Strain-responsive PH-SiBiX monolayers: computational design of multifunctional 2D materials for piezotronic and optoelectronic applications

Abstract

Two-dimensional materials with coupled electromechanical and optoelectronic functionalities are highly desirable for next-generation adaptive devices, yet their design remains challenging due to the trade-off between piezoelectricity, auxeticity, and optical tunability. Here, we propose a class of transition metal-adsorbed PH-SiBiX (X = Sc–Cd) monolayers as a multifunctional platform via first-principles calculations. Ti-adsorbed PH-SiBiTi exhibits a record-high in-plane piezoelectric coefficient (d₁₁ = 13.057 pm V⁻¹), outperforming conventional 2D materials (e.g., MoS₂) and bulk quartz, while Co/Ni/Ru/Rh-adsorbed systems demonstrate negative Poisson's ratios (ν = −0.068 to −0.206) through hinge-like lattice deformations. The strain sensitive d–p orbital hybridization between transition metals and Bi/Si atoms governs both the giant piezoelectric response and visible-light absorption anisotropy (7.5–16%), enabling mechanical tuning of optoelectronic properties. Furthermore, strong spin–orbit coupling induces band splitting (Δ ∼ 50 meV) and shifts band extrema, suggesting potential for spin-valleytronic applications. By synergizing high piezoelectricity, auxetic behavior, and strain-tunable optical absorption, PH-SiBiX monolayers bridge the gap between theoretical design and practical applications in self-powered sensors, flexible optoelectronics, and mechanically adaptive energy harvesters. This work establishes a symmetry-guided paradigm for engineering 2D materials with on-demand multifunctionality.

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1. Introduction

The quest for multifunctional two-dimensional (2D) materials capable of integrating piezoelectricity, auxeticity, and strain-tunable optoelectronics has emerged as a pivotal frontier in materials science, driven by the demands of next-generation adaptive devices. Since the discovery of graphene,^1–4 the 2D materials landscape has expanded to include transition metal dichalcogenides (TMDs, e.g., MoS₂, WS₂),^5–7 post-graphene semiconductors like InSe^8,9 and phosphorene,^10–12 each offering unique advantages.¹³ Monolayer MoS₂, for instance, exhibits moderate in-plane piezoelectricity (e₁₁ = 2.9 × 10⁻¹⁰ C m⁻¹) due to its non-centrosymmetric 2H-phase structure,^14,15 yet its positive Poisson's ratio (ν > 0) limits mechanical resilience under cyclic strain – a critical drawback for wearable electronics. Janus TMDs (e.g., MoSSe), constructed by breaking vertical symmetry,¹⁶ introduce out-of-plane polarization but lack auxetic behavior. Similarly, phosphorene's anisotropic piezoelectricity and high carrier mobility are offset by ambient instability. These trade-offs underscore a persistent challenge: no existing 2D material seamlessly combines a high piezoelectric response, negative Poisson's ratio, and strain-sensitive optoelectronics – a triad essential for adaptive interfaces in energy harvesters, self-powered sensors, and mechanically reconfigurable photodetectors.

Recent advances in materials design have sought to address these limitations through symmetry engineering and defect modulation.^17,18 Functionalized MXenes, such as oxygen-terminated Ti₃C₂T_x,¹⁹ achieve enhanced piezoelectricity via surface termination control, while single-layer graphene ribbons exhibit auxeticity.²⁰ However, these systems prioritize single functionalities: MXenes show limited optical absorption in the visible spectrum, and auxetic graphene analogs lack intrinsic piezoelectricity. Parallel efforts in strain-engineered materials, such as Mn-adsorbed SnSe₂ monolayers²¹ and BiOX (X = Cl, Br, I) systems,²² demonstrate moderate optoelectronic tunability under mechanical deformation but fail to integrate auxeticity or robust piezoelectricity. This fragmented progress highlights a critical gap in the field: the absence of a unified design strategy that leverages atomic-scale interactions to concurrently optimize electromechanical, mechanical, and optical properties.

Theoretical studies have proposed that synergistic functionality could arise from targeted symmetry breaking and orbital hybridization. In Janus monolayers like MoSTe, vertical polarization enhances out-of-plane piezoelectricity but leaves in-plane responses subpar.²³ Conversely, d–p orbital hybridization in transition metal-adsorbed systems has been shown to amplify strain-sensitive electronic transitions,²⁴ suggesting a pathway to mechanically tunable optoelectronics. However, these systems often neglect auxetic lattice dynamics or piezoelectric efficiency. A notable exception is the predicted auxetic behavior in borophane,²⁵ but its lack of piezoelectricity and weak optical absorption limits its practical utility. These insights collectively point to an uncharted opportunity: engineering 2D materials where symmetry reduction, strain-mediated orbital interactions, and lattice topology cooperate to deliver multifunctionality.

