Intrinsically asymmetric atomic character regulates piezoelectricity in two-dimensional materials

Abstract

Decreasing of layer thickness causes the decrease of polarization until it disappears due to the existence of depolarization field. Therefore, the search for strong piezoelectric materials is highly desirable for multifunctional ultra-thin piezoelectric devices. Herein, we propose a common strategy for achieving strong piezoelectric materials through the electronic asymmetry induced by the intrinsically asymmetric atomic character of different chalcogen atoms. Accordingly, in the tetrahedral lattice structures, for example, M₄X₃Y₃ (M = Pd/Ni, X/Y = S, Se or Te, X ≠ Y) monolayers are proved to display excellent out-of-plane piezoelectricity. Ni₄Se₃Te₃ possesses the largest piezoelectric coefficient d₃₃ of 61.57 pm/V, which is much larger than that of most 2D materials. Enhancing the electronic asymmetry further increases the out-of-plane piezoelectricity of Janus M₄X₃Y₃ materials. Correspondingly, the out-of-plane piezoelectricity is positively correlated with the ratio of electronegativity difference (R_ed) and the electric dipole moment (P). This work provides alternative materials for energy harvesting nano-devices or self-energized wearable devices, and supplies a valuable guideline for predicting 2D materials with strong out-of-plane piezoelectricity.

Graphical abstract

Keywords

piezoelectricity / intrinsically asymmetric atomic character / ratio of electronegativity difference / electric dipole moment / first-principles calculations

1 Introduction

Two-dimensional (2D) graphene has been successfully exfoliated experimentally [1], and it has been found that 2D graphene has good physicochemical properties such as ultra-high carrier mobility and the fermi Dirac point. Graphene doped with metal atoms can be used as a catalyst to achieve photocatalytic hydrolysis [2, 3]. In addition to graphene, other 2D materials have received a lot of attention [4-6], such as the transition metal sulfides (TMDCs) [7-11], MXenes [12], MBenes [13, 14], and sandwich-like MoSi₂N₄ monolayer [15-17]. Piezoelectric materials can realize the mutual conversion between mechanical energy and electric energy, which are widely used in the design of sensors, and microelectromechanical devices. For example, piezoelectric materials can be used to design flexible electronics, flexible screens, health monitoring devices, and electronic sensing skin [18]. Recent years, piezoelectric materials have also been applied to some blue energy harvesting, such as the wind, water power, waves and other large signal energy harvesting [19]. In addition, piezoelectric devices are also applied to the conversion of tiny mechanical energies that are often neglected, such as walking, body shaking and hand touching [20, 21]. Therefore, to achieve the collection of tiny mechanical energy, it is important to find piezoelectric materials with good stability, strong mechanical durability and strong piezoelectric response.

Materials with the out-of-plane piezoelectric properties can produce piezoelectric response to the vertical strain in the out-of-plane direction, converting mechanical stress perpendicular to the monolayer plane into electrical energy [22, 23]. Cutting-edge piezoelectric equipment such as the wearable electronics, medical blood pressure detectors and robotic bionic skin tactile sensors really utilize the out-of-plane piezoelectricity. Therefore, we make unremitting efforts to find excellent and intriguing stable 2D materials with remarkable out-of-plane piezoelectricity. Previous studies show that the transition metal dichalcogenides (TMDCs) with the in-plane piezoelectric properties can produce piezoelectric responses to internal strains, but not to vertical strains [24-28]. The symmetry of TMDCs-like materials with low symmetry C_3v space group is broken along the z axis direction, which makes them contain the out-of-plane piezoelectricity, such as the VSSe [29], MoSSe [30] or MoTO (T = S, Se, or Te) [31].

