Structural, elastic and electronic properties of intermetallics in the Pt–Sn system: A density functional investigation

Computational Materials Science 46 (2009) 921–931

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Structural, elastic and electronic properties of intermetallics in the Pt–Sn system: A density functional investigation Wei Zhou a,b,c, Lijuan Liu a,b,c, Baoling Li a,b,c, Ping Wu a,b,c,*, Qinggong Song d a

Department of Applied Physics, Faculty of Science, Tianjin University, Tianjin 300072, China Tianjin Key Laboratory of Low Dimensional Materials Physics and Preparing Technology, Tianjin 300072, China c Institute of Advanced Materials Physics, Faculty of Science, Tianjin University, Tianjin 300072, China d Department of Materials Physics, College of Science, Civil Aviation University of China, Tianjin 300300, China b

a r t i c l e

i n f o

Article history: Received 12 January 2009 Received in revised form 27 April 2009 Accepted 29 April 2009 Available online 29 May 2009 PACS: 71.15

a b s t r a c t The structural, elastic and electronic properties of intermetallics in the Pt–Sn binary system are investigated using first-principles calculations based on density functional theory (DFT). The polycrystalline elastic properties are deduced from the calculated single-crystal elastic constants. The elastic anisotropy of these intermetallics is analyzed based on the directional dependence of the Young’s modulus and its origin explained based on the electronic nature of the crystals. All the Pt–Sn intermetallics investigated are found to be mechanically stable, ductile and metallic, and some of them show high elastic anisotropy. Ó 2009 Elsevier B.V. All rights reserved.

Keywords: First principles calculations Intermetallics Elasticity Electronic structure

1. Introduction In microelectronic, optoelectronic and microelectromechanical systems (MEMS) packaging, a multilayer structure, including typically a wetting layer (e.g. Cu), a diffusion barrier layer (e.g. Ni) and an oxidation protection layer (e.g. Au), is wildly used as the contact pads for solder bonding. The necessity to obtain multiple metal layers as contact pads adds not only production cost but also design complexity, which always induces higher reliability concerns. Accordingly, to simplify the multilayer structure is critically needed in the electronic industry. In recent years, a single platinum (Pt) layer has been considered to be a good candidate to replace the traditional multilayer structure for use as contact pads in solder joints, because it has a relatively low dissolution rate, good resistance to oxidation and good wettability in relation to other Sn-based solders [1–3]. As we know, the formation of intermetallic compounds (IMCs) at the interface between the solder and the metal pad finish can affect the integrity of solder joints which is crucial in the performance reliability, especially with the size shrinking of solder joints in microelectronic devices [4,5]. Five intermetallic compounds, PtSn, PtSn4, PtSn2, Pt2Sn3 * Corresponding author. Address: Department of Applied Physics, Faculty of Science, Tianjin University, Tianjin 300072, China. Tel.: +86 022 27408599. E-mail address: [email protected] (P. Wu). 0927-0256/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2009.04.044

and Pt3Sn, have been found to form at the interface between Snbased solder and Pt pad in solder joints during reflow soldering or after thermal aging [6–8]. Although the vast majority of research has focused on the formation and evolution of Pt–Sn intermetallic compounds during interfacial reaction between the solder and the Pt pad to bridge the gap between the microstructure and mechanical property of the solder joints, surprisingly few studies about the mechanical properties of these compounds themselves exist in the literature [1–3,6–9]. Therefore, there is a strong driving force to investigate the physical properties, especially mechanical properties of intermetallic compounds in the Pt–Sn system. In the past decade, investigations on the Pt–Sn intermetallics mainly focused on their thermodynamic and electrochemical properties due to their wide applications in soldering and heterogeneous catalysis. The enthalpies of formation of different Pt–Sn intermediate phases have been investigated by Ferro et al. [10], and Schaller [11] with experimental method. And theoretical calculations were done by de Boer et al. using the model of Miedema [12]. A number of studies have been devoted to PtSn [13] and Pt3Sn [14–16] intermetallics, which indicate their good catalytic properties. And Skriver [17] has investigated the hybridization characters in Pt3Sn crystal. Recently, electronic and optical properties of PtSn2 have been reported by A. Gupta et al. [18] using the augmentedplane-wave (APW) method, and it also provided interesting information about d–d interaction as a function of primary interatomic