In this work, we bridge this gap by exploring transition metal (TM)-adsorbed PH-SiBi monolayers – a class of materials designed to harmonize piezoelectricity, auxeticity, and optoelectronic tunability. Based on the PH-silicene structure,³⁴ we have theoretically constructed the PH-SiBi structure through manual doping and confirmed its dynamic, thermodynamic, and mechanical stability. The PH-SiBi structure exhibits an indirect bandgap of 1.07 eV and features the coexistence of piezoelectric and Rashba effects within its structure. Building on prior successes in symmetry-guided 2D material design, we hypothesize that TM adsorption on a PH-SiBi substrate will simultaneously break in-plane symmetry (enhancing piezoelectricity), induce hinge-like lattice deformations (enabling auxeticity), and activate strain-sensitive d–p orbital hybridization (modulating optical absorption). Our computational approach integrates density functional theory (DFT) and piezotronic analysis to map the interplay between the atomic structure, mechanical response, and optoelectronic properties. The results unveil that PH-SiBiX monolayers not only transcend the constraints imposed by traditional TMDs and MXenes, but also pioneer a novel paradigm in the design of multifunctional 2D materials, where mechanical, electrical, and optical functionalities are inherently intertwined rather than compromised.

This study advances the field in two key ways. First, it provides a comprehensive framework for understanding how symmetry reduction and orbital interactions govern multifunctionality in 2D materials. Second, it identifies PH-SiBiX as a versatile platform for adaptive device interfaces, with potential applications ranging from fatigue-resistant piezoelectric sensors to strain-programmable photodetectors. By aligning theoretical predictions with practical device requirements, this work lays the groundwork for experimental synthesis and heterostructure integration – a critical next step toward real-world implementation.

2. Computational details

We conducted an in-depth investigation using the Vienna ab initio simulation package for first-principles calculations.^26,27 In this process, we adhered to the framework of the generalized gradient approximation and employed the Perdew Burke Ernzerhof functional to determine the exchange correlation potential.²⁸ To ensure the accuracy of our calculations, we comprehensively relaxed the atomic positions and lattice constants of each system until the maximum force acting on each atom converged below 0.01 eV Å⁻¹, while the convergence criterion for the total energy was rigorously set to 10⁻⁶ eV. When setting up the wave functions, we selected an energy cutoff value of 520 eV and utilized a 5 × 5 × 1 Monkhorst–Pack K-point grid centered at the Γ-point for momentum space integration. Additionally, to avoid interactions between adjacent slabs, we intentionally introduced a vacuum layer of 20 Å. The dynamic stability of systems was checked by phonon calculations using the force-constant method in the DS-PAW code of Devices Studio,²⁹ in which the dynamical matrices were computed using the finite differences method in a 3 × 3 × 1 large unit cell to eliminate errors in the low frequency modes.³⁰Ab initio molecular dynamics simulation was performed using the canonical ensemble with the Nosé–Hoover heat bath scheme (300 K) in a 2 × 2 × 1 supercell.³¹ During the calculation of the electronic structure, we incorporated the spin orbit coupling effect and employed the PBE+U method to correct for the Coulomb repulsion in transition metal atoms. The Hubbard parameter U is typically an empirical parameter and is not fixed. For the same element, the U value often varies across different crystal coordination environments. Therefore, to determine the most appropriate U value, we consulted relevant literature and the usage records of U values for transition metals documented from the literature.³² Through continuous adjustment and optimization, we arrived at a suitable U value. To further explore the physical properties of the materials, we also utilized the strain stress relationship method and density functional perturbation theory integrated within VASP to accurately calculate the elastic stiffness tensor and piezoelectric tensor.³³

3. Results and discussion

3.1 Structural and electronic properties

In this work, the structure of the PH-SiBi is as shown in Fig. 1(a), which is a hexagonal lattice with P31m symmetry, and the optimized lattice constants are a = b = 15.814 Å. Based on PH-silicene,³⁴ we have manually constructed a PH-SiBi structure.³⁵ PH-SiBi is a two-dimensional diatomic layer material formed by replacing the Si₁₀ and Si₁₁ atoms in PH-silicene with Bi atoms, which have similar structures to PH-silicene, called PH-SiBi. As depicted in Fig. 1(b), PH-SiBiX (X = Sc–Cd) structures are derived from the PH-SiBi structure by adsorbing transition metal atoms at the hollow adsorption sites. Through calculations of the adsorption energies at the top (T) site, bridge (B) site, and hollow (H) site of the PH-SiBi structure, it was found that the lowest energy configuration occurs when adsorption takes place at the H site. Consequently, computational studies were conducted based on the H-site adsorption structure. To assess the stability of the adsorption structure, the adsorption energy is defined as shown in the formula: E_ads = E_PH-SiBiX − (E_PH-SiBi + E_X). Here, E_PH−SiBi and E_X represent the energies of the adsorption substrate PH-SiBi structure and the adsorbed atom X (X = Sc–Cd), respectively, while E_PH-SiBiX denotes the energy of the adsorption structure. A positive adsorption energy indicates an endothermic process, implying that adsorption will not occur; in contrast, a negative value suggests that the adsorption configuration is stable. As revealed by the calculation results (Fig. S12, ESI†), the investigated adsorption structure PH-SiBiX is stable. Additionally, we have calculated the adsorption height and band gap of PH-SiBiX, which are summarized in Table 1.

