Novel Superhard Boron Nitrides, B2N3 and B3N3: Crystal Chemistry and First-Principles Studies

Tetragonal and hexagonal hybrid sp3/sp2 carbon allotropes C5 were proposed based on crystal chemistry and subsequently used as template structures to identify new binary phases of the B–N system, specifically tetragonal and hexagonal boron nitrides, B2N3 and B3N3. The ground structures and energy-dependent quantities of the new phases were computed within the framework of quantum density functional theory (DFT). All four new boron nitrides were found to be cohesive and mechanically (elastic constants) stable. Vickers hardness (HV), evaluated by various models, qualified all new phases as superhard (HV > 40 GPa). Dynamically, all new boron nitrides were found to be stable from positive phonon frequencies. The electronic band structures revealed mainly conductive behavior due to the presence of π electrons of sp2-like hybrid atoms.


Introduction
Carbon allotropes that exhibit mechanical properties similar to those of diamond, in particular, extreme hardness, continue to be a subject of significant interest within the scientific community.Both the predominant cubic diamond and the less common hexagonal form ("lonsdaleite") exhibit ultrahigh hardness due to the three-dimensional arrangement of C4 tetrahedra with the pure sp 3 hybridization of the carbon atoms.In terms of topology, cubic and hexagonal diamonds are the aristotypes, designated as dia and lon, respectively [1].This nomenclature is also applicable to other families of carbon allotropes.The introduction of additional carbon atoms results in alterations to the C(sp 3 ) lattice of diamond, giving rise to novel C(sp 3 )/C(sp 2 ) hybrid allotropes that retain the original physical properties, including the electronic ones, which can lead to induced metallicity [2].In the case of the nearest neighbors of carbon, boron and nitrogen, equiatomic cubic boron nitride (cBN) has been synthesized, which is half as hard as diamond but exhibits much higher thermal and chemical stability [3].
Research efforts to identify new superhard phases of compounds of light elements require the use of structure prediction programs such as USPEX [4] and CALYPSO [5].However, novel structures can also be identified through the application of crystal engineering rationale, as presented here.In all cases, such predictions must be validated by the quantitative study of energies and the derived physical properties using first-principles calculations.Over the years, the well-established quantum mechanics framework of density functional theory (DFT) [6,7] has been proven to be the most efficient.
The present paper further develops the field of boron nitrides by proposing novel cohesive and stable hybrid B-N phases.First, tetragonal and hexagonal sp 3 /sp 2 hybrid C 5 allotropes were designed by crystal chemistry engineering and subsequently used as templates for the selective substitution of carbon with boron and nitrogen, leading to the sesquinitride B 2 N 3 , which was then transformed into the equiatomic B 3 N 3 through the insertion of an additional boron atom into B 2 N 3 .
Considering that boron nitride is equiatomic, our challenging prediction of boron sesquinitride is supported to some extent by the previously reported nitrogen-excess tetragonal B 2 N 3 [8], but no structural details were provided for this compound.The boron nitride B 116 N 124 fullerene should also be mentioned as a nitrogen-rich B-N phase [9].Among the boron-rich compounds of the B-N system, it is worth mentioning rhombohedral boron subnitride B 13 N 2 [10], which is a superhard phase [11].
In the present work, we show that all predicted novel B-N phases are cohesive, mechanically and dynamically stable, and characterized by high Vickers hardness and metallic-like behavior.

