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**|**Materials (Basel)**|**v.9(9); 2016 September**|**PMC5457092

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Materials (Basel). 2016 September; 9(9): 726.

Published online 2016 August 25. doi: 10.3390/ma9090726

PMCID: PMC5457092

Martin O. Steinhauser, Academic Editor

Received 2016 July 14; Accepted 2016 August 22.

Copyright © 2016 by the authors;

Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

A series of carbon-based superconductors XC_{6} with high *T _{c}* were reported recently. In this paper, based on the first-principles calculations, we studied the mechanical properties of these structures, and further explored the XC

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 *sp*^{2}-hybridized framework and is ultrasoft semimetallic, whereas diamond, stable at high pressure, is superhard, insulating with a *sp*^{3} network. Recently, a novel one-dimensional metastable allotrope of carbon with a finite length was first synthesized by Pan et al. [1], 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 [11], 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 XC_{6} structure with cubic symmetry. The diamond-like B_{x}C_{y} 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-BC_{3} (Pmma-b phase) has a Vickers hardness of 64.8 GPa, showing a superhard nature, and its *T _{c}* reaches 4.9–8.8 K [5]. The P-4m2 polymorph of d-BC

As shown in Figure 1a, the structure of XC_{6} is obtained by doping the X atom into the C_{6} 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 XC_{6} structure has four C_{4} rings and eight C_{6} rings. In Table 1, the calculated lattice parameter *a* of C_{6} has a good agreement with the available result [12], and is smaller than that of the XC_{6} structures. By removing the corner atoms and only leaving the center X atom, the XC_{12} structure is obtained (Figure 1b). All of the XC_{12} phases are smaller than the corresponding XC_{6} phases, but larger than the C_{6} phase in the lattice parameter.

Unit cell of XC_{6} (**a**) and XC_{12} (**b**). The black and blue spheres represent C and X atoms, respectively.

Calculated lattice parameter *a*, elastic constants *C*_{ij} (GPa), mechanical stability, bulk modulus *B* (GPa), shear modulus *G* (GPa), Young’s modulus *E* (GPa), Poisson’s ratio ν, and *B*/*G* ratio.

The formation enthalpies of XC_{6} in [13] and XC_{12} structures are calculated reference to diamond and the most stable X phase at ambient pressure. The equations are given by $\Delta {H}_{X{C}_{6}}=({H}_{X{C}_{6}}-{H}_{X}-6{H}_{C})/7$, and $\Delta {H}_{X{C}_{12}}=({H}_{X{C}_{12}}-{H}_{X}-12{H}_{C})/13$, 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 Δ*H*, and the PC_{6} and CC_{12} have the largest Δ*H* in XC_{6} and XC_{12}, respectively. Compared to other doped elements of the second and the third periods in the XC_{6} and XC_{12}, fluorine (F) possesses the largest electronegativity difference relative to C, leading to a stronger interaction between F and C atoms; thus, FC_{6} and FC_{12} 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 [15]: ${C}_{11}>0,{C}_{44}>0,\text{}{C}_{11}\left|{C}_{12}\right|,$ and $({C}_{11}+2{C}_{12})>0.$ In Table 1, the C_{6} and HC_{6}, NC_{6}, and SC_{6} have the mechanical stability, and they are also dynamically stable [13]. The XC_{12} has ten mechanically stable phases, but only six of these phases have the dynamical stability (BC_{12}, CC_{12}, PC_{12}, SC_{12}, ClC_{12}, and KC_{12}) 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 XC_{6}, but also XC_{12}, stable.

Phonon spectra of dynamically stable phases (**a**) BC_{12}; (**b**) CC_{12}; (**c**) PC_{12}; (**d**) SC_{12}; (**e**) ClC_{12}; and (**f**) KC_{12}.

Phonon spectra of dynamically unstable phases (**a**) LiC_{12}; (**b**) BeC_{12}; (**c**) MgC_{12}; and (**d**) AlC_{12}.

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] $E=9BG/(3B+G)$ and $\mathsf{\nu}=(3B-2G)/[2(3B+G)].$ HC_{6} 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, C_{6} 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 [21], C_{6}, HC_{6}, BC_{12}, CC_{12}, and PC_{12} are brittle materials (*B*/*G* < 1.75), while NC_{6}, SC_{6}, SC_{12}, ClC_{12}, and KC_{12} 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 (*A ^{U}*), Zener anisotropy index (

Universal elastic anisotropy index (*A*^{U}), Zener anisotropy index (*A*), and percentage anisotropy in shear (*A*_{G}).

