Unveiling a New 2D Semiconductor: Biphenylene-Based InN

The two-dimensional (2D) materials class earned a boost in 2021 with biphenylene synthesis, which is structurally formed by the fusion of four-, six-, and eight-membered carbon rings, usually named 4-6-8-biphenylene network (BPN). This research proposes a detailed study of electronic, structural, dynamic, and mechanical properties to demonstrate the potential of the novel biphenylene-like indium nitride (BPN-InN) via density functional theory and molecular dynamics simulations. The BPN-InN has a direct band gap energy transition of 2.02 eV, making it promising for optoelectronic applications. This structure exhibits maximum and minimum Young modulus of 22.716 and 22.063 N/m, Poisson ratio of 0.018 and −0.008, and Shear modulus of 11.448 and 10.860 N/m, respectively. To understand the BPN-InN behavior when subjected to mechanical deformations, biaxial and uniaxial strains in armchair and zigzag directions from −8 to 8% were applied, achieving a band gap energy modulation of 1.36 eV over tensile deformations. Our findings are expected to motivate both theorists and experimentalists to study and obtain these new 2D inorganic materials that exhibit promising semiconductor properties.


INTRODUCTION
The versatility of carbon in adopting sp, sp 2 , and sp 3 hybridizations opens up an infinite array of crystalline lattices with a broad range of properties.Some examples are Tgraphene, 1 twin-graphene, 2 QPHT-graphene, 3 penta-graphene, 4 DHP-graphene, 5 TPDH-graphene, 6 graphenylene, 7 biphenylene, 8 triphenylenes, 9 naphthylenes, 10,11 and tripentaphenes. 12,13In this structure class, biphenylene features a lattice composed of four-, six-, and eight-membered carbon rings, known as the 4-6-8-biphenylene network (BPN).Another related structure, graphenylene, is characterized by a lattice containing 4-, 6-, and 12-membered carbon rings.The first one was recently obtained by surface polymer dehydrogenation (HF-zipping), 14 while the second one was produced via polymerization reactions using the 1,3,5-trihydroxybenene precursor. 15Many reports, mainly based on density functional theory (DFT), have demonstrated that both monolayers are suitable for diverse applications, including catalysis, 16 molecular membranes, 7 water purification, 17 gas sensing, 18−20 thermoelectricity, 21−23 optoelectronics, 24 energy storage, 25−27 electronics, 28 and catalysis. 29,30n the past decade, several inorganic graphenyelenes (IGPs) have been proposed based on diverse compounds such as boron nitride (BN), 31 aluminum nitride (AlN), 32 gallium nitride (GaN), 32 indium nitride (InN), 33 SiC, 34 GeC, 35,36 SiGe, 37 ZnO, 38 CdO, 39 MgO, 39 and BeO.39 Among them, IGP-ZnO showed up promising for SO 2 and NO 2 detection.40 At the same time, the IGP-SiC has proven itself to be an optimal platform for energy storage when employed in Na-based batteries 41 and hydrogen storage.42 On the other hand, the same approach has been employed for two-dimensional (2D) biphenylene.The utilization of AlN and GaN as counterparts of the biphenylene network (BPN-AlN and BPN-GaN) led to two stable monolayers with band gap energies of 2.3 and 3.2 eV, respectively.43 Both monolayers exhibit noteworthy UV activity, promising prospects as UV collectors. Futhermore, BPN-BN has structural and dynamic stability, a direct band gap of 4.5 eV, and a Young modulus between 234.4 and 273.2 GPa. 44In the current year, four novel inorganic biphenylene lattices utilizing CdS and ZnS were introduced. 45These structures are stable at 300 K and belong to the class of ultrawideband gap semiconductors, featuring band gap energies ranging from 3.59 to 4.30 eV. Paticularly noteworthy is the BPN-ZnS lattice, which displays a remarkable auxetic behavior.
In 2024, Laranjeira et al. 33 proposed the IGP-InN, showing that this monolayer is stable at 700 K and exhibits a competitive cohesive energy compared to wurtzite InN.With a band gap energy of 2.49 eV, suitable for optoelectronic applications in the UV−visible range, its band gap can be adjusted by 1.19 eV under tensile strain.This discovery has led to the design of biphenylene-like InN (BPN-InN) as a viable alternative, leveraging the similarity between graphenylene and biphenylene lattices.Furthermore, free-standing 2D hexagonal indium nitride (h-InN) is a well-consolidated material with a variety of applications, including optoelectronics, 46,47 gas sensing, 48 and thermoelectrics. 49ased on the above observations, this report introduces, for the first time, the BPN-InN structure using density functional theory (DFT) simulations, showcasing the role of simulations in anticipating properties of unknown materials.A detailed study of its electronic, structural, mechanical, and vibrational properties was performed to characterize and demonstrate the potential of BPN-InN.The electronic analysis reveals that BPN-InN exhibits semiconductor characteristics, featuring a band gap energy of 2.02 eV, falling within the visible range.Furthermore, it has a structural buckling and a negative Poisson ratio, both uncommon traits within the BPN family.To comprehend the mechanical behavior of BPN-InN, biaxial and uniaxial strains were applied in the armchair and zigzag directions, ranging from −8 to 8%, resulting in a band gap energy modulation by 1.36 eV under tensile deformations.

