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. 2026 Jan 16;11(4):6198–6206. doi: 10.1021/acsomega.5c10843

Tetra-Germanene Study from First Principles: Structure, Electronics, Mechanics, and Vibrations

Phi M Nguyen 1, Hai Hoang 2,3, Vladimir Bubanja 4,5, Tet Vui Chong 6, Hanh T T Tran 1,*
PMCID: PMC12878705  PMID: 41658074

Abstract

We present a theoretical investigation of the structural, electronic, and mechanical properties of tetra-germanene. Starting from a buckled rectangular unit cell, obtained via molecular dynamics simulations, the structure was optimized by employing density functional theory, yielding lattice constants (a = 4.01 Å, b = 4.17 Å), buckling height of 2.18 Å, and a cohesive energy of 5.14 eV/atom, exceeding that of hexagonal germanene. The analysis of the electronic band structure and density of states reveals metallic behavior without Dirac crossings. The orbital populations indicate minor participation of d orbitals and enhanced coordination compared to the hexagonal phase. The in-plane Young’s modulus is anisotropic, with values of 53.10 N/m and 68.75 N/m along the two principal directions, while the corresponding bulk modulus is 59.16 N/m, indicating moderate stiffness and directional flexibility. Phonon dispersions are mostly free of imaginary modes, and the maximum optical frequency is ∼220 cm – 1, indicating improved thermal performance at low to moderate temperatures compared with hexagonal germanene. This combination of cohesive stability, metallic conductivity, and anisotropic elasticity suggests that tetra-germanene is a promising candidate for strain-tunable conductive channels and efficient thermal pathways in the next-generation nanoelectronic, energy efficiency, and phononic devices.

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1. Introduction

The discovery of graphene has caused the rapid growth of the field of two-dimensional (2D) materials. Among the various novel structures discovered, 2D allotropes of group IV elements, particularly germanium, have garnered significant interest, resulting in the formation of germanene. Germanium can form up to four strong, directional, covalent bonds due to its four valence electrons (4s24p2). Unlike planar graphene, which features sp2 hybridization and consists of carbon atoms with a lower atomic weight, , germanene adopts a buckled honeycomb structure due to mixed sp2/sp3 hybridization and the larger atomic size of Ge atoms. , This leads to a longer Ge–Ge bond length (2.34–2.45 Å), a larger lattice constant (3.97–4.06 Å), and notable buckling (0.64–0.69 Å). , The increased surface area and structural flexibility of germanene, compared to graphene, make it well-suited for applications in surface adsorption, sensing, and energy storage.

The distinct structural characteristics influence the electronic and mechanical properties of germanene. Its electronic band structure features a Dirac point at the K point, similar to graphene. , However, due to its buckling and spin–orbit coupling, the band structure becomes slightly distorted, resulting in a small band gap of approximately 0.024 eV. Despite being narrow, the finite band gap in germanene enables tunable electronic properties, which graphene, with its highly stable and gapless nature, cannot offer. Mechanically, graphene remains superior with a Young’s modulus of around 337.14 N/m and a bulk modulus of 91.86 N/m, , reflecting its exceptional stiffness and tensile strength, which surpass those of most known natural and synthetic materials. In contrast, germanene exhibits significantly lower mechanical strength due to its buckled geometry and weaker bonding, with a Young’s modulus of only 47.87 N/m and a bulk modulus of 26.04 N/m, indicating that germanene is much softer and more deformable than graphene.

This inherent softness, combined with its flexible sp2/sp3 hybridized bonding, enables germanene to form a variety of structurally diverse allotropes. Notable allotropes are tetra-germanene and tetragonal phases, − both characterized by square-like lattice geometries. In experiments, a quasi-square superstructure of germanene has been observed on the Ag0.9Al0.1(111) surface, synthesized via atomic segregation epitaxy. This finding supports the feasibility of exotic allotropes such as tetra-germanene. In parallel with these observations, recent advances in colloidal fabrication techniques have enabled the controlled synthesis of nonclose-packed 2D square lattices. By combining crystallization in confined geometries with pH-regulated electrostatic adsorption, 2D square-lattice silica crystals have been successfully formed from negatively charged silica particles. Other square-lattice materials have also been shown to have promising properties, including the bismuth monolayer synthesized by low-temperature deposition on a Pb(111) substrate. This 2D material exhibits a Rashba-type spin-split Dirac band near the Fermi level, induced by strong spin–orbit coupling and the breaking of sublattice symmetry. Promising that 2D square-lattice systems are experimentally achievable and can host interesting electronic properties.

In computational studies, tetragonal germanene is confirmed to be mechanically stable by first-principles calculations, satisfying the Born-Huang stability criteria. Its Young’s modulus is reported to be 54.06 N/m, while the bulk modulus reaches 31.62 N/m, with both values being moderately higher than those of hexagonal germanene, indicating enhanced stiffness. In terms of electronic properties, tetragonal germanene exhibits two Dirac cones in its band structure, and an external electric field can effectively tune its band gap, leading to improved thermoelectric performance. The 4-fold configuration of germanene with tetrahedral symmetry, formed from a monovacancy defect, has also been identified as the most stable defect. These findings highlight the 2D square-lattice tetragonal phase as a structurally adaptable and electronically tunable form, enhancing the functional scope of germanene materials.

