  We review recent studies of the shot noise of spin-polarized charge currents
and pure spin currents in multiterminal semiconductor nanostructures, while
focusing on the effects brought by the intrinsic Rashba spin-orbit (SO)
coupling and/or extrinsic SO scattering off impurities in two-dimensional
electron gas (2DEG) based devices. By generalizing the scattering theory of
quantum shot noise to include the full spin-density matrix of electrons
injected from a spin-filtering electrode, we show how decoherence and dephasing
in the course of spin precession can lead to substantial enhancement of the
Fano factor (noise-to-current ratio) of spin-polarized charge currents. In
four-terminal SO-coupled nanostructures, injection of unpolarized charge
current through the longitudinal leads is responsible not only for the pure
spin Hall current in the transverse leads, but also for nonequilibrium random
time-dependent current fluctuations. The analysis of the shot noise of
transverse pure spin Hall current and zero charge current, or transverse spin
current and non-zero charge Hall current, driven by unpolarized or
spin-polarized injected longitudinal charge current, respectively, reveals a
unique experimental tool to differentiate between the intrinsic Rashba and
extrinsic SO mechanisms underlying the spin Hall effect in 2DEG devices.
Finally, we discuss the shot noise of transverse spin and zero charge currents
in the quantum-interference-driven spin Hall effect in ballistic four-terminal
Aharonov-Casher rings realized using high-mobility 2DEG with the Rashba SO
coupling.