COMPUTATIONAL SETUP
The computational simulations were carried out using the CRYSTAL17 package 50 based on the DFT in combination with the HSE06 hybrid functional, 51,52 and at the same time, the 9763111-631 53 and 6-21G* 54 basis sets were selected to represent the In and N atomic centers, respectively.
The precision of the infinite Coulomb and Hartree−Fock (HF) exchange series is controlled by five α i parameters with i = 1, 2, 3, 4, and 5, where α 1 is the overlap, α 2 is the penetration for Coulomb integrals, α 3 is the overlap for HF exchange integrals and α 4 and α 5 are the pseudo-overlaps (HF exchange series).The two-electron contributions are neglected when the overlap between atomic functions is lower than 10 −α i .For the calculations, the five α i parameters were set to 20, 20, 20, 20, and 40, respectively.The convergence criterion for SCF is 10 −6 au/cell, while for geometry optimization, it is 10 −7 au/cell, and for elastic constant calculations, it is 10 −8 au/cell.The optimization convergence was checked on the root-meansquare (RMS) and the absolute value of the largest component of both the gradients and the estimated displacements.The convergence criteria employed in the optimization for RMS and the largest component for gradient were 0.00030 and 0.00045 au and for displacement 0.00120 and 0.00180 au, respectively.The reciprocal space was sampled using Pack− Monkhorst and Gilat nets with sublattice and a shrinking factor of 6, resulting in 16 k-points in the irreducible Brillouin zone.The vibrational modes at the Γ point were evaluated using the numerical second derivatives of the total energies estimated with the coupled perturbed HF/Kohn−Sham (CPKS) algorithm. 55he quantum theory of atoms in molecules and crystals (QTAIMC) 56,57 was employed to characterize the nature of chemical bonds.This approach uses the electronic density (ρ(r)) at the bond critical points (BCPs) to obtain topological parameters, such as the laplacian (∇ 2 ρ(r)), the potential energy density (V(r)), the kinetic energy density (G(r)), and the total electronic energy density (H(r) = V(r) + G(r)).These parameters can provide valuable information regarding the type of bond interaction: shared electron pairs (shared shell or covalent bonds) or closed electron shells of one of the atoms (closed shell or ionic bonds).
The elastic constants (C ij ) were calculated as the second derivative of the energy (E) concerning the strain component (ϵ i and ϵ j ) according to the following expression To analyze the anisotropic mechanical behavior of the structures presented herein, the orientation-dependent Young modulus Y(θ), Poisson ratio ν (θ), and Shear modulus G(θ) were calculated employing the following expressions 58

