H18 Carbon: A New Metallic Phase with sp2-sp3 Hybridized Bonding Network

Design and synthesis of three-dimensional metallic carbons are currently one of the hot issues in contemporary condensed matter physics because of their fascinating properties. Here, based on first-principles calculations, we discover a novel stable metallic carbon allotrope (termed H18 carbon) in () symmetry with a mixed sp2-sp3 hybridized bonding network. The dynamical stability of H18 carbon is verified by phonon mode analysis and molecular dynamics simulations, and its mechanical stability is analyzed by elastic constants, bulk modulus, and shear modulus. By simulating the x-ray diffraction patterns, we propose that H18 carbon would be one of the unidentified carbon phases observed in recent detonation experiments. The analysis of the band structure and density of states reveal that this new carbon phase has a metallic feature mainly due to the C atoms with sp2 hybridization. This novel 3D metallic carbon phase is anticipated to be useful for practical applications such as electronic and mechanical devices.

Carbon has a variety of structural allotropes due to its ability of different hybridizations 1 . It is well known that there exist three carbon allotropes in natural materials, that is, graphite, diamond, and amorphous carbon, containing the sp 2 , sp 3 , and mixed sp 2 /sp 3 hybridized carbon atoms 2 , respectively. In the past three decades, intensive theoretical and experimental efforts have been focused on synthesizing new allotropes of carbon, among which, the zero-dimensional (0D) fullerenes 3 , one-dimensional (1D) carbon nanotubes 4 , and two-dimensional (2D) graphene 5 are the three most prototypical examples. So far, many new carbon allotropes such as 1D sp-carbyne, 2D sp-sp 2 -graphyne, and three-dimensional (3D) sp-sp 3 -yne-diamond have been experimentally fabricated or theoretically predicted 6,7 , most of them exhibit their intriguing mechanical and electronic properties. Recently, a new carbon allotrope with hardness even higher than diamond has also been observed by compressing graphite with pressure over 17 GPa 8 . Motivated by this experimental finding, several carbon crystalline phases such as monoclinic M-carbon 9 , bct-C 4 carbon 10 , W-carbon 11 , O-carbon 12 , and Z-carbon 13 were proposed to simulate this high-pressure carbon phase. These prototypical examples have given rise to significant impacts in material and information sciences, stimulating experimental and theoretical attentions on carbon allotropes [14][15][16][17][18][19][20][21][22][23] .
Among various types of carbon materials, metallic carbon allotropes exhibit more fascinating properties, e.g., a highly efficient catalytic property due to its high electronic density of states (DOS) at the Fermi level 24 . It was also identified that metallic carbon becomes magnetic when the Stoner-like criterion 25,26 is satisfied. Furthermore, metallic carbon showed a number of intriguing properties such as phonon-plasmon coupling 27 , superconductivity 28,29 and negative differential resistance 30 . Consequently, the exploration of metallic carbon candidates have attracted considerable attention in the synthesis and design of new carbon allotropes. However, all of the 3D carbon allotropes mentioned above are semiconductors or insulators.
There have so far been few progresses in search of 3D metallic carbon. Some hypothetical 3D carbon allotropes such as ThSi 2 -type tetragonal carbon 31 , hexagonal H-6 carbon 32 , and K 4 carbon 33 have been proposed to be metallic. However, all of such structures were found to be dynamically unstable 19,34,35 . In 2012, a simple cubic phase of carbon was reported to be metallic under 3 TPa, but it becomes unstable when pressure is removed 36 .
Recently, a 3,4-connected T6 carbon allotrope was proposed to be metallic, but it was also unstable at temperature about 500 K 37 . More recently, a 3D metallic K 6 carbon with pure sp 3 hybridization was reported to have a high DOS at the Fermi level 38 , however, its stability is too low to be synthesized. To our knowledge, all of these theoretical predictions of 3D metallic carbon allotropes have not been experimentally realized so far.