Previous studies have reported that Pd₂Se₃ films can be successfully fabricated experimentally, and it has interesting physicochemical properties, such as medium-sized band gap and high carrier mobility [32-34]. In addition, the Co₂Se₃ monolayer with semi-metallic properties and ultra-high mechanical stability can be obtained by substituting Pd atoms with Co atoms [35, 36]. These Pd₂Se₃ and Co₂Se₃ monolayers have similar side view to that of TMDCs monolayers with 2H phase. That is, the transition metal atoms are located in the intermediate layer, surrounded by chalcogenide Se atoms. Based on the PdSe₂, the Pd₂Se₃ was successfully fabricated in experiments using electron beam irradiation [34]. Recently, Pd₄S₃Se₃, Pd₄S₃Te₃, and Pd₄Se₃Te₃ monolayers have been proved to be excellent photocatalyst candidates with suitable bandgaps for photocatalytic water splitting [37]. However, as far as we know, the piezoelectricity of these Janus two-dimensional materials has not been studied.

In this work, therefore, we propose a common scheme to obtain strong piezoelectric materials, where different chalcogen atoms with intrinsically asymmetric atomic character can induce the electronic asymmetry. The piezoelectricity of the M₄X₃Y₃ (M = Pd/Ni, X/Y = S, Se or Te, X ≠ Y) Janus materials is theoretically predicted for the first time. Mirror electronic asymmetry leads to the introduction of the strong out-of-plane piezoelectricity. Enhancing the electronic asymmetry further increases the out-of-plane piezoelectricity. Accordingly, piezoelectric coefficients of Pd₄X₃Y₃ are positively related with the electronegative difference ratio R_ed and the electric diploe moment P, which opens a way for designing strong out-of-plane piezoelectric materials.

2 Computational details

First-principles calculations in this study are performed by Vienna ab initio Simulation Package (VASP) with the projected augmented wave (PAW) [38 39]. The Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional within generalized gradient approximations (GGA) is carried out for the structural fully optimizations of M₄X₃Y₃ monolayers [40]. The energy and atomic force convergence criteria are set to 10⁻⁷ eV and 0.0001 eV/Å, respectively. The energy cutoff is set to 550 eV, which is accurate enough to describe the outer valence electrons of Pd, Ni, S, Se and Te atoms. A higher than 20 Å of thick vacuum slab is considered to avoid neighboring-layers interaction. After a test of k-points [in Fig. S1 of the Electronic Supplementary Materials (ESM)], the Γ-centered 9 × 9 × 1 Monkhorst−Pack k-point meshs is used to sample the first Brillouin zone. Phonon spectra used to judge the dynamic stability are performed by the finite difference method [41]. Thermal stability is estimated by the ab initio molecular dynamics (AIMD) simulations [42]. In order to obtain the accurate band electronic structures for M₄X₃Y₃ monolayers, the Heyd–Scuseria–Ernzerhof (HSE06) hybrid functional [43] are performed. The Bader charge analysis technique [44] is used to analyze the uneven charge distributions. Using the small displacement methodology [45] to calculate the elastic stiffness tensors C_kj.

3 Results and discussion

Polarization usually originates from the non-overlapping of opposite charge centers, which occurs in crystals with low-symmetry space groups. Using the intrinsically asymmetric atomic character of different chalcogen atom is a method of inducing the electronic asymmetry in tetrahedral lattices (Fig.1), and then introducing polarization. Furthermore, the polarization intensity can be further increased by enhancing electronic asymmetry. In Pd₂X₃ and Ni₂X₃ (X = S, Se or Te) monolayers [Fig.1(a) and (b)], Pd atoms are positively charged by losing charges, so the positive charge center is located at the diagonal center of the rectangular lattice. While the top and bottom negatively charged X atoms are located along the x- and y-axis of the rectangular lattice. Therefore, the positive charge center is coincided with the negative charge center, and the in-plane and out-of-plane piezoelectric effect are missing in the Pd₂X₃ and Ni₂X₃ structures. However, this balance is broken by replacing different chalcogen atoms with intrinsically asymmetric atomic character (from phase I to II). In other words, replacing one side X atoms with Y atoms to obtain the Pd₄X₃Y₃ and Ni₄X₃Y₃ with piezoelectricity. Finally, further enhance the piezoelectricity by strengthening electronic asymmetry (from phase II to III). In twisted rectangular lattice Pd₄X₃Y ' ₃ monolayer, increasing the difference between the top and bottom chalcogen atoms will cause the bottom chalcogen atoms to lose electrons and show positive charge, therefore strengthen the out-of-plane piezoelectricity. While in Ni₄X₃Y ' ₃ monolayer, the bottom layer atoms still being negatively charged when further increasing the difference between X and Y ' atoms. However, this behavior increases the dipole moment between the positive and negative charge centers, thus can strengthen the piezoelectricity of Ni₄X₃Y ' ₃. In addition to rectangular lattice structure, other 2D structures (such as the TMDs with the 1T phase [46]) can also introduce high vertical polarization by following the same strategy.

