## Abstract

2D spontaneous valley polarization attracts great interest both for its fundamental physics and for its potential applications in advanced information technology, but it can only be obtained from inversion asymmetric single-layer crystals, while the possibility to create 2D spontaneous valley polarization from inversion symmetric single-layer lattices remains unknown. Here, starting from inversion symmetric single-layer lattices, a general design principle for realizing 2D spontaneous valley polarization based on van der Waals interaction is mapped out. Using first-principles calculations, we further demonstrate the feasibility of this design principle in a real material of T-FeCl_{2}. More remarkably, such design principle exhibits the additional exotic out-of-plane ferroelectricity, which could manifest many distinctive properties, for example, ferroelectricity-valley coupling and magnetoelectric coupling. The explored design-guideline and phenomena are applicable to a vast family of 2D materials. Our work not only opens up a platform for 2D valleytronic research but also promises the fundamental research of coupling physics in 2D lattices.

## Introduction

Valley pseudospin, characterizing the energy extrema of conduction or valance bands, is an emerging degree of freedom of Bloch electrons in two-dimensional (2D) crystalline solids^{1,2}. Because valleys are largely separated in momentum space, the valley pseudospin is well defined and stable^{3,4}. The manipulation of valley pseudospin for information processing and storage leads to the well-known concept of valleytronics^{5,6}. The main challenge for valleytronics lies in lifting the valley degeneracy, thereby generating valley polarization^{7,8}, and there has been much effort in creating valley polarization, either intrinsic or by extrinsic strategies^{9,10,11,12,13,14,15,16}. Among them, intrinsic valley polarization presents great and ever-increasing interest, thanks both to its relevance for fundamental physics understanding and its potential technological applications for valleytronics and spintronic devices^{17}.

Physically, there are two essential ingredients for realizing 2D spontaneous valley polarization. One is ferromagnetism with out-of-plane magnetization, and the other is inversion symmetry breaking. Accordingly, the current research on 2D spontaneous valley polarization has been mainly established in the paradigm of inversion asymmetric single layers with ferromagnetism^{18,19,20,21,22,23}. Since inversion asymmetric single-layer semiconductors with ferromagnetism are rare themselves, spontaneous valley polarization is reported in only a few single-layer systems^{18,19,20,21,22,23}. Even for these few existing systems, most of them suffer from the in-plane magnetization in nature that valley polarization disappears, and additional tuning the magnetization orientation from in-plane to out-of-plane is needed^{18,19,20,23}. This poses an outstanding challenge for the field of 2D valleytronics. To overcome this problem, it is of particular importance to go beyond the existing paradigm for creating spontaneous valley polarization.

In this work, on the basis of inversion symmetric single-layer lattices, we theoretically identify a design principle for 2D spontaneous valley polarization through van der Waals interaction. Because of interlayer interaction, antiferromagnetic coupling and out-of-plane electric polarization are admitted simultaneously, which defines spontaneous valley polarization. By means of first-principles calculations, we further predict a real material of T-FeCl_{2} to establish the feasibility of this design principle. More importantly, we also find this design principle could demonstrate many intriguing phenomena, such as the ferroelectricity-valley coupling and magnetoelectric coupling. These findings provide important insights for the fundamental research in 2D valleytronics as well as coupling physics.

