Gangtai Zhang^{a},
Yaru Zhao^{a},
Tingting Bai^{b},
Qun Wei^{c} and
Yuquan Yuan*^{d}
^{a}College of Physics and Optoelectronics Technology, Baoji University of Arts and Sciences, Baoji 721016, PR China
^{b}College of Mathematics and Information Science, Baoji University of Arts and Sciences, Baoji 721013, PR China
^{c}School of Physics and Optoelectronic Engineering, Xidian University, Xi'an 710071, PR China
^{d}School of Science, Sichuan University of Science and Engineering, Zigong 643000, PR China. E-mail: yuquan_yuan1975@suse.edu.cn

Received
6th May 2016
, Accepted 6th July 2016

First published on 6th July 2016

Using the first principles particle swarm optimization algorithm for crystal structural prediction, we have predicted a hexagonal P6_{3}/mmc structure of Tc_{2}C. The new phase is mechanically and dynamically stable, as verified by its elastic constants and phonon dispersion. The formation enthalpy–pressure curves show that the predicted P6_{3}/mmc-Tc_{2}C is more energetically favorable than the previously proposed Mo_{2}C-, anti-MoS_{2}-, Re_{2}P-, and Fe_{2}N-type structures in the considered pressure range. The calculated mechanical properties show that it is an ultra-incompressible and hard material. Meanwhile, the directional dependences of the Young's modulus, bulk modulus, and shear modulus for Tc_{2}C are systematically investigated. The analyses of the density of states and electronic localization function reveal the presence of strong covalent bonding between Tc and C atoms, which is of crucial importance in forming a hard material.

Carbides, especially TM carbides, are widely used for industrial application because of their high melting temperature, extreme hardness, and chemical stability, which makes them useful in cutting tools, dental drills, rock drills in mining, and abrasives. Experimentally, some monocarbides, such as TiC,^{8} ZrC,^{9} PtC,^{10} and WC^{5} have been synthesized, the obtained results show that they all hold very bulk modulus. A high bulk modulus is a good indicator of a superhard material, thus TM carbides are potential candidates for superhard materials. Re_{2}C belongs to the family of TM carbides, and some experimental and theoretical studies of Re_{2}C have been investigated.^{11–15} Meanwhile, these theoretical calculations and experimental data reveal that Re_{2}C is ultra-incompressible and has a high bulk module of about 400 GPa.

Technetium (Tc) lies directly above Re in the periodic table and has the same valence electron number, so it is worth studying the mechanical properties of its carbides. Experimentally, TcC has been first synthesized in 1962 and was presumed to be hexagonal and faced-centred cubic structures.^{16} Later, Giorgi and Szklarz interpreted the cubic compound on the authority of a body-centred cubic phase.^{17} Recently, an experimental synthesis has verified the existence of TcC,^{18} which also provides a useful information for understanding the crystal structure of TcC. Theoretically, the crystal structures and the related physical properties of TcC have been considerably investigated.^{19–25} These calculated results indicated that TcC with considered structures is an ultra-incompressible and hard material. However, the research works of the compounds with lower carbon contents are seldom reported so far.

In this paper, we have extensively investigated the ground state structure of Tc_{2}C by using the first principles particle swarm optimization algorithm (PSO) on crystal structural prediction.^{26} This method has been successfully applied to various systems,^{27–29} unbiased by any known information. A novel hexagonal P6_{3}/mmc structure is uncovered for Tc_{2}C, which is more energetically favorable than the earlier proposed Mo_{2}C-, anti-MoS_{2}-, Re_{2}P-, and Fe_{2}N-type structures in the pressure range of 0–120 GPa. First principles calculations are then performed to study the total energy, lattice parameters, phase stability, elastic properties, elastic anisotropy, and density of states for this novel hexagonal phase.