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distance between Pt atoms. For PtSn4, Künnen et al. [19] determined its space group to be centrosymmetric Ccca which is in contrast to the earlier report in noncentrosymmetric Aba2 [20]. Additionally, they found that PtSn4 was diamagnetic at room temperature and shown metallic conductivity. To the best of our knowledge, however, research on the elastic properties of intermetallic compounds in the Pt–Sn system in the literature is lacking until now, and the electronic nature of mechanical properties for these compounds is still not clear. The development of first-principles calculations based on density functional theory [21,22] and computing technology has made it possible to calculate fundamental physical properties of materials with minimal experimental input and high reliability. Mehl et al. performed the first DFT calculations of the elastic constants of various intermetallic compounds in a series of studies [23–25]. Recently, first-principles calculations have been employed to investigate the elastic and electronic properties of many intermetallic compounds in the interconnect structures of electronic devices, such as Cu6Sn5, Ni–Sn, Au–Sn and Al–Cu compounds, and the calculation results are in good agreement with experimental data, implying the high accuracy prediction of this calculation method [26–29]. In this work, first-principles calculations based on DFT were used to investigate the physical properties regarding the electronic nature of the intermetallic compounds in the Pt–Sn binary system. And the paper is structured as follows: after a brief summary of the methodical details, the structural, thermodynamic and elastic properties of Pt–Sn intermetallics are discussed, respectively. To further understand the electronic nature of the mechanical properties in these intermetallic compounds, the electronic structure and the bonding nature of PtSn4, as a prototypical example in the Pt–Sn system of present interest, are analyzed. Finally, a short summary and discussion of the results is present in the last section. 2. Calculation details Our calculations were performed using the Cambridge Serial Total Energy Package code (CASTEP) [30] based on DFT. Ultrasoft pseudopotentials (USPPs) [31] were used for electron–ion interactions, and the electron wave function was expanded using plane waves. Both Perdew–Zunger parametrization of the local density approximation (LDA) exchange–correlation functional [32] and the generalized gradient approximation (GGA) with the functional proposed by Perdew et al. [33] were employed to evaluate exchange–correlation energy. 5d96s1 of Pt and 5s25p2 of Sn are treated as valance electron configurations. The Broyden–Fletcher–Goldfarb–Shanno (BFGS) optimization method [34] was used to search for the ground states of the crystals, and the convergence tolerance was set to (1) energy change below 5  107 eV/atom, (2) force less than 0.01 eV/Å, (3) stress less than 0.02 GPa, and (4) change in displacement less than 5  104 Å. The Brillouin zone was sampled with the Monkhorst– Pack scheme [35]. The number of k points and the cut-off energy were increased until the calculated total energy converges within the required tolerance (an error of the total energy of at most 0.2 meV/atom and, for elastic calculations, difference of calculated bulk modulus less than 1%). And the details of the cut-off energy and the k points used in this work are shown in Table 1. 3. Results and discussion 3.1. Structural properties

Table 1 The details of the cut-off energy and the k-mesh of calculations in this work. Phase

Cut-off energy (eV)

k-Mesh

Pt Pt3Sn PtSn Pt2Sn3 PtSn2 PtSn4 b-Sn

420 420 420 420 420 420 420

16  16  16 14  14  14 14  14  9 14  14  4 10  10  10 959 9  9  16

Fig. 1. The Pt–Sn equilibrium phase diagram.

