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The elastic behavior of dense C3N4 under high pressure: First-principles calculations
Journal of Physics and Chemistry of Solids 75 (2014) 1324–1333
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The elastic behavior of dense C3N4 under high pressure: First-principles calculations Lin-Wei Ruan a, Geng-Sheng Xu a, Hai-Yu Chen a, Yu-Peng Yuan a, Xia Jiang a, Yun-Xiang Lu b, Yu-Jun Zhu a,n a b
Laboratory of Advanced Porous Materials, School of Chemistry and Chemical Engineering, Anhui University, No. 111, Jiulong Road, Hefei 230601, PR China School of Chemistry and Molecular Engineering, East China University of Science and Technology, Shanghai 200237, PR China
art ic l e i nf o
a b s t r a c t
Article history: Received 1 June 2014 Received in revised form 8 July 2014 Accepted 9 July 2014 Available online 18 July 2014
We have carried a detailed theoretical study on the geometry, density of states, elastic properties, sound velocities and Debye temperature of α-, β-, c- and p-C3N4 compounds under a maximum of pressure up to 100 GPa by using first principles calculations. The optimized lattice constants under zero pressure and zero temperature agreed well with the previous experimental and theoretical results. The band gaps of the four types of dense C3N4 were widened gradually with the increase of pressure. The calculated Poisson’s ratio γ and B/G values suggest α-, c- and p-C3N4 are brittle materials under 0–100 GPa, whereas β-C3N4 will become a ductile material as external pressure reaches 57 GPa. We found that the Debye temperature of the four dense C3N4 gradually reduces in the order of c-C3N4 4 p-C3N4 4 α-C3N4 4 β-C3N4 at 0 GPa and 0 K. However, the Debye temperature of c-C3N4 was lower than p-C3N4 when external pressure exceeds 6.3 GPa. It may hint that the results could be served as a valuable prediction for further experiments. & 2014 Elsevier Ltd. All rights reserved.
Keywords: C. High pressure D. Elastic properties
1. Introduction Super-hard materials are of great importance in scientific development and industrial production because of their applications in cutting, grind, polishing and precision instrument processing, etc [1–3]. Nowadays, diamond and cubic boron nitride (c-BN) are the most common super-hard materials. Diamond is currently known as the world's hardest industrial materials with a Vickers hardness of 97.7 GPa reported by Gao [4]. c-BN is another superhard material with slightly lower hardness compared to diamond [4,5]. Both of them have been an integral part of the functional materials in industry. Apart from the two well-known super-hard materials, many research activities have been devoted to search for new super-hard materials [6–8] or their allotropes [9–11] which have the same high mechanical properties as diamond and c-BN. For the last few decades, various studies have been performed to explore covalent compounds formed by light elements, namely, boron, carbon and nitrogen due to their ability of forming short and strong three dimensional covalent bonds [2]. Super-hard materials like BC2N [12], B4C3 [13], BC5 [14], m-CN [15] and others have been reported. Since 1989 Liu and Cohen reported a new
compound β-C3N4 which has a similar bulk modulus as that of diamond [16]. The results ignited an interest in studying carbon nitride compounds. Further theoretical calculations proposed five polymorphs of C3N4, including α-C3N4, β–C3N4, c-C3N4, p-C3N4 and g-C3N4 [17]. Among them, the first four is thought to be super-hard materials, and the c-C3N4 is expected to be a material harder than a diamond. Henceforth, numerous studies on the mechanical properties of dense C3N4 at ambient conditions have been reported [18–20]. However, there are few reports about the behavior of the properties for dense C3N4 under high pressure whether by theoretical or experimental approaches. This may be due to the difficulty in synthesizing good dense C3N4 single crystals. In this study, we preformed the first principles calculations to investigate the geometry, density of states, elastic properties, sound velocities and Debye temperature of α-, β-, c-C3N4 and p-C3N4 compounds under a wide pressure range (0–100 GPa). Besides, the elastic properties of dense C3N4 were compared with diamond and c-BN, and the results indicated that dense C3N4 could be a substitution of the diamond and c-BN.
2. Computational details n
Corresponding author. Tel.: þ 86 551 63873365. E-mail address: [email protected] (Y.-J. Zhu).
http://dx.doi.org/10.1016/j.jpcs.2014.07.010 0022-3697/& 2014 Elsevier Ltd. All rights reserved.
The alpha (α-), beta (β-) and cubic (c-) together with pseudocubic phases C3N4 (p-C3N4) were investigated based on the Density
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Functional Theory (DFT) [21,22] using the Cambridge Serial Total Energy Package (CASTEP) plane-wave code [23] in Materials Studio 7.0 software. The calculations were performed with the Generalized Gradient Approximation (GGA) in the form of Perdew–Burke–Ernzerhof (PBE) exchange correlation potential [24]. In the calculations, a hybrid semiempirical solution Grimme approach [25–27] was taken to introduce damped atom-pairwise dispersion corrections of the form C6R 6 in the DFT formalism. The semiempirical approach provides the best compromise between the cost of first principles evaluation of the dispersion terms and the need to improve non-bonding interactions in the standard DFT description. The structural optimizations were conducted using the Broyden–Fletcher–Goldfarb–Shanno (BFGS) minimization [28]. The interactions between the ionic core and valence electrons were described by the ultrasoft pseudo-potential [29]. The considered valence atomic configurations are 2s22p2 for C and 2s22p3 for N. We performed a cut-off of 700 eV, 800 eV, 820 eV and 830 eV for α-, β-, c- and p-C3N4, respectively. A Monkhorst-Pack k-mesh [30] of 3 3 4, 5 5 11, 6 6 6 and 8 8 8 were used for α-, β-, c- and p-C3N4, respectively. In order to obtain more accurate results, the Self Consistent Field (SCF) tolerance threshold was set at 5.0 10 7 eV/atom. The total energy changes during the structure optimization were less than 5.0 10 6 eV/atom, the maximum force were smaller than 0.01 eV/Å, the maximum stress were smaller than 0.02 GPa and the maximum displacement were converged to 5.0 10 4 Å. All the electronic structures, elastic properties, elastic anisotropic properties, and Debye temperatures were calculated based on the optimized crystal structures.
3. Results and discussion 3.1. Geometric and electronic structures Structure of unit cell plays a very important role in understanding the nature of solid materials. Fig. 1 shows the three-dimensional
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structure of four dense C3N4 phases. The N atoms are 3-fold coordination by three C atoms. The α-C3N4 is trigonal structure with space group of P-31c, each cell includes four units. The β-C3N4 is trigonal structure as well and belongs to space group of P-3, each cell includes two units. The c-C3N4 and p-C3N4 have higher symmetrical structure which are cubic (with I-4̄3d space group) and tetragonal (with P-4̄ 2m space group) structure respectively. One cell includes four units in c-C3N4 while one unit in p-C3N4. The calculated lattice parameters, bond overlap population and bond length as well as other available values of four dense C3N4 at 0 GPa and 0 K are shown in Table 1. And the computed results are in agreement with the experimental and previous theoretical results well, which imply that our calculation methods and results are reasonable and authentic. In order to investigate the variation of structural properties of the four types of dense C3N4 phases under different pressures, we displayed the pressure dependence of the normalized structural parameters a/a0 and c/c0 as well as V/V0 with the pressure ranging from 0 to 100 GPa as shown in Fig. 2. The values a0, c0 and V0 are the equilibrium structure parameters at 0 GPa as listed in Table 1. Fig. 2(a) shows that, as pressure increased, all the lattice parameters decreased, especially the ratio of a/a0 of α-C3N4 decreased more quickly compared to the others. The ratio of a/a0 of c-C3N4 decreases the slowest, which means that a-axis of α–C3N4 is the most easily compressed in the four types of dense C3N4. Fig. 2 (b) shows that, as pressure increased, the volume of c-C3N4 decreased more slowly than other allotropes. Besides, the volumes of α-C3N4 and p-C3N4 decreased more quickly and the both curves almost coincide with each other. The bulk modulus B is a measure of resistance against a change of volume of a material caused by external pressures [35]. Therefore, we can predict that the bulk modulus of c-C3N4 is the largest in the four dense C3N4, and the bulk modulus of α-C3N4 is very close to p-C3N4. Most covalent compounds formed by light elements (such as boron, carbon and nitrogen) are more likely to be super-hard
Fig. 1. Unit cells of four super hard C3N4 phases. (a) α-C3N4, (b) β–C3N4, (c) c-C3N4, (d) p-C3N4. Carbon and nitrogen atoms are depicted in gray and blue balls, respectively.
