  In this work, we present the analytical solution of the effective mass Pauli
equation, with Rashba and linear Dresselhaus interactions, for an electron gas
moving through a semiconductor quantum dot under a longitudinal electric field,
which is defined along the $x$-direction. We study the relative influence of
the Rashba and Dresselhaus terms on the spin-Hall effect for the first
propagating and edge channels, by analyzing the mixing between spin-up and
-down states and the zero-field spin splitting along the transverse directions.
When the spin rotation depends only on the $y$-coordinate, the spin orientation
and the spin density vary along this transverse coordinate and, in this case,
we show that the spin-Hall effect is only due to the Dresselhaus term, for
depolarized electrons. On the other hand, if the spin rotation depends on the
$z$-coordinate, the spin-Hall effect is provoked only by the Rashba
interaction.