Fig.1 Schematic diagram of induced out-of-plane polarization in the 2D rectangular structure lattice (from phase I to III). The purple arrow indicates the polarization vector point toward the −z axis direction.

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Top and side views of the fully optimized M₄X₃Y₃ (M = Pd/Ni, X/Y = S, Se or Te, X ≠ Y) monolayers with a X−X−M−Y−Y five-layer atomic structure as shown in Fig.2(a). These Pd₄X₃Y₃ and Ni₄X₃Y₃ monolayers can be constructed by replacing the chalcogen atoms on one side of the Pd₂X₃ and Ni₂X₃ (X = S, Se or Te) structures [34, 47-49] with different chalcogen atoms. Different from the Pd₂X₃ and Ni₂X₃ monolayers, Pd₄X₃Y₃ and Ni₄X₃Y₃ monolayers with mirror symmetry breaking contain the space group Pmm2 (No. 25). A unit cell of a M₄X₃Y₃ monolayer consists of four M metal atoms, three chalcogen X atoms and Y chalcogen atoms, forming a rectangular structure. In the unit cell of a M₄X₃Y₃ monolayer, six different types of bonds can be found. As shown in Fig.2(b), four 2X−M−2Y pentatomic planes form a M₄X₃Y₃ structure. The lattice parameters a and b (in Table S1) for the fully-relaxed Pd₄S₃Se₃, Pd₄S₃Te₃, Pd₄Se₃Te₃, Ni₄S₃Se₃, Ni₄S₃Te₃ and Ni₄Se₃Te₃ monolayers are 5.87 Å and 6.03 Å, 5.92 Å and 6.27 Å, 6.05 Å and 6.39 Å, 5.32 Å and 5.76 Å, 5.35 Å and 6.20 Å, 5.46 Å and 6.38 Å, respectively. Layer thickness h of Pd₄S₃Se₃, Pd₄S₃Te₃, Pd₄Se₃Te₃, Ni₄S₃Se₃, Ni₄S₃Te₃ and Ni₄Se₃Te₃ monolayers are 3.73 Å, 3.94 Å, 4.05 Å, 3.58 Å, 3.79 Å and 3.91 Å, respectively. These results are completely consistent with that of previous reports [37, 48]. As is well known that a large electron localization function (ELF) value (> 0.5) corresponds to a covalent bond, whereas the ionic bond is represented by a smaller ELF value (< 0.5). An ELF value of 0.5 represents the metallic bond. Herein, according to analysis of the ELF diagrams [Fig.2(c)], two adjacent X (Y) atoms on the upper X and lower Y planes along the y lattice form the covalent bonds in a M₄X₃Y₃ monolayer. According to the ELF map of the 2X−M−2Y pentatomic planes, each M atom form two ionic bonds with two X and Y atoms, respectively. Values of d_M-X1, d_M-X2, d_M-Y1, d_M-Y2, d_X-X and d_Y-Y (in Table S1 of the ESM) have the same order trend in M₄X₃Y₃ monolayers, i.e., Pd₄S₃Se₃ < Pd₄S₃Te₃ < Pd₄Se₃Te₃, Ni₄S₃Se₃ < Ni₄S₃Te₃ < Ni₄Se₃Te₃. Due to the different electronegativity of X and Y atoms, these monolayers contain the asymmetric structure, which is critical for the introduction of vertical polarization P pointing to the −z direction [Fig.2(a), the black arrow].