Currently, research on the physical and chemical properties of other square-like allotropes, particularly tetra-germanene, remains relatively limited. Notably, tetra-germanene’s stability has so far only been explored through molecular dynamics (MD) simulations, in which the tetra phase was obtained by melting and compressing hexa-germanene under confinement along the z-axis, followed by cooling to room temperature. By comparison, its silicon equivalent, tetra-silicene, has been investigated more extensively, with studies employing both molecular dynamics and density functional theory to confirm its mechanical and thermodynamic stability. , Tetra-silicene exhibits a highly buckled structure, semiconducting behavior with a small band gap (∼0.19 eV), and a negative Poisson’s ratio in contrast to the positive value of hexagonal silicene. Owing to its sp3 bonding configuration and absence of π states compared to hexagonal silicene, tetra-silicene emerges as a mechanically adaptive 2D material with promising potential.

Considering the currently limited first-principles studies on tetra-germanene, a comprehensive investigation using density functional theory (DFT) is essential to clarify its fundamental properties. Understanding these properties is particularly important, as the structural uniqueness of tetra-germanene may lead to distinct mechanical and electronic behaviors not observed in other germanene allotropes. This study has systematically evaluated the structural, bonding, mechanical, and electronic properties of tetra-germanene, with particular emphasis on its mechanical behavior and bonding characteristics. By comparing these results with known phases, such as hexagonal germanene and tetra-silicene, this work aims to determine whether tetra-germanene can serve as a viable and tunable 2D material. This approach advances quantum-level understanding of germanene allotropes and informs their potential use in future applications.

2. Computational Method

Establishing an initial configuration is crucial for studying tetra-germanene, as inappropriate input structures can easily relax back into hexagonal germanene during DFT optimization. For this reason, the initial structure of tetra-germanene was extracted from previous molecular dynamics simulations. These simulations employed the Stillinger-Weber potential to describe Ge–Ge interactions and modeled the compression of hexagonal germanene at a density of 28.87 g/cm3, followed by thermal relaxation at 300 K. From the equilibrated molecular dynamics trajectory, a representative 64-atom segment of tetra-germanene was selected (see Figure ) from a relatively symmetric region, away from local defects, and used as the starting point for first-principles calculations. Depending on the buckling mode, the square-lattice unit cell may contain 1 atom (planar), 2 atoms (chair-buckled), 4 atoms (zigzag- or boat-buckled), or up to 16 atoms in more complex buckled configurations. A 64-atom model was therefore used to accommodate all of these possibilities and capture the full range of potential buckling patterns and unit-cell configurations.

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Relaxed tetra-germanene model, comprising of 64 atoms, is shown with its square unit cell containing 4 atoms.

The extracted structure was relaxed using density functional theory as implemented in the SIESTA code, with a Monkhorst–Pack k-point grid of 5 × 5 × 1. The k-point mesh was chosen based on a detailed k-point convergence study, where grids ranging from 1 × 1 × 1 to 15 × 15 × 1 MP were tested to ensure accurate total energy convergence for large supercell models of germanene (See Figure S1). The interaction between ions and electrons was described using norm-conserving scalar relativistic pseudopotentials. Electrons below the Ge 3d shell are treated as frozen core states, while the 3d, 4s, and 4p electrons are included as valence, giving a total of 14 valence electrons per Ge atom. The exchange–correlation energy was treated within the generalized gradient approximation (GGA), as formulated by Perdew, Burke, and Ernzerhof (PBE), which provides a practical balance between computational efficiency and accuracy, particularly for structure optimizations and thermochemical properties. A double-ζ plus polarization (DZP) basis set was used, which has been shown to be suitable for describing the bonding characteristics and structural features of germanene-based materials. , Periodic boundary conditions were applied in the in-plane directions, while a vacuum region of 40 Å was introduced along the z-direction, perpendicular to the tetra-germanene plane, to eliminate interlayer interactions. Because the initial MD structure exhibits a wide distribution of first-neighbor bond lengths (see Figure ), three 64-atom candidate segments initially tested. Two of these segments successfully relaxed, whereas the third exhausted the maximum number of conjugate-gradient steps while exploring alternative structural arrangements. As a result, only the zigzag-buckled conformation emerged after relaxation (See Figure S3).

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Comparison of the radial distribution function g­(r) between the initial MD structure (red line) and the optimized tetra-germanene model (blue line).

Due to the structural complexity of the relaxed 64-atom supercell, the first-neighbor Ge–Ge bond lengths varied, indicating deviations from an ideal pristine configuration. To improve structural symmetry while retaining computational efficiency, the system was reduced to a four-atom unit cell (Figure ), preserving the essential characteristics of tetra-germanene. Whereas the 64-atom model was relaxed under the default optimization conditions of SIESTA, the four-atom unit cell was further optimized under zero-stress conditions. The Hellmann–Feynman force convergence criterion was gradually tightened from 0.01 eV/Å to 1 × 10 – 4 eV/Å to obtain a stable and zero-stressed structure. Furthermore, the maximum allowable atomic displacement per conjugate-gradient step was reduced from 0.05 Å to 5 × 10 – 5 Å in the final relaxation stage. With the reduced cell volume, a denser Monkhorst–Pack k-point mesh of 30 × 30 × 1 was employed to ensure accurate Brillouin zone sampling. The real-space mesh cutoff was also increased to 500 Ry to further improve numerical accuracy. In addition, a triple-ζ plus polarization (TZP) basis set with Grimme’s D3 dispersion correction was also tested for the structural optimization of tetra-germanene.

