Abstract
Spontaneous polarization and bulk photovoltaic effect (BPVE) are two concomitant physical properties in ferroelectric materials. The flipping of ferroelectric order usually accompanies the switching of BPVE in all directions because both of them are reversed under the inversion symmetry. In this study, we report the nonsynchronous BPVE in twodimensional (2D) interlayersliding ferroelectric materials featuring unswitchable inplane BPVE (lightinduced photocurrent in the xy plane) and switchable outofplane BPVE (lightinduced polarization along the zdirection). Symmetry analysis within the abstract bilayer crystal model and firstprinciples calculations validate these BPVE properties. It is because the positive and negative ferroelectric states caused by interlayer sliding are related by mirror symmetry which cannot flip all the BPVE tensor elements. This finding extends the understanding of the relationship between ferroelectricity and BPVE. On one hand, the switchable outofplane BPVE can be used to design switchable photoelectric devices. On the other hand, the inplane BPVE is robust against the ferroelectric flipping, and the unswitchable character is beneficial to construct largerscale photoelectric devices.
Introduction
Ferroelectricity and bulk photovoltaic effect (BPVE)^{1,2} are two basic physical phenomena that emerge in condensed matter materials due to symmetry breaking. Generally speaking, BPVE, an intrinsic optical rectification phenomenon, appears in materials without inversion symmetry which can convert light to electricity as solar cells, hence it is naturally fulfilled the symmetry requirement in ferroelectric materials. As a result, the ferroelectric materials with switchable electronic polarization are one of the most studied BPVE materials^{1,2,3}, and the corresponding energy conversion efficiency can exceed the ShockleyQueisser limit^{3,4,5}.
It is known to us that the BPVE usually switches with the ferroelectric order. Under the inversion operation, both ferroelectric order and BPVE in all directions switch signs, i.e., BPVE switches synchronously with the ferroelectric order. Therefore, the ferroelectric polarization is often taken as a handle to manipulate the direction of BPVE photocurrent in, e.g., perovskite oxides BiFeO_{3}^{6,7,8}, Pt/BiFeO_{3}/SrRuO_{3}^{9}, 2D (twodimensional) CuInP_{2}Se_{6}^{10}, chargetransfer complex^{11}, SnTe monolayer^{12}, MX [M = (Ge, Sn) and X = (Se, S)]^{13} etc. However, from the aspect of symmetry, the ferroelectric polarization is a polar vector, while the BPVE coefficients form a rankthree tensor. They are hence subject to distinct symmetry transformation rules, and should not always synchronize. There should exist components of the BPVE tensor that do not switch with the ferroelectric order, which was visioned^{14}, but has not been comprehensively studied before. For example, a mirror reflection perpendicular to the polarization will reverse the polarization just as the inversion operation does, while it cannot flip all the components of the BPVE. Thus, a precise understanding of the relationship between the ferroelectric order and the BPVE is of tremendous significance in both fundamental research and the design of photoelectric devices.
In this work, we think the exceptional case that nonsynchronous BPVE with ferroelectric order can be found in 2D interlayersliding ferroelectric materials^{15}. As demonstrated in recent experiments, the outofplane ferroelectricity can be achieved in bilayer WTe_{2}^{16,17,18,19}, BN^{20,21}, InSe^{22,23}, MoS_{2}^{24,25}, etc. by interlayer sliding, which effectively extends the family of 2D ferroelectric materials^{26,27,28,29}. This ferroelectric flip mechanism may lead to the BPVE property beyond the switchable scenario.
Focusing on the four most common stacking cases of 2D bilayers with interlayer sliding, we use an abstract bilayer crystal model to locate the available configurations and symmetries with nontrivial ferroelectric order. We find two cases (Case 1b and Case 2a) can have the possibility to achieve the interlayersliding ferroelectricity, moreover the opposite ferroelectric states in the two cases are all related by the mirror operator. Coincidentally, the interlayersliding ferroelectric materials found in the experiments are attributed to these two cases. Using bilayer MoS_{2} and WTe_{2} as representative examples, we performed firstprinciples calculations and found that the inplane BPVE does not change the sign with ferroelectric order, which is different from conventional ferroelectrics. In contrast to inplane BPVE, outofplane BPVE switches with the reverse of the ferroelectric order. This nonsynchronous character will lead to distinct photovoltaic phenomena in experiments.
