Intertwined Ferroelectricity and Topological State in Two-Dimensional Multilayer

The intertwined ferroelectricity and band topology will enable the non-volatile control of the topological states, which is of importance for nanoelectrics with low energy costing and high response speed. Nonetheless, the principle to design the novel system is unclear and the feasible approach to achieve the coexistence of two parameter orders is absent. Here, we propose a general paradigm to design 2D ferroelectric topological insulators by sliding topological multilayers on the basis of first-principles calculations. Taking trilayer Bi2Te3 as a model system, we show that in the van der Waals multilayer based 2D topological insulators, the in-plane and out-of-plane ferroelectricity can be induced through a specific interlayer sliding, to enable the coexistence of ferroelectric and topological orders. The strong coupling of the order parameters renders the topological states sensitive to polarization flip, realizing non-volatile ferroelectric control of topological properties. The revealed design-guideline and ferroelectric-topological coupling not only are useful for the fundamental research of the coupled ferroelectric and topological physics in 2D lattices, but also enable novel applications in nanodevices.

Unlike 3D FETIs, two-dimensional (2D) lattices with intertwined ferroelectric and topological orders are rather scare [3,[19][20][21], and the coupling of the order parameters are also quite weak in several existing cases [15,22,23]. This is partly due to that, the ferroelectricity in 2D materials has been mainly established in the single-layer asymmetric structures [24,25], while band topology is commonly seen in the materials with heavy elements and strong spin orbital coupling. There, the requirements of symmetrical structure with switchable polarization and band inversion with different parity in the revealed 2D material family have to be simultaneously satisfied, to build the 2D FETIs, which significantly restrict the possible realization of 2D FETIs [9,[26][27][28]. So far, how to expand the scope for material candidates of 2D FETIs, especially with intertwined ferroelectric and topological physics, remains an open question.
Here, based on first-principles calculations, we fill this outstanding gap by introducing a general and simple scheme to realize 2D FETIs with intertwined ferroelectric and topological physics. By studying an example system of trilayer Bi2Te3, we discover that through a specific interlayer sliding, the charge rearrangement brings the spatial electron-hole separation in van der Waals multilayer based 2D topological insulators. The resultant reversal separation leads to the appearance of both in-plane and out-of-plane ferroelectricity, while the nontrivial topological phase is well reserved, thus enabling the coexistence of ferroelectric and topological orders. The strong coupling between ferroelectric and topological orders is also observed, distinct different topological physics can be induced in such multilayer systems when the ferroelectric polarization is reversed, suggesting the ferroelectric controlled topological properties. The proposed scheme to realize 2D FETIs with intertwined ferroelectric and topological physics is schematically presented in Figure 1. Without losing generality, we start from 2D van der Waals multilayers with nontrivial topological properties, the time-reversal symmetry is preserved. Unlike band topology that links with electronic properties [1,4,10,29], ferroelectricity relates to crystal structure symmetry and electric dipole induced by electron distribution [30][31][32][33][34]. To realize ferroelectricity, the polarization has to be switchable. As illustrated in the upper part of Figure 1, if the in-plane (IP) and out-of-plane (OPP) mirror symmetries (MIP/OOP), as well as the inversion symmetry (I), of the 2D multilayer are broken, the ferroelectricity occurs as long as the polarization is switchable, yielding the 2D FETI. The polarization switching is obtained via interlayer translation. If the polarization is unswitchable, it is just a normal 2D TI, without showing ferroelectric order. On the other hand, when the 2D multilayer possesses MIP/OOP or I symmetry, as illustrated in the lower part of Figure 1, two different cases can be induced by the interlayer sliding. In first case, the systems like bilayer possess the spatial I symmetry there is no polarization, this configuration is out of our consideration. In second case, the MIP/OOP and I symmetry can be broken by interlayer sliding, the polarization thus appears, and obviously such polarization is electrically switchable. In the latter case, if the topological property is preserved, the ferroelectric-topological phases can be achieved; otherwise, it is a trivial 2D ferroelectric material. As we will show below, the obtained ferroelectric and topological orders in such systems exhibits a strong coupling. This design scheme suggests that the crystal symmetry can be utilized as one screening factor to identify 2D FETIs with intertwined ferroelectric and topological physics.  Obviously, these two polarized configurations can be switched to each other by electric field triggered middle QL sliding [ (Figure 2 (b)], and thus correlate to two ferroelectric states, suggesting the OOP ferroelectricity.
