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Elastic properties of cubic perovskite BaRuO3 from first-principles calculations
ARTICLE IN PRESS Physica B 405 (2010) 3117–3119
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Elastic properties of cubic perovskite BaRuO3 from first-principles calculations De-Ming Han, Xiao-Juan Liu, Shu-Hui Lv, Hong-Ping Li, Jian Meng n State Key Laboratory of Rare Earth Resource Utilization, Changchun Institute of Applied Chemistry, Chinese Academy of Sciences, Changchun 130022, People’s Republic of China
a r t i c l e in fo
abstract
Article history: Received 2 February 2010 Received in revised form 23 March 2010 Accepted 14 April 2010
We present first-principles investigations on the structural and elastic properties of the cubic perovskite BaRuO3 using density-functional theory within both local density approximation (LDA) and generalized gradient approximation (GGA). Basic physical properties, such as lattice constant, shear modulus, elastic constants (Cij) are calculated. The calculated energy band structures show that the cubic perovskite BaRuO3 is metallic. We have also predicted the Young’s modulus (Y), Poisson’s ratio (u), and Anisotropy factor (A). & 2010 Elsevier B.V. All rights reserved.
Keywords: First-principles calculations Elastic constants Anisotropy Perovskite
1. Introduction Perovskites materials attract much attention from physicists and material scientists because of the diverse properties, such as ferroelectricity, piezoelectricity, ferromagnetism, superconductivity, half-metallic transport, and colossal magnetoresistance. It is well-known that the electronic and elastic properties of strontium titanate, SrTiO3—one of the generic representative of d transitionmetal oxide perovskites have been investigated by Shein et al. [1] from first-principles calculations. As an example of 4d based perovskites, the cubic SrZrO3 is investigated with different theoretical approach [2–5]. In addition, the elastic and electronic properties of cubic perovskite BaHfO3 have been studied by Zhao et al. [6] using a first-principles method based on the plane-wave basis set. The properties of SrRuO3 and CaRuO3 in cubic and real orthorhombic structures have also been studied by Santi and Jarlborg [7] by means of self-consistent LMTO band calculations. The electronic properties of the hexagonal perovskite ARuO3 (A ¼Ca, Sr and Ba) have been investigated by Rama Rao et al. [8] with XPS and UPS techniques along with the tight binding linear muffin tin orbitals within atomic sphere approximation (TB–LMTO–ASA) calculations. In the earlier years, it is well-known that BaRuO3 has polytype structures [9–11] depending on how it is synthesized, i.e., the nine-layered rhombohedral (9R), the four-layered hexagonal (4H), and the six-layered hexagonal (6H), which their structure and
n
Corresponding author. Tel.: + 86 431 85262030; fax: +86 431 85698041. E-mail address: [email protected] (J. Meng).
0921-4526/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2010.04.025
physical properties have been investigated extensively by many researchers [8,12,13]. For example, the 6 H BaRuO3 is an abnormal paramagnetic metal deviation from the Fermi liquid behavior by the measurements of electrical and magnetic properties. Recently, the cubic perovskite BaRuO3 with space group Pm-3m has been synthesized by the application of high-pressure technique [14]. However, there has been no study on the elastic properties from the theoretical point of view. We have therefore performed the first-principles calculations to provide the necessary theoretical information and relationship for guidance of further experimental studies of the cubic perovskite BaRuO3.
2. Computational method We have performed the first-principles calculations to investigate the structural and elastic properties of the cubic perovskite BaRuO3 within the Vienna ab-initio simulation package (VASP) [15–17] with the ion-electron interaction described by the projector augmented wave (PAW) potential. We used both the local density approximation (LDA) [18] and generalized gradient approximation (GGA) [19] for the exchange-correlation functional. The optimization of the structural parameters is performed ˚ and the until the forces on the atoms are less than 1 meV/A, convergence criteria for energy is 10 5 eV. Calculations are converged at a cutoff energy of 400 eV for the plane-wave basis set. Brillouin zone integrations are approximated using the special k-point sampling of the Monkhorst–Pack scheme [20] with a 10 10 10 grid.
