  We propose a new scheme of spin filtering based on nanometric crossjunctions
in the presence of Spin Orbit interaction, employing ballistic nanojunctions
patterned in a two-dimensional electron gas. We demonstrate that the flow of a
longitudinal unpolarized current through a ballistic X junction patterned in a
two-dimensional electron gas with Spin Orbit coupling (SOC) induces a spin
accumulation which has opposite signs for the two lateral probes. This spin
accumulation, corresponding to a transverse pure spin current flowing in the
junction, is the main observable signature of the spin Hall effect in such
nanostructures.
  We benchmark the effects of two different kinds of Spin Orbit interactions.
The first one ($\alpha$-SOC) is due to the interface electric field that
confines electrons to a two-dimensional layer, whereas the second one
($\beta$-SOC) corresponds to the interaction generated by a lateral confining
potential.

Mesoscale and Nanoscale Physics