## Results and discussion

### Design principle for 2D spontaneous valley polarization

Going beyond the existing paradigm of exploring spontaneous valley polarization in inversion asymmetric single-layer systems, our design principle starts from inversion symmetric single-layer lattices with energy extrema of conduction or valance bands located at the K and K′ points. In principle, time-reversal symmetry breaking and inversion symmetry breaking are necessarily required for valley polarization^{18,22,23}. Concerning the first condition, it is generally believed that ferromagnetic coupling is essential. While for antiferromagnetic single layers, valley spin splitting is prohibited because of the protection of antiferromagnetic order, yielding the spin degeneracy for valleys^{24}. Such spin degeneracy limits the anomalous valley Hall effect as well as valleytronic applications. In this regard, we first impose a constraint of ferromagnetism on the single-layer lattices. To satisfy the second condition, van der Waals interaction is further introduced through constructing a bilayer lattice. In bilayer lattice, antiferromagnetism normally dominates interlayer coupling^{25}. For lifting spin degeneracy of valleys, additional constraints should be imposed on the stacking symmetry of the bilayer lattice. Only in this way, 2D spontaneous valley polarization can be realized from inversion symmetric single-layer lattices.

The proposed design principle is schematically illustrated in Fig. 1. Considering that magnetic single layers are mainly concentrated on trigonal lattices^{25}, we search through the layer groups and find out two groups with inversion symmetry, see Table 1. Without losing of generality, we assume that the inversion symmetric single layers are ferromagnetic semiconductors with energy extrema of conduction or valance bands located at the K and K′ points. Such trigonal single layers possess a rotation symmetry *C*_{3z}. By stacking two single layers together, one layer sits on the other with a \((2m - 1)\frac{{\uppi }}{3}\) rotation (*m* is any nonzero integer). In this case, inversion symmetry is broken in the bilayer lattice. To facilitate the construction of bilayer lattice, the rotation angle is selected as 180°, as shown in Fig. 1a. In detail, for a single layer with a space group \(P\bar 3/P\bar 3m1\), the lower layer can be stacked on the upper layer under the following operation:

The resultant bilayer system with *M*_{z} mirror symmetry is referred to as intermediate state (state-IM). In bilayer lattice, antiferromagnetic coupling normally dominates interlayer exchange interaction^{25}. Once we consider the magnetic space group, state-IM with interlayer antiferromagnetic ordering should break its mirror symmetry \(\hat M_z\) and time-reversal symmetry \(\hat T\). Although both \(\hat M_z\) and \(\hat T\) symmetries are broken, the joint symmetry under time-reversal and mirror reflection \((\hat O \equiv \hat M_z\hat T)\) guarantees the fact that K and K’ valleys remain degenerate in state-IM. Therefore, as shown in Fig. 1d, the K and K’ valleys in state-IM would be energetically degenerate, forbidding spontaneous valley polarization.

To break the \(\hat M_z\hat T\) symmetry, an additional operation \({{{\mathbf{t}}}}(\frac{1}{3}, - \frac{1}{3},0)/{{{\mathbf{t}}}}(\frac{1}{3},\frac{2}{3},0)/{{{\mathbf{t}}}}( - \frac{2}{3}, - \frac{1}{3},0)\) [Table 1] should be introduced, which results in state-I [Fig. 1b]. In state-I, uncompensated interlayer vertical charge transfer is induced, giving rise to an out-of-plane electric polarization pointing downwards. Also, the magnetic moments distributed on these two constituent single layers would be nonequivalent. In this case, as illustrated in Fig. 1e, spontaneous valley polarization can be realized in the bilayer lattice. In addition, there would be another energetically degenerate stacking configuration [state-II, see Fig. 1c] under the operation \({{{\mathbf{t}}}}( - \frac{1}{3},\frac{1}{3},0)/{{{\mathbf{t}}}}( - \frac{1}{3}, - \frac{2}{3},0)/{{{\mathbf{t}}}}(\frac{2}{3},\frac{1}{3},0)\) [Table 1], which exhibits an identical out-of-plane electric polarization but points upwards. In state-II, spontaneous valley polarization occurs as well, and the magnitude of valley polarization is equal to that of state-I; see Fig. 1f. Because the electrostatic potential and electric polarization are reversed, the sign of valley polarization, as well as the spin channel of valleys, are opposite to the case of state-I.

