Reversible giant out-of-plane Rashba effect in two-dimensional Ga$XY$ ($X$= Se, Te; $Y$= Cl, Br, I) compounds for persistent spin helix

First, we characterize the optimized structural parameters, symmetry of the crystal, and ferroelectricity of the Ga$XY$ ML compounds, where the atomic structure is displayed in Fig. 1(a). The crystal structure of the Ga$XY$ ML is noncentrosymmetric having a black-phosphorene-type structure belonging to $Pmn2_{1}$ space group Zhou et al. (2018); Zhang and Liu (2018); Wu et al. (2019); Absor and Ishii (2021); Kniep et al. (1983). For the convenience ofdiscussion, we choose the $x$ ($y$) axis to be along the zigzag (armchair) direction in the real space so that the reciprocal space is characterized by the FBZ as shown in Fig. 1(b). There are six atoms in the unit cell consisted of two Ga atoms, two chalcogen atoms (labeled by $X$ and $X^{\prime}$), and two halogen atoms (labeled by $Y$ and $Y^{\prime}$). These atoms are invariant under the following symmetry operations: (i) identity operation $E$, (ii) the glide reflection $\bar{M}_{xy}$ consisted of reflection about $z=0$ plane followed by $a/2$ translation along the $x$ axis and $b/2$ translation along the $y$ axis, where $a$ and $b$ is the lattice parameters of the crystal, (iii) the twofold screw rotation $\bar{C}_{2y}$ defined as $\pi/2$ rotation around $y=b/2$ line followed by $a/2$ translation along the $x$ axis, and (iv) the mirror reflection $M_{yz}$ around the $x=0$ plane. The optimized lattice parameters ($a$, $b$) for each Ga$XY$ ML compound are listed in Table 1. We find that due to the difference value between the $a$ and $b$ parameters, the crystal geometry of the Ga$XY$ ML is anisotropic, implying that these materials have different mechanical responses being subjected to uniaxial strain along the $x$- and $y$-direction similar to that observed on various group IV monochalcogenideAnshory and Absor (2020); Kong et al. (2018); Liu et al. (2019).

The atomic structure of the Ga$XY$ ML can be viewed as Ga$X(X^{\prime})$ ML surface functionalized by halogen $Y$ ($Y^{\prime}$) atoms bonded to the Ga atoms forming a sandwiched structure with $Y$-Ga$X(X^{\prime})$-$Y^{\prime}$ sequence [see Fig. 1(a)]. We then introduce a distortion vector, $\vec{r}_{0}$, defined as

where $\vec{r}_{Ga-X(X^{\prime})}$ and $\vec{r}_{Y(Y^{\prime})-Ga}$ are the vectors connected the Ga atom to chalcogen $X(X^{\prime})$ atom and the halogen $Y(Y^{\prime})$ atom to Ga atom, respectively, in the unit cell [see left side in Fig. 1(a)]. Here, the magnitude $|\vec{r}|_{Ga-X(X^{\prime})}$ and $|\vec{r}|_{Y(Y^{\prime})-Ga}$ represent the Ga-$X$($X^{\prime}$) and $Y$($Y^{\prime}$)-Ga bond lengths, respectively. Due to the $M_{yz}$ mirror symmetry operation along the $y-z$ plane, we obtain that $\vec{r}_{0}\cdot\hat{x}=0$, while the screw operation $\bar{C}_{2y}$ implies that $\vec{r}_{0}\cdot\hat{z}=0$. Accordingly, $\vec{r}_{0}$ should be parallel to the in-plane $y$ direction and induces intrinsic spontaneous polarization along the $y$ direction. Generally, the optimized structures of the Ga$XY$ ML compounds show that the $|\vec{r}|_{Ga-X(X^{\prime})}$ bond lengths are larger than the $|\vec{r}|_{Y(Y^{\prime})-Ga}$ bond lengths [see Table I]. However, the $|\vec{r}|_{Y(Y^{\prime})-Ga}$ bond lengths substantially increases for the compounds with the same chalcogen ($X$) atoms but have the heavier halogen ($Y$) atoms, thus decreasing the magnitude of the distortion vector, $|\vec{r}_{0}|$. Therefore, the decreased in magnitude of the in-plane electric polarization is expected, which is in fact confirmed by our BP calculation results shown in Table I. The existence of the in-plane ferroelectricity allows us to maintain the FRE in the Ga$XY$ ML compounds, which is expected to be observed due to the large SOC.

