The Mechanical Properties and Elastic Anisotropy of η′-Cu₆Sn₅ and Cu₃Sn Intermetallic Compounds

Abstract

Full intermetallic compound (IMC) solder joints present fascinating advantages in high-temperature applications. In this study, the mechanical properties and elastic anisotropy of η′-Cu₆Sn₅ and Cu₃Sn intermetallic compounds were investigated using first-principles calculations. The values of single-crystal elastic constants, the elastic (E), shear (G), and bulk (B) moduli, and Poisson’s ratio (ν) were identified. In addition, the two values of G/B and ν indicated that the two IMCs were ductile materials. The elastic anisotropy of η′-Cu₆Sn₅ was found to be higher than Cu₃Sn by calculating the universal anisotropic index. Furthermore, an interesting discovery was that the above two types of monocrystalline IMC exhibited mechanical anisotropic behavior. Specifically, the anisotropic degree of E and B complied with the following relationship: η′-Cu₆Sn₅ > Cu₃Sn; however, the relationship was Cu₃Sn > η′-Cu₆Sn₅ for the G. It is noted that the anisotropic degree of E and G was similar for the two IMCs. In addition, the anisotropy of the B was higher than the G and E, respectively, for η′-Cu₆Sn₅; however, in the case of Cu₃Sn, the anisotropic degree of B, G, and E was similar.

1. Introduction

The application of high-powered electronic equipment and third-generation semiconductor power devices is currently widespread and requires high service temperatures (>250 °C) for solder joints [1,2]. The two common tin-based intermetallic compounds (IMCs) are Cu₆Sn₅ and Cu₃Sn, the melting point of which is 415 and 676 °C, respectively [3], and both have excellent thermodynamic and kinetic stability [1,4,5]. As such, full IMC solder joints have fascinating advantages in high-temperature application fields. Concurrently, the volume and mass of these solder joints are reduced due to high-density packaging, which enables increasing the proportion of IMCs in solder joints and creating a full IMC solder matrix. Several studies have shown that the tensile and shear strength of solder joints will be influenced by the thickness of the IMC layer [6,7,8]. Furthermore, hundred-micron solder joints only include a small number of grains; this will cause the solder joints to exhibit obvious anisotropic mechanical properties [9]. It follows that the mechanical properties of materials, based on large-scale specimens, will not accurately characterize the mechanical behavior of microscale solder joints. Domestic and foreign research both present studies on the mechanical properties of Cu₆Sn₅ and Cu₃Sn. The Young’s modulus (E) and hardness (H) of Cu₃Sn and Cu₆Sn₅ were measured by nanoindentation experiments [10,11,12,13]. Ghosh et al. [14] identified the values of E, shear modulus (G), bulk modulus (B), and Poisson’s ratio (ν) using the pulse-echo method. However, due to the influence of experimental methods, sample manufacturing processes, and mechanical anisotropy, different experimental results were reported. For example, An et al. [15] studied E anisotropy of Cu₃Sn using the first-principles method, and the results showed that the experimental results of mechanical properties will be influenced by mechanical anisotropy. Mu et al. [16] found that Cu₆Sn₅ and (Cu, Ni)₆Sn₅ reflected high anisotropy in E and for H. Moreover, because bulk single-crystal IMCs are difficult to prepare using the current experimental devices and methods, few reports exist on directly obtaining anisotropic mechanical properties through experiments. Choudhury et al. [17] investigated anisotropic mechanical properties for single-grain Cu₆Sn₅ using nanoindentation and electron back-scattered diffraction. However, information about anisotropic mechanical properties that were obtained through experimental methods is limited because it is difficult to obtain this information from all directions.

