Anisotropic elastic properties and electronic structure of Sr–Pb compounds

Computational Materials Science 98 (2015) 311–319

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Anisotropic elastic properties and electronic structure of Sr–Pb compounds M.J. Peng a, Y.H. Duan a,b,⇑, Y. Sun a a b

School of Material Science and Engineering, Kunming University of Science and Technology, Kunming 650093, China Key Lab of Advanced Materials of Yunnan Province, Kunming University of Science and Technology, Kunming 650093, China

a r t i c l e

i n f o

Article history: Received 21 July 2014 Accepted 30 October 2014

Keywords: First-principles calculations Phase stability Elastic properties Anisotropy Electronic structure

a b s t r a c t Phase stabilities, anisotropic elastic properties and electronic structures of seven Sr–Pb binary compounds have been investigated by using first-principles based on the density functional theory. The calculated equilibrium structural parameters are in good agreement with available experimental data. The phase stabilities were obtained by formation enthalpies calculations. The bulk modulus B, shear modulus G, Young’s modulus E, Poisson’s ratios and hardness were evaluated. The mechanical anisotropy was characterized by calculating several different anisotropic indexes and describing the threedimensional (3D) surface constructions. Finally, the electronic structures, such as band structures and total density of states were discussed. Ó 2014 Elsevier B.V. All rights reserved.

1. Introduction Owing to their industrial applications as electronic materials and thermoelectric materials, a series of intermetallics in the Sr– (Si, Ge, and Sn) binary systems have been widely investigated [1–3]. A computational investigation on the lattice stability of the intermediate phases in the Sr–Si system has been performed [4]. As the same IV group element, seven intermediate phases were found to exist in the Sr–Pb system: two of these phases melt congruently: Sr2Pb (1428 K); SrPb3 (948 K); the remaining five phases form peritectically: Sr5Pb3, Sr5Pb4, SrPb, Sr2Pb3, and Sr3Pb5 [5]. The crystal structures of Sr–Pb compounds are Ti3Cu-type for SrPb3, CrB-type for SrPb, Gd5Si4-type for Sr5Pb4, Cr5B3-type for Sr5Pb3, and anti-PbCl2-type for Sr2Pb, respectively [5–7]. The band gap for Zintl phase Sr2Pb is about 100 meV by using the density functional theory (DFT) as implemented in the Vienna Ab initio Simulation Package (VASP) [8]. Only the bulk modulus of Sr2Pb has been calculated [9] and the elastic properties of the other Sr– Pb compounds are hardly found. Strontium has long-term effect on the properties of many alloys and low oxidation sensitivity. Strontium addition can intensify the creep resistance of the alloys in high temperature applications [10]. Therefore, the information on thermodynamics and mechanical properties of the Sr–Pb system is much appealing for the development of Pb alloys. Among ⇑ Corresponding author at: School of Material Science and Engineering, Kunming University of Science and Technology, Kunming 650093, China. Tel./fax: +86 871 65136698. E-mail address: [email protected] (Y.H. Duan). http://dx.doi.org/10.1016/j.commatsci.2014.10.060 0927-0256/Ó 2014 Elsevier B.V. All rights reserved.

the (Ca, Sr and Ba)–Pb binary systems, the formation enthalpies for Ca–Pb and Ba–Pb systems have been studied [11,12]. To our knowledge, there are no existing data on the formation enthalpies for Sr–Pb system. The systematical investigations on formation enthalpies, elastic properties and electronic structures for Sr–Pb intermetallic compounds are required. As a powerful tool to calculate accurately the physical and chemical properties of crystalline materials [13–16], the firstprinciples investigations provide the possibility to understand and forecast the physical properties of solids which are previously inaccessible by experiments. In order to provide some additional information to the mechanical properties and understand deeper their ground-state properties, the systematical first-principles calculations of thermodynamic and elastic properties including formation enthalpies, elastic constants and electronic structures for Sr–Pb intermetallic compounds were performed in this work. Based on the calculated elastic constants, the elastic anisotropy and the relationship between melting point and formation enthalpy were also investigated and discussed. 2. Computational method In this work, we performed the first-principles method implemented in CASTEP (Cambridge sequential total energy package) package [17], which is based on density functional theory (DFT). Ultra-soft pseudo-potentials were used to describe the core-valence interactions. The exchange–correlation energy was treated by the local density approximation (LDA) CA-PZ function [18,19]. The considered valence electrons configurations were Sr

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4s24p65s2 and Pb 5d106s26p2. The k points in the first irreducible Brillouin zone were 10  10  10, 4  4  10, 6  6  10, 10  4  10, 9  3  9, 9  9  6 and 6  10  4 for SrPb3, Sr3Pb5, Sr2Pb3, SrPb, Sr5Pb4, Sr5Pb3 and Sr2Pb, respectively. The cutoff energy for plane waves was selected as 400 eV after convergence tests. The structural optimizations were performed by using the Broyden–Fletcher–Goldfarb–Shanno (BFGS) minimization [20]. The maximum ionic Hellmann–Feynman force and maximum ionic displacement were set by 0.01 eV/Å and 5.0  104 Å, respectively. The SCF (self-consistent field) tolerance was set as 5.0  107 eV/ atom. 3. Results and discussion 3.1. Structural properties As shown in Fig. 1 and Table 1, the considered Sr–Pb intermetallic compounds are of tetragonal and orthorhombic crystal structures. In order to investigate the ground-state physical properties of Sr–Pb intermetallic compounds accurately, the geometry optimizations were performed. The optimized lattice parameters and equilibrium volume of these compounds, together with the available experimental data [5–7,21–23], are listed in Table 1. The calculated results are in good agreement with the available experimental data, indicating that the calculations we performed in the present work are reasonable and reliable. 3.2. Phase stability To determine the relative phase stability of Sr–Pb binary compounds, the formation enthalpies (DH) have been investigated by employing the following relation:

