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We analytically calculate the intrinsic spin-Hall conductivity (ISHC)($\sigma^z_{xy}$ and $\sigma^z_{yx}$) in a clean, two-dimensional system withgeneric k-linear spin-orbit interaction. The coefficients of the product of themomentum and spin components form a spin-orbit matrix $\widetilde{\beta}$. Wefind that the determinant of the spin-orbit matrix $\detbeta$ describes theeffective coupling of the spin $s_z$ and orbital motion $L_z$. The decouplingof spin and orbital motion results in a sign change of the ISHC and theband-overlapping phenomenon. Furthermore, we show that the ISHC is in generalunsymmetrical ($\sigma^z_{xy}\neq-\sigma^z_{yx}$), and it is governed by theasymmetric response function $\Deltabeta$, which is the difference inband-splitting along two directions: those of the applied electric field andthe spin-Hall current. The obtained non-vanishing asymmetric response functionalso implies that the ISHC can be larger than $e/8\pi$, but has an upper boundvalue of $e/4\pi$. We will that the unsymmetrical properties of the ISHC canalso be deduced from the manifestation of the Berry curvature at the nearlydegenerate area. On the other hand, by investigating the equilibrium spincurrent, we find that $\detbeta$ determines the field strength of the SU(2)non-Abelian gauge field.

Maximum intrinsic spin-Hall conductivity in two-dimensional systems with k-linear spin-orbit interaction