Multiferroicity, Magnetoelectricity, and Piezoelectricity in Two-Dimensional Janus VSBrI Monolayers
Qiuyue Ma Busheng Wang Guochun Yang Yong Liu







yongliu@ysu.edu.cn
State Key Laboratory of Metastable Materials Science and Technology & Key Laboratory for Microstructural Material Physics of Hebei Province, School of Science, Yanshan University, Qinhuangdao 066004, China


§ ABSTRACT 
Two-dimensional (2D) multiferroic materials that combine intrinsic ferromagnetism and ferroelectricity exhibit significant potential for applications in highly integrated magnetoelectric and multifunctional spintronic devices. Through first-principles calculations, we identify the Janus VSBrI monolayer as a promising multiferroic semiconductor material, possessing both ferromagnetism and ferroelectricity. Specifically, the VSBrI monolayer shows a large in-plane magnetic anisotropic energy (MAE) of 460 μeV/V, a significant intrinsic in-plane spontaneous ferroelectric polarization of 1.20 × 10^-10 C/m, and a high energy barrier between two ferroelectric states of 168 eV. Our findings reveal that the energy variances among different magnetic states notably correlate with polarization, hinting at the potential for sizable magnetoelectric coupling within the VSBrI. Interestingly, we find that the stability of the ferroelectric phase can be enhanced by the application of biaxial tensile strain. The calculated in-plane piezoelectric coefficient d_11 reaches 29.01 pm/V and the out-plane piezoelectric coefficient d_32 reaches 1.60 pm/V, both of which are significantly larger than those of most known 2D materials,  which is greatly desirable for practical applications in piezoelectronic devices. These intriguing properties make the Janus VSBrI monolayer a promising candidate for 2D multifunctional spintronic devices, with significant potential to advance next-generation technology and inform the design of related electronic devices.






Multiferroicity, Magnetoelectricity, and Piezoelectricity in Two-Dimensional Janus VSBrI Monolayers
    Yong Liu
    September 9, 2024
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Multiferroicity, Magnetoelectricity, and Piezoelectricity in Two-Dimensional Janus VSBrI Monolayers
    Yong Liu
    September 9, 2024
===================================================================================================



§ INTRODUCTION

Two-dimensional (2D) nanomaterials have garnered considerable interest due to their unique physical properties and atomically thin structures, especially after the successful isolation of graphene <cit.>. The advent of 2D multifunctional materials, integrating multiple advantageous properties, significantly enhances device performance and broadens the horizons for advanced optoelectronic and spintronic technologies <cit.>. The compelling nature of these multifunctional materials stems from the complex interplay between charge, spin, and lattice degrees of freedom, making them highly attractive for advanced research and technological applications <cit.>. Multiferroics are multifunctional materials that exhibit more than one ferroic ordering within a single phase, such as ferroelectricity, ferromagnetism, ferroelasticity, or ferrotoroidicity. Encouraged by the recent success of 2D ferroelectric (FE)<cit.> and ferromagnetic (FM) <cit.> materials, 2D magnetoelectric materials with coupled FM and FE orders have gained significant attention for their superior physical characteristics and potential in magnetoelectric applications <cit.>. Multiferroics with both FE and FM orders are of great importance due to the interactions between magnetic and electric polarizations, which can be highly beneficial for achieving ultra-high-speed reading and writing in data storage <cit.>. Unfortunately, 2D magnetoelectric multiferroics are rare in nature, and their design has proven unexpectedly difficult <cit.>. This scarcity arises primarily from the inherent conflict between ferroelectricity and magnetism, exemplified by the “d_0 rule" of ferroelectricity in perovskites <cit.>. Generally, the generation of atomic magnetic moments requires a partially filled d orbital, while ferroelectricity demands that the d orbital be empty. The inherent contradiction significantly limits the development of single-phase magnetoelectric materials. In recent years, tremendous efforts have been devoted to understanding and searching for magnetoelectric multiferroics <cit.>.