Importantly, the resultant state-I and state-II can be reversibly reversed under interlayer sliding, forming two ferroelectric states. The presence of additional 2D ferroelectricity in the bilayer lattice would lead to many intriguing phenomena. For example, the valley polarization would experience a reversal under ferroelectric switching [see Fig. 1e, f], yielding the ferroelectricity-valley coupling. In detail, along with ferroelectric switching, the spin channel of valleys and the sign of valley polarization are expected to be reversed. It is worth emphasizing that such reversal of valley polarization in inversion asymmetric single layers can only be induced by magnetization effect, which is rather challenging in practice^{18,20,23}. Besides, the imbalanced distribution of magnetic moments would also undergo a reversal upon switching the ferroelectric polarization, suggesting the exotic 2D magnetoelectric coupling.

### 2D spontaneous valley polarization in bilayer T-FeCl_{2}

Given the design principle, we next discuss its realization in a real material of T-FeCl_{2}. Single-layer (SL) T-FeCl_{2} with a trigonal lattice (space group \(P\bar 3m1\)) has been predicted to be capable of being exfoliated from its layered bulk^{26} and recently successfully synthesized^{27,28}. Each unit contains one Fe and two Cl atoms. As shown in Fig. 2c, under the distorted octahedral coordination, the *t*_{2g} orbitals of Fe atom split into an *e*_{g}^{*} doublet and a higher-lying *a*_{1g} singlet orbitals^{29}. The valence electronic configuration of an isolated Fe atom is 3*d*^{6}4*s*^{2}. In SL T-FeCl_{2}, after donating two electrons to Cl atoms, the left two electrons fill one orbital and four electrons half-fill the other four orbitals, resulting in a formal magnetic moment of 4 µ_{B} on each Fe atom and thus spontaneous spin polarization. The magnetic ground state of SL T-FeCl_{2} is estimated to be ferromagnetism, which is in consistent with the previous work^{29}. Supplementary Fig. 1 displays the band structure of SL T-FeCl_{2} with SOC, from which we can see the valence band maximum locating at the K points. However, as protected by inversion symmetry, valley physics is absent in SL T-FeCl_{2}.

Under the operation \(C_{2{{{\mathrm{z}}}}}(0,0,z)\;{{{\mathbf{t}}}}(0,0,{{{\mathbf{t}}}}_0)\), two SL T-FeCl_{2} are stacked together to construct the state-IM of the bilayer lattice [Fig. 2a], wherein inversion symmetry is broken, but *M*_{z} mirror symmetry is formed. Under an additional interlayer sliding of \({{{\mathbf{t}}}}(\frac{1}{3}, - \frac{1}{3},0)/{{{\mathbf{t}}}}(\frac{1}{3},\frac{2}{3},0)/{{{\mathbf{t}}}}( - \frac{2}{3}, - \frac{1}{3},0)\) state-I of bilayer T-FeCl_{2} is realized; see Fig. 2b. Its space group is *P*3*m*1, hosting neither inversion symmetry nor *M*_{z} mirror symmetry. The stability of state-I of bilayer T-FeCl_{2} is estimated by phonon spectra calculations and molecular dynamic simulations, as shown in Supplementary Fig. 2. Such a stable stacking pattern cannot be directly exfoliated from the layered bulk. In state-I, the Cl_{3} atom is right below the Cl_{1} atom, while the Cl_{2} atom locates directly above the Fe_{2} atom. Such configuration is expected to raise the separation of positive and negative charge centers along the out-of-plane direction, producing an out-of-plane electric polarization pointing downwards. This is confirmed by the differential charge density and plane averaged electrostatic potential shown in Supplementary Figs. 3 and 4. As displayed in Supplementary Fig. 3, there is a positive potential discontinuity Δ*V* = 7 × 10^{−4} eV between the vacuum levels of the upper and lower layers, which indicates the electric polarization pointing downwards. And from Supplementary Fig. 4, it can be seen that the differential charge density is obviously asymmetric. The electric polarization for state-I is calculated to be −2.7 × 10^{−3} μC cm^{−2}.