E S c S s S c s S cs
where s = sin θ, c = cos θ, and θ ∈ [0, 2π] are the angle with respect to the +x axis.S ij = C ij −1 are the elastic compliance constants.
The cohesive energy (E coh ) was utilized to confirm the structural stability of the BPN-InN by the equation , where E monolayer is the BPN-InN total energy, E In and E N are the energies of In and N isolated atoms, respectively, and n In and n N are the numbers of In and N atoms in the structure.For comparison purposes, E coh was also calculated for h-InN (hexagonal 2D InN) and IGP-InN.Molecular dynamics (MD) simulations were carried out using the extended tight-binding approximation (xTB) 59 as implemented in the DFTB+ package, 60 with the GFN1-xTB parametrization. 61The thermal stability of BPN-InN was performed using the Berendsen thermostat 62 at 300 K by 4 ps with a time step of 1 fs.The rupture temperature of BPN-InN was obtained in a second way, increasing the temperature from 300 K up to the break employing the same time step.

RESULTS AND DISCUSSION
The BPN-InN structure has a rectangular buckled lattice (Δ) of 1.04 Å, belongs to the space group Pm (no. 6) with lattice parameters a = 6.34 Å, b = 10.81Å, and α = β = 90°and possesses three different bond lengths, namely, l 1 = 2.04 Å, l 2 = 2.07 Å, and l 3 = 2.12 Å, as represented in Figure 1.The octagonal pore has a maximum diameter of 5.73 Å.The unit cell contains eight nonequivalent atoms, and its internal coordinates are detailed in the crystallographic information file (CIF) in the Supporting Information (SI).The cohesive energy of BPN-InN is −4.29 eV/atom, a value closer than the obtained for IGP-InN (−4.22 eV/atom).The magnitude of the cohesive energy demonstrates the viability of BPN-InN from an energetic point of view.
The phonon dispersion along the high-symmetry pathways was calculated to analyze the dynamical stability, as displayed in Figure 1b.Two negative phonon modes with a minimum frequency of −0.18 THz (−5 cm −1 ) are noticed.Such small imaginary frequencies could be an artifact of limited supercell size considered (4 × 4 × 1 supercell), k-points, or reflect the lattice instability over large wave undulations.Wang et al. 58 define −2 THz in the phonon vibration frequency as the limit for free-standing monolayers.Nevertheless, in phonon dispersion bands, it can be verified that the bands are generally flat, which denotes lower phonon propagation.At Γ and Y points, a quadradic dispersion is noticed, and two phononic band gaps occur (10−15 and 16−20 THz).These band gaps allow phonon propagation control, tailoring the thermal and acoustic properties for applications such as thermal insulators, phononic crystals, and energy harvesting and conversion.
Thermal stability was carried out from 300 K by 4 ps, as shown in Figure 2, which offers a top and side view of the last interaction in MD simulations with a 4 × 4 × 1 supercell with 192 atoms.It can be observed that the total energy remains in a flat pattern, with energy fluctuations in the order of meV/ atom.The MD snapshot revealed that the BPN-InN monolayer maintains its ring structure and does not undergo bond reconstruction during the simulations.The rupture temperature of BPN-InN was determined by gradually increasing the temperature, using a fixed time step of 1 fs in the heating simulation.The simulations indicate that BPN-InN is stable up to approximately 920 K, as no bond breakages or reconstructions were observed.A video animation of the MD simulations is available in the SI.
The band structure and density of states (DOS) of BPN-InN, shown in Figure 3, were analyzed to study the band dispersion and electronic state distribution.A direct band gap transition at Γ point is observed with an energy (E gap ) of 2.02 eV.The BPN-InN exhibits the lowest band gap when compared to the other BPN-based nitrides, namely, BPN-AlN, BPN-GaN, 43 and BPN-BN, 44 with band gap energies of 2.30, 3.18, and 4.50 eV, respectively.In the class of the III−IV BPN analogs, all reported via DFT simulations, only the InN and BN-based possess direct band gap transition.Their low E gap value and direct transition make the BPN-InN the most promising for optoelectronic applications.
