Crystal Chemistry and First Principles Studies of Novel Superhard Tetragonal C 7 , C 5 N 2 , and C 3 N 4

: Tetragonal C 7 , C 5 N 2 , and C 3 N 4, characterized by mixed tetrahedral and trigonal atomic hybridizations, have been devised based on crystal chemistry rationale and structural optimization calculations within density functional theory (DFT). Substitution of C(sp 2 ) and C(sp 3 ) in C 7 for nitrogen yields α -C 5 N 2 and β -C 5 N 2 , respectively, both of which are superhard, cohesive, and stable mechanically (elastic properties) and dynamically (phonon band structures). tet -C 3 N 4 with both nitrogen sites within the C 7 structure was found to be cohesive and classiﬁed as ductile with a Vickers hardness of 65 GPa. Due to the delocalization of π electrons of the sp 2 -like


Introduction
Since the 1950s, interest has focused on carbon allotropes, especially those that resemble diamond in hardness and other electronic properties, such as diamond at the nanoscale for applications in electronics [1].Expanding to carbon's neighbors, B and N, a superhard cubic boron nitride was synthesized in 1957 [2], which often replaces diamond in tools because it is more stable at high temperatures, although its hardness is less than that of diamond.As for binary compounds of the C-N system, ultrahard phases of C 3 N 4 stoichiometry have been predicted already in the 1990s [3,4], with allegedly higher bulk moduli and hardness than those of diamond due to the short and strong C-N bonds.However, the existence of such phases has been strongly questioned, and even in the case of experimentally synthesized C-N phases, their stoichiometry and structures are still controversial due to the heterogeneity of the samples obtained.In particular, it should be noted that strong N-N repulsions induced by short N-N nonbonded distances do not favor the synthesis of the nitrogen-rich C-N phases, especially when CVD and PVD methods are used.The hypothesis of gaseous carbon dinitrogen release during vapor phase deposition led to the proposal of model systems such as CN 2 [5,6], based on studies within the well-established quantum mechanics framework of density functional theory (DFT) [7,8].
Thus, the search for new C-N phases is mainly limited to the carbon-rich systems (i.e., N/C < 1), where many hypothetical binary compounds have been predicted in recent years, such as rhombohedral and cubic C 17 N 4 [9], cubic C 4 N [9], monoclinic [10] and orthorhombic [11] C 3 N, tetragonal and orthorhombic C 11 N 4 [12], cubic α-C 3 N 2 and β-C 3 N 2 [13], cubic C 4 N 3 [14], orthorhombic CN [15], etc.Among the hypothetical nitrogenrich C-N phases, only CN 2 is worth mentioning, whose stability has been predicted over a wide range of pressures [16].However, none of these many predicted structures have been synthesized, with the sole exception of the orthorhombic CN [17] and diamond-like C 2 N [18].The progress and problems in predicting novel ultrahard binary and ternary phases of the B-C-N system from first principles in conjunction with available experimental data have been recently discussed in our review paper [19].
In the present paper, we further develop the problem of carbon nitrides, which remains of great interest, as follows from recent works [20,21].In the framework of accurate DFT calculations, we have designed a new cohesive and stable (mechanically and dynamically) tetragonal heptacarbon (C 7 ) allotrope, characterized by both tetrahedral (sp 3 -like) and trigonal (sp 2 -like) carbon sites, which was used as a template for selective substitutions of carbon for nitrogen, leading to two new carbon-rich nitrides of C 5 N 2 stoichiometry, as well as to a new tetragonal C 3 N 4 (Appendix A).