Fig.2 (a) Top and side views of the M4X3Y3 monolayers. (b) Four 2X−M−2Y pentatomic planes form a M4X3Y3 structure. (c) Electron localization function (ELF) map of the top X, middle M, bottom Y layers, and the ELF map of the 2X−M−2Y pentatomic planes.

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M₄X₃Y₃ monolayers contain great dynamic stabilities demonstrated by the phonon spectra with no appreciable imaginary frequency in the first Brillouin zone (Fig.3). Except the Ni₄Se₃Te₃, the maximum frequencies in the phonon spectrum are contributed by the atoms with the largest electronegativity. For example, the maximum frequency of Pd₄S₃Se₃ is contributed by S atoms with largest electronegativity of 2.58, showing the mechanical robustness of the covalent S−S bonds. In addition, they also exhibit great thermal stability properties in ambient temperature [48] and high temperature, verifying by the results of the AIMD simulations (Fig. S2 of the ESM). From Fig. S3 of the ESM, the spin-orbit coupling (SOC) has little influence on the electronic properties for these M₄X₃Y₃ monolayers, and their non-magnetic state can be maintained. The bandgaps for M₄X₃Y₃ monolayer reduce a little, and the maximum shifting value is about 0.02 eV, accounting for 8%. Results of electronic band structures at HSE level (Fig. S4 of the ESM) suggest they are indirect semiconductors with band gaps of 1.46 eV (Pd₄S₃Se₃), 0.54 eV (Pd₄S₃Te₃) and 1.41 eV (Pd₄Se₃Te₃), 1.65 eV (Ni₄S₃Se₃), 0.66 eV (Ni₄S₃Te₃) and 0.79 eV (Ni₄Se₃Te₃), which is consistent with previous study [48]. Accordingly, the semiconductor characteristics and flexible mechanical properties are beneficial to the spontaneous introduction of excellent piezoelectricity in M₄X₃Y₃ monolayers.

Fig.3 (a−f) The Phonon spectra with the projected phonon density of states (PhDOS) for M4X3Y3 monolayers.

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Piezoelectricity is closely related with the charge distributions. The electronegativities of Pd, Ni, S, Se, and Te atoms are 2.20, 1.91, 2.58, 2.55, and 2.10, respectively, leading the uneven charge distributions in M₄X₃Y₃ monolayers. Therefore, as shown in Fig. S5 of the ESM, the vacuum energy levels of the top and bottom surfaces in M₄X₃Y₃ monolayers are unequal, with the discontinuities ∆Φ of 0.58 eV (for Pd₄S₃Se₃), 1.20 eV (for Pd₄S₃Te₃) and 0.63 eV (for Pd₄Se₃Te₃), 0.62 eV (for Ni₄S₃Se₃), 1.27 eV (for Ni₄S₃Te₃) and 0.66 eV (for Ni₄Se₃Te₃), which are larger than that of some previous studies [50-52]. The vacuum energy levels discontinuities ∆Φ allow M₄X₃Y₃ monolayers to generate a vertical built-in electric field, favoring the introduction of the out-of-plane polarization. Uneven charge distribution in the M₄X₃Y₃ monolayers can also be confirmed by the planar-average charge density (Fig.4) along the z direction. The charge density difference is calculated by [53, 54]: $\mathrm{\Delta }\rho \left(\mathit{r}\right)=$$\rho \left({\mathrm{M}}_4{\mathrm{X}}_3{\mathrm{Y}}_3\right)-{{\sum }_{\mu }\rho }_{atom}(\mathit{r}-{\mathit{R}}_{\mu })$, where the ρ(M₄X₃Y₃) is the charged density of M₄X₃Y₃ monolayers, and ${{\sum }_{\mu }\rho }_{atom}(\mathit{r}-{\mathit{R}}_{\mu })$ is the superposition of charge densities of each atom. Taking the Pd₄S₃Se₃ monolayer as example, each Pd atoms with smallest electronegativity lose electrons of 0.237 |e| and are positively charged, while S and Se atoms gain electrons and are negatively charged. Interestingly, in Pd₄S₃Te₃ and Pd₄Se₃Te₃ monolayers, the Te atoms with the smallest electronegativity lose electrons and are positively charged, while the S, Se and Pd atoms gain electrons and negatively charged.