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Table 1 Lattice parameters, bond overlap population and length of four different C3N4 phases at 0 GPa. Species
a0/Å
α–C3N4
6.4888 6.425b 6.4255 6.419b 5.4328 5.36d 3.4488 3.426e
β–C3N4 c-C3N4 p-C3N4
c0/Å 6.4665a 6.453c 6.4017a 6.394c 5.3973a 4.673c 3.4232a 3.421c
4.7288 4.715b 2.4184 2.425b
4.7097a 4.699c 2.4041a 2.397c
V0/Å3
Bond overlap population
Bond length/Å
Space group
172.4303 169.44c 86.4714 84.86c 160.3508 157.08c 41.0200 40.04c
0.78 0.77 0.73 0.79 0.81 0.85f, 0.81g 0.69 0.75g
1.46 1.4522g 1.46 1.4519g 1.47 1.461f, 1.4624g 1.48 1.4749g
P-31c P-3 I-4̄ 3d P-4̄ 2m
a
Ref. [17]. Ref. [31]. Ref. [32]. d Ref. [33]. e Ref. [34]. f Ref. [4]. g Ref. [19]. b c
Fig. 2. The normalized parameters a/a0, c/c0 and V/V0 of four dense C3N4 phases as a function of pressure.
materials [2]. α-, β-, c- and p-C3N4 satisfy the above conditions well. Fig. 3(a) shows the average C–N bond length of four dense C3N4 with pressure from 0 to 100 GPa. All the bond lengths decreased with the increasing of pressure, indicating that the dense C3N4 phases will be more “hard” under high pressures. Fig. 3 (b) shows that all the bond overlap populations of C3N4 increased with the increasing of pressure, which demonstrated that the covalent characters were enhanced over these changes. However, the overlap population of p-C3N4 remained almost unchanged in the pressure range of 40–100 GPa, indicating that the covalent character of p-C3N4 changed little as the external pressure exceeds 40 GPa. We further analyzed the electronic properties of the four dense C3N4 at different pressures in details. The total and partial density of states of the four dense C3N4 phases at zero and high pressures are shown in Fig. 4. The Fermi level is set to 0 eV. We can see that the four dense phases of C3N4 are semiconductors due to their relatively wide band gap, which is consistent with other similar theoretical reports [32–34]. We found the valence band (VB) of these compounds is mainly consisting of N-2p states. The conduction band (CB) of α-, β- and c-C3N4 are dominated by N-2s, N-2p
and C-2p states. However, the conduction band of p-C3N4 is mainly dominated by N-2p and C-2p states. The calculated band gaps of α-, β-, c- and p-C3N4 at 0 GPa are 3.808, 3.235, 2.966, 2.475 eV, accordingly, which coincide well with the previous results [17,32,34]. Furthermore, Fig. 4 shows that the band gaps of compounds are widened as the pressure increases. The valence bands and conduction bands keep away from the Fermi level and the peaks of the DOSs are dropped with the pressure increasing. 3.2. Elastic properties Elastic constants of solid compounds are important because of their close relations with various fundamental physical properties, such as elastic modulus, Debye temperature and phonon spectra [36,37]. There are six independent elastic constants for trigonal (C11, C12, C13, C14, C33, C44) and tetragonal (C11, C12, C13, C33, C44, C66) phases crystals [38]. However, for higher symmetrical cubic phase, there are only three independent elastic constants exist, i.e. C11, C12, C44 [39]. The elastic stiffness constants were calculated by a stress–strain method. In our study, the calculated independent elastic constants for four dense C3N4 as well as previous results
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The bulk modulus B, shear modulus G, Young's modulus E and Poisson's ratio γ can be directly derived with these elastic constants by the Voigte–Reusse–Hill approximations (VRH) [48]. The Voigt and Rees’s schemes represent the upper and lower bounds to the elastic moduli, respectively. In the following formulas, subscript V denotes the Voigt bound, R denotes the Reuss bound, and H denotes the Hill average. For trigonal phase, the formula can be defined as [38]: 1 BV ¼ ½2ðC 11 þC 12 Þ þ 4C 13 þ C 33 9 GV ¼
ð1Þ
1 ðM þ 12C 44 þ 12C 66 Þ 30
ð2Þ
Where M ¼ C 11 þ C 12 þ 2C 33 4C 13
ð3Þ
and 1 C 66 ¼ ðC 11 C 12 Þ 2
ð4Þ
BR ¼ C 2 =M
ð5Þ
GR ¼
5 C 2 ðC 44 C 66 C 214 Þ 2 3K V ðC 44 C 66 C 214 Þ þC 2 ðC 44 þ C 66 Þ
ð6Þ
Where C 2 ¼ ðC 11 þ C 12 ÞC 33 2C 213
ð7Þ
For cubic phase, the formula is [40]
Fig. 3. Calculated average C–N bond length and average bond overlap population of four different C3N4 as a function of pressure.
and comparable results for diamond and c-BN at ambient constants are tabulated in Table 2. The mechanical stability tests is satisfied for all systems of the four dense C3N4 [40]. As shown in Table 2, the as-obtained results are in close proximity to previous values, though there is slight disparity caused by the precision setting and method of calculations. Additionally, we find the elastic constants C11 and C44 of diamond are larger than the four types of dense C3N4, but C12 is much smaller. Interestingly, the elastic constant C14 of α-C3N4 and β-C3N4 are equal to zero, which indicate that both of the compounds are transversely isotropic [41]. For understanding the variation of elastic constants at different pressures more clearly, we use Fig. 5 to illustrate these data. It could be concluded from the figure: (1) as pressure increases, all the elastic constants increase monotonically, demonstrating the elastic properties of the four dense C3N4 can be enhanced by high pressure. We also found the lines of the elastic constants C11 and C33 located far above the other elastic constants. To the best of our knowledge, the elastic constant C11 (C22 and C33) represents the elasticity in length, which can change with longitudinal strain, and C12 (C13 and C23) and C44 (C55 and C66) are related to the elasticity in shape [47]. The results indicate the resisting axial stress abilities of the four dense C3N4 are much stronger than that of deformation. (2) C12 exceeds C44 at 73.75 and 7.98 GPa respectively, which indicate that the sensibility to pressure of the C44 of α-C3N4 and β-C3N4 is lower than that of C12. (3) The elastic constant C14 of α-C3N4 and β-C3N4 remained zero with pressure increasing, indicating the change of pressure does not affect the transversely isotropy of α-C3N4 and β-C3N4. (4) As shown in Fig. 5(d), the lines stand for elastic constants C11, C12 and C44 overlap with C33, C13 and C66, respectively, which demonstrate that the elastic properties of p-C3N4 is similar to that of c-C3N4.
1 BV ¼ ðC 11 þ 2C 12 Þ 3
ð8Þ
1 GV ¼ ðC 11 C 12 þ 3C 44 Þ 5
ð9Þ
5ðC 11 C 12 ÞC 44 4C 44 þ 3ðC 11 C 12 Þ
ð10Þ
GR ¼
and the Reuss bound on the bulk modulus is same to BV. For tetragonal phase, the Voigt bound bulk modulus is given according to Eq. (1), while the Voigt shear modulus can be defined as [38]: GV ¼
1 ðM þ 3C 11 3C 12 þ 12C 44 þ 6C 66 Þ 30
ð11Þ
and M is as the formula (3). The Reuss bulk modulus is given by Eq. (5), but the Reuss shear modulus is GR ¼ 15½18BV =C 2 þ 6=ðC 11 C 12 Þ þ6=C 44 þ 3=C 66 1
ð12Þ
and C has the same meaning of formula (7). The elastic modulus based on Hill approximation is an average of Voigt bound and Reuss bound [48]: 1 BH ¼ ðBV þ BR Þ 2
ð13Þ
1 GH ¼ ðGV þ GR Þ 2
ð14Þ
The Young's modulus E and Poisson's ratio by the following formulas [48]:
γ can be calculated
E¼
9BH GH 3BH þ GH
ð15Þ
γ¼
3BH 2GH 2ð3BH þ GH Þ
ð16Þ
The calculated elastic modulus B, shear modulus G, Young's modulus E, Poisson's and ratio γ of four dense C3N4 and diamond
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Fig. 4. Total and partial density of states for four dense C3N4 phases at different pressures. The blue lines represent the results are obtained at 0 GPa, red and green lines correspond to 60 GPa and 100 GPa, respectively. (a) α-C3N4, (b) β-C3N4, (c) c-C3N4, (d) p-C3N4. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Table 2 Calculated independent elastic constants Cij (GPa) for various dense C3N4 phases at 0 GPa. Species Diamond c-BN α–C3N4 β–C3N4 c-C3N4 p-C3N4 a
Ref. [42]. Ref. [20]. c Ref. [43]. d Ref. [44]. e Ref. [45]. f Ref. [46]. b
C12
C11 a
b
1060 , 1097 820c, 828.2d 851.21, 833.20, 834e 804.16, 861a 804.36
C13 a
C14
C33
b
150 , 125 190c, 201.2d 183.01 258.89, 279e 259.60, 300a 182.77
C44
C66 a
128.52 110.15, 138e 182.74
0 0, 0e
905.91 1049.05, 1120e 804.95
b
595 , 562 480c, 492.3d 326.18 288.94, 305e 486.24, 469a 439.09, 452f
334.10 287.16 438.78
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Fig. 5. Independent elastic constants Cij of various C3N4 versus pressures. (a) α-C3N4, (b) β-C3N4, (c) c-C3N4, (d) p-C3N4.
Table 3 The calculated results of elastic modulus B (GPa), shear modulus G (GPa), Young's modulus E (GPa), Poisson's ratio γ and Vickers hardness Hv(GPa) for Diamond, c-BN and various dense C3N4 phases at 0 GPa. Species
BV
Diamond c-BN α-C3N4 β-C3N4 c-C3N4 p-C3N4
438.8a 409f 387.60, 378.7a 408.21, 437b, 419.1a, 442f 441.12, 496b, 449.2a, 487c 390.02, 393.2a
BR
BH
GV
387.60 407.84 441.12 390.02
400e 387.60 408.02 441.12 390.02
535b, 546c 368f 341.85 322.09, 320b 400.65, 332b, 393c 387.76
GR
GH
E
338.96 311.12 369.95 376.88
409e 340.40 316.61 385.30 382.32
1178c 915.1e 789.95 754.63 895.25, 930c 864.49
BH/GH
1.14 1.29 1.14 1.02
γ
Hv
0.08
97.7d, 93.6e 79.1d, 66e 82.36, 82.7a 62.62, 60.4d 90.18, 87.1d 69.83, 79.6a
0.16 0.19 0.16, 0.18c 0.13
a
Ref. [19]. Ref. [51]. c Ref. [20]. d Ref. [52]. e Ref. [2]. f Ref. [18]. b
as well as c-BN at 0 GPa and 0 K are listed in Table 3. From the Table, we note that our results are in agreement with previous values well. Beyond that, c-C3N4 has the highest bulk modulus of 441.12 GPa, which is slightly larger than diamond. The shear modulus (385.30 GPa) and Young's modulus (895.25 GPa) of c-C3N4 are the highest in the four type of dense C3N4, which are also less than that of diamond. The Poisson's ratio γ can be judgment of ductility and brittleness, for a brittle material γ o 0.26, while for a ductile material γ Z 0.26 [49]. A conclusion could be drawn from Table 3 that the four types of dense C3N4 including diamond are brittle materials because of their Poisson's ratio are smaller than 0.26. In addition, diamond exhibits the smallest brittleness in these compounds. On the other hand, according to Pugh's criteria [50], a kind of material is brittle if the B/G ratio is less than 1.75, otherwise the material is ductile, which is consistent with the result judged by γ. Obviously,
our data further manifests the conclusions that these compounds are brittle materials. We also adopt the empirical scheme which put forwarded by Gao et al. [5] to evaluate the Vickers hardness (Hv) of four dense C3N4. The results showed that the hardness of c-C3N4 is slightly less than diamond but greater than c-BN. The bulk modulus BH, shear modulus GH, Young's modulus E and Vickers hardness Hv as a function of pressure in present work are illustrated in Fig. 6. From Fig. 6(a), we can see the bulk modulus of c-C3N4 is the largest in the four types of dense C3N4. Simultaneously, the bulk modulus of α–C3N4 is very close to that of p-C3N4, which confirms our previous predictions. Fig. 6(b) and (c) show the shear modulus and Young's modulus increase with the increasing of pressure. The shear modulus and Young's modulus of c-C3N4 are the largest at ambient temperature, however, they are smaller than that of p-C3N4 as the pressure is over 7.4 GPa and 36.0 GPa, respectively. It is well known that shear
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Fig. 6. Elastic modulus BH, shear modulus GH and Young's modulus E of various C3N4 versus pressures.