2D Janus monolayers represent an attractive new class of 2D materials with mirror asymmetry, which has brought impetus to many other fields ranging from separating membranes to 2D electronics. Graphone <cit.>, also known as single-sided hydrogenated graphene, was created by removing half of the hydrogen from the graphene and was the first proposed 2D Janus material. Since the discovery of graphone, 2D Janus materials have attracted significant interest <cit.>. Traditionally, 2D Janus materials are synthesized through bottom-up approaches, such as selective chemical functionalization of free-standing 2D materials or substitution of a surface-layer element in the layered structures of transition metal dichalcogenides (TMDs) <cit.>. 2D Janus materials show novel physical and chemical properties, including the Rashba effect <cit.>, valley spin splitting <cit.>, catalytic performance <cit.>, and piezoelectric polarization <cit.>. The TMD monolayers MXY (M = Mo, W; X ≠ Y = S, Se, Te) exhibt large Rashba spin splitting and valley spin splitting, potentially making a significant contribution to semiconductor spintronics and valleytronics <cit.>. In the realm of Janus TMDs, the Janus MoSSe monolayer, as initially synthesized by Lu et al., is derived from a monolayer of MoS_2 wherein selenium atoms have been substituted for the sulfur atoms in the top layer <cit.>. Previous studies have suggested its promising potential as an efficient photocatalyst for solar-driven water splitting across a broad spectrum due to its anticipated low carrier recombination rate <cit.>. In addition, the Janus VSSe monolayer featuring piezoelectricity, ferroelasticity, and strong valley polarization is a highly promising material for potential applications in nanoelectronics, optoelectronics, and valleytronics <cit.>. The successful synthesis and extraordinary properties of 2D Janus materials will promote the development of electronic and spintronic devices.

The piezoelectric effect is a fundamental electromechanical interaction found in semiconductors with crystal structures lacking inversion symmetry <cit.>, and it is particularly common in Janus structures. Significant advancements in 2D piezoelectric materials have created unprecedented opportunities for fascinating physics and provided a potential platform for multifunctional electronic devices <cit.>. Experimentally, the successfully synthesized Janus MoSSe monolayer exhibits vertical dipoles and a piezoelectric effect, suggesting potential for future applications <cit.>. Theoretically, the piezoelectric properties of various other 2D Janus materials have also been reported, such as Janus group-III chalcogenide monolayers <cit.>, ZnBrI <cit.>, FeGeN_3 <cit.>, and Bi_2X_2Y (X = S, Se, Te; Y = S, Se, Te; X ≠ Y) <cit.> monolayes, enriching the potential applications of 2D piezoelectric materials. However, a major issue with 2D piezoelectric materials is the absence or weakness of out-of-plane piezoelectricity. Despite many piezoelectric materials display strong in-plane piezoelectric responses, those with significant out-of-plane piezoelectric responses are rare. Therefore, the search for 2D materials with large out-of-plane piezoelectric responses has become crucial and challenging.

In this work, based on first-principles calculations, we propose a 2D multiferroic material, the Janus VSBrI monolayer, which exhibits intrinsic ferroelectricity and ferromagnetic properties. The identified Janus VSBrI monolayer shows semiconducting behavior with a bandgap of 0.76 eV. It possesses strong magnetic anisotropy (460 μeV/V) with an in-plane easy magnetic axis and an estimated Curie temperature (T_C) of 83 K. The in-plane spontaneous ferroelectric polarization is calculated as 1.20 × 10^-10 C/m. An energy barrier of 168 meV between two ferroelectric phases with opposite electronic polarizations indicates that the ferroelectricity is thermodynamically stable at room temperature and can be switched by applying a moderate external electric field <cit.>. Moreover, the calculation of the magnetoelectric coupling effect in the Janus VSBrI monolayer reveals that energy differences between various magnetic states vary significantly with polarization, confirming the presence of strong magnetoelectric coupling. It is observed that the stability of the ferroelectric phase in the VSBrI monolayer exhibits high can be further enhanced by biaxial tensile strain. The Janus VSBrI monolayer not only exhibits large in-plane piezoelectricity (d_11) but also demonstrates large out-of-plane piezoelectricity (d_32), an uncommon but highly desirable trait in 2D materials. Our findings establish a promising platform for the advancement of spintronics and multiferroic electronic devices.