Results
Polarization flipping via interlayer sliding
Now we introduce the mechanism of the polarization flipping via interlayer sliding. Figure 1a shows the outofplane polarization P_{z} in bilayer vdW (van der Waals) material without inversion nor horizontal mirror symmetry. Because the inplane polarization is usually annihilated by inplane rotational symmetry in real materials, the +P_{z} and –P_{z} polarization are denoted as +P and –P for short. The outofplane polarization can be flipped in two different ways: (i) intralayer movement, i.e., atomic relative displacements in each monolayer [Fig. 1b], and (ii) interlayer sliding^{15} where the crystal structure of each monolayer is invariant [Fig. 1c]. Although polarization flipping can be achieved in both situations, the underlying symmetries of the switch of ferroelectric states are different. For situation (i), the +P and –P ferroelectric states are correlated by the inversion operator, as indicated in unit cells in Fig. 1a, b. While for situation (ii), the +P and –P ferroelectric states are correlated by the mirror symmetry \(\hat M_{xy}\), as seen in unit cells in Fig. 1a, c (detailed analyses are presented in the Symmetry analysis section for real materials). The interlayer sliding mechanism is feasible experimentally and has been observed in bilayer vdW materials^{16,17,18,19,20,21,22,23,24,25} as stated above, which further extends the family of 2D ferroelectric materials^{26,27,28,29} effectively. Figure 1d–f shows the basis vector transformation under the inversion symmetry and mirror symmetry. The BPVEs in these interlayersliding ferroelectric materials do not always change synchronously with the ferroelectric order.
Symmetry analysis
Next, we analyze how to design the bilayer vdW materials with the outofplane polarization via interlayer sliding in Fig. 1a, and analyze how positive and negative polarizations are related by symmetry operator. Most of the discovered monolayer materials are nonpolar and nonferroelectric (such as BN, transitionmetal dichalcogenides, metal trihalide, posttransitionmetal chalcogenides), and their monolayer crystal structures belong to the following two different cases, as shown in Supplementary Figs. 1, 2 in the Supplementary Information^{30,31}:

1.
Monolayer has the inversion symmetry \(\widehat I\), but has no horizontal mirror symmetry \(\widehat M_{xy}\), such as monolayer 1TPtS_{2}, 1T’WTe_{2}, BiI_{3}, etc.;

2.
Monolayer has no inversion symmetry \(\widehat I\) but has horizontal mirror symmetry \(\widehat M_{xy}\), such as monolayer BN, 1HMoS_{2}, 1HWSe_{2}, GaSe, InSe, etc.
Besides, the bilayer can be further divided into two subcases according to the stacking manner of two layers:

(a)
A/A stacking, i.e., two adjacent layers are identical where the A represents the crystal structure of the monolayer.

(b)
A/B stacking, i.e., two adjacent layers are relatively rotated by 180° with \({{{\mathrm{B}}}} = \hat C_{2z}{{{\mathrm{A}}}}\) (A and B mean the crystal types).
The above considerations can form four cases for 2D bilayer materials with interlayer sliding, and we can analyze their symmetries with an abstract bilayer crystal model which only considers the above crystal symmetries and ignores the specific crystal structures. The complete symmetry analysis for each case is summarized in Supplementary Note 1, and the corresponding results are shown in Table 1. As can be seen, Case 1b and Case 2a host the outofplane polarization, and the two opposite interlayersliding states are correlated by \(\widehat M_{xy}\) operator, instead of inversion \(\widehat I\) operator. In contrast, the crystal structures in Case 1a and Case 2b possess the inversion symmetry whatever the interlayersliding vector, which forbids the occurrence of ferroelectricity and BPVE.
