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Theoretical investigation of the electronic structure, optical, elastic and thermodynamics properties of a newly binary boron nitride (T-B3N3)
Physica B 459 (2015) 134–139
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Theoretical investigation of the electronic structure, optical, elastic and thermodynamics properties of a newly binary boron nitride (T-B3N3) Shibo Zhao a,n, Jianping Long b a b
Network and Educational Technology Center, Chengdu University of Technology, Chengdu 610059, PR China College of Materials and Chemistry & Chemical Engineering, Chengdu University of Technology, Chengdu 610059, PR China
art ic l e i nf o
a b s t r a c t
Article history: Received 6 July 2014 Received in revised form 12 October 2014 Accepted 24 November 2014 Available online 25 November 2014
The ultrasoft pseudopotential planewave (UPPW) within density functional theory (DFT) has been used to investigate the electronic structure, optical, elastic and thermodynamics properties of newly binary boron nitride (T-B3N3). The calculated lattice parameters are in good agreement with previous theoretical results and deviated are less than 0.4%. The electronic structure showed that the T-B3N3 is metallic, and the optical spectra exhibit a noticeable anisotropy. The static dielectric constants, optical permittivity constants and the elastic properties are calculated. From our results, we observe that T-B3N3 is mechanically unstable and ductile. The entropy, enthalpy, free energy, heat capacity and Debye temperature of T-B3N3 was obtained. Up to now, there are no available experimental data about those properties. The results obtained in the present paper could provide important reference data for future studies. & 2014 Elsevier B.V. All rights reserved.
Keywords: Metallic boron nitride Density functional theory Electronic structure Elastic constants Optical properties Thermodynamic properties
1. Introduction Binary boron nitride (BN) compounds, attracting a great number of attentions due to their high temperature stability, high melting point, high mechanical strength, large thermal conductivity and useful optical properties, are widely used as a hightemperature ceramic material, insulators and abrasive materials [1–7]. It is interesting to note that BN can form many polymorphs, such as hexagonal BN (h-BN) [8], zinc-blende BN (c-BN) [9,10], wurtzite BN (w-BN) [11], one-dimensional nanostructures [12,13], hollow spheres [14], nanocages [15] and BN nanomesh [16]. At ambient pressure, the c-BN is the most energetically preferred phase. The w-BN is the second stable phase and is always more stable than h-BN. Although w-BN is energetically more stable than h-BN, the energy barrier from h-BN to w-BN is so high that the phase transformation cannot happen at room temperature. The c-BN shows extreme hardness and low dielectric constant while the h-BN is of fundamental importance to study BN nanotubes. Although B and N element appear to be metallic under high pressure, BN compounds remain an insulator with a wide band gap of around 6.0 eV [17], irrespective of their shape and structure, even when under high pressure [18–20]. The practical application of BN compounds in electronic devices is hindered because of their wide band gap. This has stimulated recent research in reducing or n
Corresponding author. Fax: þ86 2884079530. E-mail address: [email protected] (S. Zhao).
http://dx.doi.org/10.1016/j.physb.2014.11.094 0921-4526/& 2014 Elsevier B.V. All rights reserved.
even closing the band gap. Recently, Zhang et al. [21] designed a new tetragonal phase T-B3N3, and making the well-known insulating BN become metallic though changes its atomic configuration. To aid in the synthesis of the new phase of T-B3N3, Zhang et al. had analyzed of its cohesive energy, electronic structure, density of states and electron localization function. However, the elastic, optical and thermodynamic properties of T-B3N3 have not been studied at all. Therefore, the objective of the present work is to investigating these physical properties of T-B3N3.
2. Calculation methods All the calculations presented in the present work have been carried out using the CASTEP code [22], which is an implementation of the ultrasoft pseudopotential plane-wave (UPPW) method, based on the state-of-the-art of density functional theory (DFT). The electronic exchange–correlation interactions are treated with the Perdew–Burke–Ernzerhof for solid in generalized gradient approximation (GGA-PBEsol). [23] The energy with respect to k-points and the cutoff energy have been converged to less than 1 meV. In this study, the cutoff energy of plane-wave is 560 eV, which was large enough to obtain good convergence. B-2s22p1 and N-2s22p3 were explicitly treated as valence electrons. In the Brillouin zone integrations, 14 14 10 grid of Monkhorst–Pack points has been employed to ensure well convergence of the computed structures and energies. For the calculation of the
S. Zhao, J. Long / Physica B 459 (2015) 134–139
elastic and optical properties, which usually requires a dense mesh of uniformly distributed k-points, the Brillouin zone integration was performed using a 20 20 15 and 25 25 25 grid of Monkhorst–Pack points, respectively. The equilibrium crystal structures of T-B3N3 were determined using the Brodyden– Fletcher–Goldfarb–Shanno (BFGS) [24–27] minimization technique, with the threshold for converged structures: energy change per atom less than 5 10 6 eV/atom, the Hellmann–Feynman force per atom less than 0.01 eV/Å and the maximum displacement of atom is 5 10 4 Å during the geometry optimization.
135
Table 1 The lattice parameters and atomic Wyckoff positions of T-B3N3. Atom
Wyckoff positions
Cal.