Fig. 1 (a) Top and side views of the PH-SiBi, (b) top and side views of the PH-SiBiX (X = Sc–Cd), (c) two-dimensional Brillouin zone of the primitive cell, (d) and (e) phonon dispersion relationships and ab initio molecular dynamics (AIMD) of the adsorption substrate, and (f) the final structure after AIMD simulation.

Table 1 Calculated results for the adsorption height (h), band gap (E_gap), U value, M and MAE of the PH-SiBiX

Materials	h (Å)	E gap (eV)	U (eV)	M (μB)	MAE (meV)	Materials	h (Å)	E gap (eV)	U (eV)	M (μB)	MAE (meV)
PH-SiBiSc	1.77	0.17	2.90	1.56	−0.89	PH-SiBiTi	1.97	0.16	4.40	1.79	−0.21
PH-SiBiV	1.25	0.29	2.70	1.46	−0.46	PH-SiBiCr	1.70	0.40	3.50	4.81	0.03
PH-SiBiMn	1.55	0.13	4.00	4.43	0.49	PH-SiBiFe	1.25	0.23	4.60	3.36	0.15
PH-SiBiCo	1.69	0.82	5.00	1.16	0.01	PH-SiBiNi	1.57	1.00	5.10	0	0
PH-SiBiCu	0.85	Metal	4.00	0	0	PH-SiBiZn	1.78	0.95	7.50	0	0
PH-SiBiY	1.99	Metal	No	−0.27	0.34	PH-SiBiZr	1.47	0.39	No	0	0
PH-SiBiNb	1.35	0.61	2.10	0.72	−0.15	PH-SiBiMo	1.25	0.19	2.40	1.76	3.39
PH-SiBiTc	1.13	0.56	2.70	2.63	1.29	PH-SiBiRu	1.10	0.68	3.00	1.68	−1.98
PH-SiBiRh	0.98	0.41	3.30	0.70	0.29	PH-SiBiPd	0.94	0.83	3.60	0	0
PH-SiBiAg	1.61	0.23	5.80	0.39	0.02	PH-SiBiCd	2.31	1.06	2.10	0	0

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The elastic constants delineate the stiffness characteristics of a crystal in response to external strains, with their number closely tied to specific crystal systems. These constants are of paramount importance in evaluating the mechanical stability of hexagonal materials, which constitutes one of the primary focuses of this study. The mechanical stability of hexagonal crystals is predicated on satisfying a series of inequalities formulated from elastic constants, which ensure the structural robustness of the material under complex stress conditions. According to the Born–Huang stability criteria,^36,37 the stability of hexagonal crystals necessitates adherence to specific combinations of inequalities involving these constants, specifically that C₁₁ > 0, C₆₆ > 0 (where 2C₆₆ = C₁₁ − C₁₂), and C₁₁ > |C₁₂|. These stringent conditions collectively guarantee that the material will not undergo structural instability or failure under various stress conditions.

To determine the magnetic ground state of the crystalline structure, taking the PH-SiBiRu structure as an example, we considered both ferromagnetic (FM) and antiferromagnetic (AFM) configurations after expanding the unit cell to 2 × 1 × 1. Upon structural relaxation, it was found that the energy of the AFM configuration is lower than that of the FM configuration, with an energy difference (ΔE = E_FM − E_AFM) of 0.90 meV between the FM and AFM states. Therefore, the magnetic ground state of the PH-SiBiRu structure is identified as the antiferromagnetic configuration. Additionally, we computed the magnetic anisotropy energy (MAE) for the PH-SiBiX primitive cell. Here, we considered two magnetization directions: the in-plane [100] direction and the out-of-plane [001] direction. The MAE is defined as ΔE = E₁₀₀ − E₀₀₁, where E₁₀₀ and E₀₀₁ denote the total energies of the system with magnetization along the [100] and [001] directions, respectively. A positive MAE indicates that the easy magnetization axis lies along the z-axis rather than the x-axis. The MAE values are listed in Table 1, revealing that PH-SiBiX exhibits both x-axis and z-axis easy magnetization directions. Surprisingly, the MAE of monolayer PH-SiBiMo reaches 3.39 meV per cell, which is 2.47 times larger than that of recently successfully synthesized CrI₃ (1.37 meV per cell).³⁸ More intriguingly, experimental realization of magnetization direction tuning via an external magnetic field has been achieved in the monolayer Fe₃GeTe₂ system (2.76 meV per cell),³⁹ suggesting the potential for generating even more exotic physical phenomena.

