  The antiferromagnetic (AFM) CrI$_3$ bilayer is a particularly interesting
representative of van der Waals 2D semiconductors, which are currently being
studied for their magnetism and for their potential in spintronics. Using ab
initio density-functional theory calculations, we investigate the spin texture
in momentum space of the states of the (doubly degenerate) highest valence band
of the AFM CrI$_3$ bilayer with Cr-spin moments perpendicular to the layers. We
find the existence, in the main central part of the Brillouin zone, of a Rashba
in-plane spin texture of opposite signs on the two layers, resulting from the
intrinsic local electric fields acting on each layer. To study the layer
segregation of the wavefunctions, we apply a small electric field that splits
the degenerate states according to their layer occupancy. We find that the
wavefunctions of the highest valence band are layer-segregated, belonging to
only one of the two layers with opposite in-plane spin textures, and the
segregation occurs over nearly the whole Brillouin zone. The corresponding
layer locking of the in-plane-canted spin is related to the separation in
energy of the highest AFM band from the rest of the valence bands. We explain
how the band interactions destroy the layer locking at the $K$, $K'$, and
$\Gamma$ points. Furthermore, we compare the layer locking of the
in-plane-canted spin in our AFM bilayer system with the hidden spin
polarization in centrosymmetric nonmagnetic materials, pointing out the
differences in segregation mechanisms and their consequences for the layer
locking. We note that a similar Rashba effect with layer locking of
in-plane-canted spin could occur in other van der Waals AFM bilayers with
strong spin-orbit coupling and an isolated energy band.
