Xiaofeng Li* and
Junyi Du

College of Physics and Electronic Information, Luoyang Normal College, Luoyang, 471022, Henan, PR China. E-mail: dcx0828@163.com

Received
26th February 2016
, Accepted 10th May 2016

First published on 11th May 2016

Using an unbiased structure search method based on particle-swarm optimization algorithms in combination with density functional theory calculations, we investigate the phase stability and electronic properties of NbB_{3} under high pressures. By structure searching as implemented in the CALYPSO code, we obtained the most stable monoclinic phase (C2/c) and four metastable phases (P6_{3}/mmm, C2/m, Pmm2 and Im2) for NbB_{3}. Under high pressure, the C2/m structure transforms to Pmm2 phase and to Im2 phase at about 29 and 46 GPa, respectively. Surprisingly, the five phases of NbB_{3} are all dynamically and mechanically stable at ambient conditions. The high bulk and shear modulus, and low Poisson's ratio for both phases in NbB_{3}, make it a promising low compressible material. Moreover, interestingly the hardness of the five phases of NbB_{3} are very close to 40 GPa, in particular, the C2/m and Pmm2 phases have sufficient hardness (45 and 44 GPa, respectively), for it to be considered as a superhard material. All phases of NbB_{3} with high hardness are stable due to the strong covalent bonding nature via electronic density of states and electron localization function analysis.

The 4d transition metal borides have also attracted great attentions. Theoretical calculations^{14} have indicated that Amm2 and Cmcm phases of ZrB_{4} have a high hardness of about 42 GPa, which reveal that they are potentially superhard materials. It has been theoretically proved that tungsten triboride is a better candidate material than WB_{4}, because WB_{3} is are more stable. Boron atoms in tungsten borides W_{1−x}B_{3} to WB_{3+x}, space group (P6_{3}/mmc)^{15,16} exhibit a three-dimensional (3D) framework consisting of xy planar graphene-like layers with distorted sp^{2} hybridization B–B bonds and possible interstitial boron atoms, contributing to its high hardness. A stable orthorhombic phase (Imma phase) TaB_{3} (ref. 17) is predicted, which has a hardness of 41.2 GPa. The ground-state phases of ReB_{3} and IrB_{3} are predicted to have P6m2 and Amm2 phases, respectively. These crystal phases have ultra-incompressibility, and a high bulk and shear modulus.^{18} These results indicated that 3d, 4d TMs borides may be another choice for superhard material. According to the design principle, Nb-based borides have attracted attention because of their low cost, high hardness, high elastic modulus, and excellent thermal stability.^{19,20} So far, studies of Nb–B binary compounds have focused on the structure and mechanical properties of NbB_{2}.^{21,22} Pan et al.^{23} investigated the elastic properties, theoretical hardness, and chemical bonding of Nb–B compounds with different B concentrations. Among them, the calculated hardness of Nb_{2}B_{3} is 33.5 GPa and thus it can be considered as a potential superhard material. However, presently, the debate on the ground state structure of NbB_{3} has never been solved and its physical behaviors are still controversial, possibly due to the uncertainty of its crystal structure. Therefore, resolving the crystal structure of NbB_{3} has an important significance.

In the present work, the crystal structures of NbB_{3} are extensively explored over a wide range of pressures (0–150 GPa) using a specifically developed particle swarm optimization (PSO) algorithm technique for crystal structure prediction.^{24,25} A stable monoclinic phase C2/c and four metastable phases at ambient condition for NbB_{3} have been uncovered. The stabilities of these new phases at ambient condition have been determined by elastic constants and phonon spectra. Under pressure, three of the metastable phases are found to have a phase transition. Further calculations have been performed to study the mechanical and electronic properties of all phases of NbB_{3}.

Fig. 2 Crystal structures of NbB_{3} (a) C2/c (b) P6_{3}/mmm (c) C2/m (d) Pmm2 and (e) Im2. The red and green balls represent Nb and B atoms, respectively. |

