The element carbon possesses various stable and metastable allotropes; some of them have been applied in diverse fields. The experimental evidences of both carbon chain and graphdiyne have been reported. Here, we reveal the mystery of an enchanting carbon allotrope with sp-, sp2-, and sp3-hybridized carbon atoms using a newly developed ab initio particle-swarm optimization algorithm for crystal structure prediction. This crystalline allotrope, namely m-C12, can be viewed as braided mesh architecture interwoven with multigraphene and carbon chains. The m-C12 meets the criteria for dynamic and mechanical stabilities and is energetically more stable than carbyne and graphdiyne. Analysis of the B/G and Poisson’s ratio indicates that this allotrope is ductile. Notably, m-C12 is a superconducting carbon with Tc of 1.13 K, which is rare in the family of carbon allotropes.
The element carbon has always been a “star” to bring wonderful substances to the world owing to the existence of three types of hybridization: sp, sp2, and sp3, and the pursuing and exploring numerous possible carbon materials have never stopped. Over time, the carbon fullerenes, nanotubes, and graphene have been successfully fabricated one after another. Beside well-known carbons above, carbyne, the allotropic form of carbon consists of sp hybridized atoms, was experimentally evidenced inside the carbon multiwalled nanotubes in the cathode deposits prepared by hydrogen arc discharge evaporation of carbon rods1 or after the irradiation of graphene ribbons by removing carbon atoms row by row from graphene inside a transmission electron microscope2 (TEM). B.I. Yakobson et al. found carbon chain via fracturing carbon nanotubes with high strain rate by molecular dynamics using a realistic many-body potential.3 Edwin Hobi et al. obtained linear carbon atomic chain through pulling of graphene nanoribbons using ab initio molecular dynamics.4 Recently, Casari et al.5 and Ravagnan et al.6 have reported that carbon films consisting of carbyne can be fabricated at room temperature. Subsequently, graphdiyne, a carbon phase with mixed sp- and sp2-bonded structure, which is a novel, stable, and non-natural carbon allotrope, had been experimentally synthesized on the surface of copper via a cross-coupling reaction using hexaethynylbenzene.7 Lately, the carbon allotropes with a framework of sp- and sp2- (or sp3-) hybridized carbon atoms have been receiving significant attention in novel structure and excellent performance.8–19 Still, the design on theoretical predictions and synthesis on experimental methodology of new carbon allotropes is attractive and ongoing.
In this letter, we propose a superconducting carbon allotrope using first-principle calculations. This structure consists of sp-, sp2- and sp3-hybridized carbon atoms, forming a 3D monoclinic crystalline carbon that possesses the space group C2/m (No. 12). This framework with 12 carbon atoms in a unit cell is thus named as m-C12. The configuration of m-C12 is composed of corrugated layers of multigraphene connected by carbon chains, in which the sp3 hybridized atoms acted as junction. The independent elastic constants and phonon dispersion spectra verified the mechanical and dynamical stabilities of m-C12. Simultaneously, the m-C12 is thermodynamically metastable relative to graphite and diamond, but with lower ground-state energy than the graphdiyne and carbyne. Electronic band structure calculations indicate that m-C12 has superconductive characteristic, and the superconducting transition temperature (Tc) is 1.13 K. Furthermore, it is a ductile carbon allotrope deduced from the B/G and Poisson’s ratios.
II. COMPUTATIONAL DETAILS
We searched for 3D carbon structures with system sizes including up to 24 atoms/unit cell using the well-developed CALYPSO code,20 unbiased for predicting novel carbon allotropes. The structural optimization and characteristics were predicted using the CASTEP code21 based on the density functional theory22,23 (DFT). The Vanderbilt ultrasoft pseudopotential was used with a plane-wave energy cutoff of 660 eV. The exchange correlation terms were depicted using the Ceperley and Alder as parametrized by the Perdew and Zunger (CA-PZ) form of local density approximation24 (LDA). The Monkhorst−Pack grid parameters were used to generate a k-point grid25 with a separation of 2π × 0.01 Å−1. The finite displacement method was used to calculate the phonon frequencies. Primitive cells were selected to calculate the band structures, bulk modulus, shear modulus, and elastic constants. The electronic band structures and density of states (DOS) were calculated using the HSE06 functional.26,27 The Norm conserving pseudopotential was used, and the electronic minimizer was All Bands/EDFT. The plane-wave cutoff energy was 660 eV, and a k-point spacing (2π×0.01 Å−1) was used to generate Monkhorst-Pack k-point grids for Brillouin zone sampling. The electron-phonon coupling (EPC) was simulated using the Quantum-ESPRESSO package28 with the plane-wave pseudopotential method and density functional perturbation theory29,30 (DFPT).
