Layered group-IV-V-VI semiconductors with superior piezoelectricity and ferroelectricity

Abstract

Two-dimensional (2D) layered materials lacking inversion symmetry are promising candidates for nano-electromechanical and electronic technologies. Building upon the experimentally synthesized SnP{}_2Se{}_6 crystal, we computationally propose a novel series of inherently asymmetric group-IV-V-VI semiconductors through elemental substitution. This series encompasses monolayers of SiP{}_2Se{}_6, SiAs{}_2S{}_6, SiAs{}_2Se{}_6, SiAs{}_2Te{}_6, GeP{}_2S{}_6, GeP{}_2Se{}_6, GeAs{}_2S{}_6, GeAs{}_2Se{}_6, GeSb{}_2S{}_6, SnP{}_2S{}_6, SnP{}_2Se{}_6, SnAs{}_2S{}_6, SnAs{}_2Se{}_6, SnAs{}_2Te{}_6 and SnSb{}_2S{}_6. All these structures exhibit robust energetic and dynamic stability, along with excellent resistance to oxidation when exposed to air. These monolayers have moderate bandgaps ranging from 1.04 to 2.36 eV; notably, the monolayers of SiAs{}_2Se{}_6, SiAs{}_2Te{}_6, GeP{}_2S{}_6, GeAs{}_2Se{}_6 and SnAs{}_2Te{}_6 are direct-bandgap semiconductors. More importantly, these group-IV-V-VI monolayers exhibit significant in-plane piezoelectricity, with coefficients (d₁₁) as high as 22.94 pm/V. Additionally, SnP{}_2Se{}_6 bilayer exhibit significant out-of-plane piezoelectricity (d₃₃) with coefficients exceeding 5 pm/V. Moreover, the coexistence of room-temperature ferroelectricity in these monolayers is unambiguously confirmed by their intrinsic spontaneous polarization and remarkably low ferroelectric switching barriers. These findings establish a new paradigm for integrating piezoelectricity and ferroelectricity with diverse 2D materials, thereby opening exciting opportunities for designing multifunctional materials and developing novel electronic devices.

Graphical abstract

Keywords

piezoelectricity / ferroelectricity / layered group-IV-V-VI semiconductors / non-centrosymmetry

1 Introduction

Insulators or semiconductors lacking inversion symmetry exhibit the piezoelectric effect, characterized by strong electromechanical coupling that enables bidirectional energy conversion between mechanical and electrical forms [1, 2]. Such materials have found extensive applications in mechanical sensors, actuators, and energy harvesters [3−12]. Wurtzite-structured materials (e.g., α-quartz, GaN, AlN) represent typical piezoelectric candidates; however, their low piezoelectric coefficients (ranging from 2.3 to 5.1 pm/V) restrict energy conversion efficiency [13−15]. Beyond piezoelectrics, ferroelectric materials with switchable spontaneous polarization hold promise for low-power electronics. Nevertheless, traditional perovskite oxides (e.g., PbTiO₃, BaTiO₃) undergo ferroelectric degradation when scaled below a critical thickness, primarily due to depolarization fields and quantum confinement effects [16−19].

Significant interest has arisen in two-dimensional (2D) materials with built-in polarization due to their extreme thinness and distinct electronic behavior compared to bulk counterparts [20−22]. Atomic-layer control via van der Waals epitaxy techniques preserves their polarization switching capabilities [23−25]. However, existing 2D piezoelectric materials (e.g., monolayer MoS₂ [26], h-BN [27] and SnS₂ [28]) exhibit weak responses. For instance, 2H-MoS₂ possesses a piezoelectric polarization (e₁₁) of approximately 2.9 × 10⁻¹⁰ C/m [29], with energy conversion rates below 5% [26]. Additionally, theoretical predictions indicate that their piezoelectric coefficients generally remain below 10 pm/V [8, 30, 31]. For 2D ferroelectrics, although in-plane (e.g., group-IV monochalcogenides [32]), out-of-plane (e.g., In₂Se₃ [33], CuInP₂S₆ [34]), and intercalation-type (e.g., III₂–VI₃ compounds [33]) systems have been experimentally and theoretically investigated, challenges associated with large-area synthesis hinder their practical applications.

More recently, millimeter-scale SnP₂Se₆ nanosheets synthesized using chemical vapor transport [35, 36] have garnered attention due to their exceptional properties. These include layer-independent second-harmonic generation with a second-order nonlinear susceptibility of 1.3 × 10⁻⁹ m·V⁻¹ at 1550 nm [37]; high thermal conductivity in the range of 1.4−5.7 W·m⁻¹·K⁻¹ [38]; and superior electron mobility (>100 cm²·V⁻¹·s⁻¹) and on/off ratios (>10⁶ at room temperature) in field-effect transistors [39]. The robust van der Waals structure of SnP₂Se₆ renders it an ideal platform for material design through elemental substitution. Despite these intriguing properties, the piezoelectric and ferroelectric characteristics of monolayer SnP₂Se₆ and its potential derivatives remain largely unexplored.