To check the dynamical stabilities of the currently predicted phases of NbB_{3}, we have calculated their phonon dispersion curves using GGA method. A stable crystalline structure requires all phonon frequencies to be positive. As seen in Fig. 3, the absence of any imaginary phonon frequency in the whole Brillouin zone for five phases of NbB_{3} indicate the dynamical stabilities of them at ambient pressure in the whole BZ, confirming the dynamic stability of the newly proposed crystal structures of NbB_{3}. It is well known that shorter bond lengths contribute to higher phonon frequencies. The phonon frequency of all the phases of NbB_{3} (∼30 THz) shown in Fig. 3 indicates that there are short bond lengths in NbB_{3} and their stronger interactions between Nb–B and B–B. Simultaneously, the primitive cell of the C2/c, P6_{3}/mmm, C2/m, Pmm2, and Im2 phases contain 16, 8, 8, 8, and 4 atoms, which have 48, 24, 24, 24, and 12 phonon branches, respectively. The calculated zone center (Γ) phonon eigenvectors were used to deduce the symmetry labels of phonon modes. The vibrational modes at the zone center have their reducible representations constituted by infrared, Raman and hyper-Raman modes, which can be obtained via the Bilbao Crystallographic Server.^{33} Both infrared and Raman frequencies of crystal structures for Γ point can provide useful information for future experiments to identify the predicted new phases.

Fig. 3 Phonon dispersion relations of predicted new phases of NbB_{3} at ambient condition (a) C2/c (b) P6_{3}/mmm (c) C2/m (d) Pmm2 (e) Im2. |

The mechanical stability of a phase has been investigated because it is a necessary condition for the existence of a crystal. Accurate elastic constants can directly describe the response of the crystal to external stresses and are essential for many practical applications related to the mechanical properties of materials. They also provide very useful information to estimate the hardness of the material. Meanwhile, for a stable crystal, the elastic constants C_{ij} must satisfy the Born–Huang criterion.^{34} For a monoclinic crystal, the independent elastic stiffness tensor consists of thirteen components C_{11}, C_{22}, C_{33}, C_{44}, C_{55}, C_{66}, C_{12}, C_{13}, C_{23}, C_{15}, C_{25}, C_{35}, and C_{46}. Its mechanical stability of monoclinic phase is given by:

C_{ii} > 0 (i = 1, 2, 3, 4, 5, 6), [C_{11} + C_{22} + C_{33} + 2(C_{12} + C_{13} + C_{23})] > 0, (C_{33}C_{55} − C_{35}^{2}) > 0, (C_{44}C_{66} − C_{46}^{2}) > 0, (C_{22} + C_{33} − 2C_{23}) > 0, [C_{22}(C_{33}C_{55} − C_{35}^{2}) + 2C_{23}C_{25}C_{35} − C_{23}^{2}C_{55} − C_{25}^{2}C_{33}] > 0, g = C_{11}C_{22}C_{33} − C_{11}C_{23}^{2} − C_{22}C_{13}^{2} − C_{33}C_{12}^{2} + 2C_{12}C_{13}C_{23}, {2(C_{15}C_{25}(C_{33}C_{12} − C_{13}C_{23}) + C_{15}C_{35}(C_{22}C_{13} − C_{12}C_{33}) + C_{25}C_{35}(C_{11}C_{23} − C_{12}C_{13})] − [C_{15}^{2}(C_{22}C_{33} − C_{23}^{2}) + C_{25}^{2}(C_{11}C_{33} − C_{13}^{2}) + C_{35}^{2}(C_{11}C_{22} − C_{12}^{2})] + C_{55}g > 0 |

For orthorhombic phase, the mechanical stability can be judged from:

C_{ii} > 0, i = 1, 2, 3, 4, 5, 6 |

C_{11} + C_{22} + C_{33} + 2(C_{12} + C_{13} + C_{23}) > 0 |

C_{ii} + C_{jj} − 2C_{ij} > 0, (i, j = 1, 2, 3, i ≠ j) |

For a stable tetragonal phase, its elastic constants should obey the following inequalities:

C_{44} > 0, C_{66} > 0, C_{11} > |C_{12}|, 2C_{13}^{2} < C_{33}(C_{11} + C_{12}) |

Therefore, we calculated the elastic constants of the five phases of NbB_{3} by the strain–stress method, which are showed in Table 1 (both GGA and LDA), together with those of other transition metal triborides TMB_{3} (TM = V, Ir, and Re).^{17,18} We found that the elastic constants are a little larger by LDA method than GGA one. However, the differences do not affect the mechanical natures of NbB_{3}. As seen in Table 1, the predicted phases of NbB_{3} all satisfied the mechanical stability criteria, indicating that they are mechanically stable. The obtained elastic constants of the predicted new phases of NbB_{3} are competitive with those of other transition metal triborides.^{17,18} As seen in Table 1, C_{44}, C_{55}, and C_{66} for these phases are very close. C_{11}, C_{22}, and C_{33} for these phases are larger than 400 GPa, indicating strong incompressibility along the a, b, and c-axis, respectively. The values of C_{11} of C2/c, P6_{3}/mmm, and Im2 phases are larger than those of C_{33}, indicating that the bond strength along the [100] direction is much stronger than that along the [001] direction. Among them, for C2/m-NbB_{3}, C_{22} [716 GPa (GGA), 724 GPa (LDA)] is much larger than C_{11} [574 GPa (GGA), 584 GPa (LDA)] and C_{33} [577 GPa (GGA), 588 GPa (LDA)], comparable to that of c-BN, 820 GPa,^{35} which may come from the accumulation of electron density due to the shorter B–B bond (1.744 Å). C_{44} is an important indicator for the hardness of materials. All the studied structures have large C_{44} values. Notably, Pmm2-NbB_{3} possesses the largest C_{44} value, indicating its relatively strong strength against shear deformation. Unfortunately, there are no experimental data available for comparisons, therefore, our results could be a reference for future studies and applications of NbB_{3}.

Space group | Method | C_{11} |
C_{22} |
C_{33} |
C_{44} |
C_{55} |
C_{66} |
C_{12} |
C_{13} |
C_{23} |
Ref. | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

NbB_{3} |
C2/c | GGA | 644 | 624 | 513 | 239 | 233 | 258 | 103 | 174 | 184 | Present |

