Theoretical investigation on electronic structure and mechanical properties of cubic crystallographic structures with point defects in Al-based alloys

Solid State Communications 152 (2012) 1263–1269

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Theoretical investigation on electronic structure and mechanical properties of cubic crystallographic structures with point defects in Al-based alloys L.M. Dong a,b,n, Z.D. Han a,b, Y.L. Wang a, W. Li a, X.Y. Zhang a a b

College of Materials Science and Engineering, Harbin University of Science and Technology, Harbin 150040, China Key Laboratory of Engineering Dielectrics and Its Application, Ministry of Education, Harbin University of Science and Technology, Harbin 150080, China

a r t i c l e i n f o

abstract

Article history: Received 24 August 2011 Received in revised form 28 January 2012 Accepted 28 March 2012 by S. Scandolo Available online 5 April 2012

Electronic structure and mechanical properties of cubic crystallographic structures with point defects in Al-based alloys are investigated using the first-principles calculations. Equilibrium structural parameters and mechanical parameters such as bulk modulus, shear modulus, Young’s modulus, Poisson’s ratio and anisotropy are calculated and agreed well with experimental values. Effects of point defects on the electronic structures and mechanical properties of such cubic phases are further analyzed and discussed in view of the charge density and the density of states. & 2012 Elsevier Ltd. All rights reserved.

Keywords: A. Al alloys C. Defect formation D. Elastic constants

1. Introduction In order to reduce fuel consumption, preserve resources and protect the environment, the emphasis on the application of high performance and lightweight structural materials in the field of defense, aerospace, marine, automotive manufacturing is increasing. Therefore, lightweight and high performance (high mechanic strength, corrosion resistance, etc.) structural materials have been mainly researched in the domain of the structural materials. Aluminum alloys are lightweight structural materials and have great application potentials because of their low density, high mechanic strength, corrosion resistance, simple production process, low costs in comparison with other alloys, particularly the significant advantages of easy-recycling. These properties are expected to be used in the aerospace and defense industry and have been widely used in daily life [1–5]. Therefore, aluminum alloys play a decisive role in the lightweight structural materials [6]. In recent years, the properties of Al-based alloys have been extensively investigated experimentally and theoretically [7–9]. In particular, the first-principles have been employed to investigate various electronic structural parameter and related properties such as enthalpies of formation, equilibrium lattice constants, elastic constants, cohesive energy, and defect formation energies [10–14]. However, the studies on defects have mainly focused on the formation heats, and the studies on the effects of defect formation energies on mechanical properties are quiet few in the

n

Corresponding author.

0038-1098/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ssc.2012.03.037

literature. In particular, due to the effects of defects the correlation between electronic structure and elastic properties in Albased alloys has been hardly established very well. Vacancy is one of the most simple and common point defects of solid. Vacancy has an important impact on material physical and mechanical properties, For example, materials phase change, crystal growth, atomic diffusion structure evolution, etc. [7]. Therefore, it is important to study the formation and nature of vacancies in detail in order to understand the micro- and macroproperties of materials better. First-principles calculations have become a powerful tool for the investigation of the crystalline and electronic structures and mechanical properties of solids [15]. In this paper, electronic structure and mechanical properties of typical face-centered cubic structure of Al-based alloys (AlCu3 and AlNi3) with vacancy point defect have been investigated. The obtained equilibrium lattice constants, volume and point defect formation energy with vacancy point defect and no vacancy, after relaxation have been calculated by first-principles theoretical calculations, and the results are discussed in comparison with the available experimental data and other calculated value. Elastic constants and other mechanical parameters are calculated in order to study the ductility, and other mechanical properties of alloys. First, we will introduce two basic concepts: point defect formation energy and elastic constant. Point defect formation energy is a very important concept. It is an index to value the formation of the specific defects easy or not. For point defects, only the atoms with higher than the point defect formation energy can form point defects. The point formation energy in alloys relies

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on the energy of single atom or atomic chemical potentials. The formation energies of vacancy can be presented by [16] Evac ¼ ½Evac f rel þEc Erel