Here, based on first-principles total-energy and phonon calculations 39-46 , we discover a new stable 3D metallic carbon allotrope in / P mmm 6 (D h 6 1 ) symmetry with a mixed sp 2 -sp 3 hybridized bonding network. This new phase is composed of eighteen atoms per hexagonal primitive cell (hereafter termed H 18 carbon), having a larger atom density of 3.135 g/cm 3 compared to 2.28 g/cm 3 for graphite. The calculated elastic constants show that the H 18 carbon is a brittle material. From the analysis of the phonon spectra, we find that H 18 carbon does not have any unstable vibration modes. Interestingly, the simulated x-ray diffraction (XRD) pattern of H 18 carbon matches one of the unidentified carbon phases observed in recent detonation experiments 47 . The calculated band structure and DOS of H 18 carbon manifest a metallic feature mainly due to the C atoms with sp 2 hybridization. The H 18 carbon has great potential application in electronics, mechanics, and some other related fields due to its novel properties. Figure. 1 shows the structure of H 18 carbon with a 3D sp 2 -sp 3 hybridized bonding network in / P mmm 6 (D h 6 1 ) symmetry. Here, the hexagonal primitive cell contains eighteen C atoms with equilibrium lattice parameters a = b = 7.125 Å, c = 2.605 Å. The C 1 , C 2 , and C 3 atoms in Fig. 1 are bonded to four, four, and three neighboring carbon atoms, thus forming sp 3 , sp 3 and sp 2 bonds, respectively. The calculated bond lengths 3 is between the bond lengths (1.420 and 1.544 Å) of graphite and diamond which have sp 2 and sp 3 bonds, respectively. Because of a mixed bonding of sp 3 and sp 2 , H 18 carbon has five different bond angles of 107.5°, 110.1°, 110.4°, 115.2° and 129.6°, contrasting with 109.5° for sp 3 hybridized diamond and 120° for sp 2 hybridized graphite. Such several bond distortions in H 18 carbon implies the presence of strain, leading to a decrease of the relative stability compared to diamond and graphite as discussed below. In Table 1, we list the calculated lattice parameters, equilibrium densities, bond lengths, and cohesive energies of diamond, graphite, Rh6, T6, BC8, and H 18 carbons. We find that our results agree well with previous DFT calculations and experiments 21,22,37 . We note that the equilibrium bulk atom density of H 18 carbon is 3.135 g/cm 3 , larger than that (2.28 g/cm 3 ) of graphite 21 .

Results
In order to check the stability of H 18 carbon, we perform the analyses of total energy, phonon mode, and elastic constants as well as molecular dynamic (MD) simulations.
(i) Total energy: Figure 2 shows the calculated total energies of H 18 carbon, diamond, graphite, BC8, T6 and Rh6 carbon as a function of volume per atom. It is seen that graphite and diamond are more thermodynamically stable than H 18 carbon as well as other carbon phases. We note, however, that H 18 carbon is not only much more stable than BC8 carbon (which has been suggested to be the derivative of cubic diamond under pressure of ∼ 1100 GPa 48-50 ), but also more stable than T6 carbon and Rh6 20,37 . In particular, H 18 carbon has a relatively smaller 18-atom hexagonal unit cell compared to Rh6 (see Fig. 2). From the energy-volume curves of carbon allotropes (Fig. 2), one expects that Rh6 carbon can be transformed into H 18 carbon by applying a certain pressure.
(ii) Phonon mode: The calculated phonon band structure and DOS are displayed in Fig. 3. It is found that there is no negative frequencies throughout the entire Brillioun zone, indicating the dynamical stability of H 18 carbon. In Fig. 3, the highest vibrational frequency of H 18 carbon amounts to ∼ 1837 cm −1 at A point, which is higher than ∼ 1400 cm −1 of the perfectly sp 3 bonded diamond 51 as well as ∼ 1600 cm −1 of the π -conjugated graphite 52 . We note that there exists a wide band gap of ∼ 230 cm −1 between 1449 and 1679 cm −1 near the K point. These features of phonon spectra of H 18 carbon are anticipated to be measured by future experiments. In Fig. 3, we also plot the atom-resolved phonon DOS contributed by C 1 , C 2 and C 3 atoms, respectively. Obviously, the highest frequency modes around 1750 cm −1 originate from the vibrations of the strong sp 2 C 3 − C 3 bonds with a bond length of 1.317 Å, while the second highest frequency modes around 1450 cm −1 arise from the mixed sp 2 -sp 3 C 2 − C 3 bonds with a bond length of 1.475 Å. These characters of H 18 carbon are different from both graphite with bond length of 1.420 Å and diamond with bond length of 1.544 Å.
