Three-dimensional metallic carbon allotropes with superhardness

Three novel three-dimensional orthorhombic carbon phases are proposed based on first-principles calculations in this work. These phases possess dynamic stability and mechanical stability and are theoretically more favorable in energy compared to most other carbon allotropes. The hardness levels of oP-C16, oP-C20, and oP-C24 are 47.5, 49.6, and 55.3 GPa, respectively, which are greater than those of T10, T18, and O12 carbon. In addition, although oP-C16, oP-C20, and oP-C24 are metals, their ideal shear strengths are also greater than those of commonmetals such as Cu, Fe, and Al. Due to py electrons crossing the Fermi level, oP-C16, oP-C20, and oP-C24 show metallicity, and their charge densities of the band decomposition suggest that all the conductive directions of oP-C16, oP-C20, and oP-C24 are exhibited along the aand b-axis, similar to C5.


Introduction
The rapid growth of computer performance and the abundance of various experimental data have led to the intensified exploration of new energy materials [1] and novel electronic information materials. The use of computers for scientific calculations is already the third scientific method for mankind to understand and conquer nature after theoretical science and experimental science. Due to the extensive role and important influence of metal superhard carbon materials in the fields of industry, machinery, and aerospace, there is an urgent need for metal superhard carbon materials with excellent mechanical and electronic properties. However, despite theoretical predictions that carbon allotropes emerge in an endless stream, there are not many carbon materials that simultaneously exhibit metallic and superhard properties and enjoy favorable physical properties. Obviously, there is still a long way to go in designing and discovering breakthrough carbon materials with remarkable features. In this work, three superhard metal carbon materials are theoretically predicted and the results obtained show that these three carbon phases all have superhard metal characteristics and many advantageous properties, which are of great significance for enriching the gene pool of excellent materials and providing a solid theoretical basis for future experiments. More importantly, owing to their metallic and superhard properties, oP-C 16 , oP-C 20 , and oP-C 24 may possess the opportunity to play applications in industrial and electronic fields.
In recent years, a considerable number of carbon allotropes and carbon-based materials have been identified. For instance, five carbon allotropes, tP12, tP16, tP24, oP12, and oP18, with nanotubes and cage types were proposed recently [2]. There are two-dimensional (2D) TPH-I carbon and TPH-II carbon obtained from pentagraphene through Stone-Wales conversion [3]. The electronic and mechanical properties of the new superhard indirect semiconductor oP8 carbon have also been predicted [4]. Li and Xing designed a superhard semiconductor material, P2/m C 54 , which is a new type of semiconductor material that can be applied to frequency-hopping absorption filters [5]. TiO 2 /graphite carbon nanocomposite materials, which can improve photocatalytic efficiency, are synthesized under certain temperatures and conditions [6]. In light of the synthesis of graphene with benzene as the precursor, a novel all-sp 2 hybridized 2D carbon allotrope with poly-butadiene-cyclooctatetraene framework is proposed [7]. By means of molecular dynamics simulation, the interface characteristics of carbon nanotube-polyimide nanocomposites have been systematically analyzed [8].
In particular, metallic carbon has attracted much attention due to its usefulness in electronic devices, superconductivity applications, and high-performance anode materials. By cross-linking graphene sheet with sp 3 -hybridized carbon chains, Wang et al. established a new type of porous asymmetric carbon honeycomb (CHC), namely bco-C24 [9]. Using the self-assembled C8 skeleton of pentene as a building block, a series of 3D carbon allotropes (superpentalene-n family) were constructed [10]. A new type of orthogonal metal carbon phase o-C 12 with high stability and hardness has been proposed, which is expected to be applied to actual electronic and mechanical equipment [11]. Zhang et al. proposed a metal superhard carbon allotrope C 10 , which has an sp 2 + sp 3 hybrid carbon network in the orthorhombic unit cell and has a hardness of 58.70 GPa [12]. Liu et al. proposed two carbon polymorphs Orth-C10 and Orth-C10' with excellent superhard and conductive properties through first-principles calculations, which are more energetically stable than fullerene C60 at 0 GPa [13]. There are also paper carbon nanotubes that can replace aluminum foil as a current collector [14].
