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

We introduce the spinand 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.