En route to superhard and 1D conductive carbon allotropes via unusual sp 2–sp 3 bonding configuration

Carbon structures with sp 2–sp 3 hybrid bonds were expected to exhibit both excellent mechanical and electronic properties. Here, we theoretically design several unique carbon structures, which contain unusual sp 2–sp 3 hybrid bonds. We found that the introduction of sp 2 bonding units into carbon structure with sp 3 bonding can tune the electronic densities of state at Fermi level, especially resulting in 1D conductive channels in 3D structures. Further simulations indicate that Vickers hardness of these structures is close to diamond via the increase in sp 3 building blocks. Our current work provides insights into the design of carbon structures with both excellent superhard and remarkably metallic properties.


Introduction
Carbon structures have attracted great attention, since they often show a variety of hybridization modes (sp, sp 2 , sp 3 ), which enable the formation of diverse allotropes such as C 60 [1], graphite, graphene [2,3], diamond, carbon nanotubes [4], and so on [5,6].Among these carbon structures, the presence of sp 2 hybrid bonds confers metallic properties and good ductility to materials, as demonstrated in graphite, while the presence of sp 3 hybrid bonds imparts hardness and super incompressibility to materials, as demonstrated in diamond.On the basis, carbon allotropes with sp 2 -sp 3 hybrid bonding might be expected to possess both metallic and superhard properties.Of the particular interest is to design such carbon allotropes with excellent physical properties.
Currently, it is well known that two typical methods were employed for the theoretical design of sp 2 -sp 3 hybrid carbon allotropes.One approach involves utilizing state-of-the-art structural search software, such as CALYPSO, to predict the structures of materials with superhard and metallic properties [7][8][9][10][11][12][13][14], including mC28 and C14-diamond [10,11].Three predicted sp 2 -sp 3 hybridized superhard metallic carbon phases were firstly proposed through the random sampling strategy combined with space-group and graph-theory (RG 2 ) methods [15].The other approach is to predict novel materials through resembling known structures [16][17][18][19][20].For example, the T6 and T14 carbon, predicted by Zhang et al, through interlocking hexagons, exhibit metallic properties [16].Then, the O-and T-carbon, consisting of diamond nanostrips and carbon-carbon double bonds with metallic and superhard properties, were proposed by Liu et al [20].Recently, the G-D carbon, composed of diamond and graphite as fundamental building blocks, which exhibit electronic properties ranging from semiconductors to metals [18].A great number of studies show that to resemble underlying building blocks can lead to the creation of exotic structures with unusual mechanical and electrical properties.Therefore, these studies greatly motivate us to design carbon structures with excellent physical properties through resembling sp 2 and sp 3 configurations.
In this work, based on the all-sp 3 hybrid carbon allotropes proposed by cold compressed graphite and carbon nanotubes, we theoretically design several carbon structures with superhard and unique metallic properties.Via adjusting the content of sp 2 or sp 3 hybrid units, we uncover a carbon structure with one-dimensional multi-channel conduction in three-dimensional materials.With increasing the number of sp 3 building blocks, the energy favorability is greatly enhanced compared to the proposed sp 2 -sp 3 carbon allotropes, and the hardness is also improved, being closer to that of diamond.The present study presents a novel route for the subsequent design of sp 2 -sp 3 hybrid carbon allotropes.

Methods
We predicted several crystal structures with sp 2 and sp 3 hybridizing bonding.The structural relaxation and properties are calculated based on density functional theory in the VASP (Vienna Ab-initio Simulation Package) code [21,22].We used the Perdew-Burke-Ernzerhof generalized gradient approximation that coupled with the Projector Augmented Wave (PAW) potentials, in which the 2s 2 2p 2 electron of C was explicitly treated [23,24].The plane wave cutoff energy is set at 800 eV, and the Monkhorst-Pack k-point grid generates an enthalpy conversion below 1 meV/atom to ensure calculation accuracy [25].A supercell approach as implemented in the PHONONPY code was used to calculate the phonon dispersion to investigate the dynamic stability of the constructed structure [26].Elastic constants were calculated by the strain-stress method, and bulk modulus and shear modulus were thus derived from the Voigt-Reuss-Hill averaging scheme [27].The calculation of hardness is based on two empirical models proposed by Chen and Tian, respectively [28][29][30].First-principles molecular dynamics (MD) simulations under the canonical ensemble were performed to examine the thermal stability.

