α_nh_m-GeSe: a multifunctional semiconductor combining auxeticity and piezoelectricity

Abstract

Multifunctional materials with outstanding performance have enormous potential applications in the next generation of nanodevices. Using first principles calculations, we design a series of multifunctional two-dimensional materials in monolayer α_nh_m-GeSe (n, m = 1, 2) that combine auxeticity and piezoelectricity. Due to the similar local structures of α-GeSe and h-GeSe, monolayer α_nh_m-GeSe can be designed through the combination of these two materials. Elastic constants and phonon dispersion curves confirm that all the structures are mechanically and dynamically stable. Monolayer α_nh_m-GeSe exhibits auxetic properties with an in-plane negative Poisson's ratio along the diagonal direction. An out-of-plane negative Poisson's ratio effect can be observed by applying tensile strain in the x-direction, which is beneficial for mechanical devices. Only a few materials are both in-plane and out-of-plane auxetic. In addition, monolayer α_nh_m-GeSe can exhibit electric polarization because of the breaking of the central symmetry, demonstrating in-plane piezoelectric properties with strong anisotropy. The above results make monolayer α_nh_m-GeSe an interesting multifunctional material and provide a candidate material for a nanoscale device.

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Introduction

Two-dimensional materials exhibit excellent performance in various fields, such as ferroelectricity, ferromagnetism, mechanics, optics, catalysis, electrochemistry, etc.^1–3 Multifunctional two-dimensional materials have attracted considerable attention due to their highly integrated and distinct properties, which hold even greater promise to become the next generation of nanodevices.⁴ Auxeticity corresponds to the situation in which the induced strain in an orthogonal direction to the applied strain has the same sign as the applied strain. Usually, when tensile strain is applied in one direction the material is compressed in the other directions. The Poisson ratio is then defined to be positive. In an auxetic material, the Poisson ratio is negative.⁵ This unique phenomenon can enhance mechanical properties, such as shear modulus, indentation resistance and fracture toughness.^6,7 As a result, the auxetic materials have been extensively applied in foam, cubic metals and α-cristobalite.^8,9 Until today, only a few materials exhibit auxetic properties, such as BI, Be₅C₂, Ag₂S and δ-phosphorene, which implies that exploring 2D auxetic materials is meaningful.^10–12

The linear coupling between mechanical deformation and electric fields manifests as the piezoelectric effect, which plays a crucial role in the application of miniaturized and high-performance devices.¹³ Two-dimensional piezoelectric materials can endure much higher strain than brittle traditional piezoelectrics, enabling a broader range of applications such as sensors and nanoelectromechanical systems.¹⁴ Recently, various types of monolayer or few-layer two-dimensional piezoelectric materials have been successively reported, including CuInP₂S₆, transition metal dichalcogenides (TMDs), group III–IV materials, and group IV monochalcogenides (MXs, where M = Ge, Sn, and X = S, Se).^15–17 Due to their exceptional mechanical and piezoelectric properties, group-IV monochalcogenides have garnered significant attention among these piezoelectric candidate materials.^18–20

Over the past few years, a considerable number of GeSe allotropes have been theoretically discovered, and some have since been successfully synthesized experimentally.^21,22 The most stable phase, α-GeSe, exhibits a puckered structure similar to black phosphorene, displaying both in-plane ferroelectricity and out-of-plane NPR.²³ The second synthesized allotrope, β-GeSe, is transformed from α-GeSe under high pressure.²⁴ This allotrope also exhibits outstanding piezoelectric and thermoelectric properties.²⁵ Recently, a hexagonal allotrope of group-IV monochalcogenides, named γ-GeSe, has been synthesized on a SiO₂ substrate.²⁶ The synthesis of these allotropes has not only broadened the possibilities of diverse metastable allotropes but also stimulated the research and development of novel GeSe materials with intriguing attributes like mechanical and piezoelectric properties. Additionally, various other allotropes like δ-GeSe, T-GeSe, h-GeSe, etc., have been theoretically predicted using density functional theory (DFT), and these monolayers exhibit semiconductor and piezoelectric properties.^27–29

