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A series of carbon-based superconductors XC6 with high Tc were reported recently. In this paper, based on the first-principles calculations, we studied the mechanical properties of these structures, and further explored the XC12 phases, where the X atoms are from elemental hydrogen to calcium, except noble gas atoms. The mechanically- and dynamically-stable structures include HC6, NC6, and SC6 in XC6 phases, and BC12, CC12, PC12, SC12, ClC12, and KC12 in XC12 phases. The doping leads to a weakening in mechanical properties and an increase in the elastic anisotropy. C6 has the lowest elastic anisotropy, and the anisotropy increases with the atomic number of doping atoms for both XC6 and XC12. Furthermore, the acoustic velocities, Debye temperatures, and the electronic properties are also studied.
Elemental carbon exhibits a rich diversity of structures and properties, due to its flexible bond hybridization. A large number of stable or metastable phases of the pure carbon, including the most commonly known, graphite and diamond, and other various carbon allotropes [1,2,3,4] (such as lonsdaleite, fullerene, and graphene, etc.), and diversified carbides [5,6,7,8,9,10,11], have been studied in experiments and theoretical calculations. Graphite, which is the most stable phase at low pressure, has a sp2-hybridized framework and is ultrasoft semimetallic, whereas diamond, stable at high pressure, is superhard, insulating with a sp3 network. Recently, a novel one-dimensional metastable allotrope of carbon with a finite length was first synthesized by Pan et al. , called Carbyne. It has a sp-hybridized network and shows a strong purple-blue fluorescence. The successful synthesis of Carbyne is a great promotion for the further analysis on properties and applications. The 2D material MXenes as a promising electrode material, which is early transition metal carbides and carbon nitrides, is reported , owing to its metallic conductivity and hydrophilic nature. These properties of different carbides are appealing. To find superhard superconductors, researches designed some carbide superconductors, such as boron carbides and XC6 structure with cubic symmetry. The diamond-like BxCy system, which is superhard and superconductive, has also attracted much interest [5,6,7,8,9,10]. The best simulated structure of the synthesized d-BC3 (Pmma-b phase) has a Vickers hardness of 64.8 GPa, showing a superhard nature, and its Tc reaches 4.9–8.8 K . The P-4m2 polymorph of d-BC7 with a low energy also has a high Vickers hardness of 75.2 GPa . Furthermore, Wang et al.  explored more potential superhard structures of boron carbide, uncovering the stability is mainly contributed by the elemental boron at low pressure, and by the carbon at high pressure. The novel metastable carbon structure C6 bcc is predicted with a cubic symmetry . It is an indirect band gap semiconductor with 2.5 eV, calculated by the local density approximation. Recently, doped with simple metals, Lu et al.  studied a series of sodalite-based carbon structures, similar to the boron-doped diamond. Although they found these structures are all metastable, some of these structures show a superconductivity, e.g., the critical temperature of NaC6 is 116 K. In this paper, we mainly study the mechanical properties of these eleven XC6 phases (HC6, LiC6, NC6, OC6, FC6, NaC6, AlC6, SiC6, PC6, SC6, and ClC6) which is of dynamical stability and, for comparison, C6 is also calculated. In addition, the XC12 structures are systematically explored, in which the X atom is from H to Ca, except He, Ne, and Ar. The doping-induced changes in elastic constant, modulus, the anisotropy of elasticity and acoustic velocity, Debye temperature, and the electronic structures are also studied.
As shown in Figure 1a, the structure of XC6 is obtained by doping the X atom into the C6 bcc structure at (0, 0, 0). It is of Im-3m symmetry (No. 229), consisting of two formula units (f.u.) per unit cell. Each C atom has four nearest neighbors with the bond angle of 90° or 120°. The XC6 structure has four C4 rings and eight C6 rings. In Table 1, the calculated lattice parameter a of C6 has a good agreement with the available result , and is smaller than that of the XC6 structures. By removing the corner atoms and only leaving the center X atom, the XC12 structure is obtained (Figure 1b). All of the XC12 phases are smaller than the corresponding XC6 phases, but larger than the C6 phase in the lattice parameter.
The formation enthalpies of XC6 in  and XC12 structures are calculated reference to diamond and the most stable X phase at ambient pressure. The equations are given by , and , and the calculated results are shown in Figure 2. The positive values indicate these phases are metastable. The two curves of the formation enthalpy follow a similar trend, where the F-doped carbides have the lowest ΔH, and the PC6 and CC12 have the largest ΔH in XC6 and XC12, respectively. Compared to other doped elements of the second and the third periods in the XC6 and XC12, fluorine (F) possesses the largest electronegativity difference relative to C, leading to a stronger interaction between F and C atoms; thus, FC6 and FC12 phases are more stable.