ARTICLE IN PRESS 3118
D.-M. Han et al. / Physica B 405 (2010) 3117–3119
3. Results and discussion
spin up 10
9GB Gþ 3B
6 4 Energy (eV)
2 0
EF
-2 -4 -6 -8 -10 X
G
R
X
M
R
G
M
spin down 10 8 6 4 2 0
EF
-2 -4 -6 -8 -10 X
G
R
X
M
R
G
M
Fig. 2. Calculated energy band structure of BaRuO3. The Fermi energy level is at.zero.
Table 1 ˚ equilibrium volume V (A˚ 3), bulk modulus B Calculated lattice parameter a (A), (GPa), isotropic shear modulus G (GPa), and elastic constants Cij (GPa) for BaRuO3 within both LDA and GGA levels, respectively.
and Y¼
8
Energy (eV)
The considered BaRuO3 perovskite has ideal cubic structure (SG: Pm-3m) where atomic positions in the elementary cell are Ru: 1a (0, 0, 0), O: 3d (0, 0, 1/2) and Ba: 1b (1/2, 1/2, 1/2). The structure parameters optimized for cubic BaRuO3 have been listed in Fig. 1. The calculated results from LDA and GGA are very ˚ The consistent. The calculated lattice constant is a ¼4.00592 A. distance between one Ru atom and its neighbors O atom is ˚ and the Ru–O–Ru bonding angle is 180o% . Those 2.003 A, parameters are all in good agreement with the experimental values [14]. This demonstrates the reliability of the method. Although it is not our main intention here to make detailed band structure calculations, we have predicted the spin-polarized band structures for cubic BaRuO3 along the high symmetry directions from the calculated equilibrium lattice constant as shown in Fig. 2. From the results calculated, one knows that the band structure calculated using both the LDA and the GGA methods at the corresponding equilibrium lattice constants show similar patterns. Thus, as a matter of convenience, the GGA results are presented in this paper. From Fig. 2, one can see clearly that the cubic BaRuO3 exhibits metallic character. It is well-known that the accurate calculation for elasticity is important for understanding the macroscopic mechanical properties of solids and providing information on the stability and stiffness of materials. For obtaining the elastic data through the abinitio simulations of materials from the known crystal structure, we adopt an approach based on the analysis of the total energy of properly strained states of the material (volume conserving technique) [21]. Cubic lattices have three independent elastic constants [22–24], namely, C11, C12 and C44. The elastic constants obtained from LDA and GGA calculations are listed in Table 1. The traditional mechanical stability conditions in cubic crystals on the elastic constants are known as C11 C12 40, C11 40, C44 40, C11 +2C12 40, and C12 oBoC11, where B is the calculated bulk modulus of the crystal. The elastic constants in Table 1 satisfy these stability criteria, and hence, we can say that the phase is elastically stable. Unfortunately, there are no other theoretical or experimental results for comparing with the present work directly. The Poisson’s ratio u and Young’s modulus Y, which are the most interesting elastic properties for applications, are also calculated in terms of the computed data using the following relations [25]: " # 1 B 23 G ð1Þ u¼ 2 B þ 13 G
ð2Þ LDA GGA
Fig. 1. Crystal structure of cubic perovskite BaRuO3. The RuO6 octahedra are shown in blue. Red and green spheres denote the O and Ba atoms. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
a
V
B
G
C11
C12
C44
4.00592 4.00592
64.28 64.28
167.9 194.4
54.99 59.71
201.3 234.7
151.2 174.2
92.4 93.9
where G¼(GV +GR)/2 is the isotropic shear modulus, GV is Voigt’s shear modulus corresponding to the upper bound of G values, and GR is Reuss’s shear modulus corresponding to the lower bound of G values; they can be written as GV ¼(C11 C12 +3C44)/5, and 5/GR ¼4/(C11 C12)+3/C44. The calculated Poisson’s ratio (u), Young’s modulus (Y), and shear modulus (C0 ¼(C11 C12 +2C44)/4) for BaRuO3 are given in Table 2. The elastic anisotropy of crystals has an important implication in engineering science since it is highly correlated with the possibility to induce microcracks in the materials [26]. The anisotropy factor for cubic crystals [27], A¼ (2C44 + C12)/C11, has therefore been evaluated to provide insight on the elastic
ARTICLE IN PRESS D.-M. Han et al. / Physica B 405 (2010) 3117–3119
Table 2 The calculated anisotropy factor (A), Poisson’s ratio (u), Young’s modulus (Y), and Shear modulus (C0 ) for BaRuO3 within both LDA and GGA levels, respectively.
