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

We introduce the concept of bond spin currents, which describe the spin transport between two sites of the lattice model of a multiterminal spin-orbit (SO) coupled semiconductor nanostructure, and express them in terms of the spin-dependent nonequilibrium (Keldysh) Green functions for the Landauer setup where the nanostructure is attached to many semi-infinite ideal leads terminating in macroscopic thermalizing reservoirs. This formalism is applied to obtain the spatial distribution of microscopic spin currents in a clean phase-coherent two-dimensional electron (2DEG) gas with the Rashba type of SO coupling attached to four external leads. Together with the corresponding spatial profiles of the steady-state spin density, such visualization of the phase-coherent spin flow allow us to resolve several key issues for the understanding of microscopic 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: (i) while bond spin currents are nonzero locally within the SO coupled sample and neighboring region of the leads even in equilibrium (when all leads are at the same potential), the total spin currents obtained by summing the bond spin currents over any cross section within the leads are zero, so that no spin is actually transported by such equilibrium spin currents; (ii) when the device is brought into a nonequilibrium state (supporting steady-state charge current) by applying the external voltage difference between its longitudinal leads, only the wave functions (or Green functions) around the Fermi energy contribute to the total spin current through a given transverse cross section; (iii) the total spin Hall current is not conserved within the SO coupled region-however, it becomes conserved and physically well-defined quantity in the ideal leads where it is, furthermore, equal to the spin current obtained within the multiprobe Landauer-Buttiker scattering formalism in linear response regime. The spatial profiles of the local spin currents and stationary flowing spin densities 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 the static spin-independent disorder reduces the magnitude of the total spin current in the leads, the bond spin currents and spin densities remain nonzero throughout the whole diffusive 2DEG sample.