Here, in this work, we design a series of group-IV monochalcogenide hybrid α_nh_m-GeSe by combining the α-GeSe and h-GeSe structures through DFT theoretical predictions, in order to achieve coupled auxeticity and piezoelectricity with the following considerations.²⁶ Firstly, GeSe has various phases with intriguing piezoelectric and mechanical properties.^24,30–34 Secondly, two α-GeSe and h-GeSe phases share the same structural motif of threefold-coordinated Ge/Se atoms surrounded by the nearest neighbors Se/Ge atoms in a tetrahedral arrangement and this tetrahedral arrangement can even be maintained in specific intra-layer connections of different structures, which exhibit great potential to be combined with individual blocks to design new allotropes. Thirdly, the breaking of the symmetry of α_nh_m-GeSe leads to the existence of electric polarization and the puckered structure is much more flexible (softer), so that it is possible to have a negative Poisson ratio.

The phonon dispersion and molecular dynamics simulation results reveal that monolayer α_nh_m-GeSe is structurally stable. The puckered structure of α_nh_m-GeSe results in its NPR properties along both in-plane and out-of-plane directions. The underlying physics of out-of-plane NPR is related to the competition between two strain responses: the response of strain along the x-direction to the z-direction and the response of strain along the y-direction to the z-direction. Moreover, monolayer α_nh_m-GeSe displays strong in-plane anisotropy in the piezoelectricity properties. Our results demonstrate that monolayer α_nh_m-GeSe is a novel multifunctional material that holds great potential applications in mechanics and optoelectronic devices.

Method

First-principles calculations were performed based on density functional theory as implemented in the Vienna ab initio simulation package (VASP).^35–37 The electron–electron interactions were treated within a generalized gradient approximation (GGA) in the form of Perdew–Burke–Ernzerhof (PBE) for the exchange–correlation functional.^38,39 We take the energy cutoff 450 eV and the thickness of 15 Å for vacuum. The criteria were set to 10⁻⁸ eV and 10⁻⁵ eV Å⁻¹ for energy and force with the 18× 6 × 1 Monkhorst–Pack sampling for self-consistent calculations, respectively. For better evaluation of electronic and optical properties, the hybrid functional (HSE06) was adopted.⁴⁰ Besides, the phonon dispersion is performed based on density functional perturbation theory (DFPT) as embedded in the Phonopy program.⁴¹ The phonon dispersion using a 4 × 4 × 1 supercell.

Results and discussion

Fig. 1(a) shows monolayer α-GeSe with a layer-puckered structure that can be obtained by exfoliating from the lower symmetry bulk phase along the (100) plane, similar to black phosphorus. The unit cell of α-GeSe has an in-plane Ge–Se bond length of 2.63 Å, which is longer than the out-of-plane bond length (2.54 Å). The in-plane Se–Ge–Se bond angle is 96.46°, and the dihedral angle Se–Ge–Se is 97.39°. Fig. 1(b) shows the periodic structure of monolayer h-GeSe, whose unit cell consists of two atoms, similar to blue phosphorus. To better fit with monolayer α-GeSe, we reconstructed the unit cell of monolayer h-GeSe to be , which comprises four atoms in a unit cell. This structure consists of two chains: one chain is composed of germanium and selenium (Ge–Se–Ge), and the other consists of selenium and germanium (Se–Ge–Se). The bond length and angle in monolayer h-GeSe are 2.57 Å and 91.27°, respectively, similar to α-GeSe. Considering the local similarity of the two structures, we propose assembling them together. Specifically, we use the α unit cell of α-GeSe and the chain (Se–Ge–Se, Ge–Se–Ge) of h-GeSe as building blocks denoted by α and h. Fig. 1(c) demonstrates the potential to design a range of GeSe materials by combining nα and mh, forming what we refer to as α_nh_m-GeSe.

Fig. 1 Design of the α_nh_m-GeSe structures by two blocks. Top and side views of α-GeSe (a) and h-GeSe (b); (c) diagram of nα and mh assembly called α_nh_m-GeSe.