Furthermore, it is interesting to note that in BPN-InN, the valence band maximum (VBM) has a flatness aspect, characterizing heavier photogenerated holes.At the same time, the conduction band minimum (CBM) denotes a parabolic-like format that indicates lighter electrons.These features indicate a lower recombination ratio, which is an essential requisite for the photocatalytic process.The suitability of BPN-InN in photocatalysis is corroborated by its visible range E gap .
The DOS reveals significant contributions from In states in both the conduction and valence bands within the energy interval considered.At the VBM, there is a higher contribution from N atoms, which significantly decreases in the CBM.The  edge regions of the DOS support the band structure analysis.The VBM shows a high number of states, which drop abruptly to zero in the band gap, indicating highly correlated states and lower carrier mobility.Conversely, the CBM has a low DOS, indicating high band dispersion and, consequently, higher carrier mobility.
The results of the topological analysis are presented in Table 1.The ∇ 2 ρ(r) values, which are close to zero and positive, indicate transient bonds that bridge the gap between covalent and ionic interactions.A |V(r)|/G(r) ratio lower than 1 for all bonds suggests ionic interactions, while negative H/ρ(r) values indicate covalent bonds.This analysis shows that the In−N bonds cannot be strictly classified as covalent or ionic, as they lie between shared and closed-shell interactions.Notably, the l 1 , l 2 , and l 3 values are very close for all parameters, indicating nearly identical bonds from a topological perspective.
The electronic density Laplacian map was plotted as shown in Figure 4.This representation allows us to understand how the electronic density varies around the BCPs.As corroborated by Table 1, small positive values for ∇ 2 ρ(r) can be observed between the bonds, indicating interactions with a weak sharedshell character.Due to the planar representation, the l 2 and l 3 bonds appear to have lower ∇ 2 ρ(r) values compared to l 1 .The absence of ∇ 2 ρ(r) isolines reveals charge depletion centers in the N atoms, while the opposite behavior indicates charge accumulation centers in the In atoms.When interacting with other species, the N can act as an electron donor and In as a receptor.
Figure 5 represents the highest occupied crystalline orbital (HOCO) and lowest unoccupied crystalline orbital (LUCO), crystal orbital Hamilton population (COHP) for all l 1 , l 2 , and l 3 bonds, and the density of states per orbital (pDOS) for In and N.
First, the absence of orbital overlap for both HOCO and LUCO, as corroborated by the COHP, indicates that all bonds are antibonding in the VBM and CBM.
Analyzing the pDOS for In, the major contributions in the VBM mainly come from the d xz and d yz orbitals.On the other hand, the CBM is predominantly composed of d z 2 and s orbitals.Regarding the In d orbitals, the pDOS indicates a breakdown in degeneracy, resulting in higher energy for d orbitals with a z component.This finding can be related to the high buckling reported for BPN-InN, which emerges as a result of sp 2 −sp 3 mixed bonds.
The pDOS for N atoms corroborates the analysis performed for In, demonstrating total degeneracy splitting, with the p z orbitals predominating in the VBM.Due to symmetry considerations, it is expected that bonds with higher σ characters.This is in agreement with the topological parameters, which show a lower ρ(r) and ∇ 2 ρ(r) at the bond critical points (BCPs), indicating a lower orbital overlap from the perspective of a purely covalent bond.
The mechanical properties of BPN-InN were analyzed, and the results are displayed in Table 2 The anisotropy is more remarkable for the Poisson ratio, as seen in the polar diagrams in Figure 6.The Young and Shear moduli show a circular-like distribution and, therefore, reduced anisotropy.In comparison with its analogs, i.e., IGP-InN and h-InN, the anisotropy in ν on BPN-InN is a distinctive characteristic considering the almost perfectly isotropic behavior of the two first relative to Y, ν, and G. Concerning the Y max and G max , the BPN-InN is more rigid than IGP-InN and significantly softer than h-InN.
