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

First, we analyze the structural symmetry and stability of the Ga$XY$ ML compounds. As shown in Figs. 1(a)-(b) that the crystal structure of the Ga$XY$ ML compounds is noncentrosymmetric having black-phosphorus type structure, where its symmetry is isomorphic to the nonsymmorphic $Pmn2_{1}$ space groupWu et al. (2019); Kniep et al. (1983); Zhou et al. (2018); Zhang and Liu (2018). The first Brillouin zone (FBZ) corresponding to this structure is shown in Fig. 1(c). The $Pmn2_{1}$ symmetry in the Ga$XY$ ML is generated by the following symmetry operations [see Fig. 1(a)-(b)]: (i) identity operation $E$, (ii) the glide reflection $\bar{M}_{xy}$ which consists of reflection about $z=0$ plane followed by $a/2$ translation along the $x$ axis and $b/2$ translation along the $y$ axis:

(iii) the twofold screw rotation $\bar{C}_{2y}$ which consists of $\pi/2$ rotation around $y=b/2$ line followed by $a/2$ translation along the $x$ axis:

and (iv) the mirror reflection $M_{yz}$, which is reflection around the $x=0$ plane:

Here, $a$ and $b$ is the lattice parameters of the crystal as indicated in Fig. 1(a).

It is important to note here that there is an intersection site in the Ga$XY$ ML located around the $x=0$ plane, where only the $M_{yz}$ mirror symmetry operation preserves [Fig. 1(a)]. This intersection site, which belongs to $C_{s}$ point group, divides the real space into two sectors, namely $A$ and $B$ sectors. The sector $A$ is consisted of the Ga₁, $X_{1}$, and $Y_{1}$ atoms, while the sector $B$ is filled by Ga₂, $X_{2}$, and $Y_{2}$ atoms [Fig. 1(b)]. Although the global atomic site in the Ga$XY$ ML belongs to $C_{2v}$ point group symmetry, which is non-pseudo polar, the local atomic site in each sector reduces to $C_{s}$ point group, which is pseudo polar.

The nonsymmorphic $Pmn2_{1}$ space group symmetry in the Ga$XY$ ML plays an important role for generating the in-plane ferroelectricityZhang and Liu (2018); Zhou et al. (2018). Here, the orientation of the ferroelectric polarization is enforced by the $C_{2v}$ point group related to the $Pmn2_{1}$ space group, similar to that observed on various group IV monochalcogenide monolayersFei et al. (2016); Kaloni et al. (2019). Since the $C_{2v}$ point group contains mirror $xz$ and $yz$ planes, this implies that the net ferroelectric polarization vanishes along the $x$- and $z$-directions, while it is substantial along the $y$-direction. This in-plane ferroelectric polarization is originated from the polar displacements between Ga and $X$ atoms along the $y$-direction. In addition, the existence of the Ga-$Y$ bond in the Ga$XY$ ML contributes to extra dipole moments in the $y$-direction, leading to the large magnitude of the in-plane ferroelectric polarization.

In this work, we will focus on the GaTeCl ML as a representative example of the Ga$XY$ ML family since the layered GaTeCl bulk material has been experimentally synthesizedKniep et al. (1983). Here, the calculated-optimized lattice parameters, $a$ and $b$, are 4.17 Å and 5.93 Å  respectively, which is in a good agreement with previous calculationZhang and Liu (2018); Zhou et al. (2018). Our Berry phase calculation found that the calculated in-plane ferroelectric polarization is 597 pC/m, which is consistent well with previous calculation Zhang and Liu (2018); Zhou et al. (2018), but is larger than that observed on various 2D in-plane ferroelectric materials Fei et al. (2016); Ai et al. (2019). To confirm the structural stability of the GaTeCl ML, we show in Fig. 1(d) the calculated phonon dispersion bands. We can see clearly that there is no imaginary frequencies found in the phonon dispersion bands, indicating that the optimized GaTeCl ML is a dynamically stable.

