Side-Jump Mechanism for the Hall Effect of Ferromagnets

TLDR

In ferromagnets, spin‑orbit interaction produces a finite sideways displacement of a wave packet during scattering, a leading‑order effect independent of the scattering potential, while earlier nonclassical mechanisms are deemed spurious for the dc Hall effect. The authors extend the Hall‑resistivity calculation to arbitrary ω_c τ in both compensated and uncompensated metals. The side‑jump mechanism yields ρ_H ∝ ρ² in Fe, Ni, and alloys above 100 K, with predicted displacements of 10⁻¹⁰–10⁻¹¹ m using Fivaz’s effective spin‑orbit Hamiltonian, whereas asymmetric scattering dominates below 100 K.

Abstract

The center of mass of a wave packet undergoes a discontinuous and finite sideways displacement on scattering by a central potential, in the presence of spin-orbit interaction. This is the main Hall-effect mechanism (${\ensuremath{\rho}}_{H}\ensuremath{\propto}{\ensuremath{\rho}}^{2}$) for Fe, Ni, and their alloys above 100 K, while asymmetric scattering dominates below 100 K. Displacement $\ensuremath{\Delta}y$ per actual collision is calculated by partial waves. In the case of Born expansion, the leading term of $\ensuremath{\Delta}y or \frac{{\ensuremath{\rho}}_{H}}{{\ensuremath{\rho}}^{2}}$ is of zero order in the scattering potential. The magnitude is predicted correctly ($\ensuremath{\Delta}y\ensuremath{\approx}{10}^{\ensuremath{-}10}\ensuremath{-}{10}^{\ensuremath{-}11}$ m) when using the effective spin-orbit Hamiltonian derived by Fivaz from spin-orbit interband mixing. The calculation of ${\ensuremath{\rho}}_{H}$ is extended to arbitrary ${\ensuremath{\omega}}_{c}\ensuremath{\tau}$ for compensated and un-compensated metals. Other nonclassical physical mechanisms proposed by Karplus and Luttinger and by Doniach and by Fivaz are spurious for the dc Hall effect.