III. RESULTS AND DISCUSSION
The crystal structure of designed m-C12 was showed in Fig. 1. The optimized lattice parameters and Wyckoff atomic positions at ambient pressure are also listed. The m-C12 has a monoclinic system (space group: C2/m, No. 12) which is identical to the C2/m-carbon,31 but their crystal structures are different. There are three types of nonequivalent carbon atoms in the unit cell of m-C12. The sp, sp2, and sp3 hybridized atoms are colored as blue, olive, and dark grey, respectively. From the direction of  (Figure 1a), the fluctuant multigraphene layers consisted of sp2 carbon atoms are bridged by chains constructed by sp-hybridized carbon atoms, where the sp3 hybrid atoms are acted as the nodes. Also, the corresponding view of m-C12 along the  and  direction are presented in Fig. 1b and 1c, demonstrating the configuration clearly. The bond angle of sp atoms is not the perfectly 180°, but 178.31°. The bond angle of sp2 atoms is not the standard 120°, but 116.92°, and the bond angle of sp3 atoms is 109.37°, 109.61°, 107.68°, or 110.38° due to the bond bending.32,33 The m-C12 has an ambient-pressure enthalpy of 0.70eV/atom higher than that of graphite, and thus is metastable relative to graphite at ambient pressure. In spite of this, m-C12 is more stable than graphdiyne,34 with a lower ambient-pressure enthalpy of 0.19eV/atom than that of graphdiyne, indicating its viability. Its equilibrium volume of 100.354 Å3 unit cell indicates a density of 2.385 g/cm3. Inspired from the chemical synthesis of graphdiyne and theoretical analysis of structural characteristics, we assume m-C12 may be obtained by the polymerization of graphene and carbon chain under specific conditions.
To ascertain the mechanical features, the independent elastic constants (Cij), bulk (B), and shear (G) moduli of m-C12 were calculated under ambient pressure. The Born criterion35 was used to assess the mechanical stability of m-C12. The calculated elastic constants of m-C12 are C11 = 728.01 GPa, C22 = 706.39 GPa, C33 = 546.31 GPa, C44 = 206.22 GPa, C55 = 49.29 GPa, C66 = 5.75 GPa, C12 = 35.05 GPa, C13 = 103.46 GPa, C15 = 72.71 GPa, C23 = 46.35 GPa, C25 = 9.67 GPa, C35 = 34.38 GPa, and C46 = 6.66 GPa. The elastic stability criteria of monoclinic phase are:
The elastic constants of m-C12 satisfy the mechanical stability criteria for a monoclinic phase. The bulk (B) and shear (G) moduli of m-C12 are 251.70 GPa and 87.86 GPa, respectively. B/G and Poisson’s ratios are often used to assess the ductility (brittleness) of the material.36 The B/G ratio and Poisson’s ratios of m-C12 are 2.86 and 0.34, respectively, indicating that m-C12 is more ductile than diamond (0.82 and 0.07) due to the flexional sp2-hybridized bonds.
The dynamic stabilities of m-C12 at ambient pressure were investigated by the phonon dispersion (Figure 2a). Imaginary phonon frequencies were not observed throughout the entire Brillouin zone of m-C12 phase, and we confirm that there are no any imaginary vibration modes in the phonon dispersion spectra of m-C12, demonstrating the dynamic stability of m-C12. In addition, the calculated highest phonon frequency of C–C bond vibrational mode in m-C12 phase was ∼2300 cm-1, a slightly higher than that of triple yne-bond (2150 cm-1 or 2193 cm-1) in Rh12 or Rh18 phenylacetylene, indicating a relatively stronger –C≡C– bonding in m-C12. Therefore, m-C12 possesses thermodynamic, mechanical, and dynamical feasibility and the possibility of experimental synthesis.