Results and discussion
Previous theoretical investigations on cold compressed graphite and nanotubes have yielded a diverse range of carbon allotropes, which exhibit exceptional mechanical properties and remarkable stability [31][32][33][34][35][36][37][38][39][40][41][42][43][44].On the basis, our design strategy is to utilize these carbon allotropes as fundamental building blocks for sp 3 hybridization while incorporating sp 2 hybrid carbon atoms in order to construct novel carbon allotropes.After conducting a comprehensive analysis of the structural characteristics of these carbon allotropes (refer to figure S1), we can categorize them into two distinct groups: 4 + 6 + 8-membered rings and 5 + 6 + 7-membered rings.Further analysis of the 5 + 6 + 7-membered ring show that the M, C, S, O, and F carbon can be decomposed into three or more layers [34][35][36][37][38], whereas the W carbon exhibits a two-layer structure [40].The aforementioned classification results serve as the basis for our proposal of three strategies in designing sp 2 -sp 3 hybrid carbon allotropes.
The design strategy for carbon allotropes with 5 + 6 + 7-membered ring structures, consisting of three or more layers, can be categorized into three primary steps.The first step is to establish the location of the intermediate layer within the crystal structure.In the case of C carbon, observed along the [001] direction, the carbon atoms in this middle layer are positioned exactly within the crystal (depicted as blue atoms in figure 1(a)).The middle layer atoms in S carbon are situated at the periphery of the lattice, denoted by the blue atoms illustrated in figure S2(a).Therefore, in order to let S carbon's interlayer atoms locate in the center of the lattice, we have to construct a 1 × 1 × 2 supercell so that the interlayer atoms are located in the middle of the lattice (as shown in figure S2(b)).The carbon chain in the middle layer (yellow atoms in figures 1(b) and S2(c)) will then be replicated and connected to the original carbon chain, resulting in the formation of a sp 2 hybrid.The crystal structure is then optimized by utilizing the first principles methodology.The two crystal structures are derived from C carbon and S carbon, respectively.Both structures display monoclinic symmetry and are identified as P2/m space group.Consequently, we have assigned them as C-mP 8+24 (containing 8 sp 2 hybridized carbon atoms and 24 sp 3 hybridized carbon atoms per unit cell) and S-mP 4+16 (comprising of 4 sp 2 hybridized carbon atoms and 16 sp 3 hybridized carbon atoms per unit cell).The further information on crystal structure can be found in table S1.
To design a carbon structure consisting of two layers of 5 + 6 + 7 membered rings, we first construct a supercell of 1 × 2 × 1 W carbon supercell as depicted in figure 2. The next is to involve both middle layers as sp 2 hybrids or only one of the layers as a sp 2 hybrid unit.Only one of the layers chosen the layers in this example is utilized for constructing sp 2 hybrid unit blocks, as depicted in figures 2(c) and (d).The blue carbon chain within the yellow box was replicated, which could form a unit block of sp 2 hybridization.The constructed crystal structure is fully optimized, as shown in figure 2(e).Its unit cell comprises a total of 40 carbon atoms, including 8 sp 2 hybridized carbon atoms and 32 sp 3 hybridized C atoms that is denoted as W-oP 8+32 .
The design strategy employed for the crystal structure containing 4 + 6 + 8-membered rings incorporates the insertion of sp 2 hybrid carbon atoms between the carbon atoms constituting the 4-and 8-membered rings.As depicted in figure 3(a 1 ), a 2 × 2 × 1 Z carbon supercell is initially employed as the fundamental building block for sp 3 hybridization.Then, sp 2 hybridized carbon atoms (green) are intercalated between the carbon atoms of the 4-membered and 8-membered rings (blue), resulting in the final structure depicted within the orange box, which is named Z-oP 2+16 carbon.The 3D representation can be observed in figure 3(b 1 ).The average bond length of sp 3 hybridization in Z-oP 2+16 carbon is approximately 1.56 Å, which closely resembles the bond length of diamond at 1.54 Å.Additionally, the sp 2 -sp 3 bond in Z-oP 2+16 carbon is shorter than that of diamond with a bond-length of 1.50 Å.Moreover, the average bond angle of 110.89 • is close to that (109.8 • ) of diamond.The structural characteristics, similar to those of diamond, ensure the stability and hardness of the structure.
To further investigate the influence of the different combination between sp 2 and sp 3 building blocks, we designed a series of 3D metallic superhard carbon phases based on Z-oP 2+16 carbon with the aim of achieving more adjustable characteristics.By removing the atoms within the blue box of a 1 × 2 × 1 Z-oP 2+16 carbon supercell and connecting the remaining portions, we have successfully implemented a structural design strategy involving sp 2 hybrid building blocks, resulting in the formation of a novel carbon phase named Z-oP 4+24 carbon, as illustrated in figure 4. A new carbon phase is formed by adding sp 3 hybrid building blocks on both sides of Z-oP 2+16 carbon.In principle, an infinite number of these series of carbon allotropes can be achieved by adjusting the content of sp 2 or sp 3 building blocks.This study shows the design of seven carbon allotropes named Z-oP m + n carbon.The atomic proportions of sp 2 hybridized carbon atoms gradually increased from 4% to 18% in these seven structures: Z-oP 2+48 , Z-oP 2+40 , Z-oP 2+32 , Z-oP 2+24 , Z-oP 2+16 , Z-oP 4+24 , and Z-oP 6+32 (figure 5).The crystal structure information is also provided in table S1.