modulus G can be used to measure materials’ resistance to deformation and Young's modulus E can be a measurement of materials' rigidity [36,50]. Therefore, c-C3N4 has the strongest ability of resisting deformation in the range of 0–7.40 GPa, while p-C3N4 turn to have the strongest capacity as the pressure exceeds 7.40 GPa. At the same time, p-C3N4 is much stiffer than c-C3N4 as P 436 GPa. Fig. 6(d) shows the hardness of C3N4 increase monotonically with increasing pressure, demonstrating C3N4 will be more “hard” under high pressure again. Fig. 7 illustrates the change of ductility and brittleness of the four types dense C3N4 under different pressures. As shown in Fig. 7, the Poisson's ratio γ and B/G values increase monotonically, which suggest that higher pressure can weaken the brittleness of these compounds as the pressure increases. However, all the compounds are brittle materials even the pressure reaches 100 GPa except for β–C3N4. When PZ57 GPa, β–C3N4 will turn to ductility material. Actually the higher pressure is, the better the ductility of β–C3N4 is.
3.3. Elastic anisotropy Elastic anisotropy analysis is of great significance in understanding the mechanisms of materials' microcracks and durability [53]. There are a variety of methods to characterize the anisotropy of a crystal structure, for instance, the universal elastic anisotropy index AU [54], the percent compressibility of shear moduli factors (AB and AG) [55] and shear elastic anisotropic factors (A1, A2 and A3) [56]. The above anisotropic parameters can be defined as: AU ¼ 5
GV B V þ 6 GR B R
ð17Þ
Fig. 7. Poisson's ratio γ and rations of BH/GH of various C3N4 versus pressures.
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AB ¼
BV BR BV þ BR
ð18Þ
AG ¼
GV GR GV þ GR
ð19Þ
Where BV, BR, GV and GR represent bulk modulus and shear modulus in the Voigt and Reuss approximation, respectively. The shear anisotropic factors provide a measurement to the degree of anisotropy in the bonding between atoms in different planes. The shear anisotropic factor for the {1 0 0} shear planes between the 〈0 1 1〉 and 〈0 1 0〉 directions is A1 ¼
4C 44 C 11 þC 33 2C 13
ð20Þ
For the {0 1 0} shear planes between 〈1 0 1〉 and 〈0 0 1〉 directions is A2 ¼
4C 55 C 11 þC 22 2C 12
ð21Þ
and for the {0 0 1} shear planes between 〈1 1 0〉 and 〈0 1 0〉 directions is A3 ¼
4C 66 C 11 þC 22 2C 12
ð22Þ
In our paper, the calculated anisotropic parameters mentioned above at zero pressure and zero temperature are listed in Table 4. For a crystal, a value of zero represents elastic isotropy and a value of 1 (100%) represents the largest possible anisotropy [2]. Furthermore, the parameters A1, A2, and A3 must be 1 for isotropic crystals, so if any value of the three parameters is smaller or greater than 1, we could confirm that the crystal has elastic anisotropy [56]. It is obvious that the four types of dense C3N4 are anisotropic materials, and the anisotropy decreases in the following sequence: c-C3N4 4 β–C3N4 4p-C3N4 4 α-C3N4. It should be pointed out that the calculated AB of α-, c- and p-C3N4 are equal to 0, indicating the isotropy of their bulk modulus. The A3 of α-and β-C3N4 are 1, which suggests their isotropy at {0 0 1} shear planes. Moreover, since the shear anisotropic parameter A1 of α-, β- and p-C3N4 are equal to their corresponding parameter A2, so their {1 0 0} and {0 1 0} shear planes have the same anisotropy. And it is the same as {0 1 0} and {0 0 1} shear planes of c-C3N4. Table 4 The calculated universal anisotropic index (AU), percent anisotropy (AB and AG) and shear anisotropic factors (A1, A2 and A3) for various C3N4 phases at 0 GPa. Species
AU
AB (%)
AG (%)
A1
A2
A3
α–C3N4 β–C3N4 c-C3N4 p-C3N4
0.04263 0.17716 0.41492 0.14425
0 0.00045 0 0
0.00424 0.01732 0.03984 0.01422
0.86977 0.695424 3.57159 1.41206
0.86977 0.69542 1.78579 1.41206
1 1 1.78579 1.41181
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Fig. 8 shows the pressure dependence behavior of elastic anisotropy for the four dense C3N4. We can observe that, all the elastic anisotropy index AU increase monotonically with the pressure increasing. Typically, the change of elastic anisotropy index AU under high pressures is much larger than the others. Notably, Fig. 8(b) shows that AB of α-C3N4 increase monotonously with pressure increasing, whereas AB of β-C3N4 decreases with the increasing of pressure and both anisotropy values are equal to each other at 61 GPa. AB for c- and p-C3N4 is equal to 0 even at high pressure, which indicates that the external pressure has no obvious effect on the anisotropy of their bulk modulus. 3.4. Debye temperature The Debye temperature is an important parameter of solid materials, it can be used to calculated the vibrational internal energy, heat capacity and entropy. One of the standard methods for calculating Debye temperature is from elastic constant, since Debye temperature Φ is proportional to the average sound velocity vm by the equation [57]: h 3q N ρ 1=3 Θ¼ vm ð23Þ k 4π M Where h is the Planck constant, k is Boltzmann constant, q is the number of atoms in the molecule, N is Avogadro's number, ρ is the density and M is the molecular weight of the solid. The average sound velocity vm can be defined as follows [57]: " #! 1=3 1 2 1 vm ¼ þ ð24Þ 3 v3s v3l The vs and vl represent shear wave velocity and longitudinal wave velocity, respectively, and can be calculated as follows [57]: vs ¼ ðG=ρÞ1=2 vl ¼
1=2 4G =ρ Bþ 3
ð25Þ ð26Þ
Where B and G stand for isothermal bulk modulus and shear modulus, respectively, and the wave velocities and Debye temperature calculations are based on BH and GH. The calculated shear wave velocity vs, longitudinal wave velocity vl, average sound velocity vm and Debye temperature Θ of the four dense C3N4 phases at zero pressure and zero temperature are listed in Table 5. Notably, crystal with low density and high modulus may have large Debye temperature as well. The Debye temperature of the four dense C3N4 gradually decrease in order of c-C3N4 4 p-C3N4 4 α-C3N4 4 β-C3N4. Unfortunately, there are few reports on the Debye temperature of these compounds till now. Therefore, our calculated data can provide information for further experimental researches.
Fig. 8. Universal anisotropic index (AU), percent anisotropy (AB and AG) of the four types of C3N4 versus pressures. (a) AU, (b) AB, (c) AG.
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Table 5 The calculated shear wave velocity vs (km/s), longitudinal wave velocity vl (km/s), average sound velocity vm (km/s) and Debye temperature Θ (K) for various C3N4 phases at 0 GPa. Species
n
ρ
vs
vl
vm
ΦD
α-C3N4 β-C3N4 c-C3N4 p-C3N4
28 14 28 7
3.55 3.54 3.81 3.73
9.80 9.46 10.05 10.13
15.40 15.32 15.82 15.54
10.77 10.44 11.05 11.10
1750.10 1694.12 1839.81 1834.13
anisotropy decreases in the following sequence: c-C3N4 4
β–C3N4 4p-C3N4 4 α–C3N4. Besides, we calculated the sound velocities and Debye temperature of α-, β-, c- and p-C3N4 compounds. The results show that c-C3N4 has the highest Debye temperature with external pressure ranging from 0 to 6.3 GPa, while the Debye temperature of c-C3N4 is smaller than that of pC3N4 when external pressure exceed 6.3 GPa.
Acknowledgments This work was financially supported by the National Natural Science Foundation of China (21371002, 21303002), Natural Science Foundation of Anhui Province of China (1408085MB22), and Doctoral Scientific Research Foundation of Anhui University (02303319). References
Fig. 9. The Debye temperature Θ of various C3N4 versus pressures.
