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The stationary state of the spin-Hall bar is studied in the framework of a variational approach that includes non-equilibrium screening effects at the lateral edges. The minimization of the power dissipated in the system is performed with taking into account the spin-flip relaxation and the global constrains due to the electric generator and global charge conservation. The calculation is performed within both the approximations of negligible spin-flip scattering and strong spin-flip scattering.
In both cases, simple expressions the longitudinal and the transverse pure spin-currents are derived analytically. In the first case, the spin-accumulation is linear across the Spin-Hall bar, symmetric from one edge to other, and the transverse spin-current vanishes. With strong spin-flip scattering, a transverse pure spin-current flows across the sample together with with longitudinal pure spin-current.
Surprisingly, due to the small values of the screening Debye length, the effect of the spin-flip scattering seems not to change significantly the profile and the amplitude of the spin-accumulation. A quantitative analysis predicts a spin accumulation of the order of $1\%$ of the density of carriers for an applied electric field of $5 mV/\m m^2$ at $30K$, with a spin-Hall angle of $\theta_{SH} = 10^{-4}$.
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