DHðSrx Pby Þ ¼

Etotal ðSrx Pby Þ  xEbulk ðSrÞ  yEbulk ðPbÞ xþy

ð1Þ

where DH is the formation enthalpy; Etotal (SrxPby) is the total cell energy of a SrxPby primitive cell including x Sr atoms and y Pb atoms; Ebulk is the chemical potential of Sr or Pb atom in the bulk state. The calculated formation enthalpies as a function of the mole fraction of Sr are plotted in Fig. 2. In general, a negative formation enthalpy usually means an exothermic process and a more negative formation enthalpy corresponds to a better phase stability. For the investigated Sr–Pb system, more than one ordered binary compound are investigated. Our calculations of formation enthalpies are illustrated a convex hull determined by three compounds in the Sr–Pb system: SrPb3, SrPb and Sr5Pb3. The convex hull is shown in Fig. 2. It is obvious, in Fig. 2, that the convex hull is symmetric and has a maximum in formation enthalpy at SrPb. The calculated formation enthalpies for the three compounds, which located at the bottom of the convex hull in Fig. 2, are

42.719 kJ/mol for SrPb3, 62.274 kJ/mol for SrPb, and 60.752 kJ/mol for Sr5Pb3, respectively. SrPb has the best stability in the seven binary compounds in the Sr–Pb system. The deviations of the convex hull for Sr2Pb3, Sr5Pb4 and Sr2Pb are 2.7 kJ/mol, 2.5 kJ/mol and 8.0 kJ/mol, respectively. The energies lying of Sr2Pb3 and Sr5Pb4 close to the convex hull indicates that they are stable. The stability of Sr2Pb is hard to explain according to Fig. 2, however, the negative formation enthalpy for Sr2Pb (61.951 kJ/mol) reveals the observed stability of Sr2Pb. The calculated formation enthalpy for Sr3Pb5 lies above the convex hull by 22 kJ/mol in Fig. 2 and the phase should be relatively unstable comparison of the other Sr–Pb binary phases. It can be concluded that the phase stability for the Sr–Pb binary compounds follows the order: SrPb > Sr2Pb > Sr5Pb3 > Sr5Pb4 > Sr2Pb3 > SrPb3 > Sr3Pb5. Due to a peritectic formation described for Sr3Pb5, Sr2Pb3, SrPb Sr5Pb4, and Sr5Pb3, there are no definite experimental melting points for these five Sr–Pb binary compounds. We have only considered the calculated formation enthalpies (DH) compared with the congruent melting temperatures (Tm) of SrPb3 and Sr2Pb among the Sr–Pb binary compounds. The values of DH and Tm are 42.719 kJ/mol and 948 K for SrPb3, and 61.951 kJ/mol and 1428 K for Sr2Pb, respectively. It is worth noting that for these two compared compounds, the formation enthalpy of a stable compound at 0 K is more negative if it has a higher melting temperature. This is to be expected that a more negative formation enthalpy is an indicator of a greater stability. Such correlations between formation enthalpy of solids and their melting temperatures have been observed experimentally in Ca–Pb [24] and Ba–Pb binary systems [12]. 3.3. Elastic constants and polycrystalline moduli The single-crystal elastic constants can be obtained by investigating the total energy as a function of appropriate lattice deformation. A deformed cell is introduced to calculate the elastic constants Cij. The elastic strain energy is given by fellow [25]:

U ¼ DE=V 0 ¼ 1=2

6 X 6 X C ij ei ej i

ð2Þ

j

where DE is the energy difference; V0 is the volume of the original cell; Cij is the elastic constants; ei and ej are strain. The necessary number of strains is determined by the crystal symmetry [26–28]. As show in Table 2, six strains and nine strains are required for tetragonal and orthorhombic crystals, respectively. For each of the strain, seven different values of d (0.0, ±0.005, ±0.01, ±0.02) were used. From the curvature of the functional of the energy versus strain, the elastic constants were obtained. There are six and nine independent elastic constants for tetragonal and orthorhombic structures, respectively. The obtained independent elastic constants and the elastic compliance matrices calculated directly from elastic

Fig. 1. Crystal structures of the Sr–Pb binary compounds. The gray and green balls represent Pb atoms and Sr atoms, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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M.J. Peng et al. / Computational Materials Science 98 (2015) 311–319 Table 1 Calculated and experimental lattice parameters and equilibrium volume for Sr–Pb binary compounds. Phase

At% of Sr

Crystal structure

Space group

SrPb3

25.0

Tetragonal

P4/mmm

Sr3Pb5

37.5

Tetragonal

P4/mbm

Sr2Pb3

40.0

Tetragonal

P4/mbm

SrPb

50.0

Orthorhombic

Cmcm

Sr5Pb4

55.6

Orthorhombic

Pnma

Sr5Pb3

62.5

Tetragonal

I4/mcm

Sr2Pb

66.7

Orthorhombic

Pnma

Lattice parameters (Å) a

b

4.925 4.965 16.163 16.170 8.342 8.367 4.957 5.018 8.472 8.480 8.652 8.670 8.426 8.445

12.210 12.230 17.264 17.270

5.370 5.391

c

c/a

4.981 5.024 4.863 4.886 4.879 4.883 4.632 4.648 8.983 9.010 15.923 15.940 10.128 10.139

1.011 1.012 0.301 0.302 0.585 0.584 0.934 0.926 1.060 1.063 1.840 1.839 1.202 1.201

V (Å3)

Refs.