This study investigates both the monolayer and bilayer forms of SnP₂Se₆. While the monolayer serves to reveal the material’s intrinsic piezoelectric and ferroelectric properties in the 2D limit, the bilayer is examined to uncover how interlayer coupling and symmetry reduction modulate these properties, giving rise to emergent effects such as a significant out-of-plane piezoelectric response. Herein, we explore the piezoelectric and ferroelectric properties of monolayer SnP₂Se₆ and 26 related group-IV-V-VI monolayers designed through elemental substitution of group-IV (Si, Ge, Sn), group-V (P, As, Sb), and group-VI (S, Se, Te). Among them, 15 are found to be structurally stable, including the experimentally realized SnP₂Se₆, along with SiP₂Se₆, SiAs₂S₆, SiAs₂Se₆, SiAs₂Te₆, GeP₂S₆, GeP₂Se₆, GeAs₂S₆, GeAs₂Se₆, GeSb₂S₆, SnP₂S₆, SnP₂Se₆, SnAs₂S₆, SnAs₂Se₆, SnAs₂Te₆ and SnSb₂S₆. We assess their energetic and dynamic stability, and demonstrate that these materials exhibit high in-plane piezoelectric coefficients (with d₁₁ reaching up to 22.94 pm/V), significant out-of-plane piezoelectricity in bilayer SnP₂Se₆ (d₃₃ > 5.52 pm/V), and room-temperature ferroelectricity, as evidenced by intrinsic spontaneous polarization and low-energy switching barriers. These findings offer a new pathway for the development of multifunctional 2D materials for next-generation electromechanical devices.

2 Computational methods

We conducted first-principles simulations through the Vienna ab initio simulation package (VASP) [40, 41]. A plane-wave basis set incorporating an energy cutoff of 500 eV was employed, together with projector augmented wave (PAW) pseudopotentials [42] and the Perdew−Burke−Ernzerhof (PBE) generalized gradient approximation to represent the exchange-correlation functional [43, 44]. The convergence thresholds were set at 10⁻⁷ eV for total energy and 0.01 eV/Å for atomic forces. When dealing with the unit cells of 2D group-IV-V-VI systems, we sampled the Brillouin zone using a Γ-centered 13\times 13\times 1 Monkhorst-Pack k-point grid. The band structures were further refined using the HSE06 hybrid functional [45] to correct the well-known band-gap underestimation inherent in GGA calculations. The climbing-image nudged elastic band (CI-NEB) method [46] was employed to investigate O₂ adsorption kinetics and assess the chemical stability of monolayers. Each reaction pathway was mapped using five intermediate images. Relaxation of these images continued until perpendicular forces were minimized below 0.02 eV/Å. The PBE-D3 method [44] was consistently used for all structural relaxations to accurately describe van der Waals interactions, which are crucial for interlayer binding in multilayer and bulk systems as well as for the O₂ adsorption study. Dynamical stability was assessed by calculating phonon dispersion curves via the Phonopy code, utilizing density functional perturbation theory (DFPT) [47] within the VASP framework. The spontaneous polarization (P_FE) was calculated using the Berry phase approach within the modern theory of polarization [48]. This method was employed as implemented in the VASP code, which allows for the determination of the polarization difference between the ferroelectric and a reference paraelectric structure.

3 Results and discussion

3.1 Atomic structures of layered group-IV-V-VI materials

Given the successful synthesis of layered SnP₂Se₆ [36], its monolayer can be obtained via mechanical exfoliation [49]. As illustrated in Fig. 1(a), bulk SnP₂Se₆ has a pseudo-2D crystal structure: planar 2D networks of puckered hexagonal units in the xy-plane, with vdW stacking along the z-direction. Both bulk and trilayer phases crystallize in an ABC stacking, each sheet displaced by [-1/3, 1/3] (in fractional coordinates) relative to the adjacent layer. For structural models of bilayer SnP₂Se₆, the layers follow an AB stacking sequence based on this lateral shift. Note that the bilayer and trilayer models were built from the experimental bulk stacking of SnP₂Se₆. To identify the most stable configuration, we systematically tested alternative stackings via in-plane layer sliding. Full relaxation and energy comparison confirmed that the experimental-derived structures have the lowest total energy (Figs. S1 and S2). To assess the exfoliation feasibility of monolayers, we calculated the interlayer cohesive energy (E_c), defined as follows:

(1)E_{\mathrm{c}}=(E_{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}-3\times E_{\mathrm{m}\mathrm{o}\mathrm{n}\mathrm{o}\mathrm{l}\mathrm{a}\mathrm{y}\mathrm{e}\mathrm{r}})/n_0,

where E_bulk and E_monolayer denote the total energies per unit cell of bulk and monolayer SnP₂Se₆, respectively; n₀ refers to the atom count in the bulk unit cell. The calculated E_c of layered SnP₂Se₆ is -0.07 eV/atom, approximate to the values of well-established van der Waals solids, such as phosphorene (-0.055 eV/atom) [50], h-BN and graphene (~ -0.065 eV/atom) [51]. This indicates that monolayer exfoliation from the bulk crystal is highly feasible.

The unit cell of the monolayer SnP₂Se₆ is composed of one Sn atom, two P atoms, and six Se atoms, wherein each Sn atom is coordinated to six Se atoms and each P atom is bonded to three Se atoms. As depicted in Fig. 1, the optimized lattice parameters (a) of the monolayer, bilayer, trilayer, and bulk structures are 6.50 Å, 6.42 Å, 6.41 Å, and 6.41 Å, respectively (see Table 1). Population analysis [52] shows the monolayer SnP₂Se₆ consists of covalent bonds, with a bond order of 0.34 for Sn–Se bonds and 0.54 for P–Se bonds. The stability of 2D crystals is crucial for experimental synthesis and practical applications. We therefore quantitatively assess its energetic stability through formation energy (ΔH) analysis:

(2)\mathrm{\Delta }H=(E_{\mathrm{m}\mathrm{o}\mathrm{n}\mathrm{o}\mathrm{l}\mathrm{a}\mathrm{y}\mathrm{e}\mathrm{r}}-n_1\times E_1-n_2\times E_2-n_3\times E_3)/n,

where E_monolayer refers to total energy of a pristine monolayer SnP₂Se₆; E₁, E₂ and E₃ are the per-atom energies in solid Sn, P, and Se, respectively; n₁, n₂ and n₃ denote the corresponding atomic counts in the unit cell (n is the total number of atoms). The calculated ΔH of -0.29 eV/atom for the monolayer SnP₂Se₆ confirms its exothermic formation. Phonon dispersion presented in Fig. 2(a) demonstrates the dynamical stability of the monolayer SnP₂Se₆, as evidenced by the absence of any imaginary phonon modes. To evaluate the thermal stability, we performed abinitio molecular dynamics (AIMD) simulations on the monolayer SnP₂Se₆ at 500 K for 10 ps (Fig. S3). The structure remained intact throughout the simulation, confirming its robustness at room temperature.

To examine the ambient durability of monolayer SnP₂Se₆, we simulated how an O₂ molecule interacts with the monolayer across physisorption, chemisorption, and dissociative oxidation. The nature of adsorption — physisorption or chemisorption — is distinguished by several key criteria: (i) Adsorption energy: Physisorption typically involves weak van der Waals interactions with energies generally below 0.415 eV, whereas chemisorption involves stronger chemical bonding with energies often exceeding 0.415 eV [53]. (ii) Adsorbate geometry: Physisorption preserves the intrinsic bond length of the adsorbate (e.g., O−O bond in O₂), while chemisorption often involves significant bond elongation or dissociation. (iii) Electronic interaction: Negligible charge transfer is associated with physisorption, while substantial charge redistribution signifies chemisorption. Adsorption strength was quantified by the binding energy (E_bind):

(3)E_{\text{bind}}=E_{\text{total}}-E_{\text{monolayer}}-E_{{\mathrm{O}}_2},

where E_total, E_monolayer, and E_{{\mathrm{O}}_2} are the total energy of the system with adsorbed O₂, the energy of the clean monolayer, and the energy of an isolated O₂ molecule (calculated in its triplet ground state), respectively. Here, E_bind with negative value signifies favorable adsorption. The oxidation process of the monolayer SnP₂Se₆ was simulated using the minimum-energy initial and final structures identified from comprehensive adsorption site evaluations. Figure 2(b) depicts the O₂ dissociation pathway on the monolayer SnP₂Se₆, starting from physisorption and ending with two chemisorbed oxygen atoms. In light of above criteria, the initial O₂ adsorption state — with a weak E_bind of −0.10 eV and an O−O distance of 1.21 Å (close to the gas-phase value) — is identified as physisorption. Upon approach, the O−O bond length increases to 1.54 Å at the transition state, indicating bond activation. Overcoming this state requires an activation energy barrier (E_a) of 1.53 eV to reach the final chemisorbed state, where two oxygen atoms are strongly bound to the surface. Crucially, this activation energy is nearly identical to that reported for monolayer MoS₂ (1.57 eV) [54], which are benchmark materials for oxidation resistance. This comparable energy barrier suggests that the monolayer SnP₂Se₆ exhibits outstanding oxidation resistance properties. Moreover, we conducted AIMD simulations to evaluate its stability under ambient, oxygen-exposed conditions. A 3\times 3\times 1 supercell of the monolayer SnP₂Se₆ with six physisorbed O₂ molecules was simulated at 300 K for over 10 ps. Throughout the simulation, all O₂ molecules remained physisorbed on the surface, with no dissociation or chemical bond formation observed (see Fig. S4). This direct dynamical observation confirms the robust resistance to oxidation of monolayer SnP₂Se₆ at room temperature, fully consistent with the high activation barrier identified from static calculations.