With the abstract bilayer crystal model, the interlayersliding ferroelectric materials are further classified into two kinds (Case 1b and Cases 2a). Such classification outlines a potential roadmap to search for more 2D ferroelectric materials, which could greatly facilitate and advance the research and application of BPVE in 2D ferroelectric materials. Table 2 summarizes several bilayer candidates for the above four cases that can be experimentally realized by either simply mechanical exfoliation or tearandstack methods. In particular, the outofplane ferroelectricity of the bilayer WTe_{2}^{16,17,18,19} with Case 1b and BN^{20,21}, InSe^{22,23}, bilayerTMD^{24,25} with Case 2a have been discovered in experiments recently. The abstract bilayer model is convenient and efficient in symmetry analysis for vdW materials with interlayer sliding because it does not rely on the specific crystal structure. The symmetry of trilayer vdW materials with interlayer sliding can also be obtained using a similar method, and the results are shown in Supplementary Note 7, where the transformations of ferroelectricity and BPVE are also discussed there.
Now we analyze the characters of inplane and outofplane BPVE of the interlayersliding ferroelectric materials in Case 1b and Case 2a under the normal incidence of light. The inplane BPVE current density is
where \(a,b,c \in \left\{ {x,y} \right\}\), E_{b} and E_{c} are electric fields of light along b and c direction, ω is the frequency of light, and \(\sigma _{bc}^a\) is the BPVE coefficient. In contrast to x and y directions, the photoexcited shift of the wave package along the outofplane direction induces a static electric polarization instead of an electric current along the zdirection because the zdirection of 2D materials is discontinuous. The outofplane BPVE polarization is^{32}
where \(p_z\) is zcomponent polarizability.
Similar to other nonlinear optics tensors, the BPVE tensor \(\left[ {\sigma _{ab}^c} \right]_{3 \times 3 \times 3}\) in Eq. (1) [or Eq. (2)] obeys
where \([R_{ia}]\) is the 3 × 3 matrix of the symmetry operator \(\widehat R\). As shown in Fig. 1f, \(\widehat M_{xy}\) will result in the reverse of the outofplane vector with invariant inplane vectors, and the corresponding operator matrix is
The opposite states are connected by the mirror operator [\(\widehat M_{xy}( + P) =  P\)] in interlayersliding ferroelectric materials. Therefore, the inplane BPVE coefficient is invariant with the change of ferroelectric orders, i.e., \(\sigma _{ab}^c( + P) = \sigma _{ab}^c(  P),a,b,c \in \left\{ {x,y} \right\}\), according to Eq. (3). This is strikingly different from the outofplane case where the BPVE coefficient is reversed with the switch of ferroelectric orders, as indicated by \(\sigma _{bc}^{\prime z}\left( { + P} \right) =  \sigma _{bc}^{\prime z}\left( {  P} \right)\) (\({\kern 1pt} b,c \in \left\{ {x,y} \right\}\)). The calculated BPVE characters in Fig. 1a, c by a 1D effective model are also consistent with our symmetry analysis (see Supplementary Note 2).
In contrast, for the ferroelectric states correlated by inversion symmetry \(\hat I( + P) =  P\), the inversion operator \(\hat I\) will lead to the reverse of all the vectors (Fig. 1d), i.e., \(\sigma _{bc}^a( + P) =  \sigma _{bc}^a(  P)\) (\(a,b,c \in \left\{ {x,y,z} \right\}\)) according to Eq. (3). It is the reason why the BPVE coefficients reverse with ferroelectric orders in the conventional ferroelectric materials.
The inplane BPVE coefficients can be calculated by the standard secondorder Kubo formalism theory^{33,34}, and the outofplane BPVE coefficients can be obtained by a modified secondorder Kubo formalism theory^{32} that was proposed recently. BPVE coefficients are calculated by the Wannier function (see Method for details)^{35,36,37}. Recently, theory works indicate that^{38,39} BPVE coefficients are highly related to the geometry theory of electron band. In the following part, we choose a representative material for each case: bilayer MoS_{2}^{24,25} (Case 2a) and bilayer WTe_{2}^{16,17,18,19} (Case 1b) to perform the numerical firstprinciples calculations (see Methods section for calculation details).