Ref. [21]
B1 N1 B2 N2
1c 1b 2g 2g
(0.5,0.5,0.5) (0.5,0.5,0.0) (0.0,0.5,0.86037) (0.0,0.5,0.637842) 2.63 6.10
(0.5,0.5,0.5) (0.5,0.5,0.0) (0.0,0.5,0.8602) (0.0,0.5,0.6378) 2.64 6.11
a (Å) c (Å)
3. Results and discussions 3.1. Structure properties T-B3N3 (space group P 4¯ m2 no. 115) has a tetragonal primitive cell containing three formula units [21], as shown in Fig. 1. There are four chemically nonequivalent atoms in the monolayer hexagons BN by sp2 hybridization. In T-B3N3, two interlocked hexagons BN are perpendicular form the special “spiroconjugation” geometry. The optimized lattice parameters and atomic Wyckoff positions of T-B3N3 are summarized in Table 1, together with the theoretical results for comparison. The optimized lattice parameters are a ¼b ¼2.63 Å, c ¼6.10 Å, respectively. They are in good agreement with the theoretical results, and the deviations are 0.38% and 0.16%, respectively. These results show and confirm that the method used in this study is reliable thereby the optimized lattice parameters can be used for future calculations of other parameters. 3.2. Electronic properties Fig. 2 shows the calculated band structures along the high symmetry directions in the first Brillouin zone (BZ) of T-B3N3. The Fermi level (Ef) is chosen to locate at 0 eV and coincides with the top of the valence band (VB). From the Fig. 2, we see that a partially occupied band crosses the Ef in the vicinity of M point, suggesting that the T-B3N3 is metallic. To understand the clear picture of the elemental contributions to the electronic structure of T-B3N3, we have also computed the total density of states (TDOS) and partial density of states (PDOS) of T-B3N3 as given in Fig. 3. As shown in Fig. 3, the obtained value of the TDOS at the Ef is 0.96 electrons/eV. It depicts that the TDOS near the Ef for T-B3N3 is dominated by the highly localized N-2p states and a small amount of B-2p states. Our results for band structures and TDOS are similar to other calculations [21]. The VB of the T-B3N3 can be divided into two parts. The first part from 21.1 eV to 14.4 eV is mainly due to N-2 s states and minor contributions from B (2s, 2p) states. The second part from 12.4 eV to Ef is mainly due to N-2p states and small contributions of B-2p and B-2s states. The conduction band (CB), from Ef to 26.7 eV, comes primarily from B-2p, 2s states with
Fig. 2. Band structure and the total density of states of T-B3N3.
some contributions from N-2p, 2s states. 3.3. Optical properties The optical properties of T-B3N3 are studied by the frequencydependent dielectric function ε (ω) = ε′ (ω) + iε″ (ω) that is mainly connected with the electronic structures. The imaginary part ε″ (ω) of the dielectric function is calculated from the momentum matrix elements between the occupied and unoccupied electronic. The real part ε′ (ω) of the dielectric function can be derived from the imaginary part ε″ (ω) using the Kramer–Kronig relation [28]. The real part ε′ (ω) and imaginary part ε″ (ω) of the dielectric function in the energy range from 0 to 20 eV are displayed in Fig. 4. The calculations were performed for two light polarizations along [100] and [001] directions. From Fig. 4, we can see that there are five intensive peaks of absorption centered at about 0 eV, 4.20 eV, 6.02 eV, 9.50 eV and 12.21 eV for the [100] polarization.
Fig. 1. Crystal structure and the first Brillouin zone of T-B3N3.
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S. Zhao, J. Long / Physica B 459 (2015) 134–139
Table 2 Calculated static dielectric tensor ε∘ and optical permittivity tensor ε∞ of T-B3N3. Compounds T-B3N3
ε∞
ε∘
⎛18.52 0 ⎞ 0 ⎜ ⎟ 18.52 0 ⎜0 ⎟ ⎝0 0 30.09 ⎠
⎛ 4.41 0 ⎜ 4.41 ⎜0 ⎝0 0
⎞ 0 ⎟ 0 ⎟ 7.34 ⎠
Fig. 3. Total density of states and partial density of states of of T-B3N3.
These peaks are due to the electronic transitions from the VB states to the CB states. In order to calculate the static dielectric tensor ε∘ and optical permittivity tensor ε∞ of T-B3N3, we used the density functional perturbation theory (DFPT) and the results are listed in Table 2. The static dielectric constants are found to be 18.52 and 30.09 for [100] and [001] directions, respectively. The absorption coefficient α (ω) is a parameter, which characterize the decay of light intensity spreading in unit distance in medium. The calculated absorption coefficient of T-B3N3 in the energy range from 0 to 45 eV for two incident radiation polarizations along [100] and [001] are presented in Fig. 5. It is clear from Fig. 5 that the absorption spectra exhibit a noticeable anisotropy.
Fig. 5. Calculated absorption coefficient for T-B3N3.
3.4. Elastic properties To calculate the elastic constants, we have applied the nonvolume-conserving method. The complete elastic constant tensor was determined from calculations of the stresses induced by small deformations of the equilibrium primitive cell, and thus the elastic constants Cijkl are determined as [29]
Fig. 4. Calculated real part ε′ (ω) and imaginary part ε″ (ω) of the dielectric function for T-B3N3.
S. Zhao, J. Long / Physica B 459 (2015) 134–139
Table 3 The calculated elastic constants Cij (in GPa) of T-B3N3.
Table 4 Calculated bulk modulus B (all in GPa), shear modulus G (all in GPa), Young's modulus E (all in GPa), the ratio B/G and Poisson's ratio s of T-B3N3.
Compounds
C11
C22
C33
C44
C55
C66
C12
C13
C23
T-B3N3 p-BNa c-BNb w-BNc
566.2 1032 820 982
–/– 934 –/– –/–
888.2 953 –/– 1077
25.1 386 480 388
–/– 284 –/– –/–
52.5 340 –/– 424
63.4 153 190 134
116.0 90 –/– 74
–/– 114 –/– –/–
a b c
137
Ref. [8]. Ref. [31]. Ref. [32].