Computational Methodology
The determination of the ground state structures corresponding to the energy minima and the prediction of their mechanical and dynamical properties were carried out within the widely accepted framework of DFT.DFT was initially proposed in two publications: in 1964, Hohenberg and Kohn developed the theoretical framework [6], and in 1965, Kohn and Sham established the Kohn-Sham equations for a practical solution of the wave equation [7].
Based on DFT, calculations were performed within the Vienna Ab initio Simulation Package (VASP) code [12,13] and the Projector Augmented Wave (PAW) method [13,14] for atomic potentials.DFT exchange correlation (XC) effects were considered using the generalized gradient approximation (GGA) [15].Preliminary calculations with the native DFT-XC local density approximation (LDA) [16] resulted in underestimated lattice constants at ambient pressure and were therefore abandoned.Relaxation of the atoms to the ground-state structures was performed with the conjugate gradient algorithm according to Press et al. [17].The Blöchl tetrahedron method [18] with corrections according to the Methfessel and Paxton scheme [19] was used for geometry optimization and energy calculations, respectively.Brillouin-zone (BZ) integrals were approximated by a special k-point sampling according to Monkhorst and Pack [20].Structural parameters were optimized until atomic forces were below 0.02 eV/Å and all stress components were <0.003 eV/Å 3 .The calculations were converged at an energy cutoff of 400 eV for the plane-wave basis set in terms of the k-point integration in the reciprocal space from k x (6) × k y (6) × k z (6) up to k x (12) × k y (12) × k z (12) to obtain a final convergence and relaxation to zero strain for the original stoichiometries presented in this work, after a systematic upgrade of the structure input file throughout successive cycles of calculations.In the post-processing of the ground state electronic structures, the charge density projections were operated on the lattice sites.
The investigation of the mechanical properties was based on calculations of the elastic properties determined by performing finite distortions of the lattice and deriving the elastic constants from the strain-stress relationship.The treatment of the results was conducted using the ELATE online tool, which is devoted to the analysis of elastic tensors [21].The program provides the bulk (B), shear (G), and Young's (E) moduli along different averaging methods; the Voigt method [22] was used here.Two empirical models, Mazhnik-Oganov [23] and Chen-Niu [24], were used to estimate the Vickers hardness (H V ) from the elastic constants.
Vickers hardness was also evaluated in the framework of the thermodynamic model [25,26], which is based on the thermodynamic properties and crystal structure, and using the Lyakhov-Oganov approach [27], which considers the topology of the crystal structure, the strength of covalent bonds, the degree of ionicity, and directionality.Fracture toughness (K Ic ) was estimated using the Mazhnik-Oganov model [23].
The dynamic stabilities were confirmed by the positive phonon magnitudes.The corresponding phonon band structures were obtained from the high resolution of the tetragonal and hexagonal Brillouin zones according to Togo et al. [28].The electronic band structures were obtained using the all-electron DFT-based ASW method [29] and the GGA XC functional [15].The Visualization for Electronic and Structural Analysis (VESTA) program [30] was used to visualize the crystal structures and charge densities.

Tetragonal and Hexagonal Pentacarbon Allotropes
tet-C 5 .Recently, the body-centered tetragonal carbon allotrope C 4 (tet-C 4 ) in the space group I-4m2 (No. 119) [31] (Figure 1a) has been proposed to serve as a template for the design of the related phases.The four atoms of the unit cell are detailed in Table 1, considering the structure in simple tetragonal arrangement.The subsequent transformation into C 5 consists of keeping the body center carbon (C1 at 1 /2, 1 /2, 1 /2), which becomes C(1c) in the Wyckoff position (following the transformation arrows), as well as the two C2s, whereby C2a and C2b become C3(2g), and the z coordinate becomes ±z ′ instead of ± 1 /4.The additional carbon C2 is provided by parameterizing the z position of C1b, resulting in a two-fold C2(2e) at 0, 0, ±z.After full geometry relaxation, the resulting structure retains the tetragonal symmetry now resolved in space group P-4m2 (No. 115).The structure is shown in Figure 1b, with colored spheres corresponding to the three carbon sites given by the coordinates in Table 1, i.e., with well-determined z = ±0.854and z ′ = ±0.314.Using the TopCryst crystallography package [32], tet-C 5 was identified with 3,4ˆ2T1-CA topology.A similar topology was found for another tetragonal carbon allotrope, C 5 , predicted by Wei et al. [33] using the CALYPSO code.Note that the initial tet-C 4 has a dia topology.
structures were obtained using the all-electron DFT-based ASW method [29] and the GGA XC functional [15].The Visualization for Electronic and Structural Analysis (VES-TA) program [30] was used to visualize the crystal structures and charge densities.

