  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.