In the next section, we will show how the in-plane ferroelectricity plays an important role in the SOC and electronic properties of the Ga$XY$ ML compounds.

We will start our analysis by deriving the general SOC Hamiltonian in the 2D systems having in-plane ferroelectricity. The SOC Hamiltonian is further analyzed for the Ga$XY$ ML compounds within the framework of the $\vec{k}\cdot\vec{p}$ Hamiltonian model using the method of invariantWinkler et al. (2003). Finally, we discuss the important implication of the derived SOC Hamiltonian in terms of the spin splitting and spin textures involving to the in-plane ferroelectricity.

The SOC occurs in solid-state materials when an electron moving at velocity $\vec{v}$ through an electric field $\vec{E}$ experiences an effective magnetic field due to the relativistic transformation of electromagnetic fields. A general form of the SOC Hamiltonian $H_{SO}$ can be expressed as:

where $\vec{\sigma}=(\sigma_{x},\sigma_{y},\sigma_{z})$ are the Pauli matrices and $\vec{\Omega}$ is a wave-vector dependent spin-orbit field (SOF) that is simply written as

where $\alpha$ is the strength of the SOC that is proportional to the magnitude of the electric field, $|\vec{E}|$, $\hat{E}$ denotes the electric filed direction, and $\vec{k}$ is the wave vector representing the momentum electron. The $H_{SO}$ is invariant under time reversal symmetry operations, $T$, so that the following relation holds, $TH_{SO}T^{-1}=-\vec{\Omega}(-\vec{k})\cdot\vec{\sigma}=H_{SO}$. Accordingly, the SOF is a odd in wave vector $\vec{k}$, i.e. $\vec{\Omega}(-\vec{k})=-\vec{\Omega}(\vec{k})$, which also depends on the crystal symmetry of the system.

Lets us consider the general 2D systems having in-plane ferroelectricity, where we assumed that the spontaneous in-plane electric polarization being oriented along the in-plane $y$-direction. In this case, an effective electric field is induced, which is also oriented along the $y$-directions, $\vec{E}=E\hat{x}$. Due to the 2D nature of the systems, we have $\vec{k}=k_{x}\hat{x}+k_{y}\hat{y}$ for the wave vector $\vec{k}$, and by using the explicit form of the effective electric field $\vec{E}$, we find that the SOF $\vec{\Omega}$ in Eq. (4), can be expressed as

We can see that for the 2D systems having in-plane ferroelectricity, the SOF is enforced to be unidirectional in the out-of-plane direction. Inserting the Eq. (4) to the Eq. (3), we find that

The Eq. (6) clearly shows that the $H_{SO}$ is characterized only by one component of the wave vector $k_{x}$ and the out-of-plane spin vector $\sigma_{z}$, yielding a unidirectional out-of-plane Rashba effect.

The $H_{SO}$ in Eq. (6) is also obtained by considering the wave-vector symmetry group at the high symmetry points in the first-Brillouin zone. Here, we assumed that only linear terms with respect to the wave vector $\vec{k}$ contribute to the $H_{SO}$. For the case of the Ga$XY$ ML compounds, the wave-vector symmetry group of the $Pmn2_{1}$ space group at the high symmetry points such as $\Gamma$, $X$, and $Y$ points, belongs to $C_{2v}$ point groupAbsor and Ishii (2021), which has two mirror reflections about the $x-y$ plane ($M_{xy}$) and the $y-z$ plane ($M_{yz}$) as well as twofold rotation $C_{2y}$ around the $y$-axis. The transformation rules for the wave vector $\vec{k}$ and spin vector $\vec{\sigma}$ under the considered point-group symmetry operations are listed in Table II. By applying the methods of invariantWinkler et al. (2003), we list all the invariant term of the $H_{SO}$ in the form of product between the $\vec{k}$ and $\vec{\sigma}$ components (see the right column in Table II) and select those specific terms which are invariant under all symmetry operations as indicated by the underlined terms in the right column in Table II. We find that only $k_{x}\sigma_{z}$ term of the $H_{SO}$ is invariant under all symmetry operations of the $C_{2v}$, which is identical to the $H_{SO}$ shown in Eq. (6).