Fig. 9 shows the dependence of Debye temperature on the pressure range from 0 to 100 GPa. The c-C3N4 has the highest Debye temperature with external pressure ranging from 0 to 6.3 GPa. However, the Debye temperature of c-C3N4 is lower than p-C3N4 as external pressure exceeds 6.3 GPa. Besides, we find the Debye temperature of β-C3N4 changes a little with pressure increases, which indicates that the Debye temperature of β-C3N4 is less sensitive than that of the other three compounds. However, for the first time the Debye temperature of the four type of dense C3N4 under different pressure are calculated, so there are no experimental and theoretical values for comparison. Hence, our calculation results provide a very important prediction for future experiments.
4. Conclusion In short, the geometry, density of states, elastic properties, sound velocities and Debye temperature of α-, β-, c- and p-C3N4 compounds under pressure ranging from 0 to 100 GPa were investigated by first-principles calculation with generalized gradient approximation. The calculated results show that the a-axis of α–C3N4 is the most easily compressed in the four dense C3N4. The VB and CB are move away from the Fermi level, which induces the widened band gap of these compounds as external pressure increases. The elastic modulus can be obtained based on Voigt– Reuss–Hill approximations. The bulk modulus of c-C3N4 is the largest in the four dense C3N4, and the bulk modulus of α-C3N4 is very close to that of p-C3N4. The shear modulus and Young's modulus of c-C3N4 is the largest at ambient temperature, however, it is smaller than that of p-C3N4 as the pressure is over 7.4 GPa and 36.0 GPa, respectively. According to the Pugh's criterion, we conclude that α-, c- and p-C3N4 are brittle materials under 0 and even under 100 GPa. However, β–C3N4 will turn to ductile material as the external pressure exceeds 57 GPa. Elastic anisotropy analysis shows that α-, β-, c- and p-C3N4 are anisotropic materials, and the
[1] M. Wang, Y.W. Li, T. Cui, Y.M. Ma, G.T. Zou, Appl. Phys. Lett. 93 (2008) 101905. [2] Y.C. Ding, Physica B 407 (2012) 2282–2288. [3] X.Y. Zhang, J.Q. Qin, X.W. Sun, Y.N. Xue, M.Z. Ma, R.P. Liu, Phys. Chem. Chem. Phys. 15 (2013) 20894–20899. [4] F.M. Gao, Phys. Rev. B 73 (2006) 132104. [5] F.M. Gao, J.L. He, E.D. Wu, S.M. Liu, D.L. Yu, D.C. Li, S.Y. Zhang, Y.J. Tian, Phys. Rev. Lett. 91 (2003) 015502. [6] E. Gregoryanz, C. Sanloup, M. Somayazulu, J. Badro, G. Fiquet, H.K. Mao, R.J. Hemley, Nat. Mater. 3 (2004) 294–297. [7] H.Y. Chung, M.B. Weinberger, J.B. Levine, A. Kavner, J.M. Yang, S.H. Tolbert, R.B. Kaner, Science 316 (2007) 436–439. [8] H.Y. Gou, A.A. Tsirlin, E. Bykova, A.M. Abakumov, G.V. Tendeloo, A. Richter, S.V. Ovsyannikov, A.V. Kurnosov, D.M. Trots, Z. Konopkova, H.P. Liermann, L. Dubrovinsky, N. Dubrovinskaia, Phys. Rev. B 89 (2014) 064108. [9] Y.A. Kvashnina, A.G. Kvashnin, P.B. Sorokin, J. Appl. Phys. 114 (2013) 183708. [10] J.F. Li, R.Q. Zhang, Carbon 63 (2013) 571–573. [11] Z.G. Zhang, M.C. Lu, L. Zhu, L.L. Zhu, Y.D. Li, M. Zhang, Q. Li, Phys. Lett. A 378 (2014) 741–744. [12] V.L. Solozhenko, D. Andrault, G. Fiquet, M. Mezouar, D.C. Rubie, Appl. Phys. Lett. 78 (2001) 1385–1387. [13] F.M. Gao, D.D. Klug, J.S. Tse, J. Appl. Phys. 102 (2007) 084311. [14] V.L. Solozhenko, O.O. Kurakevych, D. Andrault, Y.L. Godec, M. Mezouar, Phys. Rev. Lett. 102 (2009) 015506. [15] M.G. Zhang, Q. Wei, H.Y. Yan, Y.R. Zhao, H. Wang, J. Phys. Chem. C 118 (2014) 3202-3208. [16] A.Y. Liu, M.L. Cohen, Science 245 (1989) 841–842. [17] D.M. Teter, R.J. Hemley, Science 271 (1996) 53–55. [18] M. Marqués, J. Osorio, R. Ahuja, M. Flórez, J.M. Recio, Phys. Rev. B 70 (2004) 104114. [19] J.L. He, L.C. Guo, X.J. Guo, R.P. Liu, Y.J. Tian, H.T. Wang, C.X. Gao, Appl. Phys. Lett. 88 (2006) 101906. [20] G.S. Manyali, R. Warmbier, A. Quandt, J.E. Lowther, Comput. Mater. Sci. 69 (2013) 299-303. [21] P. Hohenberg, W. Kohn, Phys. Rev 136 (1964) B864–B871. [22] W. Kohn, L.J. Sham, Phys. Rev 140 (1965) A1133–A1138. [23] M.D. Segall, P.J.D. Lindan, M.J. Probert, C.J. Pickard, P.J. Hasnip, S.J. Clark, M. C. Payne, J. Phys.: Condens. Matter 14 (2002) 2717–2744. [24] J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865–3868. [25] S. Grimme, J. Comput. Chem. 25 (2004) 1463–1473. [26] S. Grimme, J. Comput. Chem. 27 (2006) 1787–1799. [27] S. Grimme, J. Antony, S. Ehrlich, H. Krieg, J. Chem. Phys. 132 (2010) 154104. [28] T.H. Fischer, J. Almlof, J. Phys. Chem. 96 (1992) 9768–9774. [29] D. Vanderbilt, Phys. Rev. B 41 (1990) 7892–7895. [30] H.J. Monkhorst, J.D. Pack, Phys. Rev. B 13 (1976) 5188–5192. [31] D.W. He, F.X. Zhang, X.Y. Zhang, Z.C. Qin, M. Zhang, R.P. Liu, Y.F. Xu, W.K. Wang, J. Mater. Res. 13 (1998) 3458–3462. [32] B. Molina, L.E. Sansores, Mod. Phys. Lett. B 13 (1999) 193–201. [33] Z.H. Zhang, H.X. Guo, G.R. Zhong, F.W. Yu, Q.H. Xiong, X.J. Fan, Thin Solid Films 346 (1999) 96–99. [34] Y. Xu, S.P. Gao, J. Int., Hydrogen Energy 37 (2012) 11072–11080. [35] M. Mattesini, S.F. Matar, Comput. Mater. Sci. 20 (2001) 107–119. [36] H. Hou, Z.Q. Wen, Y.H. Zhao, L. Fu, N. Wang, P.D. Han, Intermetallics 44 (2014) 110–115. [37] Q.Y. Fan, Q. Wei, H.Y. Yan, M.G. Zhang, Z.X. Zhang, J.Q. Zhang, D.Y. Zhang, Comput. Mater. Sci. 85 (2014) 80–87. [38] J.P. Watt, L. Peselnick, J. Appl. Phys. 51 (1980) 1525–1531. [39] J. Haines, J.M. L'eger, G. Bocquillon, Annu. Rev. Mater. Res. 31 (2001) 1–23. [40] J.F. Nye, Physical Properties of Crystals: Their Representation by Tensors and Matrices, Oxford University Press, Oxford, 1985. [41] P.R. Heyliger, W.L. Johnson, J. Acoust. Soc. Am. 113 (2003) 1812–1825. [42] N. Mounet, N. Marzari, Phys. Rev. B 71 (2005) 205214.
L.-W. Ruan et al. / Journal of Physics and Chemistry of Solids 75 (2014) 1324–1333
[43] M. Grimsditch, E.S. Zouboulis, A. Polian, J. Appl. Phys. 76 (1994) 832–834. [44] Y.D. Guo, X.D. Yang, X.B. Li, X.S. Song, X.L. Cheng, Diamond Relat. Mater. 17 (2008) 1–6. [45] A.Y. Liu, R.M. Wentzcovitch, Phys. Rev. B 50 (1994) 10362–10365. [46] Y. Zhang, H. Sun, C.F. Chen, Phys. Rev. B 73 (2006) 064109. [47] B. Zhu, C.M. Liu, M.B. Lv, X.R. Chen, J. Zhu, G.F. Ji, Physica B 406 (2011) 3508–3513. [48] A. Reuss, Z. Angew. Math. Mech. 9 (1929) 49–58. [49] J.J. Lewandowski, W.H. Wang, A.L. Greer, Philos. Mag. Lett. 85 (2005) 77–87. [50] S.F. Pugh, XCII. Relations between the elastic moduli and the plastic properties of polycrystalline pure metals, Philos. Mag. Ser. 7 45 (1954) 823–843.
1333
[51] D.M. Teter, MRS Bull. 23 (1998) 22–27. [52] J.B. Levine, S.L. Nguyen, H.I. Rasool, J.A. Wright, S.E. Brown, R.B. Kaner, J. Am. Chem. Soc. 130 (2008) 16953–16958. [53] Y.H. Duan, Y. Sun, M.J. Peng, S.G. Zhou, J. Alloys Compd. 595 (2014) 14–21. [54] S.I. Ranganathan, M. Ostoja-Starzewski, Phys. Rev. Lett. 101 (2008) 055504. [55] D.H. Chung, W.R. Buessem, Anisotropy in Single-Crystal Refractory Compound, in: F.W. Vahldiek, S.A. Mersol (Eds.), Plenum, New York, 1968, p. 217. [56] P. Ravindran, L. Fast, P.A. Korzhavyi, B. Johansson, J. Wills, O. Eriksson, J. Appl. Phys. 84 (1998) 4891–4904. [57] O.L. Anderson, J. Phys. Chem. Solids 24 (1963) 909–917.
