  Magnetic and transport properties of a conducting layer with Rashba
spin-orbit coupling (RSOC) magnetically coupled to a layer of localized
magnetic moments are studied on strips of varying width. The localized moments
are free to rotate and they acquire an order that results from the competition
between the magnetic exchange energy and the kinetic energy of the conduction
electrons. By minimizing the total Hamiltonian within the manifold of
variational spiral orders of the magnetic moments, the phase diagram in the
space of the interlayer exchange J_{sd}, and the ratio of the Rashba coupling
to the hopping integral, lambda/t was determined. Two main phases with
longitudinal spiral order were found, one at large interlayer coupling J_{sd}
with uniform order in the transversal direction, and the other at small J_{sd}
showing a transversal staggered order. This staggered spiral order is unstable
against an antiferromagnetic (AFM) for large values of lambda/t. In both spiral
phases, the longitudinal spiral momentum that departs from the expected linear
dependence with the RSOC for large values of lambda/t. Then, various transport
properties, including the longitudinal Drude weight and the spin Hall
conductivity, inside these two phases are computed in linear response, and
their behavior is compared with the ones for the more well-studied cases of a
fixed ferromagnetic (FM) and AFM localized magnetic orders.
