Negative Piezoelectric Coefficient in Ferromagnetic 1H-LaBr₂ Monolayer

Abstract

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 , 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, where ions are clamped under applied strain. In many bulk materials, including wurtzite nitrides,^7,8e_ij^elc is negative but is dominated by positive e_ij, 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 CuInP₂S₆,⁷ which is explained in terms of its large negative e_ij^elc that is not overcome by positive e_ij. 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? To answer this question, we investigate three intrinsically piezoelectric monolayers, 1H-MoS₂, 1H-VS₂, and 1H-LaBr₂, and we discover 1H-LaBr₂ 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-LaBr₂ 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 metal-like 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 co-coefficients.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-MoS₂ monolayer,¹⁷ and the value e₁₁ (2.9 × 10^–10 C/m) is in good agreement with first-principles calculations of e₁₁ = 3.64 × 10^–10 C/m.¹⁸ Recently, the coexistence of magnetism and piezoelectricity has also been predicted in the 1H-VS₂ 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.^6,13−16,18

However, the origin of the piezoelectric co-coefficients in both magnitude and sign still remains unclear. Questions include the following: Why is the e₁₁ of the 1H-MoS₂ monolayer larger than that of the 1H-VS₂ monolayer? Why is the sign of the ionic part of e₁₁ positive in the 1H-MoS₂ monolayer but negative in the 1H-VS₂ 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 , and the bond strength. We also demonstrate that the 1H-LaBr₂ monolayer¹⁰⁻¹² can be a magnetic, piezoelectric material. Moreover, we show that antiferromagnetic ordering makes the isotropic piezoelectricity of the ferromagnetic 1H-LaBr₂ monolayer anisotropic (i.e., e₁₁ ≠ −e₁₂).

Computational Details

Our first-principles calculations are performed in the framework of spin-polarized density functional theory using projector augmented wave (PAW) potentials¹⁹ 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 4p⁶5s²6d¹ (nine electrons), 3p⁶3d⁴4s¹ (11 electrons), 4p⁶4d⁵5s¹ (12 electrons), 3s²3p⁴ (six electrons), and 4s²4p⁵ (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 z-direction (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.

Figure 1.

Rectangular unit cell. Blue and red balls represent Mo/V/La and S/O, respectively. The red arrows represent up and down collinear spin states of an atom.

Results and Discussion

Table 1 shows the lattice parameters of the monolayers. We use a rectangular unit cell, and lattice parameter a should be equal to for an ideal 1H structure. Our calculated lattice parameters are in good agreement with previously reported values.^6,10−12,18 1H-LaBr₂ 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

This is about 1.22% (0.72%) for the AFM 1H-LaBr₂ (1H-VS₂) monolayer. This deviation is also reflected in the change in angles θ^FM and θ^AFM, which are defined in Figure 1.

Table 1. Structural Information of the Monolayers: Optimized Lattice Parameters a and b^a and the Angle ∠Mo/V/La–S/Br–Mo/V/La^b.

	a (Å)	b (Å)	θFM (deg)	θAFM (deg)
1H-MoS2	5.522	3.188	–	–
1H-VS2(FM)	5.504	3.178	84.372	–
1H-VS2(AFM)	5.502	3.154	84.177	84.518
1H-LaBr2(FM)	7.298	4.214	84.189	–
1H-LaBr2(AFM)	7.344	4.189	83.624	84.556

^aSee the rectangular cell in Figure 1.

^bθ^FM (θ^AFM) is the angle ∠V↑/La↑–S/Br–V↑/La↑ (∠V↑/La↑–S/Br–V↓/La↓), where ↑ and ↓ arrows represent up and down spin polarizations. The angle ∠Mo–S–Mo is 82.537°.