Firstprinciples calculation results
Bilayer MoS_{2}
Bilayer MoS_{2} (Case 2a) that is consisted of two identical nonferroelectric monolayers shows outofplane ferroelectricity^{15}, which has been confirmed in experiments recently^{24,25}. Noting that conventional bilayer MoS_{2} is stacked in A/B form (Case 2b) with inversion symmetry, which is different from the crystal structure discussed here. As shown in Fig. 2a, the outofplane ferroelectricity is reversed by the inplane interlayer motion along the armchair direction. The monolayer possesses the \(D_{3h}\) symmetry. However, the interlayersliding bilayer breaks the \(\widehat M_{xy}\) and inplane \(\widehat C_2\) rotation symmetries, which results in the bilayer MoS_{2} only having \(C_{3v}\) symmetry (\(D_{3h} = C_{3v} + \widehat M_{xy}C_{3v}\)). Therefore, +P and –P states only have outofplane polarization. Moreover, the two ferroelectric states are connected by the \(\widehat M_{xy}\) symmetry^{24,25}, which is consistent with the above symmetry analysis. The calculated band structures of the +P and –P states are shown in Fig. 2b, which are invariant with the ferroelectric orders.
As shown in Fig. 2c, the calculated inplane BPVE coefficients with +P and –P states are equal. The inplane BPVE is invariant with the change of the ferroelectric order as predicted. In addition, \(\sigma _{xy}^x = \sigma _{xx}^y =  \sigma _{yy}^y\), and \(\sigma _{xx}^x = \sigma _{yy}^x = \sigma _{xy}^y = 0\) for each ferroelectric state due to the C_{3v} symmetry [see Supplementary Note 3]. It is noted that this unswtichable inplane BPVE character of interlayersliding ferroelectric materials is different from those of conventional ferroelectric materials^{6,7,8,9,10,11,12,40} and antiferromagnets^{41,42} where the BPVE is switched with the ferroelectric/ferromagnetic order. The inplane BPVE photocurrent as a function of the polarization direction of light (see Supplementary Note 4 for details) is shown in Fig. 2d. The inplane BPVE photocurrent shows the C_{3v} symmetry, and its magnitude is independent of the direction of light.
The calculated outofplane BPVE coefficients are shown in Fig. 2e. For each ferroelectric state, \(\sigma _{xx}^{\prime z} = \sigma _{yy}^{\prime z}\), and \(\sigma _{xy}^{\prime z} = \sigma _{yx}^{\prime z} = 0\). Moreover, the outofplane BPVE coefficients switch with the ferroelectric order following \(\sigma _{xx}^{\prime z}( + P) = \sigma _{yy}^{\prime z}( + P) =  \sigma _{xx}^{\prime z}(  P) =  \sigma _{yy}^{\prime z}(  P)\), which is consistent with our symmetry analysis. The outofplane BPVE is isotropic, and photoinduced polarization is independent of the polarization direction of light due to C_{3v} symmetry, as shown in Fig. 2f.
Similarly, experimentally prepared bilayer BN^{20,21}, InSe^{22,23}, GaSe, etc., possess similar interlayersliding ferroelectricity (Case 2a) and the same symmetry. Therefore, similar inplane and outofplane BPVE features are also expected to exhibit in these materials.
Bilayer WTe_{2}
Bilayer WTe_{2} is composed of two 180°rotated two monolayers (Case 1b), as shown in Fig. 3a. Even though each monolayer (with \(C_{2h} = \{ E,\widehat I,\widehat C_{2x},\widehat M_{yz}\}\) symmetry) has the inversion \(\widehat I\) symmetry, the bilayer breaks this symmetry. Besides, the interlayersliding vector along b axis further breaks \(\widehat M_{xy}\) and inplane \(\widehat C_2\) symmetry. Thereby the bilayer has the \(C_{1v} = \{ E,\widehat M_{yz}\}\) symmetry (\(C_{2h} = C_{1v} + \widehat M_{xy}C_{1v}\)).
