Large piezoelectric response in ferroelectric/ multiferroelectric metal oxyhalide MOX 2 (M = Ti, V and X = F, Cl and Br) monolayers †

Flexible two-dimensional (2D) piezoelectric materials are promising for applications in wearable electro-mechanical nano-devices such as sensors, energy harvesters, and actuators. A large piezo-response is required for any practical applications. Based on ﬁ rst-principles calculations, we report that ferroelectric TiOX 2 and multiferroelectric VOX 2 (X = F, Cl, and Br) monolayers exhibit large in-plane stress ( e 11 ) and strain ( d 11 ) piezoelectric coe ﬃ cients. For example, the in-plane piezo-response of TiOBr 2 (both e 11 = 28.793 × 10 − 10 C m − 1 and d 11 = 37.758 pm V − 1 ) is about an order of magnitude larger than that of the widely studied 1H-MoS 2 monolayer, and also quite comparable to the giant piezoelectricity of group-IV monochalcogenide monolayers, e.g. , SnS. Moreover, the d 11 of MOX 2 monolayers – ranging from 29.028 pm V − 1 to 37.758 pm V − 1 – are signi ﬁ cantly higher than the d 11 or d 33 of commonly used 3D piezoelec-trics such as w-AlN ( d 33 = 5.1 pm V − 1 ) and α -quartz ( d 11 = 2.3 pm V − 1 ). Such a large d 11 of MOX 2 monolayers originates from low in-plane elastic constants with large e 11 due to large Born e ﬀ ective charges ( Z ij ) and atomic sensitivity d u d η (cid:1) (cid:3) to an applied strain. Moreover, we show the possibility of opening a new way of controlling piezoelectricity by applying a magnetic ﬁ eld.


Introduction
Insulators or semiconductors that lack inversion symmetry exhibit a piezoelectric effect, which is an electromechanical coupling that allows energy conversion from mechanical to electrical, and vice versa.This effect is used in many important applications such actuators, sensors, and transducers. 1,2urrent trends in the miniaturization of devices require piezoelectricity at the nanoscale.Being at most a few atomic-layers thick, 2D piezoelectrics have potential for miniaturizing these electromechanical devices down to nanoscale.Moreover, compared with 3D piezoelectrics (e.g., bulk crystals or thin-films), few layered (typically 1-3 layers) piezoelectric materials can generally exhibit larger deformation. 1,2Importantly, nowadays these 2D materials can be grown with good crystalline quality.Hence, 2D piezoelectrics become promising for self-powered, flexible, and wearable nano-devices.These 2D piezoelectrics can also find interesting applications in new types of electronics such as piezotronics 1 where the electronic band gap is controlled by the electric potential stemming from piezoelectricityand in piezo-photonics, 2 where light is coupled with the piezoelectrically induced charges.For example, it has been predicted that the performance of MoS 2 -based solar cells can be enhanced by the coupling of semiconducting and piezoelectric properties. 3uite often, reduction in materials dimension promotes unique properties.For example, bulk 2H-MoS 2 is a non-piezoelectric due to its centrosymmetry, whereas the monolayer (also odd numbered layers e.g., trilayer) has no inversion symmetryand exhibits piezoelectric properties. 4In agreement with the theory, 4 in-plane piezoelectricity in a 1H-MoS 2 monolayer, which is comparable to the piezo-response of commercially used wurtzite nitrides, e.g., the d 33 of w-AlN (5.1 pm V −1 ), has been confirmed by recent experiments. 5However, generally speaking, a high piezo-response in these 2D materials is desired for any device-level applications.Therefore, enhancement of piezoelectricity and discovery of new 2D piezoelectrics have drawn significant research interest.Typically, 1H-type (D 3h symmetry) [6][7][8][9] (e.g., d 11 = 13.45 pm V −1 for 1H-CrTe 2 ) 6 and Janus 1T-type [10][11][12] (e.g., d 22 = 4.12 pm V −1 for 1T-MoSSe) 10 2D materials have been investigated for a large piezoelectric response.1,[16][17][18] Encouragingly, although 2D ferroelectrics are relatively rare to date, giant in-plane piezoelectric response is present in the ferroelectric monolayers of group-IV monochalcogenides 19 and MXenes (e.g., Sc 2 CS 2 ). 20A huge out-of-plane piezo-response (d 33 = 172.61pm V −1 ) is observed in buckled monolayers. 21Ferroelectric In 3 Se 3 nanoflakes 22 also show a moderate out-of-plane piezo-response.Furthermore, the co-existence of piezoelectricity and magnetism and their coupling in 2D materialsnamely vanadium dichalcogenide monolayers, 23 Janus ferromagnetic NiClI monolayers, 11 and 1H-LaBr 2 monolayers 24 have been investigated.Any strong coupling between piezoelectricity and magnetism can be utilized for making piezoelectric-based multifunctional nano-devices.In this regard, multiferroelectric materials are interesting because they usually exhibit good coupling between electric polarization and magnetic order.Piezoelectricity is linked with electric polarizationfor instance, the piezoelectric stress co-efficient (e ij ) is defined as @P i @η j ; where strain ∂η j along the j-direction induces polarization along the i-direction (∂P i ).However, how changes in the magnetic order will change the piezo-response in 2D multiferroelectricswhere polarization couples with the magnetic orderremains unanswered to the best of our knowledge.