ð1Þ

where Evac rel is the total energy of the supercell with a defect, Erel is the total energy of the perfect crystal, and Ec is the energy of single atom. Materials can restore to the previous state (original shape and size) as soon as the deforming force moved. The property of returning to an initial state is called elastic deformation. Elastic constants are the measure of the resistance of a crystal to an externally applied stress. There are the relationship between constants and mechanics of materials and dynamics. The elastic constants and other elastic modulus such as the bulk modulus, shear modulus, Young’s modulus, Poisson’s ratio, anisotropy value, etc. [17] determine the stability, stiffness and strength of crystal materials. Young’s modulus E, Poisson’s ratio g and the elastic anisotropy A of crystals are important for technological and engineering applications, e.g., Young’s modulus is defined as the ratio between stress and strain, that is used to provide a measure of the stiffness of the solid, i.e. the larger the value of E, the stiffer the material [18]; Poisson’s ratio is used to quantify the stability of the crystal against shear, which usually ranges from 1 to 0.5 and actual measured always ranges from 0 to 0.5; the elastic anisotropy is defined as A for cubic symmetric structures [19] and it is highly correlated with the possibility of inducing microcracks in materials [18,20]. If the material is completely isotropic, the value of A will be one, while values smaller or bigger than 1 measuring the degree of elastic anisotropy [18]. Elastic modulus is by nature the parameter that expresses the atomic binding force. The atomic binding force increase with elastic modulus’ increasing, results in the good alloying ability and structural stability. For small strains, Hooke’s law is applicable and the elastic strain energy is given by the crystal energy E is a quadratic function of strain [21]. Thus, to obtain the total minimum energy for calculating the elastic constants to second order, a crystal is strained and all the internal parameters relaxed. The elastic strain energy is given by U¼

DE V0

¼

6 X 6 1X C ee 2 i j ij i j

ð2Þ

where DE is the total energy difference in energy between that of the strained lattice and the unstrained lattice. V0 is the volume of equilibrium cell and Cij is the elastic constant, which is the element of n  6 elastic constant matrix in Voigt’s notation. For the cubic structure, crystal symmetry [21,22] dictates Cij ¼ Cji and all unlisted Cij ¼0, so there are three independent elastic constants, namely C11, C12 and C44. The linearly independent equations should be three, which can be obtained by considering the strain tensors: 0 1 e1 e26 e25 e6 e4 C eij ¼ B ð3Þ @ 2 e2 2 A e4 e5 e 3 2 2 In order to obtain the elastic constants C11, C12 and C44 of the cubic crystal in the present study, we applied three kinds of strains, the eij (i,j¼1,2,3), as shown in Table 1 [23,24]. If assumed (i) e1 ¼e2 ¼ g, e3 ¼(1þ g)  2  1; (ii) e1 ¼e2 ¼e3 ¼ g; (iii) e6 ¼ g, e3 ¼ g2(4 g2)  1, while the unspecified strain components are zero for each of these [21,22], the following three relations for the deformation energies are obtained:

DE V0

¼ 3ðC 11 C 12 Þg2

Table 1 The three strains used to calculate the three elastic constants of cubic crystal. The strain g is varied in steps of 0.01 from g ¼  0.02 to 0.02 or  0.03 to 0.03.

ð4Þ

Structure

Strain

Parameters (unlisted ei ¼0)

DE=V 0 to 0 (g2 )

Cubic

e(1) e(2) e(3)

e1 ¼e2 ¼ g,e3 ¼(1 þ g)  2  1 e1 ¼e2 ¼e3 ¼ g

3(C11  C12) 2 3 2 2ðC 11 þ 2C 12 Þ 1 2 2C 44

DE V0

DE V0

e6 ¼ g,e3 ¼ g2(4  g2)  1

g g

g

¼

3 ðC 11 þ 2C 12 Þg2 2

ð5Þ

¼

1 C 44 g2 2

ð6Þ

By means of polynomial fits, we extract three second-order coefficients, which is corresponding to 3(C11  C12), 32ðC 11 þ 2C 12 Þ and 12C 44 , respectively. By resolving the three equations according to the relationship above, the elastic constants C11, C12 and C44 are obtained. Further we calculate the elastic modulus, i.e., bulk modulus B, shear modulus G, Young’s modulus E, Passion’s ratio g and anisotropy value A, which can be determined by combining values of the single-crystal Cij according to the proper symmetry relations. The detailed calculations of the elastic modulus of cubic structure are as follows: The bulk modulus K K¼