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Journal of Physics and Chemistry of Solids journal homepage: www.elsevier.com/locate/jpcs
The elastic behavior of dense C3N4 under high pressure: First-principles calculations Lin-Wei Ruan a, Geng-Sheng Xu a, Hai-Yu Chen a, Yu-Peng Yuan a, Xia Jiang a, Yun-Xiang Lu b, Yu-Jun Zhu a,n a b
Laboratory of Advanced Porous Materials, School of Chemistry and Chemical Engineering, Anhui University, No. 111, Jiulong Road, Hefei 230601, PR China School of Chemistry and Molecular Engineering, East China University of Science and Technology, Shanghai 200237, PR China
art ic l e i nf o
a b s t r a c t
Article history: Received 1 June 2014 Received in revised form 8 July 2014 Accepted 9 July 2014 Available online 18 July 2014
We have carried a detailed theoretical study on the geometry, density of states, elastic properties, sound velocities and Debye temperature of α-, β-, c- and p-C3N4 compounds under a maximum of pressure up to 100 GPa by using first principles calculations. The optimized lattice constants under zero pressure and zero temperature agreed well with the previous experimental and theoretical results. The band gaps of the four types of dense C3N4 were widened gradually with the increase of pressure. The calculated Poisson’s ratio γ and B/G values suggest α-, c- and p-C3N4 are brittle materials under 0–100 GPa, whereas β-C3N4 will become a ductile material as external pressure reaches 57 GPa. We found that the Debye temperature of the four dense C3N4 gradually reduces in the order of c-C3N4 4 p-C3N4 4 α-C3N4 4 β-C3N4 at 0 GPa and 0 K. However, the Debye temperature of c-C3N4 was lower than p-C3N4 when external pressure exceeds 6.3 GPa. It may hint that the results could be served as a valuable prediction for further experiments. & 2014 Elsevier Ltd. All rights reserved.
Keywords: C. High pressure D. Elastic properties
1. Introduction Super-hard materials are of great importance in scientific development and industrial production because of their applications in cutting, grind, polishing and precision instrument processing, etc [1–3]. Nowadays, diamond and cubic boron nitride (c-BN) are the most common super-hard materials. Diamond is currently known as the world's hardest industrial materials with a Vickers hardness of 97.7 GPa reported by Gao [4]. c-BN is another superhard material with slightly lower hardness compared to diamond [4,5]. Both of them have been an integral part of the functional materials in industry. Apart from the two well-known super-hard materials, many research activities have been devoted to search for new super-hard materials [6–8] or their allotropes [9–11] which have the same high mechanical properties as diamond and c-BN. For the last few decades, various studies have been performed to explore covalent compounds formed by light elements, namely, boron, carbon and nitrogen due to their ability of forming short and strong three dimensional covalent bonds [2]. Super-hard materials like BC2N [12], B4C3 [13], BC5 [14], m-CN [15] and others have been reported. Since 1989 Liu and Cohen reported a new
compound β-C3N4 which has a similar bulk modulus as that of diamond [16]. The results ignited an interest in studying carbon nitride compounds. Further theoretical calculations proposed five polymorphs of C3N4, including α-C3N4, β–C3N4, c-C3N4, p-C3N4 and g-C3N4 [17]. Among them, the first four is thought to be super-hard materials, and the c-C3N4 is expected to be a material harder than a diamond. Henceforth, numerous studies on the mechanical properties of dense C3N4 at ambient conditions have been reported [18–20]. However, there are few reports about the behavior of the properties for dense C3N4 under high pressure whether by theoretical or experimental approaches. This may be due to the difficulty in synthesizing good dense C3N4 single crystals. In this study, we preformed the first principles calculations to investigate the geometry, density of states, elastic properties, sound velocities and Debye temperature of α-, β-, c-C3N4 and p-C3N4 compounds under a wide pressure range (0–100 GPa). Besides, the elastic properties of dense C3N4 were compared with diamond and c-BN, and the results indicated that dense C3N4 could be a substitution of the diamond and c-BN.
2. Computational details n
Corresponding author. Tel.: þ 86 551 63873365. E-mail address: [email protected] (Y.-J. Zhu).
http://dx.doi.org/10.1016/j.jpcs.2014.07.010 0022-3697/& 2014 Elsevier Ltd. All rights reserved.
The alpha (α-), beta (β-) and cubic (c-) together with pseudocubic phases C3N4 (p-C3N4) were investigated based on the Density
L.-W. Ruan et al. / Journal of Physics and Chemistry of Solids 75 (2014) 1324–1333
Functional Theory (DFT) [21,22] using the Cambridge Serial Total Energy Package (CASTEP) plane-wave code [23] in Materials Studio 7.0 software. The calculations were performed with the Generalized Gradient Approximation (GGA) in the form of Perdew–Burke–Ernzerhof (PBE) exchange correlation potential [24]. In the calculations, a hybrid semiempirical solution Grimme approach [25–27] was taken to introduce damped atom-pairwise dispersion corrections of the form C6R 6 in the DFT formalism. The semiempirical approach provides the best compromise between the cost of first principles evaluation of the dispersion terms and the need to improve non-bonding interactions in the standard DFT description. The structural optimizations were conducted using the Broyden–Fletcher–Goldfarb–Shanno (BFGS) minimization [28]. The interactions between the ionic core and valence electrons were described by the ultrasoft pseudo-potential [29]. The considered valence atomic configurations are 2s22p2 for C and 2s22p3 for N. We performed a cut-off of 700 eV, 800 eV, 820 eV and 830 eV for α-, β-, c- and p-C3N4, respectively. A Monkhorst-Pack k-mesh [30] of 3 3 4, 5 5 11, 6 6 6 and 8 8 8 were used for α-, β-, c- and p-C3N4, respectively. In order to obtain more accurate results, the Self Consistent Field (SCF) tolerance threshold was set at 5.0 10 7 eV/atom. The total energy changes during the structure optimization were less than 5.0 10 6 eV/atom, the maximum force were smaller than 0.01 eV/Å, the maximum stress were smaller than 0.02 GPa and the maximum displacement were converged to 5.0 10 4 Å. All the electronic structures, elastic properties, elastic anisotropic properties, and Debye temperatures were calculated based on the optimized crystal structures.
3. Results and discussion 3.1. Geometric and electronic structures Structure of unit cell plays a very important role in understanding the nature of solid materials. Fig. 1 shows the three-dimensional
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structure of four dense C3N4 phases. The N atoms are 3-fold coordination by three C atoms. The α-C3N4 is trigonal structure with space group of P-31c, each cell includes four units. The β-C3N4 is trigonal structure as well and belongs to space group of P-3, each cell includes two units. The c-C3N4 and p-C3N4 have higher symmetrical structure which are cubic (with I-4̄3d space group) and tetragonal (with P-4̄ 2m space group) structure respectively. One cell includes four units in c-C3N4 while one unit in p-C3N4. The calculated lattice parameters, bond overlap population and bond length as well as other available values of four dense C3N4 at 0 GPa and 0 K are shown in Table 1. And the computed results are in agreement with the experimental and previous theoretical results well, which imply that our calculation methods and results are reasonable and authentic. In order to investigate the variation of structural properties of the four types of dense C3N4 phases under different pressures, we displayed the pressure dependence of the normalized structural parameters a/a0 and c/c0 as well as V/V0 with the pressure ranging from 0 to 100 GPa as shown in Fig. 2. The values a0, c0 and V0 are the equilibrium structure parameters at 0 GPa as listed in Table 1. Fig. 2(a) shows that, as pressure increased, all the lattice parameters decreased, especially the ratio of a/a0 of α-C3N4 decreased more quickly compared to the others. The ratio of a/a0 of c-C3N4 decreases the slowest, which means that a-axis of α–C3N4 is the most easily compressed in the four types of dense C3N4. Fig. 2 (b) shows that, as pressure increased, the volume of c-C3N4 decreased more slowly than other allotropes. Besides, the volumes of α-C3N4 and p-C3N4 decreased more quickly and the both curves almost coincide with each other. The bulk modulus B is a measure of resistance against a change of volume of a material caused by external pressures [35]. Therefore, we can predict that the bulk modulus of c-C3N4 is the largest in the four dense C3N4, and the bulk modulus of α-C3N4 is very close to p-C3N4. Most covalent compounds formed by light elements (such as boron, carbon and nitrogen) are more likely to be super-hard
Fig. 1. Unit cells of four super hard C3N4 phases. (a) α-C3N4, (b) β–C3N4, (c) c-C3N4, (d) p-C3N4. Carbon and nitrogen atoms are depicted in gray and blue balls, respectively.
