Field control of anisotropic spin transport and spin helix dynamics in a modulation-doped GaAs quantum well

Electron spin transport and dynamics are investigated in a single, high-mobility, modulation-doped, GaAs quantum well using ultrafast two-color Kerr-rotation microspectroscopy, supported by qualitative kinetic theory simulations of spin diffusion and transport. Evolution of the spins is governed by the Dresselhaus bulk and Rashba structural inversion asymmetries, which manifest as an effective magnetic field that can be extracted directly from the experimental coherent spin precession. A spin-precession length ${\lambda }_{\mathrm{SOI}}$ is defined as one complete precession in the effective magnetic field. It is observed that application of (i) an out-of-plane electric field changes the spin decay time and ${\lambda }_{\mathrm{SOI}}$ through the Rashba component of the spin-orbit coupling, (ii) an in-plane magnetic field allows for extraction of the Dresselhaus and Rashba parameters, and (iii) an in-plane electric field markedly modifies both the ${\lambda }_{\mathrm{SOI}}$ and diffusion coefficient.

Figure 1

Tunability of the spin decay time in the quantum well with back-gate voltage $U_{\mathrm{BG}}$.

Figure 2

(a) Normalized spatial spin-polarization distribution $S_{\mathrm{z}}$ measured along the $y$ axis for various delay times and a constant back-gate voltage of $U_{\mathrm{BG}}=-1.4\mathrm{V}$. The three data sets are normalized and vertically shifted for clarity. The solid lines represent fits according to Eq. (1). (b) Spatiotemporal evolution of $S_{\mathrm{z}}\left(y,t\right)$ presented as false-color plot. (c) Temporal evolution of ${\lambda }_{\mathrm{SOI}}$ extracted with Eq. (1) from (b) and a solid fit line using Eq. (3).

Figure 3

Experimentally measured and simulated two-dimensional spatial maps of the spin polarization $S_z$ due to diffusive transport. Data is obtained for the back-gate voltage of $U_{\mathrm{BG}}=-1.4\mathrm{V}$ and at a delay time of 0.5 ns without external magnetic field (a) and (b) and $B_{\mathrm{y}}=57\mathrm{mT}$ (d) and (e) respectively. The pattern in the absence of the field results from the formation of a spin helix along the $y$ axis, while an applied external magnetic field allows precession to take place also along the $x$ axis. The simulated maps are obtained by numerically solving Eq. (A6) for the set of parameters extracted from measurements. (c) Schematic visualization of the map in (a) on the sample surface (inserted false-color plot magnified for improved visualization). (f) Orientation and magnitude of the effective magnetic field ${\mathit{B}}_{\mathrm{SOI}}$ in $k$ space on the Fermi surface, plotted for the ratio of $a/b=0.45$ with zero applied magnetic field, matching (a) and (b).

Figure 4

Experimental spatiotemporal maps of the evolution of $S_{\mathrm{z}}\left(y,t\right)$ for $U_{\mathrm{BG}}=-1.4\mathrm{V}$ and $B_{\mathrm{x}}$ (oriented along the $\left[1\overline{1}0\right]$ direction) equal to (a) 0 mT, (b) $115\mathrm{mT}$, and (c) $231\mathrm{mT}$. The dashed lines indicate the time dependence of the center position $y_0$, which is the position of constant phase within the spin precession. Theoretical simulations, based on Eq. (A6), are displayed in (d), (e), and (f) for magnetic field strengths corresponding to those in the experiments.

Figure 5

(a) Effective velocity of the phase offset $y_1$ for several magnetic field strengths $B_{\mathrm{x}}$. The dashed line is a linear fit. (b) Spin-diffusion coefficient for different values of $B_{\mathrm{x}}$. The dashed line is a guide to the eye.

Figure 6

In-plane electric field dependence of the electron spin evolution: (a) Experimental and (d) simulated false-color plots of $S_z(y,t)$ for $E_y=-0.9\mathrm{V}/\mathrm{cm}$. (b) Experimental and (e) simulated plots of $S_z(y,E_y)$ for $t=1\mathrm{ns}$. Experimental results recorded with $U_{\mathrm{BG}}=-1.4\mathrm{V}$ and simulations determined from Eq. (A6). (c) Electric field dependence of ${\lambda }_{\mathrm{SOI}}$ extracted by fitting the data of (b) using Eq. (1). (f) Dependence of the spin-diffusion coefficient $D_s$ on $E_y$, fit by a second-order polynomial function (red curve).

Figure 7

(a) Spatial evolution of $S_z\left(y,U_{\mathrm{BG}}\right)$ for in-plane electrical field $E_{\mathrm{y}}=-1.0\mathrm{V}/\mathrm{cm}$ and delay time of 500 ps. Extracted values of back-gate voltage-dependent (b) electron mobility $\mu$ and (c) spin-precession length ${\lambda }_{\mathrm{SOI}}$.