We then investigate the magnetic properties of state-I of bilayer T-FeCl_{2}. By considering different magnetic configurations, our calculations show that the intralayer and interlayer exchange interactions, respectively, are dominated by ferromagnetic and antiferromagnetic coupling. This magnetic configuration is energetically stable than the ferromagnetic configuration by 0.8 meV per unit cell, thus it is expected to be observed experimentally under low temperature. It should be noted that due to the existence of electric polarization pointing downwards, the magnetic moment distributed on the Fe atom from the upper layer is slightly larger than that from the lower layer, resulting in a net magnetic moment of 4 × 10^{−4} µ_{B} per unit cell. We also study the magnetocrystalline anisotropy energy (MAE) of state-I. Figure 2e illustrates the spherical plot of MAE arising from the rotating spin axis from the out-of-plane direction. The MAE is set to zero in the out-of-plane direction. Obviously, state-I favors out-of-plane magnetization, which is more stable than the in-plane magnetization by 37 meV per unit cell. It is worth emphasizing that such MAE is significantly larger than the values reported in most previous systems with spontaneous valley polarization^{19,22,23}, which is particularly promising for 2D valleytronics.

The band structures of state-IM and state-I of bilayer T-FeCl_{2} with considering SOC are shown in Fig. 3a, b, respectively. For state-IM, it is an antiferromagnetic semiconductor with an indirect bandgap of 2.59 eV. Its conduction band minimum (CBM) lies at the Γ point, and the valence band maximum (VBM) locates at the K and K′ points. Specifically, because of inversion symmetry breaking and antiferromagnetic interlayer coupling, the VBM at the K and K′ points are from the spin-down channel of the upper layer and spin-up channel of the lower layer, respectively. Due to the protection of the joint symmetry of mirror reflection and time-reversal (\(\hat M_z\hat T\)), valley polarization in state-IM is absent. When shifting the Fermi level between the K and K′ valleys and applying an in-plane electric field, the spin-up holes from the K valley and the spin-down holes from the K′ valley would accumulate, respectively, at the left and right sides of the sample, leading to the valley Hall effect, as shown in Fig. 3d. Moreover, the lowest conduction band at the K and K′ points also contributes to two valleys, but being inverted and submerged by the band at the Γ point, which is not applicable for practical applications. We, therefore, focus on the valleys in the highest valence band in the following.

When transferring state-IM to state-I, as shown in Fig. 3b, the indirect gap semiconducting character of bilayer T-FeCl_{2} is preserved, and the bandgap is found to be 2.60 eV. At the K′ valley, arising from the existence of the electric polarization pointing downwards, the spin-down band from the upper layer shifts above the spin-up band from the lower layer, while the band order at the K valley remains the same as that in state-IM. In this case, despite the antiferromagnetic interlayer coupling, both the K and K′ valleys are from spin-down channels of the upper layer, as shown in Fig. 3b. This is in consistent with the calculated nonzero net magnetic moment. And importantly, the K′ valley is higher than the K valley in energy, lifting the valley’s degeneracy. Accordingly, 2D spontaneous valley polarization is successfully realized in bilayer T-FeCl_{2}. The spontaneous valley polarization for bilayer T-FeCl_{2} is calculated to be \({\Delta}_{{{{\mathrm{K}}}}\prime - {{{\mathrm{K}}}}} = 5.3\;{{{\mathrm{meV}}}}\). This value is larger than those of the experimentally demonstrated magnetic proximity systems (0.3–1.0 meV)^{30,31}, and is close to those of previously reported doping systems, such as Cr-doped MoSSe (10 meV)^{32} and Hg-doped MnPSe_{3} (14 meV)^{33}. By shifting the Fermi level between the K and K′ valleys and applying an in-plane electric field, the anomalous valley Hall effect can be achieved. As shown in Fig. 3e, the spin-up holes from the K′ valley can transversely move to the right side of the sample in state-I. Considering that 2D materials are usually supported by substrates during device fabrication, it is likely that strain will be introduced due to lattice mismatch. To investigate the robustness of the valley feature against strain effect, the biaxial strain is applied on state-I as shown in Supplementary Fig. 5. The valley properties remain almost unchanged under strain in the range of −3 to 3%, which implies that the valley polarization is robust against the biaxial strain. Moreover, we also modulate the valley polarization by engineering the interlayer distance; see Supplementary Fig. 6. The value of valley polarization in state-I increases monotonically with the decrease of interlayer distance, which is attributed to the enhancement of interlayer interaction.