Recently, Wu et al. [15] reported a novel three-dimensional orthorhombic phase utilizing first-principles calculations. The novel three-dimensional orthorhombic phase, namely, C14-diamond, which was assembled with nanolayered sp 3 carbon in a diamond structure, was bonded with ethene-type -C]Clinks and so C14-diamond is a superhard material and metallic carbon allotrope. In addition, the conductive direction of C14-diamond is also exhibited along these sp 2 carbon atoms. Additionally, Liu et al. [16] designed O-type and T-type carbon allotropes. The diamond nanostripes in O-type and T-type carbon allotropes are similar to C14-diamond and are also bonded with ethene-type -C]Clinks. All the O-type and T-type carbon allotropes exhibited metallic properties, and the bulk modulus (B) and shear modulus (E) of O-type and T-type carbon allotropes exceeded 357 and 223 GPa, respectively.
In recent years, the focus of development has been to predict the properties of new materials and to design novel structures or some details in the new phases with function-oriented reverse design. However, compared with the huge demand in various fields such as modern industry, manufacturing, and aviation, carbon materials that are harder, more conductive, and lighter are still waiting for scientists to explore.
From ancient times to the present, carbon materials have been playing an increasingly important role in industrial applications, from initial heating materials to electronic materials and other functional materials. For example, supercapacitors based on carbon materials [17] have the same excellent characteristics as all-paper-based supercapacitor devices [18], such as fast charge and discharge speed, high power density, and long-term cycle stability. Complementary carbon nanotube transistors can even be used to build modern microprocessors [19]. Carbon nanomaterial-reinforced cement-based composites, which are stronger than traditional cement materials, smarter, and more durable, are also attracting attention [20]. Novel carbon materials and composite carbon materials [21] with excellent performance are closely related to modern engineering. The material gene bank is equivalent to the guiding light for the synthesis of new materials, so the theoretical importance of the exploration of new carbon structure is self-evident.
In this work, the three novel superhard metallic carbon allotropes (according to the number of carbon atoms in the conventional cell, denoted oP-C 16 , oP-C 20 , and oP-C 24 below) are proposed in this work and these novel metallic carbon allotropes from the space group and graph theory (RG 2 ) code [22]. The crystal structures of oP-C 16 , oP-C 20 , and oP-C 24 are similar to C14-diamond, O-type, and T-type carbon allotropes. The process is shown in graphical abstract. It can be seen that the structures of these three carbon allotropes are related and similar, and they are all composed of six-membered rings and ten-membered rings. From the ab plane, the structure of oP-C 16 is exactly identical to that of oP-C 20 . The difference exists in bc and ac planes, as shown in the red wireframe in graphical abstract, that is the structure of oP-C 20 possesses one more six-membered ring than that of oP-C 16 . Interestingly, the addition of a six-membered ring to the structure of oP-C 20 along the bc and ac planes converts it to the structure of oP-C 24 . The hardness level of oP-C 24 is higher than those of C14-diamond, O12 carbon, O20 carbon, T10 carbon, T18 carbon, and T26 carbon. Additionally, the conductive directions of oP-C 16 , oP-C 20 , and oP-C 24 are also investigated in this work.

Computational method
The crystal structures of oP-C 16 , oP-C 20 , and oP-C 24 are obtained from the random sampling strategy combining space group and graph theory (RG 2 ) [22]. For geometry optimization, ultrasoft pseudopotentials [23] are adopted in the interactions of the core electrons, and norm-conserving pseudopotentials [24] are employed for electronic band structure calculations. A generalized gradient approximation (GGA) proposed by Perdew et al. [25] and local density approximations (LDA), as developed by Ceperley and Alder and parameterized by Perdew and Zunger (CA-PZ) [26,27], is performed by utilizing the Cambridge Serial Total Energy Package [28] by means of density functional theory [29,30] calculations. The Broyden-Fletcher-Goldfarb-Shenno (BFGS) [31] minimization technique is used for geometry optimization. A plane wave basis set with an energy cut-off of 400 eV is used for oP-C 16 , oP-C 20 , and oP-C 24 . A high-accuracy grid spacing of lower than 2π × 0.025/Å is selected utilizing the Monkhorst-Pack scheme [32] (for oP-C 16 : 16 × 15 × 3, oP-C 20 : 16 × 16 × 2, and oP-C 20 : 16 × 16 × 2).