Compounds
B
G
B G
E
σ
T-B3N3 p-BNa c-BNb w-BNc h-BN
282.1 403 383 401 37d, 14e,34f
10.9 368 400
25.8
32.4 846
0.48
a
Ref. [8]. Ref. [31]. Ref. [32]. d Ref. [36]. e Ref. [19]. f Ref. [37]. b c
Cijkl =
∂σij ∂ekl
= x
1 ∂ 2E V ∂eij ∂ekl
(1)
x
E denotes the Helmholtz free energy, σij and ekl are the applied stress and Eulerian strain tensors, x is the coordinates. For the tetragonal crystals, its six independent elastic constants should satisfy the well-known stability criteria [30]
⎧ C11 > 0, C33 > 0, C44 > 0, C66 > 0 ⎫ ⎪ ⎪ ⎨ C11 − C12 > 0, C11 + C33 − 2C13 > 0⎬ ⎪ ⎪ ⎩ 2(C11 + C12 ) + C33 + 4C13 > 0 ⎭
(2)
The computed elastic constants of T-B3N3 are listed in Table 3 together with the available experimental data and theoretical results for p-BN, c-BN and w-BN. According to the criteria, it is clear that the T-B3N3 is mechanically unstable at ambient pressure. For the tetragonal system, the Voigt's (The Voigt's shear modulus or bulk modulus corresponding to the upper bound of G or B values.) and Reuss's (The Reuss's shear modulus or bulk modulus corresponding to the lower bound of G or B values.) bounds of G and B can be written as [33]
⎧ ⎫ M + 3C11 − 3C12 + 12C44 + 6C66 ⎪ GV = ⎪ 30 ⎪ ⎪ ⎪ ⎪ 2(C11 + C12 ) + C33 + 4C13 ⎪ BV = ⎪ 9 ⎪ ⎪ ⎪ 6 6 3 −1⎪ ⎪ G = 15[ 18BV + + + ] ⎪ ⎨ R C11 − C12 C44 C66 ⎬ C2 ⎪ ⎪ ⎪ ⎪ C2 ⎪ BR = ⎪ M ⎪ ⎪ ⎪ M = C11 + C12 + 2C33 − 4C13 ⎪ ⎪ ⎪ 2 ⎪ ⎪ 2 ⎩ C = (C + C ) C − 2C ⎭ 11
12
33
13
(3)
The bulk modulus B and shear modulus G can be estimated by Voigt–Reuss–Hill approximation B = 1/2(BV + BR ),G = 1/2(GV + G R ) [34]. The Young's modulus E and Poisson's ratio σ can be computed by the following equations [35], respectively:
⎧ ⎫ 9BG ⎪E = ⎪ ⎪ 3B + G ⎪ ⎨ ⎬ ⎪ σ = 3B − 2G ⎪ ⎪ 2(3B + G) ⎪ ⎩ ⎭
(4)
The calculated values of bulk, shear and Young's modulus, as well as Poisson's ratio of the T-B3N3 are given in Table 4 together with the available experimental data and theoretical results for P-BN, c-BN, w-BN and h-BN. Bulk modulus measures the resistance that material offers to changes in its volume. From Table 4, we can see that the bulk modulus of T-B3N3 is 282.1 GPa, comparable to the values of 403 GPa for p-BN [8], 383 GPa for c-BN [31] and 401 GPa for w-BN [32], indicating that T-B3N3 is more compressible than the p-BN, c-BN and w-BN. Young's modulus can give information about the stiffness of a material. The higher the
Young's modulus, the stiffer is the material. The shear modulus measures the resistance to motion of the planes of a material sliding past each other. The calculated shear modulus of 10.9 GPa is less than the corresponding values of 368 GPa for p-BN [8], 400 GPa for c-BN [31]. To the best of our knowledge, up to now, although no available experimental data and theoretical results about the elastic constants of T-B3N3 to compare with these calculated results, the results obtained in the present paper could provide important reference data for future investigation. The ratio of bulk modulus to shear modulus of crystalline phases can predict the brittle or ductile behavior of materials. If B/ G 41.75 the materials will behave in a ductile manner or else the materials demonstrates brittleness [38]. The values of B, G, E, s and the ratio B/G are given in Table 4. The obtain B/G ratio is 25.8 for T-B3N3. According to this value, the T-B3N3 behaves in a ductile manner. On the other hand, Frantsevich [39] suggested using the Poisson's ratio for this distinction, classifying compounds with so 1/3 as brittle, and those with s4 1/3 as ductile. As shown in Table 4, the value of Poisson's ratio of T-B3N3 is 0.48, indicating the ductile nature of T-B3N3. This agrees well with the estimation from the B/G ratio. The Poison's ratio s is often used to reflect the stability of a crystal against shear and provides information about the nature of the bonding forces. The Poison's ratio takes a value between 1 (B«G) and 0.5 (B»G), the lower limit implies that the material is compressible while the upper limit demonstrates that the material is incompressible [40]. Bigger Poison's ratios better the plasticity. The calculated values of the Poison's ratio show that T-B3N3 is of good plasticity. 3.5. Thermodynamic properties To study the effects of temperature on structure stabilities, we calculated the thermodynamic properties of T-B3N3 up to 1000 K by using the quasi-harmonic approximation (QHA). The variations of the entropy, enthalpy, free energy and heat capacity at various temperatures are shown in Fig. 6. From Fig. 6(a), it is noted that the enthalpy and entropy increase rapidly when the temperature increases while the free energy decrease with the temperature increasing. The temperature-dependent behavior of the constant volume heat capacity Cv is shown in Fig. 6(b). It is seen from this figure that when T o300 K, Cv is proportional to T3; when T4 300 K, Cv increases slowly with temperature and it tends to 26.2 cal/cell K (the Dulonge–Petit limit). The Debye temperature (θ D ) is not a strictly determined parameter, various estimates may be obtained through well established empirical or semi-empirical formulae. One of the semiempirical formulae can be used to estimate the Debye temperature through elastic constants, averaged sound velocity (vm ),
138
S. Zhao, J. Long / Physica B 459 (2015) 134–139
Fig. 6. Calculated entropy, enthalpy, free energy and heat capacity of T-B3N3.
Table 5 Calculated the density (r in g/cm3), transverse, longitudinal, average sound velocity (vt, vl, vm, in m/s) and the Debye temperatures (yD in K) of T-B3N3. Compounds
r
vt
vl
vm
yD
T-B3N3
2.9396
1929.3
10046.8
2205.9
343.1
longitudinal sound velocity (vl ) and transverse sound velocity (vt ) [41–43]:
θD =
1/3 h ⎡ 3n ⎛ N A ρ ⎞ ⎤ ⎟ ⎥ vm ⎜ ⎢ k ⎣ 4π ⎝ M ⎠ ⎦
(5)
⎤− (1/3) ⎡ ⎛ 1 2 1⎞ vm = ⎢ ⎜⎜ 3 + 3 ⎟⎟ ⎥ ⎢⎣ 3 ⎝ vt vl ⎠ ⎥⎦
(6)
⎛ B + (4/3) G ⎞1/2 vl = ⎜ ⎟ ρ ⎝ ⎠
(7)
⎛ G ⎞1/2 vt = ⎜ ⎟ ⎝ ρ⎠
(8)
where h is Planck constant, k is Boltzmann constant; NA is Avogadro number; ρ is the density; M is the molecular weight; and n is the number of atoms in the unit cell. The calculated values of vm , vl , vt and θ D of T-B3N3 at 0 K are given in Table 5. To our knowledge, there are no experimental or theoretical results for Debye temperature available to us for T-B3N3. The results obtained in this work could provide a useful reference for future studies.
