Numerical study of the spin Hall current in a quantum wire

We employ the scattering matrix method to numerically study the transverse pure spin current in a quasi-one-dimensional quantum wire with Rashba spin-orbital interaction (RSOI). The current and/or detection leads are considered with RSOI the same as that in the sample to preclude the surface spin scattering. The nonzero spin Hall conductance (SHC) in the infinite quantum wire with uniform RSOI confirms it to be an intrinsic property of this quasi-one-dimensional system. It is found that the SHC oscillates with the RSOI strength, Fermi energy, and system size, but keeps the same order of the universal value ($e∕8\pi$). For the leads without RSOI the spin current can reverse its direction with suitable parameters.

Figure 1

Schematic of the tight-binding model for a quasi-one-dimensional quantum wire with two longitudinal $\left(x\right)$ current leads and two transverse $\left(y\right)$ spin current probes. $V_i$ is the voltage at the corresponding lead $i$. A current flows from lead 1 to lead 3, $I_1=I_3$, and leads 2 and 4 have nonzero pure spin current but zero charge current. The leads are considered with and without RSOI.

Figure 2

SHC as a function of the Fermi energy $E_F$ for all leads without RSOI (a), leads 1 and 3 with RSOI (b), and all leads with RSOI (c); the solid line for $G_{SH}^2$ and the dotted line for $G_{SH}^4$. The scattering region size is fixed at $45\times 45$ and the RSOI strength is set as $t_{so}=0.05t.$

Figure 3

SHC $G_{SH}^{2\left(4\right)}$ as a function of the RSOI strength $t_{so}$ for system size $45\times 45,E_F=0.4t$; the solid line for $G_{SH}^2$ and the dotted line for $G_{SH}^4$. (a) Zero RSOI considered in all four leads, (b) the current leads with RSOI, and (c) all leads with RSOI.

Figure 4

The SHC vs the size $L$ of a square scattering region $45\times 45$, the solid line for $G_{SH}^2$ and the dotted line for $G_{SH}^4$. $E_F=0.4t$ and $t_{so}=0.05t$. (a), (b), and (c) correspond to three cases: all leads without RSOI, only current leads with RSOI, and all leads with RSOI, respectively. In the inset of (a), the SHC as a function of the position of the probes 2(4) with a rectangular scattering region with size $45\times 90$.