Screening effect in spin Hall devices

The stationary state of the spin Hall bar is studied in the framework of a variational approach that includes nonequilibrium screening effects at the edges. The minimization of the power dissipated in the system is performed taking into account the spin-flip relaxation and the global constraints due to the electric generator and global charge conservation. The calculation is performed within the approximations of negligible spin-flip scattering and strong spin-flip scattering. In both cases, simple expressions of the spin accumulation and the longitudinal and transverse pure spin currents are derived analytically. In usual conditions, the maximum amplitude of the spin accumulation is expected to be of the order of 1% of the equilibrium density carriers.

Figure 1

Schematic representation of the two channel model (corresponding to spin $\uparrow$ and spin $\downarrow$) for a spin Hall bar. The device is contacted to an electric generator that imposes a longitudinal stationary current $J_x^0$ (not shown). The charge accumulations $\pm \delta n_{\updownarrow }$ and the inhomogeneous part of the longitudinal currents $\delta J_{x\updownarrow }\left(y\right)=J_{x\updownarrow }\left(y\right)-J_x^0$ are represented at both edges for each channel together with the transverse current $J_{y\updownarrow }$ flowing from one edge to the other.

Figure 2

Superimposition of the two channels represented for the symmetric device and environment ($C_E=0$ in the text). The electric generator and the injected current $J_x^0$ are not shown for clarity. The system is characterized by the absence of the total charge accumulation, the spin accumulation $n^{Sp}\left(y\right)=\delta n_{\uparrow }\left(y\right)-\delta n_{\downarrow }\left(y\right)$ with opposite directions at the two edges, the inhomogeneous part of the longitudinal spin currents $J_x^{Sp}\left(y\right)=J_{x\uparrow }\left(y\right)-J_{x\downarrow }\left(y\right)$, and the transverse spin current $J_y^{Sp}=J_{y\uparrow }\left(y\right)-J_{y\downarrow }\left(y\right)$.

Figure 3

(a) Transverse pure spin current $J_y^{Sp}$ [Eq. (37) in the text] as a function the ratio $\ell /l_{\mathrm{sf}}$ of the width of the spin Hall bar over the spin-flip scattering length [Eq. (37) and Eq. (36)]. (b) Longitudinal pure spin current $J_x^{Sp}$ as a function of the temperature [Eq. (39)] and [Eq. (20)] for different values of the injected voltage $q\mathrm{\Delta }V$, expressed in $meV$. The amplitude is calculated for $y=L=\ell$. The currents are normalized by the product of the injected current $2J_x^0$ by the spin Hall angle ${\theta }_{\mathrm{so}}$ at the first order in ${\theta }_{\mathrm{so}}$.

Figure 4

Spin accumulation $n^{Sp}$ [Eq. (20) in the text] as a function the ratio $y/\ell$ for different values of the ratio $q\mathrm{\Delta }V/kT$, from $q\mathrm{\Delta }V/kT=2$ to $q\mathrm{\Delta }V/kT=12$. The spin accumulation is normalized by the product of the density of carriers at equilibrium $2n_0$ by the spin Hall angle ${\theta }_{\mathrm{so}}$. The curve for $q\mathrm{\Delta }V/kT=-12$ shows that reversing the electric polarity reverses the sign of the spin accumulation.