  A system composed of a conducting planar strip with Rashba spin-orbit
coupling (RSOC), magnetically coupled to a layer of localized magnetic moments,
at equilibrium, is studied within a microscopic Hamiltonian with numerical
techniques at zero temperature in the clean limit. In particular, transport
properties for the cases of ferromagnetic (FM) and antiferromagnetic (AFM)
coupled layers are computed in linear response on strips of varying width. Some
behaviors observed for these properties are consistent with the ones observed
for the corresponding Rashba helical currents. The case of uncoupled Rashba
strips is also studied for comparison. In the case of Rashba strips coupled to
an AFM localized order, results for the longitudinal dc conductivity, for small
strip widths, suggest the proximity to a metal-insulator transition. More
interesting, in the proximity of this transition, and in general at
intermediate values of the RSOC, it is observed a large spin-Hall conductivity
that is two orders of magnitude larger than the one for the FM order for the
same values of the RSOC and strip widths. There are clearly two different
regimes for small and for large RSOC, which is also present in the behavior of
Rashba helical currents. Different contributions to the optical and the
spin-Hall conductivities, according to a new classification of inter- or
intra-band origin proposed for planar strips in the clean limit, or coming from
the hopping or spin-orbit terms of the Hamiltonian, are examined. Finally, the
effects of different orientation of the coupled magnetic moments will be also
studied.