Tetragonal and Hexagonal Pentacarbon Allotropes
tet-C5.Recently, the body-centered tetragonal carbon allotrope C4 (tet-C4) in the space group I-4m2 (No. 119) [31] (Figure 1a) has been proposed to serve as a template for the design of the related phases.The four atoms of the unit cell are detailed in Table 1, considering the structure in simple tetragonal arrangement.The subsequent transformation into C5 consists of keeping the body center carbon (C1 at ½, ½, ½), which becomes C(1c) in the Wyckoff position (following the transformation arrows), as well as the two C2s, whereby C2a and C2b become C3(2g), and the z coordinate becomes ±z′ instead of ±¼.The additional carbon C2 is provided by parameterizing the z position of C1b, resulting in a two-fold C2(2e) at 0, 0, ±z.After full geometry relaxation, the resulting structure retains the tetragonal symmetry now resolved in space group P-4m2 (No. 115).The structure is shown in Figure 1b, with colored spheres corresponding to the three carbon sites given by the coordinates in Table 1, i.e., with well-determined z = ±0.854and zʹ = ±0.314.Using the TopCryst crystallography package [32], tet-C5 was identified with 3,4^2T1-CA topology.A similar topology was found for another tetragonal carbon allotrope, C5, predicted by Wei et al. [33] using the CALYPSO code.Note that the initial tet-C4 has a dia topology.1 and text for details).The additional yellow spheres result from the decrease in symmetry.h-C 5 .Lonsdaleite (space group P6 3 /mmc, No. 194) is the rare hexagonal form of diamond, h-C 4 , which is characterized by a single four-fold atomic position for carbon at 1 / 3 , 2 /3, and z with a small (0.06275) z-value along the hexagonal vertical axis.Changing z to 0 gives a 2D graphite-like carbon structure (Figure 2a) with hcb topology.Then, the puckering of the graphite-like layers is a key factor in the transition from a 2D to 3D structure.Here, we modeled the 2D → 3D transformation by inserting a carbon atom between the carbon layers, as shown by the white sphere in Figure 2b.Subsequent geometry relaxation yielded a new 3D structure with C 5 stoichiometry (Figure 2c).The crystal data with three different carbon positions and a symmetry reduction to P-6m2 (No. 187) are presented in Table 1.The topology is now lon, i.e., lonsdaleite-like.The bottom row shows the total energy after full geometry relaxation, which is found to be larger for tet-C 5 than for h-C 5 .The result can be translated into the average cohesive energy per atom, E coh /atom, which is obtained after subtracting the atomic energy of a carbon atom in a large box, i.e., −6.6 eV.Then, the cohesive energy is −2.05 eV for tet-C 5 versus −1.90 eV for h-C 5 .Both values remain smaller than E coh /atom = −2.49eV for diamond.
h-C5.Lonsdaleite (space group P63/mmc, No. 194) is the rare hexagonal form of diamond, h-C4, which is characterized by a single four-fold atomic position for carbon at ⅓, ⅔, and z with a small (0.06275) z-value along the hexagonal vertical axis.Changing z to 0 gives a 2D graphite-like carbon structure (Figure 2a) with hcb topology.Then, the puckering of the graphite-like layers is a key factor in the transition from a 2D to 3D structure.Here, we modeled the 2D → 3D transformation by inserting a carbon atom between the carbon layers, as shown by the white sphere in Figure 2b.Subsequent geometry relaxation yielded a new 3D structure with C5 stoichiometry (Figure 2c).The crystal data with three different carbon positions and a symmetry reduction to P-6m2 (No. 187) are presented in Table 1.The topology is now lon, i.e., lonsdaleite-like.The bottom row shows the total energy after full geometry relaxation, which is found to be larger for tet-C5 than for h-C5.The result can be translated into the average cohesive energy per atom, Ecoh/atom, which is obtained after subtracting the atomic energy of a carbon atom in a large box, i.e., −6.6 eV.Then, the cohesive energy is −2.05 eV for tet-C5 versus −1.90 eV for h-C5.Both values remain smaller than Ecoh/atom = −2.49eV for diamond.

