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. Letts. 92, 126603 (2004).] and the time-space gauge [T. Fujita et al., New J. Phys. 12, 013016 (2010).] approaches, both yielding SHE conductivity of {\sigma}_y^z=e/8{\pi} . Separate studies based on the concept of spin transverse force [S-Q Shen, Phys. Rev. Lett. 95, 187203 (2005).] provide a heuristic but not a quantifiable indication of SHE. In this paper, we provide a more complete description of the SHE using the Heisenberg approach, unifying the Yang-Mills force, the Heisenberg spin force, and the SHE {\sigma}_y^z under the classical notion of forces and accelerations. Central to this paper is the spin force equations that are satisfied by both {\sigma}_y^z and {\sigma}_x^z, the Yang-Mills, and the Heisenberg spin forces. By linking {\sigma}_y^z to the spin forces, one sees that the physics of SHE in a 2D-Rashba system is also a simple classical Lorentz force picture.