To validate the mechanical stability of PH-SiBiX, we conducted a comprehensive investigation of its linear elastic constants. Due to its hexagonal symmetry, only two independent elastic constants, C₁₁ and C₁₂ are pertinent. The computed results are summarized in Table 2. Additionally, the elastic constants of the 2H-MoS₂ monolayer structure were also calculated for comparison. The results indicate that all the elastic constants of PH-SiBiX simultaneously meet the criteria for mechanical stability, confirming their mechanical stability. Furthermore, we found that the monolayer PH-SiBiX (X = Co, Ni, Ru, and Rh) exhibits negative Poisson's ratios of −0.068, −0.206, −0.032, and −0.119, respectively. The negative Poisson's ratio characteristic refers to the material's expansion in transverse dimensions upon tension and contraction upon compression, significantly enhancing its impact and fatigue resistance.⁴⁰ The negative Poisson's ratio effect exhibits application potential in two-dimensional multifunctional materials that integrate piezoelectric and optoelectronic properties.^41,42 This negative Poisson's ratio effect renders PH-SiBiX (X = Co, Ni, Ru, and Rh) an ideal candidate for designing novel materials with specialized functionalities. For the studied PH-SiBiX, as the phonon dispersion curves, ab initio molecular dynamics simulations, and mechanical stability of the adsorbed substrate all demonstrate the substrate's stability, the studied adsorbed structure of PH-SiBiX should exhibit excellent thermodynamic stability and dynamic stability of phonon dispersion, similar to the substrate.

Table 2 Mechanical parameters for PH-SiBiX monolayers

Materials	Stiffness tensor (GPa)			Young's modulus E (GPa)	Shear modulus G (GPa)	Poisson's ratio	Mechanical stability
	C 11	C 12	C 66	X	G	X	Yes/no
PH-SiBiSc	9.167	−0.430	4.610	1.852	0.888	0.043	Yes
PH-SiBiTi	8.534	−1.570	5.153	1.926	0.974	0.644	Yes
PH-SiBiV	9.605	−0.068	4.798	2.397	1.126	0.063	Yes
PH-SiBiCr	8.770	−0.775	4.652	2.332	1.085	0.075	Yes
PH-SiBiMn	9.994	−0.098	5.454	2.847	1.381	0.030	Yes
PH-SiBiFe	10.214	−0.743	5.309	3.047	1.310	0.162	Yes
PH-SiBiCo	10.895	−0.850	5.797	2.602	1.396	−0.068	Yes
PH-SiBiNi	10.358	−0.970	5.737	2.839	1.787	−0.206	Yes
PH-SiBiZn	9.458	−0.576	4.878	2.280	1.133	0.679	Yes
PH-SiBiZr	9.030	0.521	4.380	2.660	1.275	0.043	Yes
PH-SiBiNb	10.090	−0.875	5.398	2.213	0.887	0.248	Yes
PH-SiBiMo	8.185	1.385	3.815	0.288	0.993	0.550	Yes
PH-SiBiTc	10.823	−0.241	5.582	2.295	0.992	0.156	Yes
PH-SiBiRu	10.821	−0.629	5.733	3.280	1.694	−0.032	Yes
PH-SiBiRh	8.541	1.459	4.948	3.880	2.203	−0.119	Yes
PH-SiBiPd	10.681	−0.366	5.554	2.957	1.410	0.048	Yes
PH-SiBiAg	7.652	0.181	3.766	2.158	1.026	0.052	Yes
PH-SiBiCd	9.675	−1.242	5.478	2.153	1.065	0.011	Yes

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The crux of evaluating the characteristics of novel two-dimensional light-absorbing materials lies in their electronic properties, which are directly manifested in the electronic band structures of these materials. Through first-principles calculations, we investigated the band structures and projected density of states (PDOS) of PH-SiBi substrate adsorbed with transition metal atoms. In this subsection, we present a detailed analysis of the electronic structures and PDOS for the structures adsorbed with Ti and Cr atoms as examples. The band structures for the (Ti-)Cr-adsorbed structures are illustrated in Fig. 2(b) and (d), respectively, while other PH-SiBiX (X = Sc to Cd) can be found in the Fig. S1–S3 (ESI†). Furthermore, both the PH-SiBiTi and PH-SiBiCr structures are composed of heavy atoms and transition metal atoms. Heavy atoms typically exhibit strong spin orbit coupling (SOC) effects, while transition metal atoms will introduce magnetic moments. Therefore, we calculated the band structures incorporating the influence of SOC. Upon including SOC in our calculations, it becomes evident that in the band structures of PH-SiBiTi and PH-SiBiCr, above the Fermi level at the Γ point, the originally small bandgap enlarges, and band splittings occur at the points where bands previously crossed, opening up a bandgap. This phenomenon is attributed to the presence of the heavy atom Bi and the transition metal atoms. In the PH-SiBiTi and PH-SiBiCr structures, Bi atoms account for approximately one-sixth of the total number of atoms. With a large atomic number, Bi atoms exhibit significant SOC effects. Transition metal atoms also possess strong electronic interactions. The combined action of transition metal atoms and Bi atoms leads to the splitting of the band structures. This band splitting phenomenon is of great significance for understanding the electronic properties of PH-SiBiTi and PH-SiBiCr structures, developing novel functional devices based on these materials, and exploring related physical phenomena.

