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

Abstract

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 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: (i) while bond spin currents are non-zero locally within the SO coupled region even in equilibrium (when all leads are at the same potential), the total spin current obtained by summing the bond spin currents over an arbitrary cross section is zero so that no spin can actually be transported by such equilibrium currents; (ii) when device is brought into nonequilibrium steady current state by external voltage difference applied between the 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-Büttiker scattering formalism. 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. Recent experimental observation of the spin Hall effect 1,2 opens new avenues for the understanding of fundamental role which spin-orbit (SO) couplings 3,4 can play in transport and equilibrium properties of semiconductor structures. While SO coupling effects are tiny relativistic corrections for particles moving through electric fields in vacuum, they can be enhanced in solids by several orders of magnitude due to the interplay of crystal symmetry and strong crystalline potential. 3,4 Furthermore, harnessing of spin currents induced by the spin Hall effect offers new possibilities for the envisioned all-electrical manipulation of spin for semiconductor spintronics applications 5 where electrical fields can access individual spins on s...

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