To further assess the ambient stability of SnP₂Se₆ in humid environments — a critical concern for practical applications — we systematically investigated its interaction with water molecules. The propensity for deliquescence or hydrolysis is probed by examining the adsorption behavior of H₂O on the monolayer surface. We placed a H₂O molecule at various potential adsorption sites (Fig. S5) on the SnP₂Se₆ surface with initial distances < 1 Å. After full structural relaxation, all configurations converged exclusively to states characterized by physisorption with a large equilibrium distance (~2.5 Å) between H₂O and the surface, binding energies of -0.09 to -0.13 eV, unchanged H₂O molecular geometry, and negligible charge transfer (<0.03 |e|) between the water molecule and the monolayer. The weak binding suggests that water molecules do not spontaneously dissociate on the surface nor penetrate the lattice to induce structural degradation. Therefore, the monolayer SnP₂Se₆ exhibits a high resistance to water.

In light of the stability of monolayer SnP₂Se₆, we substituted Sn, P, and Se with elements from group-IV (Si, Ge, Sn), group-V (P, As, Sb), and group-VI (S, Se, Te), respectively, resulting in a series of hypothetical group-IV-V-VI compounds with structures analogous as SnP₂Se₆. Through systematic high-throughput screening of group-IV-V-VI monolayers, we finally identified 15 stable monolayers: SiP₂Se₆, SiAs₂S₆, SiAs₂Se₆, SiAs₂Te₆, GeP₂S₆, GeP₂Se₆, GeAs₂S₆, GeAs₂Se₆, GeSb₂S₆, SnP₂S₆, SnP₂Se₆, SnAs₂S₆, SnAs₂Se₆, SnAs₂Te₆ and SnSb₂S₆. Their energetic, dynamical, and thermal stability were confirmed through calculations of formation energy (ΔH, ranging from -0.02 to -0.29 eV/atom) and phonon dispersion (Fig. S6).

Importantly, the high mechanical stability of these group-IV-V-VI monolayers can be confirmed by their elastic constants. Group-IV-V-VI monolayers crystallize in the rhombohedral structures, which gives rise to six independent elastic constants (C_ij). Material constants of group-IV-V-VI monolayers are written in contracted notation as [55]

(4)[C_{ij}]=\left(\begin{array}{cccccc}{C}_{11}& C_{12}& C_{13}& C_{14}& C_{15}& C_{16}\\ C_{21}& C_{22}& C_{23}& C_{24}& C_{25}& C_{26}\\ C_{31}& C_{32}& C_{33}& C_{34}& C_{35}& C_{36}\\ C_{41}& C_{42}& C_{43}& C_{44}& C_{45}& C_{46}\\ C_{51}& C_{52}& C_{53}& C_{54}& C_{55}& C_{56}\\ C_{61}& C_{62}& C_{63}& C_{64}& C_{65}& C_{66}\end{array}\right).

For rhombohedral structures, mechanical stability requires that the elastic constants satisfy four criteria: (i) C₁₁-C₁₂>0, (ii) C₄₄>0, (iii) C₁₃\timesC₁₃C₃₃(C₁₁+C₁₂), and (iv) C₁₄\timesC₁₄C₄₄(C₁₁-C₁₂). As listed in Table S1, all calculated elastic constants for the group-IV-V-VI monolayers fulfill these conditions, ensuring their mechanical robustness.