In agreement with previous reports,^6,10−12 we find that ferromagnetic (FM) ordering is the ground state for both 1H-LaBr₂ and 1H-VS₂ 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 VS₂ monolayer has not yet been synthesized, ferromagnetism has been recently found in its ultrathin films.^22,23

The 1H-LaBr₂(FM), 1H-VS₂(FM), and 1H-MoS₂ 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, e₁₁ (e₁₁ = −e₁₂), due to 6̅m2 point group symmetry. Table 2 shows that 1H-MoS₂ has a quite large e₁₁ value compared to those of the other two monolayers. Interestingly, FM 1H-LaBr₂ shows a negative e₁₁ which is also quite low compared to those of the other materials. To understand the origin of piezoelectric constant, e₁₁ and e₁₂ can be decomposed into two parts:^8,24

1

2

Table 2. Electronic (e₁₁^elc) and Ionic (e₁₁) Parts of the Total Piezoelectric Stress Constant e₁₁^a and Born Effective Charge Z₁₁^b,c.

	e11elc	e11ion	e11	Z11(M)	Z11(X)
1H-MoS2	315.000	56.050	371.050	–1.006	0.503	–0.037	0.018
1H-VS2(FM)d	379.025	–80.925	298.100	1.359	–0.680	–0.038	0.021
1H-LaBr2(FM)d	111.175	–205.350	–94.175	2.540	–1.269	–0.069	0.035

^aIn 2D piezoelectric unit, pC/m.

^bM = Mo, V, and La and X = S and Br in |e|, where e is the charge of an electron.

^cHere, represents the change of the position of the atoms along the a-direction under a strain along the a-direction (η₁).

^dBoth 1H-VS₂ and 1H-LaBr₂ monolayers are in the ferromagnetic (FM) state.

The clamped-ion term (e₁₁^elc or e₁₂) arises from the contributions of electrons when the ions are frozen at their zero-strain equilibrium internal atomic coordinates (u), and the internal-strain (e₁₁^ion) term arises from the contribution from internal microscopic atomic displacements in response to a macroscopic strain. In our case, the strain (η₁) is applied in the x-direction (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 (Z₁₁(k)) of the kth atom is calculated by the DFPT approach. The response of the kth atom’s internal coordinate along the x-direction (u₁(k)) in response to a macroscopic strain (η₁) is measured by . Table 2 shows that both e₁₁ and e₁₁^ion have the same sign—positive—for 1H-MoS₂, unlike the other two monolayers. This results in a large total e₁₁ for 1H-MoS₂. Note that other 1H-MX₂ (M = Mo, W, Cr and X = S, Se, Te) monolayers also exhibit positive e₁₁ and e₁₁^ion (see Table S1).¹⁸ However, due to the opposite signs of e₁₁ and e₁₁^ion in 1H-VS₂(FM), its total e₁₁ is smaller than the e₁₁ for 1H-MoS₂, even though it has a larger value of e₁₁ (see Table 2). Interestingly, 1H-LaBr₂(FM) shows a negative e₁₁^ion, which is significantly larger than its e₁₁, thus resulting in a negative total e₁₁. This is different from the recently discovered negative piezoelectric coefficient in layered ferroelectrics and wurtzite, where the negative sign comes from e₁₁^elc.⁷ Here it is important to highlight that other 2D piezoelectrics, e.g., the well-known hexagonal boron nitride (h-BN) monolayer¹⁸ and the 1H-VSe₂ monolayer,⁶ have negative e₁₁ values but the e₁₁^elc part dominates, resulting in positive e₁₁. Interestingly, we find that h-AlN and h-ZnO monolayers also exhibit negative e₁₁ values due to large negative e₁₁ values (see Table S1). We hope that our finding will inspire experimental studies of negative in-plane piezoresponse (e₁₁) in 2D materials. Deepening our understanding about the 2D piezoelectrics, this would enable discovery of 2D materials for novel electromechanical applications.