To confirm our idea, through first principles calculations, we predicted α_nh_m-GeSe (n, m = 1, 2) structures. The study primarily examines the structure of α₁h₂-GeSe (Fig. 2(a)), comprising four atomic layers with each Ge atom bonded to its three nearest Se atoms through sp³ hybridization. Fig. 2(a) illustrates that the Ge–Se bond lengths in the α₁h₂-GeSe range from 2.55 to 2.59 Å, resembling those of α-GeSe (2.54 Å) and h-GeSe (2.57 Å). The in-plane bond angle and dihedral angle of Se–Ge–Se are 94.47° and 93.46°, respectively, which closely resemble those of α-GeSe (96.46°, 97.39°) and γ-GeSe (91.27°). Therefore, the local coordination environment of α₁h₂-GeSe resembles that of monolayer α- and h-GeSe. Additionally, the α₁h₂-GeSe structure combines one unit cell of α-GeSe with two unit chains of h-GeSe. Consequently, this results in an increased layer thickness of α₁h₂-GeSe to 3.78 Å, exceeding the layer thicknesses of α-GeSe and h-GeSe at 2.6 Å and 1.45 Å, respectively. Additional structures are depicted in Fig. S1 of the ESI.†

Fig. 2 Structural stability analysis of α₁h₂-GeSe. (a) Top and side views of α₁h₂-GeSe; (b) total energy differences between the α-GeSe and other GeSe allotropes, where we define the total energy of α-GeSe as zero; (c) phonon dispersion and (d) molecular dynamics simulation for α₁h₂-GeSe.

In order to compare the energy differences between these allotropes, we calculated the total energy of α_nh_m-GeSe and the two parent phases. The results of the calculations are shown in Table 1, with the monolayer α-GeSe taken as the zero energy reference point. We notice that the total energy of α-GeSe is the most stable phase. The total energy of α_nh_m-GeSe is ∼30 meV per atom higher than that of α-GeSe, while its energy is less than that of h-GeSe, as shown in Fig. 2(b). The above results show that the total energy of the mixed phase α_nh_m-GeSe lies between those of the two sub-phases (α-GeSe and h-GeSe), indicating that the mixed phase is a metastable phase.

Table 1 The structural information and relevant parameters for monolayer α-GeSe, h-GeSe, β-GeSe and α_nh_m-GeSe, including lattice constants a and b (Å), layer thickness t (Å) and total energy ΔE differences between the α-GeSe and other GeSe allotropes which is the energy of α-GeSe taken as the zero energy reference points (meV)

Allotrope	a	b	t	ΔE
α-GeSe	3.98	4.26	2.60	0
h-GeSe	3.66	3.66	1.45	35.6
β-GeSe	3.67	5.89	1.78	33.4
α1h1-GeSe	3.79	14.57	3.48	27.8
α1h2-GeSe	3.79	10.48	3.78	15.8
α2h1-GeSe	3.79	22.77	4.01	23.4
α2h2-GeSe	3.79	14.74	4.72	29.7

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To assess the structural stability, we calculated the phonon dispersion and conducted molecular dynamics simulations. The phonon dispersion of the monolayer α₁h₂-GeSe is shown in Fig. 2(c), demonstrating no imaginary frequencies throughout the Brillouin zone, confirming the structural stability at 0 K. Additionally, the thermal stability of the monolayer α₁h₂-GeSe is confirmed by the results of an AIMD simulation, as shown in Fig. 2(d), illustrating that the structure remains stable for at least 12 ps at a temperature of 300 K. The phonon dispersions of the α₁h₂-, α₁h₂- and α₁h₂-GeSe structures are illustrated in Fig. S2 (ESI†). On the other hand, α₁h₂-GeSe belongs to space group Pmm2 (no. 25) within the orthorhombic system. For bulk materials, following Voigt notation conventions, the elastic modulus matrix can be expressed as follows:

For 2D materials, we ignore the c-direction component, so the elastic modulus matrix become a second-order tensor, and can be written as follows:

where xx = 1, yy = 2 and xy = 6 by using the Voigt notation. Thus, we only consider the C₁₁, C₂₂, C₁₂ and C₆₆ under the effect of symmetry. Furthermore, the stability of the two-dimensional material can also be established using the Born–Huang criteria,⁴² which require C₁₁, C₂₂, and C₆₆ to be greater than zero, and C₁₁ + C₂₂ − 2C₁₂ to be greater than zero. The calculated values of C₁₁ = 30.4 N m⁻¹, C₂₂ = 11.6 N m⁻¹, C₁₂ = 9.8 N m⁻¹, and C₆₆ = 12.6 N m⁻¹ for α₁h₂-GeSe confirm its stability.

