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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., Comment: 54 pages, 11 figures; Chapter 24, page 814--866 in Volume I of "The Oxford Handbook on Nanoscience and Technology: Frontiers and Advances," Eds. A. V. Narlikar and Y. Y. Fu (Oxford University Press, Oxford, 2010)

We introduce the concept of bond spin current, which describes the spin transport between two sites of the lattice model of a multiterminal spin-orbit (SO) coupled semiconductor nanostructure, and express it in terms of the spin-dependent nonequilibrium (Landauer-Keldysh) Green functions of the device. This formalism is applied to obtain the spatial distribution of microscopic spin currents in {\em clean} phase-coherent two-dimensional electron gas with the Rashba-type of SO coupling attached to four external leads. Together with the corresponding profiles of the stationary spin density, such visualization of the phase-coherent spin flow allow us to resolve several key issues for the understanding of mechanisms which generate pure spin Hall currents in the transverse leads of ballistic devices due to the flow of unpolarized charge current through their longitudinal leads. The local spin current profiles crucially depend on whether the sample is smaller or greater than the spin precession length, thereby demonstrating its essential role as the characteristic mesoscale for the spin Hall effect in ballistic multiterminal semiconductor nanostructures. Although static spin-independent disorder reduces the magnitude of the total spin current in the leads, the bond spin currents continue to flow through the whole diffusive 2DEG sample, without being localized as edge spin currents around any of its boundaries., Comment: 17 pages, 8 color EPS figures; preprint with much higher resolution figures is available from http://www.physics.udel.edu/~bnikolic/QTTG/publications.htm

We demonstrate that flow of longitudinal unpolarized current through a ballistic two-dimensional electron gas with Rashba spin-orbit coupling will induce nonequilibrium spin accumulation which has opposite sign for the two lateral edges and it is, therefore, the principal observable signature of the spin Hall effect in two-probe semiconductor nanostructures. The magnitude of its out-of-plane component is gradually diminished by static disorder, while it can be enhanced by an in-plane transverse magnetic field. Moreover, our prediction of the longitudinal component of the spin Hall accumulation, which is insensitive to the reversal of the bias voltage, offers a smoking gun to differentiate experimentally between the extrinsic, intrinsic, and mesoscopic spin Hall mechanisms., Comment: 5 pages, 3 color EPS figures; published version

We propose an all-electrical nanoscopic structure where {\em pure} spin current is induced in the transverse voltage probes attached to {\em quantum-coherent} one-dimensional ring when conventional unpolarized charge current is injected through its longitudinal leads. Tuning of the Rashba spin-orbit coupling in semiconductor heterostructure hosting the ring generates quasi-periodic oscillations of the predicted spin Hall current due to {\em spin-sensitive quantum-interference effects} caused by the difference in Aharonov-Casher phase acquired by opposite spins states traveling clockwise and counterclockwise. Its amplitude is comparable to the mesoscopic spin Hall current predicted for finite-size two-dimensional electron gases, while it gets reduced in wide two-dimensional or disordered rings., Comment: 5 pages, 4 color figures

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., Comment: 12 pages, 6 color EPS figures; expanded discussion to emphasize crucial role played by processes on the spin precession mesoscale

The conventional unpolarized current injected into a {\em quantum-coherent} semiconductor ring attached to two external leads can be modulated from perfect conductor to perfect insulator limit via Rashba spin-orbit (SO) coupling. This requires that ballistic propagation of electrons, whose spin precession is induced by the Aharonov-Casher phase, takes place through a single conducting channel ensuring that electronic quantum state remains a pure separable one in the course of transport. We study the fate of such spin interference effects as more than one orbital conducting channel becomes available for quantum transport. Although the conductance of multichannel rings, in general, does not go all the way to zero at any value of the SO coupling, some degree of current modulation survives. We analyze possible scenarios that can lead to reduced visibility of spin interference effects that are responsible for the zero conductance at particular values of the Rashba interaction: (i) the transmitted spin states remain fully coherent, but conditions for destructive interference are different in different channels; (ii) the transmitted spins end up in partially coherent quantum state arising from entanglement to the environment composed of orbital degrees of freedom of the same particle to which the spin is attached., Comment: 12 pages, 11 EPS color figures

We investigate quantum coherence of electron spin transported through a semiconductor spintronic device, where spins are envisaged to be controlled by electrical means via spin-orbit interactions. To quantify the degree of spin coherence, which can be diminished by an intrinsic mechanism where spin and orbital degrees of freedom become entangled in the course of transport involving spin-orbit interaction and scattering, we study the decay of the off-diagonal elements of the spin density matrix extracted directly from the Landauer transmission matrix of quantum transport. This technique is applied to understand how to preserve quantum interference effects of fragile superpositions of spin states in ballistic and non-ballistic multichannel semiconductor spintronic devices., Comment: 7 pages, 3 color EPS figures, prepared for Proceedings of International Symposium on Mesoscopic Superconductivity and Spintronics 2004 (Atsugi, Japan, March 1-4, 2004)

By viewing current in the detecting lead of a spintronic device as being an ensemble of flowing spins corresponding to a mixed quantum state, where each spin itself is generally described by an improper mixture generated during the transport where it couples to other degrees of freedom due to spin-orbit (SO) interactions or inhomogeneous magnetic fields, we introduce the spin density operator associated with such current expressed in terms of the spin-resolved Landauer transmission matrix of the device. This formalism, which provides complete description of coupled spin-charge quantum transport in open finite-size systems attached to external probes, is employed to understand how initially injected pure spin states, comprising fully spin-polarized current, evolve into the mixed ones corresponding to partially polarized current. We analyze particular routes that diminish spin coherence (signified by decay of the off-diagonal elements of the current spin density matrix) in two-dimensional electron gas-based devices due to the interplay of the Rashba and/or Dresselhaus SO coupling and: (i) scattering at the boundaries or lead-wire interface in ballistic semiconductor nanowires; or (ii) spin-independent scattering off static impurities in both weakly and strongly localized disordered nanowires. The physical interpretation of spin decoherence in the course of multichannel quantum transport in terms of the entanglement of spin to an effectively zero-temperature ``environment'' composed of more than one open orbital conducting channels offers insight into some of the key challenges for spintronics: controlling decoherence of transported spins and emergence of partially coherent spin states in all-electrical spin manipulation schemes based on the SO interactions in realistic semiconductor structures., Comment: 16 pages, 9 color figures