nal atoms of these compounds were fully optimized with the initial crystal structures taken from the experiments [19,37–41] (Table 2 and Fig. 2). And both LDA and GGA were used in these calculations. The calculated structure parameters of stable ground states are summarized in Table 2, which are in good agreement with the experimental results. Such agreement demonstrates that the computational methodology employed here is suitable and our calculation results are reliable. Table 2 also shows that the lattice constants calculated with the GGA functional have a smaller deviation (less than 2%) from the experimental results than the values obtained from LDA (at most 3%). Therefore, the GGA is expected to be more suitable to describe the ground states of the Pt–Sn intermetallics. And these ground state structures from GGA were used to investigate other properties. With the optimized crystal structures from GGA, the calculated mass densities of the Pt–Sn intermetallics are illustrated in Table 3 and Fig. 3. It can be seen that the mass density of Pt–Sn phases decreases linearly with Sn concentration. With increasing Sn concentration f (in mole fraction), the mass density q (in 106 g/m3) of Pt–Sn phases decreases approximately by a linear relationship q = 20.05–0.13f. 3.2. Formation energy The formation energies of Pt–Sn intermetallics at zero temperature can be evaluated relative to the composition-averaged energies of the pure elements in their equilibrium crystal structures: m Snn DEPt ¼ ½EPtm Snn  ðmEPt þ nESn Þ=ðm þ nÞ f m Snn DEPt f

The Pt–Sn equilibrium phase diagram (Fig. 1) exhibits five intermediate stoichiometric phases, Pt3Sn, PtSn, Pt2Sn3, PtSn2 and PtSn4 [36]. The geometry lattice constants and the positions of the inter-

ð1Þ

where is the atomic formation energy of a PtmSnn compound, EPtm Snn is the total energy of intermetallic PtmSnn, EPt is the total energy for an Pt atom in the pure face-centered-cubic (fcc) Pt metal with equilibrium lattice parameters and ESn is the total

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W. Zhou et al. / Computational Materials Science 46 (2009) 921–931 Table 2 Crystallographic data of the Pt–Sn intermetallics obtained from (1) experiment, (2) LDA and (3) GGA. Structure

Pearson symbol

Space group

Lattice parameters (Å)

Pt Pt3Sn PtSn

cF4 cP4 hP4

Fm-3 m (225) Pm3-m (221) P63/mmc (194)

Pt2Sn3

hP10

P63/mmc (194)

PtSn2 PtSn4

cF12 oS20

Fm-3 m (225) Ccca (68)

b-Sn

tI4

I41/amd (141)

(1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1)

a b c d e f

Ref. Ref. Ref. Ref. Ref. Ref.

a = 3.925a, (2) a = 3.921, (3) a = 3.968 a = 4.000b, (2) a = 3.967, (3) a = 4.076 a = 4.101c, (2) a = 4.031, (3) a = 4.181 c = 5.441, (2) c = 5.406, (3) c = 5.536 a = 4.325d, (2) a = 4.262, (3) a = 4.416 c = 12.934, (2) c = 12.789, (3) c = 13.221 a = 6.433c, (2) a = 6.311, (3) a = 6.554 a = 6.418e, (2) a = 6.269, (3) a = 6.542 b = 11.366, (2) b = 11.137, (3) b = 11.538 c = 6.384, (2) c = 6.250,(3) c = 6.539 a = 5.830f, (2) a = 5.700, (3) a = 5.931 c = 3.184, (2) c = 3.069, (3)c = 3.194

[37]. [38]. [39]. [40]. [19]. [41].

Fig. 2. Crystal structures of (a) Pt3Sn, (b) PtSn, (c) Pt2Sn3, (d) PtSn2 and (e) PtSn4. The big spheres in blue are Pt atoms and the small spheres in grey are Sn atoms.

energy for an Sn atom in the b-Sn metal with equilibrium lattice parameters. With Eq. (1), the formation energies of all Pt–Sn intermetallic phases were obtained from the DFT calculations. The present results along with previous experimental and theoretical ones [10–12] are shown in Table 3 and Fig. 4. In this research, the formation energies are calculated at T = 0 K. And the entropy effects are not taken into account, which is in contrast to the experimental data obtained above room temperature. Due to different measure-

ment methods used in the previous investigations, the experimental data of formation energy of Pt3Sn may exhibit large deviation (more than 20%). In the whole composition range, the formation energies calculated from GGA agree well with the experimental and theoretical values as shown in Fig. 4. To further verify the precision and reliability of the computational methodology, the calculated results from LDA are also presented here, and they exhibit larger deviation from the experimental values than the ones of GGA, indicating the common over-binding character for the LDA