Next, we characterize low energy properties of the present system involving the $H_{SO}$ term of Eq. (6). The effective $\vec{k}\cdot\vec{p}$ Hamiltonian $H$ including the kinetic and $H_{SO}$ terms can be expressed as

Solving eigenvalue problem involving the Hamiltonian of Eq. (7) leads to the eigenstates

and

corresponding to the energy dispersion,

This dispersion indicates that a strongly anisotropic spin splitting occurs around the $\vec{k}=(0,0,0)$ point, i.e., the energy bands are lifted along $k_{x}$ direction but are degenerated along the $k_{y}$ direction [Fig. 2(a)]. Importantly, this dispersion is characterized by the shifting property, $E_{\downarrow}(\vec{k})=E_{\uparrow}(\vec{k}+\vec{Q})$, where the $\vec{Q}$ is the shifting wave vector given by

The Eqs. (10) and (11) implies that a constant-energy cut shows two Fermi loops whose centers are displaced from their original point by $\pm\vec{Q}$ as schematically shown in Fig. 2(b).

The spin texture, which is $\vec{k}$-dependent spin configuration, is determined from the expectation values of the spin operators, i.e., $\vec{S}=(\hbar/2)\langle\psi_{\vec{k}}|\vec{\sigma}|\psi_{\vec{k}}\rangle$, where $\psi_{\vec{k}}$ is the electron’s eigenstates. By using $\psi_{\vec{k}}$ given in Eqs. (8) and (9), we find that

This shows that the spin configuration in the $k$-space is locked being oriented in the out-of-plane directions as schematically shown in Fig. 2(b). Such a typical spin configuration forms a persistent spin textures (PST) similar to that observed for [110] Dresselhauss modelBernevig et al. (2006). Previously, it has been reported that the PST is known to host a long-lived helical spin mode known as a persistent spin helix (PSH) Bernevig et al. (2006); Altmann et al. (2014); Schliemann (2017), enabling long-range spin transport without dissipationBernevig et al. (2006); Altmann et al. (2014); Schliemann (2017); Kohda et al. (2012); Walser et al. (2012); Koralek et al. (2009), and hence very promising for an efficient spintronic devices.

The PSH arises when the SOF is unidirectional, preserving a unidirectional spin configuration in the $k$-space. When an electron moving in the real space is accompanied by the spin precession around the SOF, a spatially periodic mode of the spin polarization is generated. According to Eq. (5), the magnitude of the effective magnetic field can be expressed as $B=2|\vec{\Omega}|/\gamma\hbar$, where $\gamma$ is the gyromagnetic ratio. Therefore, the angular frequency of the precession motion, $\omega$, can be calculated using the relation, $\omega=-\gamma B=2\alpha k_{x}/\hbar$. The spin precession angle, $\theta$, around the $y$ axis at time $t$, is obtained by   $\theta=\omega t=2\alpha k_{x}t/\hbar$. At the same time, the traveling distance of the electron is given by $l=vt=\hbar k_{x}t/m^{*}$, where $v$ is the electron velocity. By eliminating $t$, we find that $\theta=2\alpha m^{*}l/\hbar$. When $\theta=2\pi$, we then obtain the wavelength of the PSH, $l_{PSH}$Bernevig et al. (2006),

Furthermore, in term of the shifting wave vector $\vec{Q}$ defined in Eq. (11), we can write the $l_{PSH}$ as

A schematic picture of the PSH mode enforced by the unidirectional SOF is displayed in Fig. 2(c), where a spatially periodic mode of the spin polarization with the wavelength $l_{PSH}$ is shown. Such spin-wave mode protects the spins of electrons from decoherence through suppressing the Dyakonov-Perel spin relaxation mechanismDyakonov and Perel (1972) and renders an extremely long spin lifetimeBernevig et al. (2006); Altmann et al. (2014); Schliemann (2017); Kohda et al. (2012); Walser et al. (2012); Koralek et al. (2009).

Finally, we study the correlation between spin textures and ferroelectricity. Here, an important property called reversible spin textures holds, i.e., the direction of the spin textures is locked and switchable by reversing the direction of the spontaneous electric polarization. Fig. 2(d) shows a schematic view of the spin textured ferroelectric in the Ga$XY$ ML  compounds showing fully reversible out-of-plane spin textures. It is shown that switching the direction of the in-plane ferroelectric polarization from $\vec{P}$ to $-\vec{P}$ leads to reversing the direction of the out-of-plane spin textures from $z$- to $-z$-direction.

