- About this Journal ·
- Abstracting and Indexing ·
- Advance Access ·
- Aims and Scope ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

Advances in Condensed Matter Physics

Volume 2013 (2013), Article ID 136274, 7 pages

http://dx.doi.org/10.1155/2013/136274

## Structural, Elastic, and Electronic Properties of Antiperovskite Chromium-Based Carbides ACCr_{3} (A = Al and Ga)

^{1}Key Laboratory of Materials Physics, Institute of Solid State Physics, Chinese Academy of Sciences, Hefei 230031, China^{2}High Magnetic Field Laboratory, Chinese Academy of Sciences, Hefei 230031, China

Received 29 October 2012; Accepted 24 December 2012

Academic Editor: Laifeng Li

Copyright © 2013 D. F. Shao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We theoretically investigated antiperovskite chromium-based carbides ACCr_{3} through the first-principles calculation based on density functional theory (DFT). The structure optimization shows that the lattice parameter of ACCr_{3} is basically proportional to the radius of A-site elements. The calculated formation energies show that AlCCr_{3} and GaCCr_{3}
can be synthesized at ambient pressure and are stable with nonmagnetic ground states. Based on the calculation of elastic constants, some elastic, mechanical, and thermal parameters are derived and discussed. AlCCr_{3} and GaCCr_{3} show ductile natures and may have similar thermal properties. From the analysis of the electronic structures, it was found that there are electron and hole bands that cross the Fermi level for AlCCr_{3} and GaCCr_{3}, indicating multiple-band natures. The Fermi level locates at the vicinity of the density of states (DOSs) peak, which leads to a large DOS at Fermi level dominated by Cr-3d electrons. The band structures of AlCCr_{3} and GaCCr_{3} are very similar to those of the superconducting antiperovskite MgCNi_{3}. The similarity may make AlCCr_{3} and GaCCr_{3} behave superconductively, which needs to be further investigated in theoretical and experimental studies.

#### 1. Introduction

Recently, antiperovskite compounds AXM_{3} (A, main group elements; X, carbon, boron, or nitrogen; M, transition metal) have attracted considerable attention. Due to the high concentration of transition metals in a cell, it can be deduced that interesting properties will be found in the family of compounds. In the antiperovskites family, nickel-based and manganese-based antiperovskites were extensively studied. Abundant physical properties were shown in the two kinds of compounds, such as superconductivity [1–3], giant magnetoresistance (MR) [4, 5], large negative magnetocaloric effect (MCE) [6, 7], giant negative thermal expansion [8, 9], magnetostriction [10], and nearly zero temperature coefficient of resistivity [11, 12]. But there are only a few reports about other 3d-metal-based antiperovskites so far. The difficulty restricting researchers is the exploration of new materials that can be experimentally synthesized. Therefore, theoretical investigations on these potential 3d-metal-based antiperovskites are useful to find the easily prepared stable materials and explore the new physical properties.

In the earlier years, researchers have found that in chromium compounds there are varieties of interesting physical properties. Many of chromium alloys such as Cr-Ru, Cr-Rh, and Cr-Ir alloys show superconductivity [13]. And it was found there is spin density wave antiferromagnetism that coexists with superconductivity in [14], [15], and (, 0.03, 0.06, and 0.10) [16]. For chromium-based antiperovskites, Wiendlocha et al. [17] discussed the possibility of superconductivity in GaNCr_{3} and RhNCr_{3}, and recent phonon and electron-phonon coupling calculation of RhNCr_{3} [18] supports Wiendlocha et al.’s prediction. In the present work, we theoretically investigated antiperovskite chromium-based carbides ACCr_{3} through the first-principles calculation based on density functional theory (DFT). The optimized lattice parameter of ACCr_{3} is basically proportional to the radius of A-site elements. From the analysis of formation energies, we predict that only AlCCr_{3} and GaCCr_{3} can be synthesized at ambient pressure, and they are stable with nonmagnetic ground states. The elastic and electronic properties of the two compounds are specifically discussed. AlCCr_{3} and GaCCr_{3} show ductile natures and may have similar thermal properties. The electron and hole bands cross the Fermi level, implying the multiple-band nature of AlCCr_{3} and GaCCr_{3}. The Fermi level locates at the vicinity of the density of states (DOS) peak, which leads to a large DOS at Fermi level dominated by Cr-3d electrons. The bands properties of AlCCr_{3} and GaCCr_{3} are very similar to those of superconducting antiperovskite MgCNi_{3}. The similarity may make AlCCr_{3} and GaCCr_{3} show superconductivity.

