Negative Piezoelectric Coefficient in Ferromagnetic 1H-LaBr2 Monolayer

The discovery of two-dimensional (2D) magnetic materials that have excellent piezoelectric response is promising for nanoscale multifunctional piezoelectric or spintronic devices. Piezoelectricity requires a noncentrosymmetric structure with an electronic band gap, whereas magnetism demands broken time-reversal symmetry. Most of the well-known 2D piezoelectrics, e.g., 1H-MoS2 monolayer, are not magnetic. Being intrinsically magnetic, semiconducting 1H-LaBr2 and 1H-VS2 monolayers can combine magnetism and piezoelectricity. We compare piezoelectric properties of 1H-MoS2, 1H-VS2, and 1H-LaBr2 using density functional theory. The ferromagnetic 1H-LaBr2 and 1H-VS2 monolayers display larger piezoelectric strain coefficients, namely, d11 = −4.527 pm/V for 1H-LaBr2 and d11 = 4.104 pm/V for 1H-VS2, compared to 1H-MoS2 (d11 = 3.706 pm/V). 1H-MoS2 has a larger piezoelectric stress coefficient (e11 = 370.675 pC/m) than 1H-LaBr2 (e11 = −94.175 pC/m) and 1H-VS2 (e11 = 298.100 pC/m). The large d11 for 1H-LaBr2 originates from the low elastic constants, C11 = 30.338 N/m and C12 = 9.534 N/m. The sign of the piezoelectric coefficients for 1H-LaBr2 is negative, and this arises from the negative ionic contribution of e11, which dominates in 1H-LaBr2, whereas the electronic part of e11 dominates in 1H-MoS2 and 1H-VS2. We explain the origin of this large ionic contribution of e11 for 1H-LaBr2 through Born effective charges (Z11) and the sensitivity of the atomic positions to the strain (du/dη). We observe a sign reversal in the Z11 values of Mo and S compared to the nominal oxidation states, which makes both the electronic and ionic parts of e11 positive and results in the high value of e11. We also show that a change in magnetic order can enhance (reduce) the piezoresponse of 1H-LaBr2 (1H-VS2).


■ INTRODUCTION
Piezoelectric materials are used in a wide range of important devices such as microphones, medical imaging, and sensors. 1,2 Recently it has been demonstrated that the piezopotential originating from piezoelectricity can be used as a gate voltage to control the electronic band gap of a piezoelectric semiconductor, opening a new field of research named "piezotronics". 1,2 In this regard, 2D semiconductors are promising materials as they can sustain the large deformations present in piezoelectric applications. 1,2 Moreover, these 2D materials show unique optical properties, for example, valleytronics. 3,4 Hence, 2D materials can be ideal for piezophotonics where charges stemming from the piezoelectric effect can couple with light to significantly modulate the charge-carrier generation, separation, transport, and/or recombination in semiconducting nanostructures, promising better LEDs, photodetectors, and solar cells. 1,2 Piezoelectricity and valleytronics require broken inversion symmetry and a band gap. Promisingly, there already exists a wide range of noncentrosymmetric and intrinsically piezoelectric 2D materials. 3,4 On the other hand, there are only a few 2D semiconductors/insulators to date in which both time-reversal and inversion symmetry are broken. These noncentrosymmetric magnetic 2D materials, e.g., vanadium dichalcogenide monolayers, 5 exhibit spontaneous valley polarization, which can be controlled by a magnetic field. 3 Very recently, the coexistence of magnetism and piezoelectricity has also been predicted in vanadium dichalcogenide monolayers. 6 However, how the magnetic ordering impacts on their piezoelectricity remains unexplored. This understanding will allow us to couple magnetism and piezoelectricity for realizing multifunctional piezoelectric devices.
A piezoelectric stress coefficient (e ij ), defined as P i j η ∂ ∂ , where ∂P i is the induced polarization along the i-direction in response to strain ∂η j along the j-direction, can be split into two contributions: the ionic part, e ij ion , where ions are allowed to move under an applied strain, and the electronic part (also known as the clamped-ion part) e ij elc , where ions are clamped under applied strain. In many bulk materials, including wurtzite nitrides, 7,8 e ij elc is negative but is dominated by positive e ij ion , thus resulting in a positive value of e ij . Generally, a positive longitudinal piezoelectric coefficient is expected as a tensile strain is expected to increase the induced electric polarization. However, very recently an anomalous negative piezoelectric coefficient has been observed in the layered ferroelectric CuInP 2 S 6 , 7 which is explained in terms of its large negative e ij elc that is not overcome by positive e ij ion . Also, negative piezoelectric coefficientsdue to their large negative e ij elc 'shave been observed in several hexagonal ABC ferroelectrics. 9 A negative longitudinal piezoelectric coefficient would mean that the material contracts along the direction of an applied electric field rather than expands. This can enable novel nanoscale electromechanical devices, e.g., piezoelectric actuators.