For other carbon materials (C14-diamond [15], T10 carbon, T18 carbon, T26 carbon, O12 carbon, O20 carbon, O26 carbon [16], and C14 carbon [33]) for comparison, the cutoff energy of 400 eV is also used, and the k-points of the other carbon materials (T10 carbon: 16  are also used for the high-accuracy grid spacing of lower than 2π × 0.025/Å. The Heyd-Scuseria-Ernzerhof (HSE06) hybrid functional [34] is adopted to estimate the electronic properties of oP-C 16 , oP-C 20 , and oP-C 24 . The density functional perturbation theory [35] is employed to investigate the phonon spectra for oP-C 16 , oP-C 20 , and oP-C 24 . Firstprinciples calculations are favored by many researchers in the fields of condensed matter physics and computational materials science. Regarding the study of the physical properties of metallic carbon, some references [13,15,16,33] use first-principles calculations. Thus, in this work, firstprinciples calculations are adopted to study the physical properties of the three proposed carbon allotropes.

Results and discussion
The designed constructions of oP-C 16 , oP-C 20 , and oP-C 24 , and their crystal structures are shown in graphical abstract. There are four sp 2 hybridization carbon atoms in all three structures, that is, there are two -C]Cbonds in the oP-C 16 , oP-C 20 , and oP-C 24 crystal structures. The oP-C 16 , oP-C 20 , and oP-C 24 carbon phases are established by the sp 2 + sp 3 hybrid bonding network in a three-dimensional formation. These three structures also contain different numbers of inequivalent atomic positions. oP-C 16 has eight different inequivalent atomic positions, oP-C 20 has ten different inequivalent atomic positions, and oP-C 24 has twelve different inequivalent atomic positions. The optimal lattice parameters of oP-C 16 , oP-C 20 , and oP-C 24 are presented in Table 1. Furthermore, the theoretical and experimental values of the lattice parameters for diamond are displayed in Table 1 for comparison. The estimated results obtained by using the GGA method are closer to the experimental values than those obtained by using LDA. Thus, all theoretical calculations in our article are performed at the GGA level.
As described in ref. [15], the phonon modes greater than 40 THz chiefly originate from the vibration of sp 2 hybridization connected with the -C]Cbond, and the vibration frequency of sp 3 hybridization carbon atoms is mainly below 40 THz. In addition, most importantly, no hypothetical frequency is found in the whole Brillouin zone, demonstrating that the three predicted carbon allotropes are dynamically stable. Figure 1(d) shows the enthalpy per atom of oP-C 16 , oP-C 20 , and oP-C 24 , including C14-diamond [15], T26 carbon, T10 carbon, T18 carbon, O18 carbon, O12 carbon, and O20 carbon [16] in the sp 2 + sp 3 bonding network and C14 carbon [33] in all sp 3 bonding networks compared with other carbon phases reported theoretically. As shown in Figure 1(d), oP-C 24 is more stable than T10 carbon, T18 carbon, O12 carbon, O20 carbon, and C14 carbon in thermodynamics; oP-C 20 is thermodynamically more stable than those of T10, O12, O20, and C14 carbon; and oP-C 16 is thermodynamically more stable than those of T10 carbon and O12 carbon.
For the structure of a stable orthorhombic crystal system, the elastic constants C ij should meet the mechanical elasticity criteria [36]: C 11 > 0, C 11 C 22 > C 12 2 , C 11 C 22 C 33 + Table 2 shows the estimated elastic parameters for oP-C 16 , oP-C 20 , and oP-C 24 .