Fig. 2 The electronic band structures of (a) PH-SiBiTi and (b) PH-SiBiCr.

The calculated projected density of states (PDOS) for the PH-SiBiTi and PH-SiBiCr structures are presented in Fig. 3(b) and (d), respectively, while other PH-SiBiX (X = Sc to Cd) can be found in Fig. S4–S6 (ESI†). The PDOS results provide validation for further understanding the influence of Bi atoms and transition metal atoms on the band structures. It is clearly evident from the figures that in the PDOS of both PH-SiBiTi and PH-SiBiCr structures, Si atoms, Bi atoms, and Ti(Cr) atoms all contribute near the Fermi level. Among them, the Si-p_z orbital makes the most significant contribution from Si atoms, the Bi-p_x orbital contributes the most from Bi atoms, while the contributions from Ti(Cr) atoms are relatively smaller compared to those from Si and Bi atoms, with the d_z² orbital being notably prominent. The electronic distributions and interactions of these orbitals directly shape the characteristics of the band structures, endowing the materials with unique electronic properties. The d orbitals of transition metal atoms, through their distinctive electronic structures and magnetic properties, further modulate the band structures, leading to a rich diversity in the electronic transport and magnetic properties of the materials.

Fig. 3 (a), (b) PDOSs of PH-SiBiX (X = Ti, Cr) monolayers.

3.2 Piezoelectric properties

The absence of inversion symmetry in two dimensional materials leads to the emergence of piezoelectric properties, which reflect the coupling between mechanical fields and electric fields. The piezoelectric tensor e_ijk, characterizes the coupling relationship between the strain tensor e_jk and polarization P_i. The third-rank piezoelectric stress tensor e_ijk and the strain tensor d_ijk are defined as the sum of ionic and electronic contributions. e_ijk and d_ijk are defined as:andin which P_i, ε_jk, and σ_jk are the polarization vector, strain, and stress, respectively. For two-dimensional materials, in-plane stress and in-plane strain conditions are assumed, where ε_jk = σ_ij = 0 for i = 3 and j = 3.⁴³ Given the P31m point-group symmetry exhibited by PH-SiX, the piezoelectric stress and strain tensors can be simplified through the utilization of Voigt notation.The e_ijk can be calculated by DFPT and derive the values of d_ijk using the relationship with elastic constants C_mnjk.and the elastic tensor C can be expressed asThe C_ij can be attained by the strain stress relationship method. Here, d₁₁ and d₃₁ are derived by equations.and

The piezoelectric coefficients d₁₁ and d₃₁ obtained from PH-SiBiX calculations are summarized in Table 3, with the piezoelectric coefficients of the MoS₂ monolayer serving as a benchmark to validate the methodology. The calculated result (d₁₁ = 3.654 pm V⁻¹) aligns well with previous experimental and DFT studies.^44,45 For the PH-SiBiX monolayers, the computed d₁₁ values range from 2.106 to 13.057 pm V⁻¹. Except for PH-SiBiMo (d₁₁ = 2.985 pm V⁻¹) and PH-SiBiRh (d₁₁ = 2.106 pm V⁻¹), the in-plane piezoelectric coefficients of the remaining structures exceed the d₁₁ value of 2H-MoS₂. These findings are significantly larger compared to previously reported DFT calculated values for other transition metal dichalcogenides such as WS₂.^16,45 The d₁₁ values of PH-SiBiX monolayers are much higher than those of widely used bulk materials,^46–49 including α-quartz (d₁₁ = 2.3 pm V⁻¹), wurtzite aluminum nitride (d₃₃ = 5.1 pm V⁻¹), and wurtzite gallium nitride (d₃₃ = 5.1 pm V⁻¹). Furthermore, they are comparable or even superior to those of other two-dimensional materials,^23,50 such as MoS₂, WS₂, ZnO, MoSSe, and MoSTe.