4. Conclusions In the present work, the electronic structure, optical, elastic and thermodynamic properties of T-B3N3 have been studied by means of DFT. The calculated lattice parameters of T-B3N3 are in good agreement with the theoretical results and deviated are less than 0.4%. The electronic structure showed that the T-B3N3 is metallic, and the optical spectra exhibit a noticeable anisotropy for the light
polarizations along the [100] and [001] directions. The static dielectric tensor ε∘ and optical permittivity tensor ε∞ of T-B3N3 are obtained. The elastic properties such as shear modulus and Young's modulus are calculated. From our results, we observe that T-B3N3 is mechanically unstable and ductile. The entropy, enthalpy, free energy, heat capacity and Debye temperature of T-B3N3 was obtained. To the best of our knowledge, there are no available experimental data and theoretical results about the optical, elastic and thermodynamic properties of T-B3N3. The results obtained in this work could provide a useful reference for future studies.
References [1] S. Rudolph, Am. Ceram. Soc. Bull. 73 (1994) 89. [2] R. Haubner, M. Wilhelm, R. Weissenbacher, B. Lux, in: M. Jansen (Ed.), Boron Nitrides – Properties, Synthesis and Applications, Springer, Berlin, 2002. [3] G. Kern, G. Kresse, J. Hafner, Phys. Rev. B 59 (1999) 8551. [4] R.T. Paine, C.K. Narula, Chem. Rev. 90 (1990) 73. [5] Y.N. Xu, W.Y. Ching, Phys. Rev. B 44 (1991) 7787. [6] L. Vel, G. Demazeau, J. Etourneau, Mater. Sci. Eng. B 10 (1991) 149. [7] N.E. Christensen, I. Gorczyca, Phys. Rev. B 50 (1994) 4397. [8] X. Jiang, J.J. Zhao, R. Ahuja, J. Phys.: Condens. Matter 25 (2013) 122204. [9] J.L.P. Castineira1, J.R. Leite, J.L.F. da Silva, L.M.R. Scolfaro, J.L.A. Alves, H.W. L. Alves, Phys. Status Solidi B 210 (1998) 401. [10] P.B. Mirkarimi, K.F. McCarty, D.L. Medlin, Mater. Sci. Eng., R. 21 (1997) 47. [11] F.P. Bundy, R.H. Wentorf, J. Chem. Phys. 38 (1963) 1144. [12] W. Mickelson, S. Aloni, W.Q. Han, J. Cumings, A. Zettl, Science 300 (2003) 467. [13] F.L. Deepak, C.P. Vinod, K. Mukhopadhyay, A. Govindaraj, C.N.R. Rao, Chem. Phys. Lett. 353 (2002) 345. [14] L.Y. Chen, Y.L. Gu, L. Shi, Z.H. Yang, J.H. Ma, Y.T. Qian, Solid State Commun. 130 (2004) 537. [15] Y. Pan, K.F. Huo, Y.M. Hu, J.J. Fu, Y.N. Lu, Z.D. Dai, Z. Hu, Y. Chen, Small 1 (2005) 1199. [16] G. Cappellini, G. Satta, K. Tenelsen, F. Bechstedt, Phys. Status Solidi B 217 (2000) 861. [17] M. Corso, W. Auwarter, M. Muntwiler, A. Tamai, T. Greber, J. Osterwalder, Science 303 (2004) 217. [18] N. Miyata, K. Moriki, O. Mishima, M. Fujisawa, T. Hattori, Phys. Rev. B 40 (1989) 12028. [19] B. Wen, J.J. Zhao, R. Melnikc, Y.J. Tian, Phys. Chem. Chem. Phys. 13 (2011) 14565. [20] S. Botti, J.A. Flores-Livas, M. Amsler, S. Goedecker, M.A.L. Marques, Phys. Rev. B 86 (2012) 121204. [21] S.H. Zhang, Q. Wang, Y. Kawazoe, P. Jena, J. Am. Chem. Soc. 135 (2013) 18216. [22] S.J. Clark, M.D. Segall, C.J. Pickard, P.J. Hasnip, M.J. Probert, K. Refson, M. C. Payne, Z. Kristallogr. 220 (2005) 567. [23] J.P. Perdew, A. Ruzsinszky, G.I. Csonka, O.A. Vydrov, G.E. Scuseria, L. A. Constantin, X.L. Zhou, K. Burke, Phys. Rev. Lett. 100 (2008) 136406. [24] C.G. Broyden, J. Inst., Math. Comput. 6 (1970) 222.
S. Zhao, J. Long / Physica B 459 (2015) 134–139
[25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35]
R. Fletcher, Comput. J. 13 (1970) 317. D. Goldfarb, Math. Comput. 24 (1970) 23. D.F. Shanno, Math. Comput. 24 (1970) 647. S. Mamoun, A.E. Merad, L. Guilbert, Comput. Mater. Sci. 79 (2013) 125. J.H. Weiner, Statistical Mechanics of Elasticity, John Wiley, New York, 1983. S. Piskunov, E. Heifets, R.I. Eglitis, G. Borstel, Comput. Mater. Sci. 29 (2004) 165. M. Grimsditch, E.S. Zouboulis, A. Polian, J. Appl. Phys. 76 (1994) 832. K. Shimada, T. Sota, K. Suzuki, J. Appl. Phys. 84 (1998) 4951. J.P. Watt, L. Peselnick, J. Appl. Phys. 51 (1980) 1525. R. Hill, Proc. Phys. Soc. A 65 (1952) 349. J.P. Long, L.J. Yang, D.M. Li, Solid State Sci. 19 (2013) 12.
[36] [37] [38] [39]
[40] [41] [42] [43]
139
E. Knittle, R.M. Wentzcovitch, R. Jeanloz, M.L. Cohen, Nature 337 (1989) 349. Y.N. Xu, W.Y. Ching, Phys. Rev. B 44 (1991) 7787. S.F. Pugh, Philos. Mag. 45 (1954) 823. I.N. Frantsevich, F.F. Voronov, S.A. Bokuta, in: I.N. Frantsevich (Ed.), Elastic Constants and Elastic Moduli of Metals and Insulators Handbook, Naukova Dumka, Kiev, 1983, p. 60. W. Köster, H. Franz, Int. Mater. Rev. 6 (1961) 1. H.Y. Lu, L.J. Yang, W. Huang, J. Supercond. Nov. Magn. 27 (2014) 1671. H.Y. Lu, J.P. Long, L.J. Yang, W. Huang, Int. J. Mod. Phys. B 28 (2014) 1450057. J.P. Long, L.J. Yang, X.S. Wei, J. Alloys Compd. 549 (2013) 336.
