Intrinsic Spin Hall Edges

The prediction of intrinsic spin Hall currents by Murakami et al. and Sinova et al. raised many questions about methods of detection and the effect of disorder. We focus on a contact between a Rashba-type spin-orbit coupled region with a normal two-dimensional electron gas and show that the spin Hall currents, though vanishing in the bulk of the sample, can be recovered from the edges. We also show that the current-induced spin accumulation in the spin-orbit coupled system diffuses into the normal region and contributes to the spin current in the leads.

Figure 1

The schematic setup for SC generation in an R2DEG that is current biased by reservoirs R1 and R2 and equipped with 2DEG Hall contacts with vanishing spin-orbit coupling to reservoirs R3 and R4. The latter can be magnetic or nonmagnetic and voltage biased such that no charge current flows through the 2DEGs. The dashed arrows indicate the local SHC densities that are concentrated near the interfaces. The dominant part of the spin current flowing into R3 (or R4) is generated within a skin depth $L_s$ near the corresponding interface and contains a diffusion term from the SA $S_y$ in the bulk of the R2DEG. The contribution of the SHC density near R1 and R2 to the net SC [13] flowing into R3 (or R4) is exponentially small.

Figure 2

SCs and accumulations in the R2DEG ($y<0$) as a function of distance from the interface to the 2DEG. The full line is the local value of the SHC density for $\alpha k_F\tau /\hslash =0.1$. At the boundary the SHC density recovers its maximum value $j_B=eE{K}_{s-c}/2\pi \alpha$ (it approaches the universal value $eE/8\pi$ in the clean limit $\xi \gg 1$). The dashed line is the $z$ component of the diffusion current density. Inset: Corresponding local SA in the same system normalized to the magnitude of the bulk SA $eE\alpha \tau m/2\pi$. The solid and dashed lines represent the $y$ and $z$ components, respectively.