Symmetry and transport property of spin current induced spin-Hall effect

We study the spin current induced spin-Hall effect that a longitudinal spin dependent chemical potential $qV_{s=x,y,z}$ induces a transverse spin conductances $G^{ss'}$. A four terminal system with Rashba and Dresselhaus spin-orbit interaction (SOI) in the scattering region is considered. By using Landauer-B$\ddot u$ttiker formula with the aid of the Green function, various spin current induced spin-Hall conductances $G^{ss'}$ are calculated. With the charge chemical potential $qV_c$ or spin chemical potential $qV_{s=x,y,z}$, there are 16 elements for the transverse conductances $G^{\mu \nu}_p=J_{p,\mu}/V_{\nu}$ where $\mu,\nu=x,y,z,c$. Due to the symmetry of our system these elements are not independent. For the system with $C_2$ symmetry half of elements are zero, when the center region only exists the Rashba SOI or Dresselhaus SOI. The numerical results show that of all the conductance elements, the spin current induced spin-Hall conductances $G^{ss'}$ are usually much greater (about one or two orders of magnitude) than the spin Hall conductances $G^{sc}$ and the reciprocal spin Hall conductances $G^{cs}$. So the spin current induced spin-Hall effect is dominating in the present device.