#### 2. Computational Details

The calculations were performed by projected augmented-wave (PAW) [19, 20] method using the ABINIT code [21–23]. The PAW method can lead to very accurate results comparable to other all-electron methods. For the exchange-correlation functional, the generalized gradient approximation (GGA) according to the Perdew-Burke-Ernzerhof [24] parametrization was used. Electronic wavefunctions are expanded with plane waves up to an energy cutoff of 1200 eV. Brillouin zone sampling is performed on the Monkhorst-Pack (MP) mesh [25] of . The self-consistent calculations were considered to be converged when the total energy of the system was stable within Ha. Nonmagnetic, ferromagnetic (FM), and antiferromagnetic (AFM) states were tested in the study. For AFM states, we only considered the simplest case: the spins of Cr atom in [111] layers are parallel with each other in the same layer and are antiparallel with spins in the neighboring layers.

The elastic constants are evaluated according to the finite-strain continuum elasticity theory [26, 27]. The strain energy per unit mass can be expressed as a polynomial of the strain parameter [28]: where the coefficients and are combinations of second- and third-order elastic constants of the crystal, respectively. For a cubic structure, there are three independent second-order elastic constants (, , and ). Therefore, here we introduced 3 Lagrangian strain tensors labeled as in terms of . and the corresponding coefficients are listed in Table 1.

Once the single-crystal elastic constants are calculated, the related elastic and mechanic parameters may also be evaluated. Using the Voigt-Reuss-Hill approximation [29], bulk modulus and shear modulus of cubic crystal can be defined as where the subscripts and denote the Voigt, and Reuss average, respectively. In (3), Thus Young’s modulus and Poisson’s ratio can be obtained using the relations

Using the calculated bulk modulus , shear modulus , and Young’s modulus , the Debye temperature can be determined as where , , and are Planck’s, Boltzmann’s constants, and Avogadro’s number, respectively. is the mass density, is the molecular weight, and is the number of atoms in the unit cell. The mean sound velocity is defined as where and are the longitudinal and transverse sound velocities, which can be obtained from bulk modulus and shear modulus :

#### 3. Results and Discussion

##### 3.1. Ground State and Formation Energy

The structures were fully optimized with respect to lattice parameter and atomic positions. The optimized lattice parameters of ACCr_{3} for different A-site elements are listed in Table 2. The results show that the lattice parameter of ACCr_{3} is basically proportional to the radius of A-site elements (see Figure 1).

The common technique for producing antiperovskite carbides is the solid-state synthesis from the parent materials. Therefore we assumed that the antiperovskite chromium-based carbides ACCr_{3} could be synthesized in the same routine. We analyze the stability of compounds by calculating the formation energies , which is defined as the difference between the total energies of initial reagents (elements) and final compounds:
Here, , , and are the total energies of the initial reagents.

According to the definition of , negative values indicate that the formation of compounds from the initial components is energetically favorable and the compounds are stable with respect to a mixture of the initial elements. Conversely, if , the compounds are unlikely to be synthesized under normal conditions.

The calculated for the series of A-site elements with different magnetic configurations are listed in Table 2. Except for AlCCr_{3} and GaCCr_{3}, all other compounds show positive . For AlCCr_{3} and GaCCr_{3}, the values of for different magnetic configurations are very close, and the magnetic moments per Cr atom for the FM and AFM states are very small, which implies the nonmagnetic ground states of AlCCr_{3} and GaCCr_{3}. From the analysis of formation energies, it seems that ACCr_{3} can be synthesized only when A = Al and Ga. The elastic and electronic properties of the two compounds will be discussed later.

##### 3.2. Elastic and Mechanical Properties

In order to obtain accurate elastic constants, is varied with a step of 0.0025 in every case for . As an example, the strain energies for AlCCr_{3} and the fitted polynomials are shown in Figure 2. The calculated elastic constants at ambient pressure and other derived mechanical parameters are listed in Table 3. From the calculated values of the elastic constants, one can find that they satisfy the mechanical stability criteria for a cubic crystal, that is, , , and [30], which indicates that the compounds are mechanically stable.