It is noted that first-principles methods have in recent years been applied for investigating the mechanical properties of materials, including predicting the properties of metallic systems subjected to doping [18,19,20,21,22]; this has, to an extent, compensated for the shortcomings of experimental methods. In addition, Cu₆Sn₅ can exist in a variety of crystal structures, i.e., η, η′, η⁶, η⁸, η⁴⁺¹, and η″, and the η⁸ and η⁴⁺¹ superstructures are approximant structures of the incommensurate η″ phase [23]. Importantly, η′-Cu₆Sn₅ and Cu₃Sn are the most common phase during the service of solder joints [24]; therefore, they have a significant effect on the mechanical behavior of microscale solder joints. Although the anisotropic mechanical properties of Cu₃Sn were investigated by first-principles method, the anisotropy of B and G have typically been neglected. Furthermore, simulations for detecting the anisotropic mechanical properties of η′-Cu₆Sn₅ are rare. Actually, E, B, and G represent the resistance to uniaxial stretching, volume deformation, and plastic deformation capacity, respectively. Elastic anisotropy is related to the generation of microcracks. As such, there is a need for systematically analyzing the mechanical properties and elastic anisotropy of η′-Cu₆Sn₅ and Cu₃Sn, further contributing to improving the durability of solder joints.

In this paper, the elastic constants of monocrystalline η′-Cu₆Sn₅ and Cu₃Sn were calculated based on the first-principles method. Then, according to the Voigt–Reuss–Hill approximation, the E, B, G, and ν of polycrystalline could be obtained. Finally, the directional dependence and anisotropic degree of E, B, and G were investigated.

2. Computational Methods and Details

In this study, a first-principles approach, based on density functional theory integrated with the CASTEP code [25] in the Materials Studio software [26], was employed to calculate the elastic constants of single crystals. The Cu₃Sn is a large one-dimensional long-period superstructure based on a hcp unit cell. In order to improve computational efficiency, the crystal model established by Burkhardt et al. [27] was employed in first-principles calculations. The η′-Cu₆Sn₅ and Cu₃Sn are monoclinic and orthorhombic systems, respectively [27,28]. The atomic site parameters and lattice constants of these two IMCs is presented in existing research [27,28]. The crystal models of η′-Cu₆Sn₅ and Cu₃Sn were created as shown in Figure 1a, b, respectively.

The exchange-correlation energy was described by the generalized gradient approximation (GGA) with the Perdew and Wang (PW91) [29] functional for η′-Cu₆Sn₅ and Cu₃Sn. The ultrasoft pseudopotentials [30] was used to describe ionic cores for the two IMC. The Monkhorst–Pack k-points meshes were 8 × 8 × 10 and 8 × 10 × 9 for η′-Cu₆Sn₅ and Cu₃Sn, respectively. The two IMCs employed all plane-wave cutoff energy of 550 eV. The self-consistent field tolerance was set as 5.0 × 10⁻⁷ eV/atom. Once the geometric optimizations were completed, the lattice constants and the atom coordinates of the unit cells were completely relaxed, and the forces on atoms were less than 0.01 eV/Å.

3. Results and Discussion

3.1. Structural Properties

Following geometric optimization, the structural information of η′-Cu₆Sn₅ and Cu₃Sn was summarized (see

Table 1. The lattice constants of η′-Cu₆Sn₅ and Cu₃Sn.

Structure	Method	Lattice Constants
		a0 (Å)	b0 (Å)	c0 (Å)	V0 (Å3)
η′-Cu6Sn5	GGA	11.160	7.445	9.901	813.466
	Expt. [31]	11.022	7.282	9.827	779.368
	GGA [32]	11.370	7.510	10.020	846.019
Cu3Sn	GGA	5.391	4.278	4.795	110.586
	Expt. [15]	5.490	4.320	4.740	112.418
	GGA [33]	5.537	4.344	4.781	114.996

). The calculated results, obtained using the GGA, showed good agreement with the experimental value. For η′-Cu₆Sn₅ and Cu₃Sn, the lattice constant errors were lower than 2.2% and 1.8%, respectively; the volume errors of unit cells were 4.4% and 1.6%, respectively. Based on the error analysis, the results of geometry optimization were reasonable.

3.2. Elastic Properties

3.2.1. The Elastic Constants of Single Crystals

Due to the presence of anisotropy, elastic constants C_ij (the elastic stiffness matrix) and the elastic compliance matrix S_ij were important for describing the relationships between stress and strain in a single crystal. Based on the crystal symmetry, η′-Cu₆Sn₅ and Cu₃Sn had 13 and 9 independent elastic constants, respectively. The (C_ij) is an inverse matrix of, (S_ij), i.e., (C_ij) = (S_ij)⁻¹. In this paper, elastic constants C_ij and S_ij of η′-Cu₆Sn₅ and Cu₃Sn were calculated as shown in

Table 2. The C_ij of η′-Cu₆Sn₅ and Cu₃Sn (GPa).

