Nonequilibrium Spin Hall Accumulation in Ballistic Semiconductor Nanostructures

We demonstrate that the flow of a longitudinal unpolarized current through a ballistic two-dimensional electron gas with Rashba spin-orbit coupling will induce a nonequilibrium spin accumulation which has opposite signs for the two lateral edges and is, therefore, the principal observable signature of the spin Hall effect in two-probe semiconductor nanostructures. The magnitude of its out-of-plane component is gradually diminished by static disorder, while it can be enhanced by an in-plane transverse magnetic field. Moreover, our prediction of the longitudinal component of the spin Hall accumulation, which is insensitive to the reversal of the bias voltage, offers direct evidence to differentiate experimentally between the extrinsic, intrinsic, and mesoscopic spin Hall mechanisms.

Figure 1

(a) The out-of-plane component $⟨S_z(\mathbf{r})⟩$ of the nonequilibrium spin accumulation induced by nonlinear quantum transport of unpolarized charge current injected from the left lead into a two-terminal clean 2DEG (of size $L=30a>L_{\mathrm{SO}}$, $a\simeq 3\mathrm{nm}$) nanostructure with the Rashba SO coupling $t_{\mathrm{SO}}=0.1t_{\mathrm{o}}$ and spin precession length $L_{\mathrm{SO}}\approx 15.7a$. (b) Shows how lateral spin-$\uparrow$ and spin-$\downarrow$ densities will flow in opposite directions through the attached transverse ideal ($t_{\mathrm{SO}}=0$) leads to generate a linear response spin Hall current $[I_y^s{]}^z$ out of four-terminal 2DEG ($L=8a<L_{\mathrm{SO}}$) nanostructures [6], which changes sign $[I_y^s{]}^z(-V)=-[I_y^s{]}^z(V)$ upon reversing the bias voltage.

Figure 2

The one-dimensional transverse spatial profile of the spin accumulation $⟨\mathbf{S}(x=78a,y)⟩$ across the $200a\times 200a$ 2DEG with the Rashba SO coupling $t_{\mathrm{SO}}=0.02t_{\mathrm{o}}$ through which ballistic quantum transport takes place in the nonlinear regime $eV=0.4t_{\mathrm{o}}$ (solid lines) or the linear regime $eV={10}^{-3}{t}_{\mathrm{o}}$ (dotted lines). The width of the edge peaks of $⟨S_z(x=78a,y)⟩$ is $\approx L_{\mathrm{SO}}/2=\pi a{t}_{\mathrm{o}}/4t_{\mathrm{SO}}$.

Figure 3

The transverse profile $⟨S_z(x=13a,y)⟩$ of spin Hall accumulation as a function of spin-independent disorder $W$ setting the mean free path $\ell \approx 21.5a/W^2$ [the ballistic limit $W=0$ is the transverse profile of $⟨S_z(\mathbf{r})⟩$ in Fig. 1a]. The inset shows the effect of the external in-plane magnetic field $g^{*}{\mu }_B{B}_y^{\mathrm{ext}}$ on $⟨S_z(x=13a,y=y_{\min }^{\max })⟩$ at its maximum $y^{\max }=26a$ or minimum $y_{\min }=5a$ peak values.