Notably, the interlayer cohesive energies (E_c) of these monolayers are in the range of -0.06 to -0.11 eV/atom (Table 2), indicating the easy exfoliation from the bulk systems. The electronic band structures of these group-IV-V-VI monolayers, calculated via the HSE06 functional (Fig. S7), display bandgaps ranging from 1.04 to 2.36 eV. Among them, SiAs₂Se₆, SiAs₂Te₆, GeP₂S₆, GeAs₂Se₆ and SnAs₂Te₆ monolayers have direct bandgaps. Since layered SnP₂Se₆ has been successfully synthesized experimentally, we calculated the electronic properties of its bilayer, trilayer, and bulk forms, which have bandgaps of 1.34, 1.28, and 1.21 eV, respectively (Fig. 3). A gradual reduction in the bandgap is observed in two-dimensional SnP₂Se₆ as the number of layers increases.

3.2 Piezoelectricity of layered group-IV-V-VI materials

Group-IV-V-VI monolayers belong to the 32-point group (No. 149), with six independent elastic constants (C_ij) and two independent piezoelectric polarizations (e_ij) [56]. Material constants of group-IV-V-VI monolayers are written in contracted notation as

(5)[C_{ij}]=\left(\begin{array}{cccccc}{C}_{11}& C_{12}& C_{13}& C_{14}& 0& 0\\ C_{12}& C_{11}& C_{13}& -C_{14}& 0& 0\\ C_{13}& C_{13}& C_{33}& 0& 0& 0\\ C_{14}& -C_{14}& 0& C_{44}& 0& 0\\ 0& 0& 0& 0& C_{44}& C_{14}\\ 0& 0& 0& 0& C_{14}& C_{66}\end{array}\right),

(6)[e_{ij}]=\left(\begin{array}{cccccc}{e}_{11}& -e_{11}& 0& e_{14}& 0& 0\\ 0& 0& 0& 0& -e_{14}& -e_{11}\\ 0& 0& 0& 0& 0& 0\end{array}\right).

Correspondingly, the piezoelectric coefficients are derived as:

(7)[d_{ij}]=[e_{ij}][C_{ij}{]}^{\mathrm{T}-1},

where [C_ij]^T−1 is the inverse of the transpose [C_ij]^T. For these monolayer systems, we focus on the elastic constants C₁₁ and C₁₂, and the piezoelectric coefficient d₁₁. [C_ij] can be obtained via six lattice distortions (strain-stress relationships) [57]. Subsequently, we calculated the [e_ij] by using density functional perturbation theory (DFPT) [58, 59]. Finally, d₁₁ can be obtained by solving the formula provided in Eq. (7).

As listed in Table 2, the calculated elastic stiffness coefficient C₁₁ (21.11−48.23 N/m) and C₁₂ (5.85−18.10 N/m) for group-IV-V-VI monolayers are smaller than those of graphene (C₁₁ = 358.1 N/m and C₁₂ = 60.4 N/m) [60], MoS₂ (C₁₁ = 130 N/m and C₁₂ = 32 N/m) [31], and InSe (C₁₁ = 51 N/m and C₁₂ = 12 N/m) [30], indicating greater flexibility — attributed to soft group-V/VI elements and covalent bonding. Importantly, this mechanical softness is coupled with a preserved electronic structure under moderate strain, a key feature for reliable flexible electronics. Furthermore, the low stiffness implies a lower energy cost for inducing strain, which is advantageous for strain engineering of their piezoelectric and electronic properties. Thus, the discussed mechanical properties establish a crucial link between the atomic-scale structure and the macroscopic device performance, highlighting a materials design strategy that balances deformability with functional robustness.

Furthermore, the polarization vector of e₁₁ (0.23 × 10⁻¹⁰ C/m to 3.39 × 10⁻¹⁰ C/m) is comparable to 3.64 × 10⁻¹⁰ C/m of MoS₂ monolayer (e₁₁ = 3.64 × 10⁻¹⁰ C/m) [31] and larger than h-BN monolayer (e₁₁ = 1.38 × 10⁻¹⁰ C/m) [31] and InSe (0.57 × 10⁻¹⁰ C/m) [30]. According to Eq. (6), the calculated piezoelectric coefficients d₁₁ can be obtained 1.24−22.94 pm/V, as detailed in Table 2 and Fig. 4(a). A clear trend emerges in group-IV-V-VI monolayers: the piezoelectric values increase monotonically with the atomic number within the same elemental group. Notably, SiAs₂Te₆, SnP₂S₆, SnAs₂Se₆ and SnAs₂Te₆ monolayers exhibit exceptionally high piezoelectric coefficients exceeding 10 pm/V, which is one or two order of magnitude larger than the values of MoS₂ monolayer (3.74 pm/V) [31], h-BN monolayer (0.60 pm/V) [31] and InSe monolayer (1.46 pm/V) [30]. This is attributed to large spontaneous polarization and Born effective charges (BEC) [61] (12.31e−23.69e), with d₁₁ linearly correlated to both (Fig. 5), for group-IV-V-VI monolayer structures, the greater the piezoelectric coefficient, the larger the elastic tensor and effective charge, highlighting their potential for 2D piezoelectric sensors and nanogenerators. Note that BEC serves as a crucial indicator for ferroelectric and piezoelectric materials, as it directly quantifies the extent of covalent hybridization and dynamic charge transfer [62−64]. This understanding has been validated in low-dimensional polar systems like GeC and AgBiP₂Se₆ [65, 66], where a similarly strong correlation between piezoelectric coefficient and BEC is obtained. Here, the enhanced BEC directly intensifies the strain-induced polarization response, demonstrating the continuity of this fundamental structure-property relationship across dimensional scales.