Computational Framework
The identification of the ground state structures corresponding to the energy minima and prediction of their mechanical and dynamical properties were carried out via DFTbased calculations using the Vienna Ab initio Simulation Package (VASP) code [22,23] and the projector augmented wave (PAW) method [23,24] for the atomic potentials.Exchange correlation (XC) effects were considered using the generalized gradient functional approximation (GGA) [25].Relaxation of the atoms to the ground state structures was performed with the conjugate gradient algorithm according to Press et al. [26].The Blöchl tetrahedron method [27] with corrections according to the Methfessel and Paxton scheme [28] was used for geometry optimization and energy calculations, respectively.Brillouin-zone (BZ) integrals were approximated using a special k-point sampling according to Monkhorst and Pack [29].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 (4) up to k x (12) × k y (12) × k z (8) for the final convergence and relaxation to zero strains.In the post-processing of the ground-state electronic structures, the charge density projections were operated on the carbon atomic sites.The charge density was analyzed using Bader's atoms in molecules (AIM) theory [30] to assess the trends of charge transfer.The elastic constants C ij were calculated to evaluate the mechanical stability and hardness as described in detail below in the Section 3.3.The dynamic stabilities were evaluated from the phonon band structures according to Togo et al. [31].The electronic band structures were obtained using the all-electron DFT-based ASW method [32] and the GGA XC functional [25].The VESTA (Visualization for Electronic and Structural Analysis) program [33] was used to visualize the crystal structures and charge densities.

Crystal Chemistry and Structure Characterization
The calculated structures at energy minima from fully geometry-optimized setups are detailed in Table 1 and the structures are shown in Figure 1.The body-centered tetragonal 'neoglitter' C 6 [34] has been used to devise a novel carbon allotrope, tetragonal C 7 .The 'neoglitter' structure, shown in ball-and-stick and polyhedral representations, consists of C4 tetrahedra at the eight corners and body-center, connected by C-C pairs along the vertical direction.C 6 is then expressed as (C tet ) 2 (C trg ) 4 as made up of tetrahedral and trigonal sites, readily available to be used as a template for CN 2 carbon dinitride.However, such hypothesis was rejected as preliminary calculations resulted in a weakly cohesive compound (−0.62 eV/atom) compared to the magnitudes of the other dinitrides in Table 1.Furthermore, the CN 2 dinitride was characterized by unstable dynamic behavior due to negative phonons.
The novel C 7 consists of an elongated tetragonal box with tetrahedral carbon atoms occupying the eight corners as in 'neoglitter' and in-between trigonal carbons leading to a loss of the body-centered character with subsequent change of space group I-4m2 (No. 119).The fully optimized geometric structure is shown in Figure 1b, and the polyhedral representation on the right highlights the sp 3 -sp 2 hybrid structure of C 7 with tetrahedral and trigonal motifs.The first data column of Table 1 provides the numerical description of the structure inscribed as the following phases in the tetragonal P-4m2 (No. 115) space Crystals 2023, 13, 1111 3 of 11 group.Four different carbon sites build the structure.The C4 tetrahedra have an angle of 106.5 • , slightly different from the perfect 109.47 • of C(sp 3 ), and the corresponding distance is d(C-C) = 1.52 Å, also smaller than in diamond (1.54 Å).The other shorter distances correspond to the vertical C-C bonds.Subtracting the atomic C contribution of −6.6 eV from the total energy gives an atomic average cohesive energy of −2.09 eV/atom.This value is lower than the cohesive energy of diamond (−2.49eV/atom).Substitution of C for N was performed along with two working hypotheses to obtain the stoichiometry of C 5 N 2 :

•
The first hypothesis was to create a dinitride with N-N pairs as shown in Figure 1c, where the white spheres represent the N@C (2e) 0,0,z (Table 1, second data column).
After geometric optimization, the final structure shows little deviation from pristine C 7 , as can be seen from the internal parameters, while the distances show a short d(N-N) = 1.36 Å, larger than the length of the N≡N bond (~1.09Å) in the N 2 molecule.The volume is smaller, resulting in a slightly higher density.The atom-averaged cohesive energy is significantly reduced compared to C 7 , but the carbonitride labeled α-C 5 N 2 remains cohesive with E coh /atom = −1.50eV/at.

