Spin Hall effect in a simple classical picture of spin forces

Abstract

Spin Hall effect (SHE) in a 2D-Rashba system has been treated in the spin-dependent precession [J. Sinova et al.: Phys. Rev. Lett. 92 (2004) 126603] and the time-space gauge [T. Fujita et al.: New J. Phys. 12 (2010) 013016] approaches, both yielding SHE conductivity of σz y - e/8π. Separate studies based on the concept of spin transverse force [S.-Q. Shen: Phys. Rev. Lett. 95 (2005) 187203] provide a heuristic but not a quantifiable indication of SHE. We provide complete description of the SHE using the Heisenberg approach, unifying the Yang-Mills force, the Heisenberg spin force, and the SHE σz y under the classical notion of accelerations. Central to this paper is the spin force equations that are satisfied by both σz y and σz x, Yang-Mills, and Heisenberg spin forces. By linking σz y to the spin forces, one sees that the physics of SHE in a 2D-Rashba system is also a simple classical Lorentz force picture.