Universal Scaling of the Spin Hall Effect

Abstract

We study the spin Hall (SH) effect for the Dirac electrons in terms of the spin and magnetic-moment accumulation coefficients $-\hat{I}{“}^{00}{g}_{s(m)z}^{xy}$. We take short-range nonmagnetic impurities into account within the self-consistent $T$-matrix approximation. Similarly to the universal scaling for the anomalous Hall (AH) effect, we find three disctinct regimes by changing the electric conductivity $\ddot{I}{ƒ}^{yy}$; the superclean regime with $-\hat{I}{“}^{00}{g}_{s(m)z}^{xy}\propto \ddot{I}{ƒ}^{yy}$ owing to the skew scattering, moderately dirty regime with almost constant $-\hat{I}{“}^{00}{g}_{s(m)z}^{xy}$, and dirty regime with a new scaling relation $-\hat{I}{“}^{00}{g}_{s(m)z}^{xy}\propto (\ddot{I}{ƒ}^{yy}{)}^{0.6}$ whose exponent differs from that of the AH conductivity. Our results construct a unified theory of the SH effect without any ambiguity of spin current.

12 pages, 4 figures