120.82 123.85 1270.42 1277.54 339.52 341.84 280.35 285.25 1313.86 1319.51 1191.95 1198.19 458.27 461.60

Present [21] Present [5] Present [22] Present [23] Present [7] Present [5] Present [6]

Table 3 The calculated elastic constants for Sr–Pb binary compounds.

Fig. 2. The calculated formation enthalpies as a function of the mole fraction of Sr.

constants were listed in Tables 3 and 4, respectively. The mechanical stabilities of the seven Sr–Pb binary compounds could be discussed based on elastic constants. The elastic constants need to satisfy the generalized stability criteria: C11 > 0, C33 > 0, C44 > 0, C66 > 0, C11 > C12, C11 + C33  2C13 > 0, 2C11 + C33 + 2C12 + 4C13 > 0 for tetragonal crystals [29]; C11 > 0, C22 > 0, C33 > 0, C44 > 0, C55 > 0, C66 > 0, C11 + C22  2C12 > 0, C11 + C33  2C13 > 0, C22 + C33  2C23 > 0, C11 + C22 + C33 + 2C12 + 2C13 + 2C23 > 0 for orthorhombic crystals [30]. It is obvious that from Table 3, for the tetragonal crystals (SrPb3, Sr3Pb5, Sr2Pb3 and Sr5Pb3) all of the elastic constants meet the

Phase

Cij (GPa) C11

C12

C13

SrPb3 Sr3Pb5 Sr2Pb3 SrPb Sr5Pb4 Sr5Pb3 Sr2Pb

74.7 59.7 56.2 65.6 50.1 55.5 20.61

33.8 32.4 36.0 15.8 22.6 16.2 14.4

32.4 22.8 25.1 26.4 24.0 22.3 12.5

C22

C23

72.6 47.1

22.3 27.2

52.2

25.5

C33

C44

72.7 34.4 60.3 67.7 49.7 37.6 45.7

28.0 21.4 23.5 28.3 17.5 25.4 17.0

C55

34.5 17.1 15.0

C66 29.9 13.3 12.6 17.6 15.1 25.5 24.2

tetragonal crystal’s mechanical stability criteria. As for the three orthorhombic crystals (SrPb, Sr5Pb4 and Sr2Pb), the elastic constants are consistent with the restrictions to orthorhombic crystals. The results reveal that all the Sr–Pb binary compounds are mechanically stable. The elastic constant C11 represents the resistance to linear compression along x axis [31]. It is evident that the calculated C11 for SrPb3 is larger than the other Sr–Pb compounds, which indicate that SrPb3 is very incompressible under uniaxial stress along x axis. The elastic constant C33 characterizes the z direction resistance to linear compression. SrPb3 is more incompressible along z axis than the other Sr–Pb binary compounds due to the largest C33 for SrPb3 in Sr–Pb binary compounds. It can be considered that the x axis is more compressible than the z axis for Sr2Pb3, SrPb and Sr2Pb owing to their calculated C33 higher than the C11. For the SrPb3, Sr3Pb5, Sr5Pb4 and Sr5Pb3, C11 is larger than C33, which indicate that z axis is more compressible than the x axis. Sr3Pb5 is extremely compressible along z axis due to the smallest C33 (34.4 GPa). Moreover,

Table 2 Strains used to calculate elastic constants of Sr–Pb binary compounds. Structure

Strain

Parameters (unlisted ei = 0)

DE/V0 to O(d2)

Orthorhombic (SrPb, Sr5Pb4 and Sr2Pb)

e1 e2 e3 e4 e5 e6 e7 e8 e9 e1 e2 e3 e4 e5 e6

e1 = d e2 = d e3 = d e1 = 2d, e2 = d, e3 = d e1 = d, e2 = 2d, e3 = d e1 = d, e2 = d, e3 = 2d e4 = d e5 = d e6 = d e1 = 2d, e2 = e3 = d e1 = e2 = d, e3 = 2d e1 = e2 = d, e3 = 2d, e6 = 2d e1 = d e3 = d e4 = 2d

(C11d2)/2 (C22d2)/2 (C33d2)/2 (4C11  4C12  4C13 + C22 + 2C23 + C33) d2/2 (C11  4C12 + 2C13 + 4C22  4C23 + C33) d2/2 (C11 + 2C12  4C13 + C22  4C23 + 4C33) d2/2 (C44d2)/2 (C55d2)/2 (C66d2)/2 (5C11  4C12  2C13 + C33) d2/2 (C11 + C12  4C13 + 2C33) d2 (C11 + C12  4C13 + 2C33 + 2C66) d2 (C11d2)/2 (C33d2)/2 (C44d2)/2

Tetragonal (SrPb3, Sr3Pb5, Sr2Pb3 and Sr5Pb3)