Figure 2 shows the electronic band structure of the GaTeCl ML along the selected $\vec{k}$ paths in the FBZ corresponding to the density of states (DOS) projected to the atomic orbitals. One finds that the material is an indirect band-gap semiconductor with the valence band maximum (VBM) is located at the $\Gamma$ point and the conduction band minimum (CBM) is located at the $\vec{k}$ point along the $\Gamma-Y$ line [Fig. 2(a)]. Without including the SOC, the calculated band-gap is 2.17 eV under GGA level, which is in a good agreement with previous calculationsZhang and Liu (2018); Zhou et al. (2018). Since there is no magnetic ordering found in the GaTeCl ML, so the time reversal symmetry (TRS) is also preserved. Our calculated DOS projected to the atomic orbitals confirmed that the VBM is mostly dominated by contribution of the Te-$p$ orbital with a small admixture of the Ga-$p$ and Cl-$p$ orbitals, while the CBM is mainly originated from the Ga-$s$ orbital with small contribution of Ga-$p$, Te-$p$ and Cl-$p$ orbitals [Figs. 2(b)-(e)].

Turning the SOC, however, slightly reduces the band gap to about 2.05 eV and strongly modifies the electronic band structures of the GaTeCl ML. In comparison with the band structures calculated without SOC [Fig. 2(a)], one can see that a sizable band splitting produced by the SOC is observed due to the inversion symmetry breaking, except for the time-reversal-invariant $\vec{k}$ points. This splitting is particularly visible along the the $\Gamma-X$ and $X-M$ symmetry lines located in the proximity of the CBM. However, due to the protection of the glide mirror symmetry $\bar{M}_{xy}$, the band crossing appears along both the $\Gamma-X$ and $X-M$ symmetry lines, forming a hourglass-shaped band dispersions similar to that observed on bulk BiInO₃ Tao and Tsymbal (2018) and monolayer GaTeIWu et al. (2019). Along the $\Gamma-X$ line, we find that the split bands are fully spin-polarized oriented in the out-of-plane ($z$) direction, while all components of the spin polarization vanish in the split bands along the $X-M$ line [Fig. 3]. The vanishing spin polarization in the split bands along the $X-M$ line indicates that the BSVSP effect is achieved, which is similar to that observed on the SnTe MLLiu et al. (2019).

The fully out-of-plane spin-polarized bands along the $\Gamma-X$ line can be explained by using unidirectional Rashba effect induced by the in-plane ferroelectricityAbsor and Ishii (2019); Lee et al. (2020); Ai et al. (2019). The existence of the in-plane ferroelectric polarization along the $y$ direction naturally develops an in-plane electric field, $\vec{E}=E_{y}\hat{y}$, and produces a unidirectional Rashba SOC described by the following HamiltonianAbsor and Ishii (2019); Lee et al. (2020); Ai et al. (2019), $H_{\Gamma-X}=\alpha_{R}k_{x}\sigma_{z}$, where $\alpha_{R}$ is the Rashba parameter that is proportional to the magnitude of the in-plane electric field $E_{y}$, and $\sigma_{z}$ is the $z$ component of the Pauli matrices. We noted here that the similar form of $H_{\Gamma-X}$ can also be deduced by considering the little group of the wave vector $\vec{k}$ at the $\Gamma$ point belonging to the $C_{2v}$ point group similar to that reported on various 2D ferroelectric materials such as WO₂Cl₂Ai et al. (2019), SnTe Absor and Ishii (2019); Lee et al. (2020), and SnSeAnshory and Absor (2020) MLs. Therefore, the spin orientation along the $\Gamma-X$ ($k_{x}$) line is completely locked in the out-of-plane direction, forming a fully unidirectional out-of-plane spin polarization [Fig. 3]. This spin polarization is expected to maintain the persistent spin helix state Bernevig et al. (2006); Schliemann (2017) along the $\Gamma-X$ line, which suppresses the DP mechanism of the spin relaxationDyakonov and Perel (1972) and induces an extremely long spin lifetimeAltmann et al. (2014).