§ METHODS

The structural optimization and properties calculations are performed within the framework of density functional theory (DFT) <cit.>using a planewave basis set as implemented in Vienna ab initio Simulation Package (VASP) <cit.>. The projector-augmented wave (PAW) potential with the generalized gradient approximation of Perdew-Burke-Ernzerhof (GGA-PBE) is used as the exchange-correlation functional <cit.>. Considering the strong electronic correlation of 3d electrons, an effective Hubbard U (U_eff = 2 eV) is added for the V-3d orbitals <cit.>. The plane wave cutoff energy is set to 450 eV, and a Γ-centered 15 × 13 × 1 Monkhorst-Pack grid is employed to sample the first Brillouin zone <cit.>. A vacuum layer of at least 15 Å along the z direction is added to avoid unexpected interactions between periodic images. The structure undergoes full optimization until the Hellmann-Feynman forces acting on each atom are below 0.01 eV/Å, while ensuring that the total energy difference between consecutive steps remains less than 10^-6 eV. The stability dependence on temperature of 4 × 4 × 1 supercell is confirmed through ab initio molecular dynamics (AIMD) simulations <cit.>. Phonon spectrum calculations are self-consistently conducted using density functional perturbation theory (DFPT) <cit.> implemented in the PHONOPY code <cit.>. The ferroelectric polarization is calculated using the standard Berry phase method <cit.>. The elastic stiffness tensor C_ij and piezoelectric stress tensor e_ij are calculated by using the strain-stress relationship (SSR) and DFPT method, respectively.  The 2D elastic coefficients C_ij^2D and piezoelectric stress coefficients e_ij^2D have been renormalized by C_ij^2D = L_zC_ij^3D and e_ij^2D = Lze_ij^3D, where L_z is the length of unit cell along the z direction.




§ RESULTS AND DISCUSSION




Figures <ref>(a) and (b) show the top and side views of the Janus VSBrI monolayer, derived from the ground-state structure of the parent VSI_2 monolayer <cit.> by replacing the top layer of halogen atoms I with the more electronegative halogen atoms Br. The Janus nature of VSBrI arises from the presence of two outer sublayers consisting of nonequivalent halogen atoms sandwiching the central V atoms, resulting in a lower symmetry (Pm) in comparison to its parent VSI_2 monolayer (Pmm2). As shown in Fig. <ref>(c), the electron localization function (ELF) illustrates the ionic characteristics of V-S, V-I, and V-Br bonds, and bader charge analysis indicates that S, Br, and I atoms gain electrons, while V atoms lose electrons. Four magnetic configurations of V atoms (one FM state and three AFM states) with the supercell of 2 × 2 × 1 are considered to determine the magnetic ground state of the Janus VSBrI monolayer [Fig. <ref>(d)]. By comparing the energies of different magnetic configurations, it is determined that the magnetic ground state of the Janus VSBrI monolayer is FM, with energy differences of 23.18, 25.67, and 19.25 meV/f.u. relative to the three other AFM states. Spatial distribution of the spin-polarized electron density reveals that the significant FM localized magnetic moment (1 μ_B per unit cell) is primarily contributed by the V atom. The optimized lattice parameters for the FM unit cell are a = 3.85 Å and b = 4.66 Å (see Table <ref>). The V ions are located at the center of the S_2Br_2I_2 octahedron and shifts towards S ions on the one side, resulting in different bond lengths of V-S1 (2.11 Å) and V-S2 (2.55 Å), which breaking the spatial inversion symmetry along the b-axis and leading to a polar structure. For parent structure VSI_2 monolayer, the bond lengths of V-S1 and V-S2 are 2.10 Å and 2.49 Å, respectively, which are shorter than those in the Janus VSBrI monolayer. Consequently, the lattice parameter b of Janus VSBrI monolayer is larger than that of the VSI_2 (b= 4.60 Å). The V-halogen bond length depends on the monolayer's other halogen. Comparing the V-I bond lengths in VSBrI, it decreases with the addition of higher electronegative halogen atoms Br as 2.60 Å, which is lower than the V-I bond lengths in VSI_2 of 2.80 Å. The I/Br-V-S1 and I/Br-V-S2 bond angles are larger and smaller, respectively, than the I-V-S1 (96.8^∘) and I-V-S2 (83.2^∘) bond angles of the VSI_2 monolayer, resulting in the lattice parameter a of the Janus VSBrI monolayer being smaller than that of the parent structure VSI_2 (a = 3.99 Å).