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 C_{6} and XC_{6} stable phases are illustrated in Figure 5a,b, and those of XC_{12} stable phases are shown in Figure 5c,d. Particularly, the SC_{12} and ClC_{12} have the negative minimum Poisson’s ratio. It is seen that all of the structures are anisotropic and C_{6} 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 XC_{12} phases, the anisotropy of Poisson’s ratio is increasing with the atomic number. The negative minimum Poisson’s ratio of SC_{12} and ClC_{12} indicate these two phases have auxeticity [27], and ClC_{12} is more prominent than SC_{12}.

2D representations of Poisson’s ratio. (**a**) Maximum of C_{6} and XC_{6} stable phases; (**b**) minimum positive of C_{6} and XC_{6} stable phases; (**c**) maximum of XC_{12} stable phases; and (**d**) minimum positive and minimum negative of XC_{12} stable phases; particularly, **...**

The directional dependence of the Young’s modulus [28] 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 XC_{6} and XC_{12} structures, the maximum (minimum) values of SC_{6} and SC_{12} are 650 (291) and 371 (175) GPa, respectively. The *E*_{max}/*E*_{min} ratio of SC_{6} (2.23) is slightly larger than that of SC_{12} (2.12), indicating the SC_{6} is more anisotropic.

Directional dependence of the Young’s modulus of BC_{12} (**a**); CC_{12} (**b**); PC_{12} (**c**); SC_{12} (**d**); ClC_{12} (**e**); and KC_{12} (**f**).

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 [29] 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 [001], [110], and [111] 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 [30]:

- for [100], ${v}_{l}=\sqrt{{C}_{11}/\mathsf{\rho}},\text{}[010]{v}_{t1}=[001]{v}_{t2}=\sqrt{{C}_{44}/\mathsf{\rho}},$
- for [110], ${v}_{l}=\sqrt{({C}_{11}+{C}_{12}+2{C}_{44})/2\mathsf{\rho}},\text{}[1\overline{1}0]{v}_{t1}=\sqrt{({C}_{11}-{C}_{12})/2\mathsf{\rho}},\text{}[001]{v}_{t2}=\sqrt{{C}_{44}/\mathsf{\rho}},$
- for [111], ${v}_{l}=\sqrt{({C}_{11}+2{C}_{12}+4{C}_{44})/3\mathsf{\rho}},\text{}[11\overline{2}]{v}_{t1}={v}_{t2}=\sqrt{({C}_{11}-{C}_{12}+{C}_{44})/3\mathsf{\rho}}.$

where ρ is the density of the structure, *v _{l}* is the longitudinal acoustic velocity, and

All of the calculated acoustic velocities and Debye temperatures of diamond and stable XC_{6} and XC_{12} phases are shown in Table 3. Diamond is larger than C_{6} 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 NC_{6}, which has a much smaller shear modulus. Compared to C_{6}, the doping results in a decrease in the average acoustic velocity and Debye temperature. For the element S, which makes both XC_{6} and XC_{12} phases stable, the average acoustic velocity of SC_{6} decreases by 38.65% than C_{6}, and that of SC_{12} by 35.96%. Furthermore, it can be found that the Debye temperature is decreasing with the atomic number, except SC_{6}. The *Θ*_{D} 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 XC_{12} stable phases. The dash line represents the Fermi level (*E _{F}*). The electronic properties of XC

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) [33]. The Broyden-Fletcher-Goldfarb-Shanno (BFGS) minimization scheme [34] 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 XC_{6} and the further exploration of XC_{12} structures are given. The formation enthalpies of dynamically stable XC_{6} phases and all of the XC_{12} structures, and the elastic constants, are calculated. There are ten structures which have the mechanical and dynamical stability (C_{6}, HC_{6}, NC_{6}, SC_{6}, BC_{12}, CC_{12}, PC_{12}, SC_{12}, ClC_{12}, and KC_{12}). The elastic modulus and anisotropy of the ten structures are studied and, in these structures, C_{6} 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 XC_{6} and XC_{12} 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).

Author Contributions

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.

Conflicts of Interest

The authors declare no conflict of interest.

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