Novel ultrahard sp 2 /sp 3 hybrid carbon allotrope from crystal chemistry and first principles: body-centered tetragonal C 6 ('neoglitter')

: A novel ultrahard carbon allotrope, body-centered tetragonal C 6 (space group I-4 m 2) presenting mixed sp 2 /sp 3 carbon hybridizations is proposed by crystal chemistry approach and studied for the ground state structure and stability (both dynamic and mechanical) using density functional theory calculations. Given that C4 tetrahedra in-plane stacking with corner-sharing and connected out-of-plane with C–C trigonal carbon, a close relationship with so-called 'glitter', hypothetical dense carbon network devised ~30 years ago, is established, so we named the new allotrope 'neoglitter'. 'Neoglitter' is characterized by large bulk and shear moduli and very high hardness, and its metallic-like electronic structure is assigned mainly to the itinerant role of trigonal carbon π -electrons.


Introduction
Diamond, as natural gem and man-made for applications [1] is recognized as the hardest material.
The basic reason for extreme hardness is the three-dimensional arrangement of carbon in corner sharing C4 tetrahedra with d(C-C) = 1.55 Å, i.e. the sum of two r(C) = 0.77 Å atomic radii, making a perfectly covalent network.In recent decades many research efforts were focused on the identification of new ultrahard carbon allotropes mimicking structure through predicting software such as CALYPSO [2] and USPEX [3] complementarily with first principles calculations carried out mainly within the well-established quantum mechanics framework of the density functional theory (DFT) [4,5].In this context the SACADA database (SACADA 3D) [6] regroups all known carbon allotropes thus helping researchers in their endeavor.Recently we reported body-centered tetragonal C 4 (Fig. 1a) proposed in space group I-4m2 as seed to build larger carbon networks and to serve as a template for other original chemical compounds [7].Note that by considering the primitive cell (2 atoms), tet-C 4 is another expression of cubic diamond whose presentation as tet-C 4 is shown here as a starting building block to propose a novel carbon allotrope.In Table 1 showing the crystal structure parameters of tet-C 4 , the change of two-fold carbon (2b) by four-fold position leading to the separation of the C4 tetrahedra along vertical direction keeps the same body-centered tetragonal space group I-4m2 after full geometry relaxation onto ground state structure leading to tetragonal C 6 (Fig. 1b).The transformation scheme highlighting the C4 tetrahedra is proposed in Fig. 1c.Tetragonal C 6 is characterized by mixed C(sp 2 )/C(sp 3 ) hybridizations and reveals exceptional mechanical properties (very high hardness and elastic moduli) while being dynamically stable with electronic conductivity due to trigonal C(sp 2 ) atoms.
At this point, we should mention that tet-C 6 shows similarities with 'glitter' C 6 proposed theoretically as a dense tetragonal hexacarbon starting ad hoc from 1,4-cyclohexadienoid units [8].
The corresponding structure shown in Fig. 1d consists of four trigonal carbons: C(sp 2 ) and two tetrahedral carbons: C(sp 3 ).The structure was called 'glitter', i.e. shining due to the electronic conductivity.First principles investigations and full characterization of 'glitter' C 6 was recently undertaken, and the structure was used as template to propose novel equiatomic (super)hard SiCN [9].The established relationship between 'glitter' C 6 and the proposed tetragonal C 6 regarding the structural arrangement allowed us to call it a 'neoglitter'.

Computational framework
The search for the ground state energies, ground state structures and related mechanical and dynamic properties was carried out within DFT with Vienna Ab initio Simulation Package (VASP) code [10,11] using the projector augmented wave (PAW) method [11,12] for the atomic potentials with all valence states (especially in regard of such light element as carbon).The exchangecorrelation effects inherent to DFT were considered with the generalized gradient approximation (GGA) following Perdew, Burke, and Ernzerhof [13].Relaxing the atoms onto the ground state structures was carried out with the conjugate-gradient algorithm [14].The tetrahedron method was applied for both geometry relaxation and total energy calculations using Blöchl et al. corrections [15] and Methfessel-Paxton scheme [16].Brillouin-zone integrals were approximated using a special k-point sampling of Monkhorst and Pack [17].The optimization of the structural parameters was performed until the forces on the atoms were less than 0.02 eV/Å and all stress components below 0.003 eV/Å 3 .The calculations were converged at an energy cut-off of 400 eV for the planewave basis set regarding the k-point integration in the reciprocal space up to 121212 k x ,k y ,k z for the final convergence and relaxation to zero strains.In the post-treatment process of the ground state electronic structures, the charge density projections were operated onto the carbon atomic sites.The elastic constants C ij and the phonon band structures were calculated to assess the mechanical and dynamic stabilities.Calculations of phonon dispersion curves were also carried out to verify the dynamic stability of the novel carbon allotrope.The phonon modes were computed via finite displacements of the atoms off their equilibrium positions to obtain the forces from the summation over the different configurations.The phonon bands along the direction of the Brillouin zone were subsequently obtained using "phonopy" code based on Python language [18].