All the acquired elastic parameters for oP-C 16 , oP-C 20 , and oP-C 24 obey the above mechanical stability criteria. The C 22 values of oP-C 16 , oP-C 20 , and oP-C 24 are slightly lower than those of diamond, C14-diamond and C14 carbon, while they are higher than that of c-BN. Furthermore, the C 11 and C 33 of oP-C 16 , oP-C 20 , and oP-C 24 are slightly higher than those of C14-diamond, C14 carbon, c-BN, and diamond. Nevertheless, the higher values of C 11 and C 33 indicate the greater horizontal and vertical linear compression resistance of the crystal [15]. The values of B and G are estimated by adopting Voigt-Reuss-Hill approximations according to the obtained elastic constants, and Young's modulus is estimated by E = 9BG/(3B + G) based on the B and G values. The B V , B R , G V and G R are calculated as follows [37]: Three-dimensional metallic carbon allotropes  1269  of C14-diamond, C14 carbon, and diamond, they are higher than those of O12 carbon and T10 carbon, while the bulk moduli of oP-C 16 , oP-C 20 , and oP-C 24 are greater than that of c-BN. Using Chen's model [38] to estimate hardness, the related results of their carbon materials within the GGA level are also shown in Table 2. From Table 2, the hardness values of oP-C 16 , oP-C 20 , and oP-C 24 are estimated to be 47.5, 49.6, and 55.3 GPa, respectively. The hardness value of C14-diamond determined in the same way is 53.1 GPa, which is close to the previously reported hardness value (55.8 GPa [15]). In this work, the hardness values of oP-C 16 , oP-C 20 , and oP-C 24 are greater than those of O12 carbon (39.9 GPa), T10 carbon (25.6 GPa), T18 carbon (34.8 GPa), and T26 carbon (43.8 GPa). Furthermore, the hardness value of oP-C 20 is also higher than that of O20 carbon (48.8 GPa), and the hardness value of oP-C 24 is also higher than those of C14-diamond and O28 carbon (54.3 GPa). In order to verify the superhardness of the three proposed carbon allotropes, H M-O [39] and H L-O model [40] are employed. Here, H M-O can be obtained based on Poisson's ratio and Young's modulus using the following equation [39]: H M-O = 0.096 E (1-8.5v + 19.5v 2 )(1-7.5v + 12.2v 2 + 19.6v 3 ) −1 . The calculated hardness levels of oP-C 16 , oP-C 20 , and oP-C 24 are 52-64 and 66-72 GPa, respectively, which are all greater than 40 GPa. All these three methods prove the fact that oP-C 16 , oP-C 20 , and oP-C 24 have superhard characteristics.
The tensile strengths of oP-C 16 , oP-C 20 , and oP-C 24 as functions of tensile strain within the GGA functional are displayed in Figure 2(b), (d), and (e), respectively. The ideal tensile strengths along the different tensile directions of oP-C 16 /oP-C 20 /oP-C 24 are presented in Table 4. From Figure 2  direction is nearly 18 GPa greater than that of diamond, 30 GPa greater than that of O28 carbon, and 60 GPa larger than that of T10 carbon. The high values of B modulus, hardness, and ideal strength provide these new materials broad application prospects in wear-resistant metal coatings and cutting tools.
To analyze the electronic band structures of oP-C 16 , oP-C 20 , and oP-C 24 , the projected weights of each kind of orbital of oP-C 16 , oP-C 20 , and oP-C 24 using the Heyd-Scuseria-Ernzerhof (HSE06) hybrid functional are plotted in Figure 3. By showing the weights of the electrons in different orbits, it is found that the conduction band minimum between the S and X points and the valence band maximum at the Γ point both cross the Fermi surface. Similar to T10 carbon, T18 carbon, T26 carbon, O12 carbon, O20 carbon, O20 carbon, and C14-diamond, oP-C 16 , oP-C 20 , and oP-C 24 all have metallic properties. In addition, the Fermi level is chiefly provided by the p y electrons. In the Brillouin zone, the coordinates for oP-C 16 16 , oP-C 20 , and oP-C 24 , the electron localization function (ELF) [43,44] using an isosurface value of 0.75 is demonstrated in Figure 4(a-c). The calculated ELF of oP-C 16 , oP-C 20 , and oP-C 24 further indicates the strong covalent C-C bonding feature; at the same time, it can be clearly seen that the σ bond is formed by the sp 2 hybrid carbon atom. To further analyze in which direction the three metal carbon materials conduct electricity, the charge density of the band decomposition observed along the a-axis and b-axis of oP-C 16 , oP-C 20 , and oP-C 24 are shown in Figure 5(a-f), which suggests that the conductive directions of oP-C 16 , oP-C 20 , and oP-C 24 are all exhibited along the aand b-axis.