LDA | 699 | 675 | 572 | 264 | 258 | 281 | 115 | 192 | 201 | Present | ||

P6_{3}/mmm |
GGA | 583 | 438 | 206 | 110 | 128 | Present | |||||

LDA | 589 | 451 | 211 | 112 | 127 | Present | ||||||

C2/m | GGA | 574 | 716 | 577 | 237 | 272 | 273 | 108 | 126 | 135 | Present | |

LDA | 584 | 724 | 587 | 243 | 280 | 279 | 107 | 129 | 140 | Present | ||

Pmm2 | GGA | 572 | 606 | 588 | 264 | 258 | 278 | 147 | 122 | 160 | Present | |

LDA | 609 | 659 | 640 | 287 | 272 | 300 | 168 | 136 | 173 | Present | ||

Im2 | GGA | 601 | 560 | 213 | 238 | 155 | 138 | Present | ||||

LDA | 661 | 617 | 238 | 262 | 173 | 150 | Present | |||||

TaB_{3} |
Imma | 561 | 705 | 620 | 266 | 262 | 302 | 166 | 168 | 80 | 17 | |

ReB3 | P6_{3}/mmc |
665 | 809 | 217 | 128 | 123 | 18 | |||||

Pm1 | 613 | 795 | 194 | 177 | 122 | 18 | ||||||

IrB_{3} |
Amm2 | 475 | 486 | 184 | 164 | 277 | 120 | 18 | ||||

P6_{3}/mmc |
398 | 643 | 164 | 172 | 223 | 18 | ||||||

Pm1 | 556 | 666 | 131 | 176 | 147 | 18 |

Bulk and shear moduli are important indicators of the hardness of a material. The calculated elastic constants were used to estimate the bulk and shear modulus of ReB_{3} and IrB_{3} using the Voigt–Reuss–Hill (VRH) approximation.^{36} As is known, a high bulk modulus of a material illustrates its strong ability to resist volume deformation caused by an applied load. Apparently, all the predicted phases^{17,18} in Table 2 have large bulk modulus (above 255 GPa), comparable to that of TaB_{3}, ReB_{3} and IrB_{3}, which indicated that they are difficult to compress. Compared with the bulk modulus, the shear modulus (G) is a much better parameter to indicate the hardness of a material. From Table 2, all the structures of NbB_{3} possess high shear modulus. Among them, C2/m-NbB_{3} has the largest shear modulus (251 GPa by GGA, 255 GPa by LDA), indicating that it can withstand the largest extent shear strain. Besides the bulk modulus and shear modulus, Young's modulus could also provide a good measure of the stiffness of materials. The Young's modulus E is obtained by the equation: E = (9GB)/(3B + G). Young's modulus is defined as the ratio of stress and strain, and is used to provide a measure of the stiffness of materials in the range of elastic deformation. When the value of E is larger, the material is stiffer. In the same way, C2/m phase has highest Young's modulus, and therefore will be much stiffer than the other phases studied. Hence, all predicted phases could be potentially hard materials. The value of B/G is commonly used to describe the ductility or brittleness of materials, with 1.75 as critical reference point. Higher (or lower) B/G value than 1.75 indicates that the material is ductile (or brittle).considered to be ductile (or brittle). From Table 2, the B/G values of all the phases of NbB_{3} are below the critical value, implying their brittle nature. The value of the Poisson's ratio is indicative of the degree of directionality of the covalent bonds. The Poisson's ratio ν is obtained by the equation: ν = (3B − 2G)/2(3B + G). The typical ν value is 0.1 for covalent materials and 0.33 for metallic materials, respectively. The values of the Poisson's ratio of all phases for NbB_{3} are smaller than 0.33, indicating that NbB_{3} has covalent bond characterization. Especially, C2/m and Pmm2 phases have the close small Poisson ratio (0.162 and 0.167 by GGA), which indicated that the directionality degree of covalent bonding of C2/m and Pmm2 phases are stronger than the other phases of NbB_{3}. The directionality of covalent bonding plays an important role in the hardness of materials. Smaller the Poisson's ratio is, larger the hardness is. Therefore, C2/m and Pmm2 phases perhaps have higher hardness. The bulk modulus B, shear modulus G, Young modulus E, B/G, Poisson's ratio ν are all listed in Table 2. These results will provide theoretical guidance for future experimental and theoretical work.

Space group | B | G | E | ν | B/G | H_{V} |
Ref. | ||
---|---|---|---|---|---|---|---|---|---|

NbB_{3} |
C2/c | GGA | 299 | 231 | 552 | 0.193 | 1.296 | 38.8 | Present |

LDA | 329 | 253 | 590 | 0.193 | 1.296 | 40.8 | Present | ||

P6_{3}/mmm |
GGA | 258 | 211 | 497 | 0.178 | 1.223 | 37.8 | Present | |

LDA | 261 | 215 | 505 | 0.177 | 1.216 | 38.3 | Present | ||

C2/m | GGA | 287 | 251 | 581 | 0.162 | 1.148 | 43.9 | Present | |

LDA | 291 | 255 | 593 | 0.161 | 1.141 | 44.3 | Present | ||

Pmm2 | GGA | 291 | 247 | 578 | 0.169 | 1.181 | 42.7 | Present | |

LDA | 319 | 267 | 626 | 0.173 | 1.196 | 44.2 | Present | ||

Im2 | GGA | 289 | 219 | 525 | 0.197 | 1.318 | 36.8 | Present | |

LDA | 318 | 243 | 580 | 0.196 | 1.310 | 34.2 | Present | ||

TaB_{3} |
Imma | 301 | 262 | 608 | 0.16 | 1.149 | 41 | 17 | |

ReB3 | P6_{3}/mmc |
321 | 258 | 610 | 0.18 | 1.24 | 37 | 18 | |

Pm1 | 318 | 228 | 552 | 0.21 | 1.39 | 29 | 18 | ||

IrB_{3} |
Amm2 | 285 | 157 | 398 | 0.27 | 1.82 | 16 | 18 | |

P6_{3}/mmc |
297 | 143 | 370 | 0.29 | 2.08 | 13 | 18 | ||

Pm1 | 302 | 178 | 446 | 0.25 | 1.70 | 19 | 18 |

The hardness of a material is the intrinsic resistance to deformation when a force is applied, which depends on the loading force and the quality of the sample (i.e., the presence of defects such as vacancies and dislocations). Therefore, we estimate Vickers hardness (H_{V}) by an empirical model proposed by Chen et al.,^{37} H_{V} = 2(k^{2}G)^{0.585} − 3 (k = G/B). The calculated hardness of five predicted phases of NbB_{3} are listed in Table 2. The hardness determined by GGA and LDA methods is almost identical. The estimated hardness of 44 and 43 GPa for C2/m and Pmm2 phases, respectively, exceed 40 GPa, which can be considered as the superhard materials. Moreover, C2/m and Pmm2 phases have the largest hardness among other studied niobium borides.^{23}