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Theoretical investigation of the electronic structure, optical, elastic and thermodynamics properties of a newly binary boron nitride (T-B3N3) Shibo Zhao a,n, Jianping Long b a b
Network and Educational Technology Center, Chengdu University of Technology, Chengdu 610059, PR China College of Materials and Chemistry & Chemical Engineering, Chengdu University of Technology, Chengdu 610059, PR China
art ic l e i nf o
a b s t r a c t
Article history: Received 6 July 2014 Received in revised form 12 October 2014 Accepted 24 November 2014 Available online 25 November 2014
The ultrasoft pseudopotential planewave (UPPW) within density functional theory (DFT) has been used to investigate the electronic structure, optical, elastic and thermodynamics properties of newly binary boron nitride (T-B3N3). The calculated lattice parameters are in good agreement with previous theoretical results and deviated are less than 0.4%. The electronic structure showed that the T-B3N3 is metallic, and the optical spectra exhibit a noticeable anisotropy. The static dielectric constants, optical permittivity constants and the elastic properties are calculated. From our results, we observe that T-B3N3 is mechanically unstable and ductile. The entropy, enthalpy, free energy, heat capacity and Debye temperature of T-B3N3 was obtained. Up to now, there are no available experimental data about those properties. The results obtained in the present paper could provide important reference data for future studies. & 2014 Elsevier B.V. All rights reserved.
Keywords: Metallic boron nitride Density functional theory Electronic structure Elastic constants Optical properties Thermodynamic properties
1. Introduction Binary boron nitride (BN) compounds, attracting a great number of attentions due to their high temperature stability, high melting point, high mechanical strength, large thermal conductivity and useful optical properties, are widely used as a hightemperature ceramic material, insulators and abrasive materials [1–7]. It is interesting to note that BN can form many polymorphs, such as hexagonal BN (h-BN) [8], zinc-blende BN (c-BN) [9,10], wurtzite BN (w-BN) [11], one-dimensional nanostructures [12,13], hollow spheres [14], nanocages [15] and BN nanomesh [16]. At ambient pressure, the c-BN is the most energetically preferred phase. The w-BN is the second stable phase and is always more stable than h-BN. Although w-BN is energetically more stable than h-BN, the energy barrier from h-BN to w-BN is so high that the phase transformation cannot happen at room temperature. The c-BN shows extreme hardness and low dielectric constant while the h-BN is of fundamental importance to study BN nanotubes. Although B and N element appear to be metallic under high pressure, BN compounds remain an insulator with a wide band gap of around 6.0 eV [17], irrespective of their shape and structure, even when under high pressure [18–20]. The practical application of BN compounds in electronic devices is hindered because of their wide band gap. This has stimulated recent research in reducing or n
Corresponding author. Fax: þ86 2884079530. E-mail address: [email protected] (S. Zhao).
http://dx.doi.org/10.1016/j.physb.2014.11.094 0921-4526/& 2014 Elsevier B.V. All rights reserved.
even closing the band gap. Recently, Zhang et al. [21] designed a new tetragonal phase T-B3N3, and making the well-known insulating BN become metallic though changes its atomic configuration. To aid in the synthesis of the new phase of T-B3N3, Zhang et al. had analyzed of its cohesive energy, electronic structure, density of states and electron localization function. However, the elastic, optical and thermodynamic properties of T-B3N3 have not been studied at all. Therefore, the objective of the present work is to investigating these physical properties of T-B3N3.
2. Calculation methods All the calculations presented in the present work have been carried out using the CASTEP code [22], which is an implementation of the ultrasoft pseudopotential plane-wave (UPPW) method, based on the state-of-the-art of density functional theory (DFT). The electronic exchange–correlation interactions are treated with the Perdew–Burke–Ernzerhof for solid in generalized gradient approximation (GGA-PBEsol). [23] The energy with respect to k-points and the cutoff energy have been converged to less than 1 meV. In this study, the cutoff energy of plane-wave is 560 eV, which was large enough to obtain good convergence. B-2s22p1 and N-2s22p3 were explicitly treated as valence electrons. In the Brillouin zone integrations, 14 14 10 grid of Monkhorst–Pack points has been employed to ensure well convergence of the computed structures and energies. For the calculation of the
S. Zhao, J. Long / Physica B 459 (2015) 134–139
elastic and optical properties, which usually requires a dense mesh of uniformly distributed k-points, the Brillouin zone integration was performed using a 20 20 15 and 25 25 25 grid of Monkhorst–Pack points, respectively. The equilibrium crystal structures of T-B3N3 were determined using the Brodyden– Fletcher–Goldfarb–Shanno (BFGS) [24–27] minimization technique, with the threshold for converged structures: energy change per atom less than 5 10 6 eV/atom, the Hellmann–Feynman force per atom less than 0.01 eV/Å and the maximum displacement of atom is 5 10 4 Å during the geometry optimization.
135
Table 1 The lattice parameters and atomic Wyckoff positions of T-B3N3. Atom
Wyckoff positions
Cal.