The vanishing spin polarization as a manifestation of the BSVSP effect in the split bands along the $X-M$ symmetry line, on the other hand, cannot be explained in term of the unidirectional Rashba effect. Recently, based on the group theory analysis, the emergence of the BSVSP effect has been proposed for both symmorphic and non-symmorphic systems satisfying the following two-simultaneous conditionsLiu et al. (2019) : (i) the little space group associated with the wave vector $\vec{k}$ possess 1D double-value irreducible representation (IR); (ii) the little point group associated with $\vec{k}$ should be a non-pseudo polar point group (detailed analysis based on the point group theory can be found in the supplementary of Ref. 2). In our system, the crystal symmetry is isomorphic to the $Pmn2_{1}$ space group whose corresponds to the point group $C_{2v}$Wu et al. (2019). Along the $X-M$ line, the little space group of $\vec{k}$ belongs to $C_{2v}$ point group. Therefore, we find that there exists 1D double-value IR of the little space group along the $X-M$ line. At the same time, the little point group of $\vec{k}$ along the $X-M$ line also belongs to $C_{2v}$ point group, which is non-pseudo polar. All these facts confirmed that the split bands along the $X-M$ line sustains the BSVSP effect [see Fig. 3].

To clarify the origin of the BSVSP effect along the $X-M$ line, we evaluate the atomic contribution on the spin polarization in the spin-split bands. Here, we focused on the spin-split bands along the $X-M$ line at the CBM due to the large spin splitting [Fig. 2(a)]. Since there are six atoms in the unit cell of the GaTeCl ML, i.e., the Ga₁, Te₁, and Te₁ atoms in the sector $A$; and the Ga₂, Te₂, and Cl₂ atoms in the sector $B$ [Fig. 1(b)], we then decompose the spin polarization of the selected bands into these atoms. As shown in Fig. 4, we find that the spin polarizations are dominated by the $S_{x}$ component of spin, which is mainly originated from the contribution of the Ga atoms. These spin polarizations are opposite in the $x$ direction when projected into the Ga atom at the different sectors, indicating that the spin polarizations are locally emerged. Such local and opposite spin polarizations found in the bands along the $X-M$ line are analogous to the hidden spin-polarization effectZhang et al. (2014a); Yuan et al. (2019) observed on the centrosymmetric layer compoundsYuan et al. (2019); Zhang et al. (2014a); Sławińska et al. (2016); Huang et al. (2020); Yao et al. (2017); Liu et al. (2015). Since the orientations of the local spin polarization are opposite in the $x$ direction, they should cancel each other so that the net spin polarizations is zero, thus maintaining the BSVSP effect in agreement with total spin polarization shown in Fig. 3.

We noted here that the vanishing spin polarization can be understood by considering the site symmetry of the atoms in each sector in the unit cell of the GaTeCl ML. Due to the existence of the $M_{yz}$ mirror symmetry operation [Fig. 1(a)-(b)], only the $x$-component of spin polarizations preserves, while the $y$- and $z$-components of the spin polarizations are zero. However, these spin polarizations appear locally on each sector, which is suppressed by the pesudo polar $C_{s}$ point group of the intersection site symmetry. Since the atomic sites in each sector are also characterized  by the pseudo polar $C_{s}$ point group symmetry, the $x$-component of the spin polarization also maintains locally on each atom. Considering the fact that the spin-split bands along the $X-M$ line at the CBM are mainly originated from the Ga-$s$ orbital [Fig. 2(b)-(d)], their local spin polarizations are dominated by the contribution of the Ga atoms rather than Te and Cl atoms. However, these local spin polarizations have opposite signs when projected to the Ga atom in the different sectors [Fig. 4], thus resulting in vanishing of the net spin polarization as shown in Fig. 3.