Table 3 Calculated piezoelectric coefficients e₁₁, e₃₁, d₁₁, and d₃₁ for PH-SiBiX and 2H-MoS₂ monolayers

Materials	e 11 (pC m^−1)	e 31 (pC m^−1)	d 11 (pm V^−1)	d 31 (pm V^−1)
PH-SiBiSc	127.95	5.11	6.382	0.280
PH-SiBiTi	275.63	8.79	13.057	0.604
PH-SiBiV	162.63	6.14	8.047	0.308
PH-SiBiCr	88.22	13.06	4.424	0.782
PH-SiBiMn	134.85	4.09	6.397	0.198
PH-SiBiFe	100.94	13.18	4.410	0.666
PH-SiBiCo	151.21	0.61	6.162	0.029
PH-SiBiNi	141.35	1.88	5.972	0.096
PH-SiBiZn	129.35	1.78	6.171	0.096
PH-SiBiZr	153.23	9.65	8.622	0.484
PH-SiBiNb	139.53	2.95	6.088	0.153
PH-SiBiMo	42.41	0.25	2.985	0.013
PH-SiBiTc	153.65	4.91	6.648	0.222
PH-SiBiRu	164.30	6.12	6.869	0.287
PH-SiBiRh	31.15	10.36	2.106	0.496
PH-SiBiPd	167.79	9.80	7.273	0.455
PH-SiBiAg	81.47	56.57	5.219	3.455
PH-SiBiCd	154.92	5.47	6.795	0.310
2H-MoS2	364.27	—	3.654	—

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For two dimensional materials, a large out-of-plane piezoelectric response is highly desirable due to its compatibility with current bottom/top-gate technologies. Research has found that PH-SiBiX monolayers also possess out-of-plane piezoelectric coefficients, which are absent in 2H-MoS₂. For PH-SiBiCr and PH-SiBiFe, the calculated d₃₁ values are 0.782 pm V⁻¹ and 0.666 pm V⁻¹, respectively, which are much larger than those of TMXY (M = Mo and W, X/Y = S, Se, or Te) with d₃₁ values ranging from 0.007 to 0.030 pm V⁻¹.¹⁶ These values are comparable or higher than those of many known 2D materials, such as functionalized h-BN (0.13 pm V⁻¹),⁵¹ oxygen-functionalized MXenes (0.40–0.78 pm V⁻¹),⁵² Group-III Janus materials (0.46 pm V⁻¹),⁵³ Janus BiTeI/SbTeI monolayers (0.37–0.66 pm V⁻¹),⁵⁴ Janus monolayer transition metal dichalcogenides (0.03 pm V⁻¹),¹⁶ and α-In₂Se₃ (0.415 pm V⁻¹).⁵⁵ The larger d₃₁ values may be related to the larger electronegativity differences between Si, Bi, and transition metal atoms. Considering the inversion symmetry breaking along the z-direction, the out-of-plane piezoelectric coefficients (d₃₁) of PH-SiBiX monolayers are highly favored as they can provide freedom in designing diverse nanoelectromechanical devices.

The in-plane piezoelectric coefficient of PH-SiBiTi (d₁₁ = 13.057 pm V⁻¹) surpasses conventional 2D materials such as MoS₂ (d₁₁ = 3.654 pm V⁻¹) and bulk quartz (d₁₁ = 2.3 pm V⁻¹). This enhancement arises from the unique d–p orbital hybridization between Ti and Bi/Si (Fig. 3), which amplifies charge redistribution under strain. Similarly, the auxetic behavior (ν = −0.206 for PH-SiBiNi) outperforms graphene derivatives (ν = −0.07 for re-entrant honeycombs), highlighting the role of low-frequency hinge-like phonon modes (Table 2) in enabling lattice contraction under tension.

3.3 Absorption spectrum

Given the ubiquitous quantum confinement effects in low-dimensional materials, their optical properties are significantly modulated by the confined electron–hole states. When these materials absorb photons, electrons transition from lower energy states to higher energy states, a process that gives rise to the absorption spectrum. The absorption spectrum, as the most widely used spectroscopic analysis method currently, plays a pivotal role in investigating the optical properties of materials. The light absorption capacity of PH-SiBiX can be evaluated through the following formula:⁵⁶where ε₁(ω) and ε₂(ω) are the real and imaginary parts of the dielectric function ε(ω) =ε₁(ω) + iε₂(ω), respectively. The imaginary part ε₂(ω) can be calculated based on the contributions from interband transitions:⁵⁷here, Ω denotes the volume of the unit cell. The Bloch vector of the incident wave is represented by q. The symbol w_k stands for the k-point weight. The conduction band state and the valence band state are indicated by c and v, respectively. At point k, u_ck signifies the cell-periodic portion of the orbitals. The real part ε₁(ω) was derived through the application of Kramers–Kronig transformation:⁵⁸where P is the principle value, and η denotes the complex shift in the Kramers–Kronig transformation.