Fig.4 (a−f) planar-average charge density along the z direction for the M4X3Y3 monolayers. Yellow and blue regions refer to the electron accumulation and depletion with 0.006 e/bohr3 isosurfaces.

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We found that the five M₄X₃Y₃ monolayers possess great mechanical stability, C_kl (in Table S2) meet the meet the Born-Huang mechanical stability rules [55]: (C₁₁C₂₂−C₁₂C₁₂) > 0 and C₆₆ > 0). However, the Ni₄S₃Te₃ monolayer does not have the mechanical stability, so we then focus on the piezoelectric properties of other five M₄X₃Y₃ monolayers. From Fig. S6 of the ESM, M₄X₃Y₃ monolayers exhibit anisotropic elastic properties verified by the anisotropic Young’s modulus Y(θ) and Poisson’s ratio v(θ). Compared with previous reported 2D monolayers, such as the graphene [55] and MoX₂ (X = S, Se or Te) [7], M₄X₃Y₃ monolayers contain smaller Y(θ) and larger v(θ), demonstrating that M₄X₃Y₃ are soft and possess great flexibility. These flexible mechanical properties contribute to the spontaneous piezoelectricity of M₄X₃Y₃ monolayers. Especially, Ni₄Se₃Te₃ possess the smallest Y(θ) and largest v(θ), reflecting its strongest flexibility and strongest piezoelectric response. The piezoelectric stress and strain coefficients for M₄X₃Y₃ monolayers are calculated to have a visual insight into their out-of-plane piezoelectricity. The coupling between the electrical polarization P_i and the stress ε_jk and stain σ_jk tensor can be described by the third-rank tensor e_ijk = ∂P_i/∂ε_jk and d_ijk = ∂P_i/∂σ_jk, i, j and k represent the x, y and z directions. By using the contracted Voigt natation, e_ijk and d_ijk are simplified as the second-order tensors e_il and d_ik. Subscript i represents the x, y and z directions, being represented by the numbers 1, 2 and 3. And the subscripts l and k refer to the second-tensor xx, yy, zz, yz, zx and xy, which can be represented by the 1, 2, 3, 4, 5 and 6 numbers. The relationship between e_il, d_ik and the elastic stiffness tensor C_kl is [53, 54]

(1)$e_{il}=d_{ik}{C}_{kl}.$

The 2D materials essentially contain a certain thickness and can be subjected with stress and strain. Particularly, the out-of-plane stress and strain of 2D materials play a key role in practical application scenarios such as nano-self-powered devices and robot bionic skin. Herein, the in-plane and out-of-plane stress and strain for 2D material are all considered for researching the complete piezoelectric stress and strain matrix for M₄X₃Y₃ monolayers. The other works also considered the out-of-plane stress and strain for 2D material [31, 53, 54]. For these M₄X₃Y₃ monolayers with the space group Pmm2, the matrix e_il and C_kl are expressed as

(2)$e_{il}=\left(\begin{array}{cccccc}0& 0& 0& 0& e_{15}& 0\\ 0& 0& 0& e_{24}& 0& 0\\ e_{31}& e_{32}& e_{33}& 0& 0& 0\end{array}\right),$