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Table 4 The elastic compliance matrix of Sr–Pb compounds calculated from the elastic coefficients using the strain–stress method. Phase

SrPb3 Sr3Pb5 Sr2Pb3 SrPb Sr5Pb4 Sr5Pb3 Sr2Pb

Sij S11

S12

S13

0.01851 0.02640 0.03167 0.01832 0.02803 0.02376 0.07625

0.00594 0.01023 0.01766 0.00200 0.00823 0.00168 0.02496

0.00561 0.01071 0.00582 0.00648 0.00904 0.01312 0.00696

S22

S23

0.01555 0.03347

0.00435 0.01437

0.03451

0.01243

it is well known that the elastic constant C44 is the most significant parameter. A large C44 implies a strong resistance to monoclinic shear in the (1 0 0) plane [32]. The highest C44 for SrPb than that for the other Sr–Pb binary compounds means that SrPb has the strongest ability to resist shear distortion in the (1 0 0) plane. Comparing with single-crystal elastic properties, the polycrystalline elastic properties usually have higher practical application value. There are two approximation methods to calculate the polycrystalline modulus, namely, the Voigt method [33] and the Reuss method [34]. Bulk modulus B, shear modulus G, Young’s modulus E and Poisson’s ratio m are directly calculated by the Voigt–Reuss–Hill method (VRH) [35]:

1 ðBR þ BV Þ 2 1 G ¼ ðGR þ GV Þ 2 9GB E¼ G þ 3B 3B  E m¼ 6B B¼

ð3Þ ð4Þ ð5Þ ð6Þ

For tetragonal [36]:

GV ¼ ðM þ 3C 11  3C 12 þ 12C 44 þ 6C 66 Þ=30 2

GR ¼ 15=½ð18BV Þ=C þ 6=ðC 11  C 12 Þ þ 6=C 44 þ 3=C 66  BV ¼ ½2ðC 11 þ C 12 Þ þ C 33 þ 4C 13 =9

ð7Þ ð8Þ ð9Þ

BR ¼ C 2 =M

ð10Þ

C 2 ¼ ðC 11 þ C 12 ÞC 33  2C 213

ð11Þ

M ¼ C 11 þ C 12 þ 2C 33  4C 13

ð12Þ

For orthorhombic [37]: 1 1 3 ðC 11 þ C 22 þ C 33 Þ  ðC 12 þ C 13 þ C 23 Þ þ ðC 44 þ C 55 þ C 66 Þ 15 15 15 1 4 4 3 ¼ ðS11 þ S22 þ S33 Þ  ðS12 þ S13 þ S23 Þ þ ðS44 þ S55 þ S66 Þ GR 15 15 15 1 2 BV ¼ ðC 11 þ C 22 þ C 33 Þ þ ðC 12 þ C 13 þ C 23 Þ 9 9 1 ¼ ðS11 þ S22 þ S33 Þ þ 2ðS12 þ S13 þ S23 Þ BR GV ¼

ð13Þ ð14Þ ð15Þ ð16Þ

in which the subscripts V and R indicate the Voigt and Reuss averages. The results of calculated polycrystalline bulk modulus B, shear modulus G, Young’s modulus E and Poisson’s ratio m are listed in Table 5. The calculated B of Sr2Pb in the present work (22.5 GPa) is in good agreement with the GGA (generalized gradient approximation) calculated value (23.4 GPa) [9], suggesting that the calculations in the present work are reasonable. Generally speaking, the bulk modulus of a solid embodies its resistance to volume change. For the tetragonal structures (SrPb3, Sr3Pb5, Sr2Pb3 and Sr5Pb3), the order of deviation of c/a from 1 is Sr3Pb5 > Sr5Pb3 > Sr2Pb3 > SrPb3 from Table 1, which may result in the lower bulk modulus of Sr3Pb5 in these four tetragonal Sr–Pb compounds. Similar to the tetragonal structures, it can be found

S33

S44

0.01877 0.04327 0.02141 0.01873 0.03239 0.04220 0.03071

0.03578 0.04673 0.04249 0.03539 0.05703 0.03941 0.05890

S55

S66 0.03350 0.07519 0.07950 0.05670 0.06614 0.03917 0.04135

0.02900 0.05862 0.06660

Table 5 The bulk modulus B (GPa) and shear modulus G (GPa) in the Voigt and Reuss approximations, Young’s modulus E (GPa), Poisson ratio m and hardness HV (GPa) of Sr–Pb compounds calculated using the elastic constants. Phase