This raises an interesting question: can a negative total e ij be obtained due to large negative e ij ion instead of e ij elc ? To answer this question, we investigate three intrinsically piezoelectric monolayers, 1H-MoS 2 , 1H-VS 2 , and 1H-LaBr 2 , and we discover 1H-LaBr 2 as a new 2D piezoelectric monolayer that has a negative piezoelectric coefficient originating from a large negative e ij ion . Being a magnetic, semiconducting electride, 1H-LaBr 2 is a unique monolayer, although it has not been achieved experimentally yet; however, it is predicted to be feasible via chemical exfoliation from its layered bulk structure. 10 It combines peculiar features; for example, its electron density shows neither complete localization at an atomic site nor metallike delocalization, but rather it occupies the center of the hexagon from which originate localized magnetic moments. 11,12 Very recently, it has been predicted that this magnetism can be utilized for valley polarization. 10 However, its piezoelectric properties have not been investigated to date.
Recently a number of 2D materials in the 1H structure (D 3h symmetry) have been predicted to show large piezoelectric cocoefficients. 13−16 These 2D materials still remain at the stage of fundamental research; understanding the origin of piezoelectricity can promote the discovery of more 2D piezoelectrics. Encouragingly, piezoelectricity has also been experimentally confirmed in the 1H-MoS 2 monolayer, 17 and the value e 11 (2.9 × 10 −10 C/m) is in good agreement with first-principles calculations of e 11 = 3.64 × 10 −10 C/m. 18 Recently, the coexistence of magnetism and piezoelectricity has also been predicted in the 1H-VS 2 monolayer, 6 although the coupling between magnetic order and piezoelectricity was not discussed. Note that research on these 1H structured 2D piezoelectrics is mainly devoted to finding large piezoelectric coefficients, overlooking their sign as they generally show positive in-plane piezoelectric coefficients. 6,[13][14][15][16]18 However, the origin of the piezoelectric co-coefficients in both magnitude and sign still remains unclear. Questions include the following: Why is the e 11 of the 1H-MoS 2 monolayer larger than that of the 1H-VS 2 monolayer? Why is the sign of the ionic part of e 11 positive in the 1H-MoS 2 monolayer but negative in the 1H-VS 2 monolayer? In this paper, we show that the answers to these questions have their origin in the Born effective charges (BECs), the sensitivity of the atomic positions in response to a strain ( ) u d dη , and the bond strength. We also demonstrate that the 1H-LaBr 2 monolayer 10−12 can be a magnetic, piezoelectric material. Moreover, we show that antiferromagnetic ordering makes the isotropic piezoelectricity of the ferromagnetic 1H-LaBr 2 monolayer anisotropic (i.e., e 11 ≠ −e 12 ).

■ COMPUTATIONAL DETAILS
Our first-principles calculations are performed in the framework of spin-polarized density functional theory using projector augmented wave (PAW) potentials 19 to describe the core electrons and the generalized gradient approximation (GGA) of Perdew, Burke, and Ernzernhof (PBE) 20 for exchange and correlation as implemented in the Vienna Ab initio Simulation Package (VASP) based on a plane-wave basis set. 21 The valence electron configurations for La, V, Mo, S, and Br are 4p 6 5s 2 6d 1 (nine electrons), 3p 6 3d 4 4s 1 (11 electrons), 4p 6 4d 5 5s 1 (12 electrons), 3s 2 3p 4 (six electrons), and 4s 2 4p 5 (seven electrons), respectively. A cutoff energy of 500 eV for the plane-wave expansion is used in all calculations, and all structures are fully relaxed until the Hellmann−Feynman forces on all the atoms are less than 10 −3 eV/Å. An effective onsite Coulomb interaction parameter (U eff ) of 6.5 eV is used for the La f-electrons. 11 The lattice parameters and 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 zdirection (perpendicular to the monolayer) in the supercell. Note that a rectangular cell (see Figure 1) is used instead of a primitive hexagonal one for applying strain along the desired direction. This is a commonly used approach. 6,18 Geometry optimization is carried out employing the conjugated gradient technique, and the convergence for the total energy is set as 10 −7 eV. The Brillouin zone integration is sampled using a regular 6 × 8 × 1 Monkhorst−Pack k-point grid, for geometry optimizations, while a denser grid of 12 × 16 × 1 is used for density functional perturbation theory (DFPT) calculations. The elastic stiffness coefficients (C ij ) are obtained with a finite difference method as implemented in the VASP code. DFPT as implemented in the VASP code is used to calculate Born effective charges (Z ij ) and ionic and electronic parts of piezoelectric (e ij ) tensors. Table 1 shows the lattice parameters of the monolayers. We use a rectangular unit cell, and lattice parameter a should be equal to b 3 for an ideal 1H structure. Our calculated lattice parameters are in good agreement with previously reported values. 