To investigate the thermodynamic stability of the proposed structures, we have conducted simulations on its formation enthalpy relative to graphite within a pressure range of 0-60 GPa.The results of the calculations are represented in figure 6.The shaded region in figure 6 represents the energy range of Z-oP m+n carbon, wherein the highest energy state is represented by Z-oP 6+32 carbon with 18% sp 2 carbon atoms, and the lowest energy state is indicated by Z-oP 2+48 carbon with 4% sp 2 hybrid carbon atoms (figure S3).At ambient pressure, these carbon allotropes are metastable and possess energies that are 0.21-0.43eV/atom higher than graphite.However, as the pressure increases, these carbon allotropes tend to stable compared to graphite.It is worth noting that with an increase in the number of sp 3 building blocks, the stability of the structure is greatly improved and approaches that of diamond.The lowest transition pressure from graphite to Z-oP m+n carbon is 12 GPa occurred in Z-oP 2+48 carbon structure as shown in figure 6.This transition pressure is much lower than that of C14-diamond carbon, H 18 carbon [17], T6-and T14-carbon.As shown in figure 6, C-mP 8+24 , S-mP 4+16 , W-oP 8+32 and Z-oP m+n are much more stable in energy than T6, T14 and H 18 carbon as proposed in previous studies.
In order to investigate the dynamical stability of the proposed structures, we performed phonon calculations at ambient pressure (figure S4).The phonon simulations show that there is no imaginary frequency in the phonon dispersion of these structures throughout the entire Brillouin zone except for W-oP 8+32 carbon.The C-C bonds of our proposed structures, consisting of both single and double bonds, exhibit distinct phonon vibration frequencies, with the highest frequency being separated from the other frequencies.In order to investigate the thermal stability of the proposed structures at high temperatures, we performed first-principles MD simulations.The MD calculations were performed at three different temperatures of 300 K, 1000 K, and 2000 K, with each simulation consisting of 10 000 times steps and a time step length of 1 fs.During the simulation process, no C-C bond breakage was observed.As the temperature increased, we found that sp 2 hybridized carbon atoms formed sp 3 hybridized bonds with their nearest neighbors of sp 2 hybridized carbon atoms (refer to figure S5).Z-oP 2+16 and Z-oP 2+48 can withstand temperatures up to 300 K. We also simulated their x-ray diffraction (XRD) spectra and compared them to experimental data obtained from diamond coatings on stainless steel substrates (refer to figure S6) [45].The results indicate that the peaks of Z-oP 6+32 carbon coincide well with the experimental XRD features.The results of the orbital-resolved electronic band structure calculations at ambient pressure indicate that all proposed structures exhibited metallic features, as depicted in figure S7.To explore its origin of the electronic properties, we have selected Z-oP 2+16 carbon as a representative example and performed further simulations at ambient pressure.This includes detailed analysis of the orbital-resolved electronic band structure, partial density of states, electron localization function, and band decomposition charge density, as shown in figure 7. The structure of orbital-resolved electronic bands is illustrated in figure 7(a).It is clearly seen that the conduction and valence bands across the Fermi level primarily originate from the p x orbitals of carbon atoms.Further examination of the partial wave state density in figure 7(b) reveals that the p x orbital of sp 2 hybridized C 5 atom at the Fermi level significantly contributes to conductivity.We calculate the band decomposed charge density within the energy range of E F −1 to E F + 1 eV, revealing that sp 2 hybridized carbon atoms form a one-dimensional delocalization network along the [100] direction, thereby conferring electrical conductivity in this particular orientation.The interesting observation is that C-mP 8+24 and S-mP 4+16 carbon all exhibit one-dimensional multichannel conductivity in three dimensions, as illustrated in figure S8.According to the band decomposition charge density of Z-oP 6+32 carbon as shown in figure S8, the increase in the content of sp 2 hybrid building blocks could improve the conductivity of the system.The ELF of Z-oP 2+16 carbon with an isosurface value of 0.75 in figure 7(c) provides a more intuitive representation of its bonding situation [46].Table 1.Calculated elastic constants C ij (GPa), bulk moduli B (GPa), shear moduli G (GPa), and theoretical hardness Hv (GPa) of various carbon phases, experimental data are in parentheses [47].To verify the mechanical properties of these predicted structures, we calculated their elastic constants, which are presented in table 1. Z-oP m+n and W-oP 8+32 carbon satisfy the mechanical stability criteria for an orthorhombic system [48]: C-mP 8+24 and S-mP 4+16 carbon satisfy the mechanical stability criteria for a monoclinic system [48]: value of 83 GPa.The increase in the content of sp 3 building blocks tends to improve the hardness of the system, and the H Tian v of Z-oP 2+48 carbon could reach 93 GPa, which is close to that of diamond at 94 GPa.The results suggest that these structures exhibit exceptional resistance to compression.