(3)$C_{kl}=\left(\begin{array}{cccccc}{C}_{11}& C_{12}& C_{13}& 0& 0& 0\\ C_{21}& C_{22}& C_{23}& 0& 0& 0\\ C_{31}& C_{32}& C_{33}& 0& 0& 0\\ 0& 0& 0& C_{44}& 0& 0\\ 0& 0& 0& 0& C_{55}& 0\\ 0& 0& 0& 0& 0& C_{66}\end{array}\right),$

from the formulas (1), (2) and (3), then the third-order piezoelectric strain d_ik tensor is expressed by

(4)$d_{ik}=\left(\begin{array}{cccccc}0& 0& 0& 0& d_{15}& 0\\ 0& 0& 0& d_{24}& 0& 0\\ d_{31}& d_{32}& d_{33}& 0& 0& 0\end{array}\right)$

Therefore, according to Eqs. (1)−(4), the out-of-plane piezoelectric strain coefficients d₃₁, d₃₂ and d₃₃ are expressed by [53]

(5)$d_{31}={\displaystyle \frac{A{e}_{31}+B{e}_{32}+{Ce}_{33}}{G}},$

(6)$d_{32}={\displaystyle \frac{D{e}_{32}+E{e}_{33}+{Be}_{31}}{G}},$

(7)$d_{33}={\displaystyle \frac{F{e}_{33}+C{e}_{31}+{Ee}_{32}}{G}},$

where A = C₂₂C₃₃ − C₂₃C₂₃, B = C₂₃C₁₃ − C₁₂C₃₃, C = C₁₂C₂₃ − C₁₃C₂₂, D = C₃₃C₁₁ − C₁₃C₁₃, E = C₁₃C₁₂ − C₂₃C₁₁, F = C₂₂C₁₁ − C₁₂C₁₂, G = C₂₂(C₁₁C₃₃ − C₁₃C₁₃) + C₁₂(C₂₃C₁₃ − C₁₂C₃₃) + C₂₃(C₁₃C₁₂ − C₁₁C₂₃). Except the Ni₄S₃Te₃ without mechanical stability, values of piezoelectric stress e_il and strain d_ik coefficients for other five M₄X₃Y₃ monolayers are listed in Tab.1.

Tab.1 The out-of-plane piezoelectric stress eil (pC/m) and strain dik (pm/V) coefficients for five fully relaxed M4X3Y3 monolayers.

Structure	e31	e32	e33	d31	d32	d33
Pd4S3Se3	37.42	27.97	30.22	0.50	0.46	3.92
Pd4S3Te3	98.27	74.33	94.78	1.87	0.17	11.10
Pd4Se3Te3	86.78	72.34	89.28	1.54	0.32	10.45
Ni4S3Se3	77.20	61.41	64.99	0.92	0.17	2.53
Ni4Se3Te3	551.54	577.20	533.26	7.93	5.81	61.57