(iii) Elastic constants: In the linear elastic regime, the elastic constant tensor is constituted a symmetric 6 × 6 matrix with 21 independent components, where only C 11 , C 12 , C 13 , C 33 and C 44 are independent in the hexagonal lattice 53 . According to the Born stability conditions 53 , the elastic constants of the hexagonal lattice should satisfy C 11 > C 12 , C 44 > 0, C 66 > 0, and 2C 13   carbon satisfy well all of the conditions, indicating that the structure of H 18 carbon is mechanically stable. The bulk modulus and shear modulus obtained by Voigt-Reuss-Hill approximation 54 are also listed in  to that (0.83) of diamond, implying that H 18 carbon can be characterized as a brittle material according to the Paugh criterion 55 .

Table 2. Calculated elastic constants C ij (GPa), bulk modulus B (GPa), shear modulus G (GPa) and B/G value for H 18 carbon in comparison with diamond, T6, BC8, and Rh6 carbon.
(iv) MD simulations: To examine the thermal stability of H 18 carbon, we carried out the ab initio MD simulations with the canonical (NVT) ensemble at temperature of 300, 1000 and 1500 K, respectively. The system is modeling with a 2 × 2 × 3 supercell containing 216 carbon atoms and the time step of 1 fs is used. The potential energy fluctuation of H 18 carbon in MD simulation at 1000 and 1500 K are presented in Fig. (4a,b), respectively. We can see that the potential energy fluctuation are very small and geometry of H 18 carbon remains intact after heating up to 1000 K for 6 picoseconds. With the temperature increasing up to 1500 K, we find that the H 18 carbon becomes graphitization gradually (see in Fig. (4b)). These results have indicated that H 18 carbon, once synthesized, can sustain the structure even at temperature of 1000 K. For comparison, Ab initio MD simulations for T6 carbon with the same setting and a 2 × 4 × 4 supercell containing 192 carbon atoms show that it is unstable even at 500 K 37 . Based on our MD simulations, we can say that H 18 carbon is much more stable than T6 carbon at high temperature.
In addition, to evidence the experimental observation of H 18 carbon, we plot the simulated XRD spectra of graphite, diamond, BC8, T6, Rh6, and H 18 carbons in Fig. 5(a,b), together with the experimental XRD data of TNT/RDX detonation soot in Fig. 5(c). In the experimental XRD data 47 , the diffraction lines arising from graphite (g), diamond (d), and other unknown (u) carbon phases were reported. As shown in Fig. 5(b,c,i) the (001), (101) and (201) peaks of H 18 carbon match well with the experimental XRD spectra located at 34.4°, 37.4° and 45.5°, respectively, and (ii) the (111) peak of H 18 carbon that overlaps with the (111) peak of diamond can be associated with the second-strongest experimental XRD peak at 43.9°. Meanwhile, the (110) and other peaks of H 18 carbon can not be clearly identified in the experimental XRD patterns, but may be overlapped with neighboring peaks of other carbon phases. Note that the position of (100) and (101) peaks of T6 are very close to the position of (001) and (101) peaks of H 18 carbon, respectively. Since H 18 carbon is thermodynamically more stable than T6 carbon as mentioned above, it is most likely that the experimental XRD peaks near 34° and 37° would be predominantly Experimental XRD patterns for detonation soot (sample Alaska C) 47 . g, d, u indicate graphite, diamond, unknown-carbon, respectively. The X-ray wavelength we adopted is 1.54059 Å.
associated with H 18 carbon rather than T6 carbon. Therefore, we propose that H 18 carbon is one of the unidentified carbon phases which were explicitly present in recent detonation experiments 47 .
