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The newly discovered two-dimensional magnetic semiconductors such as CrI$_3$have triggered a surge of interest stemming from their exotic spin-dependentproperties and potential applications in spintronics andmagneto-optoelectronics. Using first-principle density-functional-theorycalculations, we investigate the properties of the spin-polarization texture inmomentum space in the prototype 2D centrosymmetric ferromagnetic (FM) bilayerof CrI$_3$, with perpendicular magnetization. The FM bilayer displays a richin-plane spin texture in its highest valence bands. We show the existence oftwo distinct spin canting effects, which result from the coupling of FM orderwith structural chirality and electric polarization, in establishing thein-plane spin texture. The first effect is generated by the chirality of thelayer stacking and the spin-orbit-polarized nature of the valence states, andyields the same canting on both layers. The second effect is a Rashba-relatedeffect, which in a centrosymmetric ferromagnet induces in a single electronicstate two opposite spin-canted components on the two layers. Furthermore, usingthe FM bilayer as an example, we provide general rules imposed bymagnetic-space-group symmetries on spin-polarization vectors, serving asguidelines for the shape of the spin texture in the Brillouin zone of magneticcrystals. Finally, we show that the above effects can be effectively used tomanipulate the spin texture via compressive strain, which induces in the FMbilayer valence-band-edge states with canted spins.

Chirality and Rashba-related effects in the spin texture of a 2D centro-symmetric ferromagnet: the case of CrI3 bilayer