Cracks in crystals are directly related to the anisotropy of thermal and elastic properties [31]. The elastic anisotropy factor of cubic crystal can be described as
For a complete isotropic system, the anisotropy factor takes the value of unity, and the deviation from unity measures the degree of elastic anisotropy [32]. The elastic constant represents the shear resistance in a [100] direction, which is related to the C–Cr bonding. The values of for the two compounds are almost the same, which implies the similar C–Cr bonding nature of the two compounds. turns out to be the stiffness associated with a shear in a [110] direction, which can be connected with the bonding between A–Cr bonding. Although A–M bonding is usually very weak in antiperovskite ACM_{3}, it still can influence the anisotropy of the material. The anisotropic factors are listed in Table 3. The different A–Cr bondings of the two compounds lead to different anisotropy. For AlCCr_{3}, indicates that AlCCr_{3} is almost isotropic. However, GaCCr_{3} seems like more anisotropic ().

Young’s modulus defines the ratio between the linear stress and strain. The larger the value of , the stiffer is the material. In general, as Young’s modulus increases, the covalent nature of the compound also increases, which further has an impact on the ductility of the compounds. The values of for AlCCr_{3} and GaCCr_{3} are very close to each other, suggesting the similar covalent nature of the two compounds.

The Cauchy’s pressure is defined as the difference between the two particular elastic constants . It is considered to serve as an indication of ductility: suggests that the material is expected to be ductile; otherwise, the material is expected to be brittle [33]. As shown in Table 3, the values of for AlCCr_{3} and GaCCr_{3} are positive, which is a clear indication for the compounds to be ductile. Another index of ductility is the ratio . According to Pugh [34], the ductile/brittle properties of materials could be related empirically to the ratio . If , the material would be ductile; otherwise the material behaves in a brittle manner. For ACCr_{3} (A = Al and Ga), all the calculated ratios are much larger than 1.75, which clearly highlights the ductile nature of ACCr_{3} (A = Al and Ga). Poisson’s ratio generally quantifies the stability of the crystal against shear and takes the value between −1 and 0.5, which are the lower and the upper bounds. The lower bound is where the material does not change its shape, and the upper bound is where the volume remains unchanged. The calculated of ACCr_{3} (A = Al and Ga), listed in Table 3, are very close to the typical value of 0.33 for ductile metallic materials [35]. All the parameters clearly show the ductility of ACCr_{3} (A = Al and Ga).

Using bulk modulus , shear modulus , and Young’s modulus , the Debye temperature can be calculated. As a matter of fact, a higher would imply a higher thermal conductivity associated with the material [36]. For the present calculation, the are estimated to be 532.43 K and 497.12 K for AlCCr_{3} and GaCCr_{3}, respectively. The similar values of suggest the similar thermal characteristics of the two compounds.

##### 3.3. Electronic Properties

The calculated electronic band structures along the high symmetry directions in the Brillouin zones together with the total and site-projected -decomposed DOS at equilibrium lattice parameters for ACCr_{3} (A = Al and Ga) are shown in Figures 3 and 4. The band structures of the two compounds are very similar. Bands from −13 eV to −10 eV are mainly from the C-2s and Cr-3d states with strong hybridizations characteristic. From −8 eV to 2 eV, there are hybridizations between C-2p and Cr-3d states, and the Cr-3d states contribute dominantly to the bands, which suggests the itinerant nature of Cr-3d electrons. From Figure 4, it can be seen the Fermi level locates at the vicinity of the DOS peak, which leads large DOS at the Fermi level with values of 4.79 states/eV for AlCCr_{3} and 5.62 states/eV for GaCCr_{3}.

The Fermi surfaces of the two compounds are similar (see Figure 5). There are four bands that cross the Fermi level. Hole pockets surrounding point come from the lower two bands (denoted with red and green colors in Figure 3). Electron pockets surrounding R point come from the upper two bands (denoted with blue and purple colors in Figure 3). The differences of the Fermi surface of the two compounds come from the band denoted with blue color in Figure 3. It crosses the Fermi level at the vicinity of M point for AlCCr_{3}, which makes the electron pockets surround the corners of the Brillouin zones connected to each other at M point. However, for GaCCr_{3}, it crosses the Fermi level in the place between and M points, which forms a cubic cage-like electron pocket surrounding point and connecting with the corner-centered electron pockets. The presence of electron and hole bands crossing the Fermi level indicates the multiple-band natures for AlCCr_{3} and GaCCr_{3}.