Based on state-IM, reverse interlayer sliding of \({{{\mathbf{t}}}}( - \frac{1}{3},\frac{1}{3},0)/{{{\mathbf{t}}}}( - \frac{1}{3}, - \frac{2}{3},0)/{{{\mathbf{t}}}}(\frac{2}{3},\frac{1}{3},0)\) could drive bilayer T-FeCl_{2} into another energetically degenerate stacking configuration, namely, state-II. In state-II, as shown in Fig. 2b, the Cl_{3} atom is right below the Fe_{1} atom, while the Cl_{2} atom locates directly above the Cl_{4} atom. This generates an out-of-plane electric polarization pointing upwards. The electric polarization for state-II is calculated to be 2.7 × 10^{−3} μC cm^{−2}, which is just opposite to that of state-I. Therefore, as expected, state-I and state-II could be considered as two ferroelectric states of bilayer T-FeCl_{2}. We investigated the overall stability of state-I and state-II compared to the other possible stacking patterns of bilayer T-FeCl_{2}, as shown in Supplementary Fig. 7. Note that state-I and state-II have local minimum stacking energies. The ferroelectric reversal pathway is shown in Supplementary Fig. 8. There are six high-symmetry interlayer sliding operations to reverse ferroelectric polarization. Due to the rotation symmetry *C*_{3z} of bilayer T-FeCl_{2}, under the three equivalent interlayer sliding of \({{{\mathbf{t}}}}( - \frac{1}{3}, - \frac{2}{3},0)\), \({{{\mathbf{t}}}}(\frac{2}{3},\frac{1}{3},0)\) or \({{{\mathbf{t}}}}( - \frac{1}{3},\frac{1}{3},0)\), state-I can be switched to state-II through the paraelectric state [state-PE, inset in Supplementary Fig. 8a]. The switching from state-I to state-II can also be realized by interlayer sliding along the other three equivalent directions [i.e., \({{{\mathbf{t}}}}(\frac{2}{3},\frac{4}{3},0)\), \({{{\mathbf{t}}}}( - \frac{4}{3}, - \frac{2}{3},0)\) or \({{{\mathbf{t}}}}(\frac{2}{3}, - \frac{2}{3},0)\)], which excludes state-PE but includes state-IM. As shown in Supplementary Fig. 8b, during the whole process of lateral rigid shifts, state-I and state-II are still the most stable configurations. The switching mechanism is similar to those of the recently reported van der Waals ferroelectrics^{34,35,36,37,38,39}. The state-PE with the space group *Abm*2 shows no out-of-plane electric polarization due to the existence of a glide plane in the *z*-direction. The phonon spectra of state-PE displayed in Supplementary Fig. 9 presents pronounced negative frequencies, suggesting that state-PE is unstable and would experience spontaneous transformation into state-I or state-II.

To determine the feasibility of the ferroelectric switching in bilayer T-FeCl_{2}, we investigate the energy barrier employing the nudged-elastic band (NEB) method^{40}. As shown in Fig. 2d, the lowest energy barrier for ferroelectric switching between state-I and state-II is calculated to be 4.2 meV per f.u., which is larger than that of bilayer T′-WTe_{2} (0.6 meV per f.u.)^{34}, but lower than those of bilayer h-BN (4.5 meV per f.u.)^{35}, bilayer β-GeSe (5.83 meV per f.u.)^{41} and In_{2}Se_{3} (66 meV per f.u.)^{42}. This confirms the feasibility of the ferroelectric switching between state-I and state-II.