To our knowledge, electronic structure and chemical bonding are key factors to deeper understanding the origin of hardness and elastic properties. Following, the density of states (DOS) and bond characteristic are calculated and analysed here. Fig. 4 represents the total and partial density of states of five different phases of NbB_{3}, and the black vertical dashed of DOS indicates the Fermi level (E_{F}). The obtained DOS curves show large similarities in these phases of NbB_{3} as follows. It is clear that some bands are across the E_{F}, indicating that those phases exhibit metallic behavior. It is easy to see that the peaks below-10 eV are mainly attributed to B-s states and B-p states with a slight contribution from, Nb-p and Nb-d states. The states from −10 to 0 eV mainly originate from Nb-d and B-p orbitals with slight contributions of Nb-p and B-s. Moreover, the partial DOS profiles for both Nb-4d and B-2p are very similar in the range of −10 to 0 eV, reflecting the significant hybridization between these two orbitals. This fact also shows a strong covalent interaction between the Nb and B atoms. On the other hand, the DOS profile near E_{F} comes from the 4d state of Nb. The typical feature of their DOS is to have a pseudogap near Fermi level, which is the borderline between bonding and antibonding states. It should be pointed out that the E_{F} is lying on the pseudogap in all phases, revealing the p–d bonding states started to be saturated. The nearly full occupation of bonding states and without filling on the antibonding states leads to the high bulk modulus and shear modulus, small Poisson's ratio, and also increase the structural stabilities of NbB_{3}.

To gain more detailed insight into the bonding character of NbB_{3}, we have calculated the electronic localization function (ELF),^{38} which is based on a topological analysis related to the Pauli exclusion principle. The ELF is a contour plot in real space where different contours have values ranging from 0 to 1. A region with ELF = 1 is where there is no chance of finding two electrons with the same spin. This usually occurs in places where covalent bonds or lone pairs (filled core levels) reside. An area where ELF = 0 is typical for a vacuum (no electron density) or areas between atomic orbitals. This is where electrons of like spin approach each other the closest. ELF = 0.5 for a homogeneous electron gas, with values of this order indicating regions with bonding of a metallic character. It should be noted that ELF is not a measure of electron density but is a measure of the Pauli principle, and is useful in distinguishing metallic, covalent, and ionic bonding. The contours of ELF domains for all the phases on its (100) plane except P6_{3}/mmm phase along the (001) plane are shown in Fig. 5. The high electron localization can be seen in the region between adjacent B and B atoms indicative of covalent bonding. Meanwhile, the ELF is very small at the Nb sites, whereas it attains local maximum values at the B sites, manifesting another covalent interaction between Nb and B atoms. Therefore, the strong covalent interaction between B–B bonds and B–Nb bonds is the main driving force for its higher bulk and shear modulus.