B

BV

BR

G

GV

GR

E

B/G

m

HV

SrPb3 Sr3Pb5 Sr2Pb3 SrPb Sr5Pb4 Sr5Pb3 Sr2Pb

46.6 27.5 38.3 37.2 32.7 29.8 22.5

46.6 34.4 38.3 37.2 32.7 30.0 26.0

46.6 30.5 38.2 37.1 32.7 29.5 18.9

25.0 15.6 16.8 24.8 14.6 19.7 13.7

25.4 16.3 17.7 25.5 14.8 21.1 15.3

24.7 14.8 15.8 24.0 14.3 18.2 12.1

63.7 39.4 43.9 60.8 38.0 48.3 34.1

1.86 1.76 2.29 1.50 2.25 1.52 1.64

0.27 0.26 0.31 0.23 0.31 0.23 0.25

4.44 3.38 2.64 5.62 2.44 4.72 3.34

that the bulk modulus of Sr2Pb is lower in the orthorhombic Sr2Pb, Sr5Pb4 and SrPb due to Sr2Pb’s larger value of c/a. However, the bulk moduli listed in Table 5 indicate that in all structures SrPb3 is the least compressible material. The shear modulus can be as a measurement of resistance to shape change, which is more pertinent to hardness than the bulk modulus. A larger shear modulus of a solid is mainly owing to a larger C44 [32]. Sr2Pb has the lowest shear modulus in the binary Sr–Pb compounds, which indicates Sr2Pb should possess the lowest hardness. The Young’s modulus is defined as the ratio of stress and strain and can also be used as a measure of the stiffness of a solid. When the value of Young’s modulus is large, the material is stiff [38]. In the present work, SrPb3 is stiffer than the other considered compounds due to its higher value of Young’s modulus (63.8 GPa). Sr3Pb5 has the lowest Young’s modulus (34.1 GPa), which may result in the lower stiff for Sr2Pb. Table 5 also shows the calculated B/G ratio. Recently, the quantity has been used extensively to indicate the ductility of a material [38–40]. The brittleness of a solid is related to the value of B/G. A solid with smaller B/G value (<1.75) usually is brittle while a solid with larger B/G value (>1.75) is ductile [41]. Poisson’s ratio m also refers to a ductile solid usually with large m (>0.26) [42]. Fig. 3 shows the relationship among mole fraction of Sr, B/G and m. The B/G and m values larger than 1.75 and 0.26 for SrPb3, Sr3Pb5, Sr2Pb3 and Sr5Pb4 in Fig. 3 indicate that these compounds are ductile and Sr2Pb3 is the most ductile phase. The other compounds in Sr–Pb binary system are brittle which have values of B/G and m smaller than 1.75 and 0.26, and SrPb has the most brittleness. Moreover, it can also be further supported by the classical criteria of Cauchy pressure (C12  C44). A positive Cauchy pressure reveals damage tolerance and ductility of a crystal, while a negative one demonstrates brittleness [42,43]. The calculated negative values of Cauchy pressure for SrPb, Sr5Pb3 and Sr2Pb strongly confirms their brittle nature. Hardness is generally related to the elastic and plastic properties of a material. The hardness of polycrystalline materials can be predicted by a relatively simple semi-empirical equation of hardness, which is defined as follow [44]:

 1:137 G HV ¼ 0:92 G0:708 B

ð17Þ

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expressed by the universal anisotropic index (AU) and percent anisotropy (AB and AG). The following expressions are applied for our investigation [46,47]:

GV BV þ 60 GR BR BV  BR GV  GR ; AG ¼ AB ¼ BV þ BR GV þ GR

AU ¼ 5

Fig. 3. The brittle properties of the Sr–Pb binary compounds. The dotted line denotes the dividing line between ductile and brittle solids. Solids with B/G > 1.75 and m > 0.26 are ductile, the opposite are brittle.

This formula fairly well agrees with experimental data for polycrystalline materials [45]. The hardness calculated by the formula may not be very accurate due to macroscopic concepts of bulk and shear moduli. However, a direct quantification of hardness with microscopic parameters can provide a reference to reveal the fundamental factors controlling materials hardness. The calculated hardness of all Sr–Pb binary compounds (listed in Table 5) is less than 6 GPa. The change trend among G, G/B, HV and mole fraction of Sr was plotted in Fig. 4. The HV value is multiplied by the factor of 5 for a better comparison. It can be concluded from Fig. 4 that G/B is more pertinent to hardness than the shear modulus G for Sr–Pb binary compounds (except Sr3Pb5) because of the power indexes of 1.137 and 0.708 for G/B and G, respectively. Therefore, for SrPb, it owns the largest HV (5.62 GPa) due to its maximum G/B value in Sr–Pb compounds. There are no or less available experimental and theoretical studies available on hardness for Sr–Pb compounds. Our present work could provide a useful guidance for future study. 3.4. Anisotropy of elastic moduli The elastic anisotropy of a solid is closely related to the possibility of inducing micro-cracks in materials, and can be

ð18Þ ð19Þ

where B and G are the bulk modulus and shear modulus, the subscripts V and R refer the Voigt and Reuss approximations, respectively. AU = AB = AG = 0 when a solid is isotropic. The large deviations of these three elastic anisotropic indexes from zero mean the highly anisotropic mechanical properties. Moreover, owing to the investigated crystal systems in present work including tetragonal (SrPb3, Sr3Pb5, Sr2Pb3 and Sr5Pb3) and orthorhombic (SrPb, Sr5Pb4 and Sr2Pb), the shear anisotropic factors also need consider describing the elastic anisotropy. Thus, the shear anisotropic factors A1, A2, A3 are calculated and discussed, which are defined as [48]:

A1 ¼

4C 44 C 11 þ C 33  2C 13

ð20Þ

A2 ¼

4C 55 C 22 þ C 33  2C 23

ð21Þ

A3 ¼

4C 66 C 11 þ C 22  2C 12

ð22Þ

The values of A1, A2 and A3 should equate to 1.0 for an isotropic crystal. Otherwise, it is an anisotropic crystal. The results of elastic anisotropy are listed in Table 6. It is obvious that the order of elastic anisotropy for the considered Sr–Pb compounds is Sr2Pb > Sr5Pb3 > Sr3Pb5 > Sr2Pb3 > SrPb > Sr5Pb4 > SrPb3 from the universal elastic anisotropic index. The orthorhombic Sr2Pb has the negative AU value among Sr–Pb binary compounds. The elastic moduli of Sr2Pb are strongly dependent on different directions and the calculated AG and shear anisotropic factors (A1, A2 and A3) support this conclusion due to the larger AG and A3 for Sr2Pb. Though AG, A1, A2 and A3 determine the anisotropy of the shear modulus, the values of A1, A2 and A3 are quite different from AG [46]. The calculated A1, A2 and A3 values seem to support the surmise that the shear modulus of Sr–Pb intermetallics compounds has a strong directional dependence. However, the universal anisotropic index AU can provide unique and consistent results for the mechanical anisotropic properties of Sr–Pb binary compounds. An elastic anisotropy diagram in the (BV/BR, GV/GR) space was constructed and shown in Fig. 5. The region with BV/BR 6 1 and GV/GR 6 1 (i.e. AU 6 0) is not considered physically. The increment along the GV/ GR axis affects the anisotropy AU much more than along BV/BR axis due to the multiplied factor for GV/GR larger than that for BV/BR. The larger AU represents the more anisotropic. Sr2Pb has the largest BV/ BR and GV/GR among the six Sr–Pb binary compounds considering in Fig. 5. Therefore, Sr2Pb is the most anisotropic, followed by Sr5Pb3 and Sr3Pb5. Although the BV/BR of the tetragonal Sr2Pb3 is Table 6 The universal anisotropic index (AU), percent anisotropy (AG and AB) and shear anisotropic factors (A1, A2 and A3) for Sr–Pb compounds.

Fig. 4. The change trend among G, G/B and HV as a function of the mole fraction of Sr. Note that HV value is magnified by 5 times to the initial value.

Phase

BV/BR

GV/GR

AU

AG

AB

A1

A2

A3

SrPb3 Sr3Pb5 Sr2Pb3 SrPb Sr5Pb4 Sr5Pb3 Sr2Pb

1.0004 1.1279 1.0023 1.0030 1.0015 1.0186 1.3701

1.0267 1.1014 1.1209 1.0629 1.0342 1.1605 1.2695

0.134 0.635 0.607 0.317 0.173 0.821 1.717

0.013 0.048 0.057 0.031 0.017 0.074 0.119

0.0002 0.060 0.0012 0.0015 0.0008 0.0092 0.1562

1.355 1.765 1.417 1.403 1.357 2.093 1.645

1.355 1.765 1.417 1.442 1.612 2.093 1.281

1.460 0.974 1.241 0.662 1.164 1.299 2.847

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smaller than that of SrPb, Sr2Pb3 has a larger GV/GR of 1.1209 than SrPb, and Sr2Pb3 is more anisotropic than SrPb. One can draw the similar conclusion for Sr5Pb4 and SrPb3 that Sr5Pb4 has the most anisotropy due to the larger BV/BR and GV/GR. It should be noted in Fig. 5 that the orthorhombic Sr2Pb with the largest AU = 1.717 is the most anisotropic in Sr–Pb binary compounds. As a valid method to describe the elastic anisotropic behavior of a crystal completely, the three-dimensional (3D) surface construction is depicted to further illustrate the elastic anisotropic features of the Sr–Pb binary compounds. The 3D figures of directional dependences of the reciprocal of Young‘s modulus for the Sr–Pb binary compounds can be defined by the following equations due to their various crystal structures [49]. For tetragonal system,

in shape from the sphere in Sr2Pb3 is obviously smaller than that in Sr3Pb5. The same conclusion can be drawn from the ratio of C11/C33 or C33/C11 for SrPb and Sr5Pb4. For the tetragonal SrPb3, the 3D directional dependence of the Young’s modulus is only slightly deviate in shape from the sphere, which is the z axis is somewhat compressible than the x axis due to the C33 very smaller than the C11. It indicates that SrPb3 has a smallest anisotropy comparing to the other Sr–Pb binary compounds. From the 3D directional dependence of the Young’s modulus, it can be concluded that the degree of the elastic anisotropy for the investigated Sr–Pb compounds follows the order Sr2Pb > Sr5Pb3 > Sr3Pb5 > Sr2Pb3 > SrPb > Sr5Pb4 > SrPb3. This result is consistent with the results from the analysis of the universal elastic anisotropy indexes.

    1 4 4 2 2 2 2 4 ¼ S11 l1 þ l2 þ ð2S13 þ S44 Þ l1 l3 þ l2 l3 þ S33 l3 þ ð2S12 E

3.5. Anisotropy of acoustic velocities and Debye temperature

2 2

þ S66 Þl1 l2

ð23Þ

For orthorhombic system, 1 4 4 4 2 2 2 2 2 2 2 2 2 2 ¼ l S11 þ l2 S22 þ l3 S33 þ 2l1 l2 S12 þ 2l1 l3 S13 þ 2l2 l3 S23 þ l2 l3 S44 þ l1 l3 S55 E 1 2 2