6,10−12,18 ACS Applied Electronic Materials pubs.acs.org/acsaelm Article 1H-LaBr 2 has significantly larger lattice parameters compared to these of the other two monolayersmainly because the ionic radius of La (Br) is larger than that of Mo/V (S) according to the Database of Ionic Radii (http://abulafia.mt.ic.ac.uk/shannon/ ptable.php). However, we notice that strip antiferromagnetic (AFM) ordered structures shown in Figure 1 deviate from the ideal relationship, shrinking along the zigzag, or b-axis, direction and expanding along the armchair, or a-axis, direction. We quantify this deviation as i

■ RESULTS AND DISCUSSION
This is about 1.22% (0.72%) for the AFM 1H-LaBr 2 (1H-VS 2 ) monolayer. This deviation is also reflected in the change in angles θ FM and θ AFM , which are defined in Figure 1.
In agreement with previous reports, 6,10−12 we find that ferromagnetic (FM) ordering is the ground state for both 1H-LaBr 2 and 1H-VS 2 monolayers, lying 51.520 and 88.545 meV lower in energy compared to the strip AFM state. However, the magnetic order of these monolayers has not been clearly identified in experiments to date. Although the VS 2 monolayer has not yet been synthesized, ferromagnetism has been recently found in its ultrathin films. 22, 23 The 1H-LaBr 2 (FM), 1H-VS 2 (FM), and 1H-MoS 2 monolayers belong to the nonmagnetic space group P6m2 (157), that is, considering their structures but without their magnetic order. Unlike the corresponding bulk materials, this structure has no inversion symmetry and is intrinsically piezoelectric. We calculated their piezoelectric stress coefficients, which are shown in Table 2. The piezoelectric coefficients that involve strain along the z-direction are ill-defined for the monolayers. Our 1H monolayers have only one independent piezoelectric coefficient, e 11 (e 11 = −e 12 ), due to 6̅ m2 point group symmetry. Table 2 shows that 1H-MoS 2 has a quite large e 11 value compared to those of the other two monolayers. Interestingly, FM 1H-LaBr 2 shows a negative e 11 which is also quite low compared to those of the other materials. To understand the origin of piezoelectric constant, e 11 and e 12 can be decomposed into two parts: 8 The clamped-ion term (e 11 elc or e 12 elc ) arises from the contributions of electrons when the ions are frozen at their zero-strain equilibrium internal atomic coordinates (u), and the internal-strain (e 11 ion ) term arises from the contribution from internal microscopic atomic displacements in response to a macroscopic strain. In our case, the strain (η 1 ) is applied in the xdirection (see Figure 1). Here, k runs over all the atoms in the unit cell, a is the in-plane lattice constant, e is the electron charge, and A is the area as the 2D unit is used. The Born effective charge (Z 11 (k)) of the kth atom is calculated by the DFPT approach. The response of the kth atom's internal coordinate along the xdirection (u 1 (k)) in response to a macroscopic strain (η 1 ) is measured by u k  Table S1). 18 However, due to the opposite signs of e 11 elc and e 11 ion in 1H-VS 2 (FM), its total e 11 is smaller than the e 11 for 1H-MoS 2 , even though it has a larger value of e 11 elc (see Table 2). Interestingly, 1H-LaBr 2 (FM) shows a negative e 11 ion , which is significantly larger than its e 11 elc , thus resulting in a negative total e 11 . This is different from the recently discovered negative piezoelectric coefficient in layered ferroelectrics and wurtzite, where the negative sign comes from e 11 elc . 7 Here it is important to highlight that other 2D piezoelectrics, e.g., the well-known hexagonal boron nitride (h-BN) monolayer 18 and the 1H-VSe 2 monolayer, 6 have negative e 11 ion values but the e 11 elc part dominates, resulting in positive e 11 . Interestingly, we find that h-AlN and h-ZnO monolayers also exhibit negative e 11 values due to large negative e 11 ion values (see Table S1). We hope that our finding will inspire experimental studies of negative in-plane piezoresponse (e 11 ) in 2D materials. Deepening our understanding about the 2D piezoelectrics, this would enable discovery of 2D materials for novel electromechanical applications. a See the rectangular cell in Figure 1   (see eq 1). Providing microscopic insight into the piezoelectric coefficients, BEC is a dynamical charge that is directly related to the change of electric polarization or dipole moment (for molecules) in response to an atomic displacement. 