To get more insight into the OOP ferroelectricity, we investigate the underlying physics for the electric polarization. In β1-Bi2Te3, as displayed in Figure 2( directions, which leads to a zero net IP polarization. When introducing the substrate effect, the threefold rotation symmetry is broken to realize the IP ferroelectricity, which has been well demonstrated in experiment [34,[42][43][44]. Accordingly, both IP and OOP ferroelectricity can be realized in trilayer Bi2Te3. Schematic representation of 2D FETI with two polar TI states. Next, we study the electronic properties of trilayer Bi2Te3 in the ferroelectric phase. As β1-and β2-Bi2Te3 are linked as two equivalent ferroelectric states, here we take β1-Bi2Te3 as an example. Figure   S1(a) shows the band structure of β1-Bi2Te3 without including spin-orbit coupling (SOC), from which we see that it is an indirect gap semiconductor with a global gap of 0.51 eV near the Γ point. By analyzing orbital contributions, we find the highest valence bands (VB) near the Fermi level is mainly contributed by Te-p orbital, while Bi-p orbital makes the dominate contribution to the lowest conduction bands (CB). Upon taking SOC into account, the VB and CB bands near the Γ point experience a significant Rashba spin splitting [ Figure S1(b)], which can be attributed to the existence of OOP electric polarization in β1-Bi2Te3. The corresponding Rashba parameter is calculated to be = 2 0 = 0.67 eVÅ. When SOC effect is considered, it is interesting to notice that the CBM and VBM move closer and the band gap is reduced to 9 meV. Such band narrowing and M-shaped VBM normally indicates a nontrivial topological phase.
To confirm the nontrivial topological order in β1-Bi2Te3, we calculate the topological invariant Z2.
Due to its broken inversion symmetry, the Z2 invariant is calculated by tracing the Wannier charge center (WCC) using non-Abelian Berry connection [45]. The Wannier functions (WFs) related with lattice vector R can be written as: Here, a WCC is defined by the mean value of < 0 |̂|0 > , where the ̂ represent the position operator and 0 is the state corresponding to a WF in the cell with R = 0. Then we obtain: Assuming ∑ ̅ = 1 2 ∫ with S = I or II, the summation in α is the occupied states and A is Berry connection. So we get the Z2 invariant following The calculated evolution of WCC is shown in Figure 3(a). As expected, the WCC is crossed by any arbitrary horizontal reference lines an odd number of times, indicating Z2 = 1. This firmly confirms the nontrivial topological phase of β1-Bi2Te3. As the existence of the localized metallic helical edge channels is the prominent feature for 2D TI, we calculate the armchair edge states by using a tightbinding (TB) Hamiltonian in the maximally localized WF. As shown in Figure 3 In the following, we discuss their coupling of ferroelectricity and topological orders in trilayer  [46][47][48]. In addition, utilizing the either IP or OOP external electric field, such topological p-n junctions are controllable.
Meanwhile, for such multilayer, the two ferroelectric states with opposite polarizations are linked to each other through an inversion operation. As a result, as shown in Figure 3(c), the chirality as well as the direction of the spin-locked currents at boundaries are closely associated with the direction of ferroelectric polarization. In other words, the direction of topological spin current can be fully controlled by ferroelectricity, which would promote novel applications in conceptually new devices.
Moreover, due to the direction of spin-locked current can be viewed as a ferroic order, such multilayers can also be treated as multiferroic systems, see Figure 3(d), holding potential for novel multiferroic devices. We wish to stress that these coupled ferroelectric and topological physics are not limited to trilayer Bi2Te3, but applicable for all 2D FETIs designed by this scheme.
In summary, we introduce a general scheme to realize coupled ferroelectric and topological physics in multilayer systems. Taking trilayer Bi2Te3 as an example, we show that through a specific interlayer sliding, both IP and OOP ferroelectricity can be realized in van der Waals multilayer based 2D topological insulators, resulting in the coexistence of ferroelectric and topological orders. We further show that the ferroelectric and topological orders exhibit a strongly coupling. Under the ferroelectric switching, distinct different topological physics can be induced in such multilayer systems.