•
The second hypothesis was to replace N@C (2g) 1 2 , 0, z (Table 1, third data column), thus creating CN4 tetrahedra at the eight corners as schematically shown in Figure 1d.Therefore, d(C-N) = 1.49Å is shorter than in C 7 and α-C 5 N 2 .The remarkable result of such β-C 5 N 2 is the higher (by 0.23 eV) cohesive atomic energy of this configuration, which is clearly more favorable than that of α-C 5 N 2 .Finally, the density is higher at 3.252 g/cm 3 .Therefore, the tetrahedral configuration prevails.
Finally, we found that the simultaneous N@C (2g) and N@C (2e) substitutions resulted in the known C 3 N 4 nitrogen-rich stoichiometry.The structure thus constructed was then calculated for the sake of completeness.The results given in the last column of Table 1 are also presented in Figure 1e, with N shown as red spheres.The structure shows a small d(N-N) and the volume is the smallest, resulting in the highest density.The cohesive energy is smaller than those of both α-C 5 N 2 and β-C 5 N 2 .C 3 N 4 , proposed by Teter and Hemley [4], calculated for comparison, gives a cohesive energy of −1.29 eV/atom, i.e., it is more cohesive than tet-C 3 N 4 (Table 1).and Hemley [4], calculated for comparison, gives a cohesive energy of −1.29 eV/atom, i.e., it is more cohesive than tet-C3N4 (Table 1).Simulated X-ray diffraction patterns of three new phases are shown in Figure 2. It is easy to see that they all have the same topology, which is significantly different from diamond topology.

Trends of Charge Transfer
For Pauling negativities χN = 3.04 and χC = 2.55, charge transfers from C to N of different magnitudes are expected.From the high BZ integration, the charge density output (CHGCAR) was analyzed based on the AIM theory according to Bader [30].The analyses of the carbon nitrides studied showed a small amount of charge transfer.
In α-C5N2, QC = 0.676 and QN = −1.69; in β-C5N2, QC = 0.588 and QN = −1.47;and in tet-C3N4, QC = 2.22 and QN = −1.67.Regarding C7, despite the different atomic positions and chemical behaviors from sp 3 /sp 2 , the charge transfers were found to be negligible, letting the allotrope be considered as a covalent chemical system.Both C5N2 polymorphs can be considered polar covalent, but the large amount of charge transfer in tet-C3N4 allows it to be characterized as ionocovalent, reinforced by the large number of electrons transferred from C to N. Mechanical and dynamic properties of the novel phases were further analyzed.

Elastic Constants
The mechanical properties were obtained through determining the elastic constants from the strain-stress relationship.For the tetragonal crystal systems, the terms of mechanical stability are defined by elastic constants Cij as follows: The calculated Cij of C7 and new carbon nitrides are given in Table 2.All values are positive, and their combinations given above obey the rules of mechanical stability.The bulk (BV) and shear (GV) moduli were calculated via Voigt averaging the single-crystal elastic constants using ELATE software [35].As a general trend, the bulk moduli are close to 390 GPa for β-C5N2 and tet-C3N4, proportional to their densities, and close to 360 GPa for C7 and α-C5N2, which have lower densities.The shear moduli of the new phases are close to 230 GPa, except for a slightly lower value of 205 GPa for tet-C3N4.There is a

Trends of Charge Transfer
For Pauling negativities χ N = 3.04 and χ C = 2.55, charge transfers from C to N of different magnitudes are expected.From the high BZ integration, the charge density output (CHGCAR) was analyzed based on the AIM theory according to Bader [30].The analyses of the carbon nitrides studied showed a small amount of charge transfer.
In  3 /sp 2 , the charge transfers were found to be negligible, letting the allotrope be considered as a covalent chemical system.Both C 5 N 2 polymorphs can be considered polar covalent, but the large amount of charge transfer in tet-C 3 N 4 allows it to be characterized as ionocovalent, reinforced by the large number of electrons transferred from C to N. Mechanical and dynamic properties of the novel phases were further analyzed.