Conclusions
In summary, we theoretically design a series of metal superhard carbon phases with sp 2 -sp 3 hybrid bonding, denoted as C-mP 8+24 , S-mP 4+16 , and W-oP 8+32 , and Z-oP m+n carbon.In these structures, the metallic properties are tuned via the introduction of sp 2 hybrid bonds while their superhard properties are retained due the presence of a great amount of sp 3 hybrid bonds in the hybrid system.Z-oP m+n carbon exhibit tunable mechanical properties through systematic variations in the number of sp 3 building blocks, and Z-oP 2+48 carbon achieves a Vickers hardness of 93 GPa, approaching that of diamond at 94 GPa.More significantly, the adjustment of sp 2 building blocks allows for adjustable 1D multi-channel conduction in a 3D framework.The present results offer a feasible strategy to design carbon structures with excellent hardness and unusual electronic properties.

Figure 1 .
Figure 1.The design strategy of C-mP 8+24 is based on C carbon.(a) The crystal structure of C carbon, (b) the diagram illustrating the insertion of a sp 2 hybrid atom, and (c) the crystal structure of C-mP 8+24 .The blue spheres represent the carbon chain in the middle layer, while the yellow spheres represent the duplicated and inserted atom in the C carbon.The gray and blue spheres represent sp 3 -hybridized carbon atom, the green sphere represents sp 2 -hybridized carbon atoms.

Figure 2 .
Figure 2. The design strategy of W-oP 8+32 is based on W carbon.(a) The crystal structure of W carbon, (b) the four-layer structure was obtained by constructing a 1 × 2 × 1 W carbon supercell.(c) the construction of a 1 × 2 × 2 W carbon supercell was carried out with the intention of positioning the single-layer carbon chain at the center of the unit cell, (d) the schematic illustration depicts the incorporation of sp 2 hybrid atoms, while (e) showcases the crystal structure of W-oP 8+32 .The blue atom symbol represents the carbon chain in the middle layer, while the yellow atom represents the duplicated and inserted atom in the W carbon.The gray and blue spheres represent sp 3 -hybridized carbon atoms, and the green sphere represents sp 2 -hybridized carbon atoms.

Figure
Figure (a) The design strategy is based on a 2 × 2 × 1 supercell of the Z carbon, (b) a view of the crystal structure of Z-oP 2+16 carbon from various orientations, (b1) the 3D view, (b2) and (b3) the front and top views of Z-oP 2+16 carbon, respectively.The gray and blue spheres represent sp 3 -hybridized carbon atoms, and the green sphere represents sp 2 -hybridized carbon atoms.

Figure 4 .
Figure 4.The design strategy of incorporating sp 2 hybridized carbon atoms.

Figure 6 .
Figure 6.The enthalpies per atom of various carbon phases relative to graphite, the shaded region represents the energy range of Z-oP m + n carbon.

Figure 7 .
Figure 7. electronic properties of Z-oP 2+16 carbon at 0 GPa.(a) The orbital-resolved electronic band structure, (b) orbital-resolved partial density of states, (c) the electron localization function (ELF) distribution with an isosurface value of 0.75, (d) the band decomposed charge density between EF −1 and EF +1 eV, isosurface level is 0.003 eÅ −3 .
C 11 C 22 C 33 − C 11 C 2 23 − C 22 C 2 13 − C 33 C 2 12 + 2C 12 C 13 C 23The shear modulus, bulk moduli, and Vickers hardness were computed by using the H Chen v and H Tian v models for hardness.The carbon allotropes designed in this work, as presented in table 1, all estimated to show exceptional superhard properties.Remarkably, Z-oP 2+16 carbon exhibited a H Tian v