Mechanical stabilities of K4 carbon and K4-like NaC2
Highlights
► Mechanical stability of K4 carbon and K4-like NaC2 has been studied. ► K4 carbon is not a metastable phase. ► K4-like NaC2 is a metastable phase.
Introduction
Distinguished by the type of orbital hybridization, carbon can form three well known allotropes, i.e., diamond, graphite and carbine, with sp3, sp2 and sp1 hybridizations, respectively [1]. Then the bond length strongly depends on the hybridization, for example, the bond lengths are 1.42 Å and 1.54 Å for graphite sp2 and diamond sp3 bonds, respectively. Although the bond length of the graphite sp2 bond is shorter than that of the diamond sp3 bond, graphite is known to be an excellent lubricant, and diamond, in contrast, is known to be the hardest material. The reason for graphite's intrinsically soft nature is its two-dimensional structure [2]. If an sp2 bond connects into an all sp2 network, this structure may be harder even than diamond. Therefore, many researchers have tried to build an all sp2 network three-dimensional structure [2]. In 2008, based on Sunada's mathematical analysis [3], a new carbon polytype, K4 carbon, which can be regarded as a twin of cubic diamond, was theoretically predicted [2], [4], and K4 carbon has been described as an sp2-hybridized three-dimensional crystal structure [4], [5]. In addition to K4 carbon, new kinds of K4-like metal-doped crystal MC2 (M: Na and Mg) compounds, which have a SrSi2 structure [6], have also been theoretically predicted by first-principle calculations [7]. Because K4 carbon and K4-like MC2 have different bonding arrangements from those of ordinary crystals, their physical and chemical properties are of great interest. Ab initio calculation results reveal that both K4 carbon and K4-like MC2 exhibit metallic properties with respect to electrical conductivity [4], [7], which suggests that they may be used in a variety of applications in the future.
However, very recently, Yao et al. and Liang et al. investigated the structural stability of K4 carbon by analyzing its phonon-band structure [8] and by analyzing its elastic constants [9], respectively. They argued that the metallic K4 polymorphic form of carbon is unstable and should not exist at ambient pressure. Although the mechanical stability of crystal can be determined by analyzing the corresponding phonon-band structure, the results are usually dependent on the computational numerical accuracy, because low numerical accuracy can reduce to the existence of imaginary frequency modes [10]. Murrieta et al. have proposed a method [11] for investigating the mechanical stability of cubic crystals that involves analyzing the relationship between total energy and deformation [12]. To confirm the results about structural stability of K4 carbon by Yao et al. and Liang et al., and further determine the mechanical stability of K4 carbon and K4-like NaC2, in this work, we use Murrieta's method to investigate the mechanical stabilities of K4 carbon and K4-like NaC2. The total energy is calculated as a function of isotropic deformations and volume-conserving tetragonal and trigonal deformations. Our first-principle computational results indicate that K4 carbon is mechanically unstable, but that K4-like NaC2 is mechanically stable.
Section snippets
Computational method
As is known, the mechanical stabilities of crystal structures can be estimated from their elastic constants. A cubic crystal has three independent elastic moduli: the bulk modulus B=(C11+2C12)/3 and two shear moduli, G′=(C11C12)/2 and G=C44. In this way, the mechanical stability imposes restrictions on the elastic constants, which for cubic crystals are [12]
In other words, for a cubic lattice to be mechanically stable, the total energy surface should have a
Results and discussion
The lattice parameters and internal coordinates of K4 carbon and K4-like NaC2 were optimized by utilizing the crystallographic data of these compounds reported in Refs. [4], [7] for the initial configurations. Table 1 shows the optimized lattice parameters and the corresponding atomic positions and heat of formation. The total energies as a function of isotropic, tetragonal, and trigonal deformations were calculated for K4 carbon and K4-like NaC2 by setting the atomic fractional coordinates to
Conclusions
In summary, the mechanical stabilities of K4 carbon and K4-like NaC2 have been studied by calculating the total energy as a function of isotropic, tetragonal, and trigonal deformations. Our calculations indicate that the total energy of K4 carbon has a minimum for isotropic and trigonal deformations, but that it exhibits maxima for tetragonal deformation. For K4-like NaC2, the total energy has a minimum under all three deformations (i.e., isotropic deformations and volume-conserving tetragonal
Acknowledgments
The authors acknowledge financial support from the Japan Science and Technology Agency—Core Research for Evolutional Science and Technology (JST—CREST) program and a Grant-in-Aid for Scientific Research (KAKENHI; Grant no. 21656203). They acknowledge the staff of the Center for Computational Materials Science, Institute for Materials Research, Tohoku University for assistance with using the computing facilities.
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