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Table 1 Lattice parameters, bond overlap population and length of four different C3N4 phases at 0 GPa. Species
a0/Å
α–C3N4
6.4888 6.425b 6.4255 6.419b 5.4328 5.36d 3.4488 3.426e
β–C3N4 c-C3N4 p-C3N4
c0/Å 6.4665a 6.453c 6.4017a 6.394c 5.3973a 4.673c 3.4232a 3.421c
4.7288 4.715b 2.4184 2.425b
4.7097a 4.699c 2.4041a 2.397c
V0/Å3
Bond overlap population
Bond length/Å
Space group
172.4303 169.44c 86.4714 84.86c 160.3508 157.08c 41.0200 40.04c
0.78 0.77 0.73 0.79 0.81 0.85f, 0.81g 0.69 0.75g
1.46 1.4522g 1.46 1.4519g 1.47 1.461f, 1.4624g 1.48 1.4749g
P-31c P-3 I-4̄ 3d P-4̄ 2m
a
Ref. [17]. Ref. [31]. Ref. [32]. d Ref. [33]. e Ref. [34]. f Ref. [4]. g Ref. [19]. b c
Fig. 2. The normalized parameters a/a0, c/c0 and V/V0 of four dense C3N4 phases as a function of pressure.
materials [2]. α-, β-, c- and p-C3N4 satisfy the above conditions well. Fig. 3(a) shows the average C–N bond length of four dense C3N4 with pressure from 0 to 100 GPa. All the bond lengths decreased with the increasing of pressure, indicating that the dense C3N4 phases will be more “hard” under high pressures. Fig. 3 (b) shows that all the bond overlap populations of C3N4 increased with the increasing of pressure, which demonstrated that the covalent characters were enhanced over these changes. However, the overlap population of p-C3N4 remained almost unchanged in the pressure range of 40–100 GPa, indicating that the covalent character of p-C3N4 changed little as the external pressure exceeds 40 GPa. We further analyzed the electronic properties of the four dense C3N4 at different pressures in details. The total and partial density of states of the four dense C3N4 phases at zero and high pressures are shown in Fig. 4. The Fermi level is set to 0 eV. We can see that the four dense phases of C3N4 are semiconductors due to their relatively wide band gap, which is consistent with other similar theoretical reports [32–34]. We found the valence band (VB) of these compounds is mainly consisting of N-2p states. The conduction band (CB) of α-, β- and c-C3N4 are dominated by N-2s, N-2p
and C-2p states. However, the conduction band of p-C3N4 is mainly dominated by N-2p and C-2p states. The calculated band gaps of α-, β-, c- and p-C3N4 at 0 GPa are 3.808, 3.235, 2.966, 2.475 eV, accordingly, which coincide well with the previous results [17,32,34]. Furthermore, Fig. 4 shows that the band gaps of compounds are widened as the pressure increases. The valence bands and conduction bands keep away from the Fermi level and the peaks of the DOSs are dropped with the pressure increasing. 3.2. Elastic properties Elastic constants of solid compounds are important because of their close relations with various fundamental physical properties, such as elastic modulus, Debye temperature and phonon spectra [36,37]. There are six independent elastic constants for trigonal (C11, C12, C13, C14, C33, C44) and tetragonal (C11, C12, C13, C33, C44, C66) phases crystals [38]. However, for higher symmetrical cubic phase, there are only three independent elastic constants exist, i.e. C11, C12, C44 [39]. The elastic stiffness constants were calculated by a stress–strain method. In our study, the calculated independent elastic constants for four dense C3N4 as well as previous results
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The bulk modulus B, shear modulus G, Young's modulus E and Poisson's ratio γ can be directly derived with these elastic constants by the Voigte–Reusse–Hill approximations (VRH) [48]. The Voigt and Rees’s schemes represent the upper and lower bounds to the elastic moduli, respectively. In the following formulas, subscript V denotes the Voigt bound, R denotes the Reuss bound, and H denotes the Hill average. For trigonal phase, the formula can be defined as [38]: 1 BV ¼ ½2ðC 11 þC 12 Þ þ 4C 13 þ C 33 9 GV ¼
ð1Þ
1 ðM þ 12C 44 þ 12C 66 Þ 30
ð2Þ
Where M ¼ C 11 þ C 12 þ 2C 33 4C 13
ð3Þ
and 1 C 66 ¼ ðC 11 C 12 Þ 2
ð4Þ
BR ¼ C 2 =M
ð5Þ
GR ¼
5 C 2 ðC 44 C 66 C 214 Þ 2 3K V ðC 44 C 66 C 214 Þ þC 2 ðC 44 þ C 66 Þ
ð6Þ
Where C 2 ¼ ðC 11 þ C 12 ÞC 33 2C 213
ð7Þ
For cubic phase, the formula is [40]
Fig. 3. Calculated average C–N bond length and average bond overlap population of four different C3N4 as a function of pressure.
and comparable results for diamond and c-BN at ambient constants are tabulated in Table 2. The mechanical stability tests is satisfied for all systems of the four dense C3N4 [40]. As shown in Table 2, the as-obtained results are in close proximity to previous values, though there is slight disparity caused by the precision setting and method of calculations. Additionally, we find the elastic constants C11 and C44 of diamond are larger than the four types of dense C3N4, but C12 is much smaller. Interestingly, the elastic constant C14 of α-C3N4 and β-C3N4 are equal to zero, which indicate that both of the compounds are transversely isotropic [41]. For understanding the variation of elastic constants at different pressures more clearly, we use Fig. 5 to illustrate these data. It could be concluded from the figure: (1) as pressure increases, all the elastic constants increase monotonically, demonstrating the elastic properties of the four dense C3N4 can be enhanced by high pressure. We also found the lines of the elastic constants C11 and C33 located far above the other elastic constants. To the best of our knowledge, the elastic constant C11 (C22 and C33) represents the elasticity in length, which can change with longitudinal strain, and C12 (C13 and C23) and C44 (C55 and C66) are related to the elasticity in shape [47]. The results indicate the resisting axial stress abilities of the four dense C3N4 are much stronger than that of deformation. (2) C12 exceeds C44 at 73.75 and 7.98 GPa respectively, which indicate that the sensibility to pressure of the C44 of α-C3N4 and β-C3N4 is lower than that of C12. (3) The elastic constant C14 of α-C3N4 and β-C3N4 remained zero with pressure increasing, indicating the change of pressure does not affect the transversely isotropy of α-C3N4 and β-C3N4. (4) As shown in Fig. 5(d), the lines stand for elastic constants C11, C12 and C44 overlap with C33, C13 and C66, respectively, which demonstrate that the elastic properties of p-C3N4 is similar to that of c-C3N4.
1 BV ¼ ðC 11 þ 2C 12 Þ 3
ð8Þ
1 GV ¼ ðC 11 C 12 þ 3C 44 Þ 5
ð9Þ
5ðC 11 C 12 ÞC 44 4C 44 þ 3ðC 11 C 12 Þ
ð10Þ
GR ¼
and the Reuss bound on the bulk modulus is same to BV. For tetragonal phase, the Voigt bound bulk modulus is given according to Eq. (1), while the Voigt shear modulus can be defined as [38]: GV ¼
1 ðM þ 3C 11 3C 12 þ 12C 44 þ 6C 66 Þ 30
ð11Þ
and M is as the formula (3). The Reuss bulk modulus is given by Eq. (5), but the Reuss shear modulus is GR ¼ 15½18BV =C 2 þ 6=ðC 11 C 12 Þ þ6=C 44 þ 3=C 66 1
ð12Þ
and C has the same meaning of formula (7). The elastic modulus based on Hill approximation is an average of Voigt bound and Reuss bound [48]: 1 BH ¼ ðBV þ BR Þ 2
ð13Þ
1 GH ¼ ðGV þ GR Þ 2
ð14Þ
The Young's modulus E and Poisson's ratio by the following formulas [48]:
γ can be calculated
E¼
9BH GH 3BH þ GH
ð15Þ
γ¼
3BH 2GH 2ð3BH þ GH Þ
ð16Þ
The calculated elastic modulus B, shear modulus G, Young's modulus E, Poisson's and ratio γ of four dense C3N4 and diamond
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Fig. 4. Total and partial density of states for four dense C3N4 phases at different pressures. The blue lines represent the results are obtained at 0 GPa, red and green lines correspond to 60 GPa and 100 GPa, respectively. (a) α-C3N4, (b) β-C3N4, (c) c-C3N4, (d) p-C3N4. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Table 2 Calculated independent elastic constants Cij (GPa) for various dense C3N4 phases at 0 GPa. Species Diamond c-BN α–C3N4 β–C3N4 c-C3N4 p-C3N4 a
Ref. [42]. Ref. [20]. c Ref. [43]. d Ref. [44]. e Ref. [45]. f Ref. [46]. b
C12
C11 a
b
1060 , 1097 820c, 828.2d 851.21, 833.20, 834e 804.16, 861a 804.36
C13 a
C14
C33
b
150 , 125 190c, 201.2d 183.01 258.89, 279e 259.60, 300a 182.77
C44
C66 a
128.52 110.15, 138e 182.74
0 0, 0e
905.91 1049.05, 1120e 804.95
b
595 , 562 480c, 492.3d 326.18 288.94, 305e 486.24, 469a 439.09, 452f
334.10 287.16 438.78
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Fig. 5. Independent elastic constants Cij of various C3N4 versus pressures. (a) α-C3N4, (b) β-C3N4, (c) c-C3N4, (d) p-C3N4.
Table 3 The calculated results of elastic modulus B (GPa), shear modulus G (GPa), Young's modulus E (GPa), Poisson's ratio γ and Vickers hardness Hv(GPa) for Diamond, c-BN and various dense C3N4 phases at 0 GPa. Species
BV
Diamond c-BN α-C3N4 β-C3N4 c-C3N4 p-C3N4
438.8a 409f 387.60, 378.7a 408.21, 437b, 419.1a, 442f 441.12, 496b, 449.2a, 487c 390.02, 393.2a
BR
BH
GV
387.60 407.84 441.12 390.02
400e 387.60 408.02 441.12 390.02
535b, 546c 368f 341.85 322.09, 320b 400.65, 332b, 393c 387.76
GR
GH
E
338.96 311.12 369.95 376.88
409e 340.40 316.61 385.30 382.32
1178c 915.1e 789.95 754.63 895.25, 930c 864.49
BH/GH
1.14 1.29 1.14 1.02
γ
Hv
0.08
97.7d, 93.6e 79.1d, 66e 82.36, 82.7a 62.62, 60.4d 90.18, 87.1d 69.83, 79.6a
0.16 0.19 0.16, 0.18c 0.13
a
Ref. [19]. Ref. [51]. c Ref. [20]. d Ref. [52]. e Ref. [2]. f Ref. [18]. b
as well as c-BN at 0 GPa and 0 K are listed in Table 3. From the Table, we note that our results are in agreement with previous values well. Beyond that, c-C3N4 has the highest bulk modulus of 441.12 GPa, which is slightly larger than diamond. The shear modulus (385.30 GPa) and Young's modulus (895.25 GPa) of c-C3N4 are the highest in the four type of dense C3N4, which are also less than that of diamond. The Poisson's ratio γ can be judgment of ductility and brittleness, for a brittle material γ o 0.26, while for a ductile material γ Z 0.26 [49]. A conclusion could be drawn from Table 3 that the four types of dense C3N4 including diamond are brittle materials because of their Poisson's ratio are smaller than 0.26. In addition, diamond exhibits the smallest brittleness in these compounds. On the other hand, according to Pugh's criteria [50], a kind of material is brittle if the B/G ratio is less than 1.75, otherwise the material is ductile, which is consistent with the result judged by γ. Obviously,
our data further manifests the conclusions that these compounds are brittle materials. We also adopt the empirical scheme which put forwarded by Gao et al. [5] to evaluate the Vickers hardness (Hv) of four dense C3N4. The results showed that the hardness of c-C3N4 is slightly less than diamond but greater than c-BN. The bulk modulus BH, shear modulus GH, Young's modulus E and Vickers hardness Hv as a function of pressure in present work are illustrated in Fig. 6. From Fig. 6(a), we can see the bulk modulus of c-C3N4 is the largest in the four types of dense C3N4. Simultaneously, the bulk modulus of α–C3N4 is very close to that of p-C3N4, which confirms our previous predictions. Fig. 6(b) and (c) show the shear modulus and Young's modulus increase with the increasing of pressure. The shear modulus and Young's modulus of c-C3N4 are the largest at ambient temperature, however, they are smaller than that of p-C3N4 as the pressure is over 7.4 GPa and 36.0 GPa, respectively. It is well known that shear
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Fig. 6. Elastic modulus BH, shear modulus GH and Young's modulus E of various C3N4 versus pressures.
