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

We analytically calculate the intrinsic spin-Hall conductivities (ISHCs) (sigma(z)(xy) and sigma(z)(yx)) in a clean, two-dimensional system with generic k-linear spin-orbit interaction. The coefficients of the product of the momentum and spin components form a spin-orbit matrix (beta) over bar. We find that the determinant of the spin-orbit matrix det (beta) over bar describes the effective coupling of the spin s(z) and orbital motion L-z. The decoupling of spin and orbital motion results in a sign change of the ISHC and the band-overlapping phenomenon. Furthermore, we show that the ISHC is in general unsymmetrical (sigma(z)(xy) not equal sigma(z)(yx)), and it is governed by the asymmetric response function Delta(beta) over bar, which is the difference in band-splitting along two directions: those of the applied electric field and the spin-Hall current. The obtained non-vanishing asymmetric response function also implies that the ISHC can be larger than e/8 pi, but has an upper bound value of e/4 pi. We will show that the unsymmetrical properties of the ISHC can also be deduced from the manifestation of the Berry curvature in the nearly degenerate area. On the other hand, by investigating the equilibrium spin current, we find that det (beta) over bar determines the field strength of the SU(2) non-Abelian gauge field.