By switching state-I to state-II, because ferroelectric polarization is reversed, the magnetic moment distributed on the Fe atom from the lower layer is slightly larger than that from the upper layer, which is opposite to the case of state-I. As a consequence, the magnetic properties of bilayer T-FeCl_{2} can be controlled by ferroelectricity, thereby producing exotic magnetoelectric coupling. In addition to the magnetoelectric coupling, the ferroelectricity-valley coupling is also realized in bilayer T-FeCl_{2}. Figure 3c shows the band structure of state-II. In state-II, the spin orientations for both K and K′ valleys are reversed, namely, from spin-down channels of the upper layer, as shown in Fig. 3c. In state-II, the K′ valley is lower than the K valley in energy. This lifts the valley degeneracy, leading to the 2D spontaneous valley polarization in state-II. The spontaneous valley polarization for state-II is calculated to be \({\Delta}_{{{{\mathrm{K}}}}\prime - {{{\mathrm{K}}}}} = - 5.3\;{{{\mathrm{meV}}}}\), which is opposite to the case of state-I. The anomalous valley Hall effect of state-II of bilayer T-FeCl_{2} is illustrated in Fig. 3f. With proper hole doping, the spin-down holes from the K valley flow to the left side of the sample under an in-plane electric field, which generates a voltage opposite to the case of state-I. Therefore, the electrical permanent control of valley physics of bilayer T-FeCl_{2} can be realized by the ferroelectric switching, resulting in the ferroelectricity-valley coupling.

### 2D spontaneous valley polarization in triple-layer T-FeCl_{2}

From above, we successfully establish a general design principle for spontaneous valley polarization from inversion symmetric single layers. Actually, this design principle is also applicable for multilayer lattices. Here, we take triple-layer T-FeCl_{2} as an example to address it. Starting from AAA stacking pattern, interlayer sliding \({{{\mathbf{t}}}}( \!\mp\! \frac{1}{3}, \pm\! \frac{1}{3},0)\) of the top or bottom layer can break the inversion symmetry as well as *M*_{z} mirror symmetry, leading to state-I or state-II of triple-layer T-FeCl_{2}, as shown in Fig. 4a. The out-of-plane electric polarizations for state-I and state-II are calculated to be 5.4 × 10^{−3} and −5.4 × 10^{−3} μC cm^{−2}, respectively. These two states are ferroelectric states of triple-layer T-FeCl_{2}, which can be reversed under a relative interlayer sliding of the middle layer. The energy barrier for transferring state-I to state-II is calculated to be 26.8 meV per f.u. Although such barrier is higher than that of the bilayer case, it is still much lower as compared with traditional ferroelectrics^{43,44}. Akin to the case of bilayer T-FeCl_{2}, the inequivalent magnetic moment distribution for state-I is opposite to state-II, which correlates to the opposite electric polarization. This yields magnetoelectric coupling in triple-layer T-FeCl_{2} as well. Figure 4b, c present the band structures of state-I and state-II with considering SOC, respectively. For both cases, 2D spontaneous valley polarization is successfully achieved. In state-I, the K and K’ valleys are from spin-down channels of the bottom layer [Fig. 4b], and the K′ valley is higher than the K valley in energy, generating a valley polarization of 4.7 meV. While in state-II, as shown in Fig. 4c, the K and K′ valleys are from spin-down channels of the top layer, and the K′ valley is lower than the K valley in energy, leading to a valley polarization of −4.7 meV. In this regard, when shifting the Fermi level between the K and K′ valleys and applying an in-plane electric field, the spin-up holes from the K′ and K valleys will accumulate at the same side of the sample in the state-I and state-II, leading to the anomalous valley Hall effect. As a consequence, 2D spontaneous valley polarization and magnetoelectric coupling can be obtained in triple-layer FeCl_{2} as well.

