Gate control of the spin mobility through the modification of the spin-orbit interaction in two-dimensional systems

Spin drag measurements were performed in a two-dimensional electron system set close to the crossed spin helix regime and coupled by strong intersubband scattering. In a sample with an uncommon combination of long spin lifetime and high charge mobility, the drift transport allows us to determine the spin-orbit field and the spin mobility anisotropies. We used a random walk model to describe the system dynamics and found excellent agreement for the Rashba and Dresselhaus couplings. The proposed two-subband system displays a large tuning lever arm for the Rashba constant with gate voltage, which provides a new path towards a spin transistor. Furthermore, the data show large spin mobility controlled by the spin-orbit constants setting the field along the direction perpendicular to the drift velocity. This work directly reveals the resistance experienced in the transport of a spin-polarized packet as a function of the strength of anisotropic spin-orbit fields.

Figure 1

(a) Longitudinal ($R_{xx}$) and Hall ($R_{xy}$) magnetoresistance of the two-subband QW. From the SdH periodicity, one can obtain the subbands density $n_{\nu }$ in the lower inset. The top inset shows the potential profile and subbands charge density calculated from the self-consistent solution of Schrödinger and Poisson equations for $E_z=0$. (b) Subband energy levels and (c) electron concentration dependence on $V_g$ and $E_z$. (d) Geometry of the device and contacts configuration.

Figure 2

(a)–(c) Calculated SOCs for the Rashba (${\alpha }_{\nu }$), linear (${\beta }_{1,\nu }$), and cubic (${\beta }_{3,\nu }$) Dresselhaus for each subband $\nu =\{1,2\}$, as well as intersubband SOCs $\eta$ and $\mathrm{\Gamma }$ as a function of $E_z$. The purple lines give the sum of ${\alpha }_{\nu }$ and ${\beta }_{\nu }^{*}$. (d) The ratio ${\alpha }_{\nu }/{\beta }_{\nu }=\pm 1$ when the subband $\nu$ is set to the ${\mathrm{PSH}}^{\pm }$ regime. The insets show the single-subband magnetization maps on the $xy$ plane for the ${\mathrm{PSH}}^{\pm }$ regimes, and the self-consistent potentials and subband densities for the respective $E_z$. (e) Two-subband magnetization maps in the strong ISS regime for different $E_z$. At $E_z=0$ the well is symmetric (${\alpha }_{\nu }=0$) and the magnetization shows an isotropic Bessel pattern. For finite $E_z$ the broken symmetry leads to the stripped PSH pattern in accordance with the positive ratio $\sum {\alpha }_{\nu }/\sum {\beta }_{\nu }$ [purple line in (d)]. The arrows in the Fermi circle show the first harmonic component of $\sum {\mathbf{\text{B}}}_{\text{SO},\nu }\left(\mathit{k}\right)$, illustrating the transition from isotropic to uniaxial field with increasing $E_z$. All the $xy$ maps are frames of the spin pattern at $t=13$ ns.

Figure 3

Calculated coefficients $\mathbf{b}$ with v${}_{dr}$ parallel to (a) $x$ and (b) $y$ for each subband (colored) and total field (black). (c) Amplitude of the drifting spin polarization in space showing, for example, the center of the packet $d_c$ for 75 mV. (d) Linear dependence of $v_{dr}$ with the channel $V_{\text{ip}}$. The slope gives the spin mobility along $v_{dr}$ in $x$ or $y$. (e) Field scan of ${\varphi }_K$ for several $V_{\text{ip}}$ measured at $d_c$. (f) $B_{\text{SO}}^{y\left(x\right)}$ as a function of $v_{dr}^{x\left(y\right)}$ and the current flowing in that channel. The slopes $b^{x\left(y\right)}$ give the strength of the SOCs that generate the field along $y\left(x\right)$ for drift in $x\left(y\right)$. The solid lines are Gaussian (c) and linear [(d) and (f)] fittings. Scans taken at $t=13$ ns.

Figure 4

(a) Spin mobility and $B_{\text{SO}}$ as a function of the gate-tunable $E_z$. (b) Ratio $b^{x\left(y\right)}$ from (a), showing a crossing at $E_z=0$. (c) SOCs obtained from the addition and subtraction of $b^x$ and $b^y$ in (b). (d) Spin mobility as function of the SOCs that define the $B_{\text{SO}}$ strength along the direction perpendicular to $v_{dr}$. The solid lines are linear fittings and the dashed lines [(b) and (c)] are the theoretical results from the RWM combined with the self-consistent calculation of the SOCs.