Ref. [21]
B1 N1 B2 N2
1c 1b 2g 2g
(0.5,0.5,0.5) (0.5,0.5,0.0) (0.0,0.5,0.86037) (0.0,0.5,0.637842) 2.63 6.10
(0.5,0.5,0.5) (0.5,0.5,0.0) (0.0,0.5,0.8602) (0.0,0.5,0.6378) 2.64 6.11
a (Å) c (Å)
3. Results and discussions 3.1. Structure properties T-B3N3 (space group P 4¯ m2 no. 115) has a tetragonal primitive cell containing three formula units [21], as shown in Fig. 1. There are four chemically nonequivalent atoms in the monolayer hexagons BN by sp2 hybridization. In T-B3N3, two interlocked hexagons BN are perpendicular form the special “spiroconjugation” geometry. The optimized lattice parameters and atomic Wyckoff positions of T-B3N3 are summarized in Table 1, together with the theoretical results for comparison. The optimized lattice parameters are a ¼b ¼2.63 Å, c ¼6.10 Å, respectively. They are in good agreement with the theoretical results, and the deviations are 0.38% and 0.16%, respectively. These results show and confirm that the method used in this study is reliable thereby the optimized lattice parameters can be used for future calculations of other parameters. 3.2. Electronic properties Fig. 2 shows the calculated band structures along the high symmetry directions in the first Brillouin zone (BZ) of T-B3N3. The Fermi level (Ef) is chosen to locate at 0 eV and coincides with the top of the valence band (VB). From the Fig. 2, we see that a partially occupied band crosses the Ef in the vicinity of M point, suggesting that the T-B3N3 is metallic. To understand the clear picture of the elemental contributions to the electronic structure of T-B3N3, we have also computed the total density of states (TDOS) and partial density of states (PDOS) of T-B3N3 as given in Fig. 3. As shown in Fig. 3, the obtained value of the TDOS at the Ef is 0.96 electrons/eV. It depicts that the TDOS near the Ef for T-B3N3 is dominated by the highly localized N-2p states and a small amount of B-2p states. Our results for band structures and TDOS are similar to other calculations [21]. The VB of the T-B3N3 can be divided into two parts. The first part from 21.1 eV to 14.4 eV is mainly due to N-2 s states and minor contributions from B (2s, 2p) states. The second part from 12.4 eV to Ef is mainly due to N-2p states and small contributions of B-2p and B-2s states. The conduction band (CB), from Ef to 26.7 eV, comes primarily from B-2p, 2s states with
Fig. 2. Band structure and the total density of states of T-B3N3.
some contributions from N-2p, 2s states. 3.3. Optical properties The optical properties of T-B3N3 are studied by the frequencydependent dielectric function ε (ω) = ε′ (ω) + iε″ (ω) that is mainly connected with the electronic structures. The imaginary part ε″ (ω) of the dielectric function is calculated from the momentum matrix elements between the occupied and unoccupied electronic. The real part ε′ (ω) of the dielectric function can be derived from the imaginary part ε″ (ω) using the Kramer–Kronig relation [28]. The real part ε′ (ω) and imaginary part ε″ (ω) of the dielectric function in the energy range from 0 to 20 eV are displayed in Fig. 4. The calculations were performed for two light polarizations along [100] and [001] directions. From Fig. 4, we can see that there are five intensive peaks of absorption centered at about 0 eV, 4.20 eV, 6.02 eV, 9.50 eV and 12.21 eV for the [100] polarization.
Fig. 1. Crystal structure and the first Brillouin zone of T-B3N3.
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S. Zhao, J. Long / Physica B 459 (2015) 134–139
Table 2 Calculated static dielectric tensor ε∘ and optical permittivity tensor ε∞ of T-B3N3. Compounds T-B3N3
ε∞
ε∘
⎛18.52 0 ⎞ 0 ⎜ ⎟ 18.52 0 ⎜0 ⎟ ⎝0 0 30.09 ⎠
⎛ 4.41 0 ⎜ 4.41 ⎜0 ⎝0 0
⎞ 0 ⎟ 0 ⎟ 7.34 ⎠
Fig. 3. Total density of states and partial density of states of of T-B3N3.
These peaks are due to the electronic transitions from the VB states to the CB states. In order to calculate the static dielectric tensor ε∘ and optical permittivity tensor ε∞ of T-B3N3, we used the density functional perturbation theory (DFPT) and the results are listed in Table 2. The static dielectric constants are found to be 18.52 and 30.09 for [100] and [001] directions, respectively. The absorption coefficient α (ω) is a parameter, which characterize the decay of light intensity spreading in unit distance in medium. The calculated absorption coefficient of T-B3N3 in the energy range from 0 to 45 eV for two incident radiation polarizations along [100] and [001] are presented in Fig. 5. It is clear from Fig. 5 that the absorption spectra exhibit a noticeable anisotropy.
Fig. 5. Calculated absorption coefficient for T-B3N3.
3.4. Elastic properties To calculate the elastic constants, we have applied the nonvolume-conserving method. The complete elastic constant tensor was determined from calculations of the stresses induced by small deformations of the equilibrium primitive cell, and thus the elastic constants Cijkl are determined as [29]
Fig. 4. Calculated real part ε′ (ω) and imaginary part ε″ (ω) of the dielectric function for T-B3N3.
S. Zhao, J. Long / Physica B 459 (2015) 134–139
Table 3 The calculated elastic constants Cij (in GPa) of T-B3N3.
Table 4 Calculated bulk modulus B (all in GPa), shear modulus G (all in GPa), Young's modulus E (all in GPa), the ratio B/G and Poisson's ratio s of T-B3N3.
Compounds
C11
C22
C33
C44
C55
C66
C12
C13
C23
T-B3N3 p-BNa c-BNb w-BNc
566.2 1032 820 982
–/– 934 –/– –/–
888.2 953 –/– 1077
25.1 386 480 388
–/– 284 –/– –/–
52.5 340 –/– 424
63.4 153 190 134
116.0 90 –/– 74
–/– 114 –/– –/–
a b c
137
Ref. [8]. Ref. [31]. Ref. [32].
