Spin-Hall Effect and Diamagnetism of Anisotropic Dirac Electrons in Solids

Spin-Hall conductivity $\sigma_{{\rm s}xy}$ and orbital susceptibility $\chi$ are investigated for the anisotropic Wolff Hamiltonian, which is an effective Hamiltonian common to Dirac electrons in solids. It is found that, both for $\sigma_{{\rm s}xy}$ and $\chi$, the effect of anisotropy appears only in the prefactors, which is given as the Gaussian curvature of the energy dispersion, and their functional forms are equivalent to those of the isotropic Wolff Hamiltonian. As a result, it is revealed that the relationship between the spin Hall conductivity and the orbital susceptibility in the insulating state, $\sigma_{{\rm s}xy}=(3mc^2/\hbar e)\chi$, which was firstly derived for the isotropic Wolff Hamiltonian, is also valid for the anisotropic Wolff Hamiltonian. Based on this theoretical finding, the magnitude of spin-Hall conductivity is estimated for bismuth and its alloys with antimony by that of orbital susceptibility, which has good correspondence between theory and experiments. The magnitude of spin-Hall conductivity turns out to be as large as $e\sigma_{{\rm s}xy} \sim 10^4 {\Omega}^{-1}{\rm cm}^{-1}$, which is about 100 times larger than that of Pt.