For experimentally synthesized SnP₂Se₆, we examined the impact of the number of layers on the piezoelectric effects. Bilayer, trilayer, and bulk SnP₂Se₆ belong to the 3-point group, with material constants detailed as follows:

(8)[C_{ij}]=\left(\begin{array}{cccccc}{C}_{11}& C_{12}& C_{13}& C_{14}& -C_{25}& 0\\ C_{12}& C_{22}& C_{13}& -C_{14}& C_{25}& 0\\ C_{13}& C_{13}& C_{33}& 0& 0& 0\\ C_{14}& -C_{14}& 0& C_{44}& 0& 0\\ -C_{25}& C_{25}& 0& 0& C_{44}& C_{14}\\ 0& 0& 0& C_{25}& C_{14}& C_{66}\end{array}\right),

(9)[e_{ij}]=\left(\begin{array}{cccccc}{e}_{11}& -e_{11}& 0& e_{14}& e_{15}& e_{22}\\ -e_{22}& e_{22}& 0& e_{15}& -e_{14}& -e_{11}\\ e_{31}& e_{31}& e_{33}& 0& 0& 0\end{array}\right).

Similarly, we calculated elastic stiffness coefficients and piezoelectric coefficients using Eqs. (8) and (9). As presented in Table 1, elastic constants and piezoelectric effects are dependent on the number of layers. C₁₁ and C₂₂ (72.87−115.88 N/m) increase with layer number, exceeding the monolayer (~30.25 N/m), indicating enhanced stiffness of bilayer, trilayer and bulk SnP₂Se₆ in comparison with that of monolayer. Multilayer SnP₂Se₆ also appears strong piezoelectric effect. The reduced symmetry in the bilayer, trilayer, and bulk systems results in a more typical array of piezoelectric coefficients, including d₁₁, d₂₂, d₃₁ and d₃₃, compared with one coefficient d₁₁ of monolayer SnP₂Se₆. As listed in Table 1 and Fig. 4(b), bilayer, trilayer and bulk SnP₂Se₆ possess d₁₁ values (5.22, 3.94 and 2.07 pm/V, respectively), which are smaller than the monolayer (6.80 pm/V), likely due to in-plane depolarizing fields from adjacent layers. Notably, multilayer SnP₂Se₆ generates strong out-of-plane piezoelectricity. Especially, piezoelectric coefficient d₃₁ for the multilayer is in the range of 0.20−1.28 pm/V, which are approximate to the typical piezoelectric materials, i.e., 1.9 pm/V for bulk AlN and 1.6 pm/V for bulk GaN [13−15]. Moreover, piezoelectric coefficient d₃₃ of layered SnP₂Se₆ can reach 5.52 pm/V ― exceeding bulk AlN (5.1 pm/V) and GaN (2.6 pm/V) [13−15].

The emergence of significant out-of-plane piezoelectric coefficients (d₃₁ and d₃₃) in SnP₂Se₆ bilayer originates from the symmetry reduction upon stacking and the resultant interlayer electromechanical coupling. While the monolayer (point group 3 m) exhibits in-plane polarization, the stacking into a bilayer (point group 3) lowers the symmetry, which permits new independent components in the piezoelectric tensor, including those for out-of-plane responses. Within this symmetry-allowed framework, the microscopic mechanism is governed by interlayer coupling. Specifically, d₃₁ arises as in-plane strain modulates the intralayer polar bonds; this altered polarization state is then coupled across the van der Waals gap via the interlayer interaction, inducing out-of-plane polarization. Conversely, d₃₃ directly probes the sensitivity of the interlayer coupling itself to out-of-plane strain: vertical deformation changes the interlayer orbital overlap, efficiently driving charge redistribution along the stacking direction. The considerable magnitudes of both d₃₁ and d₃₃ thus highlight the efficiency of this symmetry-enabled, interlayer charge-response pathway in bilayers SnP₂Se₆.

