Transverse Spin-Orbit Force in the Spin Hall Effect in Ballistic Semiconductor Wires

TL;DR

This paper introduces a spin and momentum dependent force operator in ballistic semiconductor wires with Rashba spin-orbit coupling, explaining the transverse deflection and spin accumulation characteristic of the spin Hall effect.

Contribution

It presents a novel force operator framework to analyze spin Hall phenomena in clean semiconductor wires with Rashba coupling, highlighting spin coherence decay without impurities.

Findings

Center of wave packet deflects transversely due to the force operator.

Out-of-plane spin accumulation occurs at the wire edges.

Spin coherence diminishes along the wire even without impurities.

Abstract

We introduce the spin and momentum dependent {\em force operator} which is defined by the Hamiltonian of a {\em 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 {\em 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-$\uparrow$ and spin-$\downarrow$ 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.