The electronic band structures for m-C12 were calculated using the HSE06 functional as illustrated in Figure 2b, and overall it shows that m-C12 is a conductor. Moreover, several flat bands crossing M and A points as well as G and Z points appeared near the Fermi level. Meanwhile, some steep conduction bands along M point toward L or A point as well as G point toward A or Z point and V point toward Z point passed through the Fermi level, indicating the electron velocities and additional electronic transfer channels. The existence of flat and steep bands near the Fermi level can be considered as an index for a superconductor,37 indicating the potential superconductivity of m-C12. The superconducting transition temperatures Tc were assessed utilizing the Allen and Dynes modified McMillan equation, , where ωlog is the logarithmic average characteristic frequency, λ is the EPC coefficient, and μ* is the Coulomb pseudopotential.38 The calculated EPC coefficient λ was 0.40 for m-C12. An empirical value of μ* = 0.139 was used to estimate the Tc of m-C12, consistent with the experimental result in measuring the Tc of boron-doped diamond. Under the same value of Coulomb pseudopotential, a Tc value of 1.13 K for m-C12 was obtained. The Tc of 1.13 K is much less than the GT-8 and GT-1640 (5.2 and 14.0 K, respectively) under the same value of Coulomb pseudopotential μ*, which are the superconducting carbon that have been proposed before. However, the Tc of 1.13 K is comparable to a Tc of 1.5 K corresponding to B-doped n-type diamond as the estimated dopant concentration dc (%) = 3.0.41,42 Doped diamond43–47 and graphane48 as a superconductor are more explored than the pure carbon allotropes.
To better understand the entire conducting behavior of m-C12, the electronic projected density of state (PDOS) was further calculated, which is the key to understanding which atoms would offer contributions near the Fermi level and give rise to conductivity in a crystal. As shown in Fig. 3, the C1 and C2 hybrid atoms significantly contribute to the electrical conductivity near the Fermi surface, where the influence of C3 hybrid atoms on conductivity is lesser. The conductive electrons of m-C12 derived from 2px, 2py, and 2pz orbitals are depicted to investigate the hybridized state of each type of carbon atoms. The PDOS of C1 atoms; 2s orbital intersects 2px orbital above the Femi level, indicating that C1 has the sp hybridized state, in which the contribution of electrical conductivity stems mainly from the 2py orbital. The PDOS from 2s and 2px orbitals/2py and 2pz orbitals of sp blue atoms are similar, respectively, indicating that the 2s and 2px orbitals with low PDOS are hybridized to form σ bonds and the hybridization of 2py and 2pz orbitals with high PDOS forms the π orbitals (Figure 3a). The PDOS of C2 (Figure 3b) shows that C2 atom has the sp2-hybridized state due to the intersection of 2s, 2py, and 2pz orbitals, as well as the 2px orbital make a major contribution for conductivity. The 2s, 2pz, and 2py orbitals of sp2 olive atoms have the similar PDOS, indicating that they are hybridized to form the σ bonds and the 2px orbitals with high PDOS also forms the π orbitals. The C3 atoms are classified as unconventionality sp3-hybridized states in Figure 3c according to the hybridization of 2s, 2px, 2py, and 2pz orbitals; meanwhile, it is like the C1 atom that the 2py orbital primarily devote to the conduction of electricity. The mention of the unconventionality is mainly due to that the 2s and 2py orbitals/2px and 2pz orbitals of sp3 dark grey atoms have the similar PDOS, suggesting that the 2s and 2py orbitals showing higher PDOS are tend to form the π bonds; conversely, the 2px and 2pz orbitals with lower PDOS are apt to come into being the σ bonds, which is owing to the buckling of the sp3 bonds result in the change of bond state.
To elucidate the conductive mechanism, the electron orbitals of m-C12 derived from the bands across the Fermi level, i.e. the electron orbitals of top valence band and bottom conduction band, were calculated. It is known that each energy band is defined by the position of its eigenvalue with ordered electronic energy at each k-point, and the corresponding electron orbital is the square of the absolute value of the wave function for this given electronic band, summed over all k-points. Here, our drawn electron orbitals in the side view are the summations from the red bands (Fig. 2b) near the Fermi level along the x-axis (Figure 4a), y-axis (Figure 4b), and z-axis (Figure 4c). The electron orbitals appearing in sp, sp2, and sp3 carbon atoms all provide the access to electricity, which is consistent with the previous analysis of PDOS in the conductive contribution.
In summary, a superconducting sp-sp2-sp3 monoclinic carbon, called m-C12, was proposed from evolutionary particle-swarm structural search. It has a network configuration from the wave layers consisting of multigraphene intertwining the sp-hybridized carbon chains. The lower ground-state energy and mechanical and dynamical stabilities demonstrate its viability. The calculated B/G and Poisson’s ratios of m-C12 were 2.86 and 0.34, respectively, suggesting that it is ductile. The m-C12 also exhibited a peculiar superconducting character with Tc of 1.13 K, which is unusual in carbon allotropes. This sp-sp2-sp3 carbon with superconductive and nice mechanical properties possesses potential practical value in electronics and machinery industry fields.
This work was supported by National Natural Science Foundation of China (NSFC) (51421091, 51472213, 51332005, 51525205, 51672238, and 51722209).