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Table 3 The mass density and formation energy of the Pt–Sn intermetallics from GGA. Phase

Mass density (106 g/m3)

Reference

Formation energy (KJ/mol atoms)

References

Pt3Sn

17.26 18.02

This work [38]

PtSn

12.38 13.15

This work [39]

Pt2Sn3

11.03 11.75

This work [40]

PtSn2

10.21 10.79

This work [39]

PtSn4

9.02 9.20

This work [19]

43.0 41.4 50.2 42.0 61.8 58.6 57.0 54.5 54.4 50.0 50.0 52.3 43 32.86 27.2

This work [11] Experiment [10] Experiment [12] Calculation This work [10] Experiment [12] Calculation This work [10] Experiment [12] Calculation This work [10] Experiment [12] Calculation This work [10] Experiment

ture [8], and metastable phase PtSn4 mainly forms under non-equilibrium condition due to low dissolution rate of Pt in Sn according to the Pt–Sn phase diagram. In addition, all the calculated formation energies of Pt–Sn compounds are larger than 30 kJ/mol atoms, implying a relatively strong chemical interaction between Pt and Sn atoms. 3.3. Elastic properties

Fig. 3. Calculated mass density for the Pt–Sn intermetallics (references are shown in Table 3).

The elastic stiffness determines the response of a crystal to an imposed strain and provides information about bonding characteristics near the equilibrium state. Therefore, it is essential to investigate the elastic stiffness to understand the mechanical properties of Pt–Sn intermetallics. A stress–strain approach was employed to calculate elastic properties in this study. Based on the generalized Hooke’s law, a linear relationship exists between two tensors, i.e. stress (r) and strain (e). Thus, a proportional elastic stiffness Cij could be simply written as:

ri ¼

6 X

C ij ej

ð2Þ

j¼1

Fig. 4. Calculated formation energy compared to experimental and other theoretical values for the Pt–Sn intermetallics.

functional. Based on these discussions, the GGA functional is further considered to be more suitable to investigate the properties of Pt–Sn binary system. Both the calculated and the experimental data of formation energies reveal the tendency of Pt–Sn mixtures to form stable intermetallic compounds. In the pure binary system, PtSn phase is more stable than any other stoichiometry. However, the Pt/Sn interface does not exhibit the full phase sequence at low tempera-

and the elastic compliance matrices, {Sij}, equal to {Cij}1. In order to obtain each independent elastic constant, an appropriate number of strain patterns were imposed on crystal cell with a maximum strain value of 0.006 in the current calculations. Before the calculations for the Pt–Sn intermetallics, the elastic constants of pure fcc-Pt and b-Sn were calculated using this method to validate the computational procedure. And the obtained results (Table 4) agree with the previous theoretical and experimental data [42,43], implying that our method for calculating the elastic properties is reasonable and reliable. The calculated independent Cij values of single-crystal for the Pt–Sn intermetallics are summarized in Table 5. As for the mechanical stability of a structure, one condition is that its strain energy must be positive against any homogeneous elastic deformation. For different crystal structures, the mechanical stability criterion can be expressed as [44]: Cubic system:

C 11 þ 2C 12 > 0; C 44 > 0; C 11  C 22 > 0

ð3Þ

Hexagonal system:

C 11 > 0; C 44 > 0; C 11  C 22 > 0; ðC 11 þ C 12 ÞC 33 > 2C 213

ð4Þ

Orthorhombic system:

C 11 þ C 12 þ C 33 þ 2C 12 þ 2C 13 þ 2C 23 > 0 C 11 þ C 22 > 2C 12 ; C 22 þ C 33 > 2C 23 ; C 11 þ C 33 > 2C 13 C ii > 0 ði ¼ 1—6Þ

ð5Þ

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W. Zhou et al. / Computational Materials Science 46 (2009) 921–931 Table 4 A comparison between calculated and experimental single-crystal elastic constants (Cij) of fcc-Pt and b-Sn (units in GPa).

fcc-Pt

b-Sn

a b c

Cij

This work (GGA)

Ab initio (previous) (USPP-GGA)

Experiment

C11 C12 C44 C11 C33 C44 C66 C12 C13

320.4 225.1 63.3 73.2 90.6 21.9 23.8 59.8 39.0

– – – 63.0a 79.9 20.3 23.3 51.2 25.7

346.7b 250.7 76.5 82.7c 103.1 27.0 28.2 57.9 34.2

Ref. [27]. Ref. [42]. Ref. [43].