In order to understand the bonding nature among the atoms in AlCCr_{3} and GaCCr_{3}, we plot the contour maps of the charge density of the two compounds in Figure 6. The C–Cr bonds are very strong, which coincides with the strong hybridization between C-2p and Cr-3d electrons shown in DOS figures. The similarity of the bonding nature for the two compounds coincides with the similar Young’s modulus discussed before.

The electronic properties of AlCCr_{3} and GaCCr_{3} are very similar to those of the superconducting antiperovskite MgCNi_{3} [37], for which the Fermi level locates at the vicinity of the DOS peak as well, and there is multiple-band nature present. The similarity possibly makes AlCCr_{3} and GaCCr_{3} potential superconductors. According to McMillan’s coupling theory [38], the electron-phonon coupling constant can be calculated from the following expression , where is the atomic mass and is the averaged phonon frequency and can be approximated as . For superconducting MgCNi_{3}, there is a peak (a van Hove singularity) in DOS just below Fermi level, which leads to a large . The is not large enough to lead magnetic instability, but it can lead to a large electron-phonon coupling constant [39]. To the best of our knowledge, except for the superconducting nickel antiperovskites, there are no other antiperovskites that have such a nature. Wiendlocha et al. [17] pointed out that in chromium systems, chromium atoms have very large values of the McMillan-Hopfield parameters [38], which may lead to a very strong electron-phonon coupling. The recently reported work about RhNCr_{3} [18] supports Wiendlocha et al.’s prediction. The phonon and electron-phonon coupling calculations show that RhNCr_{3}, which has a large , is a strong coupling superconductor with Tc above 16 K. Therefore, we consider that there is possibility for superconductivity appearing in AlCCr_{3} and GaCCr_{3}. Phonon and electron-phonon coupling calculations will be carried out in the future to confirm the possibility.

#### 4. Conclusion

In conclusion, we theoretically investigated the antiperovskite chromium-based carbides ACCr_{3} through the first-principles calculation based on density functional theory. The optimized lattice parameter of ACCr_{3} is basically proportional to the radius of A-site elements. Only AlCCr_{3} and GaCCr_{3} have negative formation energies, implying that the two compounds can be synthesized at ambient pressure and are stable with nonmagnetic ground states. AlCCr_{3} and GaCCr_{3} show ductile natures, and they may have similar thermal properties. Similar to superconducting antiperovskite MgCNi_{3}, there are electron and hole bands that cross the Fermi level for AlCCr_{3} and GaCCr_{3}, indicating multiple-band natures. The Fermi level locates at the vicinity of the DOS peak, which leads to a large dominated by Cr-3d electrons. These similarities possibly make AlCCr_{3} and GaCCr_{3} show superconductivity, which needs to be further investigated from theoretical and experimental studies.

#### Acknowledgments

This work was supported by the National Key Basic Research under Contract no. 2011CBA00111 and the National Nature Science Foundation of China under Contract nos. 91222109, 11274311, 51171177, 11174295, U1232139, and 50701042. The calculations were partially performed at the Center for Computational Science, CASHIPS.