þ l1 l2 S66

ð24Þ

where Sij is the usual elastic compliance constant (see Table 4). l1, l2 and l3 are the direction cosines. The results are shown in Fig. 6. The surface in each graph denotes the magnitude of B and E along different directions. The 3D figure appears as a spherical shape for an isotropic structure, while the deviation from the spherical shape exhibits the content of anisotropy [16]. It can be seen from Fig. 6 that the 3D figure of the Young’s modulus for the orthorhombic Sr2Pb with the biggest AU value (1.717) has the largest deviation in shape from the sphere, which indicates that the Young’s modulus for the considered orthorhombic structure Sr2Pb shows a very large anisotropy. The AU value is 0.821 for Sr5Pb3, which means the Young’s modulus for Sr5Pb3 shows a large anisotropy. The 3D figure of the Young’s modulus for the tetragonal Sr3Pb5 is characterized by more compressible along the z axis than along the x axis. The AU value (0.635) for Sr3Pb5 obviously smaller than that for Sr5Pb3, the deviate in shape from the sphere in Sr3Pb5 is significantly smaller than that in Sr5Pb3. As for the tetragonal Sr2Pb3, the 3D directional dependence of the Young’s modulus along the x axis is more compressible than along the z axis. Owing to the ratio of C33/C11 in Sr2Pb3 (=1.07) significantly smaller than that of C11/C33 in Sr3Pb5 (=1.74), the deviate

Using the single crystal elastic constants, the phase velocities of pure transverse and longitudinal modes of the Sr–Pb compounds can be calculated by following the procedure of Brugger [50]. The sound velocities are determined by the symmetry of the crystal and propagation direction. In the principal directions the acoustic velocities can be expressed by: Tetragonal pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi ½100ml ¼ ½010ml ¼ C 11 =q; ½001mt1 ¼ C 44 =q; ½010mt2 ¼ C 66 =q pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi ½001ml ¼ C 33 =q; ½100mt1 ¼ ½010mt2 ¼ C 66 =q pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi ½110ml ¼ ðC 11 þ C 12 þ 2C 66 Þ=2q; ½001mt1 ¼ C 44 =q; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  mt2 ¼ ðC 11  C 12 Þ=2q; ½1 10

ð25Þ Orthorhombic

pffiffiffiffiffiffiffiffiffiffiffiffiffi C 11 =q; pffiffiffiffiffiffiffiffiffiffiffiffiffi ½0 10 ml ¼ C 22 =q; pffiffiffiffiffiffiffiffiffiffiffiffiffi ½0 0 1mt ¼ C 33 =q;

½1 0 0ml ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffi C 66 =q; pffiffiffiffiffiffiffiffiffiffiffiffiffi ½1 0 0mt1 ¼ C 66 =q; pffiffiffiffiffiffiffiffiffiffiffiffiffi ½1 0 0mt1 ¼ C 55 =q; ½0 1 0mt1 ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffi C 55 =q pffiffiffiffiffiffiffiffiffiffiffiffiffi ½0 0 1mt2 ¼ C 44 =q pffiffiffiffiffiffiffiffiffiffiffiffiffi ½0 1 0mt2 ¼ C 44 =q ½0 0 1mt2 ¼

ð26Þ where q is the density of Sr–Pb binary compounds; vl is the longitudinal sound velocity; vt1 and vt2 represent the first and the second transverse modes, respectively. The calculated sound velocities in the directions for Sr–Pb compounds are listed in Table 7. In general, the solid with a small density q and large elastic constants has large sound velocities. It can find that the longitudinal waves of tetragonal crystals are the fastest along [1 1 0] direction, while the longitudinal waves of orthorhombic crystals are the fastest along [0 0 1] direction. The anisotropic sound velocities also indicate the elastic anisotropy of these Sr–Pb binary compounds. As an important physical parameter of a solid, the Debye temperature HD is related to thermal expansion, specific heat and elastic constants. The Debye temperature can be evaluated from the average sound velocity tm by the following equation [51]:

HD ¼

  1 h 3n NA q 3 tm k 4p M

ð27Þ

where h and k are Planck constant and Boltzmann’s constant, respectively; NA is the Avogadro number, M is the molecular weight, q is the density, and n is the number of atoms per formula unit. The average sound velocity tm can be estimated from the following expression [52]:

tm ¼ Fig. 5. The elastic anisotropy diagram for the binary compounds in the Sr–Pb systems.

  13 1 2 1 þ ; 3 t3t t3l

tl ¼

  12 4G Bþ q ; 3

mt ¼

 12 G

q

ð28Þ

where B and G are isothermal bulk modulus and shear modulus. tl and tt are the longitudinal velocity and the transverse sound

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Fig. 6. The directional dependence of the Young’s modulus for the Sr–Pb compounds. The magnitude of Young’s modulus at different directions is represented by the contour. The units are in GPa.

Table 7 The density (in g/cm3), elastic wave velocity (in m/s) and the Debye temperature (in K) for the polycrystalline compounds in Sr–Pb system. Phase

SrPb3

Sr3Pb5

Sr2Pb3

SrPb

Sr5Pb4

Sr5Pb3

Sr2Pb

q

9.51 2973 1714 1466 2802 1772 1714

7.92 2737 1644 1313 2746 1296 1644

7.74 2753 1744 1144 2695 1275 1744

6.87

6.38

5.50

2761 1772 1772 2855 1598 1778 170.35

2084 1296 1296 2470 1403 1560 143.31

2795 1275 1275 2798 1471 1645 150.88

3091 1602 2241 3250 1602 2028 3139 2241 2028 3195 1898 2102 190.14

2801 1539 1635 1883 1539 1658 2790 1635 1658 2858 1511 1689 151.36

5.87 3235 2079 1830 3076 2085 2079

[1 1 0]