Table 5 Calculated Elastic Stiffness (Cij) of the Pt–Sn intermetallics from GGA. Cij(GPa)

Pt3Sn

PtSn

Pt2Sn3

PtSn2

PtSn4

C11 C22 C33 C44 C55 C66 C12 C13 C23

268.7 – – 76.8 – – 175.4 – –

175.5 – 290.3 61.4 – – 115.6 81.3 –

142.4 – 179.2 25.7 – – 92.9 68.9 –

133.7 – – 24.4 – – 71.7 – –

105.9 111.4 100.0 22.2 19.1 47.3 50.6 41.9 42.0

As shown in Table 5, the values of calculated elastic constants satisfy the above corresponding criteria, indicating that all of Pt– Sn intermetallics have elastically stable structures. The bulk modulus (K), shear modulus (G), Young’s modulus (E) and Poisson’s ratio (v) for polycrystalline crystal can be obtained from independent single-crystal elastic constants. For all crystal structures, the polycrystalline modulus is estimated with two approximation methods, namely, the Voigt method and the Reuss method [45], and they are given by:

KV ¼

1 2 ðc11 þ c22 þ c33 Þ þ ðc12 þ c13 þ c23 Þ 9 9

1 ¼ ðs11 þ s22 þ s33 Þ þ 2ðs12 þ s13 þ s23 Þ KR GV ¼

1 1 ðc11 þ c22 þ c33 Þ  ðc12 þ c13 þ c23 Þ 15 15 1 þ ðc44 þ c55 þ c66 Þ 15

1 4 4 ¼ ðs11 þ s22 þ s33 Þ  ðs12 þ s13 þ s23 Þ GR 15 15 3 ðs44 þ s55 þ s66 Þ þ 15

1 1 ðK V þ K R Þ G ¼ ðGV þ GR Þ 2 2

KV KR K GV GR G EV ER E v vm

HD

Pt3Sn

PtSn

Pt2Sn3

PtSn2

PtSn4

206.5 206.5 206.5 64.7 61.0 62.8 262.3 249.6 256.1 0.29 2148.1 249.0

133.1 130.4 131.7 54.8 45.2 50.0 144.6 121.6 133.1 0.33 2253.9 243.0

102.8 102.7 102.8 30.8 28.2 29.5 84.0 77.5 80.8 0.37 1843.5 194.4

92.4 92.4 92.4 27.0 26.7 26.8 73.8 73.1 73.3 0.37 1826.0 189.9

65.2 64.8 65.0 29.9 27.1 28.5 77.8 71.4 74.6 0.31 1987.7 203.2

ð6Þ

ð7Þ

ð8Þ

Fig. 5. Calculated bulk modulus for the Pt–Sn intermetallics.

ð9Þ

where the subscripts V and R indicate the Voigt and Reuss averages. They provide the upper (Voigt) and lower (Reuss) bounds to the polycrystalline elastic modulus. The arithmetic average of the Voigt and the Reuss bounds are called the Hill approximation or Voigt–Reuss–Hill (VRH) average [45], and it is considered as the best estimate of the theoretical polycrystalline elastic modulus. It can be expressed as:

K¼

Table 6 Voigt (index V), Reuss (index R), and averaged macroscopic VRH modulus including the Debye temperature for the polycrystalline Pt–Sn intermetallics; all in GPa except for v (dimensionless), vm (m/s), and HD (K).