#### References

- T. He, Q. Huang, A. P. Ramirez et al., “Superconductivity in the non-oxide perovskite MgCNi
_{3},”*Nature*, vol. 411, no. 6833, pp. 54–56, 2001. View at Publisher · View at Google Scholar · View at Scopus - M. Uehara, T. Yamazaki, T. Kôri, T. Kashida, Y. Kimishima, and I. Hase, “Superconducting properties of CdCNi
_{3},”*Journal of the Physical Society of Japan*, vol. 76, no. 3, Article ID 034714, 2007. View at Publisher · View at Google Scholar · View at Scopus - M. Uehara, A. Uehara, K. Kozawa, and Y. Kimishima, “New antiperovskite-type superconductor ZnN
_{y}Ni_{3},”*Journal of the Physical Society of Japan*, vol. 78, no. 3, 2009. View at Publisher · View at Google Scholar · View at Scopus - K. Kamishima, T. Goto, H. Nakagawa et al., “Giant magnetoresistance in the intermetallic compound Mn
_{3}GaC,”*Physical Review B*, vol. 63, no. 2, Article ID 024426, 2001. View at Scopus - Y. B. Li, W. F. Li, W. J. Feng, Y. Q. Zhang, and Z. D. Zhang, “Magnetic, transport and magnetotransport properties of Mn
_{3+x}Sn_{1-x}C and Mn_{3}Zn_{y}Sn_{1-y}C compounds,”*Physical Review B*, vol. 72, no. 2, Article ID 024411, 2005. View at Publisher · View at Google Scholar - T. Tohei, H. Wada, and T. Kanomata, “Negative magnetocaloric effect at the antiferromagnetic to ferromagnetic transition of Mn
_{3}GaC,”*Journal of Applied Physics*, vol. 94, no. 3, pp. 1800–1802, 2003. View at Publisher · View at Google Scholar · View at Scopus - M. H. Yu, L. H. Lewis, and A. R. Moodenbaugh, “Large magnetic entropy change in the metallic antiperovskite Mn
_{3}GaC,”*Journal of Applied Physics*, vol. 93, no. 12, pp. 10128–10130, 2003. View at Publisher · View at Google Scholar · View at Scopus - K. Takenaka and H. Takagi, “Giant negative thermal expansion in Ge-doped anti-perovskite manganese nitrides,”
*Applied Physics Letters*, vol. 87, no. 26, Article ID 261902, pp. 1–3, 2005. View at Publisher · View at Google Scholar · View at Scopus - K. Takenaka, K. Asano, M. Misawa, and H. Takagi, “Negative thermal expansion in Ge-free antiperovskite manganese nitrides: tin-doping effect,”
*Applied Physics Letters*, vol. 92, no. 1, Article ID 011927, 2008. View at Publisher · View at Google Scholar · View at Scopus - K. Asano, K. Koyama, and K. Takenaka, “Magnetostriction in Mn
_{3}CuN,”*Applied Physics Letters*, vol. 92, no. 16, Article ID 161909, 2008. View at Publisher · View at Google Scholar · View at Scopus - E. O. Chi, W. S. Kim, and N. H. Hur, “Nearly zero temperature coefficient of resistivity in antiperovskite compound CuNMn
_{3},”*Solid State Communications*, vol. 120, no. 7-8, pp. 307–310, 2001. View at Publisher · View at Google Scholar · View at Scopus - J. C. Lin, B. S. Wang, P. Tong et al., “Tunable temperature coefficient of resistivity in C- and Co-doped CuNMn
_{3},”*Scripta Materialia*, vol. 65, no. 5, pp. 452–455, 2011. View at Publisher · View at Google Scholar · View at Scopus - B. T. Matthias, T. H. Geballe, V. B. Compton, E. Corenzwit, and G. W. Hull, “Superconductivity of chromium alloys,”
*Physical Review*, vol. 128, no. 2, pp. 588–590, 1962. View at Publisher · View at Google Scholar · View at Scopus - Y. Nishihara, Y. Yamaguchi, T. Kohara, and M. Tokumoto, “Itinerant-electron antiferromagnetism and superconductivity in bcc Cr-Re alloys,”
*Physical Review B*, vol. 31, no. 9, pp. 5775–5781, 1985. View at Publisher · View at Google Scholar · View at Scopus - Y. Nishihara, Y. Yamaguchi, M. Tokumoto, K. Takeda, and K. Fukamichi, “Superconductivity and magnetism of bcc Cr-Ru alloys,”
*Physical Review B*, vol. 34, no. 5, pp. 3446–3449, 1986. View at Publisher · View at Google Scholar · View at Scopus - H. L. Alberts, D. S. McLachlan, T. Germishuyse, and M. Naidoo, “Superconductivity and antiferromagnetism in Cr-Mo-Ru alloys,”
*Journal of Physics*, vol. 3, no. 12, pp. 1793–1800, 1991. View at Publisher · View at Google Scholar · View at Scopus - B. Wiendlocha, J. Tobola, S. Kaprzyk, and D. Fruchart, “Electronic structure, superconductivity and magnetism study of Cr
_{3}GaN and Cr_{3}RhN,”*Journal of Alloys and Compounds*, vol. 442, no. 1-2, pp. 289–291, 2007. View at Publisher · View at Google Scholar · View at Scopus - H. M. Tütüncü and G. P. Srivastava, “Phonons and superconductivity in the cubic perovskite Cr