Now to understand the origin of negative/positive e₁₁^ion, we expressed e₁₁ in terms of Z₁₁ and (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.²⁵Z₁₁(k) is proportional to , where ∂P₁ is the change of the dipole moment in the x-direction induced by a small displacement of atom k in the same direction (∂τ₁(k)).²⁵ The negative slope will result in a negative BEC, which is the case for 1H-MoS₂. 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 e₁₁^ion in 1H-MoS₂ 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-MoS₂.²⁶ 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 e₁₁ values positive; thus, both e₁₁^ion and e₁₁ positively contribute to the total e₁₁. We also observe the previously reported trend that the larger the chalcogen is, the larger the e₁₁. This is mainly because the larger the chalcogen is, the larger the e₁₁^ion due to larger Z₁₁ and (see Table S1). Compared to 1H-LaBr₂(FM), the magnitude of e₁₁ in 1H-MoS₂ and 1H-VS₂(FM) is smaller because of smaller BECs and . We find that the large negative e₁₁^ion in 1H-LaBr₂(FM) originates from its large Z₁₁ and ; both terms are almost 2 times larger than those of 1H-MoS₂ or 1H-VS₂(FM). The large of 1H-LaBr₂(FM) can be due to its weaker La–Br bond (see 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 d₁₁ with electronegativity or Bader charges, whereas d₁₁ 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

Table 3. Elastic Constants (C₁₁ and C₁₂), ICOHP of a Bond between Cation (Mo/V/La) and Anion (S/Br), Poisson’s ratio ν (=C₁₂/C₁₁), and Piezoelectric Strain Coefficient d₁₁.

	C11 (N/m)	C12 (N/m)	ICOHP (eV/bond)	ν	d11 (pm/V)
1H-MoS2	133.214	33.105	–3.113	0.249	3.706
1H-VS2(FM)	101.421	28.785	–2.510	0.284	4.104
1H-LaBr2(FM)	30.338	9.534	–1.919	0.314	–4.527

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-LaBr₂ monolayer is checked according to the criteria for a 2D hexagonal crystal structure:²⁷C₁₁ > C₁₂ and C₆₆ > 0. Considering the two independent elastic constants (only two independent elastic constants due to space group P6m2 and two-dimensionality) that we obtain, namely, C₁₁ = 30.34 N/m and C₁₂ = 9.53 N/m (notice that C₆₆ = (C₁₁ – C₁₂)/2), it can be concluded that the monolayer is mechanically stable. The dynamic stability of 1H-LaBr₂ 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-MoS₂ and 1H-VS₂(FM) monolayers, its lower C₁₁ and C₁₂ values but larger ν indicate that the 1H-LaBr₂(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-LaBr₂(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 d₁₁ (again the only independent coefficient due to symmetry and dimensionality; ) of 1H-LaBr₂(FM) is about 22% larger than that of the well-known 2D piezoelectric 1H-MoS₂ because the former has quite low elastic constants. The origin of the negative sign in d₁₁ of 1H-LaBr₂(FM) is in its negative e₁₁, 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 (d₁₁ = 2.3 pm/V)²⁹ and wurtzite nitrides such as AlN (d₃₃ = 5.1 pm/V)³⁰ and GaN (d₃₃ = 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 e₁₁ and e₁₂ are shown in Table 4. Interestingly, we find that e₁₁ is not equal to −e₁₂ for AFM, whereas e₁₁ = −e₁₂ for FM. Moreover, e₁₁ in AFM is quite different from e₁₁ in FM (see Table 4). For example, e₁₁ of AFM 1H-LaBr₂ is almost double compared to that of FM; however, e₁₁ is still negative. To understand the origin of e₁₁ ≠ – e₁₂ 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 e₁₁ ≠ – e₁₂ 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 e₁₁^ion ≠ −e₁₂ and e₁₁^elc ≠ −e₁₂ for AFM) in response to the change in magnetic order. Table 4 also shows how the BECs and change, resulting in changes to e^ion. We notice that the magnitude of Z₁₁ for both La (V) and Br (S) in AFM order has increased (decreased), promoting enhancement in total e₁₁ or e₁₂.

