Intrinsic spin-Hall effect in the presence of an in-plane magnetic field

The intrinsic spin-Hall effect (SHE) induced by a driving electric field E$_{x}$ in the presence of an in-plane magnetic field $\vec {B}$ in a 2D semiconductor strip is studied. In the diffusive regime, the spatial distribution of spin densities S$_{i}$ (\textit{i=x, y ,z}) is calculated from a spin diffusion equation derived from nonequilibrium Green's function. For the case of Rashba spin-orbit interaction (SOI), we find that the spin polarization S$_{z}$ normal to the 2D strip remains zero with or without the in-plane magnetic field. For the case of Dresselhaus SOI, where cubic term is included, the symmetry of S$_{z}$ with respect to the in-plane magnetic field depends on the orientation of the $\vec {B}$ field. With $\vec {B}$ along $\hat {x}$, S$_{z}$ exhibits symmetric dependence on B$\hat {x}$. However, with a transverse in-plane magnetic field, along $\hat {y}$, at the edge of the strip exhibits asymmetric dependence on B$\hat {y}$. These results lead to a possible diagnostic tool for the identification of the SOI in the system.