To further demonstrate the microscopic origin of the BSVSP effect along the $X-M$ line at the CBM, we construct $\vec{k}\cdot\vec{p}$ effective Hamiltonian describing the band structure around the high symmetry $X$ point. Here, the $\vec{k}\cdot\vec{p}$ Hamiltonian can be derived based on the theory of invariantLuttinger (1956); Winkler et al. (2003). For the particular high symmetry point in the first Brillouin zone, the little group of the wave vector $\vec{k}$ is characterized by a point group $G$, where the matrix representation for the chosen basis functions is given by $\left\{D(g)\mathrel{\mathop{\ordinarycolon}}g\in G\right\}$, where $g$ is the symmetry operations in the point group. The derived Hamiltonian should satisfy the following invariant conditionWinkler et al. (2003):

At the $X$ point, the little group of the wave vector $\vec{k}$ belongs to the $C_{2v}$ point group. Therefore, the wave vector $\vec{k}$ and spin vector $\vec{\sigma}$ can be transformed according to the symmetry operations in this point group. Taking into account the spin and sub lattice degree of freedom in the $X$ point, the corresponding transformations are given in Table I. Collecting all terms which are invariant under these symmetry operations, we obtain the following effective Hamiltonian:

where $E_{0}=\frac{\hbar^{2}k^{2}_{x}}{2m_{x}}+\frac{\hbar^{2}k^{2}_{y}}{2m_{y}}$ is the free electron contribution with $m_{x,y}$ being the effective mass along the $k_{x}$ and $k_{y}$ directions. Here, $\lambda$, $\alpha$, $\beta$, $\gamma$, and $\delta$ are independent parameters, $\sigma_{0}$ and $\tau_{0}$ are the $2\times 2$ identity matrices. Along the $X-M$ line, we have $k_{x}=0$, so the Hamiltonian of Eq. (6) can be written as

where $E_{0_{y}}=\frac{\hbar^{2}k^{2}_{y}}{2m_{y}}$. At the $X$ point, the $\lambda\tau_{z}\sigma_{z}$ term in Eq. (7) splits the states into two doublets ($\Psi_{1}$,$\Psi_{2}$) separated by $\Delta=2\lambda$. Solving eigenvalue problem involving the Hamiltonian of Eq. (7) leads to the solution for each doublets as follows:

By fitting the band dispersion of Eq. (8) to the DFT energy band along the $X-M$ in the CBM, we obtain the following parameters: $\lambda=0.09$ eV and $\alpha=2.27$ eVÅ. Importantly, the calculated $\alpha$, which represents the spin-orbit strength parameter, is much larger than that observed on various 2D materialsZhuang et al. (2015); Popović et al. (2015); Absor et al. (2018); Affandi and Ulil Absor (2019); Absor et al. (2017), which is sufficient to support the room temperature spintronics functionality.

The BSVSP effect can be further confirmed by calculating the spin polarization projected to the atoms in each sector in the unit cell. As previously mentioned that there are two local sectors in the unit cell of the GaTeCl ML, where the intersection site is characterized by only the $M_{yz}$ mirror symmetry operation [Fig. 1(a)]. Therefore, we define the projected wave function onto each sector as $\Psi^{A(B)}(\vec{k})=P^{A(B)}\Psi(\vec{k})$, where $P^{A(B)}$ is the projection operator onto $A(B)$ sectors and $\Psi(\vec{k})$ is the global wave function given in Eq. (8). The expectation value of spin operator projected to the sectors $A(B)$ can be written as

By using the fact that $\Psi(\vec{k})$ is related to the $\Psi(-\vec{k})$ by the mirror symmetry $M_{yz}$ and time reversal symmetry $T$ operations, we obtain that

We can clearly see that the expectation value of the spin operator shows an opposite sign between $A$ and $B$ sectors, indicating that the local spin polarizations have opposite orientation.

Now, we focused on the expectation value of the spin operator projected to the Ga atom since the spin-split bands along the $X-M$ line is mainly dominated by the Ga-$s$ orbitals [Fig. 2(b)]. By evaluating Eq. (10) together with the wave functions given in Eq. (8), we find that the expectation value of the spin operator projected to the Ga atoms can be written as

This shows that both Ga₁ and Ga₂ atoms contributes to the local spin polarization having opposite direction along the $x$ direction, which is consistent well with our DFT results shown in Fig 4. Therefore, the net spin polarization becomes zero, giving rise to the BSVSP effect that is in agreement with our DFT results presented in Fig. 3.