- 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 .
- R. B. Kaner, J. J. Gilman and S. H. Tolbert, Science, 2005, 308, 1268 CrossRef CAS PubMed .
- B. H. Chu, D. Li, F. B. Tian, D. F. Duan, X. J. Sha, Y. Z. Lv, H. D. Zhang, B. B. Liu and T. Cui, Sci. Rep., 2015, 5, 10500 CrossRef PubMed .
- V. V. Brazhkin, A. G. Lyapin and R. J. Hemley, Philos. Mag. A, 2002, 82, 231 CAS .
- N. Mounet and N. Marzari, Phys. Rev. B: Condens. Matter Mater. Phys., 2005, 71, 205214 CrossRef .
- J. C. Zheng, Phys. Rev. B: Condens. Matter Mater. Phys., 2005, 72, 052105 CrossRef .
- A. L. Ivanovskii, Prog. Mater. Sci., 2012, 57, 184 CrossRef CAS .
- B. L. Jonathan, H. T. Sarah and B. K. Richard, Adv. Funct. Mater., 2009, 19, 3519 CrossRef .
- R. W. Cumberland, M. B. Weinberger, J. J. Gilman, S. M. Clark, S. H. Tolbert and R. B. Kaner, J. Am. Chem. Soc., 2005, 127, 7264 CrossRef CAS PubMed .
- X. F. Hao, Y. H. Xu, Z. J. Wu, D. F. Zhou, X. J. Liu, X. Q. Cao and J. Meng, Phys. Rev. B: Condens. Matter Mater. Phys., 2006, 74, 224112 CrossRef .
- H. Y. Chung, M. B. Weinberger, J. M. Yang, S. H. Tolbert and R. B. Kaner, Appl. Phys. Lett., 2008, 92, 261904 CrossRef .
- M. G. Zhang, H. Y. Yan, G. T. Zhang and H. Wang, J. Phys. Chem. C, 2010, 114, 6722 CAS .
- Q. Tao, D. Zheng, X. Zhao, Y. Chen, Q. Li, Q. Li, C. Wang, T. Cui, Y. M. Ma, X. Wang and P. Zhu, Chem. Mater., 2014, 26, 5297 CrossRef CAS .
- X. Y. Zhang, J. Q. Qin, X. W. Sun, Y. N. Xue, M. Z. Ma and R. P. Liu, Phys. Chem. Chem. Phys., 2013, 15, 20894 RSC .
- R. Mohammadi, A. T. Lech, M. Xie, B. E. Weaver, M. T. Yeung, S. H. Tolbert and R. B. Kaner, Proc. Natl. Acad. Sci. U. S. A., 2011, 108, 10958 CrossRef CAS PubMed .
- I. Zeiringer, P. Rogl, A. Grytsiv, J. Polt, E. Bauer and G. Giester, J. Phase Equilib. Diffus., 2014, 35, 384 CrossRef CAS .
- X. Z. Zhang, E. J. Zhao and Z. J. Wu, J. Alloys Compd., 2015, 632, 37 CrossRef CAS .
- Q. Yan, Y. X. Wang, B. Wang, J. M. Yang and G. Yang, RSC Adv., 2015, 5, 25919 RSC .
- A. Pallas and K. Larsson, J. Phys. Chem. B, 2006, 110, 5367 CrossRef CAS PubMed .
- E. Deligoz, K. Colakoglu and Y. O. Ciftci, Solid State Commun., 2010, 150, 405 CrossRef CAS .
- P. Vajeeston, P. Ravindran, C. Ravi and R. Asokamani, Phys. Rev. B: Condens. Matter Mater. Phys., 2001, 63, 045115 CrossRef .
- I. R. Shein and A. L. Lvanovskii, J. Phys.: Condens. Matter, 2008, 20, 415218 CrossRef .
- Y. Pan and Y. H. Lin, J. Phys. Chem. C, 2015, 119, 23175 CAS .
- Y. Wang, J. Lv, L. Zhu and Y. Ma, Phys. Rev. B: Condens. Matter Mater. Phys., 2010, 82, 094116 CrossRef .
- Y. C. Wang, J. Lv, L. Zhu and Y. M. Ma, Comput. Phys. Commun., 2012, 183, 2063 CrossRef CAS .
- D. M. Ceperley and B. J. Alder, Phys. Rev. Lett., 1980, 45, 566 CrossRef CAS .
- G. Kresse and J. Joubert, Phys. Rev. B: Condens. Matter Mater. Phys., 1999, 59, 1758 CrossRef CAS .
- P. Blaha, K. Schwarz, P. Sorantin and S. B. Trickey, Comput. Phys. Commun., 1990, 59, 399 CrossRef CAS .
- K. Parlinski, Z. Q. Li and Y. Kawazoe, Phys. Rev. Lett., 1997, 78, 4063 CrossRef CAS .
- A. Togo, F. Oba and I. Tanaka, Phys. Rev. B: Condens. Matter Mater. Phys., 2008, 78, 134106 CrossRef .
- M. Wang, Y. Li, T. Cui, Y. Ma and G. Zou, Appl. Phys. Lett., 2008, 93, 101905 CrossRef .
- K. Umenoto, R. M. Wentzcovitch, S. Saito and T. Miyake, Phys. Rev. Lett., 2010, 104, 125504 CrossRef PubMed .
- E. Kroumova, M. I. Aroyo, J. M. Perez-Mato, A. Kirov, C. Capillas, S. Ivantchev and H. Wondratschek, Phase Transform., 2003, 76, 155 CrossRef CAS .
- Z. J. Wu, E. J. Zhao, H. P. Xiang, X. F. Hao, X. J. Liu and J. Meng, Phys. Rev. B: Condens. Matter Mater. Phys., 2007, 76, 054115 CrossRef .
- M. Grimsditch, E. S. Zouboulis and A. Polian, J. Appl. Phys., 1994, 76, 832 CrossRef CAS .
- R. Hill, Proc. Phys. Soc., London, Sect. A, 1952, 65, 349 CrossRef .
- X. Q. Chen, H. Y. Niu, D. Z. Li and Y. Y. Li, Intermetallics, 2011, 19, 1275 CrossRef CAS .
- A. D. Becke and K. E. Edgecombe, J. Chem. Phys., 1990, 92, 5397 CrossRef CAS .

## Footnote |

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c6ra05162f |

This journal is © The Royal Society of Chemistry 2016 |