Fig. 1 Crystal structure of the P6_{3}/mmc-Tc_{2}C structure. Large and small spheres represent Tc and C atoms, respectively. |

To explore the thermodynamic stability for further experimental synthesis, the formation enthalpy of the Tc_{2}C with respect to the separate phases is examined by the reaction route ΔH = H_{Tc2C} − 2H_{Tc} − H_{c}, here ΔH is the formation enthalpy, the hexagonal Tc (space group: P6_{3}/mmc) and the graphite C are chosen as the reference phases. The calculated formation enthalpy under pressure is shown in Fig. 4. As shown in this figure, the stabilities of Tc_{2}C with different structures are gradually enhanced with increasing the pressure, therefore, the pressure is helpful to their stabilities. At ambient condition, the negative values of the formation enthalpies for the predicted P6_{3}/mmc-Tc_{2}C and anti-ReB_{2}-type structures indicate that they are thermodynamically stable, whereas the positive formation enthalpies for other structures show that they are not thermodynamically stable. Moreover, it is can be seen that the enthalpy–pressure curves of the P6_{3}/mmc- and anti-ReB_{2}-type structures merge together in our considered pressure range, which further proves that they belong to the same structure. The present calculated results suggest that the predicted P6_{3}/mmc-Tc_{2}C can be synthesized at ambient condition, thus further experimental synthesis is highly desirable.

Fig. 4 The calculated formation enthalpy–pressure curves for Tc_{2}C with five different structures. |