To assess the structural integrity of Janus VSBrI monolayer, we calculated its cohesive energy, defined as E_coh = (E_V + E_S + E_Br + E_I - E_VSBrI)/4, where E_V, E_S, E_Br, E_I, and E_VSBrI represent the total energies of a single V atom, a single S atom, a single Br atom, a single I atom, and VSBrI monolayer, respectively. The number 4 signifies the total number of atoms in each unit cell. With a cohesive energy value of 3.46 eV/atom for Janus VSBrI monolayer, it demonstrates a level of stability comparable to the material phosphorene (3.48 eV/atom) <cit.>. As shown in Fig. <ref>(e) for the phonon dispersion, the absence of imaginary frequencies confirms the dynamic stability of Janus VSBrI monolayer. Its good thermal stability is evidenced by the little energy fluctuation and  slight structural deformation observed in the AIMD simulation conducted at a constant room temperature of 300 K [see Fig.S1 of the Supplemental Material]. The calculated elastic constants (C_11 = 37.45 N/m , C_12 = 7.49 N/m, C_22 = 31.48 N/m, and C_66 = 9.96 N/m) satisfy the Born-Huang criteria for mechanical stability <cit.>: C_11 > 0, C_66 > 0, C_11C_22 - C_12^2 > 0,  indicating that the Janus VSBrI monolayer is mechanically stable. Besides stability, it is important to consider material strength and anisotropy for practical applications. Therefore, we calculated the angular-dependent elastic properties: Young's modulus Y_2D(θ) and Poisson's ratio ν_2D(θ), using the two following formulas <cit.>:

    Y_2D(θ) = C_11C_22 - C_12^2/C_11sin^4 θ + C_22cos^4 θ + (B - 2C_12)sin^2 θcos^2 θ


    ν_2D(θ) = (C_11 + C_22 - B)sin^2 θcos^2 - C_12(sin^4 θ + cos^4 θ)/C_11sin^4 θ + C_22cos^4 θ + (B - 2C_12)sin^2 θcos^2 θ

Where the θ is the angle of the direction with the x direction as 0^∘ and the y direction as 90^∘, B = (C_11C_22 - C_12^2)/C_66. The results show that the Young's modulus and Poisson's ratio exhibt obvious anisotropic characters [Fig. <ref>]. The Young's modulus peaks along the x and y directions, with values of 35.67 N/m and 29.98 N/m, respectively. It is noteworthy that the Young's modulus of VSBrI is lower than that of many other 2D materials, suggesting it can be easily tuned by strain. This makes the Janus VSBrI monolayer highly suitable for novel flexible piezotronics and electronics applications. In addition, the Poisson's ratio is 0.24 and 0.20 along the x and y directions, respectively.




The electrical properties of the Janus VSBrI monolayer were investigated by calculating its electronic band structure and density of states (DOS). As illustrated in Fig. <ref>(a), the VSBrI monolayer is identified as a semiconductor with a band gap of 0.76 eV, as calculated using the revised PBE + U functional. The band gap of the Janus VSBrI monolayer is larger than that of the parent VSI_2 monolayer (0.63 eV) <cit.>. The valence band maximum and the conduction band minimum are located at the X and Y high symmetry point, respectively, indicating a characteristic indirect band gap (0.76 eV) of the Janus VSBrI monolayer. It is important to note that the hybrid Heyd-Scuseria-Ernzerh (HSE06) functional exhibits similar band structure to the PBE + U functional. However, the values of band gaps (1.58 eV in spin-up, and 3.21 eV in spin-down) are larger than those (0.78 eV in spin-up, and 1.83 eV in spin-down) of the PBE + U functional [see Fig. <ref>(b)], as the PBE functional often underestimate the band gap. The DOS of the Janus VSBrI monolayer is shown in Fig. S2 of the Supplemental Material, which indicates that the S and I ions contribute to the valence band maximum, while the conduction band minimum is primarily influenced by the V and S ions. Additionally, near the Fermi level, the spin-up states are  primarily contributed by the electronic states of V and I ions.




