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We predict that unpolarized charge current driven through the longitudinalleads 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 voltageprobes. Its magnitude can be tuned by the Rashba spin-orbit (SO) interactionand, moreover, it is resilient to weak spin-independent scattering offimpurities within the metallic diffusive regime. While the polarization vectorof the spin transported through the transverse leads is not orthogonal to theplane of 2DEG, we demonstrate that only two components (out-of-plane andlongitudinal) of the transverse spin current are signatures of the spin Halleffect in four-probe Rashba spin-split semiconductor nanostructures. The linearresponse spin Hall current, obtained from the multiprobe Landauer-B\" uttikerscattering formalism generalized for quantum transport of spin, is theFermi-surface determined nonequilibrium quantity whose scaling with the 2DEGsize $L$ reveals the importance of processes occurring on the spin precession{\em mesoscale} $L_{\rm SO}$ (on which spin precesses by an angle $\pi$)--theout-of-plane component of the transverse spin current exhibits quasioscillatorybehavior for $L \lesssim L_{\rm SO}$ (attaining the maximum value in 2DEGs ofthe size $L_{\rm SO} \times L_{\rm SO}$), while it reaches the asymptotic valuein the macroscopic regime $L \gg L_{\rm SO}$. Furthermore, these values of thespin Hall current can be manipulated by the measuring geometry defined by theattached leads.Comment: 12 pages, 6 color EPS figures; expanded discussion to emphasize  crucial role played by processes on the spin precession mesoscal

Mesoscopic spin Hall effect in multiprobe ballistic spin-orbit-coupled semiconductor bridges