modulus G can be used to measure materials’ resistance to deformation and Young's modulus E can be a measurement of materials' rigidity [36,50]. Therefore, c-C3N4 has the strongest ability of resisting deformation in the range of 0–7.40 GPa, while p-C3N4 turn to have the strongest capacity as the pressure exceeds 7.40 GPa. At the same time, p-C3N4 is much stiffer than c-C3N4 as P 436 GPa. Fig. 6(d) shows the hardness of C3N4 increase monotonically with increasing pressure, demonstrating C3N4 will be more “hard” under high pressure again. Fig. 7 illustrates the change of ductility and brittleness of the four types dense C3N4 under different pressures. As shown in Fig. 7, the Poisson's ratio γ and B/G values increase monotonically, which suggest that higher pressure can weaken the brittleness of these compounds as the pressure increases. However, all the compounds are brittle materials even the pressure reaches 100 GPa except for β–C3N4. When PZ57 GPa, β–C3N4 will turn to ductility material. Actually the higher pressure is, the better the ductility of β–C3N4 is.
3.3. Elastic anisotropy Elastic anisotropy analysis is of great significance in understanding the mechanisms of materials' microcracks and durability [53]. There are a variety of methods to characterize the anisotropy of a crystal structure, for instance, the universal elastic anisotropy index AU [54], the percent compressibility of shear moduli factors (AB and AG) [55] and shear elastic anisotropic factors (A1, A2 and A3) [56]. The above anisotropic parameters can be defined as: AU ¼ 5
GV B V þ 6 GR B R
ð17Þ
Fig. 7. Poisson's ratio γ and rations of BH/GH of various C3N4 versus pressures.
L.-W. Ruan et al. / Journal of Physics and Chemistry of Solids 75 (2014) 1324–1333
AB ¼
BV BR BV þ BR
ð18Þ
AG ¼
GV GR GV þ GR
ð19Þ
Where BV, BR, GV and GR represent bulk modulus and shear modulus in the Voigt and Reuss approximation, respectively. The shear anisotropic factors provide a measurement to the degree of anisotropy in the bonding between atoms in different planes. The shear anisotropic factor for the {1 0 0} shear planes between the 〈0 1 1〉 and 〈0 1 0〉 directions is A1 ¼
4C 44 C 11 þC 33 2C 13
ð20Þ
For the {0 1 0} shear planes between 〈1 0 1〉 and 〈0 0 1〉 directions is A2 ¼
4C 55 C 11 þC 22 2C 12
ð21Þ
and for the {0 0 1} shear planes between 〈1 1 0〉 and 〈0 1 0〉 directions is A3 ¼
4C 66 C 11 þC 22 2C 12
ð22Þ
In our paper, the calculated anisotropic parameters mentioned above at zero pressure and zero temperature are listed in Table 4. For a crystal, a value of zero represents elastic isotropy and a value of 1 (100%) represents the largest possible anisotropy [2]. Furthermore, the parameters A1, A2, and A3 must be 1 for isotropic crystals, so if any value of the three parameters is smaller or greater than 1, we could confirm that the crystal has elastic anisotropy [56]. It is obvious that the four types of dense C3N4 are anisotropic materials, and the anisotropy decreases in the following sequence: c-C3N4 4 β–C3N4 4p-C3N4 4 α-C3N4. It should be pointed out that the calculated AB of α-, c- and p-C3N4 are equal to 0, indicating the isotropy of their bulk modulus. The A3 of α-and β-C3N4 are 1, which suggests their isotropy at {0 0 1} shear planes. Moreover, since the shear anisotropic parameter A1 of α-, β- and p-C3N4 are equal to their corresponding parameter A2, so their {1 0 0} and {0 1 0} shear planes have the same anisotropy. And it is the same as {0 1 0} and {0 0 1} shear planes of c-C3N4. Table 4 The calculated universal anisotropic index (AU), percent anisotropy (AB and AG) and shear anisotropic factors (A1, A2 and A3) for various C3N4 phases at 0 GPa. Species
AU
AB (%)
AG (%)
A1
A2
A3
α–C3N4 β–C3N4 c-C3N4 p-C3N4
0.04263 0.17716 0.41492 0.14425
0 0.00045 0 0
0.00424 0.01732 0.03984 0.01422
0.86977 0.695424 3.57159 1.41206
0.86977 0.69542 1.78579 1.41206
1 1 1.78579 1.41181
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Fig. 8 shows the pressure dependence behavior of elastic anisotropy for the four dense C3N4. We can observe that, all the elastic anisotropy index AU increase monotonically with the pressure increasing. Typically, the change of elastic anisotropy index AU under high pressures is much larger than the others. Notably, Fig. 8(b) shows that AB of α-C3N4 increase monotonously with pressure increasing, whereas AB of β-C3N4 decreases with the increasing of pressure and both anisotropy values are equal to each other at 61 GPa. AB for c- and p-C3N4 is equal to 0 even at high pressure, which indicates that the external pressure has no obvious effect on the anisotropy of their bulk modulus. 3.4. Debye temperature The Debye temperature is an important parameter of solid materials, it can be used to calculated the vibrational internal energy, heat capacity and entropy. One of the standard methods for calculating Debye temperature is from elastic constant, since Debye temperature Φ is proportional to the average sound velocity vm by the equation [57]: h 3q N ρ 1=3 Θ¼ vm ð23Þ k 4π M Where h is the Planck constant, k is Boltzmann constant, q is the number of atoms in the molecule, N is Avogadro's number, ρ is the density and M is the molecular weight of the solid. The average sound velocity vm can be defined as follows [57]: " #! 1=3 1 2 1 vm ¼ þ ð24Þ 3 v3s v3l The vs and vl represent shear wave velocity and longitudinal wave velocity, respectively, and can be calculated as follows [57]: vs ¼ ðG=ρÞ1=2 vl ¼
1=2 4G =ρ Bþ 3
ð25Þ ð26Þ
Where B and G stand for isothermal bulk modulus and shear modulus, respectively, and the wave velocities and Debye temperature calculations are based on BH and GH. The calculated shear wave velocity vs, longitudinal wave velocity vl, average sound velocity vm and Debye temperature Θ of the four dense C3N4 phases at zero pressure and zero temperature are listed in Table 5. Notably, crystal with low density and high modulus may have large Debye temperature as well. The Debye temperature of the four dense C3N4 gradually decrease in order of c-C3N4 4 p-C3N4 4 α-C3N4 4 β-C3N4. Unfortunately, there are few reports on the Debye temperature of these compounds till now. Therefore, our calculated data can provide information for further experimental researches.
Fig. 8. Universal anisotropic index (AU), percent anisotropy (AB and AG) of the four types of C3N4 versus pressures. (a) AU, (b) AB, (c) AG.
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Table 5 The calculated shear wave velocity vs (km/s), longitudinal wave velocity vl (km/s), average sound velocity vm (km/s) and Debye temperature Θ (K) for various C3N4 phases at 0 GPa. Species
n
ρ
vs
vl
vm
ΦD
α-C3N4 β-C3N4 c-C3N4 p-C3N4
28 14 28 7
3.55 3.54 3.81 3.73
9.80 9.46 10.05 10.13
15.40 15.32 15.82 15.54
10.77 10.44 11.05 11.10
1750.10 1694.12 1839.81 1834.13
anisotropy decreases in the following sequence: c-C3N4 4
β–C3N4 4p-C3N4 4 α–C3N4. Besides, we calculated the sound velocities and Debye temperature of α-, β-, c- and p-C3N4 compounds. The results show that c-C3N4 has the highest Debye temperature with external pressure ranging from 0 to 6.3 GPa, while the Debye temperature of c-C3N4 is smaller than that of pC3N4 when external pressure exceed 6.3 GPa.
Acknowledgments This work was financially supported by the National Natural Science Foundation of China (21371002, 21303002), Natural Science Foundation of Anhui Province of China (1408085MB22), and Doctoral Scientific Research Foundation of Anhui University (02303319). References
Fig. 9. The Debye temperature Θ of various C3N4 versus pressures.