Based on first-principles calculations, several approaches such as defect engineering, 25 doping/charging, 26 and chemical functionalization 27,28 have been proposed for combining ferroelectricity and magnetism in 2D materials.There are also a limited number of intrinsically multiferroelectric 2D materials discovered recentlyincluding the metal phosphorus chalcogenides family, 29,30 buckled CrN and CrB 2 monolayers, 31 and MXene Hf 2 VC 2 F 2 monolayers. 32Multiferroelectricity in the monolayers of the metal oxyhalide VOX 2 family [33][34][35][36][37] has been predicted with interesting violation of the d 0 rule. 38In VOX 2 monolayers, the ferroelectric polarization direction is perpendicular to the partially occupied d xy orbital that is the origin of magnetism.As a result, the partially occupied d orbital does not suppress the ferroelectric atomic displacement.Moreover, compared to ferroelectric TiOX 2 monolayers with the empty d orbital, the presence of an electron in the d xy of VOX 2 monolayers rather positively contributes to the total electric polarization. 34Initially, the ground state of the VOI 2 monolayer was predicted as ferromagnetic and ferroelectric. 34However, later it has been predicted that the ferroelectric VOI 2 monolayer can exhibit spiral magnetism for a short period due to iodine's strong (compared to other halogens) spin-orbit coupling (SOC). 36,37Alternatively, ferroelectricity in the VOI 2 monolayer can also be suppressed by on-site strong Coulomb interaction making it a ferromagnetic metal. 37The coexistence of ferroelectricity and ferromagnetism is predicted in the VOF 2 monolayer, 35 whereas VOCl 2 and VOBr 2 monolayers have a ferroelectric ground state with antiferromagnetic (AFM) spin order. 33,34ote that the VOCl 2 monolayer can be exfoliated experimentally from its bulk layered van der Waals structure (space group: Immm). 33Generally, ferroelectric materials exhibit good piezo-electricity.Although ferroelectricity and multiferroelectricity of MOX 2 monolayers have been investigated, [33][34][35][36][37] their piezoelectric properties remain unknown to date.In this paper, we investigate the piezoelectric properties of both TiOX 2 and VOX 2 monolayers and how the piezo-response changes with magnetic order, which remain unexplored to date.We find that these monolayers exhibit a remarkably large piezo-response compared to most of the known 2D piezoelectrics, and they are promising materials for nanoscale electromechanical applications.