For comparative purposes, a conventional germanene model with a hexagonal unit cell containing two Ge atoms was also relaxed using identical computational parameters. Phonon calculations were performed at the Γ-point using the finite displacement method as implemented in SIESTA. A 3 × 3 supercell and a displacement amplitude of 0.04 Bohr were used for these calculations to capture the dynamic stability and vibrational properties of the system. Electronic charge density was obtained from SIESTA outputs using the Denchar utility and visualized with the VESTA program to analyze the spatial distribution of electrons in tetra-germanene.

3. Results and Discussion

3.1. Tetra-Germanene Unit Cell Structural

The optimized lattice parameters of the final model are a = 4.012 Å, b = 4.171 Å, and c = 42.069 Å, with lattice angles α = 89.977°, β = 90.013°, and γ = 90° (see Table ). Despite the relaxation, the out-of-plane lattice parameter c was maintained as a large vacuum spacing and the orthorhombic symmetry of the in-plane lattice was preserved. Symmetry analysis concluded tetra-germanene belongs to the orthorhombic space group Pmm2 (No. 25). The lattice parameters of tetra-germanene are close to those reported for the complex, layered intermetallic compounds LuGe3 (a = 3.976 Å; b = 3.870 Å) and LuGe2 (a = 3.982 Å; b = 3.848 Å) both of which feature a layer of two-dimensional square-net composed of germanium atoms.

1. Comparison of Structural Parameters and Binding Energy between Germanene Models.

parameter tetra-germanene germanene MD model
bond length (Å) 2.960; 3.018 2.440 2.480
lattice constant a (Å) 4.012 4.010  
lattice constant b (Å) 4.171    
angle γ (°) 90 120  
buckling height (Å) 2.182 0.620 0.737
cohesive energy (eV) 5.135 4.763  
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The optimized structure was subsequently verified by refining the calculation using the TZP basis set with Grimme’s D3 dispersion corrections, with the maximum force tolerance criterion progressively tightened from 0.01 to 0.005 eV/Å. The results from the TZP-D3 calculation are consistent with those from DZP, with structural and energetic differences of only around 1% (see Table S1). Therefore, DZP was retained as the primary basis set to ensure computational efficiency. For further comparison, a hexagonal germanene model tested under the same computational setup showed good agreement with previous theoretical studies, , exhibiting optimized lattice constants of a = b = 4.087 Å and an in-plane angle of γ = 120.068°. This result confirms that the simulation setup and the pseudopotential were sufficiently reliable to reproduce the established structure of hexagonal germanene.

The relaxed tetra-germanene model exhibits a buckled structure similar to hexagonal germanene, but with a much larger vertical buckling height of 2.18 Å, compared with 0.64–0.695 Å observed in pristine hexagonal-germanene , and significant deviations from the initial molecular dynamics model. In particular, the 64-atom system shows an increased first-nearest-neighbor Ge–Ge bond length peak of 2.91 Å, compared with 2.48 Å in the MD-relaxed structure (Figure ). The out-of-plane buckling also increased substantially from 0.737 Å in the initial MD model to 2.18 Å after relaxation, highlighting the structural reconfiguration upon relaxation without external constraints. This difference is primarily attributed to the boundary conditions applied during the MD simulation, where an elastic reflection was imposed along the z-axis, limiting out-of-plane relaxation. In contrast, the DFT optimization allowed for full relaxation of both atomic coordinates and lattice vectors, enabling the structure to reach a lower-energy, stress-free configuration.

The final bond lengths between neighboring Ge atoms in the 4-unit-cell system increased to 2.96 Å and 3.018 Å, significantly longer than the 2.44 Å observed in hexagonal germanene (Table ). These elongated bonds have been observed in LuGe3, where Ge–Ge bond lengths range from 2.475 to 2.774 Å, forming a distorted square-buckled layer. This structure allows each germanium atom to be in closer proximity to additional neighboring Ge atoms within the unit cell and its periodic images, effectively increasing the coordination number compared to that of pristine germanene. As a consequence of this enhanced coordination and altered hybridization, the electrons in tetra-germanene may occupy higher energy states than those in sp2/sp3 hybridization of hexagonal germanene, , thus implying a denser electronic environment. Furthermore, owing to its denser atomic packing and pronounced buckling, tetra-germanene exhibits a reduced surface area per atom of 4.184 Å2/atom, in contrast to the higher value of 7.232 Å2/atom found in hexagonal germanene. This reduction may limit its effectiveness in applications requiring high surface accessibility, such as gas adsorption or surface catalysis.

The stability of germanene models can be evaluated by calculating the cohesive energy:

Ecoh=(nEGe−Egermanene)n 1

where n is the number of germanium atoms in the system, EGe is the total energy of a single germanium atom in vacuum (eV), and Egermanene is the total energy of the germanene structures (eV). Based the calculation results, the cohesive energy of tetra-germanene is 5.135 eV, which is notably higher than that of hexagonal germanene, calculated to be 4.763 eV. In contrast, silicene structures does not exhibit this behavior, as its tetra and hexagonal forms possess nearly identical cohesive energies (∼3.90 eV/atom). This highlights that phase reconstruction into tetra-germanene leads to a more distinct energetic benefit than in the case of silicene. The lower total energy in tetra-germanene attribute to the increased number of bonding interactions within a single unit cell, with each atom potentially exhibiting 4-fold coordination, in contrast to the typical 3-fold coordination observed in hexa-germanene.