The monolayer PH-SiBiX materials exhibit a broad bandgap range from 0.13 to 1.06 eV, rendering them worthy of exploration for their potential in optical absorption capabilities. We have calculated the absorption coefficients across the ultraviolet-visible (UV-VIS) to near-infrared (NIR) spectral regions, with the results presented in Fig. 4. Due to the structural symmetry of the PH-SiBiX monolayers, their optical absorption characteristics exhibit isotropy, meaning that the absorption spectra along the in-plane x and y directions are fully overlapped (only the results for the x direction are shown here), which is highly advantageous for the application of isotropic optical devices. Unfortunately, however, there is almost no light absorption in the infrared region due to its smaller energy range. Despite this, the intensity of light absorption in the visible region is particularly important, as it accounts for nearly half of the solar energy. PH-SiBiX demonstrates exceptional optical absorption performance within the visible energy range of 1.64 to 3.19 eV. Thanks to the varying bandgaps of the PH-SiBiX monolayers, their optical absorption can cover multiple spectral regions including infrared, visible, and ultraviolet. Specifically, PH-SiBiX primarily absorbs light in the visible region, with an absorption rate ranging from 7.5% to 16%. Meanwhile, it also exhibits good light absorption performance in the ultraviolet region, with an absorption rate of approximately 7% to 17%, which is slightly lower than that of Bi₂SSe₂ and Bi₂S₂Se monolayers (approximately 20%),⁵⁹ but significantly higher than that of MoSi₂N₄ monolayers (5% to 10%),^60,61 and comparable to that of commonly used solar energy films. This absorption coefficient is comparable to those of some high-performance two-dimensional transition metal dichalcogenides and phosphorus-based II, III, and IV binary compound photocatalytic materials.^62,63 These rich optical properties bring great convenience to the application of PH-SiBiX monolayers in the field of optoelectronic devices.

Fig. 4 Optical absorption coefficient α(ω) of the PH-SiBiX. The pale orange, orange and purple regions represent the infrared, visible light and ultraviolet, respectively.

To investigate the effect of strain modulation on the optical absorption coefficient, we selected the PH-SiBiCr structure as an example and calculated the absorption spectra under various applied strains. The calculated absorption spectra are shown in Fig. 5. The figure displays the absorption spectra of PH-SiBiCr under biaxial strains of −4%, −2%, 0%, 2%, and 3%. Under −4% and −2% strain, the light absorption rate of PH-SiBiCr in the visible region increased from 15.10% in the unstrained state to 16.28% and 15.44%, respectively. Whereas, under 2% and 3% tensile strain, the light absorption rate in the visible region decreased to 14.63% and 14.04%, respectively. This demonstrates that under biaxial strain modulation, compressive strain is beneficial for improving the light absorption rate of PH-SiBiCr in the visible region.

Fig. 5 Consideration of the optical absorption coefficient of PH-SiBiCr under strain.

To investigate the relationship between the optical properties and the thickness of the vacuum layer, we took the PH-SiBiRu structure as an example. Initially, the vacuum layer thickness in the crystal structure used for calculating the optical properties of PH-SiBiRu was 20 Å. We modified this thickness to 22 Å and 18 Å, respectively, and then re-calculated the optical properties. As shown in Fig. S13 (ESI†), the results obtained were identical to those before altering the vacuum layer thickness. Consequently, we conclude that the thickness of the vacuum layer does not influence the optical properties of the PH-SiBiRu structure during the calculation of its optical characteristics.

The visible-light absorption rate of PH-SiBiX (7.5–16%) is comparable to high-performance TMDs (monolayer MoS₂ absorbs less than 8%⁶⁴) yet uniquely combines this with piezoelectric and auxetic functionality. This multifunctional synergy is absent in Janus TMDs (e.g., MoSSe), d₁₁ = 0.03 pm V⁻¹,¹⁶ which prioritize directional polarization at the expense of auxeticity. Furthermore, the strain-tunable absorption (Fig. 5) demonstrates active spectral control – a feature critical for adaptive photodetectors and solar cells.

3.4 Potential device applications

The multifunctional properties of PH-SiBiX monolayers, including giant piezoelectricity, auxetic behavior, and strain-tunable optical absorption, position them as a transformative platform for next-generation adaptive devices. Here, we propose three promising application scenarios based on their unique characteristics:

1. Self-powered piezotronic sensors: the record-high in-plane piezoelectric coefficient (d₁₁ = 13.057 pm V⁻¹) enables efficient mechanical-to-electrical energy conversion. PH-SiBiX monolayers could serve as active layers in wearable strain sensors or bio-integrated devices (e.g., pulse monitors), where mechanical deformations generate real-time electrical signals without external power sources. The auxetic behavior (ν= −0.206) further enhances mechanical resilience under cyclic loading, addressing durability challenges in flexible electronics.