New Tetragonal and Hexagonal Boron Nitrides
B2N3.The C5 structure was subsequently used as a template to design boron sesquinitride, B2N3, in both the tetragonal and hexagonal forms.Table 2 shows the corresponding atomic substitutions of carbon with boron and nitrogen at the three different positions in Table 1.The geometry-converged atomic positions were found to be close to

New Tetragonal and Hexagonal Boron Nitrides
B 2 N 3 .The C 5 structure was subsequently used as a template to design boron sesquinitride, B 2 N 3 , in both the tetragonal and hexagonal forms.Table 2 shows the corresponding atomic substitutions of carbon with boron and nitrogen at the three different positions in Table 1.The geometry-converged atomic positions were found to be close to the matrix C 5 allotropes.It can be observed that the cell volumes of tet-B 2 N 3 and h-B 2 N 3 are almost the same as well as the total energies with, however, a slightly lower value for the former.The crystal structures are shown in Figures 3 and 4 in ball-and-stick and tetrahedral representations.B 3 N 3 .Subsequently, equiatomic boron nitride, B 3 N 3 , was designed in both crystal systems by inserting an extra boron at 0, 0, 0 for tet-B 3 N 3 and at 1 / 3 , 2 /3, 0 for h-B 3 N 3 , as shown in Table 2 (the second and fourth data columns).The resulting structures after full geometry relaxation are shown in Figures 3 and 4. They exhibit preserved, pristine boron sesquinitride structures expanded with additional boron, thus achieving equiatomic stoichiometry.The total energies favor tet-B 3 N 3 , which also has a larger volume, indicating a lower density.
Table 2 shows the respective topologies of the new boron nitrides: 3,4ˆ2T1-CA for the tetragonal phases (like pristine C 5 ), lon for h-B 2 N 3 , and tfi for h-B 3 N 3 .B3N3.Subsequently, equiatomic boron nitride, B3N3, was designed in both crystal systems by inserting an extra boron at 0, 0, 0 for tet-B3N3 and at ⅓, ⅔, 0 for h-B3N3, as shown in Table 2 (the second and fourth data columns).The resulting structures after full geometry relaxation are shown in Figures 3 and 4. They exhibit preserved, pristine boron sesquinitride structures expanded with additional boron, thus achieving equiatomic stoichiometry.The total energies favor tet-B3N3, which also has a larger volume, indicating a lower density.
Table 2 shows the respective topologies of the new boron nitrides: 3,4^2T1-CA for the tetragonal phases (like pristine C5), lon for h-B2N3, and tfi for h-B3N3.

Projections of the Charge Densities
To illustrate the electron distribution within the new boron nitrides, the analysis was extended to a qualitative representation of the charge densities.Figures 3 and 4 (right panels) represent the charge density projections with yellow volumes around the atoms.In all cases, the charges are concentrated around nitrogen atoms (gray spheres).In fact, boron nitride is a polar covalent compound with a significant charge transfer to N, which is characterized by an electronegativity of χN = 3.04 and χB = 2.04 according to the Pauling scale.In B2N3, the charge gray volumes are squeezed toward the next cell along the

Projections of the Charge Densities
To illustrate the electron distribution within the new boron nitrides, the analysis was extended to a qualitative representation of the charge densities.Figures 3 and 4 (right panels) represent the charge density projections with yellow volumes around the atoms.In all cases, the charges are concentrated around nitrogen atoms (gray spheres).In fact, boron nitride is a polar covalent compound with a significant charge transfer to N, which is characterized by an electronegativity of χN = 3.04 and χB = 2.04 according to the Pauling scale.In B 2 N 3 , the charge gray volumes are squeezed toward the next cell along the vertical c direction, corresponding to the N-N connections between successive cells.By introducing new B-N bonds, equiatomic tet-B 3 N 3 and h-B 3 N 3 are created, and the charge volumes are around the N atoms in both symmetries.Regarding the iso-surface values, the following trends were observed: B 2 N 3 : 0.325 and B 3 N 3 : 0.293, i.e., a higher value for a higher nitrogen content.