C 11 þ2C 12 3

ð7Þ

The shear modulus G is obtained from the following formula: G¼

C 11 C 12 2

ð8Þ

Young’s modulus E and Poisson’s ratio n are expressed according to the following formulas: E¼

ð3C 11 4C 44 ÞC 44 C 11 C 44

n¼

3BE 6B

ð9Þ

ð10Þ

The anisotropy value A has been obtained according to the following formula [19]: A¼

2C 44 þ C 12 C 11

ð11Þ

2. Calculation method The Vienna ab initio Simulation Package (VASP) [25–27] was used in our calculations to perform the electronic structure calculations based on DFT [28,29] and the projector augmented wave (PAW) [30,31] method. The exchange and correlation energy was treated within the generalized gradient approximation of Perdew–Wang 91 version (GGA-PW91) [32] and were described by the local density approximation (LDA) of Perdew and Wang [33]. The PAW potential used for Al treats 3s, 3p states as valence states, and the other electron–ion interaction is described by 3d, 4s, valence states for Cu, 3p, 3d, 4s and 3p, 3d, 4s valence states for Ni. Convergence test showed that increasing the energy cut-off to 300 eV and it was used to calculate the electronic structures of all the compounds. The electron–ion interaction was described by pseudo-potential with the s, p valence electrons for Al, and s, p, d valence electrons for Cu. The Brillouin Zone were performed using the Monkhorst–Pack [34] k-point meshes, e.g.,

L.M. Dong et al. / Solid State Communications 152 (2012) 1263–1269

the k-point meshes for bulk calculations of AlCu3, and AlNi3 were, and 7  7  7 for the geometry optimization. After the optimized k-point meshes for the surface calculations, the structural parameters (lattice constants and atom positions) were optimized via a conjugate gradient method [35], and the total energy and density of state (DOS) calculations were performed with the linear tetrahedron method. The coordinates of internal atoms were allowed to relax until the forces on unconstrained atoms ˚ converged to less than 0.001 eV/A. Here it should be pointed out in general, the effect of spin polarization should be considered. But in fact, it was found that spin-polarization effects in all compounds of the Al–Ni–Y system are negligible [36]; moreover, the previous studies on Al–Ni system indicated that the spin-polarized and nonpolarized PDs (phase diagrams) were almost identical [37]. Based on these studies, in the present work, all calculations are assumed no spin polarizations and performed using the ‘‘accurate’’ setting within VASP to avoid wrap-around errors. AlCu3 and AlNi3 belong to the cubic structure with the space group Pm-3m [38,39] (space number 221) and unit cell with four atoms. Calculations for imperfect crystal were performed using the supercell with 32 atoms, which was eight times greater than the number of atoms in the primitive cell with four atoms. This perfect supercell has 32 atoms, for Al8Cu24, eight of Al atoms and 24 of Cu atoms and for Al8Ni24, eight of Al atoms and 24 of Ni atoms as show in Fig. 1. Construction models with single-atom vacancy point defect namely imperfect crystal models has 31atoms, that is to say, remove an Al atom out of supercell with 32 atoms. This is reasonable to construct the model to calculate imperfect crystal.

˚ respectively. These values were 7.214, 7.354, 7.351 and 7.094 A, ˚ agree well with their experimental values of 7.214 and 7.143 A. 3.2. Formation energy of point defects For Al-based alloys, the total energies of perfect crystal (Al8Cu24) and imperfect crystal (Al7Cu24) were calculated respectively, and were  124.9 eV and 119.4 eV. The energy of an Al atom was 3.7 eV. The formation energies of Al vacancies in Al8Cu24 and Al8Ni24 according to the formula (1) are 2.1025 eV and 3.6025 eV, respectively, as shown in Table 3. It is found that Al7Ni24 is easy to form single-atom Al vacancy point defect than Al8Cu24. Therefore, Al-based cubic structure alloys with singleatom vacancy point defect can increase the crystal’s volume and e, the total energy. 3.3. Mechanical properties We also think cubic structure with single-atom vacancy point defect as cubic structure, on account of crystal structure cannot be changed by single-atom vacancy point defect, so we use Al7Cu24 and Al7Ni24 to calculate the elastic constants and elastic modulus. According to the introduction of elastic constants above, the strain d is varied in steps of 0.01 from d ¼  0.02 to 0.02. In order to analyze the difference of mechanical properties between perfect crystal and imperfect crystal with deformations of the lattice, we calculate the elastic constants and elastic modulus of AlCu3 and AlNi3. The strain d is from d ¼  0.03 to 0.03 in steps of 0.01. The corresponding E  d curve was obtained, as shown in Fig. 2 and the following three relations for the deformation energies are obtained:

DE

3. Results and discussion

V0

3.1. Crystal structure

DE V0

The crystal structures for the Al-based alloys (AlCu3 and AlNi3) are based on the available experimental and crystallographic data as shown in Fig. 1 and Table 2. The perfect and imperfect crystal structures were first optimized to determine the accurate internal positions of the atoms. There were three parameters in the cubic crystal structure, i.e., total energy, lattice parameters, equilibrium volume. The total energy as a function of the lattice constant and primitive cell volume was calculated by the Murnaghan equation of state (EOS). From that, the equilibrium volume corresponding to minimum total energy and the corresponding lattice constants were obtained as shown in Table 2. The theoretical values of the lattice parameters, a, for Al8Cu24, Al7Cu24, Al8Ni24 and Al7Ni24

1265

¼ 3ðC 11 C 12 Þg2

¼

ð12Þ

3 ðC 11 þ 2C 12 Þg2 2

ð13Þ

Table 2 ˚ of Crystallographic data and calculated and experimental lattice constants a (A) Al-based alloys. Compound

Lattice constants

Space group

AlCu3

7.214 7.214 [29] 7.354 7.351 7.143 [30] 7.094

Pm-3m

Al7Cu24 AlNi3 Al7Ni24

Fig. 1. (Color online) Model of crystal structure for AlCu3 (a) and AlNi3 (b).

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DE V0

¼

L.M. Dong et al. / Solid State Communications 152 (2012) 1263–1269

1 C 44 g2 2

ð14Þ

By means of polynomial fits, we extract three second-order coefficients, which is corresponding to 3(C11 C12), 32ðC 11 þ2C 12 Þ and 12C 44 , respectively. By resolving the three equations according to the relationship above, the obtained elastic constants C11, C12 and C44 are tabulated in Table 4. In the present study, we also calculate the elastic modulus, i.e., bulk modulus B, shear modulus G, Young’s modulus E, Passion’s ratio g and anisotropy value A, which can be determined by combining values of the singlecrystal Cij according to the proper symmetry relations. The elastic constants and elastic modulus of Al7Cu24 and Al7Ni24 were got by the same way and also listed in Table 4. The mechanical stability criterion for cubic crystals leads to the following restrictions on the elastic constants [40]: ðC 11 C 12 Þ 4 0,

C 11 4 0,

C 44 4 0,

ðC 11 þ2C 12 Þ 4 0:

ð15Þ

Table 3 Calculated and experimental volumes V (A˚ 3), bulk modulus B0 (GPa), total energy E (eV) of point defects of AlCu3 and AlNi3. (eV) and formation energy Evac f E