LDA GGA
A
u
Y(GPa)
C0 (GPa)
1.669 1.542
0.35 0.36
148.73 162.49
58.72 62.07
u1 (m/s)
ut (m/s)
um (m/s)
TD (K)
5768.764 6120.823
2817.329 2896.579
3164.752 3259.175
640 659
anisotropy of the cubic BaRuO3. For a completely isotropic material, the A factor takes the value of 1, when the value of A is smaller or greater than unity it is a measure of the degree of elastic anisotropy. Seen from Table 2, the values of anisotropy factor for the BaRuO3 are 1.542 and 1.669 within both LDA and GGA levels, greater than 1. The BaRuO3 is showing a certain amount of elastic anisotropy, which might lead to a higher probability to develop microcracks or structural defects during the growing process. The Debye temperature (TD) is a fundamental parameter of a material which is linked to many physical properties, such as specific heat, elastic constants, and melting point. At low temperatures, the vibrational excitations arise solely from acoustic vibrations. Hence, at low temperatures, the Debye temperature calculated from elastic constants is the same as that determined from specific heat measurements. It can be obtained from the average wave velocity by use of the following equation [28]: _ 3n NA r 1=3 um ð3Þ TD ¼ k 4p M Here _ is the Planck’s constant; k the Boltzmann’s constant; NA the Avogadro’s number; r the density; M the molecular mass per formula unit; n the number of atoms per formula unit. The average wave velocity um is approximately estimated by the equation: " !#1=3 1 2 1 um ¼ þ ð4Þ 3 u3t u3l where u1 and ut are longitudinal and transverse elastic wave velocities, respectively, which are obtained from Navier’s equations [29] 3B þ4G 1=2 ð5Þ ul ¼ 3r and ut ¼
1=2 G
r
BaRuO3 [13]. Our results indicated that the TD value from LDA is smaller than that obtained from the GGA, owing to the smaller isotropic shear modulus and bulk modulus calculated from LDA.
4. Conclusions
Table 3 The longitudinal, transverse, average elastic wave velocities, and Debye temperature for cubic BaRuO3 within both LDA and GGA levels, respectively.
LDA GGA
3119
ð6Þ
The calculated average, longitudinal and transverse elastic wave velocities and Debye temperature for BaRuO3 are given in Table 3. Unfortunately, no other theoretical or experimental data exist for comparison with the present values. However, our values for Debye temperatures are higher than those for 9R, 4H and 6H
In this work we have reported, for the first time, some theoretical results on the structural and elastic properties for the cubic perovskite BaRuO3 based on the first-principles calculations. The calculated lattice constant is in good agreement with experimental value in the level of LDA and GGA. Unfortunately, for the other properties calculated in this work, there are no theoretical and experimental results for comparison. Certainly, we believe that more experimental and theoretical work is required on the high-pressure phase of BaRuO3 to clarify these properties in all aspects.
Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant Nos.20831004, 20671088, 20601026 and 20771100).