From the symmetry point of view, switching the electric polarization direction $\vec{P}$ is equivalent to the space inversion symmetry operation $I$, which changes the wave vector from $\vec{k}$ to $-\vec{k}$, but preserves the spin vector $\vec{S}$Kim et al. (2014). Now, suppose that $|\psi_{\vec{p}}(\vec{k})\rangle$ is the Bloch wave function of the crystal with electric polarization $\vec{P}$. The inversion symmetry operation $I$ on the Bloch wave function hold the following relation, $I|\psi_{\vec{P}}(\vec{k})\rangle=|\psi_{-\vec{P}}(-\vec{k})\rangle$. Applying the time-reversal symmetry $T$ brings $-\vec{k}$ back to $\vec{k}$ but flip the spin vector $\vec{S}$, thus $TI|\psi_{\vec{P}}(\vec{k})\rangle=|\psi_{-\vec{P}}(\vec{k})\rangle$. The expectation values of spin operator $\expectationvalue{S}$ can be further expressed in term of $\vec{P}$ and $\vec{k}$ vectors as

which clearly shows that the spin directions is fully reversed by switching the direction of the electric polarization $\vec{P}$.

In the next section, we implement these general description of the spin-orbit coupled ferroelectric to discuss our results from the first-principles DFT calculations on various Ga$XY$ ML compounds.

Figure 3 shows the electronic band structure of the Ga$XY$ ML compounds calculated along the selected $\vec{k}$ paths in the FBZ corresponding to the density of states (DOS) projected to the atomic orbitals. It is found that the Ga$XY$ ML compounds are semiconductors with direct or indirect band gaps depending on the chalcogen ($X$) atoms. In the case of the GaSe$Y$ MLs, the electronic band structure shows a direct bandgap where the valence band maximum (VBM) and conduction band minimum (CBM) is located at the $\Gamma$ point [Figs. 3(a)-3(c)]. The VBM at the $\Gamma$ point retains for the case of the GaTe$Y$ MLs but the CBM shifts to the $k$ point along the $\Gamma-Y$ line, resulting in an indirect bandgap [Figs. 3(d)-3(f)]. We find that the band gap significantly decreases for the compounds with the same chalcogen $X$ atoms but has the larger $Z$ number of the halogen ($Y$) atoms. For an instant, the calculated bandgap for the GaTeCl ML is 2.17 eV under GGA level, which is much larger than that for the GaTeI ML (1.10 eV). Our calculated DOS projected to the atomic orbitals confirmed that the VBM is mostly dominated by the contribution of the chalcogen $X$-$p$ orbital with a small admixture of the Ga-$p$ and halogen $Y$-$p$ orbitals, while the CBM is mainly originated from the Ga-$s$ orbital with a small contribution of Ga-$p$, chalcogen $X$-$p$ and halogen $Y$-$p$ orbitals [Figs. 3(a)-(f)].

Introducing the SOC, however, strongly modifies the electronic band structures of the Ga$XY$ ML compounds. Here, we observed a significant band splitting produced by the SOC due to the lack of the inversion symmetry, which is mainly visible at the $k$ bands along the $\Gamma-X-M$ symmetry lines [Figs. 3(a)-(f)]. However, along the $\Gamma-Y$ line in which the wave vector $\vec{k}$ is parallel to the effective electric field associated with the in-plane ferroelectric polarization, the bands are double degenerated. Since the electronic states near the Fermi level are important for transport carriers, we then focused our attention on the bands near the VBM. Fig. 4(a) shows the calculated band structure along the $Y-\Gamma-X$ line around the VBM for GaTeCl ML as a representative example of the Ga$XY$ ML compounds. At the $\Gamma$ point, the electronic states at the $\Gamma$ point are double degenerated due to time reversibility. This doublet splits into singlet when considering the bands $\vec{k}$ along the $\Gamma-X$ line. However, the doublet remains for the $\vec{k}$ along the $\Gamma-Y$ line, which is protected by the $\bar{C}_{2y}$ screw rotation and the $\bar{M}_{xy}$ glide mirror reflection. Accordingly, a strongly anisotropic splitting is clearly observed around the $\Gamma$ point as highlighted by the red line in Fig. 4(a), which is in good agreement with the energy dispersion shown in Eq. (10) as well as Fig. 2(a).