At last, we wish to stress that such 2D spontaneous valley polarization induced by interlayer translation and rotation from inversion symmetric lattices is predicted to exist widely in many other van der Waals systems, such as CrF_{3}, T-CoCl_{2} and T-CoBr_{2}, etc., as these ferromagnetic single-layers exhibit the space group \(P\bar 3/P\bar 3m1\) similar to T-FeCl_{2} and possess the energy extrema of conduction or valance bands at the K and K′ points as well^{28,45}.

In conclusion, we introduce a general design principle for 2D spontaneous valley polarization from inversion symmetric lattices and predict a real material to realize it. Moreover, this design principle exhibits the exotic out-of-plane ferroelectricity, which could demonstrate many distinctive properties, for example, ferroelectricity-valley coupling and magnetoelectric coupling. Our work greatly enriches the physics and expands the family of 2D spontaneous valley polarization, which is expected to draw immediate experimental interest.

## Methods

### Density functional theory calculations

First-principles calculations are performed based on density functional theory^{46} as implemented in Vienna ab initio simulation package (VASP)^{47}. The generalized gradient approximation (GGA) in the form of Perdew–Burke–Ernzerhof (PBE)^{48} is applied. Structures are fully relaxed until the force on each atom is less than 0.02 eV Å^{−1}, and the electronic iteration convergence criterion is set to 10^{−5} eV. A Monkhorst–Pack k-point mesh of 15 × 15 × 1 is used to sample 2D Brillouin zone. The vacuum space of at least 20 Å is introduced to avoid spurious interactions between neighboring sheets. The cutoff energy is set to 450 eV. Grimme’s DFT-D3 method is employed for taking van der Waals interaction into account^{49}. According to previous work^{50}, we use the GGA + *U* approach with *U*_{eff} = (*U* − *J*) = 4.0 eV for Fe-*d* orbitals, but it is not included in the structural optimization. Ab initio molecular dynamics (AIMD) simulation is performed to examine the thermal stability at 500 K for 5 ps with a time step of 1 fs^{51}. The calculation of phonon dispersion is based on a supercell approach using PHONOPY code^{52}. The energy barrier of ferroelectric switching is investigated using the nudged-elastic band (NEB) method^{40}. The ferroelectric polarization is evaluated using the Berry phase approach^{53}, and the dipole correction is used to meet the convergent criterion^{54}.

## Data availability

The authors declare that the data supporting the findings of this study are available within the paper and its supplementary information files.

## Code availability

The central codes used in this paper are VASP. Detailed information related to the license and user guide are available at https://www.vasp.at.

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## Acknowledgements

This work is supported by the National Natural Science Foundation of China (No. 12074217), Shandong Provincial Natural Science Foundation (Nos. ZR2019QA011 and ZR2019MEM013), Shandong Provincial Key Research and Development Program (Major Scientific and Technological Innovation Project) (No. 2019JZZY010302), Shandong Provincial Key Research and Development Program (No. 2019RKE27004), Shandong Provincial Science Foundation for Excellent Young Scholars (No. ZR2020YQ04), Qilu Young Scholar Program of Shandong University, and Taishan Scholar Program of Shandong Province.

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T.Z. and X.X. performed calculations and data analysis. Y.M. supervised the project. T.Z. and Y.M. co-wrote the paper. All authors discussed the results and commented on the manuscript at all stages.

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Zhang, T., Xu, X., Huang, B. *et al.* 2D spontaneous valley polarization from inversion symmetric single-layer lattices.
*npj Comput Mater* **8**, 64 (2022). https://doi.org/10.1038/s41524-022-00748-0

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DOI: https://doi.org/10.1038/s41524-022-00748-0