3D hybrid carbon composed of multigraphene bridged by carbon chains

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.

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 theory^22,23 (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 theory^29,30 (DFPT).

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 [010] (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 [110] and [100] 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 ų 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 (C_ij), 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, $Tc=ωlog1.2exp-1.041+λλ-μ*1+0.62λ$⁠, where ω_log 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 diamond^43–47 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 2p_x, 2p_y, and 2p_z orbitals are depicted to investigate the hybridized state of each type of carbon atoms. The PDOS of C₁ atoms; 2s orbital intersects 2p_x orbital above the Femi level, indicating that C₁ has the sp hybridized state, in which the contribution of electrical conductivity stems mainly from the 2p_y orbital. The PDOS from 2s and 2p_x orbitals/2p_y and 2p_z orbitals of sp blue atoms are similar, respectively, indicating that the 2s and 2p_x orbitals with low PDOS are hybridized to form σ bonds and the hybridization of 2p_y and 2p_z 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, 2p_y, and 2p_z orbitals, as well as the 2p_x orbital make a major contribution for conductivity. The 2s, 2p_z, and 2p_y orbitals of sp² olive atoms have the similar PDOS, indicating that they are hybridized to form the σ bonds and the 2p_x 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, 2p_x, 2p_y, and 2p_z orbitals; meanwhile, it is like the C₁ atom that the 2p_y orbital primarily devote to the conduction of electricity. The mention of the unconventionality is mainly due to that the 2s and 2p_y orbitals/2p_x and 2p_z orbitals of sp³ dark grey atoms have the similar PDOS, suggesting that the 2s and 2p_y orbitals showing higher PDOS are tend to form the π bonds; conversely, the 2p_x and 2p_z 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).