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It has been generally accepted that the spin-orbit coupling effect innoncentrosymmetric materials leads to the band splitting and non-trivial spinpolarization in the momentum space. However, in some cases, zero net spinpolarization in the split bands may occurs, dubbed as the band splitting withvanishing spin polarization (BSVSP) effect, protected by non-pseudo-polar pointgroup symmetry of the wave vector in the first Brillouin zone [Liu et. al.,Nat. Commun. \textbf{10}, 5144 (2019)]. In this paper, by usingfirst-principles calculations, we show that the BSVSP effect emerges intwo-dimensional (2D) nonsymmorphic Ga$XY$ ($X$= Se, Te; $Y$= Cl, Br, I) family,a new class of 2D materials having in-plane ferroelectricity. Taking the GaTeClmonolayer as a representative example, we observe the BSVSP effect in the splitbands along the $X-M$ line located in the proximity of the conduction bandminimum. By using $\vec{k}\cdot\vec{p}$ Hamiltonian derived based on thesymmetry analysis, we clarify that such effect is originated from thecancellation of the local spin polarization, enforced by non-pseudo-polar$C_{2v}$ point group symmetry of the wave vector along the $X-M$ line.Importantly, we find that the spin polarization can be effectively induced byapplying an external out-of-plane electric field, indicating that anelectrically tunable spin polarization for spintronic applications isplausible.

Large band splitting with tunable spin polarization in two-dimensional ferroelectric GaXY (X= Se, Te; Y= Cl, Br, I) family