Asymmetries in intrinsic spin-Hall effect in low in-plane magnetic field

Effects of low in-plane magnetic field on bulk spin densities and edge spin accumulations of a diffusive two-dimensional semiconductor stripe are studied. Focusing upon the Dresselhaus-type intrinsic spin-orbit interaction (SOI), we look for the symmetry, or asymmetry, characteristics in two magnetic-field orientations: along and transverse to the stripe. For longitudinal field, the out-of-plane spin density $S_z$ exhibits odd parity across the stripe and even parity in the magnetic field and is an edge accumulation. For transverse field, the out-of-plane $S_z$ becomes asymmetric in both spatial and field dependencies and has finite bulk values for finite magnetic fields. Our results support utilizing low in-plane magnetic fields for the probing of the underlying SOI.

Figure 1

Top-view schematic illustration of the 2D stripe. The 2D stripe has a width $d$. In the system, electric field $\mathbf{E}$ and in-plane magnetic field $\mathbf{B}$ are applied. The direction along the stripe, or $\widehat{x}$, is denoted longitudinal and that along $\widehat{y}$, transverse.

Figure 2

(Color online) Spin densities $S_i$ versus $y$, in units of $l_{\text{so}}$, for the case of a longitudinal in-plane magnetic field. Spin densities $S_x$, $S_y$, and $S_z$ in units of $\mu {\text{m}}^{-2}$ are shown in (a), (b), and (c), respectively, for magnetic fields $B_x=-300\text{mT}$ (black/ triangles), $B_x=0\text{mT}$ (blue/solid curve) and $B_x=300\text{mT}$ (red/dashed curve). The edges of the stripe are at $y=\pm 5l_{\text{so}}$.

Figure 3

(Color online) Spin densities $S_i$ versus $y$ for the case of a transverse in-plane magnetic field. Spin densities $S_x$ and $S_z$ in units of $\mu {\text{m}}^{-2}$ are shown in (a) and (b), respectively, for magnetic fields $B_y=-300\text{mT}$ (black/triangles), $B_y=0\text{mT}$ (blue/solid curve), and $B_y=300\text{mT}$ (red/dashed curve). $S_y$ remains zero in all these cases.

Figure 4

(Color online) Edge spin densities $S_i^{\pm }$ versus magnetic field for both field orientations: longitudinal $B_x$ cases in (a) and (b) and transverse $B_y$ cases in (c) and (d). $S_i^{\pm }$ denotes spin densities at the edges $y=\mp d/2$. $S_x$ is labeled by the dashed curve, $S_y$ by the dashed-dotted curve, and $S_z$ by the solid curve.

Figure 5

(Color online) Contour plot of $S_z$ on the $B_i\text{−}y$ plane for (a) the longitudinal and (b) the transverse field orientations.