ð10Þ

Additionally, the polycrystalline elastic modulus and the Poisson’s ratio can be calculated from the values of elastic modulus with the relationships:

E¼

9KG 3K þ G

v¼

3K  2G 2ð3K þ GÞ

ð11Þ

And the results of our calculations are summarized in Table 6 and Fig. 5. The calculated bulk modulus of fcc-Pt and b-Sn are close to the experimental values, and the remaining differences from the experimental data may be associated with the thermal expansion. The DFT calculations refer to T = 0, whereas the experimental data are obtained at finite temperature. Due to the large thermal expansion coefficients of both fcc-Pt and b-Sn, a certain deviation can be expected between the calculation result and the experimental value, which is similar to the case in V–Si system reported by Thieme et al. [46]. On the other hand, the influence of elastic anisotropy on the experimental measurement may also induce the deviation.

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W. Zhou et al. / Computational Materials Science 46 (2009) 921–931

Table 7 The anisotropy factors of the Pt–Sn intermetallics.

AG AE

Pt3Sn

PtSn

Pt2Sn3

PtSn2

PtSn4

2.9% 2.5%

9.6% 8.6%

4.4% 4.0%

0.56% 0.48%

4.9% 4.3%

According to our calculations, the bulk modulus of Pt–Sn intermetallics decreases with Sn concentration, as shown in Fig. 5. And this trend of negative deviation of bulk modulus at the Sn-rich side has also been reported by Ghosh in Au–Sn and Ni–Sn systems [27,28]. It implies that the decrease of the bulk modulus with Sn concentration should be an intrinsic property of X–Sn (X = Au, Ni, Pt) phases. The bulk modulus of PtSn4 (65 GPa) is much smaller than that of Pt3Sn (206.5 GPa) which has he largest value in the five intermetallics, and it implies that PtSn4 has high compressibility. The Pt3Sn phase also has the largest Young’s modulus (256.1 GPa) and shear modulus (62.8 GPa) in the Pt–Sn system. The quotient of shear modulus to bulk modulus, G/K, can be considered as an indication of the extent of fracture range in metals [47]. A low value and a high value of G/K are associated with ductility and brittleness, respectively. From the calculations, the G/K values are 0.287, 0.290, 0.304, 0.380 and 0.439 for Pt2Sn3, PtSn2, Pt3Sn, PtSn and PtSn4,

respectively. Although the above evaluation method is highly simplistic, it nevertheless indicates the tendency of ductility for the five Pt–Sn intermetallics. As we know, a material is regarded as brittle if the value of G/K is above 0.57. This suggests that all Pt–Sn intermetallic compounds have ductile character. It is known that the elastic anisotropy can be measured using two dimensionless quantities AG and AE, defined as AG = (GV  GR)/(GV + GR) and AE = (EV  ER)/(EV + ER). The subscripts V and R represent the Voigt and Reuss approximation. The value of A is always positive, and is zero for crystals which are elastically isotropic. For the Pt–Sn intermetallics, the calculated anisotropy factors, A, are listed in Table 7. The results demonstrate that the PtSn phase is much more anisotropic than the other four intermetallics, and it is also in agreement with the following calculation results of anisotropy. To further investigate the elastic anisotropy of Pt–Sn intermetallics, a three-dimensional surface representation of the elastic anisotropy was employed to show the variation of the elastic modulus with the crystallographic direction. And the degree of deviation in shape from a sphere indicates the degree of anisotropy in the system, since for an isotropic system one would see a spherical shape. Pt–Sn intermetallic compounds investigated in this work belong to three kinds of crystal systems, and the directional dependence of

Fig. 6. (a) Directional dependence of Young’s modulus and (b) plane projections of the directional dependence of Young’s modulus in Pt3Sn (units GPa).

Fig. 7. (a) Directional dependence of Young’s modulus and (b) plane projections of the directional dependence of Young’s modulus in PtSn (units GPa).

W. Zhou et al. / Computational Materials Science 46 (2009) 921–931

Fig. 8. (a) Directional dependence of Young’s modulus and (b) plane projections of the directional dependence of Young’s modulus in Pt2Sn3 (units GPa).

Fig. 9. (a) Directional dependence of Young’s modulus and (b) plane projections of the directional dependence of Young’s modulus in PtSn2 (units GPa).