[1 0 0]

[0 1 0]

[0 0 1]

[1 1 0]ml [0 0 1]mt1  0mt2 ½1 1 [1 0 0]ml [0 1 0]mt1 [0 0 1]mt2 [0 1 0]ml [1 0 0]mt1 [0 0 1]mt2 [0 0 1]ml [1 0 0]mt1 [0 1 0]mt2

ml mt mm HD

velocity, respectively. The calculated Debye temperatures for Sr–Pb compounds are listed in Table 7. Owing to small mechanical moduli and large densities for most Sr–Pb compounds, the elastic wave velocities of these compounds are relatively small. The largest HD is 190.14 K for SrPb while the lowest one is 143.31 K for Sr3Pb5. The Debye temperatures for Sr–Pb compounds follow the order which is SrPb > Sr5Pb3 > SrPb3 > Sr2Pb > Sr5Pb4 > Sr2Pb3 > Sr3Pb5. The Debye temperature reflects the thermal conductivity for a solid, which is a higher Debye temperature responding to a higher correlative thermal conductivity. The highest Debye temperature of SrPb indicates that SrPb has the best thermal conductivity relative to other Sr–Pb binary compounds. We have also investigated the calculated Debye temperatures (HD) compared with the formation enthalpy (DH) of the Sr–Pb intermetallic compounds. It is interesting to note that for these seven compounds, the Debye temperature is larger if it has a more negative formation enthalpy. This is to be expected since a larger Debye temperature is an indicator of a stronger interatomic bonding. Stronger bonding leads to more negative formation enthalpy. In the considered Sr–Pb system, for example, SrPb has the largest Debye temperature than the other Sr–Pb compounds and SrPb should have the most negative formation enthalpy. However, there is no experimental and theoretical

2530 2085 20858 3088 1830 2026 180.27

1936 2097 1653 3081 2097 1757 2884 1653 1757 2720 1577 1750 154.32

data available for the Debye temperatures of Sr–Pb compounds so far. Therefore, our calculated results are of academic importance and can serve as a guide for future works on Sr–Pb compounds. 3.6. Electronic structure The band structures and total densities of states (TDOS) near Fermi level of the considered Sr–Pb compounds in the present work have also been calculated, and the results are shown in Fig. 7. The zero energy means the Fermi level (EF). It can be seen from Fig. 7 that the energies of band structures near Fermi level for most compounds (except SrPb3) are concentrated in the range of 10 to 5 eV. Sr2Pb has the band gap of 0.102 eV, which both the top valence band and the lowest conduction band are situated at G point in Fig. 7. Therefore, Sr2Pb is a direct band gap semiconductor. The investigated result of the band gap of Sr2Pb (0.102 eV) is in good agreement with the VASP calculation (about 0.100 eV) [8]. As for other compounds considered in the present work, their valence bands overlap the conduction bands at the Fermi surface, which indicates that they are conductors. Furthermore, the TDOS for the Sr–Pb compounds were discussed in Fig. 7. It is found from the TDOS ranging from 5 eV to

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Fig. 7. The band structures and total densities of states near Fermi level of Sr–Pb binary compounds. The dotted lines at zero energy denote the Fermi level EF.

0 eV that the bonding electron numbers of these Sr–Pb binary compounds per atom are 1.941 for SrPb3, 1.904 for Sr3Pb5, 1.955 for Sr2Pb3, 2.001 for SrPb, 1.982 for Sr5Pb4, 1.987 for Sr5Pb3, and 1.992 for Sr2Pb, respectively. The order of bonding number per atom is SrPb > Sr2Pb > Sr5Pb3 > Sr5Pb4 > Sr2Pb3 > SrPb3 > Sr3Pb5. Larger bonding electron numbers correspond to the stronger charge interaction, and the phase stability of the compound will be better [53]. Thus, SrPb has the best phase stability while Sr3Pb5 possesses the worst stability among the Sr–Pb binary compounds.

4. Conclusions We have performed first-principles calculations to investigate the phase stability, elastic moduli, hardness, elastic anisotropy properties, Debye temperatures, and electronic structures of Sr– Pb binary compounds. The formation enthalpies indicate that the phase stability follows the order: SrPb > Sr2Pb > Sr5Pb3 > Sr5Pb4 > Sr2Pb3 > SrPb3 > Sr3Pb5. The calculated elastic constants reveal these compounds (except Sr3Pb5) are mechanically stable. The G/ B is more pertinent to hardness than the shear modulus G for all the considered compounds. The elastic anisotropic results indicate that the Sr–Pb compounds show anisotropic mechanical properties and the order of elastic anisotropy is Sr2Pb > Sr5Pb3 > Sr3Pb5 > Sr2Pb3 > SrPb > Sr5Pb4 > SrPb3. The predicted Debye temperatures follow the order which is SrPb > Sr5Pb3 > SrPb3 > Sr2Pb > Sr5Pb4 > Sr2Pb3 > Sr3Pb5. The analysis on the electronic structures shows that Sr2Pb is a direct band gap semiconductor with the band gap of 0.102 eV, and the higher phase stability of SrPb relative to other Sr–Pb compounds can be attributed to its higher bonding electron numbers. Acknowledgments This work was supported by the National Natural Science Foundation of China under Grant No. 51201079 and the Scientific

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