Interlayer sliding^{43,44} induced outofplane ferroelectricity in bilayer WTe_{2} has been verified experimentally^{16,17,18,19}. The –P state is the mirror image of +P state, as shown in Fig. 3a. Figure 3b shows the band structures of +P and –P states, where the valence and conduction bands overlap in our calculation.
The calculated inplane BPVE coefficients are shown in Fig. 3c. There are three independent tensor elements \(\sigma _{xy}^x\), \(\sigma _{xx}^y\), \(\sigma _{yy}^y\), and \(\sigma _{xx}^x = \sigma _{yy}^x = \sigma _{xy}^y = 0\) due to the C_{1v} symmetry. The calculated inplane BPVE coefficients are the same in two opposite ferroelectric orders: \(\sigma _{xy}^x( + P) = \sigma _{xy}^x(  P)\), \(\sigma _{xx}^y( + P) = \sigma _{xx}^y(  P)\), \(\sigma _{yy}^y( + P) = \sigma _{yy}^y(  P)\), which is consistent with the above symmetry analysis. The inplane BPVE photocurrent with the direction of polarization of light is shown in Fig. 3d, which shows \(\widehat M_{yz}\) symmetry, and its magnitude is dependent on the light (see Supplementary Note 4 for details). Recently, it was shown that bilayer WTe_{2} exhibits unswitchable inplane nonlinear anomalous Hall effect^{14,18}.
On the contrary, the outofplane BPVE coefficients switch with the change of ferroelectric order, \(\sigma _{xx}^{\prime z}( + P) =  \sigma _{xx}^{\prime z}(  P)\), \(\sigma _{yy}^{\prime z}( + P) =  \sigma _{yy}^{\prime z}(  P)\) [Fig. 3e]. Unlike the bilayer MoS_{2}, the outofplane BPVE coefficients show anisotropy \(\sigma _{xx}^{\prime z} \,\ne\, \sigma _{yy}^{\prime z}\) (\(\sigma _{xy}^{\prime z} = \sigma _{yx}^{\prime z} = 0\)) for each ferroelectric state due to the absence of inplane C_{3} symmetry, leading to the outofplane BPVE polarization is dependent on the polarization of light as shown in Fig. 3f. Recently, ZrI_{2}, a sister material of polar semimetals WTe_{2}, is also demonstrated^{45,46,47} to have interlayersliding ferroelectricity. Similar BPVE behaviors with bilayer WTe_{2} are expected to be observed due to the same symmetry and interlayerstacking way (Case 1b).
The calculated values of the inplane BVPE coefficient of bilayer MoS_{2} and bilayer WTe_{2} are comparable to the reported monolayer MX [M = (Ge, Sn) and X = (Se, S)]^{13,48} and larger than 3D conventional ferroelectric materials (such as BiFeO_{3}^{49}, BiTiO_{3}, and PbTiO_{3}^{2}). The outofplane BPVE coefficients are comparable to twisted bilayers graphene^{32}, which induces voltage are detectable experimentally. We can distinguish the photoinduced polarization and intrinsic polarization of interlayersliding ferroelectric materials in two subsequent detections with/without light in experiments.
We can calculate the lightinduced outofplane electric dipole moment and voltage caused by the outofplane BPVE, and estimate the possibility to flip the ferroelectric order. With light power of 2.65 × 10^{11} W/cm^{2} (equivalent electric field E=1 V/nm) and a typical value of \(\sigma _{xx}^{\prime z} = \sigma _{yy}^{\prime z}\)=1 e/V^{2} for the outofplane BPVE coefficients, the lightinduced polarization p_{z} = 1 e/nm^{2}, meanwhile the lightinduced voltage difference between the two layers (\(V = p_zd/\varepsilon _0\), d = 1 nm) is found to be 1.8 V. The photoinduced polarization is already bigger than intrinsic polarization of interlayersliding ferroelectric materials (about 0.01 e/nm^{2})^{15}, and the photoinduced outofplane voltage is also bigger than the flipping voltage (about 0.1 V/nm) in experiments^{16,17,18,19,20,21,22,23,24,25}. Therefore, we think the photoinduced polarization can flip the ferroelectric order when the laser intensity is larger enough. Recently, experiment^{50} shows that the stacking order in bulk WTe_{2} can be manipulated by ultrafast optical excitation, which indicates the possibility for the flip the ferroelectric order by optical irradiation. The kinetic process of ferroelectric phase transition under light is beyond the scope of current work, which needs to be further studied.