Evac f

Compound

V

B0

B00

AlCu3

397.2 375.4 [29] 392.5 361.3 363.2 [30] 357.8

125.0

6.9

 124.9

–

130.5 177.1

5.5 5.9

 119.4  174.9

2.1 –

169.0

4.3

 167.6

3.6

Al7Cu24 AlNi3 Al7Ni24

from calculation, we found that the whole set of Cij of the perfect crystal and imperfect crystal obey well all the above conditions, which is an important requisite for (meta)stable materials eligible for the requirement of mechanical stability for cubic crystals [41–43]. From the calculated values we find that Young’s modulus of AlNi3 is 73.8 GPa larger than AlCu3. This result may indicate that AlNi3 is better stiffer than AlCu3. All the calculated Poisson’s ratios are very close to 0.25, which means that they are all materials with predominantly central interatomic forces. On the solid basis of facts fracture criterion: the high Poisson’s ratio, the better ductility. Poisson’s ratios of cubic structure with single-atom vacancy point defect increase, this indicates they will show good ductility. The ratio of the shear modulus to bulk modulus of polycrystalline phases introduced by Mattesini et al. [42] can predict the brittle and ductile behavior of materials. This theory considers K as the resistance to fracture and G as the resistance to plastic deformation. A high (low) G/K value was associated to brittleness (ductility); the critical value which separates ductility and brittleness is about 0.57. The values of K/G for AlCu3 and AlNi3, Al7Cu24 and Al7Ni24 are 0.15 and 0.22, 0.11 Table 4 The elastic constants and elastic modulus (GPa) of Al-based alloys. Compound

C11

C12

C44

K

G

G/K

E

n

A

Al7Cu24 AlCu3 Cal. [33] Al7Ni24 AlNi3

142.9 159.1 176.0 219.5 232.3

115.0 119.8 117.4 155.1 153.8

75.7 83.3 92.4 114.2 118.5

124.3 132.9 136.9 176.6 180.0

14.0 19.7 29.3 32.2 39.3

0.11 0.15 0.24 0.18 0.22

141.8 158.3 173.0 218.8 232.1

0.8 0.2 0.3 0.5 0.3

5.4 4.2 3.1 3.6 3.0

Fig. 2. Crystal energy E as function of the lattice distortion parameters d for the three different strains defined in the text. AlCu3 (a) and AlNi3 (b), Al7Cu24 (c) and Al7Ni24 (d), e (1), e (2) and e (3) correspond to the distortions given in Eqs. (1)–(3), respectively.

L.M. Dong et al. / Solid State Communications 152 (2012) 1263–1269

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Fig. 3. (Color online) The total and partial density of states (DOS) of Al8Cu24 (a), Al7Cu24 (b) and Al8Ni24(c), Al7Ni24 (d), and the Fermi level is set at zero energy and marked by the vertical lines.

and 0.18, respectively, and implying that they are slightly ductile. The values of K/G with single-atom vacancy point defect are smaller than before, which imply that the imperfect materials have better ductile. In order to predict the elastic anisotropy of these Al-based alloys, the anisotropy constant was calculated. For a completely isotropic material, the value of A will be 1 [19], while values smaller or bigger than 1 measuring the degree of elastic anisotropy. It is interesting to note that the values of A we measured (Table 4) were bigger than before, which may suggest that the imperfect Al-based fcc alloys were slightly anisotropy. As discussed above, the calculated values of Poisson’s ratio are not very accurate and out-of-range, but the calculated values of elastic constants and elastic modules agree well with previous experimental and theoretical values. In addition, they meet the requirement of mechanical stability for cubic crystals. We can draw a conclusion that cubic structures with single-atom point defect get better stability and scalability. 3.4. Density of states (DOS) In order to further analysis on electronic structure and elastic properties of Al-based alloys with single-atom point defect, the total and partial DOS of Al7Cu24 and Al7Ni24 are performed and the results are shown in Fig. 4(b) and (d). For understanding the change of cubic structure with single-atom point defect, the total and partial density of States of Al8Cu24 and Al8Ni24 are also

calculated, as shown in Fig. 4(a) and (c). For Al8Cu24 and Al7Cu24 from Fig. 3(a) and (b), it is found that the total between  10.3 and 6.6 eV are predominantly derived from the Cu 3d orbits, while below the Fermi level originate mainly from the Cu 3d orbits and Al 3p orbits. Using the same method, the total and partial DOS of Al8Ni24 and Al7Ni24 have also been studied as shown in Fig. 3(c) and (d). 4s and 3d states of Ni, 3s and 3p states of Al primarily contribute to the total DOS above the Fermi level, while the 3d state of Ni and 3p state of Al contributes to the total DOS below the Fermi level. The total DOS of cubic structure with single-atom vacancy point defect below the Fermi level is predominantly derived from Cu 3d orbits and Ni 3d orbits and the bonding electron numbers below the Fermi level of Al8Cu24, Al7Cu24 and Al8Ni24, Al7Ni24 per atom are 3.44, 3.46 and 3.14, 3.21, respectively. It has a better structural stability when charge interactions become stronger with the number of bonding electrons increasing [44–46]. Hence, Al7Cu24 and Al7Ni24 have the stronger structural stability. Consequently, there is the better structural stability in cubic structure with single-atom vacancy point defect of Al-based alloys. 3.5. Charge density Charge density distribution maps on the (001) plan for Al8Cu24, Al7Cu24, Al8Ni24 and Al7Ni24 are shown in Fig. 4 the contour lines