In light of its distinct puckered configuration, α₁h₂-GeSe is anticipated to display intriguing mechanical characteristics. In order to examine its mechanical behavior, we evaluated the Young's modulus Y(θ) and Poisson's ratio υ(θ) via the following expressions:⁴³

where A = (C₁₁C₂₂ − C²₁₂)/C₆₆ − 2C₁₂, B = C₁₁ + C₂₂ − (C₁₁C₂₂ − C²₁₂)/C₆₆ and θ ∈ [0, 2π]. Here, θ is the polar angle with respect to the lattice constant x of the material. The results are plotted in Fig. 3(a). The Young's modulus reflects the rigidity or flexibility of materials, and α₁h₂-GeSe exhibits notable anisotropy in its Young's modulus, with varying values at different angles. Here we consider the x-direction to be 0°. The minimum value of 9.4 N m⁻¹ at θ = 90° and 270° along with its maximum value of 30.4 N m⁻¹ at θ = 58°, 148°, 238° and 328°, suggest that α₁h₂-GeSe is significantly softer along the y-direction. The Poisson's ratio is another crucial parameter that reflects the system's mechanical response to uniaxial strains. Fig. 3(b) illustrates the clear anisotropy in the Poisson's ratio υ(θ) of α₁h₂-GeSe. The Poisson's ratio attains its peak value of 0.84 at θ = 0° and 90°, and its lowest value of −0.048 at θ = 58°, 148°, 238° and 328°, indicating an in-plane NPR value. The Young's modulus Y(θ) and Poisson's ratio υ(θ) for α₁h₁-, α₂h₁- and α₂h₂-GeSe structures are presented in Fig. S5 and S6 in the ESI.†

Fig. 3 Mechanical property diagram of α₁h₂-GeSe. Young's modulus Y(θ) (a) and Poisson's ratio (b) of α₁h₂-GeSe. θ = 0° means in the x-direction and the blue line is NPR. Mechanical response of α₁h₂-GeSe during uniaxial strain in the (c) x-direction and (d) y-direction. The evolution of local structure of monolayer α₁h₂-GeSe under uniaxial tensile strain along the (e) x-direction and (f) y-direction.

Previous studies have reported that α-groupV exhibits out-of-plane NPR.⁴⁴ The presence of a partial α phase structure exists in the mixed phase α₁h₂-GeSe structure; it is worthwhile to investigate whether α₁h₂-GeSe also possesses out-of-plane NPR. For studying the out-of-plane response mechanism, strain (ε) is defined as ε_i = (l_i − l_i0)/l_i0, where i refers to x, y, z directions, and l_i and l_i0 denote the lattice parameter along the i-direction with and without strain, respectively. The Poisson's ratio can be determined by fitting ε = −υ₁ε_i + υ₂ε_i² + υ₃ε_i³, where ε and υ₁ represent the resultant strain and Poisson's ratio, respectively. This study examines the strain response of α₁h₂-GeSe ranging from −5% to +5%, as illustrated in Fig. 3(c) and (d). Under y-direction tension, ε_z decreases, leading to a reduction in the z-direction thickness of α₁h₂-GeSe with increasing ε_y strain, as denoted by the PPR value of υ_z = 0.24. Conversely, when subjected to x-direction tension, ε_z increases, causing an augmentation in the z-direction thickness of α₁h₂-GeSe with increasing ε_x strain, as shown by the NPR value of υ_z = −0.207, which is higher than that of α-phosphorene (−0.027)⁴⁵ and SnSe (−0.17),⁴⁶ but less than that of δ-GeSe (−0.32)⁴⁷ and Ag₂S (−0.52)¹²Table 2.