Table 4. Electronic (e₁₁^elc and e₁₂) and Ionic (e₁₁^ion and e₁₂) Parts of the Total Piezoelectric Stress Constants e₁₁ and e₁₂ of Antiferromagnetic 1H-VS₂ and 1H-LaBr₂ Monolayers^a and Born Effective Charge Z₁₁^b,c.

	e11elc	e11ion	e11	Z11(M)	Z11(X)			e12elc	e12ion	e12
1H-VS2	221.900	–38.375	183.525	0.662	–0.210	–0.031	0.015	–284.150	12.175	–271.975	0.025	–0.013
1H-VS2*d	218.075	–14.525	203.550	0.657	–0.203	–0.033	0.016	–280.825	40.200	–240.625	0.026	–0.013
1H-LaBr2	43.650	–269.025	–225.375	2.765	–1.381	–0.079	0.040	–96.225	199.300	103.075	0.066	–0.033
1H-LaBr2*d	57.275	–281.000	–223.725	2.744	–1.370	–0.083	0.041	–100.075	197.700	97.625	0.068	–0.034

^aIn 2D piezoelectric unit, pC/m.

^bM = Mo, V, and La and X = S and Br in |e|, where e is the charge of an electron.

^cHere, represents the change of the position of the atoms along the a-direction under a strain along the b-direction (η₂).

^d1H-VS₂* and 1H-LaBr₂* represent antiferromagnetic 1H-VS₂ and 1H-LaBr₂ monolayers in their ferromagnetic structures (i.e., just the magnetic order is changed, no structural relaxation).

We also calculate the d_ij coefficients (see Table 5) using the relation

We notice that C₁₁ is not equal to C₂₂ for AFM structures. As e₁₁ ≠ −e₁₂ and C₁₁ ≠ −C₂₂ for AFM structures, d₁₁ ≠ −d₁₂ for AFM as shown in Table 5 is expected. We find that the d₁₁ (also e₁₁) of AFM 1H-LaBr₂ 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.

Table 5. Elastic Constants (C₁₁, C₂₂, and C₁₂) and Piezoelectric Strain Coefficients (d₁₁ and d₁₂).

	C11 (N/m)	C22 (N/m)	C12 (N/m)	d11 (pm/V)	d12 (pm/V)
1H-VS2(AFM)	94.586	105.378	32.867	2.832	–3.359
1H-LaBr2(AFM)	28.005	31.198	8.981	–10.343	6.137

Conclusion

We show that the 1H-LaBr₂ monolayer exhibits an unusual in-plane 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 x-direction. Here the origin of the negative piezoelectric coefficient is because of a large negative e₁₁^ion that cannot be compensated by e₁₁; this is different from hitherto observed negative piezocoefficients in some bulk materials due to large e_ij^elc values.^7,9 The 1H-LaBr₂ monolayer is a promising 2D piezoelectric, having a large piezoelectric d₁₁ (−4.527 pm/V) coefficient, which is comparable to those of well-known 2D piezoelectric 1H-MoS₂ and 1H-VS₂ monolayers and is larger than that of bulk wurtzite GaN (d₃₃ ∼ 3.1 pm/V). We also explain the origin—both sign and magnitude—of the piezoelectric coefficients of three monolayers (1H-LaBr₂, 1H-MoS₂, and 1H-VS₂) 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 high-throughput 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.

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsaelm.1c01214.

• Electronic (e₁₁^elc) and ionic (e₁₁) parts of total piezoelectric stress constant e₁₁; Born effective charge Z₁₁ and du₁/dη₁ of 1H-MoS₂, 1H-MoSe₂, 1H-MoTe₂, 1H-WS₂, 1H-WSe₂, 1H-WTe₂, 1H-CrS₂, 1H-CrSe₂, 1H-CrTe₂, h-BN, h-AlN, and h-ZnO (PDF)

The authors declare no competing financial interest.