The time-averaged spin-Hall conductivity in two-dimensional spin-orbit coupled systems: the role of spin dynamics

We investigate the spin-Hall conductivity (SHC) in the two-dimensional spin-orbit coupled systems by taking spin precession into account. The influence of spin precession on the spin-Hall current in the presence of disorder is investigated. Perturbation method leads to the result that the spin dynamics is composed of Larmor (periodic) and non-Larmor (non-periodic) precession. The non-Larmor component of spin is found to be the same result obtained from the Kubo formula in the short time limit. The Larmor component of spin, i.e., spin precession in the unperturbed system, was neglected in the previous studies, but it plays an important role in the linear response theory. The Larmor precession of spin should be zero after time average over several complete periods because there is no specific orientation perpendicular to the plane. By using the requirement of vanishing time-averaged Larmor precession of spin, we calculate the time-averaged non-Larmor SHC for k-cubic Rashba system by using the conventional definition of the spin current. Our result is 2.1(q/8 pi) which is very close to the experimental value 2.2(q/8 pi) by Wunderlich et al [2005 Phys. Rev. Lett. 94, 047204], where -q is the electric charge.