25 ∂P 1 is the change of the dipole moment in the x-direction induced by a small displacement of atom k in the same direction (∂τ 1 (k)). 25 The negative slope ( ) will result in a negative BEC, which is the case for 1H-MoS 2 . This proportionality (i.e., the slope) is the origin of BECs and has the dimensionality of an electric charge. This charge is a well-defined and experimentally measurable quantityowing to the fact that the BECs are related to LO−TO splitting, which is the frequency difference between the longitudinal (LO) and transverse (TO) optical phonon modes. 25 From Table 2, it is clear that the positive e 11 ion in 1H-MoS 2 is due to unusual BECs of Mo and S as we see a negative (positive) sign for cation Mo (anion S) in the BECs. Such counterintuitive BECsMo (S) shows a negative (positive) dynamical charge, opposite to its static positive chargeare also reported for bulk 2H-MoS 2 . 26 Interestingly, we notice that other good 2D piezoelectric transition metal (Mo, W, and Cr) dichalcogenide (S, Se, and Te) monolayers also exhibit counterintuitive BECs (see Table S1), making their e 11 ion values positive; thus, both e 11 ion and e 11 elc positively contribute to the total e 11 . We also observe the previously reported trend that the larger the chalcogen is, the larger the e 11 . This is mainly because the larger the chalcogen is, the larger the e 11 ion due to larger Z 11 and u d d 1 1 η (see Table S1). Compared to 1H-LaBr 2 (FM), the magnitude of  Table 3) indicated by the integrated crystal orbital Hamilton population (ICOHP). In addition, compared to the other two monolayers, its larger lattice parameters (see Table 1) can promote larger displacement of atoms in response to strain as atoms have more space to move. We propose that BECs and lattice parametersrather than static charges like Bader chargecan be ideal descriptors for searching for improved 2D piezoelectrics as they are directly related to the e ij and can be routinely computed, allowing for automated and high-throughput screening. Our results also explain the previous observation that there is no significant correlation of d 11 with electronegativity or Bader charges, whereas d 11 shows a strong correlation with polarizabilities of anions and cations. 13 Note that BECs can also be considered as a manifestation of local polarizabilities of atoms. 25 Now we calculate the piezoelectric stress constants (d ij ) using e ij and elastic constants (C ij ) (see Table 3). First, the mechanical/elastic stability of the ferromagnetic (FM) 1H-LaBr 2 monolayer is checked according to the criteria for a 2D hexagonal crystal structure: 27 C 11 > C 12 and C 66 > 0. Considering the two independent elastic constants (only two independent elastic constants due to space group P6m2 and twodimensionality) that we obtain, namely, C 11 = 30.34 N/m and C 12 = 9.53 N/m (notice that C 66 = (C 11 − C 12 )/2), it can be concluded that the monolayer is mechanically stable. The dynamic stability of 1H-LaBr 2 in terms of phonon modes has already predicted. 10 Our calculated elastic coefficients for the monolayers are in good agreement with the previously reported values. 6,10,18 Compared to the 1H-MoS 2 and 1H-VS 2 (FM) monolayers, its lower C 11 and C 12 values but larger ν indicate that the 1H-LaBr 2 (FM) monolayer is much softer. This is also expected because of its larger lattice parameters. This softening of elastic coefficients can also be understood from bond strength analysis. For that, we use the ICOHP approach, 28 which allows us to quantify the strength of the covalency of a bond. The more negative ICOHP, the stronger the covalent bonding. Here we emphasize that ICOHP is a reasonable qualitative estimation of the bond strength but it is not the bond enthalpy. We see in Table 3 that the 1H-LaBr 2 (FM) monolayer has a significantly weaker La−Br bond, with an ICOHP of −1.919 and a La−Br bond length of 3.14 Å, compared to those of Mo−S, with an ICOHP of −3.113 and a Mo−S distance of 2.417 Å, or V−S, with an ICOHP of −2.51 and a V−S distance of 2.366 Å. Table 3 shows that d 11 (again the only independent coefficient due to symmetry and dimensionality; d e C C 11 11 11 12 = − ) of 1H-LaBr 2 (FM) is about 22% larger than that of the well-known 2D piezoelectric 1H-MoS 2 because the former has quite low elastic constants. The origin of the negative sign in d 11 of 1H-LaBr 2 (FM) is in its negative e 11 , which is discussed above. As also previously reported, 18 despite being ultrathin, the piezocoefficients of these 2D piezoelectrics are comparable with those of well-known bulk piezoelectrics, e.g., α-quartz (d 11 = 2.3 pm/V) 29 and wurtzite nitrides such as AlN (d 33 = 5.1 pm/V) 30 and GaN (d 33 = 3.1 pm/ V). 