We emphasized here that the observed BSVSP effect in the present system is located in the spin-split bands along the $X-M$ line, which is slightly higher in energy (0.15 eV) than the CBM at the $k$ point along the $\Gamma-Y$ line [Fig. 5(a)]. However, one has to note that the CBM can be effectively tuned by a small amount of strain [Fig. 5(b)-(c)], which is realized by choosing appropriate substrates. For example, under compressive strain of -8% along the $x$ direction (zigzag), the CBM shifts significantly (with shifting energy of 0.5 eV) being located at the $k$ point along the $X-M$ line [see Fig. 5(c)] without losing the BSVSP effect [Fig. 5(d)]. Therefore, it is possible to access the BSVSP effect experimentally by introducing electron doping. Recently, the electron doping techniques in various 2D materials are under rapid development realized by using ion liquid gating technique Mak et al. (2013); Zhang et al. (2014b), thus application of the electron doping in GaTeCl ML is also plausible. Since the BSVSP effect in the strained GaTeCl ML is observed in the proximity of the CBM, introducing n-type electron doping can move the interesting bands close to the Fermi level, which can be further resolved by using spin-polarized angle-resolved photo-electron spectroscopy.

Finally, we highlight the main difference of the BSVSP effect found in the present system with that observed on SnTe ML as previously reported by Liu et. al.Liu et al. (2019). In the SnTe ML, the BSVSP effect is predicted to appear on the spin-split bands located at valley near the $X$ point in both the CBM and VBM Liu et al. (2019). However, due to the orthorhombic structure, there is another valley, namely $Y$ valley, characterized the electronic band structures of the SnTe ML Absor and Ishii (2019); Kim et al. (2019). Since the energy band of both the $X$ and $Y$ valleys are very close, the observed BSVSP effect near the $X$ valley may be interfered by the spin-polarized states in the spin-split bands at $Y$ valley. Therefore, it is difficult to exclusively detected the BSVSP effect without mixing with the spin-polarized states. In contrast, the observed BSVSP effect in the present system is located at the non-valley bands in the proximity of the CBM where the mixing with spin-polarized states can be minimized, for example, by introducing the small-compressive uniaxial strains [Fig.5(b)-(c)]. Moreover, compared with the SnTe ML, the observed BSVSP effect in the present system exhibits the larger spin splitting, which is rationalized by the fact that the spin-orbit strength parameter predicted in the present system (2.27 eVÅ) is much larger than that observed in the SnTe ML (1.2 eVÅ)Absor and Ishii (2019). Therefore, the present system provides an advantage to higher possibility for realization spintronic devices operating at room temperature.

Although the observed BSVSP effect exhibits the large bands splitting along the $X-M$ line, which is beneficial for spintronics, the vanishing spin polarization in these bands may induce the undesired effect of losing the spin information. Therefore, inducing the spin polarization is the important task for realization spintronic devices. Since the BSVSP effect along the $X-M$ line is protected by the non-pseudo-polar $C_{2v}$ point group, reducing this point group symmetry may exhibits the non-zero spin polarization. For this propose, we apply an external out-of-plane electric field $\vec{E}=E_{z}\hat{z}$ perpendicular to the GaTeCl ML thin film, which can be realized by introducing a gate voltage. Introducing the out-of-plane external electric field is expected to break both the glide mirror plane $\bar{M}_{xy}$ and two-fold screw rotation axis $\bar{C}_{2y}$ in the crystal of the GaTeCl ML, implying that the point group symmetry reduces to the pseudo-polar $C_{s}$ point group. Therefore, the non-zero spin-polarization is allowed in the bands along the $X-M$ line.