Magnetic anisotropy energy (MAE) describing the magnetic anisotropy can resist disruptions to magnetic ordering caused by external magnetic fields or thermal agitation. The larger the MAE, the better resisting thermal disturbance and keeping long-range magnetic ordering. The relative energy associated with different magnetization directions indicates that the Janus VSBrI monolayer tends to favor in-plane magnetization (x-axis) [Fig. <ref>(c)]. The calculated MAE of the Janus VSBrI monolayer is 460 μeV/V, which is comparable to that of the Mn_2PSb monolayer (450 μeV/Mn) <cit.> and larger than those of Cr_1.5Br_1.5I (356 μeV/Cr) <cit.>, VP (101.4 μeV/V), and VAs (262.6 μeV/V) <cit.>. Then, we determined the magnetic coupling parameters (J) of the Janus VSBrI monolayer using the classical Heisenberg model. The Hamiltonian is defined as:

    H =  - ∑_ < i,j > J_ijS_iS_j- A(S_i^Z)^2

where the S and A represent the spin quantum number (|S|=1/2) and the MAE, respectively. The magnetic coupling parameters of the nearest-neighbor J_1, next-nearest-neighbor J_2, and third-nearest-neighbor are calculated via:

    E(FM) = E_0- (4J_1+4J_2+8J_3)S^2


    E(AFM1) = E_0 - (4J_1-4J_2-8J_3)S^2


    E(AFM2) = E_0 - (-4J_1+4J_2-8J_3)S^2


    E(AFM3) = E_0 - (-4J_1-4J_2+8J_3)S^2

The calculated J_1, J_2, and J_3 are 21.77, 16.79 and 14.82 meV, respectively. Due to the positive values of J, the magnetic configuration of the Janus VSBrI monolayer favors a FM arrangement. With the above-mentioned magnetic coupling parameters and MAE, T_C of the Janus VSBrI monolayer is estimated by using the Monte Carlo simulation based on the Heisenberg model. Figure <ref>(d) shows the magnetic moment and specific heat of the V atoms as a function of temperature. The calculated T_C of Janus VSBrI monolayer is 83 K, which is higher than those of magnetoelectric multiferroics, such as VOClBr (43 K) <cit.>, NbSCl_2 (25 K) <cit.>, and  VOF_2 (15 K) <cit.> monoalyers.




