Spin Hall effect of a 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 (Zhang et al., cond-mat/0503505) 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 nonzero spin Hall current are clarified. In particular, it is found that (i) the spin Hall current is nonzero 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.

Figure 1

Diagrammatic representation of the self-energy $\underset{̱}{\Sigma }$ in the present self-consistent approximation. Doubled line is the dressed Keldysh Green’s function $\underset{̱}{G}$. Dashed line with a cross denotes the average over the impurity positions in the second order for (a) and in the third order (b), respectively.

Figure 2

(Color online) Contour map of the SHC ${\sigma }_H^s$. It has the maximum value nearly along $\alpha =2\lambda p_F\tau =1$. The momentum cutoff is set to be $\Lambda \approx 3p_F$, and $U∕{\epsilon }_F=0.02$.