Fig. 9 shows the dependence of Debye temperature on the pressure range from 0 to 100 GPa. The c-C3N4 has the highest Debye temperature with external pressure ranging from 0 to 6.3 GPa. However, the Debye temperature of c-C3N4 is lower than p-C3N4 as external pressure exceeds 6.3 GPa. Besides, we find the Debye temperature of β-C3N4 changes a little with pressure increases, which indicates that the Debye temperature of β-C3N4 is less sensitive than that of the other three compounds. However, for the first time the Debye temperature of the four type of dense C3N4 under different pressure are calculated, so there are no experimental and theoretical values for comparison. Hence, our calculation results provide a very important prediction for future experiments.
4. Conclusion In short, the geometry, density of states, elastic properties, sound velocities and Debye temperature of α-, β-, c- and p-C3N4 compounds under pressure ranging from 0 to 100 GPa were investigated by first-principles calculation with generalized gradient approximation. The calculated results show that the a-axis of α–C3N4 is the most easily compressed in the four dense C3N4. The VB and CB are move away from the Fermi level, which induces the widened band gap of these compounds as external pressure increases. The elastic modulus can be obtained based on Voigt– Reuss–Hill approximations. The bulk modulus of c-C3N4 is the largest in the four dense C3N4, and the bulk modulus of α-C3N4 is very close to that of p-C3N4. The shear modulus and Young's modulus of c-C3N4 is the largest at ambient temperature, however, it is smaller than that of p-C3N4 as the pressure is over 7.4 GPa and 36.0 GPa, respectively. According to the Pugh's criterion, we conclude that α-, c- and p-C3N4 are brittle materials under 0 and even under 100 GPa. However, β–C3N4 will turn to ductile material as the external pressure exceeds 57 GPa. Elastic anisotropy analysis shows that α-, β-, c- and p-C3N4 are anisotropic materials, and the
[1] M. Wang, Y.W. Li, T. Cui, Y.M. Ma, G.T. Zou, Appl. Phys. Lett. 93 (2008) 101905. [2] Y.C. Ding, Physica B 407 (2012) 2282–2288. [3] X.Y. Zhang, J.Q. Qin, X.W. Sun, Y.N. Xue, M.Z. Ma, R.P. Liu, Phys. Chem. Chem. Phys. 15 (2013) 20894–20899. [4] F.M. Gao, Phys. Rev. B 73 (2006) 132104. [5] F.M. Gao, J.L. He, E.D. Wu, S.M. Liu, D.L. Yu, D.C. Li, S.Y. Zhang, Y.J. Tian, Phys. Rev. Lett. 91 (2003) 015502. [6] E. Gregoryanz, C. Sanloup, M. Somayazulu, J. Badro, G. Fiquet, H.K. Mao, R.J. Hemley, Nat. Mater. 3 (2004) 294–297. [7] H.Y. Chung, M.B. Weinberger, J.B. Levine, A. Kavner, J.M. Yang, S.H. Tolbert, R.B. Kaner, Science 316 (2007) 436–439. [8] H.Y. Gou, A.A. Tsirlin, E. Bykova, A.M. Abakumov, G.V. Tendeloo, A. Richter, S.V. Ovsyannikov, A.V. Kurnosov, D.M. Trots, Z. Konopkova, H.P. Liermann, L. Dubrovinsky, N. Dubrovinskaia, Phys. Rev. B 89 (2014) 064108. [9] Y.A. Kvashnina, A.G. Kvashnin, P.B. Sorokin, J. Appl. Phys. 114 (2013) 183708. [10] J.F. Li, R.Q. Zhang, Carbon 63 (2013) 571–573. [11] Z.G. Zhang, M.C. Lu, L. Zhu, L.L. Zhu, Y.D. Li, M. Zhang, Q. Li, Phys. Lett. A 378 (2014) 741–744. [12] V.L. Solozhenko, D. Andrault, G. Fiquet, M. Mezouar, D.C. Rubie, Appl. Phys. Lett. 78 (2001) 1385–1387. [13] F.M. Gao, D.D. Klug, J.S. Tse, J. Appl. Phys. 102 (2007) 084311. [14] V.L. Solozhenko, O.O. Kurakevych, D. Andrault, Y.L. Godec, M. Mezouar, Phys. Rev. Lett. 102 (2009) 015506. [15] M.G. Zhang, Q. Wei, H.Y. Yan, Y.R. Zhao, H. Wang, J. Phys. Chem. C 118 (2014) 3202-3208. [16] A.Y. Liu, M.L. Cohen, Science 245 (1989) 841–842. [17] D.M. Teter, R.J. Hemley, Science 271 (1996) 53–55. [18] M. Marqués, J. Osorio, R. Ahuja, M. Flórez, J.M. Recio, Phys. Rev. B 70 (2004) 104114. [19] J.L. He, L.C. Guo, X.J. Guo, R.P. Liu, Y.J. Tian, H.T. Wang, C.X. Gao, Appl. Phys. Lett. 88 (2006) 101906. [20] G.S. Manyali, R. Warmbier, A. Quandt, J.E. Lowther, Comput. Mater. Sci. 69 (2013) 299-303. [21] P. Hohenberg, W. Kohn, Phys. Rev 136 (1964) B864–B871. [22] W. Kohn, L.J. Sham, Phys. Rev 140 (1965) A1133–A1138. [23] M.D. Segall, P.J.D. Lindan, M.J. Probert, C.J. Pickard, P.J. Hasnip, S.J. Clark, M. C. Payne, J. Phys.: Condens. Matter 14 (2002) 2717–2744. [24] J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865–3868. [25] S. Grimme, J. Comput. Chem. 25 (2004) 1463–1473. [26] S. Grimme, J. Comput. Chem. 27 (2006) 1787–1799. [27] S. Grimme, J. Antony, S. Ehrlich, H. Krieg, J. Chem. Phys. 132 (2010) 154104. [28] T.H. Fischer, J. Almlof, J. Phys. Chem. 96 (1992) 9768–9774. [29] D. Vanderbilt, Phys. Rev. B 41 (1990) 7892–7895. [30] H.J. Monkhorst, J.D. Pack, Phys. Rev. B 13 (1976) 5188–5192. [31] D.W. He, F.X. Zhang, X.Y. Zhang, Z.C. Qin, M. Zhang, R.P. Liu, Y.F. Xu, W.K. Wang, J. Mater. Res. 13 (1998) 3458–3462. [32] B. Molina, L.E. Sansores, Mod. Phys. Lett. B 13 (1999) 193–201. [33] Z.H. Zhang, H.X. Guo, G.R. Zhong, F.W. Yu, Q.H. Xiong, X.J. Fan, Thin Solid Films 346 (1999) 96–99. [34] Y. Xu, S.P. Gao, J. Int., Hydrogen Energy 37 (2012) 11072–11080. [35] M. Mattesini, S.F. Matar, Comput. Mater. Sci. 20 (2001) 107–119. [36] H. Hou, Z.Q. Wen, Y.H. Zhao, L. Fu, N. Wang, P.D. Han, Intermetallics 44 (2014) 110–115. [37] Q.Y. Fan, Q. Wei, H.Y. Yan, M.G. Zhang, Z.X. Zhang, J.Q. Zhang, D.Y. Zhang, Comput. Mater. Sci. 85 (2014) 80–87. [38] J.P. Watt, L. Peselnick, J. Appl. Phys. 51 (1980) 1525–1531. [39] J. Haines, J.M. L'eger, G. Bocquillon, Annu. Rev. Mater. Res. 31 (2001) 1–23. [40] J.F. Nye, Physical Properties of Crystals: Their Representation by Tensors and Matrices, Oxford University Press, Oxford, 1985. [41] P.R. Heyliger, W.L. Johnson, J. Acoust. Soc. Am. 113 (2003) 1812–1825. [42] N. Mounet, N. Marzari, Phys. Rev. B 71 (2005) 205214.
L.-W. Ruan et al. / Journal of Physics and Chemistry of Solids 75 (2014) 1324–1333
[43] M. Grimsditch, E.S. Zouboulis, A. Polian, J. Appl. Phys. 76 (1994) 832–834. [44] Y.D. Guo, X.D. Yang, X.B. Li, X.S. Song, X.L. Cheng, Diamond Relat. Mater. 17 (2008) 1–6. [45] A.Y. Liu, R.M. Wentzcovitch, Phys. Rev. B 50 (1994) 10362–10365. [46] Y. Zhang, H. Sun, C.F. Chen, Phys. Rev. B 73 (2006) 064109. [47] B. Zhu, C.M. Liu, M.B. Lv, X.R. Chen, J. Zhu, G.F. Ji, Physica B 406 (2011) 3508–3513. [48] A. Reuss, Z. Angew. Math. Mech. 9 (1929) 49–58. [49] J.J. Lewandowski, W.H. Wang, A.L. Greer, Philos. Mag. Lett. 85 (2005) 77–87. [50] S.F. Pugh, XCII. Relations between the elastic moduli and the plastic properties of polycrystalline pure metals, Philos. Mag. Ser. 7 45 (1954) 823–843.
1333
[51] D.M. Teter, MRS Bull. 23 (1998) 22–27. [52] J.B. Levine, S.L. Nguyen, H.I. Rasool, J.A. Wright, S.E. Brown, R.B. Kaner, J. Am. Chem. Soc. 130 (2008) 16953–16958. [53] Y.H. Duan, Y. Sun, M.J. Peng, S.G. Zhou, J. Alloys Compd. 595 (2014) 14–21. [54] S.I. Ranganathan, M. Ostoja-Starzewski, Phys. Rev. Lett. 101 (2008) 055504. [55] D.H. Chung, W.R. Buessem, Anisotropy in Single-Crystal Refractory Compound, in: F.W. Vahldiek, S.A. Mersol (Eds.), Plenum, New York, 1968, p. 217. [56] P. Ravindran, L. Fast, P.A. Korzhavyi, B. Johansson, J. Wills, O. Eriksson, J. Appl. Phys. 84 (1998) 4891–4904. [57] O.L. Anderson, J. Phys. Chem. Solids 24 (1963) 909–917.