Elastic Constants
The mechanical properties were obtained through determining the elastic constants from the strain-stress relationship.For the tetragonal crystal systems, the terms of mechanical stability are defined by elastic constants C ij as follows: The calculated C ij of C 7 and new carbon nitrides are given in Table 2.All values are positive, and their combinations given above obey the rules of mechanical stability.The bulk (B V ) and shear (G V ) moduli were calculated via Voigt averaging the single-crystal elastic constants using ELATE software [35].As a general trend, the bulk moduli are close to 390 GPa for β-C 5 N 2 and tet-C 3 N 4 , proportional to their densities, and close to 360 GPa for C 7 and α-C 5 N 2 , which have lower densities.The shear moduli of the new phases are close to 230 GPa, except for a slightly lower value of 205 GPa for tet-C 3 N 4 .There is a general trend of the Pugh modulus ratio (G V /B V ) < 1 with the lowest value of 0.52 for tet-C 3 N 4 .This is indicative of ductile systems with low brittleness, probably due to the presence of trigonal atomic fragments in the structures of all phases.

Hardness
To estimate the Vickers hardness (H V ) of the proposed new phases, two contemporary theoretical models were used.The thermodynamic (T) model [36], which is based on thermodynamic properties and crystal structure, generally shows perfect agreement with experimental data and is therefore recommended for the hardness evaluation of superhard phases [19].The Lyakhov-Oganov (LO) model [37] considers the topology of the crystal structure, the strength of covalent bonding, the degree of ionicity, and directionality.As shown earlier [19], empirical models using elastic properties are not reliable in the case of superhard compounds of light elements, so we have not considered these models.Fracture toughness (K Ic ) was evaluated using the Mazhnik-Oganov model [38].results for the currently proposed phases compared to the properties of diamond and hypothetical cubic C 3 N 4 [3] are summarized in Tables 3 and 4.
The hardness of tetragonal C 7 is the highest of all the new phases, but it is ~15% less than that of diamond.This is to be expected since the density of this phase is significantly lower (by a factor of 1.2).The hardness of the three new C-N phases is even lower and is ~65 GPa for all of them, closer to the hardness of cubic C 3 N 4 [3].
The elastic moduli of all four new phases are close, with the Pugh modulus ratio (G/B) being less than 1 and varying from 0.52 for tet-C 3 N 4 to 0.64 for C 7 .
These trends in mechanical properties are consistent with the fact that nitrogen plays a major role in the structure softening of the three C-N compounds, while the presence of trigonal carbon in all new phases further reinforces this tendency.

Dynamical Properties with the Phonons
To verify the dynamic stability of the new phases, the phonon band structures were cal- The methodology consists of calculating the phonon modes by finite displacements of the atoms from the equilibrium position found through the preliminary optimization of the crystal geometry-another heavier method exists as density functional perturbation theory (DFPT).This leads the forces on all atoms constituting the crystal structure to raise off their equilibrium positions.Analysis of the forces associated with a systematic set of displacements yields a series of phonon frequencies.The phonon dispersion curves along the main lines of the tetragonal Brillouin zone are then obtained using the Python code "Phonopy" (see [31] for further details).
Figure 3 shows the phonon band structures for the new carbon allotrope and carbon nitride phases that develop along the main lines of the primitive tetragonal Brillouin zone.
The hardness of tetragonal C7 is the highest of all the new phases, but it is ~15 than that of diamond.This is to be expected since the density of this phase is signifi lower (by a factor of 1.2).The hardness of the three new C-N phases is even lower ~65 GPa for all of them, closer to the hardness of cubic C3N4 [3].
The elastic moduli of all four new phases are close, with the Pugh modulu (G/B) being less than 1 and varying from 0.52 for tet-C3N4 to 0.64 for C7.
These trends in mechanical properties are consistent with the fact that nitrogen a major role in the structure softening of the three C-N compounds, while the prese trigonal carbon in all new phases further reinforces this tendency.