References [1] I.R. Shein, V.L. Kozhevnikov, A.L. Ivanovskii, Solid State Sci. 10 (2008) 217. [2] M. Yoshino, K. Nakatsuka, H. Yukawa, M. Morinaga, Solid State Ionics 127 (2000) 109. [3] Y.S. Lee, J.S. Lee, T.W. Noh, D.Y. Byun, K.S. Yoo, K. Yamaura, E. TakayamaMuromachi, Phys. Rev. B 67 (2003) 113101. [4] R.A. Evarestov, A.V. Bandura, V.E. Aleksandrov, E.A. Kotomin, Phys. Status Solidi 242 (2005) R11. [5] R.A. Evarestov, I.I. Tupitsyn, A.V. Bandura, V.E. Aleksandrov, Int. J. Quantum Chem. 106 (2006) 2191. [6] Zhao Hongsheng, Chang Aimin, Wang Yunlan, Physica B 404 (2009) 2192. [7] G. Santi, T. Jarlborg, J. Phys.: Condens. Matter 9 (1997) 9563. [8] M.V. Rama Rao, V.G. Sathe, D. Sornadurai, B. Panigrahi, T. Shripathi, J. Phys. Chem. Solids 62 (2001) 797. [9] P.C. Donohue, L. Katz, R. Ward, Inorg. Chem. 4 (1965) 306. [10] S.T. Hong, A.W. Sleight, J. Solid State Chem. 128 (1997) 251. [11] J.T. Rijssenbeek, R. Jin, Yu. Zadorozhny, Y. Liu, B. Batlogg, R.J. Cava, Phys. Rev. B 59 (1999) 4561. [12] T.W. Noh, Y.S. Lee, J.S. Lee, K.W. Kim, J. Yu, G. Cao, J.E. Crow, M.K. Lee, C.B. Eom, Physica C 364–365 (2001) 480. [13] J.G. Zhao, L.X. Yang, Y. Yu, F.Y. Li, R.C. Yu, Z. Fang, L.C. Chen, C.Q. Jin, J. Solid State Chem. 180 (2007) 2816. [14] C.Q. Jin, J.S. Zhou, J.B Goodenough, Q.Q. Liu, J.G. Zhao, L.X. Yang, Y. Yu, R.C. Yu, T. Katsura, A. Shatskiy, E. Ito, Proc. Natl., Acad. Sci. USA 105 (2008) 7115. [15] G. Kresse, J. Hafner, Phys. Rev. B 47 (1993) 558. ¨ [16] G. Kresse, J. Furthmuller, Comput. Mater. Sci. 6 (1996) 15. ¨ [17] G. Kresse, J. Furthmuller, Phys. Rev. B 54 (1996) 11169. [18] D.M. Ceperley, B.J. Alder, Phys. Rev. Lett. 45 (1980) 566. [19] J.P. Perdew, J.A. Chevary, S.H. Vosko, K.A. Jackson, M.R. Pederson, D.J. Singh, C. Fiolhais, Phys. Rev. B 46 (1992) 6671. [20] H.J. Monkhorst, J.D. Pack, Phys. Rev. B 13 (1976) 5188. [21] S.Q. Wu, Z.F. Hou, Z.Z. Zhu, Solid State Commun. 143 (2007) 425. [22] J.F. Nye, Physical Properties of Crystals: Their Representation by Tensors and Matrices, Oxford University Press, Oxford, 1985. [23] M. Mattesini, R. Ahuja, B. Johansson, Phys. Rev. B 68 (2003) 184108. [24] M. Mattesini, J.M. Soler, F. Yndura´in, Phys. Rev. B 73 (2006) 094111. [25] B. Mayer, H. Anton, E. Bott, M. Methfessel, J. Sticht, P.C. Schmidt, Intermetallics 11 (2003) 23. [26] V. Tvergaard, J.W. Hutchinson, J. Am. Ceram. Soc. 71 (1988) 157. [27] B.B. Karki, L. Stixrude, S.J. Clark, M.C. Warren, G.J. Ackland, J. Crain, Am. Mineral. 82 (1997) 51. [28] O.L. Anderson, J. Phys. Chem. Solids 24 (1963) 909. [29] E. Schreiber, O.L. Anderson, N. Soga, in: Elastic Constants and their Measurements, McGraw-Hill, New York, 1973.
