Anisotropic Rashba splitting dominated by out-of-plane spin polarization in two-dimensional Janus $XA_{2}Y$ ($A$= Si, Sn, Ge; $X,Y$= Sb, Bi) with surface imperfection

The anisotropic Rashba effect allows for the manipulation of electron spins in a more precise and tunable manner since the magnitude of the Rashba splitting and orientation of the spin textures can be simply controlled by tuning the direction of the externally applied electric field. Herein, we predict the emergence of the anisotropic Rashba effect in the two-dimensional (2D) Janus $XA_{2}Y$ constructed from the group IV ($A$= Si, Sn, Ge) and group V ($X,Y$ = Sb, Bi) elements having trigonal prismatic structures but lacking in-plane mirror symmetry. Due to the lowering point group symmetry of the crystal, the Rashba spin splitting is enforced to becomes anisotropic around certain high symmetry points in the Brillouin zone and preserves the out-of-plane spin textures. We illustrate this behavior using density functional theory calculations supplemented with $\vec{k}\cdot\vec{p}$ analysis on the Janus SbSi$_{2}$Bi monolayer as a representative example. Specifically, we observed large and anisotropic Rashba splitting with prominence contribution of the out-of-plane spin textures in the conduction band minimum around the $M$ point and valence band maximum around the $\Gamma$ point. More importantly, the anisotropic spin splitting and out-of-plane spin polarization are sensitively affected by surface imperfections, depending on the concentration and configuration of the $X$ and $Y$ elements in the 2D Janus $XA_{2}Y$ surface. Our study offer the possibility to realize the present systems for spintronics applications.