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

Branislav K. Nikolic·Liviu P. Zarbo·Sven Welack
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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.

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Branislav K. Nikolic, Liviu P. Zarbo, Sven Welack. 2005-06-23. Transverse Spin-Orbit Force in the Spin Hall Effect in Ballistic Semiconductor Wires. https://doi.org/10.1103/physrevb.72.075335

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