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Elastic properties of cubic perovskite BaRuO3 from first-principles calculations De-Ming Han, Xiao-Juan Liu, Shu-Hui Lv, Hong-Ping Li, Jian Meng n State Key Laboratory of Rare Earth Resource Utilization, Changchun Institute of Applied Chemistry, Chinese Academy of Sciences, Changchun 130022, People’s Republic of China
a r t i c l e in fo
abstract
Article history: Received 2 February 2010 Received in revised form 23 March 2010 Accepted 14 April 2010
We present first-principles investigations on the structural and elastic properties of the cubic perovskite BaRuO3 using density-functional theory within both local density approximation (LDA) and generalized gradient approximation (GGA). Basic physical properties, such as lattice constant, shear modulus, elastic constants (Cij) are calculated. The calculated energy band structures show that the cubic perovskite BaRuO3 is metallic. We have also predicted the Young’s modulus (Y), Poisson’s ratio (u), and Anisotropy factor (A). & 2010 Elsevier B.V. All rights reserved.
Keywords: First-principles calculations Elastic constants Anisotropy Perovskite
1. Introduction Perovskites materials attract much attention from physicists and material scientists because of the diverse properties, such as ferroelectricity, piezoelectricity, ferromagnetism, superconductivity, half-metallic transport, and colossal magnetoresistance. It is well-known that the electronic and elastic properties of strontium titanate, SrTiO3—one of the generic representative of d transitionmetal oxide perovskites have been investigated by Shein et al. [1] from first-principles calculations. As an example of 4d based perovskites, the cubic SrZrO3 is investigated with different theoretical approach [2–5]. In addition, the elastic and electronic properties of cubic perovskite BaHfO3 have been studied by Zhao et al. [6] using a first-principles method based on the plane-wave basis set. The properties of SrRuO3 and CaRuO3 in cubic and real orthorhombic structures have also been studied by Santi and Jarlborg [7] by means of self-consistent LMTO band calculations. The electronic properties of the hexagonal perovskite ARuO3 (A ¼Ca, Sr and Ba) have been investigated by Rama Rao et al. [8] with XPS and UPS techniques along with the tight binding linear muffin tin orbitals within atomic sphere approximation (TB–LMTO–ASA) calculations. In the earlier years, it is well-known that BaRuO3 has polytype structures [9–11] depending on how it is synthesized, i.e., the nine-layered rhombohedral (9R), the four-layered hexagonal (4H), and the six-layered hexagonal (6H), which their structure and
n
Corresponding author. Tel.: + 86 431 85262030; fax: +86 431 85698041. E-mail address: [email protected] (J. Meng).
0921-4526/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2010.04.025
physical properties have been investigated extensively by many researchers [8,12,13]. For example, the 6 H BaRuO3 is an abnormal paramagnetic metal deviation from the Fermi liquid behavior by the measurements of electrical and magnetic properties. Recently, the cubic perovskite BaRuO3 with space group Pm-3m has been synthesized by the application of high-pressure technique [14]. However, there has been no study on the elastic properties from the theoretical point of view. We have therefore performed the first-principles calculations to provide the necessary theoretical information and relationship for guidance of further experimental studies of the cubic perovskite BaRuO3.
2. Computational method We have performed the first-principles calculations to investigate the structural and elastic properties of the cubic perovskite BaRuO3 within the Vienna ab-initio simulation package (VASP) [15–17] with the ion-electron interaction described by the projector augmented wave (PAW) potential. We used both the local density approximation (LDA) [18] and generalized gradient approximation (GGA) [19] for the exchange-correlation functional. The optimization of the structural parameters is performed ˚ and the until the forces on the atoms are less than 1 meV/A, convergence criteria for energy is 10 5 eV. Calculations are converged at a cutoff energy of 400 eV for the plane-wave basis set. Brillouin zone integrations are approximated using the special k-point sampling of the Monkhorst–Pack scheme [20] with a 10 10 10 grid.
