Dynamic spin-Hall effect and driven spin helix for linear spin-orbit interactions

We derive boundary conditions for the electrically induced spin accumulation in a finite, disordered two-dimensional semiconductor channel. While for dc electric fields these boundary conditions select spatially constant spin profiles equivalent to a vanishing spin-Hall effect, we show that an in-plane ac electric field results in a nonzero ac spin-Hall effect, i.e., it generates a spatially nonuniform out-of-plane polarization even for linear intrinsic spin-orbit interactions. Analyzing different geometries in [001]- and [110]-grown quantum wells, we find that although this out-of-plane polarization is typically confined to within a few spin-orbit lengths from the channel edges, it is also possible to generate spatially oscillating spin profiles which extend over the whole channel. The latter is due to the excitation of a driven spin-helix mode in the transverse direction of the channel. We show that while finite frequencies suppress this mode, it can be amplified by a magnetic field tuned to resonance with the frequency of the electric field. In this case, finite-size effects at equal strengths of Rashba and Dresselhaus SOI lead to an enhancement of the magnitude of this helix mode. We comment on the relation between spin currents and boundary conditions.

Figure 1

Left: conducting channel infinite in the ${\mathbf{e}}_{x_2}$ direction and of width $L$ in the ${\mathbf{e}}_{x_1}$ direction. ac electric field $\mathbf{E}\left(\omega \right)\parallel {\mathbf{e}}_{x_2}$ induces boundary spin accumulation. An external magnetic field ${\mathbf{b}}_0$, applied parallel to $\mathbf{E}$, gives rise to EDSR (see Sec. 7). Right, (a): a standard [001]-grown quantum well with the [110] crystal axis taken along the $x_2$ direction. The bulk polarization $\mathbf{\Omega }\left(\mathbf{e}\mathbf{E}\tau \right)\propto {\mathbf{e}}_{x_1}\left(\alpha +\beta \right)$ points along ${\mathbf{e}}_{x_1}$ [cf. Eqs. (2, 3)]. Right, (b): a [110]-grown quantum well with $\mathbf{E}\parallel \left[\overline{1}10\right]$ along ${\mathbf{e}}_{x_2}$. The internal field [see Eq. (4)] $\mathbf{\Omega }\left(\mathbf{e}\mathbf{E}\tau \right)$ has both in-plane (due to the Rashba SOI) and out-of-plane (due to the Dresselhaus SOI) components.

Figure 2

(Color online) Characteristic wave numbers $\theta$ of the homogeneous solutions $s_5\left(r_1\right)=s_{5,0}{e}^{{\theta }_5{r}_1}$ and $s_6=s_6\left(r_1\right)=s_{6,0}{e}^{{\theta }_6{r}_1}$ of Eq. (7) as a function of $\alpha /\beta$ for $\omega \tau ={10}^{-3}$ and ${\xi }_{\beta }=0.1$ in a [001]-grown quantum well. For $\alpha =-\beta$ (indicated by arrow), the wave numbers have small real parts $\left|\text{Re}{\theta }_5\right|\approx \left|\text{Re}\sqrt{-2i\omega \tau }\right|⪡1$, which implies a nearly undamped oscillatory mode. Under EDSR conditions, $\text{Re}{\theta }_5$ vanishes identically at $\alpha +\beta =0$ [cf. Eq. (32)]. The presence of modes with almost imaginary wave numbers leads to an oscillating spin profile, as shown in Fig. 3.

Figure 3

(Color online) Real (solid line) and imaginary parts (dashed line) of the out-of-plane spin density $S^3\left(x\right)/S_b$ [solution of Eqs. (8, 9, 10, 19)] in the standard QW [Fig. 1a] are shown for $\omega \tau ={10}^{-4}$, ${\xi }_{\alpha }=0.1$, ${\xi }_{\beta }=-0.095$, and $L=500l$. As Rashba and Dresselhaus SOI interfere destructively for this case, the bulk polarization is much smaller than the value discussed in Sec. 6: all other parameters being equal, $S_b=0.005\mu {\text{m}}^{-2}$ instead of $1.1\mu {\text{m}}^{-2}$. Inset: in-plane polarization $S^1\left(x\right)$ [along the internal field $\mathbf{\Omega }\left(e\mathbf{E}\tau \right)$] in the same situation.

Figure 4

Real (upper panel) and imaginary parts (lower panel) of the out-of-plane spin polarization $S^3\left(x\right)/S_b$ for frequencies $\omega \tau =0.1,0.3,0.5,0.7,0.9,1.1\times {10}^{-5}$ (gray to black lines), ${\xi }_{\beta }=0.1$, ${\xi }_{\alpha }=0.003$, $L=2800l$, and $S_b=1\mu {\text{m}}^{-2}$. The geometry with $\mathbf{E}\parallel \left[\overline{1}10\right]$ is shown in Fig. 1b. Horizontal dashed lines mark ac bulk polarization according to Eq. (26) for the same parameters.

Figure 5

Polarization in the EDSR geometry $\mathbf{E},\mathbf{B}\parallel \mathbf{y}$ for case shown in Fig. 1a with ${\mathbf{S}}_b=0.02\mu {\text{m}}^{-2}$. Upper panel: $\text{Im}{S}^3\left(x=20l\right)$ (black) and $\text{Im}{S}^1\left(x=20l\right)$ (gray curve) are shown as a function of ${\omega }_L\tau$. Resonance is seen at $\pm {\omega }_L\tau =\omega \tau ={10}^{-3}$. Parameters of the [001]-grown QW: ${\xi }_{\beta }=-0.08$, ${\xi }_{\alpha }=0.1$, and $L=100l$. Lower Panel: density plot of $\text{Re}{S}^3\left(x\right)$ and $\text{Im}{S}^3\left(x\right)$ as a function of $x$ and ${\omega }_L\tau /{\xi }_{\alpha }^2$ for the same parameters.