_{3}RhN,”*Journal of Applied Physics*, vol. 112, no. 9, Article ID 093914, 2012. View at Publisher · View at Google Scholar - P. E. Blöchl, “Projector augmented-wave method,”
*Physical Review B*, vol. 50, no. 24, pp. 17953–17979, 1994. View at Publisher · View at Google Scholar · View at Scopus - M. Torrent, F. Jollet, F. Bottin, G. Zérah, and X. Gonze, “Implementation of the projector augmented-wave method in the ABINIT code: application to the study of iron under pressure,”
*Computational Materials Science*, vol. 42, no. 2, pp. 337–351, 2008. View at Publisher · View at Google Scholar · View at Scopus - X. Gonze, J. M. Beuken, R. Caracas et al., “First-principles computation of material properties: the ABINIT software project,”
*Computational Materials Science*, vol. 25, no. 3, pp. 478–492, 2002. View at Publisher · View at Google Scholar · View at Scopus - X. Gonze, B. Amadon, P. M. Anglade et al., “ABINIT: first-principles approach to material and nanosystem properties,”
*Computer Physics Communications*, vol. 180, no. 12, pp. 2582–2615, 2009. View at Publisher · View at Google Scholar · View at Scopus - X. Gonze, G. M. Rignanese, M. Verstraete et al., “A brief introduction to the ABINIT software package,”
*Zeitschrift fur Kristallographie*, vol. 220, no. 5-6, pp. 558–562, 2005. View at Publisher · View at Google Scholar · View at Scopus - J. P. Perdew, K. Burke, and M. Ernzerhof, “Generalized gradient approximation made simple,”
*Physical Review Letters*, vol. 77, no. 18, pp. 3865–3868, 1996. View at Scopus - H. J. Monkhorst and J. D. Pack, “Special points for Brillouin-zone integrations,”
*Physical Review B*, vol. 13, no. 12, pp. 5188–5192, 1976. View at Publisher · View at Google Scholar · View at Scopus - F. Birch, “Finite elastic strain of cubic crystals,”
*Physical Review*, vol. 71, no. 11, pp. 809–824, 1947. View at Publisher · View at Google Scholar · View at Scopus - F. D. Murnaghan,
*Finite Deformation of An Elastic Solid*, Dover Publications, New York, NY, USA, 1951. - J. Zhao, J. M. Winey, and Y. M. Gupta, “First-principles calculations of second- and third-order elastic constants for single crystals of arbitrary symmetry,”
*Physical Review B*, vol. 75, no. 9, Article ID 094105, 2007. View at Publisher · View at Google Scholar · View at Scopus - R. Hill, “The elastic behaviour of a crystalline aggregate,”
*Proceedings of the Physical Society A*, vol. 65, no. 5, pp. 349–354, 1952. View at Publisher · View at Google Scholar · View at Scopus - D. C. Wallace,
*Thermodynamics of Crystals*, John Wiley & Sons, New York, NY, USA, 1972. - D. J. Green,
*An Introduction to the Mechanical Properties of Ceramics*, Cambridge University Press, Cambridge, UK, 1998. - R. E. Newnham,
*Properties of Materials; Anisotropy, Symmetry, Structure*, Oxford University Press, New York, NY, USA, 2005. - D. G. Pettifor, “Theoretical predictions of structure and related properties of intermetallics,”
*Materials Science and Technology*, vol. 8, no. 4, pp. 345–349, 1992. View at Scopus - S. Pugh, “Relations between the elastic moduli and the plastic properties of polycrystalline pure metals,”
*Philosophical Magazine Series*, vol. 7, no. 45, pp. 823–843, 1954. - J. Haines, J. M. Léger, and G. Bocquillon, “Synthesis and design of superhard materials,”
*Annual Review of Materials Research*, vol. 31, pp. 1–23, 2001. View at Publisher · View at Google Scholar - V. Kanchana, “Mechanical properties of Ti
_{3}AlX (X = C, N): Ab initio study,”*Europhysics Letters*, vol. 87, no. 2, p. 26006, 2009. - S. Mollah, “The physics of the non-oxide perovskite superconductor MgCNi
_{3},”*Journal of Physics Condensed Matter*, vol. 16, no. 43, pp. R1237–R1276, 2004. View at Publisher · View at Google Scholar · View at Scopus - W. L. McMillan, “Transition temperature of strong-coupled superconductors,”
*Physical Review*, vol. 167, no. 2, pp. 331–344, 1968. View at Publisher · View at Google Scholar · View at Scopus - J. H. Shim, S. K. Kwon, and B. I. Min, “Electronic structures of antiperovskite superconductors MgXNi
_{3}(X = B, C, and N),”*Physical Review B*, vol. 64, no. 18, Article ID 180510, 2001. View at Scopus