ARTICLE IN PRESS 3118
D.-M. Han et al. / Physica B 405 (2010) 3117–3119
3. Results and discussion
spin up 10
9GB Gþ 3B
6 4 Energy (eV)
2 0
EF
-2 -4 -6 -8 -10 X
G
R
X
M
R
G
M
spin down 10 8 6 4 2 0
EF
-2 -4 -6 -8 -10 X
G
R
X
M
R
G
M
Fig. 2. Calculated energy band structure of BaRuO3. The Fermi energy level is at.zero.
Table 1 ˚ equilibrium volume V (A˚ 3), bulk modulus B Calculated lattice parameter a (A), (GPa), isotropic shear modulus G (GPa), and elastic constants Cij (GPa) for BaRuO3 within both LDA and GGA levels, respectively.
and Y¼
8
Energy (eV)
The considered BaRuO3 perovskite has ideal cubic structure (SG: Pm-3m) where atomic positions in the elementary cell are Ru: 1a (0, 0, 0), O: 3d (0, 0, 1/2) and Ba: 1b (1/2, 1/2, 1/2). The structure parameters optimized for cubic BaRuO3 have been listed in Fig. 1. The calculated results from LDA and GGA are very ˚ The consistent. The calculated lattice constant is a ¼4.00592 A. distance between one Ru atom and its neighbors O atom is ˚ and the Ru–O–Ru bonding angle is 180o% . Those 2.003 A, parameters are all in good agreement with the experimental values [14]. This demonstrates the reliability of the method. Although it is not our main intention here to make detailed band structure calculations, we have predicted the spin-polarized band structures for cubic BaRuO3 along the high symmetry directions from the calculated equilibrium lattice constant as shown in Fig. 2. From the results calculated, one knows that the band structure calculated using both the LDA and the GGA methods at the corresponding equilibrium lattice constants show similar patterns. Thus, as a matter of convenience, the GGA results are presented in this paper. From Fig. 2, one can see clearly that the cubic BaRuO3 exhibits metallic character. It is well-known that the accurate calculation for elasticity is important for understanding the macroscopic mechanical properties of solids and providing information on the stability and stiffness of materials. For obtaining the elastic data through the abinitio simulations of materials from the known crystal structure, we adopt an approach based on the analysis of the total energy of properly strained states of the material (volume conserving technique) [21]. Cubic lattices have three independent elastic constants [22–24], namely, C11, C12 and C44. The elastic constants obtained from LDA and GGA calculations are listed in Table 1. The traditional mechanical stability conditions in cubic crystals on the elastic constants are known as C11 C12 40, C11 40, C44 40, C11 +2C12 40, and C12 oBoC11, where B is the calculated bulk modulus of the crystal. The elastic constants in Table 1 satisfy these stability criteria, and hence, we can say that the phase is elastically stable. Unfortunately, there are no other theoretical or experimental results for comparing with the present work directly. The Poisson’s ratio u and Young’s modulus Y, which are the most interesting elastic properties for applications, are also calculated in terms of the computed data using the following relations [25]: " # 1 B 23 G ð1Þ u¼ 2 B þ 13 G
ð2Þ LDA GGA
Fig. 1. Crystal structure of cubic perovskite BaRuO3. The RuO6 octahedra are shown in blue. Red and green spheres denote the O and Ba atoms. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
a
V
B
G
C11
C12
C44
4.00592 4.00592
64.28 64.28
167.9 194.4
54.99 59.71
201.3 234.7
151.2 174.2
92.4 93.9
where G¼(GV +GR)/2 is the isotropic shear modulus, GV is Voigt’s shear modulus corresponding to the upper bound of G values, and GR is Reuss’s shear modulus corresponding to the lower bound of G values; they can be written as GV ¼(C11 C12 +3C44)/5, and 5/GR ¼4/(C11 C12)+3/C44. The calculated Poisson’s ratio (u), Young’s modulus (Y), and shear modulus (C0 ¼(C11 C12 +2C44)/4) for BaRuO3 are given in Table 2. The elastic anisotropy of crystals has an important implication in engineering science since it is highly correlated with the possibility to induce microcracks in the materials [26]. The anisotropy factor for cubic crystals [27], A¼ (2C44 + C12)/C11, has therefore been evaluated to provide insight on the elastic
ARTICLE IN PRESS D.-M. Han et al. / Physica B 405 (2010) 3117–3119
Table 2 The calculated anisotropy factor (A), Poisson’s ratio (u), Young’s modulus (Y), and Shear modulus (C0 ) for BaRuO3 within both LDA and GGA levels, respectively.