We noted here that the remaining band degeneracy at the wave vector $\vec{k}$ along the $\Gamma-Y$ line can be explained in term of the symmetry analysis. Since the wave vector $\vec{k}$ at the $\Gamma-Y$ line is invariant under $\bar{C}_{2y}$ and $\bar{M}_{xy}$ symmetry operations, the folowing relation holds, $\bar{M}_{xy}\bar{C}_{2y}=-e^{-ik_{x}}\bar{C}_{2y}\bar{M}_{xy}$, where the minus sign comes from the fact that both $\bar{C}_{2y}$ and $\bar{M}_{xy}$ operators are anti-commutative, $\left\{\bar{C}_{2y},\bar{M}_{xy}\right\}=0$ due to the anti-commutation between $\sigma_{y}$ and $\sigma_{z}$ spin rotation operators, $\left\{\sigma_{y},\sigma_{z}\right\}=0$. Supposed that $\left|\psi_{g}\right\rangle$ is an eigenvector of  $\bar{M}_{xy}$ operator with the eigenvalue of $g$, we obtain that $\bar{M}_{xy}(\bar{C}_{2y}\left|\psi_{g}\right\rangle)=-g(\bar{C}_{2y}\left|\psi_{g}\right\rangle)$. This evident shows that both $\left|\psi_{g}\right\rangle$ and $\bar{C}_{2y}\left|\psi_{g}\right\rangle)$ states are distinct states degenerated at the same energy, thus ensuring the double degeneracy of the states at the wave vector $\vec{k}$ along the $\Gamma-Y$ line. To further clarify the observed anisotropic splitting around the $\Gamma$ point, we show in Fig. 4(b) momentum-resolved map of the spin-splitting energy calculated along the entire of the FBZ. Consistent with the band structures, we identify the non-zero spin-splitting energy except for the bands $\vec{k}$ along the $\Gamma-Y$ line. Here, the largest splitting is observed at the $\vec{k}$ closed to the $\Gamma$ point at along the $\Gamma-X$ line, where the splitting energy up to 0.25 eV is achieved. Such value is comparable with the splitting energy observed on various 2D transition metal dichalcogenides $MX_{2}$ ($M$= Mo, W; $X$ = S, Se, Te) MLs [0.15 eV - 0.46 eV] Zhu et al. (2011); Affandi and Ulil Absor (2019); Absor et al. (2017); Yao et al. (2017); Absor et al. (2016). The large splitting energy observed in the present system is certainly sufficient to ensure proper function of spintronic devices operating at room temperatureYaji et al. (2010).

The nature of the anisotropic splitting around the $\Gamma$ point at the VBM is further analyzed by identifying the spin textures of the spin-split bands. As shown in Fig. 4(c), it is found that a uniform pattern of the spin textures is observed around the $\Gamma$ point, which is mostly characterized by fully out-of-plane spin components $S_{z}$ rather than the in-plane spin components ($S_{x},S_{y}$). These spin textures are switched from $S_{z}$ to $-S_{z}$ when crossing at $k_{x}$=0 along the $\Gamma-Y$ line. Although we identified large in-plane spin components ($S_{x},S_{y}$) in the $\Gamma-Y$ line, the net in-plane spin polarization vanishes, which is due to the equal population of the opposite in-plane spin polarization between the outer and inner branches of the spin split bands [see black arrows in Fig. 4(c)].  Such a pattern of the spin textures, which is similar to that observed on several 2D ferroelectric materials such as WO₂Cl₂ Ai et al. (2019) and various group IV monochalcogenide MLs Absor and Ishii (2019a); Lee et al. (2020); Anshory and Absor (2020); Absor et al. (2021), is strongly different from the in-plane Rashba-like spin textures reported on the widely studied 2D materialsAbsor et al. (2018); Affandi and Ulil Absor (2019); Absor et al. (2017); Yao et al. (2017). Moreover, the fully out-of-plane spin texture becomes clearly visible when measured at the constant energy cut of 1 meV below the degenerated states at the VBM around the $\Gamma$ point [Fig. 4(d)].  Here, two circular loops of the Fermi lines with the opposite $S_{z}$ spin components are observed, which are shifted along the $\Gamma-X$ ($k_{x}$) direction. The observed spin textures, as well as Fermi lines, are all consistent well with our $\vec{k}\cdot\vec{p}$ Hamiltonian model presented in Eq. (7) and the schematic pictures shown in Figs. 2(a)-(b). Remarkably, the observed unidirectional out-of-plane spin textures in the present system lead to the PSTBernevig et al. (2006); Schliemann (2017), which can host a long-lived helical spin-wave mode through the PSH mechanism Bernevig et al. (2006); Altmann et al. (2014); Schliemann (2017); Kohda et al. (2012); Walser et al. (2012); Koralek et al. (2009).