Fig. 10. (a) Directional dependence of Young’s modulus and (b) plane projections of the directional dependence of Young’s modulus in PtSn4 (units GPa).

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Fig. 11. Total density of states for the Pt–Sn intermetallics.

Fig. 12. The projected density of states (PDOS) of Pt and Sn atoms in the PtSn4 crystal.

W. Zhou et al. / Computational Materials Science 46 (2009) 921–931

the Young’s modulus for different crystal structures can be written as [48]: Cubic system:

E ¼ 1=ðs11 þ 2ðs11  s12 

2 2 0:5s44 Þðl1 l2

þ

2 2 l2 l3

þ

2 2 l3 l1 ÞÞ

ð12Þ

Hexagonal system: 2

4

2

2

E ¼ 1=ðð1  l3 Þ2 s11 þ l3 s33 þ l3 ð1  l3 Þð2s13 þ s44 Þ

ð13Þ

Monoclinic system: 4

2 2

2 2

4

2 2

4

þ

2 2 l2 l3 s44

þ

2 2 l1 l3 s55

þ

2 2 l1 l2 s66 Þ

3.4. Debye temperature As the temperature of a crystal’s highest normal mode of vibration, Debye temperature gives some insight into the thermodynamics of material from the elastic properties. For simplification, the Debye temperature can be estimated from the elastic constants, which has been validated compared with the heat capacity measurements [49]. The Debye temperature can be calculated by the following equation [45]:

HD ¼

E ¼ 1=ðl1 s11 þ 2l1 l2 s12 þ 2l1 l3 s13 þ l2 s22 þ 2l2 l3 s23 þ l3 s33 ð14Þ

where l1, l2 and l3 are the direction cosines to the a, b and c axes in the crystal, respectively. Then, the directional dependence of the Young’s modulus can be obtained from calculated compliance constants, and the results are shown in Figs. 6–10. Taking the PtSn crystal, for example, the representation of directional dependence of Young’s modulus shows a high degree of anisotropy as illustrated in Fig. 7. The difference between the maximum and minimum modulus is 148.6 GPa, which is the largest value among the five intermetallic compounds consistent with above discussion about elastic anisotropy factors. A high degree of anisotropy among these Pt–Sn intermetallics also implies that the deviation of bulk modulus between the theoretical calculation and the experimental measurement may be due to the influence of strong elastic anisotropy.

929

 1=3 hvm 3n NA q Þ ð k 4p M

ð15Þ

where h is Planck’s constant, k is Boltzmann’s constant, NA is Avogadro’s number, n is the number of atoms in unit cell, q is the density of material, M is the mass of unit cell, and vm is the averaged elastic wave velocity defined as:

vm ¼

  1=3 1 2 1 þ 3 v3t v3l

ð16Þ

where vt and vl are the transverse and longitudinal elastic waves of the crystal which can be obtained by:

vl ¼

 1=2 B þ 4G=3

q

vt ¼ ðG=qÞ1=2

ð17Þ ð18Þ

Fig. 13. Total and difference charge density distribution of PtSn4 (a) in the (0 4 0) plane including Pt atoms, (b) in the (0 8 0) plane including Sn atoms, (c) in the plane including Pt atoms with nearest Sn atoms.

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W. Zhou et al. / Computational Materials Science 46 (2009) 921–931

Fig. 14. (a) Total and (b) difference charge density distribution of PtSn4 in the (0 0 12) plane.

