In the beginning of the “Spin Hall pdf” embedded below we analyze, according to Quantum field theory the scattering of spin 1/2 charged particles on heavy spinless charged particles, as it happens for electrons from a charge current along a thin plane conductor plate scattering on impurity nodes from the conductor crystal lattice grid. The motion of the electrons is confined to the plane of the conductor plate and the impurity nodes appear (because of the difference in valence electrons between impurity atoms and majoritar atoms from the plate) as heavy charged diffusion centers for the electrons in the charge current.
We consider two relevant cases: first, when the measurable spin angular momentum direction of the electrons is normal to the motion plane and second, when this direction is parallel to the motion plane and notice that only in the second case the electrons experience spin flipping during the scattering process.
Then we show, as for the classical Coulomb scattering, how the deflection angle of the trajectory of the electrons in the scattering on a diffusion center depends on a chosen shock parameter which determines the differential cross section for the scattering process, when the electrons scatter in presence of the Coulomb field of the diffusion center and of the spin-orbit coupling which comes in principal from the Larmor interaction energy corresponding to the spin magnetic moment of the electron moving in the electric field produced by the heavy diffusion center, considering also the Thomas precession which generates an inertial forces field that combines its energy with the Larmor interaction energy. We show that the non-conservative part of this inertial forces field, appearing in the second of the two considered relevant cases, is the origin of spin flipping.
For the first relevant case, we show how a spin current, normal to the charge current, appears in the plate as a consequence of spin polarization dependent scattering angles. Thus we conclude that up-spin electrons are deflected mostly to the right of the charge current and down-spin electrons are deflected mostly to the left of the charge current direction, resulting a accumulation of up-spin on the right edge and of down-spin on the left edge of the sample.
Finally, based on referenced experimental data, we verify that assumptions we made about the shock parameter and the drift velocity of electrons in the computations for the scattering process analysis, are actually experimentally available (in our case for a Copper with Iridium impurities plate).
This is a chapter you can also find in the post
Notes on … quantum field theory
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