Our results show that the strong piezoelectric materials can be obtained by the electron asymmetry through the intrinsically asymmetric atomic character of different chalcogen atoms. And it has also been confirmed that stronger piezoelectric materials can be obtained by further enhancing electronic asymmetry. As shown in Fig.5(a), in Pd₂S₃ monolayer, when the top S bottom layer S atoms are replaced by Se atoms, the Pd₄S₃Se₃ monolayer is constructed, which contains the out-of-plane piezoelectricity. Importantly, the piezoelectricity is increased with the enhancing electron asymmetry, which increased by the introduction of Te atoms with smaller electronegativity of 2.1. Therefore, the Pd₄S₃Te₃ and Pd₄Se₃Te₃ monolayers possess stronger piezoelectricity than Pd₄S₃Se₃. As shown in Tab.1, for Pd₄X₃Y₃ monolayers, their out-of-plane piezoelectric strain coefficients d₃₁ (ranges from 0.5 to 1.54 pm/V), d₃₂ (0.46 to 70.32 pm/V) and d₃₃ (3.92 to 10.45 pm/V), which are comparable to and even larger than many reported 2D materials [22, 56-59]. It is worth mentioning that outmost d₃₃ = 11.10 pm/V is found in Pd₄S₃Te₃ due to its strongest electron asymmetry among Pd₄X₃Y₃ monolayers. Moreover, the out-of-plane piezoelectricity is also introduced in Ni₄S₃Se₃ via replacing the top layer S atoms with Se atoms in Ni₂S₃ [Fig.5(b)]. We further replace bottom layer S atoms with the Te atom to enhance electron asymmetry, thereby obtaining a strongest piezoelectric Ni₄Se₃Te₃ monolayer. Benefitting from the smallest Y(θ) and largest v(θ) among all M₄X₃Y₃ monolayers, the Ni₄Se₃Te₃ has the strongest out-of-plane piezoelectricity, with the d₃₁ = 7.93 pm/V, d₃₂ = 5.81 pm/V and d₃₃ = 61.57 pm/V (Tab.1). The out-of-plane piezoelectricity of M₄X₃Y₃ monolayers is much larger than that of many 2D materials (such as the oxygen functionalized MXenes (0.40−0.78 pm/V) [60], Janus TMDCs monolayers (0.03 pm/V) [30], Janus group-III materials (0.46 pm/V) [61] and In₂Se₃ (0.415 pm/V) [62]), and is comparable to that of MoSTe multilayers (5.7−13.5 pm/V) [30]. These strong out-of-plane polarization properties are originating from the uneven charge distribution due to electronegativity difference between X and Y atoms.

Fig.5 (a, b) Schematic diagram of induced out-of-plane polarization in Pd2S3 and Ni2S3. The purple arrow indicates the polarization vector point toward the −z axis direction.

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We then use the electronegativity difference (R_ed) to analyze piezoelectric changes of the five mechanically stable M₄X₃Y₃ monolayers. The ratio of electronegativity difference (R_ed) for M₄X₃Y₃ monolayers can be expressed as

(8)$R_{\mathrm{e}\mathrm{d}}={\displaystyle \frac{|X_{\mathrm{e}\mathrm{l}\mathrm{n}}-M_{\mathrm{e}\mathrm{l}\mathrm{n}}|}{|Y_{\mathrm{e}\mathrm{l}\mathrm{n}}-M_{\mathrm{e}\mathrm{l}\mathrm{n}}|}},$

where X_eln, M_eln, and Y_eln represent the electronegativity of X (upper), Pd/Ni (middle) and Y (lower) atoms, respectively. The electronegativity of Pd, Ni, S, Se and Te atoms are 2.20, 1.91, 2.58, 2.55 and 2.10, respectively. Therefore, the calculated R_ed for Pd₄S₃Se₃, Pd₄S₃Te₃, Pd₄Se₃Te₃, Ni₄S₃Se₃ and Ni₄Se₃Te₃ monolayers is 1.09, 3.80, 3.50, 1.05 and 3.37, respectively. In Fig.6 and Fig. S7 of the ESM, we find that the out-of-plane piezoelectric stress (e₃₃) and strain coefficients (d₃₃) increases with the increase of the differential charge density difference, which is consistent with previous studies, such as MoTO (T = S, Se or Te) monolayers [31], and the group-III(A) Janus hydrofluoride monolayers [53]. In other words, the intrinsically asymmetric atomic character of X and Y atoms can effectively regulate the piezoelectricity of M₄X₃Y₃ monolayers.

Fig.6 The out-of-plane piezoelectric stress (e33) and strain coefficients (d33) as a function of the electronegativity difference ratio Red for Pd4X3Y3 monolayers.