In our work, we only studied BPVE caused by the electronic shift current mechanism. While recently studies^{51,52,53} showed that excitons can enhance BPVE even above the band gap. The exciton effect on the BPVE in interlayersliding ferroelectric materials is deserved further study, however, the BPVE behavior with the ferroelectric order should not be changed because it is constrained by symmetry.
Discussion
According to the above two specific examples, the inplane BPVE is invariant, while the outofplane BPVE reverses with the change of ferroelectric order in interlayersliding ferroelectric materials. The ferroelectric order and BPVE in bilayer interlayer sliding ferroelectric materials are subject to different symmetry transformation rules, and do not always synchronize. The study of nonsynchronous BPVE with the ferroelectric order in mirror symmetryrelated ferroelectric materials allows the comprehension of the relationship between them in another view.
Moreover, these BPVE characters in 2D interlayersiding ferroelectric materials will lead to distinctive photovoltaic phenomena and applications in photoelectric devices. The inplane BPVE photocurrents in a single crystal of interlayersiding ferroelectric materials are invariant with two opposite ferroelectric domains, thus the photovoltaic currents between different domains superimpose rather than cancel, as shown in Fig. 4. Since ferroelectric domains are often unavoidable in real materials, especially for the interlayersliding ferroelectric materials fabricated via the tearandstack method^{20,21,25,54}. Therefore, this unswitchable BPVE feature boosts the development of largescale photoelectric devices that are immune to the polarization of ferroelectric domains. Nevertheless, the outofplane BPVE inverses between the opposite ferroelectric orders (Fig. 4), which can be utilized to construct switchable photoelectric devices.
In conclusion, we use the abstract bilayer crystal model to analyze the symmetries of the four most common cases of bilayer vdW materials with interlayer sliding and found two cases (Case 1b and Case 2a) can have the possibility to achieve the interlayersliding ferroelectricity, which the interlayersliding ferroelectric materials found in experiments fall into. We revealed that the opposite ferroelectric states in the interlayersliding ferroelectric materials are linked by the horizontal mirror symmetry, leading to the inplane BPVE being invariant while the outofplane BPVE switches with opposite ferroelectric states. BPVE in all directions of interlayersliding ferroelectric materials are not flipped synchronously with the ferroelectric orders, which is different from the scenario occurs in traditional ferroelectric materials. Our theoretical study provides a strategy for a comprehensive understanding of the relationship between the ferroelectric order and BPVE in 2D interlayersliding ferroelectric materials. Moreover, the switchable outofplane BPVE can be used to design switchable photoelectric devices. On the other hand, the inplane BPVE is robust against the ferroelectric order, and the unswitchable character is beneficial to construct largerscale photoelectric devices.
Methods
Inplane and outofplane BPVE theory
According to the secondorder Kubo formalism theory^{33,34}, \(\sigma _{bc}^a\) in the Eq. (1) of the main text can be expressed as
where
\(\delta = \hbar /\tau\), and τ is the lifetime, ω is the frequency of the light, \(f_{lm}\) and \(E_{ml}\) are the difference of occupation number and band energy between bands l and m.