Computational details
Our first-principles calculations are performed in the framework of spin-polarized density functional theory as implemented in the Vienna Ab initio Simulation Package (VASP) based on a plane-wave basis set. 39The projector augmented wave (PAW) potentials 40 are used for describing the core electrons.The generalized gradient approximation (GGA) of Perdew, Burke, and Ernzernhof (PBE) 41 is employed for treating the exchange and correlation.The valence electron configurations considered for Ti, V, O, F, Cl, and Br are 3d 3 4s 1 (4 electrons), 3d 4 4s 1 (5 electrons), 2s 2 2p 4 (6 electrons), 2s 2 2p 5 (7 electrons), 3s 2 3p 5 (7 electrons), and 4s 2 4p 5 (7 electrons), respectively.A cutoff energy of 500 eV is used for the planewave expansion in all calculations.All structures are fully relaxed until the Hellmann-Feynman forces on all the atoms are less than 10 −3 eV Å −1 .The lattice parameters a and b are relaxed, keeping c fixed as required for 2D materials, and the internal coordinates of the 2D structures are fully relaxed to achieve the lowest energy configuration using the conjugate gradient algorithm.To prevent the interaction between the periodic images in the calculations, a vacuum layer with a thickness of approximately 25 Å is added along the z-direction ( perpendicular to the monolayer) in the supercell.Note that previous reports [33][34][35] employed about 15-20 Å vacuum layers, and also considered the van der Waals interaction between the layers. 33,35However, we have not considered the van der Waals interaction as we simulate an isolated monolayer.The convergence for the total energy is set as 10 −7 eV.For a 1 × 1 × 1 unit cell, the Brillouin zone integration is sampled using a regular 12 × 12 × 1 Monkhorst-Pack k-point grid for geometry optimizations, while a denser grid of 18 × 18 × 1 is used for density functional perturbation theory (DFPT) calculations.To study magnetic ordering, 1 × 2 × 1, 2 × 1 × 1, and 2 × 2 × 1 VOX 2 supercells (shown in Fig. 1(c)) with 12 × 6 × 1, 6 × 12 × 1, and 6 × 6 × 1 Monkhorst-Pack k-point grids, respectively are used.The elastic stiffness coefficients (C ij ) are obtained using a finite difference method as implemented in the VASP code.DFPT is used to calculate the Born effective charges (Z ij ) and ionic and electronic parts of piezoelectric (e ij ) tensors.A 4 × 4 × 1 supercell is used for the phonon dispersion calculations of the monolayers, which is obtained with PHONOPY code 42 using the DFPT method.Recently it has been found that the Hubbard effective U (U eff ) correction does not alter the magnetic and ferroelectric properties of VOF 2 . 35However, to confirm the lack of impact of the Hubbard + U correction on the piezoelectric response of VOBr 2 , we apply the GGA + U eff (U eff ranging from 1 eV to 3 eV) approach 43 for the 3d orbitals of V. We find that the Hubbard U eff correction increases both e 11 and d 11 (see the ESI †).This further supports our conclusion that VOBr 2 has a large piezoelectric response.

Results and discussion
We start with the fully optimized centrosymmetric paraelectric (and also ferromagnetic for VOX 2 ) phase (space group: Pmmm) of MOX 2 monolayers and calculated their phonon dispersion.We find that there is an imaginary (soft) optical vibration mode at the center of the Brillouin zone (Γ-point) for the PE phase (see Fig. 1(a) for TiOBr 2 and also the ESI † for other MOX 2 monolayers).The frequency (iω Γ) associated with the polar soft mode is given in Table 1.This suggests that there is a spontaneous atomic displacement of Ti(V) along the Ti-O (V-O) chain, breaking the inversion symmetry, thus producing a spontaneous in-plane (along the a-direction) electric polarization.This can also be understood in terms of long and short Ti-O (V-O) bonds along the a-direction in the FE phase (space group: Pmm2), whereas all M-O bonds are the same in the PE phase.Therefore, the a lattice parameter of the FE phase becomes slightly larger than that of the PE phase, although the b lattice parameter remains almost unchanged (see Table 1).As Ti 4+ and V 4+ have almost the same ionic radii, their lattice parameters are close.We see an increase in b as the radius of X increases from F to Br, which is expected because X atoms are only along the b-direction.With the exception of TiOF 2 (FE), the phonon dispersion of FE MOX 2 monolayers shows no appreciable soft mode, indicating their stability.Interestingly, we find that the ground state of the TiOF 2 monolayer is not the FE (Pmm2) phaserather the nonpolar (Pmma) phase (see the ESI †), which is 26.620 meV per atom lower in energy than the FE phasetherefore, we will not discuss its properties in the main paper.