As is well known, the mechanical stability is a necessary condition for the stabilization of a crystal to exist, and the mechanical properties, such as elastic constants and elastic moduli, are important for potential technological and industrial applications. Thus we perform further the studies on the mechanical properties of the P6_{3}/mmc-Tc_{2}C phase. By using a strain-energy method, we obtain the zero-pressure elastic constants (C_{ij}) of the P6_{3}/mmc-Tc_{2}C and P-6m2-TcC, respectively. The calculated elastic constants C_{ij} are listed in Table 1, along with the theoretical values and available experimental data of other TM carbides (WC,^{5} PtC,^{10} Re_{2}C,^{11,12,14,15} Ru_{2}C,^{37,38} and Os_{2}C^{36,39}) for comparison. For a stable hexagonal crystal, C_{ij} should satisfy the following criteria:^{40} C_{33} > 0, C_{44} > 0, C_{12} > 0, C_{11} > |C_{12}|, (C_{11} + 2C_{12})C_{33} > 2C_{13}^{2}. As shown in Table 1, the elastic constants of the predicted P6_{3}/mmc-Tc_{2}C satisfy completely the elastic stability criteria for a hexagonal crystal, thus it is mechanically stable at ambient condition. The large values of C_{11} and C_{33} for the compound suggest that it is extremely difficult to be compressed along a-axis (or b-axis) and c-axis, respectively. Moreover, the C_{33} value of the P6_{3}/mmc phase (781 GPa) is close to those for WC,^{5} TcC,^{19–21} and Re_{2}C^{11} but much larger than Ru_{2}C^{38} and Os_{2}C,^{39} suggesting its high linear incompressibility along the c-axis. Based on the obtained elastic constants C_{ij}, the polycrystalline bulk modulus B and shear modulus G are thus determined by the Voigt–Reuss–Hill approximation. The Young's modulus E and Poisson's ratio v can be derived by the following equations E = 9BG/(3B + G) and v = (3B − 2G)/(6B + 2G). The calculated bulk modulus, shear modulus, Young's modulus, and Poisson's ratio of the P6_{3}/mmc phase together with the reference materials mentioned above are listed in Table 1. As shown in Table 1, the bulk modulus of the P6_{3}/mmc phase is 343 GPa, which is comparable to the experimental data of WC (439 GPa),^{5} PtC (301(±15)),^{10} and Re_{2}C (405(30) and 386(10))^{14,15} but much more higher than that of Ru_{2}C (178(4) GPa),^{37} indicating its ultra-incompressible structure nature. Moreover, the calculated bulk modulus (B = 343 GPa) for the P6_{3}/mmc-Tc_{2}C agrees well with that directly obtained from the fitting results (B_{0} = 344 GPa) of the third-order Birch–Murnaghan equation of states, further verifying the reliability of the present elastic calculations. In order to further compare the incompressibility of the P6_{3}/mmc-Tc_{2}C, WC, PtC, Re_{2}C, and Ru_{2}C under pressure, the volume compressions as a function of the pressure are presented in Fig. 5. It can be seen that the volume incompressibility of the P6_{3}/mmc-Tc_{2}C is comparable to those of WC and Re_{2}C but exceeds those of PtC and Ru_{2}C. Compared with the bulk modulus, the shear modulus of a material quantifies its resistance to the shear deformation and acts as a better indicator of the potential hardness. Obviously, the predicted P6_{3}/mmc phase has a large shear modulus of 216 GPa, thus it is expected to withstand the shear strain to a large extent. In addition, Young's modulus can also provide a good measure of the stiffness of materials except the bulk modulus and shear modulus. The larger Young's modulus the material has, the harder it is to deform. The calculated Young's modulus of the P6_{3}/mmc-Tc_{2}C is 536 GPa, indicating that it is a hard material. Poisson's ratio v is a crucial parameter to describe the degree of directionality of the covalent bonding. Generally speaking, the typical v value is 0.1 for covalent materials and 0.33 for metal materials, respectively. From Table 1, it can be seen that the v value of the P6_{3}/mmc-Tc_{2}C phase is below 0.33 and at the same time the P6_{3}/mmc-Tc_{2}C has the smallest Poisson's ratio 0.239 in contrast to other considered TM carbides, such as Ru_{2}C,^{38} Os_{2}C,^{39} and TcC.^{19–21} Consequently, the P6_{3}/mmc-Tc_{2}C has a strong degree of covalent boding. Furthermore, the B/G ratio is commonly used to predict the brittle or ductile behavior of materials. According to the Pugh criterion,^{41} if B/G > 1.75, the material behaves in a ductile way, otherwise, the material behaves in a brittle manner. For the case of the P6_{3}/mmc-Tc_{2}C, our predicted value of B/G is 1.59, suggesting that it has a little brittle nature. Using the empirical formula for the hardness prediction proposed by Chen et al.,^{42} the estimated Vickers hardness for the P6_{3}/mmc-Tc_{2}C is 24.11 GPa, which is comparable to the known hard materials of SiO_{2} (30.6 GPa)^{43} and B_{4}C (32.8 GPa).^{44} All of these excellent mechanical properties strongly support that the P6_{3}/mmc-Tc_{2}C is an ultra-incompressible and hard material.

Structure | C_{11} |
C_{12} |
C_{13} |
C_{33} |
C_{44} |
C_{66} |
B | B_{0} |
G | E | B/G | v | Θ | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

a Our calculated result from ref. 39 on the basis of v = (3B − 2G)/(6B + 2G). | ||||||||||||||

P6_{3}/mmc-Tc_{2}C |
This work (GGA) | 612 | 201 | 172 | 781 | 199 | 206 | 343 | 344 | 216 | 536 | 1.59 | 0.239 | 675 |

P6_{3}/mmc-Re_{2}C |
Theor. (GGA)^{11} |
748 | 201 | 219 | 939 | 252 | 273 | 410 | 400 | 273 | 671 | 0.227 | 564 | |

Theor.^{12} |
388.9 | |||||||||||||

Expt.^{14} |
405(30) | |||||||||||||

Expt.^{15} |
386(10) | |||||||||||||

P-3m1-Ru_{2}C |
Expt.^{37} |
178(4) | ||||||||||||

P-31m-Ru_{2}C |
Theor. (GGA)^{38} |
500 | 198 | 239 | 481 | 151 | 315 | 139 | 2.27 | 0.307^{a} |
||||