Compounds
B
G
B G
E
σ
T-B3N3 p-BNa c-BNb w-BNc h-BN
282.1 403 383 401 37d, 14e,34f
10.9 368 400
25.8
32.4 846
0.48
a
Ref. [8]. Ref. [31]. Ref. [32]. d Ref. [36]. e Ref. [19]. f Ref. [37]. b c
Cijkl =
∂σij ∂ekl
= x
1 ∂ 2E V ∂eij ∂ekl
(1)
x
E denotes the Helmholtz free energy, σij and ekl are the applied stress and Eulerian strain tensors, x is the coordinates. For the tetragonal crystals, its six independent elastic constants should satisfy the well-known stability criteria [30]
⎧ C11 > 0, C33 > 0, C44 > 0, C66 > 0 ⎫ ⎪ ⎪ ⎨ C11 − C12 > 0, C11 + C33 − 2C13 > 0⎬ ⎪ ⎪ ⎩ 2(C11 + C12 ) + C33 + 4C13 > 0 ⎭
(2)
The computed elastic constants of T-B3N3 are listed in Table 3 together with the available experimental data and theoretical results for p-BN, c-BN and w-BN. According to the criteria, it is clear that the T-B3N3 is mechanically unstable at ambient pressure. For the tetragonal system, the Voigt's (The Voigt's shear modulus or bulk modulus corresponding to the upper bound of G or B values.) and Reuss's (The Reuss's shear modulus or bulk modulus corresponding to the lower bound of G or B values.) bounds of G and B can be written as [33]
⎧ ⎫ M + 3C11 − 3C12 + 12C44 + 6C66 ⎪ GV = ⎪ 30 ⎪ ⎪ ⎪ ⎪ 2(C11 + C12 ) + C33 + 4C13 ⎪ BV = ⎪ 9 ⎪ ⎪ ⎪ 6 6 3 −1⎪ ⎪ G = 15[ 18BV + + + ] ⎪ ⎨ R C11 − C12 C44 C66 ⎬ C2 ⎪ ⎪ ⎪ ⎪ C2 ⎪ BR = ⎪ M ⎪ ⎪ ⎪ M = C11 + C12 + 2C33 − 4C13 ⎪ ⎪ ⎪ 2 ⎪ ⎪ 2 ⎩ C = (C + C ) C − 2C ⎭ 11
12
33
13
(3)
The bulk modulus B and shear modulus G can be estimated by Voigt–Reuss–Hill approximation B = 1/2(BV + BR ),G = 1/2(GV + G R ) [34]. The Young's modulus E and Poisson's ratio σ can be computed by the following equations [35], respectively:
⎧ ⎫ 9BG ⎪E = ⎪ ⎪ 3B + G ⎪ ⎨ ⎬ ⎪ σ = 3B − 2G ⎪ ⎪ 2(3B + G) ⎪ ⎩ ⎭
(4)
The calculated values of bulk, shear and Young's modulus, as well as Poisson's ratio of the T-B3N3 are given in Table 4 together with the available experimental data and theoretical results for P-BN, c-BN, w-BN and h-BN. Bulk modulus measures the resistance that material offers to changes in its volume. From Table 4, we can see that the bulk modulus of T-B3N3 is 282.1 GPa, comparable to the values of 403 GPa for p-BN [8], 383 GPa for c-BN [31] and 401 GPa for w-BN [32], indicating that T-B3N3 is more compressible than the p-BN, c-BN and w-BN. Young's modulus can give information about the stiffness of a material. The higher the
Young's modulus, the stiffer is the material. The shear modulus measures the resistance to motion of the planes of a material sliding past each other. The calculated shear modulus of 10.9 GPa is less than the corresponding values of 368 GPa for p-BN [8], 400 GPa for c-BN [31]. To the best of our knowledge, up to now, although no available experimental data and theoretical results about the elastic constants of T-B3N3 to compare with these calculated results, the results obtained in the present paper could provide important reference data for future investigation. The ratio of bulk modulus to shear modulus of crystalline phases can predict the brittle or ductile behavior of materials. If B/ G 41.75 the materials will behave in a ductile manner or else the materials demonstrates brittleness [38]. The values of B, G, E, s and the ratio B/G are given in Table 4. The obtain B/G ratio is 25.8 for T-B3N3. According to this value, the T-B3N3 behaves in a ductile manner. On the other hand, Frantsevich [39] suggested using the Poisson's ratio for this distinction, classifying compounds with so 1/3 as brittle, and those with s4 1/3 as ductile. As shown in Table 4, the value of Poisson's ratio of T-B3N3 is 0.48, indicating the ductile nature of T-B3N3. This agrees well with the estimation from the B/G ratio. The Poison's ratio s is often used to reflect the stability of a crystal against shear and provides information about the nature of the bonding forces. The Poison's ratio takes a value between 1 (B«G) and 0.5 (B»G), the lower limit implies that the material is compressible while the upper limit demonstrates that the material is incompressible [40]. Bigger Poison's ratios better the plasticity. The calculated values of the Poison's ratio show that T-B3N3 is of good plasticity. 3.5. Thermodynamic properties To study the effects of temperature on structure stabilities, we calculated the thermodynamic properties of T-B3N3 up to 1000 K by using the quasi-harmonic approximation (QHA). The variations of the entropy, enthalpy, free energy and heat capacity at various temperatures are shown in Fig. 6. From Fig. 6(a), it is noted that the enthalpy and entropy increase rapidly when the temperature increases while the free energy decrease with the temperature increasing. The temperature-dependent behavior of the constant volume heat capacity Cv is shown in Fig. 6(b). It is seen from this figure that when T o300 K, Cv is proportional to T3; when T4 300 K, Cv increases slowly with temperature and it tends to 26.2 cal/cell K (the Dulonge–Petit limit). The Debye temperature (θ D ) is not a strictly determined parameter, various estimates may be obtained through well established empirical or semi-empirical formulae. One of the semiempirical formulae can be used to estimate the Debye temperature through elastic constants, averaged sound velocity (vm ),
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Fig. 6. Calculated entropy, enthalpy, free energy and heat capacity of T-B3N3.
Table 5 Calculated the density (r in g/cm3), transverse, longitudinal, average sound velocity (vt, vl, vm, in m/s) and the Debye temperatures (yD in K) of T-B3N3. Compounds
r
vt
vl
vm
yD
T-B3N3
2.9396
1929.3
10046.8
2205.9
343.1
longitudinal sound velocity (vl ) and transverse sound velocity (vt ) [41–43]:
θD =
1/3 h ⎡ 3n ⎛ N A ρ ⎞ ⎤ ⎟ ⎥ vm ⎜ ⎢ k ⎣ 4π ⎝ M ⎠ ⎦
(5)
⎤− (1/3) ⎡ ⎛ 1 2 1⎞ vm = ⎢ ⎜⎜ 3 + 3 ⎟⎟ ⎥ ⎢⎣ 3 ⎝ vt vl ⎠ ⎥⎦
(6)
⎛ B + (4/3) G ⎞1/2 vl = ⎜ ⎟ ρ ⎝ ⎠
(7)
⎛ G ⎞1/2 vt = ⎜ ⎟ ⎝ ρ⎠
(8)
where h is Planck constant, k is Boltzmann constant; NA is Avogadro number; ρ is the density; M is the molecular weight; and n is the number of atoms in the unit cell. The calculated values of vm , vl , vt and θ D of T-B3N3 at 0 K are given in Table 5. To our knowledge, there are no experimental or theoretical results for Debye temperature available to us for T-B3N3. The results obtained in this work could provide a useful reference for future studies.
