Transverse spin-orbit force in the spin Hall effect in ballistic semiconductor wires

We introduce the spin- and momentum-dependent force operator, which is defined by the Hamiltonian of a clean semiconductor quantum wire with homogeneous Rashba spin-orbit (SO) coupling attached to two ideal (i.e., free of spin and charge interactions) leads. Its expectation value in the spin-polarized electronic wave packet injected through the leads explains why the center of the packet gets deflected in the transverse direction. Moreover, the corresponding spin density will be dragged along the transverse direction to generate an out-of-plane spin accumulation of opposite signs on the lateral edges of the wire, as expected in the phenomenology of the spin Hall effect, when spin-↑ and spin-↓ polarized packets (mimicking the injection of conventional unpolarized charge current) propagate simultaneously through the wire. We also demonstrate that spin coherence of the injected spin-polarized wave packet will gradually diminish (thereby diminishing the “force”) along the SO coupled wire due to the entanglement of spin and orbital degrees of freedom of a single electron, even in the absence of any impurity scattering.

Figure 1

(Color online) The expectation value of the transverse component of the SO force operator (upper panel) in the quantum state of propagating spin-wave packet along the two-probe nanowire. The middle panel shows the corresponding transverse position of the center of the wave packet as a function of its longitudinal coordinate. The initial state in the left lead is fully spin-polarized wave packet Eq. (3), which is injected into the SO region of the size $L_x\times L_y\equiv 100a\times 31a$ $(a\simeq 3\mathrm{nm})$ with strong Rashba coupling $t_{\mathrm{SO}}=\alpha ∕2a=0.1t_{\mathrm{o}}$ and the corresponding spin precession length $L_{\mathrm{SO}}=\pi t_{\mathrm{o}}a∕2t_{\mathrm{SO}}\approx 15.7a<L_x$ (middle panel) or weak SO coupling $t_{\mathrm{SO}}=0.01t_{\mathrm{o}}$ and $L_{\mathrm{SO}}\approx 157a>L_x$ (lower panel).

Figure 2

(Color online) The dynamics of spin density $\mathbf{S}\left(\mathbf{r}\right)\equiv [S_x\left(\mathbf{r}\right),S_y\left(\mathbf{r}\right),S_z\left(\mathbf{r}\right)]$ induced by simultaneous propagation of two electrons through quantum wire $100a\times 31a$ with the Rashba SO coupling $t_{\mathrm{SO}}=0.1t_{\mathrm{o}}$. Both electrons are injected at $t=0$ from the left lead, one as spin-↑ and the other one as spin-↓ polarized (along the $z$ axis) wave packet Eq. (3). The different snapshots of the sum of their spin densities are taken at the points (a), (b), (c), where the transverse SO force and the $y$ coordinate of the center of these wave packets have values shown in the middle panel of Fig. 1.

Figure 3

(Color online) Spin precession, as signified by oscillations of the spin polarization vector $(P_x,P_y,P_z)$, and spin decoherence (as measured by decrease of the purity $\mid \mathbf{P}\mid$ below one) of the spin state of a single electron propagating along the Rashba quantum wire $100a\times 31a$ with the SO coupling strength $t_{\mathrm{SO}}=0.1t_{\mathrm{o}}$ $(L_{\mathrm{SO}}\approx 15.7a)$. The electron is injected from the left lead as a spin-↑ polarized wave packet, whose spin subsystem is therefore fully coherent $\mid \mathbf{P}\mid =1$ at $t=0$. The bottom panel shows the $z$ component of the spin density $S_z\left(\mathbf{r}\right)$ at different values of $(\hslash ∕2)P_z=\int d\mathbf{r}{S}_z\left(\mathbf{r}\right)$ (selected in the upper panel) along the wire.

Figure 4

(Color online) The spin accumulation $[S_x\left(\mathbf{r}\right),S_y\left(\mathbf{r}\right),S_z\left(\mathbf{r}\right)]$ induced by the ballistic flow of unpolarized charge current, simulated by injecting one after another 600 pairs of spin-↑ and spin-↓ polarized (along the $z$ axis) wave packets from the left lead, through quantum wire $100a\times 31a$ with the Rashba SO couplings: (a) $t_{\mathrm{SO}}=0.1t_{\mathrm{o}}$ $(L_{\mathrm{SO}}\approx 15.7a)$ and (b) $t_{\mathrm{SO}}=0.01t_{\mathrm{o}}$ $(L_{\mathrm{SO}}\approx 157a)$.