To achieve additional information for probable future experimental examination, the simulation of x-ray diffraction (XRD) patterns for oP-C 16 , oP-C 20 , and oP-C 24 along with that of diamond are shown in Figure 6. Diamond possesses two diffraction peaks, which appear at 43.93 and 75.29°of (111) and (220), respectively, and in the present work, they are in good agreement with the experimental XRD spectra of 43.9 and 75.2°for (111) and (220) of diamond [45], indicating that the performed simulation method is reliable. Because of the same space groups of oP-C 16 , oP-C 20 , and oP-C 24 , their XRD spectra are also very close, which is reflected in the fact that all their three strong peaks appear between 35°and 48°. The peak intensities in the XRD patterns of oP-C 16 , oP-C 20 , and oP-C 24 are not equivalent. The strongest peaks are (104), (105), and (106), which are located around 43.97, 43.48, and 43.55°for oP-C 16 , oP-C 20 , and oP-C 24 , respectively. For oP-C 16 and oP-C 20 , their stronger peaks both contain the two peaks (013) and (105), while the stronger peaks of oP-C 24 are (015), (106), and (017). Apart from these major diffraction peaks, several other weak diffraction peaks appeared in the XRD spectrum, such as, for (101) peaks, the diffraction angles are 36.19, 35.99, and 35.87°for oP-C 16 , oP-C 20 , and oP-C 24 , respectively; for (111) peaks, the diffraction angles are 50.86, 50.89, and 50.91°for oP-C 16 , oP-C 20 , and oP-C 24 , respectively. These XRD features have important significance and guidance for identifying the structures of oP-C 16 , oP-C 20 , and oP-C 24 in future experiments.

Conclusion
In summary, three new carbon allotropes, oP-C 16 , oP-C 20 , and oP-C 24 , with metallic and superhard properties are proposed theoretically through first-principles calculations. These three new types of carbon structures possess many outstanding characteristics. oP-C 16 is energetically more favorable than T10 and O12 carbon, while oP-C 20 and oP-C 24 are more favorable in energy than T18, O12, and O20 carbon. More importantly, oP-C 16 , oP-C 20 , and oP-C 24 enjoy hardness levels of 47.5, 49.6, and 55.3 GPa, respectively. For oP-C 16 /oP-C 20 /oP-C 24 , the peak shear stresses are 11.5-18.  [1][2][3][4][5][6][7][8][9][10][11], and (111) [1][2][3][4][5][6][7][8][9][10] directions, respectively. Although the three carbon allotropes proposed in this work are metals, their ideal shear strengths are also greater than those of common metals such as Cu, Fe, and Al. The p y electrons cross the Fermi level, revealing that oP-C 16 , oP-C 20 , and oP-C 24 exhibit metallicity. Superhard, conductive, and other excellent physical properties make oP-C 16 , oP-C 20 , and oP-C 24 have great application potential in multifunctional devices under extreme conditions, and they are also potential materials for electronic devices and mechanical tools. The theoretical research in this article is of great significance for enriching the gene pool of excellent materials and providing a solid theoretical basis for future experiments.   Figure 6: Simulated XRD patterns of oP-C 16 , oP-C 20 , oP-C 24 , and diamond, using an x-ray wavelength (1.5406 Å) with a copper source for simulation in this work.