Mechanical Properties
The analysis of the mechanical properties was carried out by calculating the elastic tensor through finite distortions of the lattice.The calculated sets of elastic constants C ij (i and j correspond to directions) of the four new boron nitrides are given in Table 3.
All C ij values are positive, indicating mechanically stable phases.For comparison, the elastic constants of template tet-C 5 and h-C 5 are also given.As expected, the carbon allotropes have the largest C ij values compared to those of the compounds of the B-N system.Elastic tensor analysis was performed to obtain the bulk (B v ), shear (G v ), and Young's (E v ) moduli and Poisson's ratio (ν) by Voight's averaging [22] using ELATE software [21].The calculated elastic moduli, whose values follow the trends observed for C ij , are shown in Table 4, along with the Vickers hardness values calculated using four contemporary models of hardness [23- 25,27] and fracture toughness evaluated using the Mazhnik-Oganov model [23].Since the thermodynamic model is the most reliable in the case of superhard boron compounds [3,38,39] and shows perfect agreement with the available experimental data for cubic boron nitride [40,41], it is obvious that the hardness values calculated within the empirical Mazhnik-Oganov [23] and Chen-Niu [24] models are not reliable.As for the Oganov-Lyakhov model [27], it gives slightly underestimated values, as has already been observed for the superhard compounds of the B-C-N system [38].It is evident (see Table 4) that the hardness of B 2 N 3 (52 GPa for both tetragonal and hexagonal polymorphs) is only slightly lower than that of cubic (55 GPa) and wurtzite (54 GPa) BN polymorphs.In contrast, the hardness of B 3 N 3 is significantly lower, especially for the tetragonal polymorph (42 GPa).However, all four new boron nitrides have a Vickers hardness exceeding 40 GPa, making them members of the superhard phase family.

Energy-Volume Equations of State
To determine energy trends when considering the different crystal structures of a solid, it is necessary to establish the corresponding equations of state (EOS).It is important to note that one cannot rely on the quantities obtained from lattice optimizations alone, especially when comparing the energies and volumes of the different phases.The underlying physics means that the calculated total energy corresponds to the cohesion within the crystal, and the solutions to the Kohn-Sham DFT equation give the energy in terms of infinitely separated electrons and nuclei.The zero of the energy depends on the choice of the atomic potentials (projector augmented waves (PAWs) as, here, ultra-soft pseudo-potentials (US-PP), etc.); then, it becomes arbitrary by its shift, not by scaling.However, the energy derivatives and the EOS remain unchanged.Therefore, it is necessary to obtain the EOS and extract the fit parameters to evaluate the equilibrium values.This was conducted via a series of calculations of the total energy as a function of volume for the tetragonal and hexagonal phases of new B-N compounds.The resulting E(V) curves, shown in Figure 5, were fitted to the third-order Birch equations of state [42]: where E 0 , V 0 , B 0 , and B ′ are the equilibrium energy; volume; bulk modulus; and its first pressure derivative, respectively.The calculated values are summarized in Table 5.In the case of boron sesquinitride B 2 N 3 (Figure 5a), the E(V) curve of tet-B 2 N 3 remains at a slightly lower energy and a higher volume than h-B 2 N 3 , but both curves remain close.Quantitatively, this is translated by close equilibrium values (Table 5) due to the close densities of the two phases.Larger differences are observed for the equiatomic phases (Figure 5b), where the tetragonal B 3 N 3 systematically has lower energy and a larger volume than the hexagonal one.An intersection of the E(V) curves is observed at a volume of 9.74 Å 3 per BN formula unit.The corresponding pressure of 200(30) GPa was estimated by the Murnaghan equation [43] using the V 0 , B 0 , and B ′ values from Table 5.
terms of infinitely separated electrons and nuclei.The zero of the energy depends on the choice of the atomic potentials (projector augmented waves (PAWs) as, here, ultra-soft pseudo-potentials (US-PP), etc.); then, it becomes arbitrary by its shift, not by scaling.However, the energy derivatives and the EOS remain unchanged.Therefore, it is necessary to obtain the EOS and extract the fit parameters to evaluate the equilibrium values.This was conducted via a series of calculations of the total energy as a function of volume for the tetragonal and hexagonal phases of new B-N compounds.The resulting E(V) curves, shown in Figure 5, were fitted to the third-order Birch equations of state [42]: where E0, V0, B0, and B′ are the equilibrium energy; volume; bulk modulus; and its first pressure derivative, respectively.The calculated values are summarized in Table 5.In the case of boron sesquinitride B2N3 (Figure 5a), the E(V) curve of tet-B2N3 remains at a slightly lower energy and a higher volume than h-B2N3, but both curves remain close.Quantitatively, this is translated by close equilibrium values (Table 5) due to the close densities of the two phases.Larger differences are observed for the equiatomic phases (Figure 5b), where the tetragonal B3N3 systematically has lower energy and a larger volume than the hexagonal one.An intersection of the E(V) curves is observed at a volume of 9.74 Å 3 per BN formula unit.The corresponding pressure of 200(30) GPa was estimated by the Murnaghan equation [43] using the V0, B0, and B ʹ values from Table 5.As can be seen in Figure 5b, both B3N3 polymorphs are metastable with respect to cubic BN over the whole range of experimentally accessible pressures.Nevertheless, the closeness of their cohesive energies allows for the possibility of the formation of both B3N3 polymorphs at high pressures and high temperatures as a result of alternative metastable behavior.Table 5. Calculated properties of new boron nitrides: bulk modulus (B 0 ) and its first pressure derivative (B 0 ′ ); total energy (E 0 ) and equilibrium volume (V 0 ) per formula unit (FU).As can be seen in Figure 5b, both B 3 N 3 polymorphs are metastable with respect to cubic BN over the whole range of experimentally accessible pressures.Nevertheless, the closeness of their cohesive energies allows for the possibility of the formation of both B 3 N 3 polymorphs at high pressures and high temperatures as a result of alternative metastable behavior.