Structure	C11	C22	C33	C44	C55	C66	C12	C13	C15	C23	C25	C35	C46	Note
η′-Cu6Sn5	136.2	164.5	146.3	38.2	44.1	44.2	65.8	58.3	−10.6	63.0	0.4	−3.0	1.1	Present
	156.4	165.2	155.8	42.3	51.9	48.0	62.2	69.4	-	60.6	-	-	-	Theo. [32]
Cu3Sn	213.6	218.6	185.0	54.2	52.3	63.1	96.8	99.4	-	108.1	-	-	-	Present
	154.6	173.7	148.2	50.2	44.2	55.0	78.9	76.5	-	95.1	-	-	-	Theo. [35]
	207.0	226.0	194.0	58.0	47.0	57.0	93.0	94.0	-	94.0	-	-	-	Theo. [15]

and

Table 3. The S_ij of η′-Cu₆Sn₅ and Cu₃Sn (×10⁻³, GPa⁻¹).

Structure	S11	S22	S33	S44	S55	S66	S12	S13	S15	S23	S25	S35	S46
η′-Cu6Sn5	10.13	8.19	8.92	26.20	23.20	22.63	−3.03	−2.69	2.27	−2.34	−0.95	−0.02	−0.66
Cu3Sn	6.64	6.85	8.62	18.45	19.13	15.85	−1.65	−2.60	-	−3.11	-	-	-

. The calculated C_ij was in good agreement with previously calculated values and demonstrated the reasonability of the results in the present work. To ensure stable η′-Cu₆Sn₅ and Cu₃Sn crystal structures, C_ij should comply with the corresponding mechanical stability criteria given by Equations (1) and (2) [34], respectively. By substituting the data of Table 2 into Equations (1) and (2), the above two crystal structures were shown to be mechanically stable.

$$\begin{array}{c}{C}_{11}>0,C_{22}>0,C_{33}>0,C_{44}>0,C_{55}>0,C_{66}>0,[C_{11}+C_{22}+C_{33}+2(C_{12}+C_{13}+C_{23})]>0,\\ (C_{33}{C}_{55}-C_{35}^2)>0,(C_{44}{C}_{66}-C_{46}^2)>0,(C_{22}+C_{33}-2C_{23})>0,\\ [C_{22}(C_{33}{C}_{55}-C_{35}^2)+2C_{23}{C}_{25}{C}_{35}-C_{23}^2{C}_{55}-C_{25}^2{C}_{33}]>0,\\ \{2[C_{15}{C}_{25}(C_{33}{C}_{12}-C_{13}{C}_{23})+C_{15}{C}_{35}(C_{22}{C}_{13}-C_{12}{C}_{23})+C_{25}{C}_{35}(C_{11}{C}_{23}-C_{12}{C}_{13})]\\ -[C_{15}^2(C_{22}{C}_{33}-C_{23}^2)+C_{25}^2(C_{11}{C}_{33}-C_{13}^2)+C_{35}^2(C_{11}{C}_{22}-C_{12}^2)]+C_{55}g\}>0\\ g=C_{11}{C}_{22}{C}_{33}-C_{11}{C}_{23}^2-C_{22}{C}_{13}^2-C_{33}{C}_{12}^2+2C_{12}{C}_{13}{C}_{23}.\end{array}$$

(1)

$$\begin{array}{l}{C}_{11}>0,C_{44}>0,C_{55}>0,C_{66}>0,C_{11}{C}_{22}>C_{12}^2,\\ C_{11}{C}_{22}{C}_{33}+2C_{12}{C}_{13}{C}_{23}-C_{11}{C}_{23}^2-C_{22}{C}_{13}^2-C_{33}{C}_{12}^2>0.\end{array}$$

(2)

