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We introduce the concept of bond spin current, which describes the spintransport between two sites of the lattice model of a multiterminal spin-orbit(SO) coupled semiconductor nanostructure, and express it in terms of thespin-dependent nonequilibrium (Landauer-Keldysh) Green functions of the device.This formalism is applied to obtain the spatial distribution of microscopicspin currents in {\em clean} phase-coherent two-dimensional electron gas withthe Rashba-type of SO coupling attached to four external leads. Together withthe corresponding profiles of the stationary spin density, such visualizationof the phase-coherent spin flow allow us to resolve several key issues for theunderstanding of mechanisms which generate pure spin Hall currents in thetransverse leads of ballistic devices due to the flow of unpolarized chargecurrent through their longitudinal leads. The local spin current profilescrucially depend on whether the sample is smaller or greater than the spinprecession length, thereby demonstrating its essential role as thecharacteristic mesoscale for the spin Hall effect in ballistic multiterminalsemiconductor nanostructures. Although static spin-independent disorder reducesthe magnitude of the total spin current in the leads, the bond spin currentscontinue to flow through the whole diffusive 2DEG sample, without beinglocalized as edge spin currents around any of its boundaries.

physical review. b, condensed matter and materials physics

Imaging mesoscopic spin Hall flow: Spatial distribution of local spin currents and spin densities in and out of multiterminal spin-orbit coupled semiconductor nanostructures