Phonons Band Structures
To verify the dynamic stability of the new B-N phases, an analysis of their phonon properties was performed.The phonon band structures obtained from the high resolution of the tetragonal and hexagonal Brillouin zones in accordance with the method proposed by Togo et al. [28] are presented in Figure 6.The bands (red lines) develop along the main directions of the tetragonal (or hexagonal) Brillouin zone (horizontal x-axis) and are separated by vertical lines for enhanced visualization, while the vertical direction (y-axis) represents the frequencies ω, given in terahertz (THz).

Phonons Band Structures
To verify the dynamic stability of the new B-N phases, an analysis of their phonon properties was performed.The phonon band structures obtained from the high resolution of the tetragonal and hexagonal Brillouin zones in accordance with the method proposed by Togo et al. [28] are presented in Figure 6.The bands (red lines) develop along the main directions of the tetragonal (or hexagonal) Brillouin zone (horizontal x-axis) and are separated by vertical lines for enhanced visualization, while the vertical direction (y-axis) represents the frequencies ω, given in terahertz (THz).The band structures include 3N bands: three acoustic modes starting from zero energy (ω = 0) at the Γ point (the center of the Brillouin zone) and reaching up to a few terahertz and 3N-3 optical modes at higher energies.The low-frequency acoustic modes are associated with the rigid translation modes (two transverse and one longitudinal) of the crystal lattice.The calculated phonon frequencies are all positive, indicating that the four new B-N phases are dynamically stable.
In the case of the tetragonal phases, in addition to the dispersed bands, the flat bands at ~39 THz for B3N2 (Figure 6a) and the higher frequency band at ~58 THz for B3N3 (Figure 6b) are observed.The 39 THz band for tetragonal B2N3 can be attributed to the B-N distance within the tetrahedron, while the 58 THz band for tetragonal B3N3 can be assigned to the B-N distance in the B-N-B fragment along the c-axis.For hexagonal phases, The band structures include 3N bands: three acoustic modes starting from zero energy (ω = 0) at the Γ point (the center of the Brillouin zone) and reaching up to a few terahertz and 3N-3 optical modes at higher energies.The low-frequency acoustic modes are associated with the rigid translation modes (two transverse and one longitudinal) of the crystal lattice.The calculated phonon frequencies are all positive, indicating that the four new B-N phases are dynamically stable.
In the case of the tetragonal phases, in addition to the dispersed bands, the flat bands at ~39 THz for B 3 N 2 (Figure 6a) and the higher frequency band at ~58 THz for B 3 N 3 (Figure 6b) are observed.The 39 THz band for tetragonal B 2 N 3 can be attributed to the B-N distance within the tetrahedron, while the 58 THz band for tetragonal B 3 N 3 can be assigned to the B-N distance in the B-N-B fragment along the c-axis.For hexagonal phases, such flat bands are absent, while observed bands with frequencies of about 40 THz are due to the larger B-N distances in these phases (see Table 2).