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In Pd₄X₃Y₃ monolayers, the position of positive and negative charge centers along the z direction is remarked in Fig.7(a). The vertical distance (characterized by l) between positive and negative charge centers and the total polarized charge value of centers (characterized by q) are both important factors affecting the electric dipole moment P (P = ql). Pd₄S₃Se₃, Pd₄S₃Te₃ and Pd₄Se₃Te₃ monolayers have l₁ = 0.535 Å, l₂ = 2.093 Å and l₃ = 2.971 Å, respectively. And the q for Pd₄S₃Se₃, Pd₄S₃Te₃ and Pd₄Se₃Te₃ monolayers is 0.947 e/cell, 0.700 e/cell and 0.345 e/cell, respectively. Therefore, Pd₄S₃Se₃ monolayer possesses the smallest electric dipole moment P of 0.507 eÅ/cell, corresponding to its smallest piezoelectric coefficients e₃₃ and d₃₃. Compared to Pd₄Se₃Te₃ with the P of 1.025 eÅ/cell, the Pd₄S₃Te₃ contains the largest electric dipole moment P of 1.464 eÅ/cell and the largest e₃₃ and d₃₃. Moreover, as shown in Fig. S7 of the ESM, the Ni₄Se₃Te₃ with P of 0.814 eÅ/cell has lager e₃₃ and d₃₃ than that of Ni₄S₃Se₃ monolayer with P of 0.379 eÅ/cell. Obviously, the out-of-plane piezoelectricity of Pd₄X₃Y₃ monolayers is positively correlated with the electric dipole moment P [Fig.7(b)]. The interesting relationship between piezoelectricity and electronegativity difference ratio R_ed, as well as the interesting relationship with electric dipoles P, provides us with new ideas for the design of excellent piezoelectric nanomaterials. In short, the intrinsically asymmetric atomic character is important in regulating piezoelectricity of 2D materials. The ultra-high out-of-plane piezoelectricity makes Pd₄X₃Y₃ monolayers possessing significant potential applications in piezoelectric nano-devices, such as the collection of blue energy (wind, waves, etc.) and the conversion of tiny mechanical energy (human walking, body shaking, and hand touch).

Fig.7 (a) Position of the positive and negative charge centers in Pd4X3Y3 along the z direction. Blue arrows are the vertical distance l between positive and negative charge centers; black arrows are the electric dipole moments P; H1, H2, H3, and H4 represent the vertical distances between two adjacent atomic layers, respectively. (b) The e33 and d33 as a function of the electric dipole moment P for Pd4X3Y3 monolayers.

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4 Conclusions

In summary, we systematically investigated a family of 2D materials with strong out-of-plane piezoelectricity, namely M₄X₃Y₃ (X/Y = S, Se or Te, X ≠ Y) monolayers. Uneven charge distribution in favors of the generation of out-of-plane polarization. Ni₄Se₃Te₃ monolayer possesses the remarkable out-of-plane piezoelectricity with largest d₃₃ of 61.57 pm/V, which is much larger than that of most known 2D materials. Importantly, we proposed an achieving strategy for introducing strong piezoelectricity in 2D materials via the electronic asymmetry induced by the intrinsically asymmetric atomic character of different chalcogen atoms. And this strategy has been validated in M₄X₃Y₃ materials, proving that the piezoelectricity is greatly regulated by enhancing the electronic asymmetry. Electronegativity difference ratio R_ed and electric dipole moment P effectively predict the strength of out-of-plane piezoelectricity for 2D materials because piezoelectric coefficients are related with the R_ed and P. These M₄X₃Y₃ nanomaterials with strong out-of-plane piezoelectricity can serve as alternative materials for energy-harvesting nanodevices or self-powered wearables. Our work offers useful design guidelines for the discovery of 2D strong piezoelectric materials.

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Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Electronic supplementary materials

The online version contains supplementary material available at https://doi.org/10.1007/s11467-023-1348-5 and https://journal.hep.com.cn/fop/EN/10.1007/s11467-023-1348-5.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 11474123).

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