Results and Discussion
As a first assessment, the trend of cohesive energy per atom obtained after full geometry relaxation onto ground state is a decrease from E total/atom (tet-C 4 ) = -2.4eV to E total/atom (tet-C 6 ) = -1.7 eV.Such trend is explained by the loss of the homogeneous 3D corner sharing C4 tetrahedra observed in diamond-like tet-C 4 in contrast to tet-C 6 .Nevertheless, tet-C 6 remains cohesive, and as shown below, it is stable both mechanically (elastic properties) and dynamically (phonons).
Table 1 shows the crystal parameters of tet-C 6 ; the parameters of tet-C 4 [7] are also presented for the sake of comparison.Expectedly, a larger volume characterizes tet-C 6 due to the spacing between C4 tetrahedra as schematized in Fig. 1c.Common C1 positioned at corners and body center features both carbon structures but the introduction of C2-C2 pairs is represented by the change from twofold (2d) in tet-C 4 to four-fold (4f) in tet-C 6 .The interatomic distances are very close in both structures.
Comparing the two tetragonal C 6 allotropes, we should say that whereas 'glitter' was devised from modeling an organic molecule inscribed in a tetragonal cage (ad hoc), presently proposed 'neoglitter' tet-C 6 is structurally derived from diamond expressed as tet-C 4 .
Simulated X-ray diffraction patterns of three carbon allotropes ('glitter', 'neoglitter' and diamond) are shown in Fig. 2. The observed differences of diffraction lines intensities of 'glitter' and 'neoglitter' are likely related to the different topologies of these two structures identified by TopCryst analysis program [19] respectively as "tfi" and "tfa" in RCSR (Reticular Chemistry Structure Resource) nomenclature.

Charge density analyses.
As further illustration of the above mentioned energy trends, we analyzed difference between tet-C 4 and tet-C 6 considering the charge density resulting from the calculations.
The corresponding projections of the charge density onto the atoms and bonds are shown in Fig. 3 with yellow volumes using multicell for the sake of extended views with the structure sketches and with a polyhedral view in left-hand-side and right-hand side representations, respectively.The trigonal character of C2-C2 pairs is also shown with a substantial charge density.But the most representative of the charge density difference between the two structures is observed in the change from a purely covalent system in tet-C 4 where the charge density is localized on each C atom with tetrahedra shape, to tet-C 6 where there is a continuous density charge from C1 (tetrahedral) to C2 (trigonal) allowing us to suggest a transition from the insulator (diamond) to a conductor.Indeed, the analysis of the electronic band structure in the subsection below will show such a result.

Mechanical properties (i) Elastic constants
The investigation of mechanical characteristics was based on the calculations of elastic properties determined by performing finite distortions of the lattice and deriving the elastic constants (C ij ) from the strain-stress relationship.It is well-known that most of solids are polycrystalline and can be considered to a large extent as randomly oriented single-crystalline grains.Therefore, they may be described by bulk (B) and shear (G) moduli obtained by averaging the single-crystal elastic constants.The method used here is Voigt's [20] which is based on a uniform strain.The calculated sets of elastic constants are given in Table 2 for both 'glitter' and 'neoglitter'.
All C ij values are positive.Their combinations obeying the rules pertaining to the mechanical stability of the phase, and the equations providing bulk B V and shear G V moduli are as follows for the tetragonal system: and G V moduli are somewhat larger than those of 'glitter', but still smaller than moduli accepted for diamond i.e.B V =445 GPa and G V = 530 GPa (cf.[1] and references therein).