3.2. Bonding Characteristics

The electronic density of states (DOS) provides critical insight into the electronic structure of germanene and can be directly obtained from first-principles calculations. In conventional hexagonal germanene, the occupied states below the Fermi level are dominated by Ge 4p and 4s orbitals, consistent with sp2/sp hybridization (Figure a). In contrast, the DOS of tetra-germanene reveals an additional contribution from the 4d orbitals below the Fermi level (see Figure b). Tetra-germanene, however, lacks the Dirac cone and semimetallic characteristics of hexagonal germanene, , instead displaying a fully metallic electronic structure. Additionally, the spin-resolved density of states calculation reveals no asymmetry between spin-up and spin-down channels, indicating that the material is nonmagnetic (see Figure S2).

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Total and partial electronic density of states for (a) hexagonal germanene and (b) tetra-germanene.

The bonding characteristics are further analyzed using Mulliken charge distribution, which quantifies the contribution of valence orbitals to the total electron density (see Table ). No significant charge transfer is observed between Ge atoms within the tetra-germanene unit cell, similar to the behavior in hexagonal germanene, indicating that bonding is predominantly covalent or metallic in nature. As a result, Table reports the Mulliken population for a representative Ge atom, as all atoms exhibit nearly identical charge distributions. Compared to hexagonal germanene, the tetra-germanene structure exhibits a reduced electron population in the 4p orbital, while the charges in the 4s and 4d orbitals increase. These findings suggest that the geometric reconstruction of the germanene lattice in the tetra phase promotes greater involvement of higher-energy orbitals, particularly the 4d orbital, in bonding and electronic conduction, leading to the longer Ge–Ge bond lengths and increased buckling height observed in tetra-germanene compared with its hexagonal counterpart.

2. Mulliken Charge Distribution on Atomic Orbitals of Germanene.

structure 3d (e) 4s (e) 4p (e) 4d (e)
tetra-germanene 10.009 1.582 2.117 0.290
hexa-germanene 10.015 1.512 2.319 0.151
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The COOP/COHP analysis was further performed to provide a comprehensive understanding of the bonding characteristics and the distribution of bonding and antibonding interactions in the tetra-germanene structure. The COOP/COHP results (See Figure ) shows that the Ge–Ge bonding interactions in both germanene structures are dominated by 4p–4p and 4s–4p hybridizations, which form the dominant σ-bonding framework throughout the valence region. A small 4d–4s bonding component is detected; however, the 4d–4p and 4d-4d COHP signals remains essentially negligible, indicating that the Ge 4d states contribute minimally to directional bonding. Thus, although weak d–s interactions are present, they are vastly overshadowed by the dominant p–p bonding character. Overall, the bonding environment in tetra-germanene is therefore best described as a distorted sp3 configuration rather than any higher-order promoted hybridization scheme like dsp2 or dsp3.

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COOP/COHP analyses for (a) tetra-germanene and (b) germanene.

To gain deeper insight into the bonding directional of the tetra-germanene structure, we examined the charge density difference by plotting the electron redistribution relative to isolated atoms. This analysis was conducted using the following relation:

Δρ=ρtetra−Ge−∑iρisolated,i 2

where ρ tetra‑Ge is the total charge density of the tetra-germanene system (eV/Å3), and ρ isolated represents the charge density of each noninteracting germanium atom (eV/Å3), calculated using the same computational settings. In pristine germanene, the charge difference plot shows a clear accumulation of electron density between adjacent Ge atoms, which is proof of strong covalent bonding (see Figure ). In contrast, the tetra-germanene model exhibits significantly less electron accumulation between neighboring Ge atoms. Charge sharing occurs preferentially between Ge atoms from different buckling heights, rather than between atoms at the same height. This spatial redistribution suggests that each Ge–Ge bond in tetra-germanene is weaker and more delocalized compared to the strong covalent bonds in hexagonal germanene. The reduced interatomic charge accumulation is consistent with a more metallic bonding environment and the presence of multicentered bonding characteristics of tetra-germanene.

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Differential charge density distribution of germanene structures: (a) hexa-germanene and (b) tetra-germanene. The isosurface value is indicated at the top right of each panel. The red line indicates the plane of the cross-sectional charge density slice.