P6_{3}/mmc-Os_{2}C |
Theor. (GGA)^{36} |
418.48 | ||||||||||||

Theor. (GGA)^{39} |
639 | 142 | 329 | 579 | 111 | 381 | 152 | 402 | 2.51 | 0.32 | ||||

P-6m2-TcC | This work (GGA) | 682 | 200 | 166 | 940 | 169 | 241 | 371 | 372 | 225 | 562 | 1.65 | 0.247 | 756 |

Theor. (LDA)^{19} |
749 | 244 | 211 | 1031 | 199 | 252 | 426 | 270 | 1.59 | 0.24 | 831 | |||

Theor. (LDA)^{20} |
752 | 228 | 199 | 1030 | 198 | 416 | 252 | 629 | 0.249 | |||||

Theor. (LDA)^{21} |
759 | 232 | 188 | 1044 | 188 | 414 | 416 | 249 | 623 | 1.67 | 0.249 | 864 | ||

P-6m2-WC | Expt.^{5} |
720 | 254 | 267 | 972 | 328 | 439 | |||||||

Fm-3m-PtC | Expt.^{10} |
301(±15) |

Fig. 5 The calculated volume compression as a function of pressure for the P6_{3}/mmc-Tc_{2}C compared with WC, PtC, Re_{2}C, and Ru_{2}C. |

For engineering applications that make use of single crystals, it is necessary to know the values of the Young's modulus, bulk modulus, and shear modulus as a function of crystal orientation. For the hexagonal P6_{3}/mmc-Tc_{2}C, the Young's modulus and bulk modulus are expressed by:

E^{−1} = s_{11}(α^{2} + β^{2})^{2} + s_{33}γ^{4} + (2s_{13} + s_{44})(β^{2}γ^{2} + α^{2}γ^{2}),
| (1) |

B^{−1} = (s_{11} + s_{12} + s_{13}) − (s_{11} + s_{12} − s_{13} − s_{33})γ^{2},
| (2) |

G^{−1} = 4s_{11}(α_{1}^{2}α_{2}^{2} + β_{1}^{2}β_{2}^{2}) + 4s_{33}γ_{1}^{2}γ_{2}^{2} + 8s_{12}α_{1}α_{2}β_{1}β_{2} + 8s_{13}(α_{1}α_{2} + β_{1}β_{2})γ_{1}γ_{2} + s_{44}[(β_{1}γ_{2} + β_{2}γ_{1})^{2} + (α_{1}γ_{2} + α_{2}γ_{1})^{2}] + s_{66}(α_{1}β_{2} + α_{2}β_{1})^{2},
| (3) |