Next, the ferroelectric properties of the Janus VSBrI monolayer are studied in detail. Due to the anisotropic octahedral crystal field in the VSBrI monolayer, the cation 3d orbitals split into four energy levels: d_z^2, d_x^2-y^2, d_xz/d_yz, d_xy orbitals, respectively [Fig. <ref>(a)]. The V^4+ ion has a 3d^1 electronic configuration, with the single unpaired d electron occupying the lowest energy d_xy orbital, which is distributed in the plane perpendicular to the V-S chain, as illustrated by the charge density plot in Fig. <ref>(b). Typically, magnetism arises from the partially filled d orbitals of transition-metal cations. However, it is widely accepted that this phenomenon hinders the emergence of ferroelectricity, a principle known as the d_0 rule in multiferroics <cit.>. Here, although V^4+ is not d_0, it behaves like d_0 along the z direction (the b-axis) since the d_yz and d_xz orbitals are empty. This unique anisotropic orbital ordering is responsible for the violation of the d_0 rule, resulting in the emergence of proper ferroelectricity, referred to as the “anisotropic d_1 rule" <cit.>. In the Janus VSBrI monolayer, the displacement of the V ions from the middle of the octahedron disrupts inversion symmetry, leading to the generation of a reversible in-plane electric polarization along the b-axis. The calculated in-plane spontaneous ferroelectric polarization (P_s) is 1.20 × 10^-10 C/m, which is larger than those of the CrNCl (0.60 × 10^-10 C/m) <cit.>, SnTe (0.22 × 10^-10 C/m) monolayers <cit.>. Figure <ref>(c) shows the energy versus structural parameter h for the Janus VSBrI monolayer, which resembles a double-well potential curve. The h is defined as the difference between the two V-S bond lengths along the b-axis. Here, h = 0.45 and -0.45 Å represent the two equivalent stable FE phases with opposite electric polarizations, and h = 0 represents the centrosymmetric paraelectric (PE) phase. For the Janus VSBrI monolayer, the calculated energy barrier (E_G) for FE switching is 169 meV/f.u, which is higher than that of the parent VSI_2 monolayer (140 meV/f.u), indicating that its FE phase is relatively more stable. Then, we simulated an in-plane ferroelectric polarization switching pathway through the antiferroelectric (AFE) phase for the Janus VSBrI monolayer [Fig. <ref>(d)]. The energy barrier (ΔE) of the FE-AFE-FE pathway significantly decreases to 88 meV per formula unit. Despite the close energy of the AFE and FE states, the energy barrier between them is significantly larger than the thermal energy at room temperature (25 meV), ensuring that the FE phase remains stable and does not transition to the AFE phase due to thermal vibrations at room temperature. The magnetoelectric coupling effect of the Janus VSBrI monolayer was preliminarily assessed by observing the magnetic response to the variation of ferroelectric polarization. We displaced the V atom along the V-O chain away from the ground state to monotonically change the polarization. The energy dependence of the three magnetic configurations on polarization was calculated using the ferromagnetic phase as the energy reference [Fig. S3 of the Supplemental Material]. The results indicate that the energy differences between the different magnetic states vary significantly with polarization, highlighting the influence of polarization on magnetic properties.  Therefore, it becomes possible for the Janus VSBrI monolayer to exhibit a notable magnetoelectric coupling.


Biaxial strain engineering is an effective approach to tuning the properties of 2D materials. As shown in Fig. <ref>, we investigated the impact of biaxial tensile strain ranging from 0% to 6% on the magnetic, electronic, and ferroelectric properties of the Janus VSBrI monolayer. The energy differences between three AFM and FM configurations (ΔE = E_AFM -  E_FM) were calculated at varying percentages of biaxial strain [Fig. <ref>(a)]. The positive energy differences indicate that there is no magnetic transition resulting from the application of biaxial tensile strain. Therefore, it can be concluded that the Janus VSBrI monolayer exhibits strong ferromagnetic robustness under the biaxial strain from 0% to 6%. As shown in Fig. <ref>(b), with increasing biaxial tensile strain, J_1 exhibits a monotonically increasing trend, indicating that the ferromagnetic coupling strength between nearest-neighbor V ions gradually strengthens. Both J_2 and J_3 decrease under biaxial tensile strain, suggesting that the biaxial strain weakens the ferromagnetic coupling strength between the next-nearest-neighbor and third-nearest-neighbor V ions. Since the number of J_3 is twice that of J_1 and J_2, J_3 predominantly determines the change in T_C. Figure S4 of the Supplemental Material shows the variation of T_C and MAE under biaxial strain ranging from 0% to 6%. Despite the gradual decrease in T_C with increasing biaxial tensile strain, it remains higher than the temperature of CrI_3 (45 K) <cit.>. The MAE initially decreases and then increases with increasing biaxial tensile strain. Calculations reveal that when a 6% biaxial tensile strain is applied, the MAE value along the z-axis is the lowest, causing the magnetic easy axis to shift from the x-axis to the z-axis. We calculated the electronic band structures of the Janus VSBrI monolayer under biaxial tensile strain [Fig. S5 of the Supplemental Material]. The Janus VSBrI monolayer maintains its semiconductor behavior within the range of 0 to 6% biaxial tensile strain, and the spin gap gradually increases with the tensile strain, reaching 0.92 eV at 6% strain. As illustrated in Fig. <ref>(c), the polar displacement and spontaneous ferroelectric polarization are evaluated under the application of biaxial tensile strain. The polar displacement is defined as the variation in the V ion's fractional coordinates along the polar axis (b-axis) during the transition from the PE state to the FE state. The results show that the polar displacement increases monotonically with the application of biaxial tensile strain. Due to the direct relationship between polar displacement and spontaneous ferroelectric polarization, the trend of ferroelectric polarization is similar to the trend of polar displacement under biaxial tensile strain. The spontaneous ferroelectric polarization reaches its maximum value of 1.43 × 10^-10 C/m under 6% biaxial tensile strain. Additionally, the activation energy barrier E_G and ΔE for electric polarization switching can be effectively adjusted by applying biaxial tensile strain [Fig. <ref>(d)]. E_G values are nearly twice the ΔE values under the biaxial strain changing from 0% to 6%, leading to the conclusion that the path through the AFE phase consistently exhibits the minimum activation energy, irrespective of the strain magnitude. The E_G and Δ E values increase with biaxial tensile strain, ranging from 168 to 378 meV/f.u. and 89 to 194 meV/f.u., respectively. It is known that high-energy barriers can ensure the stability of the ferroelectric phase. By studying the effect of biaxial tensile strain on ferroelectric properties, it was found that the stability of the ferroelectric phase in Janus VSBrI monolayer is further enhanced with increasing biaxial tensile strain.