Poisson ratio (ν) measures the ratio of lateral strain to axial strain in a material.A Poisson ratio of precisely zero would Where ρ(r) is the charge density, ∇ 2 ρ(r) is the Laplacian of the charge density, |V(r)|/G(r) is the ratio between the virial (V(r)) and the kinetic density energy (G(r)), and (H(r)/ρ(r)) is the bond degree all calculated on the bond critical points (BCPs).imply that the material does not change in lateral dimensions when stretched or compressed, which is a rare and idealized case.Interestingly, both BPN-InN and IGP-InN exhibit ν closer to zero, which is not reported for other inorganic BPN and IGP-based structures.For example, BPN-AlN and BPN-GaN possess ν max = 0.91 and 0.82 and ν min = 0.20 and 0.11, respectively. 43Abdullahi and Ersan reported ν max = 0.54, 0.87, 0.90, and 1.00 and ν min = 0.35, 0.43, 0.54, and 0.54 for BPN-BeO, BPN-MgO, BPN-ZnO, and BPN-CdO, respectively. 39In the same work, the authors also provide ν = 0.33, 0.45, 0.55,    32 showed ν = 0.38 for IGP-GaN, and Martins et al., ν = 0.33 for IGP-GeC. 35On the other hand, h-InN exhibits higher ν (0.58), as verified by Peng et al. 64 that obtained ν = 0.59.These findings demonstrate that, in particular, InN biphenylene-based structures result in a closer to zero Poisson ratio.
To investigate the behavior of BPN-InN under mechanical deformations, biaxial strains (ε) as well as uniaxial strains in the armchair (ε x ), and zigzag (ε y ) directions ranging from −8 to 8% were applied to the monolayer.Figure 7 shows the E gap and the buckling (Δ) as functions of the strain.For tensile strains, i.e., positive ε, ε x , and ε y values, a notable decrease of E gap is verified for BPN-InN.The biaxial strain reinforces this decrease, while when uniaxial strains act, it is softened.
The E gap achieved their minimum for ε = 8% (E gap = 0.66 eV) and the maximum for ε x = −8% (E gap = 2.36 eV).A variation of 1.36 eV is noticed for tensile ε in the E gap , greater than the 1.19 eV reported by Laranjeira et al. 33 for IGP-InN at WC1LYP/DFT level.In this sense, Li et al. 65 showed a decrease of 1.20 eV for the MoS 2 monolayer for biaxial tensile strains.Mortazavi and Rabczuk 66 achieved E gap decreases of 1.02, 0.95, 0.99, and 1.32 eV for SiP, SiAs, GeP, and GeAs monolayers, respectively.These findings show that strain engineering is a suitable strategy to fine-tune the electronic properties of BPN-InN.
The buckling (Δ) increases under compressive strains and decreases under tensile strains.The Δ increase is higher for biaxial strains than for unaxial.Buckling reaches its maximum value at ε = −8% (2.43 Å) and decreases until it disappears at ε = ε x = ε y = 4%.Beyond 4%, Δ it remains zero.Li et al. 67 reported a similar phenomenon for penta-B 2 C, observing buckling values of 2.24 and 0.00 Å for ε = −20 and 20%, respectively.This indicates that planarity can be achieved in BPN-InN with significantly lower deformations compared to penta-B 2 C. For IGP-InN employing the same strain range, a buckling variation of approximately 2.6 Å was observed, consistent with the behavior of BPN-InN.Importantly, when BPN-InN becomes planar, a new symmetry emerges, indicating a phase transition induced by mechanical strain.Under biaxial strain, this new structure belongs to space group Pmma (no.51) with lattice parameters a = 11.028Å and b = 6.467Å.It contains four nonequivalent atoms, as detailed in the CIF file in the SI.Similarly, Laranjeira et al. 33 observed that tensile strains in IGP-InN (space group P3, no.147) decrease the buckling to zero, resulting in a planar structure with increased symmetry (space group P6/m, no.175).
From the perspective of bond lengths, as shown in Figure 8, when subjected to biaxial strain, the BPN-InN bonds (l 1 , l 2 , and l 3 ) exhibit approximately equal behavior, with all l 1 , l 2 , and l 3 describing linear curves with similar inclination concerning.This result is consistent with the topological analysis, which showed that the l 1 , l 2 , and l 3 bonds are practically identical.
When evaluating the armchair deformations ε x , the l 1 and l 2 bonds do not show significant modifications, while the l 3 bond distributes the tension along the lattice.In contrast, for ε y , an inverse behavior to ε x is observed, with l 1 and l 2 being most influenced by the strain, while l 3 remains constant.