Table 2 The Poisson's ration of some monolayer puckered structures. The υ_xy (x ≠ y) represents the Poisson's ratio for the stretching strain along the x direction and the corresponding response in the y direction

Materials	υ yx	υ xy	υ zx	υ xz
α1h2-GeSe	0.830	0.240	−0.207	0.940
SnS^5	0.422	0.961	−0.004	0.404
SnSe^5	0.423	0.851	−0.210	0.352
BP^48	0.400	0.930	0.046	−0.027
As^49	0.350	1.070	0.130	−0.093
W–Sb^50	0.273	0.689	0.656	0.315

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To enhance our understanding of the NPR effect in the z-direction, we propose that the Poisson's ratio of monolayer α₁h₂-GeSe in the z-direction results from the superposition of changes in strain ε_z caused by strain ε_x in the x-direction and changes in strain ε_z caused by strain ε_y in the y-direction. The tensile strain in the x-direction leads to the compression of the material along the y- and z-directions, whereas tensile strain in the y-direction additionally influences the compression strain in the z-direction. Hence, the z-direction strain can be expressed as ε_z = ε_z1 + ε_z2 = −Aε_x − Bε_y = −Cε_x (A, B, C > 0). When there is tension strain (ε_x > 0) along the x-direction (refer to Fig. 3(e)), α₁h₂-GeSe will experience compression along the y-direction (ε_y < 0). Taking into account the impact of Poisson's ratio, when A > B × υ_yx, ε_z = −Cε_x with C being positive. This results in a z-direction thickness decrease with increasing x-direction tension strain, resulting in a positive Poisson's ratio (PPR). Conversely, under tensile strain (ε_y > 0) along the y-direction (refer to Fig. 3(f)), α₁h₂-GeSe will undergo compression in the x-direction (ε_x < 0). If the A < B × υ_yx, thus ε_z = −Cε_y and C < 0. This results in a z-direction thickness increase with increasing y-direction tension strain, results in NPR. In this context, we can calculate the values of A and B utilizing density functional theory (DFT). By applying strain ε_x/ε_y in the x/y-direction and eliminating the Poisson effect along the y/x-direction, the strain response coefficients for A and B in the z-direction are approximately 0.97 and 1.11.

To evaluate the electronic structure of α₁h₂-GeSe more accurately, we performed band structure calculations at the HSE06 level. Our results, depicted in Fig. 4(a), show that a monolayer of α₁h₂-GeSe is a semiconductor with a bandgap of 2.306 eV. The valence band maximum (VBM) is at the Γ point along the Γ M path, while the conduction band minimum (CBM) is at the M point.

Fig. 4 Electronic structure and light absorption. The band structure (a) and light absorption (b) of α₁h₂-GeSe at the HSE06 level. Here, the 0 means the Fermi level for the band structure.

In general, the dielectric function is divided into a real part ε^r and an imaginary part εⁱ. Without considering electron–hole or exciton effects, we calculate it based on the independent particle approximation.⁵¹ The imaginary part εⁱ is obtained through VASP calculations, while the real part ε^r is derived using the Kramers–Kronig relation. The formula below establishes a direct relationship between light absorption w and the dielectric function:

Fig. 4(b) presents the optical absorption spectra of α₁h₂-GeSe, which reveals a strong light absorption between the in-plane w_x and w_y absorption coefficients near the ultraviolet range, reaching a maximum value of 22.5% near 280 nm and 19% near 230 nm for the w_x and w_y directions, respectively. We compared the absorption of α₁h₂-GeSe to other transition metal dichalcogenides, such as α-GeSe and δ-GeS and found that α₁h₂-GeSe has a greater absorption in the ultraviolet range (23%), indicating its potential for use in ultraviolet applications.