Fig. 6(a) and (c) are the three-dimensional surface representations which show the variation of the Young's modulus and bulk modulus, respectively. For an isotropic crystal, one would see a spherical shape, while a deviation from a spherical shape can directly reflect the degree of the elastic anisotropy in the crystal. From these two figures, one can see that the Young's modulus exhibits a high degree of the elastic anisotropy due to a large deviation from a spherical shape, whereas the bulk modulus is nearly isotropic because of a small deviation from a spherical shape. The projections of the Young's modulus and bulk modulus on the ab and bc planes are also plotted in Fig. 6(b) and (d) for comparison. As can be seen, in-plane anisotropy in the ab plane is not existent, whereas in-plane anisotropy in the bc plane is directly revealed. A deeper insight of the change of the Young's modulus alone different directions can be obtained by investigating the directional dependences of the Young's modulus alone tensile axis in the (0001), (100), (112), and (012) planes, respectively, and the obtained results are given in Fig. 7(a). For the (0001) plane, since the direction cosines are α = cosθ, β = sinθ, and γ = 0 (θ is the angle between tensile stress and [0001]), one obtains E^{−1} = s_{11} from eqn (1). This implies that the Young's modulus on the basal plane is independent of the tensile stress direction for the P6_{3}/mmc-Tc_{2}C, which is the reason of the elastic isotropy in the basal plane for hexagonal crystal. For the directional dependences of the Young's modulus from [0001] to [20] in the (100) plane, the P6_{3}/mmc-Tc_{2}C possesses a maximum of E_{[0001]} = 708.2 GPa and a minimum of E_{[20]} = 527.8 GPa. For the (112) plane, the Young's modulus alone the [100] direction has the minimal value (E_{min} = 527.8 GPa) and the Young's modulus alone the [21] direction has the maximal one (E_{max} = 653.6 GPa). For the change of the Young's modulus in the (012) plane for the quadrant of directions between [20] and [101], the P6_{3}/mmc-Tc_{2}C exhibits a maximum of E_{[101]} = 591.7 GPa and a minimum of E_{[20]} = 527.8 GPa. Consequently, the order of the Young's modulus as a function of the principal crystal tensile [uvw] for the P6_{3}/mmc-Tc_{2}C is: E_{[0001]} > E_{[21]} > E_{[101]} > E_{[100]}. To understand the plastic deformation for the predicted P6_{3}/mmc-Tc_{2}C phase, we investigate the dependence of the shear modulus on the stress direction, and the corresponding result is presented in Fig. 7(b). For the (0001) shear plane with the shear stress direction varied from [100] to [20], since the direction cosines are α_{1} = β_{1} = 0, γ_{1} = 0, α_{2} = cosθ, β_{2} = sinθ, and γ_{2} = 0 (θ is the angle between the shear direction and [100]), one can obtain G^{−1} = s_{44} from eqn (3). This implies that the shear modulus is isotropic in the basal plane, as shown by the green curve in Fig. 7(b). For the shear plane (100) with the shear stress direction changed from [0001] to [20], α_{1} = 1, β_{1} = γ_{1} = 0, α_{2} = 0, β_{2} = sinθ, and γ_{2} = cosθ (θ is the angle between the shear direction and [0001]), so the shear module can be reduced to G^{−1} = s_{66} + (s_{44} − s_{66})cos^{2}θ. Since s_{44} > s_{66} in our calculations, the shear module on the (100) plane is the highest when θ = 90° and the lowest when θ = 0, corresponding to G_{max} = 205.5 GPa for the [20] direction and G_{min} = 199 GPa for the [0001] direction. For the shear plane (112) with the shear stress direction rotated from [100] to [21], the maximum of the shear modulus is 212.6 GPa for the [21] direction and the minimum of the shear modulus is 204.9 GPa for the [100] direction. When the pyramidal plane (012) is the shear plane and the shear stress direction is changed from [20] to [101], the shear modulus is the largest alone the [101] direction (G_{max} = 232.4 GPa) and the shear modulus is the smallest along the [20] direction (G_{min} = 204.2 GPa). Through the above analysis, we draw conclusions: (1) the Young's modulus and shear modulus are constant on the basal plane and orientation-dependent on both the pyramidal and prismatic planes; (2) the bulk modulus is almost isotropic.

Fig. 7 Directional dependences of Young's modulus (a) and the shear modulus (b) in the (0001), (100), (012), and (112) planes. |

The elastic anisotropy of crystals can exert great effects on the properties of the physical mechanism, such as anisotropic plastic deformation, crack behavior, and elastic instability. Therefore, it is important and necessary to investigate the elastic anisotropy to improve their mechanical durability. The compression and the shear anisotropic factors provide the measures of the degrees of anisotropy in atomic bonding in different crystallographic planes. For hexagonal crystal, the anisotropies in compressibility and shear are defined as:

(4) |

(5) |

For an isotropic crystal, the factors A_{comp} and A_{shear} must be one, while any departure from one is a measure of the degree of elastic anisotropy possessed by the crystal. According to the above definitions, the values of A_{comp} and A_{shear} are calculated to be 0.77 and 0.968, respectively. This indicates that the P6_{3}/mmc-Tc_{2}C phase has a certain degree elastic anisotropy.

