Quasiclassical approach to the spin Hall effect in the two-dimensional electron gas

We study the spin-charge coupled transport in a two-dimensional electron system using the method of quasiclassical ($\xi$-integrated) Green’s functions. In particular we derive the Eilenberger equation in the presence of a generic spin-orbit field. The method allows us to study spin and charge transport from ballistic to diffusive regimes and continuity equations for spin and charge are automatically incorporated. In the clean limit we establish the connection between the spin Hall conductivity and the Berry phase in momentum space. For finite systems we solve the Eilenberger equation numerically for the special case of the Rashba spin-orbit coupling and a two-terminal geometry. In particular, we calculate explicitly the spin Hall induced spin polarization in the corners, predicted by Mishchenko et al. [Phys. Rev. Lett. 93, 226602 (2004)]. Furthermore we find universal spin currents in the short-time dynamics after switching on the voltage across the sample, and calculate the corresponding spin Hall polarization at the edges. Where available, we find perfect agreement with analytical results.

Figure 1

The two-terminal geometry under consideration: A rectangular strip of a two-dimensional electron gas is connected to two reservoirs. We assume that the strip and the reservoirs are made of the same material, i.e., the spin-orbit field exists also in the reservoirs.

Figure 2

Spin polarization in the presence of an electrical current flowing in the $x$ direction for a strip of length $L_x=20l$ and $L_y=10l$. The spin-orbit coupling strength is $\alpha ={10}^{-3}{v}_F$ and the elastic scattering rate is $1∕\tau =\alpha p_F∕2$. The spin polarization is given in units of the bulk value, $S_0=-\mid e\mid E\alpha \tau N_0$.

Figure 3

$S_y$ in units of $S_0$ as a function of $x$ for $L_x=200l$, $L_y=100l$, $\alpha ∕v_F={10}^{-3}$, and $\alpha p_F\tau =0.005,0.01,0.02,0.05,0.1,1$ (from bottom to top).

Figure 4

Spin relaxation length $L_s$ in units of $l$ as a function of disorder, obtained by fitting the spatial dependence of the electric field induced spin polarization (shown in Fig. 3) using $S_y=a+b\left\{\exp \right(-x∕L_s)+\exp [-(L_x-x)∕L_s\left]\right\}$. The diffusive limit expression is shown as a dashed line.

Figure 5

Time evolution of the spin Hall current at the interface to the leads and in the bulk. In the bulk we compare our numerical result (data points) with the analytical result (full line) of Eq. (48). Near the leads, only numerical data are available (dashed curve). $j_{s,y}^z$ is evaluated at $y=L_y∕2$, $x=0$ (boundary) and $x=L_x∕2$ (bulk) for $L_x=20l$, $L_y=10l$, $\alpha ∕v_F={10}^{-3}$ and $\alpha p_F\tau =2$.

Figure 6

Spin Hall effect induced spin polarization $S_z$ in units of $S_0$ as a function of $y$ and $t$ at $x=L_x∕2$ for $L_x=20l$, $L_y=10l$, $\alpha ∕v_F={10}^{-3}$, and $\alpha p_F\tau =0.25,2,5$ (from bottom to top).