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L.M. Dong et al. / Solid State Communications 152 (2012) 1263–1269

Fig. 4. (Color online) The contour plots of charge density on the (0 0 1) plane for Al-based alloys of perfect crystal (Al8Cu24, Al8Ni24) and imperfect crystal (Al7Cu24, Al7Ni24).

are plotted from 0.0 to 0.06 e/A˚ 3 with 0.0012 e/A˚ 3 intervals. Higher density region corresponds to the core electron distribution of Al, Ni and Cu atoms that contributes relatively little to the bonding. In Fig. 4(a) the obvious overlap of electron densities between Al and Al indicates a covalent bonding between them. The electron distribution around Cu atoms exhibits a metallic bonding between Cu and Cu and can be well described by the nearly free electron model. The charge density distribution maps exhibit mainly ionic bonding between Cu and Al atoms due to gain and loss of free electron. While in Fig. 4(b), the change of charge density distribution maps of Cu atoms attributes to single Al atom vacancy point defect. It makes a covalent bonding between Cu and Cu atoms around single Al vacancy point defect. In the same way, the charge density distribution maps exhibit a covalent bonding between Al and Al atoms, a metallic bonding between Ni and Ni atoms and an ionic bonding between Ni and Al atoms in Fig. 4(c). In Fig. 4(d), the change of charge density distribution maps of Ni atoms attributes to single Al atom vacancy point defect. It also forms the covalent bonding between Ni and Ni. In a word, a chemical bond of atoms has changed from the ionic bond to the covalent bond attribute to single-atom vacancy point defect. This results in a good alloying ability and structural stability of Al-based alloys. These results are consistent with the DOS analysis above.

4. Conclusions An analysis of the structure, electronic and mechanical properties of AlCu3 and AlNi3 with cubic structure and single-atom vacancy point defect have been studied using the density functional theory within a GGA approximation. The effects of the alloying ability, structural stability and mechanical properties with single-atom vacancy point defect could be reasonably described and explained in terms of the electronic structure.

The conclusions from the theory analysis and discussion can be drawn as follows: (1) AlNi3 is easier to form single-atom vacancy point defect than AlCu3. (2) Investigation of the mechanical properties shows that cubic structures with single-atom vacancy point defect obey well the requirement of mechanical stability for cubic crystals. (3) Cubic structure with single-atom vacancy point defect can make the crystal size smaller and especially, has a good alloying ability and structural stability due to an increase in the bonding electron numbers below the Fermi level.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]