Fig. 6(a) shows the calculated spin polarization projected to the bands along the $X-M$ line under the influenced of 1.0 V/Å out-of-plane electric field. By comparing the bands with and without an external electric field [see Fig. 3 and Fig. 6(a)], we find that the band crossing at the $\vec{k}$ along the $X-M$ line breaks due to the external electric field, so that the Hourglass band dispersion splits into two pair of the split bands. In contrast to the system without the external electric field, the split bands along the $X-M$ line exhibit significant spin polarization, which is dominated by the $S_{x}$ component of spin [Fig. 6(a)]. Moreover, by evaluating the spin polarization at certain $\vec{k}$ along the $X-M$ line, one can observe that the magnitude of the spin polarization enhances linearly by increasing the magnitude of the external electric field [Fig. 6(b)], indicating that the spin polarization can be effectively tuned by an external electric field. Interestingly, we find that the orientation of the spin polarizations can be fully reversed by switching the direction of the external electric field. These tunable and reversible spin polarization by the external electric field enable the controlled of the spin-dependent properties, which is the hint for realization spintronic devices.

The large spin polarizations in the split bands along the $X-M$ line induced by the out-of-plane external electric field can further be understood by considering the $\vec{k}\cdot\vec{p}$ Hamiltonian around the $X$ point. By including the out-of-plane electric field contribution, the total $\vec{k}\cdot\vec{p}$ Hamiltonian can rewritten as

where $H_{X}$ is given in Eq. (6) and $\alpha_{E_{z}}=E_{z}\left\langle\phi(\vec{k},\vec{r})|z|\phi(\vec{k},\vec{r})\right\rangle$. Here, $E_{z}$ is the magnitude of the out-of-plane external electric field. Along the $X-M$ line, the Hamiltonian of Eq. (12) can be simplified to

This Hamiltonian leads to the solutions:

For the case of the external electric field $E_{z}=1.0$ V/Å  the parameters $\lambda$, $\alpha$, and $\alpha_{E_{z}}$ can be obtained by fitting the energy dispersion of Eq. (14) to the DFT energy band of Fig. 6(a), and we find that $\lambda=0.11$ eV, $\alpha=2.12$ eVÅ  and $\alpha_{E_{z}}=0.03$ eV.

Furthermore, by using the wave function given on Eq. (14), the calculated expectation values of the spin operators can be expressed as

It is clearly seen from Eq. (15) that the electric field induces the significant spin polarization along the $x$ direction, which is consistent well with our DFT results shown in Fig. 6(a). Moreover, switching the direction of the electric field from $E_{z}$ to $-E_{z}$ revers the sign of $\alpha_{E_{z}}$ to $-\alpha_{E_{z}}$, leading to reversing the spin polarization in the $x$ direction that is also in agreement with our DFT results shown in Fig. 6(b).

Now, we discuss the possible applications of the electric field-driven spin polarization in GaTeCl ML. Here, we propose a possibility to observe a Hall effect similar to the valley Hall effect previously reported on the 2D transition metal dichalcogenidesMak et al. (2014). As mentioned previously that introducing an external electric field in the GaTeCl ML produces the large spin polarization in the split bands along the $X-M$ line at the CBM [Fig. 6(a)-(b)]. Accordingly, such spin polarization is expected to occur in the two states located at the $\vec{k}$ and $-\vec{k}$ near the CBM. This implies that the Berry curvatures should be observed with opposite sign. By using polarized optical excitation, it is possible to create imbalance population of the electron in these two states, and hence a charge Hall current can be detected.

Finally, we discuss another possible application of the GaTeCl ML in term of the reversible spin polarization effect induced by switching the external electric field. Such reversible spin polarization may be implemented in the magnetic tunnel junctionsTsymbal et al. (2003), where at the interface of a magnetic tunnel junction, the Rashba SOC induces a tunneling spin Hall effect and tunneling anomalous Hall effect (AHE)Vedyayev et al. (2013). Since the magnitude of the tunneling AHE is linearly depends on the SOC parameter, the tunneling AHE effect is more experimentally accessible for the systems having larger the SOC parameter. In our system, we found that large the spin-orbit parameter ($\alpha=2.12$ eVÅ) is observed when the electric field $E_{z}=1.0$ V/Å  indicating that this material is a favorable candidate for detecting the AHE effect experimentally. Since, in our system, the spin polarization can be reversed by switching the direction of the electric field, it is expected that the reversible AHE effect by the electric field is also achieved. Therefore, we conclude that the present system is promising candidate for spintronic applications.