To provide more physical quantities that are accessible from experiments, we also simulated the Raman spectra of H 18 carbon and compared the results with graphite, diamond and T6 carbon structures. The simulation results are presented in Fig. 6. From Fig. 6, we can see that the E 2g mode in graphite is estimated to be 1586 cm −1 , which is well agreement with the experimental data of 1581 cm −1 56 . The T 2g mode in diamond is estimated to be 1326 cm −1 , which is close to the experimental data of 1333 cm −1 57 . Although the main XRD peaks of T6 and H 18 carbon are close to each other, their Raman spectra show rather different characters. For T6 carbon, there is only one main peak A 1g presents at 1762 cm −1 . However, we find that, for H 18 , there are two main peaks A 1g at 1035 and 1849 cm −1 , respectively. In addition, both T6 and H 18 carbon also show some weaker peaks (E g and B 2g for T6, E 1g and E 2g for H 18 ). All the features above may be helpful for identifying this new carbon phase in further experiments.  Finally, we discuss the electronic properties of H 18 carbon. The band structures and partial density of states (PDOS) are displayed in Fig. 7(a,b), respectively. It is seen that the Fermi level crosses the bands near the A and Γ points, giving rise to the presence of the electron (hole) pocket around A (Γ). Thus, H 18 carbon is metallic. It is noted that the metallic feature obtained using the semilocal DFT calculation with the Perdew-Burke-Ernzerhof (PBE) functional is preserved in the hybrid DFT calculation with the Heyd, Scuseria, and Ernzerhof (HSE06) functional 58,59 [see Fig. 7(a)]. From the PDOS projected onto C 1 , C 2 and C 3 atoms [ Fig. 7(b)], we find that the electronic states near E F dominantly involve the p y character of C 3 . In order to elucidate the origin of the metallic feature in H 18 carbon, we calculate the charge density of the partially occupied bands in the energy windows of E F − 0.5 to E F + 0.5 eV [see Fig. 7(c)]. It is seen that the p y orbital of C 3 atoms largely contribute to the charge density, forming a delocalized network. On the basis of the PDOS [ Fig. 7(b)] and the charge character near E F [Fig. 7(c)], we can say that the metallicity of H 18 carbon is attributed to a large delocalization of the p y orbital of C 3 atoms with sp 2 hybridization.

Conclusion
Our first-principles DFT total energy and phonon calculations discover a novel stable carbon allotrope (termed H 18 carbon) which is metallic. The analyses of the total energy, phonon mode, and elastic constants as well as molecular dynamic simulations obviously show that this new carbon allotrope exists as a stable structure. More importantly, we demonstrate that the H 18 carbon may be one of the candidates of the unidentified carbon phases which appeared in the XRD spectrum analysis of a recent detonation experiment. In particular, H 18 carbon has a metallic feature mainly due to the p y orbitals of sp 2 hybridized carbon atoms. Unlike previously reported 3D metallic carbon allotropes, H 18 carbon can not only keep its metallicity at ambient pressure but also can likely maintain its structure at high temperatures. This novel 3D metallic carbon phase is anticipated to be useful for practical applications such as electronics and mechanics devices. Our findings will attract immediate broad interest and stimulate further experimental and theoretical studies for this new carbon allotrope.

Methods
The present total-energy and phonon calculations were carried out by using the density functional theory. Both local density approximation (LDA) in the form of Ceperley-Alder 39 and the generalized gradient approximation (GGA) developed by Perdew, Burke, and Ernzerhof (PBE) 40 are adopted for the exchange-correlation potential as implemented in the Vienna ab initio simulation package (VASP) [41][42][43] . All the discussions in this paper are based on the results got by GGA-PBE method, except for special notations. The all-electron projector augmented wave (PAW) 44 method is adopted with C 2s 2 2p 2 treated as valence electrons. A plane-wave basis set with an energy cutoff of 800 eV is used. The Gaussian smearing with a smearing factor of 0.05 eV is used in the calculations. The Brillouin zone (BZ) is sampled by a 9 × 9 × 21 Monkhorst-Pack (MP) special k-point grid including Γ-point. The geometries are optimized with no symmetry constraints, the convergence criteria employed for both the electronic self-consistent relaxation and the ionic relaxation are set to 10 −7 eV and 10 −3 eV/Å for electronic energy and Hellmann-Feynman force, respectively. Phonon dispersion curves are calculated by using the package phonopy 45,46 with the forces calculated with VASP code. The first-principles molecular dynamics simulations are performed in the canonical (NVT) ensemble with the Nosé thermostat 60 . Each simulation lasted for 6 ps, with a time step of 1 fs. All the calculations in this work are performed at zero pressure.