3.2.2. Elastic Constants for Polycrystalline Aggregates

Mechanical properties of polycrystalline aggregate, such as B and G could be obtained by Voigt–Reuss–Hill approximation [36,37]. The Voigt, Reuss, and Hill approximations obtain the maximum, minimum, and average value of B and G, respectively. Voigt bulk modulus B_V, Voigt shear modulus G_V, Reuss bulk modulus B_R, and Reuss shear modulus G_R can be calculated by Equations (3)–(6) [38].

$$B_{\mathrm{V}}=\frac{1}{9}(C_{11}+C_{22}+C_{33})+\frac{2}{9}(C_{12}+C_{13}+C_{23})$$

(3)

$$G_{\mathrm{V}}=\frac{1}{15}(C_{11}+C_{22}+C_{33}-C_{12}-C_{13}-C_{23})+\frac{1}{5}(C_{44}+C_{55}+C_{66})$$

(4)

$$B_{\mathrm{R}}={[(S_{11}+S_{22}+S_{33})+2(S_{12}+S_{13}+S_{23})]}^{-1}$$

(5)

$$G_{\mathrm{R}}=15{[4(S_{11}+S_{22}+S_{33})-4(S_{12}+S_{13}+S_{23})+3(S_{44}+S_{55}+S_{66})]}^{-1}$$

(6)

According to empirical formulas proposed by Hill [39], the B and G of polycrystalline aggregates satisfy Equations (7) and (8), respectively.

$$B=(B_{\mathrm{V}}+B_{\mathrm{R}})/2$$

(7)

$$G=(G_{\mathrm{V}}+G_{\mathrm{R}})/2.$$

(8)

The E and ν of polycrystalline aggregates are defined as:

$$E=9BG/(3B+G)$$

(9)

$$\nu =(3B-2G)/(6B+2G).$$

(10)

The theoretical results of E, G, B, and ν for η′-Cu₆Sn₅ and Cu₃Sn are summarized in

Table 4. The E (GPa), G (GPa), B (GPa), ν, and G/B of η′-Cu₆Sn₅ and Cu₃Sn.

Structure	E	GV	GR	G	BV	BR	B	ν	G/B	Note
η′-Cu6Sn5	109.8	42.6	42.0	42.3	91.2	90.0	90.6	0.30	0.47	Present
	94.4	-	-	35.9	-	-	84.4	0.31	0.43	Expt. [14]
	116.7	-	-	45.0	-	-	95.5	-	0.47	Theo. [32]
	107.9	-	-	42.4	-	-	79.4	0.27	0.53	Theo. [20]
Cu3Sn	143.9	54.8	53.9	54.4	136.2	135.6	135.9	0.32	0.40	Present
	123.2	-	-	46.7	-	-	113.8	0.32	0.41	Expt. [14]
	143.0	-	-	-	-	-	-	-	-	Expt. [10,40]
	147.0	-	-	56.0	-	-	132.0	0.32	0.42	Theo. [15,41]

. The theoretical values and the reported experimental results showed excellent consistency for Cu₃Sn. Although reports on experimental results are rare for η′-Cu₆Sn₅, the present work is supported by previously computed values (see Table 4). Therefore, in this paper, the results calculated using the first-principles method is reasonable.

Generally, G/B and ν [19] are employed to evaluate the brittleness of materials. If G/B > 0.57 and ν < 0.26, the materials will be brittle, or, alternatively, will show ductility. Table 4 shows that η′-Cu₆Sn₅ and Cu₃Sn all conformed to the following criterion: G/B < 0.57, ν > 0.26; as such, they were ductile materials.

