  This chapter of "The Oxford Handbook of Nanoscience and Technology: Frontiers
and Advances" reviews nonequilibrium Green function (NEGF) approach to modeling
spin current generation, transport, and detection in semiconductor
nanostructures containing different types of spin-orbit (SO) couplings. Its
tutorial style--with examples drawn from the field of the spin Hall effects
(SHEs) and with treatment of the Rashba, Dresselhaus and extrinsic SO
couplings--offers practical recipes to compute total spin and charge currents
flowing out of the device, as well as the nonequilibrium local spin densities
and spin fluxes within the multiterminal nanostructure. These quantities, which
are obtained from the knowledge of spin-resolved NEGFs that can describe both
ballistic and diffusive transport regimes while handling phase-coherent effects
or dephasing mechanisms relevant at room temperature, can be employed to
understand recent experiments on all-electrical detection of mesoscopic and
quantum SHE in complicated low-dimensional nanostructures, as well as to model
a multitude of "second generation" spintronic devices exploiting coherent spin
dynamics. The chapter also provides extensive coverage of relevant technical
and computational details, such as: (i) the construction of retarded and lesser
Green functions for SO-coupled nanostructures attached to many electrodes; (ii)
computation of self-energies introduced by different types of electrodes
attached to the central region; and (iii) accelerated algorithms for NEGF
evaluation that make possible spin transport modeling in devices of the size
comparable to the spin precession length (typically few hundreds of nanometers)
that sets the scale where mesoscopic SHE effect in ballistic SO-coupled
nanostructures is expected to reach its optimal magnitude.
