Current-Controlled Spin Precession of Quasistationary Electrons in a Cubic Spin-Orbit Field

Space- and time-resolved measurements of spin drift and diffusion are performed on a GaAs-hosted two-dimensional electron gas. For spins where forward drift is compensated by backward diffusion, we find a precession frequency in the absence of an external magnetic field. The frequency depends linearly on the drift velocity and is explained by the cubic Dresselhaus spin-orbit interaction, for which drift leads to a spin precession angle twice that of spins that diffuse the same distance.

Figure 1

Measurement of drifting spins after a local spin excitation at time $t=0$. (a) Measured spin polarization $S_z$ vs $y$ for different $t$ at an electric field $E_y=1.56\mathrm{kV}/\mathrm{m}$. The data are offset according to $t$ and normalized to the maximum spin polarization $S_z^0$. Circles are experimental data and solid lines are fits with Eq. (1). (b) Color scale plot of $S_z(y,t)$ for $E_y=1.56\mathrm{kV}/\mathrm{m}$. The violet dashed line marks the center of the spin packet. The gray solid lines are contour lines of a global fit as explained in the text. The solid green line indicates the slope of the lines of equal spin phase. It is tilted because spin precession from drift is different from that from diffusion owing to cubic SOI. (c) Color scale plot of $S_z(y,t)$ for $E_y=-1.8\mathrm{kV}/\mathrm{m}$, where the slope of the green line is reversed. Inset: schematic layout of the cross-shaped mesa structure. Four Ohmic contacts allow the application of electric fields along the $y||[110]$ and the $x||[1\overline{1}0]$ direction.

Figure 2

Fit results. (a) Drift velocity $v_{\text{dr}}$ plotted against the applied electric field. Dots are the fit values obtained from the measured $S_z(y,t)$. The solid line is the drift velocity calculated from the measured current $I$ via $v_{\text{dr}}=I/(e{n}_{\mathit{s}}w)$. (b) Values for the spatial wave number $q$. Dots are the fit values and the red line is the model of Eq. (5) with $\alpha +{\beta }^{*}=6.2\times {10}^{-13}\mathrm{eV}\mathrm{m}$. (c) Values for the precession frequency $\omega$. Dots are fit values and the red line is the model of Eq. (6) with ${\beta }_3=8.5\times {10}^{-14}\mathrm{eV}\mathrm{m}$. Confidence intervals in all plots are defined as a 5% increase of the fit error.

Figure 3

Model of drift and diffusion. (a) Scattering events lead to diffusive trajectories of individual electrons. Shown are two trajectories of electrons that travel the same distance $\hslash {\mathbf{k}}_{\text{di}}t/m^{*}$. (b) The Fermi circle is shifted by the drift vector ${\mathbf{k}}_{\text{dr}}$. (c) Exemplary map of $S_z(y,t)$ generated from Eq. (1). Electrons with an average ${\mathbf{k}}_{\text{di}}=0$ drift along the violet dashed line. Electrons measured away from this line additionally experience a diffusive motion. Because of the unequal contributions of drift and diffusion to the spin precession, the phase of quasistationary electron spins (for example, those on the solid green line) depends on how the travel is divided between drift and diffusion. This leads to a precession in time, as seen in the lower panel (shown for spins at $y=y_0$).

Figure 4

Drift along $x$. (a) Measured $S_z(x,t)$ for $E_x=-3\mathrm{kV}/\mathrm{m}$. No precession is visible, although for our sample we expect $3\times {10}^{-14}<({\beta }^{*}-\alpha )<7\times {10}^{-14}\mathrm{eV}\mathrm{m}$, for which our model predicts a precession (green dashed lines mark $S_z=0$). (b) Numerical Monte Carlo simulation data of $S_z(x,t)$ for $\alpha =2.9\times {10}^{-13}\mathrm{eV}\mathrm{m}$ and ${\beta }^{*}=3.1\times {10}^{-13}\mathrm{eV}\mathrm{m}$. As in (a), no precession pattern is observed although it is predicted by the model (green solid line). (c) Numerical Monte Carlo simulation data of $S_z(x,t)$ for $\alpha =0.5\times {10}^{-13}\mathrm{eV}\mathrm{m}$ and ${\beta }^{*}=5.5\times {10}^{-13}\mathrm{eV}\mathrm{m}$. Here, the SOI is almost isotropic and the model (green solid lines) describes the precession pattern well. (d) When $\alpha$ is gradually increased from zero, the $q_x$ observed in the simulation (blue circles) initially follows $q_x=2m^{*}({\beta }^{*}-\alpha )/{\hslash }^2$ (red line). In a finite range around $\alpha ={\beta }^{*}$ (PSH), precession along $x$ is suppressed ($q_x=0$). The total strength of the SOI in all simulations (b)–(d) is kept constant at $\alpha +{\beta }^{*}=6\times {10}^{-13}\mathrm{eV}\mathrm{m}$.