Temperature Dependence of the Heat Capacity
The thermodynamic properties of the new B-N phases were calculated from the phonon frequencies using the statistical thermodynamic approach [44] on a high-precision sampling mesh in the Brillouin zone.The temperature dependencies of the heat capacity at constant volume (C V ) for all new boron nitrides are presented in Figure 7 and for B 3 N 3 in comparison with the available experimental C p data for cubic BN [45].The heat capacity of tetragonal B 2 N 3 is slightly higher than that of the hexagonal phase (Figure 7a), while the opposite is observed for B 3 N 3 (Figure 7b).The heat capacities of both B 3 N 3 polymorphs are slightly higher than that of cubic BN, which is consistent with their more open structures compared to the dense cubic structure.The observed excellent agreement between the calculated and experimental data for cBN supports the validity of the method used to estimate the thermodynamic properties of new boron nitrides.
Molecules 2024, 29, x FOR PEER REVIEW 11 of 14 such flat bands are absent, while observed bands with frequencies of about 40 THz are due to the larger B-N distances in these phases (see Table 2).

Temperature Dependence of the Heat Capacity
The thermodynamic properties of the new B-N phases were calculated from the phonon frequencies using the statistical thermodynamic approach [44] on a high-precision sampling mesh in the Brillouin zone.The temperature dependencies of the heat capacity at constant volume (CV) for all new boron nitrides are presented in Figure 7 and for B3N3 in comparison with the available experimental Cp data for cubic BN [45].The heat capacity of tetragonal B2N3 is slightly higher than that of the hexagonal phase (Figure 7a), while the opposite is observed for B3N3 (Figure 7b).The heat capacities of both B3N3 polymorphs are slightly higher than that of cubic BN, which is consistent with their more open structures compared to the dense cubic structure.The observed excellent agreement between the calculated and experimental data for cBN supports the validity of the method used to estimate the thermodynamic properties of new boron nitrides.

Electronic Band Structures
The electronic band structures of the new boron nitrides were calculated using the all-electron DFT-based augmented spherical wave (ASW) method [29] using the crystal structure data from Table 1.The results are shown in Figure 8.The bands (blue lines) develop along the main directions of the respective tetragonal and hexagonal Brillouin zones.The zero energy along the vertical axis is considered with respect to the Fermi level EF.For both B2N3 polymorphs (Figure 8a,b), the bands cross the Fermi level with a tendency toward a weakly metallic behavior.Obviously, a semiconducting behavior is observed for tet-B3N3, where a small gap develops (Figure 8c); hence, the energy reference is now at EV, i.e., at the top of the valence band.Finally, h-B3N3 is clearly metallic with several bands crossing EF (Figure 8d).The different behaviors observed between the two equiatomic phases can be attributed to the fact that tet-B3N3 has considerably lower energy than h-B3N3, as shown in Table 2.As a result, its electronic structure is closer to that of cubic BN.Thus, the new B-N phases exhibit different electronic properties.

Electronic Band Structures
The electronic band structures of the new boron nitrides were calculated using the all-electron DFT-based augmented spherical wave (ASW) method [29] using the crystal structure data from Table 1.The results are shown in Figure 8.The bands (blue lines) develop along the main directions of the respective tetragonal and hexagonal Brillouin zones.The zero energy along the vertical axis is considered with respect to the Fermi level E F .For both B 2 N 3 polymorphs (Figure 8a,b), the bands cross the Fermi level with a tendency toward a weakly metallic behavior.Obviously, a semiconducting behavior is observed for tet-B 3 N 3 , where a small gap develops (Figure 8c); hence, the energy reference is now at E V , i.e., at the top of the valence band.Finally, h-B 3 N 3 is clearly metallic with several bands crossing E F (Figure 8d).The different behaviors observed between the two equiatomic phases can be attributed to the fact that tet-B 3 N 3 has considerably lower energy than h-B 3 N 3 , as shown in Table 2.As a result, its electronic structure is closer to that of cubic BN.Thus, the new B-N phases exhibit different electronic properties.