LDA GGA
A
u
Y(GPa)
C0 (GPa)
1.669 1.542
0.35 0.36
148.73 162.49
58.72 62.07
u1 (m/s)
ut (m/s)
um (m/s)
TD (K)
5768.764 6120.823
2817.329 2896.579
3164.752 3259.175
640 659
anisotropy of the cubic BaRuO3. For a completely isotropic material, the A factor takes the value of 1, when the value of A is smaller or greater than unity it is a measure of the degree of elastic anisotropy. Seen from Table 2, the values of anisotropy factor for the BaRuO3 are 1.542 and 1.669 within both LDA and GGA levels, greater than 1. The BaRuO3 is showing a certain amount of elastic anisotropy, which might lead to a higher probability to develop microcracks or structural defects during the growing process. The Debye temperature (TD) is a fundamental parameter of a material which is linked to many physical properties, such as specific heat, elastic constants, and melting point. At low temperatures, the vibrational excitations arise solely from acoustic vibrations. Hence, at low temperatures, the Debye temperature calculated from elastic constants is the same as that determined from specific heat measurements. It can be obtained from the average wave velocity by use of the following equation [28]: _ 3n NA r 1=3 um ð3Þ TD ¼ k 4p M Here _ is the Planck’s constant; k the Boltzmann’s constant; NA the Avogadro’s number; r the density; M the molecular mass per formula unit; n the number of atoms per formula unit. The average wave velocity um is approximately estimated by the equation: " !#1=3 1 2 1 um ¼ þ ð4Þ 3 u3t u3l where u1 and ut are longitudinal and transverse elastic wave velocities, respectively, which are obtained from Navier’s equations [29] 3B þ4G 1=2 ð5Þ ul ¼ 3r and ut ¼
1=2 G
r
BaRuO3 [13]. Our results indicated that the TD value from LDA is smaller than that obtained from the GGA, owing to the smaller isotropic shear modulus and bulk modulus calculated from LDA.
4. Conclusions
Table 3 The longitudinal, transverse, average elastic wave velocities, and Debye temperature for cubic BaRuO3 within both LDA and GGA levels, respectively.
LDA GGA
3119
ð6Þ
The calculated average, longitudinal and transverse elastic wave velocities and Debye temperature for BaRuO3 are given in Table 3. Unfortunately, no other theoretical or experimental data exist for comparison with the present values. However, our values for Debye temperatures are higher than those for 9R, 4H and 6H
In this work we have reported, for the first time, some theoretical results on the structural and elastic properties for the cubic perovskite BaRuO3 based on the first-principles calculations. The calculated lattice constant is in good agreement with experimental value in the level of LDA and GGA. Unfortunately, for the other properties calculated in this work, there are no theoretical and experimental results for comparison. Certainly, we believe that more experimental and theoretical work is required on the high-pressure phase of BaRuO3 to clarify these properties in all aspects.
Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant Nos.20831004, 20671088, 20601026 and 20771100).
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