Frequency- and transverse wave-vector-dependent spin Hall conductivity in two-dimensional electron gas with disorder

We determine wave number $q$ and frequency $\omega$ dependent spin Hall conductivity $\sigma_{yx}^s(q, \omega)$ for a disordered two dimensional electron system with Rashba spin orbit interaction when $\q$ is {\it transverse} to the electric field. Both the conventional definition of spin current and its new definition which takes care of the conservation of spins, have been considered. The spin Hall conductivitivities for both of these definitions are qualitatively similar. $\sigma_{yx}^s(q, \omega)$ is zero at $q=0, \omega =0$ and is maximum at $q=0$ and at small but finite $\omega$ whose value depends on different parameters of the system. Interestingly for $\omega \to 0$, $\sigma_{yx}^s(q)$ resonates when $\Lambda \simeq L_{so}$ which are the wavelength $(\Lambda = 2\pi/q)$ of the electric field's spatial variation and the length for one cycle of spin precession respectively. The sign of the out-of-plane component of the electrons' spin flips when the sign of electric field changes due to its spatial variation along transverse direction. It changes the mode of spin precession from clockwise to anti-clockwise or {\it vice versa} and consequently a finite spin Hall current flows in the bulk of the system.