Debye temperature is related to many physical properties of materials, such as specific heat, elastic constants, and melting temperature.^{46} It is used to differentiate between high and low temperature regions for a solid. When the temperature T > Θ_{D}, all modes are expected to have the energy of k_{B}T; when T < Θ_{D}, the high-frequency modes are expected to be frozen, namely the vibrational excitations origin only from the acoustic vibrations. The Debye temperature Θ_{D} for the studied P6_{3}/mmc-Tc_{2}C is estimated from the average sound velocity (v_{m}) by the following equation:

(6) |

(7) |

The electronic structure is the key to understand the mechanical properties of the P6_{3}/mmc-Tc_{2}C, the total and partial densities of states (DOS) are calculated and given in Fig. 9, where the vertical dashed line denotes the Fermi level. Distinctly, the P6_{3}/mmc-Tc_{2}C exhibits a good metallic behavior due to the adequately large total DOS at the Fermi level. This metallicity might make it a better candidate for hard conductors. From partial DOS, it can be clearly seen that the peaks from −13 eV to −10.8 eV are mainly contributed by C-2s and Tc-4d states with small contributions from the 4p and 5s electrons of Tc. The states from −7.3 eV to the Fermi energy (0 eV) mainly originate from the Tc-4d and C-2p orbitals with small contributions of Tc-4p and Tc-5s. In addition, the partial DOS profiles of Tc-4d and C-2p are very similar in the energy region from −7.3 eV to 0 eV, which reflects that Tc-4d orbital has a significant hybridization with C-2p orbital. This fact also shows the existence of a strong covalent bonding between the Tc and C atoms. For the total DOS, the typical feature is the presence of so-called pseudogap, which is regarded as the borderline between the bonding states and antibonding states. Since the Fermi energy is located below the pseudogap for the P6_{3}/mmc-Tc_{2}C phase, its bonding states are partially occupied and full antibonding states are unoccupied. This property also increases the structural stability of Tc_{2}C. Fig. 10 presents the calculated electronic localization function (ELF) of the predicted P6_{3}/mmc-Tc_{2}C phase on the (110) plane. From this figure, one can see that the large ELF value between the Tc and C atoms indicates the partially Tc–C covalent bonding interaction in the P6_{3}/mmc-Tc_{2}C phase. This also explains the reason of the high bulk modulus and large shear modulus of Tc_{2}C.

Fig. 9 Total (a) and partial (b) densities of states of the P6_{3}/mmc-Tc_{2}C. The vertical dashed line denotes the Fermi level E_{F}. |