As electric polarization (P 1 ) arises due to the polar distortion from the PE phase, we calculated P 1 in the 2D unit (C m −1 ) using Z 11 and the atomic displacement (Δu k,1 ) of the k-th atom along the a-direction as P  find anomalously large Z 11 , which decreases after the PE-to-FE transition (see Table 1).Similar anomalous BECs have been observed for other well-known ferroelectric materials. 44In agreement with previous reports, 35 we find that MOX 2 monolayers have quite large P 1 , which is comparable with that of group-IV monochalcogenide orthorhombic monolayers, e.g., SnS (P 1 = 2.47 × 10 −10 C m −1 ). 45,46We also estimated the energy barrier for FE polarization switching.We take the difference in energy (ΔE) between FE and PE phases; lattice parameters a and b are fully relaxed in both phases.Our ΔE values are in good agreement with the reported values.We see a general trend that P 1 , ΔE, and iω Γ decrease as the ionic radius of X increases from F to Br.Interestingly, we also observed that magnetic VOX 2 monolayers have significantly larger P 1 than non-magnetic TiOCl 2 or TiOBr 2 .This is in line with the previous report that the presence of an electron in the d xy orbital of V does not suppress but rather enhances ferroelectric polarization. 34This is also confirmed by the larger iω Γ of VOX 2 (see Table 1).
To examine the impact of magnetic configuration on VOX 2 monolayers, we consider four (1 FM and 3 AFM) collinear magnetic spin configurations (see Fig. 1(c) and Table 2).Each V 4+ contributes 1μ B , which comes from an unpaired electron in the d xy orbital. 34Comparing the energy difference of an AFM configuration with respect to the FM order, in agreement with previous reports, we find that the FE VOF 2 monolayer has an FM ground state. 35However, we find that the AFM3-type AFM order (see Fig. 1(c)) is more stable than other configurations in VOCl 2 and VOBr 2 monolayers. 33,34The alternating up and down collinear spin configuration of V atoms along the b-direction (see Fig. 1(c)) shortens the b lattice parameter, compared with that of the FM state (shown in Table 1).By applying an external magnetic field in an experiment, the AFM order can be changed to FM.This will also lead a change in P 1 with a reduction of 7.49% and 6.13% for AFM1-to-FM transition in VOCl 2 and VOBr 2 monolayers, respectively.The AFM1-to-FM transition slightly hardens the soft mode (iω Γ ; see Table 1; also see the ESI † for AFM3), and consequently reduces the ferroelectric switching barrier (ΔE) slightly.This indicates that there is a weak coupling between the magnetic and ferroelectric orders in VOX 2 .
All ferroelectrics exhibit piezoelectricity.It is interesting to know the piezo-response of our FE MOX 2 monolayers as strong in-plane piezoelectricity has already been predicted in 2D FE group-IV monochalcogenides. 19Our calculated piezoelectric stress coefficients (e ij ) are shown in Table 3. e ij are important coefficients for estimating the figure-of-merit of a piezoelectric thin-film (TFFOM); usually the larger the e ij , the higher the figure-of-merit.Because strain along the z-direction (vacuum) is ill-defined in 2D materials, we have only three independent piezoelectric coefficients: e 11 , e 12 , and e 16 .There is a mirror symmetry along the b-direction, which does not allow any polarization in that direction, thus e 22 = 0.However, strain along the b-direction can induce polarization along the a-direction, which results in a non-zero e 12 coefficient.The FE MOX 2 monolayer (space group: Pmm2) due to the mm2 point group has a symmetry of reflection with reference to the M-O atomic plane.This prohibits an out-of-plane electric polarization, thus e 31 = 0. We mainly focus on the piezo-response related to uniaxial strain along the a-direction (η 1 ) and the b-direction (η 2 ), which are e 11 (d 11 ) and e 12 (d 12 ), respectively.e 16 is associated with shear strain (η 12 ), 19 and we exclude it for simplicity.