3.3. In-Plane Young’s and Bulk Modulus

To evaluate the mechanical stiffness of the tetra-germanene structure, the in-plane Young’s modulus was calculated using a thickness-independent formulation. This method avoids the ambiguities associated with defining an effective thickness in two-dimensional materials, which often result in significant discrepancies in reported Young’s modulus values. In this work, the in-plane Young’s modulus Ys (N/m) is calculated using the following expression:

Ys=1A0∂2Es∂ε2 3

where a0 is the equilibrium cross-sectional area of the unit cell (Å2), es is the strain energy (eV), and ε is the applied uniaxial strain. The second derivative of the strain energy with respect to strain quantifies the material’s resistance to deformation. To evaluate ∂2Es/∂ε2, the relaxed system was subjected to uniaxial strains from – 0.5% (compressive) to +0.5% (tensile) in 0.1% increments, with the ± 0.1% cases excluded since the total energy approached the numerical limit. These small strains ensure that the system remains within the linear elastic regime, where Hooke’s law is applicable. For each strain step, total energy calculations were performed, and the corresponding strain energies were recorded. The mesh cutoff was increased to 900 Ry to minimize numerical artifacts in the strained states. These energy values were plotted as a function of strain, and a second-order polynomial was fitted to the data (see Figure ). All quadratic fittings yielded coefficients of determination R2 > 0.99, reflecting the high quality of the strain–energy fits used to determine elastic constants. The second derivative of this polynomial with respect to strain yields the ∂2es/∂ϵ2 value, which was then substituted into the above equation to compute the in-plane Young’s modulus.

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Stress–strain energy curve under uniaxial deformation applied in (a) the a-direction and (b) the b-direction of tetra-germanene.

Similar to Young’s modulus calculation, we further compute the bulk modulus by evaluating the second derivative of the strain energy to the instantaneous cross-sectional area (Å2). Bulk modulus B (N/m) can be derived from the energy–area relation, using the following formula. ,

B=A0∂2Es∂A2 4

The relaxed system was subjected to biaxial strains ranging from – 0.5% to +0.5%, applied in 0.1% increments, and ∂2Es/∂A2 was obtained by second-order polynomial fitting with coefficients of determination R2 > 0.99 (see Figure ). The bulk and Young’s modulus of hexagonal-germanene were also obtained using the same computational approach.

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Stress–strain energy curve under biaxial deformation of tetra-germanene.

According to the Young’s modulus results (see Table ), tetra-germanene shows anisotropic in-plane stiffness with longitudinal (a-direction) and transverse (b-direction) values of 53.10 N/m and 68.75 N/m, respectively. These values are slightly higher than those of pristine germanene of our calculation (∼40.09–42.20 N/m) and other studies (∼42.05–47.87 N/m), , as well as tetragonal germanene (54.06 N/m) but still considerably lower than graphene, whose exceptional stiffness (∼309–342 N/m). ,, Compared to silicene (63 N/m), , and tetra-silicene (65–75 N/m), the Young’s modulus of tetra-germanene falls within a comparable range, reaching up to 68 N/m. Moreover, similar to tetra-silicene, tetra-germanene also exhibits anisotropic Young’s modulus, as evidenced by the noticeable difference between its longitudinal and transverse values. This anisotropy likely stems from the unique tetra-germanene lattice (a = 4.01 Å, b = 4.17 Å), whose deviation from a perfect square leads to direction-dependent stiffness under strain.

3. Mechanical Modulus Comparison of Tetra-Germanene and Other 2D Materials .

material bulk modulus (N/m) longitudinal Young modulus (N/m) transverse Young’s modulus (N/m)
tetra-germanene (This work) 59.16 ± 1.69 53.10 ± 1.51 68.75 ± 1.77
germanene (this work) 31.83 ± 0.72 42.20 ± 1.59 40.09 ± 0.30
germanene , 26.04; 13.36 47.87; 42.05 47.86; 42.05
graphene ,, 91.86 337.14; 309; 342.20 337.14; 309; 342.20
silicene , 20.17 63 63
tetra-silicence 66.98 65.26 75.54
tetragonal germanene 31.62 54.06 54.06
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a

All uncertainties are expressed as standard errors (SE) derived from the fitting procedure.

The bulk modulus of tetra-germanene is calculated to be 59.16 N/m, indicating moderate in-plane stiffness. This value is higher than those of hexagonal-germanene in our calculation (31.83 N/m) and previous studies including germanene (13.36–26.04 N/m), , tetragonal-germanene (31.62 N/m) and silicene (20.17 N/m). A similar trend is observed in tetra-silicene, whose bulk modulus, estimated from the published elastic constants using B = (C11+C22+2C12)/2, is ∼ 66.98 N/m, notably higher than that of hexagonal silicene (20.17 N/m). This observation is consistent with the notion that four-membered rings enhance the stiffness of 2D hexagonal lattices.

Overall, tetra-germanene demonstrates moderate mechanical strength stiffer than hexagonal germanene and tetragonal germanene. This behavior originates from differences in bonding characteristics between hexagonal and tetra structures. Tetra-germanene configuration imposes higher coordination number, leading to a higher bulk modulus and in-plane stiffness compared to hexagonal germanene. From the COOP/COHP comparison between tetra-germanene and hexagonal germanene (see Figure ), the antibonding σ* states appear significantly higher above the Fermi level in hexagonal germanene, whereas in tetra-germanene these antibonding states emerge immediately after the Fermi level. This indicates that in tetra-germanene, small perturbations can more easily populate these antibonding states, which enhances resistance to deformation and contributes to its greater mechanical stiffness compared to the hexagonal phase. With anisotropy in Young’s modulus, tetra-germanene can be exploited in applications that require direction-dependent mechanical response, such as anisotropic strain sensors, flexible nanoelectromechanical systems, or directional energy dissipation and wave propagation control in 2D mechanical metamaterials.