The observed spin splitting and spin textures can be quantified by the strength of the SOC, $\alpha$, which is obtained from the unidirectional out-of-plane Rashba model given by Eq. (7). Here, we can rewrite the energy dispersion of Eq. (10) in the following form:

where $E_{R}$ and $k_{0}$ are the shifting energy and the wave vector evaluated from the spin-split bands along the $\Gamma-X$ ($k_{x}$) line as illustrated in Fig. 4(a). Accordingly, the following relation holds,

Both $E_{r}$ and $k_{0}$ are important parameters to stabilize spin precession and achieve a phase offset for different spin channels in the spin-field effect transistor deviceDatta and Das (1990). In table III, we summarize the calculated result of the SOC strength $\alpha$ in Table III, and compare this result with a few selected PST systems previously reported on several 2D materials. It is found that the calculated value of $\alpha$ for the GaTeCl ML is 2.65 eVÅ, which is the largest among the Ga$XY$ ML compounds. This value is comparable with that observed on the PST systems reported for several 2D group IV monochalcogenide including Ge$XY$ ($X,Y$ = S, Se, Te) MLs (3.10 - 3.93 eVÅ) Absor et al. (2021), layered SnTe (1.28 - 2.85 eVÅ) Lee et al. (2020). However, the calculated value of $\alpha$ is much larger than that observed on the PST systems found in other class of 2D materials such as WO₂Cl₂ ML (0.90 eVÅ) Ai et al. (2019) and transition metal dichalcogenide $MX_{2}$ MLs with line defect such as PtSe₂ (0.20 - 1.14 eVÅ) Absor et al. (2020) and (Mo,W)$X_{2}$ ($X$=S, Se) (0.14 - 0.26 eVÅ) Li et al. (2019).

The emergence of the PST with large SOC strength $\alpha$ predicted in the present system indicates that the formation of the PSH mode with a substantially small wavelength $l_{PSH}$  of the spin polarization is achieved. Here, the wavelength $l_{PSH}$ can be estimated by using Eq. (13) evaluated from the band dispersion along the $\Gamma-X$ line in the VBM [see the insert of Fig. 4(a)]. The resulting wavelength $l_{PSH}$ for all members of Ga$XY$ ML compounds are shown in Table III. In particular, we find a very small wavelength $l_{PSH}$ of the PSH mode for the GaTeCl ML (1.20 nm), which is the smallest over of all known 2D materials so far [see Table III]. Importantly, the small wavelength of the PSH mode observed in the present system is typically on the scale of the lithographic dimension used in the recent semiconductor industryFiori et al. (2014), which is possible to access the features down to the nanometers scale with sub-nanosecond time resolution by using near-field scanning Kerr microscopy. Thus, we concluded that that the present system is promising for miniaturization spintronics devices.

Before summarizing, we highlighted the interplay between the in-plane ferroelectricity, spin splitting, and the spin textures in the Ga$XY$ ML compounds. Fig. 5(a) displayed the in-plane electric polarization as a function of the ferroelectric distortion, $\tau$. Here, $\tau$ is defined as the magnitude of the distortion vector $|\vec{r}|$ of the systems defined by Eq. (1), which is normalized by the magnitude of the distortion vector of the optimized ferroelectric phase, $|\vec{r}_{0}|$. Therefore, $\tau=0$ represents the paraelectric phase, while $\tau=1$ shows the optimized ferroelectric phase as shown by the insert of Fig. 5(a). We can see that it is possible to manipulate the in-plane electric polarization $\vec{P}$ by distorting the atomic position [see Fig. 1(a)]. The dependence of the in-plane polarization on the ferroelectric distortion $\tau$ sensitively affects the spin-split bands at the VBM around the $\Gamma$ point as shown in Fig. 5(b). It is found that the splitting energy and the position of the VBM around the $\Gamma$ point strongly depend on the ferroelectric distortion, i.e., a decrease in $\tau$ substantially reduces the spin splitting energy while the position of the VBVM shifts up to be higher in energy around the $\Gamma$ point. Accordingly, the significant change of the SOC strength $\alpha$ is achieved, in which a linear trend of $\alpha$ as a function of $\tau$ is observed as shown in Fig. 5(c). Importantly, our results also show that the SOC strength $\alpha$ changes sign when the direction of the in-plane ferroelectric polarization $\vec{P}$ is switched, resulting in a full reversal of the out-of-plane spin textures shown in Figs. 5(d)-(e). Such reversible spin textures are agreed well with our symmetry analysis given by Eq. (15), putting forward Ga$XY$ ML compounds as a candidate of the FER class of 2D materials exhibiting the PST, which is useful for efficient and non-volatile spintronic devices.