Mesoscopic Spin Hall Effect in Multiprobe Ballistic Spin-Orbit Coupled Semiconductor Bridges

We predict that unpolarized charge current driven through the longitudinal leads attached to ballistic quantum-coherent two-dimensional electron gas (2DEG) in semiconductor heterostructure will induce a {\em pure} spin current, which is not accompanied by any net charge flow, in the transverse voltage probes. Its magnitude can be tuned by the Rashba spin-orbit (SO) interaction and, moreover, it is resilient to weak spin-independent scattering off impurities within the metallic diffusive regime. While the polarization vector of the spin transported through the transverse leads is not orthogonal to the plane of 2DEG, we demonstrate that only two components (out-of-plane and longitudinal) of the transverse spin current are signatures of the spin Hall effect in four-probe Rashba spin-split semiconductor nanostructures. The linear response spin Hall current, obtained from the multiprobe Landauer-B\" uttiker scattering formalism generalized for quantum transport of spin, is the Fermi-surface determined nonequilibrium quantity whose scaling with the 2DEG size $L$ reveals the importance of processes occurring on the spin precession {\em mesoscale} $L_{\rm SO}$ (on which spin precesses by an angle $\pi$)--the out-of-plane component of the transverse spin current exhibits quasioscillatory behavior for $L \lesssim L_{\rm SO}$ (attaining the maximum value in 2DEGs of the size $L_{\rm SO} \times L_{\rm SO}$), while it reaches the asymptotic value in the macroscopic regime $L \gg L_{\rm SO}$. Furthermore, these values of the spin Hall current can be manipulated by the measuring geometry defined by the attached leads.