for $I=\pm 200$ $\mu \mathrm{A}$ and $I=0$ $\mu \mathrm{A}$, respectively. The external magnetic field is $B_{\mathrm{ext}}=235$ mT, with $\beta ={45}^{\circ }$ for (c) and $\beta ={225}^{\circ }$ for (d).

Figure 2

(a), (b) Current-dependent SO energy splitting in the presence of different $B_{\mathrm{ext}},B_{\mathrm{ext}}=156$ mT (circles), $B_{\mathrm{ext}}=235$ mT (triangles), and $B_{\mathrm{ext}}=313$ mT (squares), with currents along the $\left[1\overline{1}0\right]$ direction in (a) and along the [001] direction in (b). The solid line in (a) is a linear fit to all the symbols. (c) Dependence of the SO energy splitting $\hslash {\omega }_{\mathrm{SO}}$ on the direction of an introduced current of 100 $\mu \mathrm{A}$ (cross symbols), 200 $\mu \mathrm{A}$ (plus symbols), and 300 $\mu \mathrm{A}$ (star symbols) with $B_{\mathrm{ext}}=235$ mT. Solid lines are fits. Inset: Schematic description of rotation of the current with respect to the crystal axis.

Figure 3

(a), (b) Dependence of the angle $\alpha$ on the current along the [001] direction, as determined experimentally from the Kerr rotation amplitudes $S_{\parallel }$ and $S_{\perp }$. (c), (d) Dependence of $\alpha$ (symbols) on the current along the $\left[1\overline{1}0\right]$ direction. The lines are calculated from the experimentally determined values of ${\omega }_{\mathrm{SO}}$ and the current-dependent $g$-factor tensor. For (a)–(d), the external magnetic field is $B_{\mathrm{ext}}=156$ mT for circles and $B_{\mathrm{ext}}=313$ mT for squares.

Figure 4

(a) Two-dimensional plot of experimentally measured spin dynamics in real space obtained by scanning the pump beam along the $\left[1\overline{1}0\right]$ direction (upper panel) and the [001] direction (lower panel). The external magnetic field applied along the $\left[1\overline{1}0\right]$ direction is from left to right sequentially, $B_{\mathrm{ext}}=0$, 100, 200, and 400 mT. (b) Corresponding Monte Carlo numerical simulation results.