By applying stress or strain to a non-centrosymmetric material, the piezoelectric effect can be induced. A third-rank piezoelectric stress tensors e_ijk and strain tensor d_ijk can be used to describe the piezoelectric response of a material. The relaxed piezoelectric tensor (e_ijk and d_ijk) include the electronic and ionic contributions,

andwhere σ_jk, ε_ij and P_i represent piezoelectric stress, strain and vector, respectively, and the superscripts elc and ion can be described the electronic and ionic contributions. Thus, e^ion_ijk and d^ion_ijk mean relaxed-ion piezoelectric coefficients, while e^elc_ijk and d^elc_ijk are also called clamped-ion cases. The elastic tensor C_mnjk can be used to associate e_ijk and d_ijk,

Due to the difference in electronegativity and the broken symmetry, monolayer α₂h₁-GeSe has in-plane electric polarization, akin to the characteristics of α-GeSe. Other hybrid phases also do not possess center inversion symmetry and can exhibit piezoelectricity by applying in-plane strain. Here the two independent d₂₂(y) and d₂₁ can be expressed as follows:

which can be calculated by first principles, listed in Table 3. The predicted e₂₂/e₂₁ is 2.5–4.5 × 10⁻¹⁰/0.2–0.4 × 10⁻¹⁰ C m⁻¹ with electronic part 0.6–1.2 × 10⁻¹⁰/0.2–0.4 × 10⁻¹⁰ C m⁻¹ and ionic part 1.8–3.2 × 10⁻¹⁰/0.14–0.43 × 10⁻¹⁰ C m⁻¹. For both e₂₂ and e₂₁, the electronic and ionic parts have superposed contributions. Based on eqn (9), the predicted in-plane piezoelectric coefficient d₂₂ of α_nh_m-GeSe is in the range 40–50 pm V⁻¹, with electronic part 10–15 pm V⁻¹ and ionic part 30–40 pm V⁻¹, which is significantly higher than h-BN (0.6 pm V⁻¹)⁵² and MoS₂ (3.73 pm V⁻¹).⁵³ For d₂₁, the calculated value is about −10 to −16 pm V⁻¹ for α_nh_m-GeSe. Additionally, it is noteworthy that the d₂₂ is three times larger than d₂₁, indicating that α_nh_m-GeSe have a strong anisotropy in piezoelectricity. This might be owing to the change of the structural properties of different materials in α_nh_m-GeSe.

Table 3 The piezoelectric strain tensors and relevant parameters for monolayer α_nm₂-GeSe (n, m = 1, 2), including d_ij (pm V⁻¹), e_ik (10⁻¹⁰ C m⁻¹) and C_jk (N m⁻¹)

					Clamped-ion					Relaxed-ion
Materials	C 11	C 22	C 12	C 66	e ^elc22	e ^elc21	d ^elc22	d ^elc21	e ^ion22	e ^ion21	d ^ion22	d ^ion21
α1h1-GeSe	37.7	11.0	13.5	13.0	0.82	0.15	12.4	−4.0	2.18	0.25	33.9	−11.4
α1h2-GeSe	30.4	11.6	9.8	12.6	1.22	0.06	14.2	−4.3	3.18	0.14	37.1	−11.5
α2h1-GeSe	39.6	11.3	14.7	16.0	0.78	0.17	12.2	−4.1	2.22	0.43	35.2	−12.0
α2h2-GeSe	32.2	8.4	9.0	12.9	0.66	0.06	10.9	−2.8	1.84	0.15	30.5	−8.0

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Summary

In conclusion, we design a new class of stable α_nh_m-GeSe (n, m = 1, 2), which are characterized by combining with α- and h-GeSe structures. These materials have thermodynamic, kinetic and mechanical stability. α₁h₂-GeSe demonstrates auxetic properties, with both in-plane and out-of-plane negative Poisson's ratios (NPR). The underlying physics for out-of-plane NPR is related to the competition of two the responses i.e. ε_z to ε_x and ε_y. Moreover, α₁h₂-GeSe also exhibits a strong anisotropy in mechanical and piezoelectric properties due to its unique structure. These findings suggest that α₁h₂-GeSe may have potential applications in mechanics and optoelectronics.

Data availability

The data that support the findings of this study are available with the corresponding author and can be obtained upon reasonable request.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

This work was supported by the Natural Science Foundation of China (Grant No. 22175150 and U2002217).

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Footnote

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d4cp04045g

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