- F. Occelli, D. L. Farber and R. L. Toullec, Nat. Mater., 2003, 2, 151 CrossRef CAS PubMed.
- Y. Zhang, H. Sun and C. F. Chen, Phys. Rev. B: Condens. Matter Mater. Phys., 2006, 73, 144115 CrossRef.
- D. W. He, Y. S. Zhao, L. Daemen, J. Qian, T. D. Shen and T. W. Zerda, Appl. Phys. Lett., 2002, 81, 643 CrossRef CAS.
- H. Y. Chung, M. B. Weinberger, J. B. Levine, A. Kavner, J. M. Yang, S. H. Tolbert and R. B. Kaner, Science, 2007, 316, 436 CrossRef CAS PubMed.
- M. Lee and R. S. Gilmore, J. Mater. Sci., 1982, 17, 2657 CrossRef CAS.
- J. C. Crowhurst, A. F. Goncharov, B. Sadigh, C. L. Evans, P. G. Morrall, J. L. Ferreira and A. J. Nelson, Science, 2006, 311, 1275 CrossRef CAS PubMed.
- J. S. Tse, D. D. Klug, K. Uehara, Z. Q. Li, J. Haines and J. M. Léger, Phys. Rev. B: Condens. Matter Mater. Phys., 2000, 61, 10029 CrossRef CAS.
- N. A. Dubrovinskaia, L. S. Dubrovinsky, S. K. Saxena, R. Ahuja and B. Johansson, J. Alloys Compd., 1999, 289, 24 CrossRef CAS.
- R. Chang and L. J. Graham, J. Appl. Phys., 1966, 37, 3778 CrossRef CAS.
- S. Ono, T. Kikegawa and Y. Ohishi, Solid State Commun., 2005, 133, 55 CrossRef CAS.
- H. Ozisik, E. Deligoz, K. Colakoglu and G. Surucu, Phys. Status Solidi RRL, 2010, 4, 347 CrossRef CAS.
- Z. S. Zhao, L. Cui, L. M. Wang, B. Xu, Z. Y. Liu, D. L. Yu, J. L. He, X. F. Zhou, H. T. Wang and Y. J. Tian, Cryst. Growth Des., 2010, 10, 5025 Search PubMed.
- E. A. Juarez-Arellano, B. Winkler, A. Friedrich, D. J. Wilson, M. Koch-Müller, K. Knorr, S. C. Voge, J. J. Wall, H. Reiche, W. Crichton, M. Ortega-Aviles and M. Avalos-Borja, Z. Kristallogr., 2008, 223, 492 CAS.
- E. A. Juarez-Arellano, B. Winkler, A. Friedrich, L. Bayarjargal, V. Milman, J. Y. Yan and S. M. Clark, J. Alloys Compd., 2009, 481, 577 CrossRef CAS.
- N. Yasui, M. Sougawa, M. Hirai, K. Yamamoto, T. Okada, D. Yamazaki, Y. Kojima, H. Ohfuji, S. Kunitsugu and K. Takarabe, Cogent Physics, 2015, 2, 1076702 CrossRef.
- W. Trzebiatowski and J. Z. Rudzinski, Z. Chem., 1962, 2, 158 CrossRef CAS.
- A. L. Giorgi and E. G. Szklarz, J. Less-Common Met., 1966, 11, 455 CrossRef CAS.
- V. F. Peretrukhin, K. N. Gedgovd, M. S. Grigoriev, A. V. Tarasov, Y. V. Plekhanov, A. G. Maslennikov, G. S. Bulatov, V. P. Tarasov and M. Lecomteb, J. Nucl. Radiochem. Sci., 2005, 6, 211 CrossRef.
- Y. X. Wang, Phys. Status Solidi RRL, 2008, 2, 126 CrossRef CAS.
- Y. C. Liang, C. Li, W. L. Guo and W. Q. Zhang, Phys. Rev. B: Condens. Matter Mater. Phys., 2009, 79, 024111 CrossRef.
- Y. C. Zou, J. Zhu, Y. J. Hao, G. Xiang, X. C. Liang and J. R. Wang, Phys. Status Solidi B, 2014, 251, 1372 CrossRef CAS.
- X. W. Sun, Y. D. Chu, Z. J. Liu, T. Song, J. H. Tian and X. P. Wen, Chem. Phys. Lett., 2014, 614, 167 CrossRef CAS.
