  An asymmetric triangular potential well provides the simplest model for the
confinement of mobile electrons at the interface between two insulating oxides,
such as LaAlO_3 and SrTiO_3 (LAO/STO). These electrons have been recently shown
to exhibit a large spin-orbit coupling of the Rashba type, i.e., linear in the
in-plane momentum. In this paper we study the intrinsic spin Hall effect due to
Rashba coupling in an asymmetric triangular potential well. Besides splitting
each subband into two branches of opposite helicity, the spin-orbit interaction
causes the wave function in the direction perpendicular to the plane of the
quantum well (the z direction) to depend on the in-plane wave vector k. In
contrast to the extreme asymmetric case, i.e., the wedge-shaped quantum well,
for which the intrinsic spin Hall effect vanishes due to vertex corrections, we
find that the asymmetric well supports a non-vanishing intrinsic spin Hall
conductivity, which increases in magnitude as the well becomes more symmetric.
The spin Hall conductivity is found to be proportional to the square of the
spin-orbit coupling constant and, in the limit of low carrier density, depends
only on the effective mass renormalization associated with the k-dependence of
the wave functions in the z direction. Its origin lies in the non-vanishing
matrix elements of the spin current between subbands corresponding to different
states of quantized motion perpendicular to the plane of the well.