Dynamical Properties with the Phonons
To verify the dynamic stability of the new phases, the phonon band structure calculated.The methodology consists of calculating the phonon modes by fini placements of the atoms from the equilibrium position found through the prelim optimization of the crystal geometry-another heavier method exists as density tional perturbation theory (DFPT).This leads the forces on all atoms constituti crystal structure to raise off their equilibrium positions.Analysis of the forces asso with a systematic set of displacements yields a series of phonon frequencies.The p dispersion curves along the main lines of the tetragonal Brillouin zone are then ob using the Python code "Phonopy" (see [31] for further details).
Figure 3 shows the phonon band structures for the new carbon allotrope and c nitride phases that develop along the main lines of the primitive tetragonal Br zone.Frequencies (ω) expressed in units of THz are along the vertical direction.In all four panels the frequencies are positive, providing the signature of dynamically stable systems.Between 0 and ~10 THz, the first three bands correspond to lattice rigid translations: two inplane and one out-of-plane; they are labeled as acoustic modes.At higher energies and up to the highest ω, the remaining bands are the optical modes.Knowing that diamond has the highest band observed using Raman spectroscopy at ω~40 THz [41], all frequency maxima of the four phases in Figure 3a-d are above this magnitude, up to ω~49 THz for α-C 5 N 2 (Figure 3b), assigned to N-N stretching.Then, while diamond has pure sp 3 hybridization, the currently studied C 7 and new carbon nitrides have mixed sp 3 /sp 2 hybridization, the latter leading to high frequencies for very short distances (Table 1).

Electronic Band Structures
Figure 4 shows the electronic band structure plots obtained using the all-electron DFT-based augmented spherical method (ASW) [32] and the GGA exchange correlation DFT functional.In all four panels along the vertical direction of the energy, there is a continuous development of bands.Consequently, there is no separation (gap) between the valence band (VB) filled with electrons and the empty conduction band (CB), indicating metallic-like behavior.Then, zero energy is with respect to the Fermi level (E F ).Such electronic behavior can be attributed to the delocalization induced by the π electrons of the sp 2 -like hybridized atoms present in all systems.Insulating behavior requires electron localization as found in diamond, which is known to be an electronic insulator with a wide band gap of ~5 eV between VB and CB.In fact, diamond is characterized only by tetrahedral C(sp 3 ).
tems.Between 0 and ~10 THz, the first three bands correspond to lattice rigid translations: two in-plane and one out-of-plane; they are labeled as acoustic modes.At higher energies and up to the highest ω, the remaining bands are the optical modes.Knowing that diamond has the highest band observed using Raman spectroscopy at ω~40 THz [41], all frequency maxima of the four phases in Figure 3a-d are above this magnitude, up to ω~49 THz for α-C5N2 (Figure 3b), assigned to N-N stretching.Then, while diamond has pure sp 3 hybridization, the currently studied C7 and new carbon nitrides have mixed sp 3 /sp 2 hybridization, the latter leading to high frequencies for very short distances (Table 1).

Electronic Band Structures
Figure 4 shows the electronic band structure plots obtained using the all-electron DFT-based augmented spherical method (ASW) [32] and the GGA exchange correlation DFT functional.In all four panels along the vertical direction of the energy, there is a continuous development of bands.Consequently, there is no separation (gap) between the valence band (VB) filled with electrons and the empty conduction band (CB), indicating metallic-like behavior.Then, zero energy is with respect to the Fermi level (EF).Such electronic behavior can be attributed to the delocalization induced by the π electrons of the sp 2 -like hybridized atoms present in all systems.Insulating behavior requires electron localization as found in diamond, which is known to be an electronic insulator with a wide band gap of ~5 eV between VB and CB.In fact, diamond is characterized only by tetrahedral C(sp 3 ).

Density of States (DOS)
For a better assessment of the role of each site and chemical constituent in the electronic structure and, further, on the band structures, the density of states (DOS) was calculated and plotted.The energy reference is now along the x-axis.As in the band structures, zero energy is considered with respect to the Fermi level EF, visualized with a straight vertical red line.Along the y-axis, the DOSs are in units of 1/eV. Figure 5a, showing C7's DOS, exhibits a continuous skyline of the different constituents' valence

Density of States (DOS)
For a better assessment of the role of each site and chemical constituent in the electronic structure and, further, on the band structures, the density of states (DOS) was calculated and plotted.The energy reference is now along the x-axis.As in the band structures, zero energy is considered with respect to the Fermi level E F , visualized with a straight vertical red line.Along the y-axis, the DOSs are in units of 1/eV. Figure 5a