The calculated averaged elastic wave velocities and Debye temperatures for the five kinds Pt–Sn intermetallics are summarized in Table 6. The results indicate that the averaged elastic wave velocities of these intermetallics are relatively small which are around 2000 m/s, and the Debye temperature decreases with Sn concentration expect for PtSn4. At low temperature, specific heat mainly contributes to the vibration of lowest temperature acoustic mode. Therefore, our calculations imply that the Pt–Sn intermetallics have small specific heats compared to other intermetallics [45]. 3.5. Nature of bonding The total density of states (DOS) of all investigated compounds is obtained with density functional band-structure calculations (Fig. 11). All Pt–Sn intermetallic compounds are metallic because of the finite DOS at the Fermi level. Band structure analysis shows that the sum of DOS in the zone from 11.5 eV to about 5.5 eV are mostly contributed by the 5s state of Sn. With increasing Pt concentration, the width of this band decreases, indicating a sequential location of the Sn-s state from PtSn4 to Pt3Sn. To obtain further insight into the nature of the bonding and analyze the electronic nature associated with the elastic properties of the Pt–Sn intermetallics, PtSn4 was taken as a model candidate for analysis. The projected density of states (PDOS) of Pt and Sn atoms in the PtSn4 crystal are depicted in Fig. 12. It is evident that the ground state properties of this intermetallic compound are determined by 3d bands of Pt. Below the Fermi level, the states can be roughly divided into three parts. At low energy part, the Sn-s band dominates the separated part between 11.5 and 5.5 eV, and from about 5.5 to 3 eV, the strong hybridization between Sn-p and Pt-d states can be observed in the region. Finally, from about 3 eV up to the Fermi level, bands have predominately Sn-p character. And the low-lying Pt-d band hybridizes weakly with the Sn-p states. The dominant bonding mechanism of the hybridization between Sn-p and Pt-d states has also been found in the Ni–Sn system by Ghosh [28]. And the similarity may be due to the same group of Ni and Pt in the periodic table. In addition, some hybridization of Pt-d with the Sn-s band exists. Besides Pt–Sn bonding, considerable Sn–Sn p–p bonding occurs. This is indicated in the shallow well at the Fermi level (arising form a p band bonding–antibonding splitting). The charge density distribution between neighbor atoms can give important information about bonding. Therefore, the total charge density and difference charge density (bonding charge density) distribution in the planes including neighbor Pt–Pt, Pt–Sn or Sn–Sn atoms in the PtSn4 crystal were simulated and represented

in Figs. 13 and 14. The core regions of Pt atoms have large values of charge density, while these values are smaller in the interstitial area, and directional bonding features can be clearly seen between Pt–Pt, Pt–Sn and Sn–Sn atoms in different configurations as shown in Figs. 13 and 14. As presented in Fig. 2, the PtSn4 structure can be geometrically described as a stacking of layers along the h0 1 0i direction. And each layer includes interconnected polyhedron which is the fundamental building block of this compound formed by a Pt atom and its eight nearest neighbor Sn atoms. As discussed above, a mixture of Pt–Sn and Pt–Pt bondings exists in each layer, and the neighbor layers are coupled with Sn–Sn bonding. These directional bondings enhance the abilities against uniaxial strains along the h0 1 0i direction and the plane perpendicular this direction in the PtSn4 crystal. It also results in relatively high degree of anisotropy of elastic properties, which is consistent with our calculations on elastic anisotropy of PtSn4 in Section 3.3. 4. Conclusions In summary, we have investigated the structural, elastic and electronic properties of all Pt–Sn intermetallics based on the Pt– Sn binary phase diagram with the DFT method. The calculated lattice constants of the five intermetallics are in good agreement with previous experimental data. The bulk modulus of Pt–Sn intermetallic compounds decreases with Sn concentration. Polycrystalline elastic properties were also obtained from single-crystal elastic constants, and a three-dimensional surface representation of the elastic anisotropy was employed to show the variation of the elastic modulus with the crystallographic direction. The calculated Debye temperatures indicate that Pt–Sn intermetallics are soft with small wave velocities. Furthermore, the electronic nature of PtSn4 was employed to analyze the origin of the elastic anisotropy. Acknowledgments The authors would like to acknowledge the support of the National Natural Science Foundation of China (50674071), the Tianjin Natural Science Foundation (06YFJZJC01300), and the Program for New Century Excellent Talents in University (NCET-06-0245). References [1] M. Klein, B. Wiens, M. Hutter, H. Oppermann, R. Aschenbrenner, H. Reichl, in: Proceedings of 50th Electronic Components and Technology Conference, IEEE, Las Vegas, NV, 2000, p. 40.

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