Table 1 Structural information of the monolayers: optimized lattice parameters (a and b; see the rectangular cells in Fig. 1).M-O (M-X) represents the bond length between metal (M) and oxygen (halogen; X) atoms.Z 11 is the Born effective charge in |e| unit.The values in the parentheses are for paraelectric phases.P 1 and ΔE are the in-plane electric polarization in 2D unit (C m −1 ) and the energy difference between the ferroelectric and paraelectric phases (the positive ΔE value suggests that the FE phase is lower in energy compared to the PE phase).iω Γ stands for the lowest imaginary frequency of the PE phase at the Γ-point Table 3 shows that the TiOCl 2 or TiOBr 2 monolayer has quite large e 11 but small e 12 , compared to those of VOX 2 monolayers.We also notice that unlike 1H-type monolayers, e.g., 1H-MoS 2 where e 11 = −e 12 due to the 6 ˉm2 point group symmetry, MOX 2 monolayers exhibit a highly anisotropic piezoresponse, where e 11 is significantly larger than e 12 .This is also expected as the monolayers have a strong in-plane electric polarization P 1 , hence atomic displacement in response to strain along the a-direction can change P 1 directly.Interestingly, we observe a general trend that the in-plane piezo-response (e 11 ) decreases as the in-plane polarization increases (see Tables 1 and 3).To understand the origin of the large/small piezoelectric constant, we split e 11 and e 12 into two terms -(i) the clamped-ion term (e elc 11 or e elc 12 ), which is the electronic contribution where the atoms are fixed at their equilibrium internal coordinates (u) and (ii) the ionic contribution term (e ion 11 or e ion 12 ), due to the atomic displacements in response to a macroscopic strain η 1 (η 2 ) along the a-direction (b-direction).The e ion 11 of TiOCl 2 and TiOBr 2 monolayers is almost twice larger than that of VOX 2 .Interestingly, we notice that both ionic and electronic parts of e 11 are positive (see Table 3), thus they contribute positively to the total e 11similar to 1H-MoS 2 . 24That is why the TiOCl 2 or TiOBr 2 monolayer has significantly large e 11 , compared to that of VOX 2 , although the e elc 11 of TiOCl 2 and TiOBr 2 monolayers is slightly smaller than that of VOX 2 .On the other hand, the ionic and electronic parts of e 12 are opposite in sign, hence they reduce the total e 12 .We see that because of their small positive e elc 12 but large negative e ion 12 , TiOCl 2 and TiOBr 2 monolayers have quite small e 12 (see Table 3).We further split the ionic part: e ion 11 or e ion 12 involves summation running over all the atoms (k) in a cell, e is the charge of an electron, and A is the area of the cell of the 2D unit.The response of the k-th atom's internal coordinate along the a-direction (u 1 (k)) in response to a macro-scopic strain (η 1 ) in the same direction is measured by du 1 ðkÞ dη 1 .
Similarly, du 1 ðkÞ dη 2 represents the change in the k-th atom's internal coordinate along the a-direction (u 1 (k)) in response to a macroscopic strain (η 2 ) along the b-direction.Relaxing the atomic positions in response to the strains η 1 and η 2 , we obtain the slopes du 1 ðkÞ dη 1 and du 1 ðkÞ dη 2 , respectively.We notice that the large e ion 11 of TiOCl 2 and TiOBr 2 monolayers comes from their large Z 11 (see Table 1) and du 1 dη 1 (see Table 3).Also, we see that the du 1 dη 1 of Ti/V/O is an order of magnitude larger than du 1 dη 2 i.e., the uniaxial strain η 1 can displace atoms along the a-direction more than η 2 .This also gives the large difference between e ion 11 and e ion 12 .Moreover, we observe that the AFM1 order of VOCl 2 and VOBr 2 marginally enhances e 11 because of a slight increase in both e elc 11 and e ion 11 (see Table 3).Note that a change in the magnetic order also changes the e ij of other magnetic 2D piezoelectrics. 24Piezoelectric constants for AFM3 of VOCl 2 and VOBr 2 are presented in the ESI †.