The calculated elastic constants and moduli are listed in Table 1. The generalized Born’s mechanical stability criteria of cubic phase are given by : and In Table 1, the C6 and HC6, NC6, and SC6 have the mechanical stability, and they are also dynamically stable . The XC12 has ten mechanically stable phases, but only six of these phases have the dynamical stability (BC12, CC12, PC12, SC12, ClC12, and KC12) due to the absence of the imaginary frequency in the whole Brillouin zone (see Figure 3 and Figure 4). The S is the only element that is capable to make not only XC6, but also XC12, stable.
By Voigt-Reuss-Hill approximations [16,17,18], the bulk modulus B and shear modulus G can be obtained, and the Young’s modulus E and Poisson’s ratio ν are defined as [19,20] and HC6 has the largest bulk modulus of 346 GPa, showing the best ability to resist the compression. The shear modulus is often used to qualitatively predict the hardness, and Young’s modulus E is defined as the ratio between stress and strain to measure the stiffness of a solid material. In Table 1, C6 is the largest in shear modulus and Young’s modulus, which means that doping leads to a weakening in mechanical properties. The Poisson’s ratio exhibits the plasticity; usually, the larger the value, the better the plasticity. According to Pugh , C6, HC6, BC12, CC12, and PC12 are brittle materials (B/G < 1.75), while NC6, SC6, SC12, ClC12, and KC12 are ductile materials (B/G > 1.75). This conforms the calculated results of Poisson’s ratio.
The elastic anisotropy is important for the analysis on the mechanical property and, thus, the universal elastic anisotropy index (AU), Zener anisotropy index (A), and the percentage anisotropy in compressibility and shear are calculated. For the cubic phase, the universal elastic anisotropy index  is defined as: , the nonzero value suggests an anisotropy characteristic. Furthermore, it is known that C44 represents the resistance to deformation with respect to a shear stress applied across the (100) plane in the  direction, and represents the resistance to shear deformation by a shear stress applied across the (110) plane in the direction. For an isotropic crystal, the two shear resistances turn to identical. Therefore, Zener  introduced to quantify the extension of anisotropy. The value of 1.0 represents the isotropy, and any deviation from 1.0 indicates the degree of the shear anisotropy. The percentage anisotropy in compressibility and shear are given by: and . The AB is always 0.0 for a cubic phase. As shown in Table 2, C6 has the lowest anisotropy. The universal elastic anisotropy index and the percentage anisotropy in shear is increasing with the atomic number of doped element for both XC6 and XC12, and the anisotropy which obtains from the shear anisotropic factor is also increasing, except SC6 and KC12. Furthermore, owing to the percentage anisotropy in shear of C6, BC12, and CC12 being slight, they are almost isotropic.
The elastic anisotropies are calculated with the elastics anisotropy measures (ElAM) code [25,26] which makes the representations of non-isotropic materials easy and visual. For the cubic phase, the representation in xy, xz, and yz planes are identical, as a result, only the xy plane is presented. The 2D figures of the differences in each direction of Poisson’s ratio are shown in Figure 5. The maximum value curves and minimum positive value curves of C6 and XC6 stable phases are illustrated in Figure 5a,b, and those of XC12 stable phases are shown in Figure 5c,d. Particularly, the SC12 and ClC12 have the negative minimum Poisson’s ratio. It is seen that all of the structures are anisotropic and C6 has the lowest anisotropy, suggesting the doping increase the elastic anisotropy. The largest value of maximum curve is in the same direction of the lowest value of minimum positive value curve for each structure. Furthermore, for XC12 phases, the anisotropy of Poisson’s ratio is increasing with the atomic number. The negative minimum Poisson’s ratio of SC12 and ClC12 indicate these two phases have auxeticity , and ClC12 is more prominent than SC12.
The directional dependence of the Young’s modulus  are demonstrated in Figure 6 and Figure 7. The distance from the origin of system of coordinate to the surface equals the Young’s modulus in this direction, and thus any departure from the sphere indicates the anisotropy. As shown, all of the phases are anisotropic, and the anisotropy of Young’s modulus is increasing with the doping atomic number. For the S-doped phases, which have stable XC6 and XC12 structures, the maximum (minimum) values of SC6 and SC12 are 650 (291) and 371 (175) GPa, respectively. The Emax/Emin ratio of SC6 (2.23) is slightly larger than that of SC12 (2.12), indicating the SC6 is more anisotropic.