3.4. Phonon and Band Structure

To further assess the physical properties of tetra-germanene, its electronic and vibrational characteristics were investigated. The electronic band structure (Figure a) indicates that tetra-germanene exhibits metallic behavior. The Fermi level intersects multiple energy bands, with a dense distribution of electronic states, confirming the presence of free carriers and the absence of a band gap. Moreover, tetra-germanene shows no Dirac points or linear band crossings between the conduction and valence bands. These features, in contrast, are characteristic of hexagonal germanene, , highlighting that the change in lattice symmetry fundamentally modifies the electronic structure. Spin–orbit coupling (SOC), which is known to open a bandgap in hexagonal germanene [5] and to induce spin-split Dirac states in square-lattice Bi(110) monolayers, was also included in the band-structure calculations for tetra-germanene (see Figure S4), but it does not introduce any significant changes to the bands near the Fermi level.

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(a) Electronic band structure and (b) phonon spectrum of tetra-germanene.

Meanwhile, the phonon dispersion (Figure b) plot shows that all phonon modes lie above zero frequency, except for a minor imaginary branch near the Γ point. This indicates that tetra-germanene is overall dynamically stable. However, the presence of a low-frequency soft mode near Γ implies that the structure is close to a mechanical instability threshold, which may relate to its tendency to deform under external perturbations in that phonon direction. Based on these results, the germanene-tetra phase is unlikely to be stabilized as a freestanding monolayer and is more likely to exist only when supported on a substrate or within a confined environment that suppresses the unstable out-of-plane distortions. The observed splitting between longitudinal and transverse optical phonon modes near the Γ point, which is not present in hexagonal germanene, reflects the underlying anisotropy in interatomic restoring forces. This behavior is consistent with the directional dependence of the Young’s modulus in tetra-germanene. The maximum phonon frequency reaches approximately 220 cm – 1, which is lower than that of hexagonal germanene (∼300 cm – 1). A similar reduction in maximum phonon frequency is also observed in the tetra-silicene structure compared to hexagonal silicene. This means tetra-germanene potentially has a larger heat capacity at low-to-moderate temperatures and reaches the Dulong-Petit (3R) limit at a lower temperature than normal germanene.

In summary, tetra-germanene demonstrates stable metallic characteristics with a high electronic density of states at the Fermi level and no significant phonon instabilities. The reduction in maximum phonon frequency suggests that tetra-germanene could be a promising candidate for applications requiring good electrical and thermal conductivity at room temperature.

4. Conclusions

In this study, the structural, electronic, and mechanical properties of tetra-germanene were explored using first-principles density functional theory calculations. Geometry optimization confirmed its thermodynamic stability, revealing a buckled structure with significant out-of-plane distortion (2.18 Å) and a near-square lattice featuring slightly asymmetric Ge–Ge bonds (2.89–3.01 Å). Tetra-germanene has a higher cohesive energy (5.135 eV/atom) than hexagonal germanene, attributed to its increased atomic coordination despite weaker individual bonds. Mulliken analysis indicates a small contribution from 4d orbitals, while COOP/COHP analysis confirms predominantly sp3-type bonding with minor d-orbital participation. Differential charge density further suggests delocalized, multicenter bonding. Mechanically, tetra-germanene exhibits moderate in-plane anisotropy, with Young’s moduli of 53.10 N/m and 68.75 N/m and a bulk modulus of 59.16 N/m. Electronic band analysis shows clear metallic behavior with a high density of states at the Fermi level and no Dirac cones. Phonon dispersion analysis confirms the dynamic stability of tetra-germanene, with low-frequency modes indicating mechanical softness and a maximum phonon frequency of ∼ 220 cm–1, suggesting enhanced heat capacity and improved thermal stability performance at low to moderate temperatures compared to hexagonal germanene. These findings suggest that tetra-germanene is a stable, flexible, and conductive 2D material with potential for use in strain-tunable sensors, flexible nanoelectronics, or electrode materials in next-generation devices.

Supplementary Material

ao5c10843_si_001.pdf (254.3KB, pdf)

Acknowledgments

This research is funded by Vietnam National University HoChiMinh City (VNU-HCM) under grant number: DS2025-20-13. We acknowledge Ho Chi Minh City University of Technology (HCMUT) and VNU-HCM for supporting this study. The authors would like to thank the New Zealand eScience Infrastructure (NeSI) for the use of supercomputing facilities.

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsomega.5c10843.

  • Additional figures, computational details, and supplementary data that complement the results presented in the main manuscript, including the following: (Figure S1) total energy convergence of the 64-atom tetra-germanene model as a function of Monkhorst–Pack k-point grid density; (Figure S2) spin-resolved electronic density of states of tetra-germanene; (Figure S3) side view of the relaxed tetra-germanene structure along a rotated z-axis showing the zigzag buckling pattern; (Figure S4) bandstructure of tetra-germanene with considerate of spin–orbit coupling effect; (Table S1) comparison of tetra-germanene models calculated using DZP and TZP-D3 basis sets (PDF)

H.T.T.T. conceived the idea of developing the research. P.M.N. conducted the simulation. H.T.T.T. and P.M.N. analyzed the data and wrote the manuscripts. H.H., V.B., and T.V.C. contributed to the review of the manuscript and approved the final version of the manuscript.

The authors declare no competing financial interest.