3.3. Elastic Anisotropy

Elastic anisotropy is related to the generation of microcracks; it also causes poor consistency in the results of tests reviewing the mechanical properties of materials. Therefore, the investigation of elastic anisotropy related to η′-Cu₆Sn₅ and Cu₃Sn is favorable for better understanding the failure behaviors in these materials and, accordingly, improving their mechanical durability. In this paper, three anisotropic indexes were applied to assess the elastic anisotropy of η′-Cu₆Sn₅ and Cu₃Sn, i.e., universal anisotropic index A^U, as well as the anisotropic percentage in the compression (A_comp) and shear (A_shear) modes [36]. The three anisotropic indexes can be expressed as follows:

$$A^{\mathrm{U}}=5G_{\mathrm{V}}/G_{\mathrm{R}}+B_{\mathrm{V}}/B_{\mathrm{R}}-6$$

(11)

$$A\mathrm{comp}=(B_{\mathrm{V}}-B_{\mathrm{R}})/(B_{\mathrm{V}}+B_{\mathrm{R}})$$

(12)

$$A_{\mathrm{shear}}=(G_{\mathrm{V}}-G_{\mathrm{R}})/(G_{\mathrm{V}}+G_{\mathrm{R}})$$

(13)

Furthermore, shear anisotropy in different crystal planes can be evaluated by shear anisotropy factors, i.e., A₁, A₂, and A₃ denote the anisotropic degree in the (100), (010), and (001) crystal planes, respectively [33]. The three shear anisotropy factors for the monoclinic, and orthorhombic structures can be defined as follows [33,42]:

$$A_1=4C_{44}/(C_{11}+C_{33}-2C_{13})$$

(14)

$$A_2=4C_{55}/(C_{22}+C_{33}-2C_{23})$$

(15)

$$A_3=4C_{66}/(C_{11}+C_{22}-2C_{12})$$

(16)

Generally, A^U = A_comp = A_shear = 0, and A₁ = A₂ = A₃ = 1, which represent isotropy; otherwise, A^U ≠ 0, A_comp ≠ 0, A_shear ≠ 0, A₁ ≠ 1, A₂ ≠ 1, and A₃ ≠ 1, which represent anisotropy. For A^U, A_comp, and A_shear, the deviation between 0 and actual values was used to describe the anisotropic degree. For A₁, A₂, and A₃, the deviation of from 1 represent the anisotropic degree. The calculated A^U, A_comp, A_shear, A₁, A₂, and A₃ of η′-Cu₆Sn₅ and Cu₃Sn are summarized in

Table 5. The anisotropic indexes of η′-Cu₆Sn₅ and Cu₃Sn.

Structure	AU	Acomp	Ashear	A1	A2	A3
η′-Cu6Sn5	0.0848	0.0066	0.0071	0.9210	0.9545	1.0455
Cu3Sn	0.0838	0.0022	0.0079	1.0847	1.1164	1.0574

. The values of A^U were 0.0848 and 0.0838 for η′-Cu₆Sn₅ and Cu₃Sn, respectively, which showed that the elastic anisotropy of η′-Cu₆Sn₅ was higher. Meanwhile, the A_comp of Cu₃Sn were all close to 0; this indicated modest compression anisotropy. It is worth noting that the A_shear of η′-Cu₆Sn₅ and Cu₃Sn were 0.0071 and 0.0079, respectively, and the shear anisotropy of η′-Cu₆Sn₅ was lower. Moreover, according to the calculated values of A₁, A₂, and A₃, the shear anisotropy of η′-Cu₆Sn₅ was low and similar in three crystal planes and was the highest and lowest in the (010) and (001) planes for Cu₃Sn, respectively.

To reveal the elastic anisotropy more concretely, the directional dependences of E, B, and G were obtained, and the results are exhibited as three-dimensional (3D) surfaces in Cartesian coordinates. Furthermore, the deviation between the 3D surfaces and the sphere shape denotes the degree of anisotropy. The 3D surfaces of E can be obtained by Equation (17) [43] as follows:

$$\begin{array}{ll}1/E(\mathbf{n})& =S_{11}{n}_1^4+S_{22}{n}_2^4+S_{33}{n}_3^4+(S_{44}+2S_{23})n_2^2{n}_3^2+(S_{55}+2S_{31})n_3^2{n}_1^2+(S_{66}+2S_{12})n_1^2{n}_2^2\\ & +2n_2{n}_3[(S_{14}+S_{56})n_1^2+S_{24}{n}_2^2+S_{34}{n}_3^2]+2n_3{n}_1[S_{15}{n}_1^2+(S_{25}+S_{46})n_2^2+S_{35}{n}_3^2]\\ & +2n_1{n}_2[S_{16}{n}_1^2+S_{26}{n}_2^2+(S_{36}+S_{45})n_3^2]\end{array}$$