- T. Song, Q. Ma, J. H. Tian, X. B. Liu, Y. H. Ouyang, C. L. Zhang and W. F. Su, Mater. Res. Bull., 2015, 61, 58 CrossRef CAS.
- M. Kavitha, G. S. Priyanga, R. Rajeswarapalanichamy and K. Iyakutti, Int. J. Refract. Met. Hard Mater., 2015, 52, 219 CrossRef CAS.
- Q. G. Wang, K. E. German, A. R. Oganov, H. F. Dong, O. D. Feya, Y. V. Zubavichus and V. Y. Murzin, RSC Adv., 2016, 6, 16197 RSC.
- Y. C. Wang, J. Lv, L. Zhu and Y. M. Ma, Phys. Rev. B: Condens. Matter Mater. Phys., 2010, 82, 094116 CrossRef.
- J. Lv, Y. C. Wang, L. Zhu and Y. M. Ma, Phys. Rev. Lett., 2011, 106, 015503 CrossRef PubMed.
- L. Zhu, H. Wang, Y. C. Wang, J. Lv, Y. M. Ma, Q. L. Cui, Y. M. Ma and G. T. Zou, Phys. Rev. Lett., 2011, 106, 14550 Search PubMed.
- Z. S. Zhao, B. Xu, L. M. Wang, X. F. Zhou, J. L. He, Z. Y. Liu, H. T. Wang and Y. J. Tian, ACS Nano, 2011, 5, 7226 CrossRef CAS PubMed.
- Y. M. Ma, Y. C. Wang, J. Lv and L. Zhu, http://nlshm-lab.jlu.edu.cn/%7Ecalypso.html.
- G. Kresse and D. Joubert, Phys. Rev. B: Condens. Matter Mater. Phys., 1999, 59, 1758 CrossRef CAS.
- P. E. Blöchl, Phys. Rev. B: Condens. Matter Mater. Phys., 1994, 50, 17953 CrossRef.
- H. J. Monkhorst and J. D. Pack, Phys. Rev. B: Solid State, 1976, 13, 5188 CrossRef.
- A. Togo, F. Oba and I. Tanaka, Phys. Rev. B: Condens. Matter Mater. Phys., 2008, 78, 134106 CrossRef.
- R. Hill, Proc. Phys. Soc., London, Sect. A, 1952, 65, 349 CrossRef.
- N. R. Sanjay Kumar, S. Chandra, S. Amirthapandian, N. V. Chandra Shekar and P. C. Sahu, Mater. Res. Express, 2015, 2, 016503 CrossRef.
- N. R. Sanjay Kumar, N. V. Chandra Shekar, S. Chandra, J. Basu, R. Divakar and P. Ch Sahu, J. Phys.: Condens. Matter, 2012, 24, 362202 CrossRef CAS PubMed.
- J. Lu, F. Hong, W. J. Lin, W. Ren, Y. W. Li and Y. F. Yan, J. Phys.: Condens. Matter, 2015, 27, 175505 CrossRef PubMed.
- L. P. Ding, P. Shao, F. H. Zhang, X. F. Huang and T. L. Yuan, J. Phys. Chem. C, 2015, 119, 21639 CAS.
- M. Born, Proc. Cambridge Philos. Soc., 1940, 36, 160 CrossRef CAS.
- S. F. Pugh, Philos. Mag., 1954, 45, 823 CrossRef CAS.
- X. Q. Chen, H. Y. Niu, D. Z. Li and Y. Y. Li, Intermetallics, 2011, 19, 1275 CrossRef CAS.
- F. M. Gao, J. L. He, E. D. Wu, S. M. Liu, D. L. Yu, D. C. Li, S. Y. Zhang and Y. J. Tian, Phys. Rev. Lett., 2003, 91, 015502 CrossRef PubMed.
- G. Kresse and D. Joubert, Phys. Rev. B: Condens. Matter Mater. Phys., 1999, 59, 1758 CrossRef CAS.
- J. F. Nye, Physical properties of Crystals: Their Representation by Tensors and Matrices, Oxford University Press, New York, 1985 Search PubMed.
- P. Ravindran, L. Fast, P. A. Korzhavyi, B. Johansson, J. Wills and O. Eriksson, J. Appl. Phys., 1998, 84, 4891 CrossRef CAS.
- R. Ahmed, F. E. Aleem, S. J. Hashemifar and H. Akbarzadeh, Phys. B, 2007, 400, 297 CrossRef CAS.

This journal is © The Royal Society of Chemistry 2016 |