References

  1. Geim A. K., Novoselov K. S.. The rise of graphene. Nat. Mater. 2007;6:183–191. doi: 10.1038/nmat1849. [DOI] [PubMed] [Google Scholar]
  2. Urade A. R., Lahiri I., Suresh K.. Graphene Properties, Synthesis and Applications: A Review. JOM. 2023;75:614–630. doi: 10.1007/s11837-022-05505-8. [DOI] [PMC free article] [PubMed] [Google Scholar]
  3. Makhsin S. R.. Physical Properties of Graphene. J. Mech. Eng. 2023;SI 12:225–267. doi: 10.24191/jmeche.v12i1.24649. [DOI] [Google Scholar]
  4. Chegel R., Behzad S.. Tunable Electronic, Optical, and Thermal Properties of two-dimensional Germanene via an external electric field. Sci. Rep. 2020;10:704. doi: 10.1038/s41598-020-57558-x. [DOI] [PMC free article] [PubMed] [Google Scholar]
  5. Bechstedt F., Gori P., Pulci O.. Beyond graphene: Clean, hydrogenated and halogenated silicene, germanene, stanene, and plumbene. Prog. Surf. Sci. 2021;96:100615. doi: 10.1016/j.progsurf.2021.100615. [DOI] [Google Scholar]
  6. Scalise E., Houssa M., Pourtois G., van den Broek B., Afanas’ev V., Stesmans A.. Vibrational properties of silicene and germanene. Nano Res. 2013;6:19–28. doi: 10.1007/s12274-012-0277-3. [DOI] [Google Scholar]
  7. Sosa A. N., Santana J. E., Miranda Á., Pérez L. A., Rurali R., Cruz-Irisson M.. Transition metal-decorated germanene for NO, N2 and O2 sensing: A DFT study. Surf. Interfaces. 2022;30:101886. doi: 10.1016/j.surfin.2022.101886. [DOI] [Google Scholar]
  8. John R., Merlin B.. Theoretical Investigation of Structural, Electronic, and Mechanical Properties of Two Dimensional C, Si, Ge, Sn. Cryst. Struct. Theory Appl. 2016;5:43–55. doi: 10.4236/csta.2016.5304. [DOI] [Google Scholar]
  9. Andrew R. C., Mapasha R. E., Ukpong A. M., Chetty N.. Mechanical properties of graphene and boronitrene. Phys. Rev. B. 2012;85:125428. doi: 10.1103/PhysRevB.85.125428. [DOI] [Google Scholar]
  10. Goli M., Mozvashi S. M., Aghdasi P., Yousefi S., Ansari R.. A DFT study on the mechanical properties of hydrogenated and fluorinated germanene sheets. Superlattices Microstruct. 2021;152:106854. doi: 10.1016/j.spmi.2021.106854. [DOI] [Google Scholar]
  11. Ngoc L. N., Nguyen H. T. T., Hoang V. V., Ngoc Thanh Thuy T.. Compressioninduced hexa-to-tetra phase transition of confined germanene. J. Mol. Graph. Model. 2023;124:108553. doi: 10.1016/j.jmgm.2023.108553. [DOI] [PubMed] [Google Scholar]
  12. Mondal N. S., Nath S., Jana D., Ghosh N. K.. Band engineering of nonhexagonal 2D tetragonal-silicene sheet and nanoribbons: A theoretical approach. J. Phys. Chem. Solids. 2021;150:109801. doi: 10.1016/j.jpcs.2020.109801. [DOI] [Google Scholar]
  13. Chegel R.. Tunable band gap and enhanced thermoelectric performance of tetragonal Germanene under bias voltage and chemical doping. Sci. Rep. 2023;13:12023. doi: 10.1038/s41598-023-39318-9. [DOI] [PMC free article] [PubMed] [Google Scholar]
  14. Ghosal S., Jana D.. Mechanical characteristics and electric field-influenced thermoelectric and optical responses of tetragonal germanene. J. Phys. D: Appl. Phys. 2021;54:405303. doi: 10.1088/1361-6463/ac09b8. [DOI] [Google Scholar]
  15. Yuhara J., Matsuba D., Ono M., Ohta A., Miyazaki S., Araidai M., Takakura S. i., Nakatake M., Le Lay G.. Formation of germanene with free-standing lattice constant. Surf. Sci. 2023;738:122382. doi: 10.1016/j.susc.2023.122382. [DOI] [Google Scholar]
  16. Hoang V. V., Giang N. H., Dong T. Q.. Confined tetra-silicene obtained by cooling from the melt. Comput. Mater. Sci. 2019;158:406–413. doi: 10.1016/j.commatsci.2018.11.034. [DOI] [Google Scholar]
  17. Qiao M., Wang Y., Li Y., Chen Z.. Tetra-silicene: A Semiconducting Allotrope of Silicene with Negative Poisson’s Ratios. J. Phys. Chem. C. 2017;121:9627–9633. doi: 10.1021/acs.jpcc.7b02413. [DOI] [Google Scholar]
  18. Soler J. M., Artacho E., Gale J. D., García A., Junquera J., Ordejón P., Sánchez-Portal D.. The SIESTA method for ab initio order-N materials simulation. J. Phys.: Condens. Matter. 2002;14:2745–2779. doi: 10.1088/0953-8984/14/11/302. [DOI] [Google Scholar]