(17)

where n₁, n₂, and n₃ denote the directional cosines of loading-direction n with three principal directions. For the two IMCs, the 3D surfaces and their cross-sections of E are illustrated in Figure 2 and Figure 3, respectively. In this study, anisotropy ratio E_max/E_min was employed to quantify the amount of anisotropy; the larger the anisotropy ratio, the higher the anisotropy [44]. For η′-Cu₆Sn₅, E_max = 124.6 GPa, E_min = 89.5 GPa, and the anisotropy ratio was 1.39. The directions of E_max and E_min were (1, 0, −0.76) and (1, 0, 0.42), respectively. The anisotropic ratios, maximum and minimum values of E is listed

Table 6. The anisotropic ratios, maximum and minimum values of E for η′-Cu₆Sn₅ and Cu₃Sn.

	η′-Cu6Sn5				Cu3Sn
	Whole	yz	xz	xy	Whole	yz	xz	xy
Emax (GPa)	124.6	122.1	124.6	122.1	153.9	149.7	150.6	177.8
Emin (GPa)	89.5	103.2	89.5	98.7	116.0	116.0	116.0	125.1
Anisotropic ratios	1.39	1.18	1.39	1.24	1.33	1.29	1.30	1.42

for η′-Cu₆Sn₅ and Cu₃Sn. Obviously, E anisotropy in the xz plane was the highest for η′-Cu₆Sn₅. For Cu₃Sn, the results showed that the 3D plot of E was non-spherical (see Figure 2b); moreover, E_max and E_min were 153.9 GPa and 116.0 GPa, respectively. The anisotropy ratio was 1.33, and the directions of E_max and E_min were (1, 0.84, 0) and (0, 0, 1), respectively. It is noted that E anisotropy of the yz and xz planes was similar for Cu₃Sn. Overall, the anisotropy ratio of η′-Cu₆Sn₅ was higher, which showed that E anisotropy of η′-Cu₆Sn₅ was higher. This conclusion is also supported by the A^U values shown in Table 5. Furthermore, the 3D surfaces of G could be obtained by the following formula [43]:

$$\begin{array}{cc}1/G(\mathbf{n},\mathbf{m})\hfill & =4[2S_{12}-(S_{11}+S_{22}-S_{66})]n_1{m}_1{n}_2{m}_2+4[2S_{23}-(S_{22}+S_{33}-S_{44})]n_2{m}_2{n}_3{m}_3\hfill \\ & +4[2S_{31}-(S_{33}+S_{11}-S_{55})]n_3{m}_3{n}_1{m}_1+4(n_1{m}_2+n_2{m}_1)[(S_{16}-S_{36})n_1{m}_1+(S_{26}-S_{36})n_2{m}_2]\hfill \\ & +4(n_2{m}_3+n_3{m}_2)[(S_{24}-S_{14})n_2{m}_2+(S_{34}-S_{14})n_3{m}_3]\hfill \\ & +4(n_3{m}_1+n_1{m}_3)[(S_{35}-S_{25})n_3{m}_3+(S_{15}-S_{25})n_1{m}_1]\hfill \\ & +S_{44}{(n_2{m}_3-n_3{m}_2)}^2+S_{55}{(n_3{m}_1-n_1{m}_3)}^2+S_{66}{(n_1{m}_2-n_2{m}_1)}^2\hfill \\ & +2S_{45}(n_2{m}_3+n_3{m}_2)(n_3{m}_1+n_1{m}_3)\hfill \\ & +2S_{56}(n_3{m}_1+n_1{m}_3)(n_1{m}_2+n_2{m}_1)+2S_{64}(n_1{m}_2+n_2{m}_1)(n_2{m}_3+n_3{m}_2)\hfill \end{array}$$

(18)

where m₁, m₂, and m₃ are the directional cosines of measurement direction m, and m is perpendicular to n. In a particular direction n, G will be changed with a change in m. For the two IMCs, the directional dependences of maximum and minimum G are plotted in Figure 4. In addition, the cross-sections of maximum and minimum G in the yz, xz, and xy planes are shown in Figure 5. In this study, anisotropy ratio G_max/G_min was employed to describe the anisotropic degree of G. For η′-Cu₆Sn₅, the directions of G_max = 48.3 GPa and G_min = 36.6 GPa were (0.71, 1, −0.71) and (−0.67, 0.60, −1), respectively; hence, the anisotropy ratio was 1.32. The anisotropic ratios, maximum and minimum values of G is listed

Table 7. The anisotropic ratios, maximum and minimum values of G for η′-Cu₆Sn₅ and Cu₃Sn.