Furthermore, our calculations indicate that the exceptional out-of-plane piezoelectricity is not limited to SnP₂Se₆. For instance, few-layer structures of SiAs₂Te₆ — whose monolayer already possesses the highest in-plane coefficient (d₁₁ ≈ 22.94 pm/V) — also exhibit enhanced out-of-plane responses (d₃₁, d₃₃, Table S2) compared to their SnP₂Se₆ counterparts at equivalent thickness. This suggests that the superior piezoelectric performance identified in the monolayer screening can translate effectively into few-layer systems.

3.3 Ferroelectricity of layered group-IV-V-VI materials

Ferroelectricity in group-IV-V-VI monolayers requires not only broken inversion symmetry but also switchable spontaneous polarization. Given that layered SnP₂Se₆ has been prepared experimentally, it would be employed as a prime example for exploring ferroelectric properties. We identified a centrosymmetric paraelectric (PE) phase [SnSe₆ cluster at the unit cell center, Fig. 6(a)] with 0.41 eV higher energy than the ferroelectric (FE) phase, indicating PE instability and a tendency to transition to FE below the Curie temperature (T_C). Within Landau-Ginzburg theory [67], free energy (E) and polarization (P) satisfy:

(10)E={\displaystyle \sum _i[{\displaystyle \frac{\alpha }{2}}{(P_i)}^{{}^2}+{\displaystyle \frac{\beta }{2}}{(P_i)}^4]+{\displaystyle \frac{\gamma }{2}}{\displaystyle \sum _{i,j}{(P_i-P_j)}^2.}}

Here, P_i and P_j denote the local polarization vectors at lattice sites i and j, respectively. The first sum represents the local anharmonic potential (with coefficients α and β), while the second sum accounts for the energy cost of polarization gradients between neighboring sites, characterized by the coupling constant γ. This effective Hamiltonian form, which can be directly parametrized from first-principles data, captures the essential double-well landscape and inter-site correlations governing the ferroelectric transition. This approach of using a coupled double-well Hamiltonian (with terms up to P⁴ on-site) for analyzing energy barriers in ferroelectrics is well-established in the literature. Our analysis directly follows the methodology presented in previous work [68], which provides a rigorous framework for mapping first-principles results onto such an effective model to assess ferroelectric stability and switching. The corresponding double-well potential of the monolayer SnP₂Se₆ [Fig. 6(b)] reveals a 0.41 eV FE-to-PE switching barrier (E_B), a pivotal indicator of thermodynamic stability and operational feasibility for ferroelectric monolayers. The allowed polarization directions and switching paths are dictated by the 3 m point group symmetry of the monolayer. The spontaneous polarization vector is an intrinsic property aligned along specific symmetry-equivalent directions within the plane. Consequently, a switch between two energetically degenerate states must be mediated by a symmetry operation inherent to the crystal. The only such operation yielding a distinct polarization state is a 180° reversal (from +P to ‒P), which corresponds to inversion in the context of the polar axis. Other rotation angles, such as 60°, are not symmetry-allowed and would require breaking the crystal symmetry, leading to prohibitively high energy barriers.

To further elucidate the microscopic mechanism of the ferroelectric–paraelectric (FE-PE) phase transition and address its kinetic feasibility, we performed phonon spectrum calculations for the high-symmetry paraelectric phase of SnP₂Se₆. As shown in Supplementary Fig. S8, the phonon dispersion exhibits imaginary frequencies around the Γ point. These soft modes correspond to collective vibrations involving a cooperative displacement of the SnSe₆ octahedral unit along the in-plane diagonal direction, coupled with a minor rotational distortion — a pattern that aligns precisely with the structural pathway illustrated in Fig. 6(a). This confirms that the PE structure is dynamically unstable and naturally distorts towards the FE ground state, reinforcing that the computed energy barrier is associated with a structurally natural symmetry-lowering process. While direct experimental observation of ferroelectric switching in monolayer SnP₂Se₆ remains a target for future research, the parent bulk crystal has been experimentally synthesized in a non-centrosymmetric structure [28, 29], providing the essential foundational symmetry breaking. This is further corroborated by the measured robust second-harmonic generation (SHG) response [30], a direct optical signature of the non-centrosymmetric character that is prerequisite for ferroelectricity. Therefore, our theoretical predictions, now reinforced by the phonon analysis, aim to motivate and guide future experimental probes (e.g., piezoelectric force microscopy, variable-temperature SHG/Raman spectroscopy) to directly detect and switch the polarization.

