Proper Definition of Spin Current in Spin-Orbit Coupled Systems

The conventional definition of spin current is incomplete and unphysical in describing spin transport in systems with spin-orbit coupling. A proper and measurable spin current is established in this study, which fits well into the standard framework of near-equilibrium transport theory and has the desirable property to vanish in insulators with localized orbitals. Experimental implications of our theory are discussed.

Figure 1

An example demonstrating the irrelevance of the conventional spin current to spin transport. Left: A macroscopic system consisting of a uniform distribution of microscopic boxes (gray rectangles) in which an electron is confined. Right: The inside structure of the box, where the walls flips the spin of electron upon collision. When the system is driven to such a state schematized at the right, the macroscopic average of the conventional spin current is nonzero. However, because all electrons are localized, no spin transport or boundary spin accumulation can occur. ${\mathcal{T}}_B$ denotes the total boundary spin torque due to the spin-flip scattering. It is easy to show the corresponding spin torque dipole $P_{\tau }=-{\mathcal{T}}_B{l}_B$ exactly cancels the conventional spin current. As a result, the spin current ${\mathcal{J}}_s$ defined in Eq. (4) is zero.