4. Conclusions In the present work, the electronic structure, optical, elastic and thermodynamic properties of T-B3N3 have been studied by means of DFT. The calculated lattice parameters of T-B3N3 are in good agreement with the theoretical results and deviated are less than 0.4%. The electronic structure showed that the T-B3N3 is metallic, and the optical spectra exhibit a noticeable anisotropy for the light
polarizations along the [100] and [001] directions. The static dielectric tensor ε∘ and optical permittivity tensor ε∞ of T-B3N3 are obtained. The elastic properties such as shear modulus and Young's modulus are calculated. From our results, we observe that T-B3N3 is mechanically unstable and ductile. The entropy, enthalpy, free energy, heat capacity and Debye temperature of T-B3N3 was obtained. To the best of our knowledge, there are no available experimental data and theoretical results about the optical, elastic and thermodynamic properties of T-B3N3. The results obtained in this work could provide a useful reference for future studies.
References [1] S. Rudolph, Am. Ceram. Soc. Bull. 73 (1994) 89. [2] R. Haubner, M. Wilhelm, R. Weissenbacher, B. Lux, in: M. Jansen (Ed.), Boron Nitrides – Properties, Synthesis and Applications, Springer, Berlin, 2002. [3] G. Kern, G. Kresse, J. Hafner, Phys. Rev. B 59 (1999) 8551. [4] R.T. Paine, C.K. Narula, Chem. Rev. 90 (1990) 73. [5] Y.N. Xu, W.Y. Ching, Phys. Rev. B 44 (1991) 7787. [6] L. Vel, G. Demazeau, J. Etourneau, Mater. Sci. Eng. B 10 (1991) 149. [7] N.E. Christensen, I. Gorczyca, Phys. Rev. B 50 (1994) 4397. [8] X. Jiang, J.J. Zhao, R. Ahuja, J. Phys.: Condens. Matter 25 (2013) 122204. [9] J.L.P. Castineira1, J.R. Leite, J.L.F. da Silva, L.M.R. Scolfaro, J.L.A. Alves, H.W. L. Alves, Phys. Status Solidi B 210 (1998) 401. [10] P.B. Mirkarimi, K.F. McCarty, D.L. Medlin, Mater. Sci. Eng., R. 21 (1997) 47. [11] F.P. Bundy, R.H. Wentorf, J. Chem. Phys. 38 (1963) 1144. [12] W. Mickelson, S. Aloni, W.Q. Han, J. Cumings, A. Zettl, Science 300 (2003) 467. [13] F.L. Deepak, C.P. Vinod, K. Mukhopadhyay, A. Govindaraj, C.N.R. Rao, Chem. Phys. Lett. 353 (2002) 345. [14] L.Y. Chen, Y.L. Gu, L. Shi, Z.H. Yang, J.H. Ma, Y.T. Qian, Solid State Commun. 130 (2004) 537. [15] Y. Pan, K.F. Huo, Y.M. Hu, J.J. Fu, Y.N. Lu, Z.D. Dai, Z. Hu, Y. Chen, Small 1 (2005) 1199. [16] G. Cappellini, G. Satta, K. Tenelsen, F. Bechstedt, Phys. Status Solidi B 217 (2000) 861. [17] M. Corso, W. Auwarter, M. Muntwiler, A. Tamai, T. Greber, J. Osterwalder, Science 303 (2004) 217. [18] N. Miyata, K. Moriki, O. Mishima, M. Fujisawa, T. Hattori, Phys. Rev. B 40 (1989) 12028. [19] B. Wen, J.J. Zhao, R. Melnikc, Y.J. Tian, Phys. Chem. Chem. Phys. 13 (2011) 14565. [20] S. Botti, J.A. Flores-Livas, M. Amsler, S. Goedecker, M.A.L. Marques, Phys. Rev. B 86 (2012) 121204. [21] S.H. Zhang, Q. Wang, Y. Kawazoe, P. Jena, J. Am. Chem. Soc. 135 (2013) 18216. [22] S.J. Clark, M.D. Segall, C.J. Pickard, P.J. Hasnip, M.J. Probert, K. Refson, M. C. Payne, Z. Kristallogr. 220 (2005) 567. [23] J.P. Perdew, A. Ruzsinszky, G.I. Csonka, O.A. Vydrov, G.E. Scuseria, L. A. Constantin, X.L. Zhou, K. Burke, Phys. Rev. Lett. 100 (2008) 136406. [24] C.G. Broyden, J. Inst., Math. Comput. 6 (1970) 222.
S. Zhao, J. Long / Physica B 459 (2015) 134–139
[25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35]
R. Fletcher, Comput. J. 13 (1970) 317. D. Goldfarb, Math. Comput. 24 (1970) 23. D.F. Shanno, Math. Comput. 24 (1970) 647. S. Mamoun, A.E. Merad, L. Guilbert, Comput. Mater. Sci. 79 (2013) 125. J.H. Weiner, Statistical Mechanics of Elasticity, John Wiley, New York, 1983. S. Piskunov, E. Heifets, R.I. Eglitis, G. Borstel, Comput. Mater. Sci. 29 (2004) 165. M. Grimsditch, E.S. Zouboulis, A. Polian, J. Appl. Phys. 76 (1994) 832. K. Shimada, T. Sota, K. Suzuki, J. Appl. Phys. 84 (1998) 4951. J.P. Watt, L. Peselnick, J. Appl. Phys. 51 (1980) 1525. R. Hill, Proc. Phys. Soc. A 65 (1952) 349. J.P. Long, L.J. Yang, D.M. Li, Solid State Sci. 19 (2013) 12.
[36] [37] [38] [39]
[40] [41] [42] [43]
139
E. Knittle, R.M. Wentzcovitch, R. Jeanloz, M.L. Cohen, Nature 337 (1989) 349. Y.N. Xu, W.Y. Ching, Phys. Rev. B 44 (1991) 7787. S.F. Pugh, Philos. Mag. 45 (1954) 823. I.N. Frantsevich, F.F. Voronov, S.A. Bokuta, in: I.N. Frantsevich (Ed.), Elastic Constants and Elastic Moduli of Metals and Insulators Handbook, Naukova Dumka, Kiev, 1983, p. 60. W. Köster, H. Franz, Int. Mater. Rev. 6 (1961) 1. H.Y. Lu, L.J. Yang, W. Huang, J. Supercond. Nov. Magn. 27 (2014) 1671. H.Y. Lu, J.P. Long, L.J. Yang, W. Huang, Int. J. Mod. Phys. B 28 (2014) 1450057. J.P. Long, L.J. Yang, X.S. Wei, J. Alloys Compd. 549 (2013) 336.