As shown in Fig. S6 of the Supplemental Material, we calculated the planar averaged electrostatic potential energy variation along the z direction for VSI_2 and VSBrI monolayers. Due to horizontal mirror symmetry, the electrostatic potential energy difference and intrinsic polar field of the VSI_2 monolayer are zero. Due to the larger electronegativity of the halogen atom Br (2.96) compared to the halogen atom I (2.66), there is an out-of-plane polarization in the Janus VSBrI monolayer with a value of 0.04 × 10^-10 C/m. The existence of out-of-plane polarization results in an electrostatic potential energy difference and an intrinsic polar field in the Janus VSBrI monolayer. The electrostatic potential energy difference between the sides is 0.44 eV, which can be interpreted as a surface dependent work function. As illustrated in Fig. S6(b) of the Supplemental Material, we determined the strength of the intrinsic polar field from the slope of the curve, approximately 1.60 eV/Å, which is comparable to the reported value in the 2D piezoelectric material Janus Si_2SeTe (1.79 eV/Å) <cit.>. An intrinsic polar field may give rise to the piezoelectricity of 2D materials, suggesting that the Janus VSBrI monolayer possesses potential piezoelectricity. Accordingly, the piezoelectric response of the VSBrI was then estimated. In noncentrosymmetric materials, the piezoelectric effect leads to the generation of electric dipole moments and electric charge in response to applied mechanical strain or stress. The 2D piezoelectric effect is described by the piezoelectric stress coefficients e_ij and the piezoelectric strain coefficients d_ij. The relaxed piezoelectric tensors (e_ij and d_ij) are obtained as the sum of ionic and electronic contributions:

    e_ij = ∂P_i/∂ε_j = e_ij^elc + e_ij^ion


    d_ij = ∂P_i/∂σ_j = d_ij^elc + d_ij^ion

where the p_i, ε_j, and σ_j represent the piezoelectric polarizations, strains, and stresses, respectively. The piezoelectric strain coefficients d_ij can be derived by piezoelectric stress coefficients e_ij and elastic stiffness coefficients C_ij:

    d_11 = e_11C_22 - e_12C_12/C_11C_22 - C_12^2


    d_12 = e_12C_22 - e_12C_12/C_11C_22 - C_12^2


    d_31 = e_31C_22 - e_32C_12/C_11C_22 - C_12^2


    d_32 = e_32C_11 - e_31C_12/C_11C_22 - C_12^2

The calculated in-plane piezoelectric stress coefficients e_11 and e_12 are 11.10 × 10^-10 C/m and 3.17 × 10^-10 C/m, respectively, while the out-of-plane piezoelectric stress coefficients e_31 and e_32 are 0.40  × 10^-10 C/m and 0.56 × 10^-10 C/m, respectively. According to equations (8)-(11), the calculated in-plane piezoelectric strain coefficients d_11 and d_12 are 29.01 pm/V and 3.17 pm/V, respectively. The d_11 value for the VSBrI monolayer is much larger than that of the well-known piezoelectric material MoS_2 (3.7 pm/V) <cit.>. In contrast to the centrosymmetric VSI_2 with only in-plane piezoelectricity, the absence of inversion symmetry in VSBrI results in two distinctive out-of-plane piezoelectric responses characterized by d_31 and d_32. The d_31 (0.75 pm/V) of the Janus VSBrI monolayer is much larger than that of Janus group-III chalcogenide monolayers (0.07-0.46 pm/V) <cit.> and is larger than that of the Janus VOFI monolayer (-0.68 pm/V) <cit.> with the same space group as the VSBrI. Interestingly, the d_32 is more than twice d_31, with a value of 1.60 pm/V, which is larger than many known 2D materials, making it highly desirable for multifunctional piezoelectric devices.

Considering the valence electron is from a 3d orbital for Janus VSBrI monolayer, the Hubbard U correction was reassessed using the GGA+U method (U_eff = 0 ∼ 2 eV) to re-check its magnetic, ferroelectric, structural, and electronic properties. We listed the energy differences between the FM and three AFM states across the U_eff range of 0 to 2 eV in Table SI of the Supplemental Material. The results indicate that the energy of the FM configuration is always lower than that of the three AFM configurations, regardless of the U_eff value, suggesting that the ground state of the Janus VSBrI monolayer is still FM. As shown in Fig. S7 of the Supplemental Material, the polar displacement decreases as the U_eff increase, indicating that the FE state approaches the PE state, resulting in a reduction in the polarization value. The calculation results indicate that the spontaneous ferroelectric polarization decreases from 1.33 × 10^-10 C/m to 1.20 × 10^-10 C/m.  The physical reason for the diminishing ferroelectricity is that the increased U_eff weakens the d-p orbital hybridization between V and S, thereby reduce the driving force of its proper ferroelectricity. As shown in Fig. S8 of the Supplemental Material, the lattice constants a and b do not change significantly as the U_eff value increases. The band structures of Janus VSBrI monolayer under different U_eff value show no obvious alterations, and its semiconductor characteristics remain unaltered [Fig.S9 of the Supplemental Material]. Therefore, it can be confidently concluded that the magnetic ground state of the Janus VSBrI monolayer remains both FM and FE, unaffected by the choice of the Hubbard U in DFT calculations, and the selected U_eff value have little effect on the structural and electronic properties.









§ CONCLUSION

In summary, we predict a 2D intrinsic semiconductor Janus VSBrI monolayer with a band gap of 0.76 eV. The Janus VSBrI monolayer exhibits ferroelasticity, ferroelectricity, and strong piezoelectric behavior. MC simulations based on the Heisenberg model show that the magnetic T_C reaches up to 83 K. The VSBrI monolayer possesses a large MAE (460 μeV/V) with an in-plane easy magnetization direction. Additionally, the calculated in-plane spontaneous ferroelectric polarization along the b-axis is about 1.20 × 10^-10 C/m. In-plane ferroelectric switching pathways involve a PE intermediate phase with an energy barrier of 169 meV/f.u. and an AFE intermediate phase with an energy barrier of 88 meV/f.u. Interestingly, the VSBrI monolayer exhibits a notable magnetoelectric coupling effect, driven by the substantial variation in energy discrepancies among different magnetic states with polarization. We find that the stability of the ferroelectric phase is further enhanced under increasing biaxial tensile strain. The predicted in-plane d_11 (29.01 pm/V) and out-of-plane d_32 (1.60 pm/V) are higher than or comparable with ones of many 2D known materials, making it highly desirable for ultrathin piezoelectric devices. Our work provides a promising platform for exploring multiferroics and piezoelectric materials, which is important for the development of multifunctional spintronic devices.









§ ACKNOWLEDGEMENT

This work was supported by the Natural Science Foundation of China under Grants (No.22372142), the Innovation Capability Improvement Project of Hebei province (22567605H), the Natural Science Foundation of Hebei Province of China (No. B2021203030), the Science and Technology Project of Hebei Education Department (No. JZX2023020). The numerical calculations in
this paper have been done on the supercomputing system in the High Performance Computing Center of Yanshan University.



§ DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.









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