  19. Perdew J. P., Burke K., Ernzerhof M.. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett. 1996;77:3865–3868. doi: 10.1103/PhysRevLett.77.3865. [DOI] [PubMed] [Google Scholar]
  20. Bursch M., Mewes J., Hansen A., Grimme S.. Best-Practice DFT Protocols for Basic Molecular Computational Chemistry. Angew. Chem., Int. Ed. 2022;61:e202205735. doi: 10.1002/anie.202205735. [DOI] [PMC free article] [PubMed] [Google Scholar]
  21. Thi Thu Hanh T., Minh Phi N., Van Hoa N.. Hydrogen adsorption on twodimensional germanene and its structural defects: an ab initio investigation. Phys. Chem. Chem. Phys. 2020;22:7210–7217. doi: 10.1039/D0CP00016G. [DOI] [PubMed] [Google Scholar]
  22. Momma K., Izumi F.. VESTA 3 for three-dimensional visualization of crystal, volumetric and morphology data. J. Appl. Crystallogr. 2011;44:1272–1276. doi: 10.1107/S0021889811038970. [DOI] [Google Scholar]
  23. Shiraz A. K., Goharrizi A. Y., Hamidi S. M.. Structural stability and electron density analysis of doped germanene: a first-principles study. Mater. Res. Express. 2019;6:1050c2. doi: 10.1088/2053-1591/ab21f1. [DOI] [Google Scholar]
  24. Jomehpour Zaveh S., Roknabadi M. R., Morshedloo T., Modarresi M.. Electronic and thermal properties of germanene and stanene by first-principles calculations. Superlattices Microstruct. 2016;91:383–390. doi: 10.1016/j.spmi.2016.01.033. [DOI] [Google Scholar]
  25. Cao K., Feng S., Han Y., Gao L., Hue Ly T., Xu Z., Lu Y.. Elastic straining of free-standing monolayer graphene. Nat. Commun. 2020;11:284. doi: 10.1038/s41467-019-14130-0. [DOI] [PMC free article] [PubMed] [Google Scholar]
  26. Man Q., An Y., Shen H., Wei C., Xiong S., Feng J.. Two-dimensional silicene/silicon and its derivatives: Properties, synthesis and frontier applications. Mater. Today. 2023;67:566–591. doi: 10.1016/j.mattod.2023.06.022. [DOI] [Google Scholar]
  27. Zhao J., Ma D., Wang C., Guo Z., Zhang B., Li J., Nie G., Xie N., Zhang H.. Recent advances in anisotropic two-dimensional materials and device applications. Nano Res. 2021;14:897–919. doi: 10.1007/s12274-020-3018-z. [DOI] [Google Scholar]
  28. Aoyama Y., Toyotama A., Okuzono T., Ishikawa T., Hyodo K., Nishida M., Yamanaka J.. Two-Dimensional Square Lattice of Colloidal Particles Formed by Electrostatic Adsorption in Confined Space. Langmuir. 2025;41:1948–1956. doi: 10.1021/acs.langmuir.4c04480. [DOI] [PubMed] [Google Scholar]
  29. Oh E., Jin K.-H., Yeom H. W.. Realizing a Superconducting Square-Lattice Bismuth Monolayer. ACS Nano. 2023;17:7604–7610. doi: 10.1021/acsnano.2c12884. [DOI] [PubMed] [Google Scholar]
  30. Hastuti D. P., Nurwantoro P., Sholihun. Stability Study of Germanene Vacancies: The First-Principles Calculations. Mater. Today Commun. 2019;19:459–463. doi: 10.1016/j.mtcomm.2019.04.003. [DOI] [Google Scholar]
  31. Rivera-Julio J., González-García A., González-Hernández R., López-Pérez W., Peeters F. M., Hernández-Nieves A. D.. Vibrational Properties of Germanane and Fluorinated Germanene in the Chair, Boat, and Zigzag-Line Configurations. J. Phys.: Condens. Matter. 2019;31:075301. doi: 10.1088/1361-648X/aaf45f. [DOI] [PubMed] [Google Scholar]
  32. Hübner J.-M., Bobnar M., Akselrud L., Prots Y., Grin Y., Schwarz U.. Lutetium Trigermanide LuGe3: High-Pressure Synthesis, Superconductivity, and Chemical Bonding. Inorg. Chem. 2018;57:10295–10302. doi: 10.1021/acs.inorgchem.8b01510. [DOI] [PubMed] [Google Scholar]
  33. Zhang J., Ding F., Lee J.-J., Shan G., Bobev S.. One Structure, Two Elements–LuGe2 Superconductor vs Ordinary Metallic Conductor LuSn2: A Case Study on How Site-Selective Germanium for Tin Atom Substitution Leads to Modulating of the Charge Distribution. Inorg. Chem. 2020;59:16853–16864. doi: 10.1021/acs.inorgchem.0c01062. [DOI] [PubMed] [Google Scholar]
  34. Dronskowski R., Blöchl P. E.. Crystal Orbital Hamilton Populations (COHP): Energy-Resolved Visualization of Chemical Bonding in Solids Based on Density-Functional Calculations. J. Phys. Chem. 1993;97:8617–8624. doi: 10.1021/j100135a014. [DOI] [Google Scholar]

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