The acoustic velocity is a fundamental parameter to measure the chemical bonding characteristics, and it is determined by the symmetry of the crystal and propagation direction. Brugger  provided an efficient procedure to calculate the phase velocities of pure transverse and longitudinal modes from the single crystal elastic constants. The cubic structure only has three directions , , and  for the pure transverse and longitudinal modes and other directions are for the qusi-transverse and qusi-longitudinal waves. The acoustic velocities of a cubic phase in the principal directions are :
where ρ is the density of the structure, vl is the longitudinal acoustic velocity, and vt1 and vt2 refer the first transverse mode and the second transverse mode, respectively. It should be noted that there is a misprint for equation of in . Here, the correct expression is given. Based on the elastic constants, the anisotropic properties of acoustic velocities indicate the elastic anisotropy in these crystals. As a fundamental physical parameter which correlates with many physical properties of solids, the Debye temperature can be obtained from the average acoustic velocity: , where h and kB are the Planck and Boltzmann constants, respectively; NA is Avogadro’s number; n is the total number of atoms in the formula unit; M is the mean molecular weight, and is the density. The average acoustic velocity is , where is the average longitudinal acoustic velocity, and is the average transverse acoustic velocity.
All of the calculated acoustic velocities and Debye temperatures of diamond and stable XC6 and XC12 phases are shown in Table 3. Diamond is larger than C6 and doped structures in anisotropic and average acoustic velocity. The densities are increasing and the average acoustic velocities are decreasing with the atomic number, except NC6, which has a much smaller shear modulus. Compared to C6, the doping results in a decrease in the average acoustic velocity and Debye temperature. For the element S, which makes both XC6 and XC12 phases stable, the average acoustic velocity of SC6 decreases by 38.65% than C6, and that of SC12 by 35.96%. Furthermore, it can be found that the Debye temperature is decreasing with the atomic number, except SC6. The ΘD characterizes the strength of the covalent bond in solids, so the strength of the covalent bond is lower for the phase which has the larger atomic number of doping atom.
Figure 8 shows the electronic band structure and density of state (DOS) of XC12 stable phases. The dash line represents the Fermi level (EF). The electronic properties of XC6 have been studied in . For XC12, all of the band structures cross the Fermi level in the Brillouin zone, showing the metallic nature. The conduction band and valence band are mainly characterized by the contributions of C-p states, whereas the DOS near the Fermi level originated from the p orbital electrons of the doped element, except the ClC12 and KC12.
The calculations are performed with the first-principles calculations. The structural optimizations are using the density functional theory (DFT) [31,32] with the generalized gradient approximation (GGA), which is parameterized by Perdew, Burke, and Ernzerrof (PBE) . The Broyden-Fletcher-Goldfarb-Shanno (BFGS) minimization scheme  was used in the geometry optimization, and the total energy convergence tests are within 1 meV/atom. When the total energy is 5.0 × 10−6 eV/atom, the maximum ionic Hellmann-Feynman force is 0.01 eV/Å, the maximum stress is 0.02 GPa and the maximum ionic displacement is 5.0 × 10−4 Å, the structural relaxation will stop. The energy cutoff is 400 eV, and the K-points separation is 0.02 Å−1 in the Brillouin zone.
By using the first-principles calculations, the analyses on the mechanical properties of XC6 and the further exploration of XC12 structures are given. The formation enthalpies of dynamically stable XC6 phases and all of the XC12 structures, and the elastic constants, are calculated. There are ten structures which have the mechanical and dynamical stability (C6, HC6, NC6, SC6, BC12, CC12, PC12, SC12, ClC12, and KC12). The elastic modulus and anisotropy of the ten structures are studied and, in these structures, C6 has the lowest elastic anisotropy and the anisotropy increases with the atomic number. The doping leads to the weakening in mechanical properties and the increase in the elastic anisotropy. In addition, Debye temperatures and the anisotropy of acoustic velocities are also studied. The electronic properties studies show the metallic characteristic for XC6 and XC12 phases.
This work was financially supported by the Natural Science Foundation of China (No. 11204007), Natural Science Basic Research plan in Shaanxi Province of China (grant No.: 2016JM1026, 20161016), and Education Committee Natural Science Foundation in Shaanxi Province of China (grant No.: 16JK1049).
Qun Wei and Meiguang Zhang designed the project; Quan Zhang and Qun Wei performed the calculations, Qun Wei and Quan Zhang prepared the manuscript, Meiguang Zhang revised the paper, all authors discussed the results and commented on the manuscript.
The authors declare no conflict of interest.