Density of States (DOS)
For a better assessment of the role of each site and chemical constituent in the electronic structure and, further, on the band structures, the density of states (DOS) was calculated and plotted.The energy reference is now along the x-axis.As in the band structures, zero energy is considered with respect to the Fermi level EF, visualized with a straight vertical red line.Along the y-axis, the DOSs are in units of 1/eV. Figure 5a Upon introduction of nitrogen (C5N2, Figure 5b,c), additional features appear with a broadening of VB by almost 5 eV due to the larger electronegativity of N compared to C. VB is divided into s-like VB (from −27 eV to −20 eV) dominated by N DOS, and the part with p-states (from −20 eV to EF), which also shows dominant N states.Less intense states are found at EF, mainly originating from C2, indicating s-like conductivity.Additionally, in Figure 5d, the C3N4 DOSs show increasingly larger nitrogen contribution throughout VB and above EF in CB.

Conclusions
The presence of mixed sp 3 /sp 2 carbon hybridization in a novel tetragonal C7 allotrope allowed us to devise three new carbon nitrides along hypotheses related to the selective occupations of either atomic site.Two C5N2 phases were found to be mechanically and dynamically stable, with the tetrahedral CN4 phase being more cohesive than the phase

Conclusions
The presence of mixed sp 3 /sp 2 carbon hybridization in a novel tetragonal C 7 allotrope allowed us to devise three new carbon nitrides along hypotheses related to the selective occupations of either atomic site.Two C 5 N 2 phases were found to be mechanically and dynamically stable, with the tetrahedral CN4 phase being more cohesive than the phase with linear N-N motifs.Tetragonal C 3 N 4 was similarly devised through allowing N to occupy both sp 3 and sp 2 carbon sites in C 7 , resulting in a ductile metallic phase.C 7 was found to be cohesive and classified as ductile with a Vickers hardness of 65 GPa.The ductility was found to be enhanced with a decrease of the Pugh modulus ratio G/B along the series from 0.65 to 0.52 with a resultant decrease in hardness.Due to the delocalization of π electrons of the sp 2 -like hybridized atoms, metallic behavior characterizes all four phases.If synthesized, either in bulk via solid-state synthesis at high pressures and high temperatures, or through sputtering and other deposition techniques, the new phases are likely to improve our understanding of the long-standing problem of nitrogen chemistry in carbon nitrides.

Figure 2 .
Figure 2. Simulated X-ray diffraction patterns of new tetragonal phases.

Figure 2 .
Figure 2. Simulated X-ray diffraction patterns of new tetragonal phases.

Figure 3 .
Figure 3. Phonon band structures of tetragonal C7 (a), α-C5N2 (b), tet-C5N2 (c), and tet-C3N4 ( Frequencies (ω) expressed in units of THz are along the vertical direction.In a panels the frequencies are positive, providing the signature of dynamically stab
, showing C 7 s DOS, exhibits a continuous skyline of the different constituents' valence states from the lowest VB energy (−22 eV-−15 eV), where s states predominate, up to −5 eV, where p states predominate.Such similar shapes of the site-resolved DOS are indicative of quantum mixing of the respective valence states describing the chemical bonding.From −5 eV up to E F , lower-intensity DOS of C1 is observed with almost zero intensity at E F , while the other C DOSs show large intensity at E F .This illustrates the different behavior of tetrahedral C1 versus trigonal C2.

Figure 5 .
Figure 5. Site-projected density of states (DOS) of tetragonal C 7 (a), α-C 5 N 2 (b), β-C 5 N 2 (c), and tet-C 3 N 4 (d).Upon introduction of nitrogen (C 5 N 2 , Figure 5b,c), additional features appear with a broadening of VB by almost 5 eV due to the larger electronegativity of N compared to C. VB is divided into s-like VB (from −27 eV to −20 eV) dominated by N DOS, and the part with p-states (from −20 eV to E F ), which also shows dominant N states.Less intense states are found at E F , mainly originating from C2, indicating s-like conductivity.Additionally, in Figure 5d, the C 3 N 4 DOSs show increasingly larger nitrogen contribution throughout VB and above E F in CB.

Table 2 .
Elastic constants and bulk (B V ) and shear (G V ) moduli (in GPa) of new tetragonal phases.G/B is the Pugh modulus ratio.