W. Bouaeshi, D. Li, Tribol. Int. 40 (2007) 188. J. Nie, B. Muddle, Acta. Mater. 56 (2008) 3490. R. Uyyuru, M. Surappa, S. Brusethaug, Wear 260 (2006) 1248. L. Yanbin, L. Zhiyi, L. Yuntao, X. Qinkun, Z. Jie, Mater. Sci. Eng.: A 492 (2008) 333. S. Zhang, F. Wang, J. Mater. Process. Technol. 182 (2007) 122. J. Guo, K. Ohtera, Acta. Mater. 46 (1998) 3829. K. Rzyman, Z. Moser, Prog. Mater. Sci. 49 (2004) 581. R.X. Hu, P. Nash, J. Mater. Sci. 40 (2005) 1067. R. Arroyave, D. Shin, Z.K. Liu, Acta Mater. 53 (2005) 1809. S. Yu, C.Y. Wang, Physica B 396 (2007) 138. I. Ansara, B. Sundman, P. Willemin, Acta. Metall. 36 (1988) 977. A. Pasturel, C. Colinet, A.T. Paxton, J. Phy.: Condens. Matter 4 (1992) 945. R.E. Watson, M. Weinert, Phys. Rev. B 58 (1998) 5981. D. Shi, B. Wen, J Solid State Chem. 182 (2009) 2664. J. Jordan, S. Deevi, Intermetallics 11 (2003) 507. H. Baltache, R. Khenata, M. Sahnoun, M. Driz, B. Abbar, B. Bouhafs, Phys. B: Phys. Condens. Matter 344 (2004) 334. T. Maeda, T. Wada, J. Phys. Chem. Solids 66 (2005) 1924. P. Ravindran, L. Fast, P. Korzhavyi, B. Johansson, J. Wills, O. Eriksson, J. Appl. Phys. 84 (1998) 4891. M. Mattesini, R. Ahuja, B. Johansson, Phys. Rev. B 68 (2003) 184108. B. Karki, L. Stixrude, S. Clark, M. Warren, G. Ackland, J. Crain, Amer. Mineral. 82 (1997) 51. V. Tvergaard, J. Hutchinson, J. Am. Ceram. Soc. 71 (1988) 157.

L.M. Dong et al. / Solid State Communications 152 (2012) 1263–1269

[22] J. Nye, Physical Properties of Crystals: Their Representation by Tensors and Matrices, Oxford University Press, USA, 1985. [23] J.H. Westbrook, R.L. Fleischer (Eds.), Intermetallic Compounds: Principles and Practice, Vol I: Principles, Wiley, London, 1995. (Chapter 9). [24] M. Mehl, J. Osburn, D. Papaconstantopoulos, B. Klein, Phys. Rev. B 42 (1990) 5362. [25] O. Beckstein, J.E. Klepeis, G.L.W. Hart, O. Pankratov, Phys. Rev. B 63 (2001) 134112. [26] G. Kresse, J. Hafner, Phys. Rev. B 49 (1994) 14251. [27] G. Kresse, J. Furthmuller, Comput. Mater. Sci. 6 (1996) 15. ¨ [28] G. Kresse, J. Furthmuller, Phys. Rev. B 54 (1996) 11169. [29] W. Kohn, L. Sham, Phys. Rev. 140 (1965) A1133. [30] P. Hohenberg, W. Kohn, Phys. Rev. 136 (1964) B864. ¨ [31] P.E. Blochl, Phys. Rev. B 50 (1994) 17953. [32] J. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865. [33] J. Perdew, Y. Wang, Phys. Rev. B 45 (1992) 13244. [34] J.P. Perdew, A. Zunger, Phys. Rev. B 23 (1981) 5048.

1269

[35] H.J. Monkhorst, J.D. Pack, Phys. Rev. B 13 (1976) 5188. [36] W. Press, S. Teukolsky, W. Vetterling, B. Flannery, Numerical Recipes in FORTRAN 90: The Art of Scientific Computing, Cambridge University Press, New York, 1992. [37] W.J. Golumbfskie, R. Arroyave, D. Shin, Acta. Mater. 54 (2006) 2291. [38] H.Y. Geng, N.X. Chen, Phys. Rev. B 70 (2004) 094203. [39] M. Debili, N. Boukhris, M. Lallouche, Vch Verlagsgesellschaft Mbh: 2006, p. 65. [40] P. Rao, S. Suryanarayana, K. Murthy, S. Naidu, J. Phys.: Condens. Matter 1 (1989) 5357. [41] D. Wallace, Thermodynamics of Crystals, Dover Pubns, 1998. [42] M. Mattesini, R. Ahuja, B. Johansson, Phys. Rev. B 68 (2003) 184108. [43] W. Zhou, L. Liu, B. Li, Q. Song, P. Wu, J. Electron. Mater. 38 (2009) 356. [44] S. Pugh, Philosophical Mag. Ser. 45 (7) (1954) 823. [45] C. Fu, X. Wang, Y. Ye, K. Ho, Intermetallics 7 (1999) 179. ¨ ¨ [46] J. Nyle´n, F.J. Garcıa Garcıa, B.D. Mosel, R. Pottgen, U. Haussermann, Solid State Sci. 6 (2004) 147.