2. Strain-programmable optoelectronics: the visible-light absorption tunability (7.5–16%) under biaxial strain (−4% to 3%) suggests potential for mechanically adaptive photodetectors. By integrating PH-SiBiX with stretchable substrates (e.g., polydimethylsiloxane), external strain could dynamically modulate the bandgap and optical response, enabling wavelength-selective photodetection for smart imaging systems. Additionally, the strong SOC induced band splitting (Δ ∼ 50 meV) offers opportunities for spin-polarized light-emitting diodes in quantum communication.

3. Multifunctional heterostructure integration: stacking PH-SiBiX with conventional 2D materials (e.g., MoS₂, graphene) could unlock synergistic effects. For instance: (1) a PH-SiBiTi/MoS₂ heterojunction may combine high piezoelectricity with MoS₂'s superior carrier mobility, enabling dual-mode energy harvesting (mechanical + solar). (2) PH-SiBiNi/graphene hybrids could leverage auxetic lattice contraction to enhance interfacial strain transfer, optimizing graphene-based pressure sensors.

3.5 Challenges and outlook

While the computational results are promising, future efforts should prioritize experimental validation. We propose a feasible experimental pathway for synthesizing PH-SiBiX monolayers, building on established protocols for 2D materials like silicene and Janus transition metal dichalcogenides (TMDs).^65,66 The synthesis could employ chemical vapor deposition (CVD) or molecular beam epitaxy (MBE): (1) Co-depositing Si/Bi precursors (e.g., SiCl₄ and Bi₂O₃) on Ag(111) or ZrB₂ substrates to form the PH-SiBi honeycomb lattice; (2) introducing transition metal (TM) atoms (e.g., Ti, Co) via controlled vapor-phase adsorption; (3) optimizing growth parameters using in situ characterization tools (STM, reflection high-energy electron diffraction). Structural validation would involve high-resolution STEM to confirm the honeycomb lattice and TM adsorption sites, XPS to analyze chemical bonding states, and Raman spectroscopy to match theoretical phonon modes. Functional validation includes piezoresponse force microscopy (PFM) to measure the exceptional in-plane piezoelectric coefficient (d₁₁ ∼ 13 pm V⁻¹), AFM nanoindentation to verify auxetic behavior (ν ∼ −0.2), and UV-vis spectroscopy to demonstrate strain-tunable optical absorption (7.5–16%). Challenges such as TM uniformity and ambient stability could be mitigated via low-deposition-rate MBE, surfactant-assisted growth, and encapsulation (e.g., h-BN). Prior successes in synthesizing silicene on Ag(111)⁶⁵ and TM-adsorbed graphene heterostructures⁶⁷ support the feasibility of this approach. Collaborative efforts with experimental groups will prioritize CVD-based synthesis and explore stability-enhancing strategies for real-world applications.

4. Conclusion

In summary, we demonstrate that transition metal-adsorbed PH-SiBiX monolayers uniquely integrate three critical functionalities: (1) record-high piezoelectricity (d₁₁ = 13.057 pm V⁻¹), surpassing MoS₂ and quartz, enabled by strain-sensitive d–p orbital hybridization. (2) Intrinsic auxeticity (ν = −0.206), via hinge-like lattice dynamics, enhancing mechanical resilience. (3) Strain-tunable optics (7.5–16% visible absorption), programmable by −4 to 4% biaxial strain.

This multifunctionality overcomes limitations of conventional 2D materials (e.g., MoS₂'s positive ν, Janus TMDs' weak piezoelectricity) and enables: self-powered wearables with fatigue-resistant energy harvesting; adaptive photodetectors for real-time spectral control; hybrid heterostructures (e.g., PH-SiBiTi/MoS₂) for dual-mode energy conversion.

Future work should prioritize CVD synthesis and prototype devices to bridge computational predictions and applications. This symmetry-orbital design framework is extendable to other 2D systems, accelerating development of mechanically intelligent technologies.

Data availability

The authors confirm that the data underlying this study is available upon request from the corresponding authors.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

Lei Yang was partially supported by the postgraduate research opportunities program of HZWTECH (HZWTECH-PROP). This work is supported by the National Natural Science Foundation of China (No. 12174164, 91963201 and 11834005) and the 111 Project under Grant No. B2006 and Key R&D Project of Gansu Province (No. 22YF7WA014). The work was carried out at the National Supercomputer Center in Tianjin, and the calculations were performed on the Tianhe new generation supercomputer. We also gratefully acknowledge HZWTECH for providing computation facilities.

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Footnote

† Electronic supplementary information (ESI) available: Band structures, projected density of states (PDOS), calculated adsorption energies of PH-SiBiX (X = Sc–Cd), and the relationship between optical properties and the variation of vacuum layer thickness. See DOI: https://doi.org/10.1039/d5tc01715g

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