CONCLUSIONS
In this work, DFT/HSE06 simulations were performed to characterize the novel biphenylene-like InN in terms of its structural, electronic, mechanical, and vibrational properties.Molecular dynamics demonstrated that this monolayer is stable up to 920 K without reconstructions and bond breakages.Phonon dispersion bands reveal small imaginary modes that are also discussed in the literature and that do not denote the instability of the material.The electronic analysis showed a direct band gap transition of 2.02 eV, promising for visiblerange optoelectronic applications.The contribution of the In states is greater than N in both the conduction and valence bands.From the mechanical properties, it was noticed that BPN-InN fulfills the Born−Huang criteria.The Young modulus, Poisson ratio, and shear modulus ranged values of 22.716 and 22.063 N/m, 0.018 and -0.008, and 11.448 and 10.860 N/m, respectively.Biaxial and uniaxial strains in armchair and zigzag directions from −8 to 8% were applied, and a phase transition induced by strain was observed when the structure became utterly planar.An expressive band gap modulation of 1.36 eV over tensile deformations was reported.Our findings are expected to motivate both theorists and experimentalists to explore and synthesize biphenylene-based InN structures, which exhibit promising semiconductor properties as demonstrated herein.

Figure 1 .
Figure 1.(a) Representation of BPN-InN with the lattice vectors, structural buckling (Δ), and nonequivalent bonds (l 1 , l 2 , and l 3 ) and (b) phonon bands dispersion along the high-symmetry pathway in the Brillouin zone.

Figure 2 .
Figure 2. (a) Total energy profile along thermal stability (at 300 K) molecular dynamics simulations and (b) last iteration snapshot.

Figure 3 .
Figure 3. (a) Band structure and (b) density of states of BPN-InN.
. The calculated BPN-InN elastic constants are C 11 = 22.07 N/m, C 12 = 0.41 N/m, C 22 = 22.18 N/m, and C 66 = 11.45N/m.The Born−Huang criteria 63 plays a crucial role in determining the mechanical stability of 2D materials.For rectangular lattices, the key conditions, namely C 11 > 0, C 66 > 0, and C 11 C 22 > (C 12 ) 2 , were fulfilled, confirming the mechanical stability of BPN-InN.BPN-InN exhibits maximum and minimum Young modulus (Y max and Y min , respectively) of 22.716 and 22.063 N/m, Poisson ratio (ν max and ν min , respectively) of 0.018 and −0.008, and shear modulus (G max and G min , respectively) of 11.448 and 10.860 N/m.

Figure 9 .
Figure 9. (a) Raman and (b) Infrared (IR) spectra obtained through the CPKS approach, along with corresponding vibrational modes for the most intense peaks in (c) IR and (d) Raman.

Table 1 .
Topological Parameters Based on the QTAIMC Analysis for l 1 , l 2 , and l 3 Bonds in the BPN-InN a