Spin Hall effect of conserved current: Conditions for a nonzero spin Hall current

We study the spin Hall effect taking into account the impurity scattering effect as general as possible with the focus on the definition of the spin current. The conserved bulk spin current (Shi et al. [Phys. Rev. Lett. 96, 076604 (2006)]) satisfying the continuity equation of spin is considered in addition to the conventional one defined by the symmetric product of the spin and velocity operators. Conditions for non-zero spin Hall current are clarified. In particular, it is found that (i) the spin Hall current is non-zero in the Rashba model with a finite-range impurity potential, and (ii) the spin Hall current vanishes in the cubic Rashba model with a $\delta$-function impurity potential.