Current-Induced Spin Polarization in Anisotropic Spin-Orbit Fields

The magnitude and direction of current-induced spin polarization and spin-orbit splitting are measured in ${\mathrm{In}}_{0.04}{\mathrm{Ga}}_{0.96}$ As epilayers as a function of in-plane electric and magnetic fields. We show that, contrary to expectation, the magnitude of the current-induced spin polarization is smaller for crystal directions corresponding to larger spin-orbit fields. Furthermore, we find that the steady-state in-plane spin polarization does not align along the spin-orbit field, an effect due to anisotropy in the spin relaxation rate.

Figure 1

(a) InGaAs epilayer (blue) is etched into cross patterns with four electrical contacts (orange) on the GaAs substrate (gray). Kerr rotation measures the component of spin polarization along the laser axis ($\widehat{z}$). (b) Voltages applied to the contacts determine the electron drift momentum $\mathrm{k⃗}$, at angle $\varphi$ with respect to the [100] crystal direction. ${\vec{B}}_{\text{ext}}$ is oriented at angle $\theta$ by rotating the cryostat. (c) Total SO field as a function of $\vec{k}$ from SIA and BIA components with relative strength $\alpha /\beta =2$ and $\alpha >0$. (d) The SO field ${\mathrm{B⃗}}_{\text{int}}$ makes an angle $\xi$ with respect to the external magnetic field ${\mathrm{B⃗}}_{\text{ext}}$. The steady-state in-plane spin polarization ${\mathrm{S⃗}}_{xy}$ is shifted from ${\mathrm{B⃗}}_{\text{int}}$ by angle $\zeta$. The measured CISP is maximized when $\zeta +\xi ={90}^{\circ }$. We take $\vec{\gamma }\parallel {\mathrm{B⃗}}_{\text{int}}$.

Figure 2

(a) Kerr rotation as a function of ${\mathbf{B}}_{\text{ext}}$ at a fixed pump-probe time delay for external voltages of 0 (black squares), 1 (red circles), and 2V (blue triangles) fit to Eq. (2). (b) The location of the spin packet center from the fit amplitude (${\theta }_{op}$) gives the electron drift velocity as a function of voltage. (c),(d) Components of the SO field perpendicular and parallel to ${\mathbf{B}}_{\text{ext}}$ as a function of pump-probe separation. (e) The slope ($\kappa$) of ${\mathbf{B}}_{\text{int}}$ evaluated at the spin packet center is used to characterize the strength of the SO field. (f),(g) The SO field coefficient ($\kappa$) and direction ($\xi$) are measured as a function of the electron momentum direction ($\varphi$) for samples $A$ (black squares) and $B$ (red circles). In (a)–(e), the data set is taken from sample $A$ for $\theta =\varphi =15^{\circ }$. Voltage measurements are taken at linear spacings between 0 (black) and 2 V (blue).

Figure 3

(a) CISP measurements at 0.2 (black squares), 1 (red circles), and 2 V (blue triangles). The lines are fits to Eq. (7). (b) The magnitude of the spin orientation per unit time $\gamma$ as a function of drift velocity for several $\mathbf{k}$ directions. The slope $\eta$ is used to characterize the CISP magnitude. (c) CISP amplitude as a function of angle ($\xi$) between ${\mathbf{B}}_{\text{ext}}$ and ${\mathbf{B}}_{\mathrm{int}}$ for $\varphi =-{45}^{\circ }$ (black squares), 0° (red circles), and $+{45}^{\circ }$ (blue triangles) all taken at an applied voltage of 2 V. Solid lines are fits to a sine curve. ${\theta }_{\mathrm{el}}$ is maximized when ${\mathrm{S⃗}}_{xy}\perp {\mathbf{B}}_{\mathrm{e}\mathrm{x}\mathrm{t}}$. The dashed red line is the expected curve for $\varphi =0^{\circ }$ if ${\mathrm{S⃗}}_{xy}\parallel \vec{\gamma }$; it is shifted from the experimental curve by $\sim {37}^{\circ }$. (d) Deviation ($\zeta$) of in-plane spin polarization (${\mathrm{S⃗}}_{xy}$) from the CISP alignment vector ($\vec{\gamma }$) extracted from measurements in (c) (red circles) and from fits to Eq. (7) (black squares). The blue line is a curve expected from our model given by Eq. (6).

Figure 4

(a) CISP for various SO field magnitudes for different directions of $\mathbf{k}$ in samples $A$ and $B$. A negative differential relationship is observed. (b) The same negative differential relationship is observed in samples $C$–$E$ which have two channels patterned along [110] and $[1\overline{1}0]$.