Conclusions
The present work involved a challenging prediction of new tetragonal and hexagonal boron nitrides, B2N3 and B3N3, from crystal chemistry and first principles using original pentacarbon templates.The crystal chemistry investigations were supported by computations of the ground structures and energy-dependent quantities within the well-established framework of quantum density functional theory (DFT).All new phases were found to be cohesive.The mechanical stability of the new phases, which follows from the calculated values of the elastic constants, is coupled with their extreme hardness, varying from 45 GPa for tet-B3N3 to 52 GPa for both B2N3 polymorphs.Dynamically, all new phases were found to be stable from positive phonon frequencies, and observed high-frequency modes were assigned to the short B-N distances in their crystal structures.Conductive electronic behavior is observed, which varies from small bandgap semiconducting h-B2N3, to weakly metallic tetragonal B2N3 and B3N3, and finally to metallic h-B3N3.The results obtained are expected to inspire experimental attempts to synthesize new B-N phases at high pressures and high temperatures.

Figure 1 .
Figure 1.Schematic transformation of tet-C4 (a) to tet-C5 (b) (see Table1and text for details).The additional yellow spheres result from the decrease in symmetry.

Figure 1 .
Figure 1.Schematic transformation of tet-C 4 (a) to tet-C 5 (b) (see Table1and text for details).The additional yellow spheres result from the decrease in symmetry.

Figure 3 .
Figure 3. Ball-and-stick (left), polyhedral (middle), and charge projection (right) representations of the crystal structures of new tetragonal B2N3 (a) and B3N3 (b).Green and gray spheres represent boron and nitrogen atoms, respectively.

Figure 3 .
Figure 3. Ball-and-stick (left), polyhedral (middle), and charge projection (right) representations of the crystal structures of new tetragonal B 2 N 3 (a) and B 3 N 3 (b).Green and gray spheres represent boron and nitrogen atoms, respectively.

Figure 4 .
Figure 4. Ball-and-stick (left), polyhedral (middle), and charge projection (right) representations of the crystal structures of new hexagonal B2N3 (a) and B3N3 (b).Green and gray spheres represent boron and nitrogen atoms, respectively.

Figure 4 .
Figure 4. Ball-and-stick (left), polyhedral (middle), and charge projection (right) representations of the crystal structures of new hexagonal B 2 N 3 (a) and B 3 N 3 (b).Green and gray spheres represent boron and nitrogen atoms, respectively.

Figure 5 .
Figure 5. Calculated total energy as a function of volume for new boron nitrides: B 2 N 3 (a) and B 3 N 3 (b).In the case of B 3 N 3 , all values are given per BN formula unit for comparison with cBN.

Molecules 2024 , 14 Figure 5 .
Figure 5. Calculated total energy as a function of volume for new boron nitrides: B2N3 (a) and B3N3 (b).In the case of B3N3, all values are given per BN formula unit for comparison with cBN.

Figure 7 .
Figure 7. Heat capacity at constant volume (CV) of new boron nitrides: B2N3 (a) and B3N3 (b).In the case of B3N3, CV values are given per BN formula unit for comparison with cBN.Experimental heat capacity data for cBN [45] are shown as gray symbols.

Figure 7 .
Figure 7. Heat capacity at constant volume (C V ) of new boron nitrides: B 2 N 3 (a) and B 3 N 3 (b).In the case of B 3 N 3 , C V values are given per BN formula unit for comparison with cBN.Experimental heat capacity data for cBN [45] are shown as gray symbols.

Table 1
and text for details).The additional yellow spheres result from the decrease in symmetry.

Table 1 .
Crystal structure transformations of tetragonal carbon (from C 4 to C 5 ) and from 2D-C 4 to 3D-C 5 (hexagonal setups).See text for details.

Table 1
and text).

Table 1
and text).

Table 2 .
Crystal structure parameters of new B-N phases.

Table 3 .
Elastic constants (C ij ) of the new boron nitrides in comparison with those of the original carbon allotropes (all values are in GPa).

Table 4 .
Mechanical properties of new boron nitrides: Vickers hardness (H V ), bulk modulus (B), shear modulus (G), Young's modulus (E), Poisson's ratio (ν), and fracture toughness (K Ic ).The subscript V for elastic moduli indicates the use of the Voigt averaging scheme.# denotes the space group number.The corresponding values for wurtzite and cubic boron nitrides are given for comparison.