Note that the e 11 of MOX 2 monolayers is significantly (about 6-10 times) larger than that of the well-known 1H-type piezoelectric monolayers e.g., 1H-MoS 2 (e 11 = 3.64 × 10 −10 C m −1 ). 4,6,8We notice that the e ion 11 of MOX 2 monolayers is an order of magnitude larger than that of 1H-MoS 2 or 1H-VS 2 , 24 although their electronic parts are quite comparable. 24Both the Z 11 and du 1 dη 1 of MOX 2 monolayers are remarkably higher than those of 1H-MoS 2 or 1H-VS 2 . 24Our e 11 is quite comparable with that of group-IV monochalcogenide monolayers such as SnS, 19 although the difference between e 11 and e 12 in group-IV monochalcogenides is not as pronounced as in MOX 2 monolayers.Note that the large piezo-response of our MOX 2 is very similar to that of ferroelectric niobium oxyhalide monolayers. 49r piezoelectric thin-film-based applications, e 2 11 ε 0 ε 11 , where ε 0 and ε 11 are the vacuum permittivity and static dielectric constant, respectively, is a key figure-of-merit (TFFOM). 49A recent high-throughput calculation has found that niobium oxyhalide Table 3 The electronic (e elc 11 and e elc 12 ) and ionic (e ion 11 and e ion 12 ) parts of the total piezoelectric stress constants e 11 and e 12 in the 2D piezoelectric unit of 10 −10 C m −1 of the MOX 2 monolayers, and the Born effective charges (Z 11 ) of metals (Ti and V), O, and halogens (X = F, Cl, and Br) in the charge of an electron (|e|) unit.du 1 dη 1 or du 1 dη 2 represents the change of the atomic coordinates along the a-direction in response to a strain along the a-direction  49 We find that the TFFOM of MOX 2 monolayers is remarkably higher than that of niobium oxyhalide monolayers, 49   ). 49Interestingly, in comparison to bulk piezoelectric materials, we find that the piezo-response of MOX 2 monolayers is remarkably strong.For example, the d 11 (37.758pm V −1 ) of TiOBr 2 is an order of magnitude larger than that of α-quartz (d 11 = 2.3 pmV −1 ) 52 or the d 33 of w-GaN (3.1 pm V −1 ); 53 and also about 7 times higher than the d 33 of w-AlN (5.1 pm V −1 ). 53Note that group-IV monochalcogenide monolayers 19  As VOCl 2 and VOBr 2 monolayers have an AFM ground state, we also study how their piezo-response will change in response to the AFM-to-FM phase transition, which can be experimentally possible under an external magnetic field. 54Note that the FM-to-AFM transition can be a challenge in experiments.We find that the AFM1-to-FM transition somewhat increases the elastic constantsespecially C 22thus slightly decreases d 11 .Interestingly, such hardening of C 22 is intrinsic to the AFM1-to-FM transition as we see that it comes from the mere magnetic order change even if the lattice parameters and atomic positions are fixed at AFM1 (see Table 4).There is a significant decrease in d 12 (see Table 4).Interestingly, d 12 changes its sign during the AFM1-to-FM transition for the VOBr 2 monolayer, indicating that subject to an external electric field the monolayer can shrink or expand depending on the presence of a magnetic field.This can allow us to control piezoelectricity by magnetism, which may find applications in realizing multifunctional nano-devices.We believe that other magnetic piezoelectrics, especially 2D multiferroelectric, can also exhibit such coupling between piezo-response and magnetic order.

Conclusion
Our first principles calculations demonstrate that FE MOX 2 monolayers have a strong in-plane piezoelectric response, which is not only significantly larger than that of the wellknown 1H-type 2D piezoelectricse.g., both the e 11 and d 11 of MOX 2 are about an order of magnitude larger than those of 1H-MoS 2but also remarkably stronger than some of bulk piezoelectrics such as w-AlN or w-GaN.These monolayers also exhibit a remarkably large anisotropy in their piezo-responsei.e., piezo-response due to strain along the a-direction is about an order of magnitude larger than that of along the b-direction.We also show that a change in the magnetic order can change the piezo-response in multiferroelectric VOX 2 monolayers, which can potentially couple piezoelectricity and magnetism.We believe that this work will inspire more research in searching for new piezoelectric materials that can couple strongly with magnetism.Also, such a large in-plane piezoresponse can particularly be beneficial for 2D nanoscale flexible piezo-devicese.g., actuators purely based on in-plane displacement.

Fig. 1
Fig. 1 As a representative of MOX 2 monolayers, top and side views of the TiOBr 2 monolayer in (a) the paraelectric and (b) ferroelectric phases are shown.Beside the structure, the phonon band structure is also shown.We see an imaginary phonon (soft) mode at the Γ-point for paraelectric TiOBr 2 ; the vibration mode is indicated by the black arrow, whereas yellow arrows represent the direction of atomic displacement associated with the imaginary mode.In the phonon band structure, Γ(0,0,0), X(1/2,0,0), S(1/2,1/2,0), and Y(0,1/2,0) are the high symmetric points in the Brillouin zone.Blue, red, and green balls represent Ti/V, O, and F/Cl/Br, respectively.(c) The four magnetic configurations for VOX 2 monolayers are shown; yellow arrows represent the collinear spin direction (up or down).The dashed lines represent the rectangle simulation cells.