Within a modified secondorder Kubo formalism theory^{32}, the outofplane BPVE coefficient in Eq. (2) of the main text can be expressed as
and
where \((p_z)_{nm} = \left\langle n \right\hat p_z\left m \right\rangle\) is the outofplane dipole matrix element, and \(\hat p_z\) is the outofplane dipole operator. The interatomic electric dipoles due to the overlapping of wave functions should be very small, which was ignored in our calculation. \(\hat p_z\) contributed by interatomic electric dipole only has nonzero diagonal components which can be expressed as
where \(r_{i,z}\) is the zcomponent position of the ith atom. We define the middle of the 2D materials as the origin: \(\mathop {\sum}\limits_i {r_{i,z}} = 0\). Equation (6) is very similar to Eq. (8), except for the position operator \(\hat p_{i,z}\) instead of the velocity operator.
Firstprinciples calculations
The firstprinciples calculations based on density functional theory (DFT) are performed by using the VASP package. General gradient approximation (GGA) according to the Perdew–Burke–Ernzerhof (PBE) functional is used. The energy cutoff of the plane wave basis is set to 400 eV. The Brillouin zone is sampled with a 12 × 12 × 1 (12 × 8 × 1) mesh of kpoints for MoS_{2} (WTe_{2}). To simulate the monolayers, vacuum layers (~15 Å) are introduced. The vdW force with DFTD2 correction is considered in the main text. The calculation results and corresponding discussions using vdW force with DFTD3 correction are shown in Supplementary Note 6. Spinorbital coupling effects are considered for bilayer MoS_{2} and WTe_{2} in the band structure and BPVE calculations.
The DFT Bloch wave functions are iteratively transformed into maximally localized Wannier functions by the Wannier90 code^{55,56}. Mod and Sp (Wd and Tep) orbitals are used to construct the Wannier functions for MoS_{2} (WTe_{2}). The inplane and outofplane BPVE coefficients are calculated by our own program WNLOP (Wannier Nonlinear Optics Package) based on effective tightbinding (TB) Hamiltonian. A convergence test of kmesh is performed, and 500 × 500 × 1 (600 × 300 × 1) kmesh is sufficient in BPVE calculations of MoS_{2} (WTe_{2}). We adopt δ = 0.02 eV for Eq. (6) and Eq. (8) in our calculation to take into account various relaxation processes.
The 3Dlike BPVE coefficients are obtained assuming an active singlelayer with a thickness of \(L_{{{{\mathrm{active}}}}}\)^{13}
where \(\sigma _{{{{\mathrm{slab}}}}}\) is the calculated BPVE coefficient, and \(L_{{{{\mathrm{slab}}}}}\) (\(L_{{{{\mathrm{active}}}}} \,<\, L_{{{{\mathrm{slab}}}}}\)) is the effective thickness slab.
Data availability
The authors declare that all source data supporting the findings of this study are available within the article and the Supplementary Information file.
Code availability
The calculating codes are available from the corresponding authors upon reasonable request.
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Acknowledgements
The authors acknowledge the Highperformance Computing Platform of Anhui University for providing computing resources. We thank Bin Xu, DingFu Shao, YuanJun Jin, and Xue Liu for their useful discussions. This work is supported by the National Natural Science Foundation of China under No. 11947212, Natural Science Foundation of Anhui Province under No. 110158162022, in part by the Joint Funds of the National Natural Science Foundation of China and the Chinese Academy of Sciences LargeScale Scientific Facility under Grant No. U1932156, and in part by the Natural Science Foundation of Anhui Province under Grant No. 2008085QA29. Y.G. and R.C.X. acknowledge the startup foundation from USTC and AHU, respectively.
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R.C.X. conceived the project with H.L. and edited the code and performed the firstprinciples calculations. R.C.X. performed the symmetry analysis and Y.G. checked it. R.C.X., Y.G., and H.L. discussed the results and the writing. The manuscript was written through the contributions of all authors. All authors have approved the final version of the manuscript.
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Xiao, RC., Gao, Y., Jiang, H. et al. Nonsynchronous bulk photovoltaic effect in twodimensional interlayersliding ferroelectrics. npj Comput Mater 8, 138 (2022). https://doi.org/10.1038/s41524022008281
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DOI: https://doi.org/10.1038/s41524022008281