Other group-IV-V-VI monolayers exhibit E_B values range from 0.34 eV (GeP₂S₆) to 0.80 eV (SiAs₂Se₆ and SiAs₂Te₆), exhibiting distinct trends correlated with the group-IV, -V, and -VI elemental compositions. Notably, Si-based monolayers consistently display the highest E_B values (0.69−0.80 eV), with SiAs₂Se₆ and SiAs₂Te₆ achieving the maximum of 0.80 eV. Such a trend implies that silicon substitution enhances the energetic stability of the ferroelectric phase, likely attributed to stronger covalent bonding and more pronounced dipole−dipole interactions within the lattice, which hinder the transition to the centrosymmetric paraelectric state. In contrast, Ge-based systems show broader variability: GeP₂S₆ exhibits the lowest E_B (0.34 eV), while GeAs₂Se₆ reaches 0.69 eV, indicating that replacing P with As in Ge-centered structures reinforces the ferroelectric ordering. Sn-based monolayers maintain intermediate E_B values (0.39−0.46 eV), with SnAs₂Te₆ (0.46 eV) slightly outperforming its S/Se analogs, reflecting the influence of heavier group-VI elements in stabilizing polarized configurations. Crucially, these E_B values a exceed those of conventional perovskite ferroelectrics by approximately an order of magnitude (e.g., BaTiO₃ and PbTiO₃) [69], addressing the critical challenge of ferroelectric degradation in ultrathin films.

AIMD simulations [70] using 3\times 3\times 1 supercells demonstrate that monolayer SnP₂Se₆ retains spontaneous polarization (P_FE = 2.28 × 10⁻⁹ C/m) up to ~260 K [Fig. 6(c)]. Fitting polarization-temperature data to a sigmoid function P(T)=a/[1+{\mathrm{e}}^{-k(T-T_{\mathrm{C}})}] (where a and k are constants) and pyroelectric response (p = -dP/dT) can also yield an estimated T_C. Since T_C correlates with E_B and dipole-dipole coupling, monolayers with E_B ≥ 0.4 eV are projected to exhibit T_C that are approximately or even significantly higher than room temperature. This capability undoubtedly paves the way for advancing the development of ferroelectric devices. With its combination of high quality, substantial size, and unique intrinsic properties, group-IV-V-VI film hold great promise for a wide array of applications extending beyond logic devices. Our systematic screening further reveals that compositional engineering within this family can yield candidates with exceptional integrated performance. A notable example is the monolayer SiAs₂Te₆, which simultaneously exhibits: (i) a high in-plane piezoelectric coefficient (d₁₁ ≈ 22.94 pm/V), (ii) a substantial ferroelectric switching barrier (E_B ≈ 0.80 eV), and (iii) a Curie temperature of ~280 K from double-well and AIMD analysis (see S9, Fig. S9). This estimated T_C indicates robust stability of the ferroelectric phase at ambient conditions.

Piezoelectricity and ferroelectricity in these 2D material are intrinsically linked via symmetry breaking and polarization mechanisms. Ferroelectric domain walls induce strain fields that boost piezoelectric sensitivity, while piezoelectric strain conversely stabilizes ferroelectric domains. This synergy enables innovative applications: self-powered memory uses piezoelectric charges to write/read ferroelectric states without external power [71, 72] and flexible sensors combine piezoelectric signals with nonvolatile ferroelectric memory for adaptive sensing and efficient storage [73, 74]. Such integration emphasizes the transformative capability of 2D materials in boosting the progress of next-generation energy-efficient electronics.

4 Conclusions

Motivated by experimentally synthesized SnP₂Se₆ thin films, we identified 15 stable group-IV-V-VI monolayers via elemental substitution, including SiP₂Se₆, SiAs₂S₆, SiAs₂Se₆, SiAs₂Te₆, GeP₂S₆, GeP₂Se₆, GeAs₂S₆, GeAs₂Se₆, GeSb₂S₆, SnP₂S₆, SnP₂Se₆, SnAs₂S₆, SnAs₂Se₆, SnAs₂Te₆ and SnSb₂S₆ monolayers. These materials exhibit bandgaps (1.04−2.36 eV), with five being direct-bandgap semiconductors. Their in-plane piezoelectric coefficients (d₁₁: 1.24−22.94 pm/V) surpass common 2D materials (MoS₂, h-BN and InSe), while bilayers SnP₂Se₆ show large out-of-plane d₃₃ (5.52 pm/V). Meanwhile, low ferroelectric switching barriers enable room-temperature operation, supporting non-volatile memory applications. Notably, our study identifies specific candidates such as SiAs₂Te₆ that exemplify the performance ceiling of this family, showcasing how targeted elemental substitution can simultaneously enhance piezoelectric strength, ferroelectric stability, and dimensional adaptability. Combining strong piezoelectric/ferroelectric effects with semiconducting properties, these materials hold transformative potential for energy, sensing, and information processing.

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