30 Now we discuss how the magnetic ordering can affect the piezoelectric response. We consider simple strip-type antiferromagnetic (AFM) order (see Figure 1). The calculated values of e 11 and e 12 are shown in Table 4. Interestingly, we find that e 11 is not equal to −e 12 for AFM, whereas e 11 = −e 12 for FM. Moreover, e 11 in AFM is quite different from e 11 in FM (see Table 4). For example, e 11 of AFM 1H-LaBr 2 is almost double compared to that of FM; however, e 11 is still negative. To understand the origin of e 11 ≠ − e 12 for AFM, we consider two cases: (i) the structures (lattice parameters a and b and atomic positions) are relaxed and (ii) AFM order is used, keeping the lattice parameters a and b and atomic positions fixed in their FM structures, which are represented by asterisks in Table 4. We see that it is the change in magnetic order that intrinsically causes e 11 ≠ − e 12 for AFM, not the structural changes associated with this magnetic order change, although the structural relaxation changes the values, too. Both e elc and e ion change (both e 11 ion ≠ −e 12 ion and e 11 elc ≠ −e 12 elc for AFM) in response to the change in magnetic order.  change, resulting in changes to e ion . We notice that the magnitude of Z 11 for both La (V) and Br (S) in AFM order has increased (decreased), promoting enhancement in total e 11 or e 12 .
We also calculate the d ij coefficients (see Table 5) using the relation We notice that C 11 is not equal to C 22 for AFM structures. As e 11 ≠ −e 12 and C 11 ≠ −C 22 for AFM structures, d 11 ≠ −d 12 for AFM as shown in Table 5 is expected. We find that the d 11 (also e 11 ) of AFM 1H-LaBr 2 is about 2 times larger than that of its FM. We believe that such a change in piezoresponse induced by magnetic order can also be observed in other magnetic 2D piezoelectrics. In experiments, the magnetic direction (noncollinear magnetic order) can play a vital role, which is beyond the scope of this paper. Note that changing the magnetic order in a controlled way experimentally might be a challengeespecially for the change from FM to AFM. However, a transition from the AFM state to the FM state can be achieved by applying an external magnetic field to the AFM ordered samples.

■ CONCLUSION
We show that the 1H-LaBr 2 monolayer exhibits an unusual inplane negative piezoelectric coefficient, unlike many other 1H structured 2D piezoelectrics. 13−16 This would mean that the monolayer contracts along the x-direction (armchair direction) rather than expands, when an electric field is applied in the xdirection. Here the origin of the negative piezoelectric coefficient is because of a large negative e 11 ion that cannot be compensated by e 11 elc ; this is different from hitherto observed negative piezocoefficients in some bulk materials due to large e ij elc values. 7,9 The 1H-LaBr 2 monolayer is a promising 2D piezoelectric, having a large piezoelectric d 11 (−4.527 pm/V) coefficient, which is comparable to those of well-known 2D piezoelectric 1H-MoS 2 and 1H-VS 2 monolayers and is larger than that of bulk wurtzite GaN (d 33 ∼ 3.1 pm/V). We also explain the originboth sign and magnitudeof the piezoelectric coefficients of three monolayers (1H-LaBr 2 , 1H-MoS 2 , and 1H-VS 2 ) in terms of their dynamical charges (BECs) and atomic sensitivity (du/dη) to an applied strain. Being directly linked with e ij , we propose that BECs, rather than a static charge like the Bader charge, which relate to atom polarizability, can be good descriptors for searching new 2D piezoelectrics, also providing insight into the underlying mechanism. The calculation of BECs can be automated to allow for highthroughput screening. Additionally, we show that a change in magnetic order can have an effect on their piezoresponse quite significantly, which can be a unique way for coupling magnetism and electromechanical properties in 2D magnets.  represents the change of the position of the atoms along the a-direction under a strain along the b-direction (η 2 ). d 1H-VS 2 * and 1H-LaBr 2 * represent antiferromagnetic 1H-VS 2 and 1H-LaBr 2 monolayers in their ferromagnetic structures (i.e., just the magnetic order is changed, no structural relaxation).