(ii) Hardness
Vickers hardness (H V ) was predicted using three modern theoretical models: (i) thermodynamic model based on crystal structure and thermodynamic properties [21], (ii) Lyakhov-Oganov model which considers the strength of covalent bonding, degree of ionicity and topology of the crystal structure [22] and empirical Chen-Niu model that uses the elastic properties [23].The fracture toughness (K Ic ) was evaluated within Mazhnik-Oganov model [24].Table 3 presents Vickers hardness and bulk moduli (B 0 ) calculated in the framework of thermodynamic model of hardness; other mechanical properties such as shear modulus (G), Young's modulus (E), the Poisson's ratio () and fracture toughness (K Ic ) are given in Table 4.As has been shown earlier, in the case of ultrahard compounds of light elements thermodynamic model shows surprising agreement with available experimental data [26].Moreover, its use is preferable in the case of hybrid dense carbon allotropes, for which Lyakhov-Oganov model gives underestimated hardness values whereas the empirical Chen-Niu model does not work at all.Since Vickers hardness of both 'glitter' and 'neoglitter' exceeds 80 GPa, they should be attributed to the family of ultrahard phases [27].The higher hardness of 'neoglitter' is apparently due to the higher density.

Dynamic stability from the phonons.
Besides the structural stability criteria observed for 'neoglitter' from the positive magnitudes of the elastic constants and their combinations, another stability criterion is obtained from the phonons.Fig. 4 shows the phonon bands.Along the horizontal direction, the bands run along the main lines of the tetragonal Brillouin zone (reciprocal k-space).The vertical direction shows the frequencies given in units of terahertz (THz).Since no negative frequency magnitudes are observed, tet-C 6 is considered as dynamically stable.There are 3N-3 optical modes at higher energy than three acoustic modes that can be counted between  and R.They start from zero energy (ω = 0) at the  point, center of the Brillouin zone, up to a few terahertz.Acoustic modes describe the lattice rigid translation modes with two transverse modes and one longitudinal mode.The remaining bands correspond to the optic modes culminating at ω = 40 THz, a magnitude close to the observed for diamond by Raman spectroscopy: ω ~ 40 THz [28].

Electronic band structures.
Fig. 5 shows the electronic band structure and the site projected density of states (DOS) of tet-C 6 obtained using the all-electrons DFT-based augmented spherical method (ASW) [29] using GGA exchange correlation functional [13].In Fig. 5a the energy level along the vertical line is with respect to the Fermi level (E F ) crossed by bands with subsequent metallic behavior.The DOS shown beside the band structure (Fig. 5b) provide further interpretation for the electronic structure with the zero energy along the x-axis at E F .The projected DOS along C1 (tetrahedral) and C2 (trigonal) sites show large similarities in the energy range {-20; -5 eV} ensuring for the chemical bonding between the two sites, but large differences appear above -5 eV and above E F within the (empty) conduction band.The prevailing C2-DOS at E F and the vanishingly small C1 magnitude of C1 DOS clearly show that the conductivity is selectively caused by trigonal carbon as discussed in the charge density subsection.Further support is also found from the weighted bands in Fig. 5c where the bold bands relative to C1 are found deeper in energy whereas C2 bands are found responsible of the states crossing E F , thus responsible of the metallic character and resulting from the π-electrons having more itinerant behavior than the less extended  electrons.

Conclusions
In the present work a novel ultrahard carbon allotrope, body-centered tetragonal C 6 , was proposed with the particularity of presenting mixed sp 2 -sp 3 carbon hybridizations featuring tetrahedral carbon in C4 tetrahedra and the stacking of in-plane corner-sharing C4 being connected by trigonal carbon.
The closeness of this configuration to that of 'glitter' C 6 proposed almost 30 years ago allows us to name the new allotrope showing very high hardness and metallic-like electronic structure a 'neoglitter'.b Lyakhov-Oganov model [22] c Chen-Niu model [23] d E and  values calculated using isotropic approximation e Mazhnik-Oganov model [24] f Ref.

Figure 2 Figure 3 .
Figure 2 Simulated X-ray diffraction patterns of three dense carbon allotropes.

Table 1 .
Crystal structure parameters of two tetragonal carbon allotropes

Table 3
[21]ers hardness (H V ) and bulk moduli (B 0 ) of carbon allotropes calculated in the framework of the thermodynamic model of hardness[21]