Spin Hall effect and spin filtering in ballistic nanojunctions

We discuss the behaviour of a four- probe ballistic cross nanojunction patterned in a two- dimensional electron gas ( 2DEG) by taking into account the spin - orbit ( SO) coupling due to the in- plane electric field. We propose a new scheme of spin filtering based on this SO interaction in the regime of the so- called quenching of the spin Hall effect by discussing the geometrical effects on the transport with a semiclassical approach. We show that a pure spin current is induced in the transverse probes when an unpolarized charge current is injected through their longitudinal leads. We relate the spin conductance in the transverse probes to the charge conductance through the longitudinal leads. We also demonstrate that the flow of a longitudinal unpolarized current through the junction will induce a spin accumulation which has opposite signs for the two lateral probes.