in the range of 105.43 nN for VOBr 2 -203.43 nN for TiOBr 2 , indicating their potential for flexible piezoelectric nano-devices.The TFFOMs of TiOCl 2 , VOF 2 , and VOCl 2 are 201.57nN, 187.57nN, and 118.47 nN, respectively, which are huge compared to the TFFOM of 1H-MoS 2 (3.45 nN; note that our calculated ε 11 of 1H-MoS 2 is 4.51, which is consistent with the previous report of 4.20 50 ).Such high TFFOMs of MOX 2 monolayers are the result of their low dielectric constants (ε 11 ) and large e 11 values.The ε 11 values of TiOCl 2 , TiOBr 2 , VOF 2 , VOCl 2 , and VOBr 2 are 3.82, 4.60, 2.32, 2.42, and 2.80, respectively, whereas the ε 11 values of niobium oxyhalide monolayers are in the range of 12-15.49Our piezoelectric strain constants (d ij )another important figure of merit for many piezoelectric applicationsare obtained using e ij and elastic constants (C ij ) (see Table4

): d 11 ¼ 12 .
C 22 e 11 À C 12 e 12 C 11 C 22 À C 2 12 and d 12 ¼ C 11 e 12 À C 12 e 11 C 11 C 22 À C 2 The nonzero and independent C ij in the Voigt notation of FE MOX 2 monolayers are given in Table 4, and they also are positive (i.e., C 11 , C 22 , C 12 , and C 66 > 0), indicating their mechanical stability; our orthorhombic monolayers clearly satisfy the Born elastic stability criterion: 51 C 11 C 22 − C 12 2 > 0. Unlike 1H-type monolayers, MOX 2 are anisotropic elastically (i.e., C 11 ≠ C 22 -Young's modulus (Y) and Poisson's ratio (ν) along the a-direction are also different from those along the b-direction; these are presented in the ESI.† Note that Y quantifies how easily a material can be stretched and deformed, whereas ν quantifies the deformation in the material in a direction perpendicular to the applied force's direction).We find large d 11 for MOX 2 monolayersand small d 12 .However, the d 12 of TiOCl 2 or TiOBr 2 is quite comparable with that of 1H-MoS 2 (3.73 pm V −1 ) 4 or 1H-VS 2 (4.104 pm V −1 ). 24TiOBr 2 has the largest d 11 (37.758pm V −1 ), which is 2-10 times larger than those of 1Htype monolayers 4,6,8 (e.g., d 11 of 1H-MoS 2 and 1H-CrTe 2 is 3.65 pm V −1 and 13.45 pm V −1 , respectively 6 ).This is because compared to 1H-type piezoelectrics, MOX 2 have significantly larger e 11 and relatively smaller elastic constants (e.g., the C 11 of 1H-MoS 2 is 130 N m −14 ).Note that the d 11 of MOX 2 is very similar to that of niobium oxyhalide monolayers (27.4 pm V −1 to 42.20 pm V −1

Table 2
Energy difference (ΔE AFM = E AFM − E FM ; E AFM and E FM are the energy per unit-formula of fully-relaxed structures in AFM and FM magnetic orders, respectively) in meV per unit formula of 3 magnetic configurations with respect to the FM order; negative means the AFM configuration is more stable than the FM order η 1 ) or the b-direction (η 2 ), respectively have a significantly large TFFOM (in the range of 59.60 nN-71.70 nN) compared to other 2D piezoelectrics (e.g., the TFFOM of CuInP 2 Se 6 is 3.10 nN). monolayers 19ve relatively smallerindicating their softness -C 11 and C 22 (e.g., C 11 = 20.87Nm−1 and C 22 = 53.40Nm−1forGeS monolayer19) than MOX 2 monolayers.That is why group-IV monochalcogenide monolayers have larger d 11 (e.g., d 11 = 75.43pmV−1 of the GeS monolayer)19than that of MOX 2 .

Table 4
Elastic constants (C 11 , C 22 , C 12 , and C 66 ) in the 2D unit of N m −1 and the piezoelectric strain coefficient in d 11 and d 12 in pm V −1 .*VOCl 2 (FM) and *VOBr 2 (FM) represent the structures with the FM order but their lattice parameters and atomic positions are fixed at those of their AFM1 configurations