	η′-Cu6Sn5				Cu3Sn
	Whole	yz	xz	xy	Whole	yz	xz	xy
Gmax (GPa)	48.3	46.6	46.6	47.0	63.1	63.1	63.1	63.1
Gmin (GPa)	36.6	38.0	38.0	37.2	46.1	48.9	48.9	52.3
Anisotropic ratios	1.32	1.23	1.23	1.26	1.37	1.29	1.29	1.21

for η′-Cu₆Sn₅ and Cu₃Sn. The anisotropic degree of η′-Cu₆Sn₅ was similar in the yz, xz, and xy planes. For Cu₃Sn, G_max = 63.1 GPa, G_min = 46.1 GPa, and the G_max/G_min was 1.37 for Cu₃Sn. The directions of G_max and G_min were (0, 1, 0) and (0, 1, 1), respectively. Obviously, G anisotropy of the yz, xz, and xy planes was similar for Cu₃Sn.

In summary, for the two IMCs, G anisotropy of Cu₃Sn was higher. This conclusion is also supported by the A_shear values shown in Table 5. Finally, the directional dependences of B are described by formula (19) [45].

$$\begin{array}{cc}1/B(\mathbf{n})\hfill & =(S_{11}+S_{12}+S_{13})n_1^2+(S_{16}+S_{26}+S_{36})n_1{n}_2+(S_{15}+S_{25}+S_{35})n_3{n}_1\hfill \\ & +(S_{12}+S_{22}+S_{23})n_2^2+(S_{14}+S_{24}+S_{34})n_2{n}_3\hfill \\ & +(S_{13}+S_{23}+S_{33})n_3^2\hfill \end{array}$$

(19)

The directional dependence of B is shown in Figure 6. The cross-sections of B in the yz, xz, and xy planes are shown in Figure 7. Anisotropy ratio B_max/B_min was performed to describe the anisotropic degree of B. For η′-Cu₆Sn₅, B_max = 354.2 GPa was located on the yz and xy planes, and the B_min = 206.1 GPa was located on the xz planes. The anisotropy ratio was 1.72, and the directions of B_max and B_min were (0, 1, 0) and (1, 0, 0.67), respectively. The anisotropic ratios, maximum and minimum values of B is listed

Table 8. The anisotropic ratios, maximum and minimum values of B for η′-Cu₆Sn₅ and Cu₃Sn.

	η′-Cu6Sn5				Cu3Sn
	Whole	yz	xz	xy	Whole	yz	xz	xy
Bmax (GPa)	354.2	354.2	289.9	354.2	480.7	480.7	419.0	480.7
Bmin (GPa)	206.1	257.3	206.1	226.4	343.9	343.9	343.9	418.9
Anisotropic ratios	1.72	1.38	1.41	1.56	1.40	1.40	1.22	1.15

for η′-Cu₆Sn₅ and Cu₃Sn. The B anisotropy of η′-Cu₆Sn₅ was found to be the highest in the xy plane. For Cu₃Sn, B_max = 480.7 GPa, B_min = 343.9 GPa, and the anisotropy ratio was 1.40. The directions of B_max and B_min were (0, 1, 0) and (0, 0, 1), respectively. The anisotropic degree of B was highest in the yz plane of Cu₃Sn. It is noted that B anisotropy of η′-Cu₆Sn₅ was higher than Cu₃Sn, which can be confirmed by A_comp (see Table 5).

4. Conclusions

Clarifying the mechanical properties of IMCs is helpful to provide basic material properties for finite element analysis and improve the reliability of microscale solder joints. In this work, the mechanical properties and elastic anisotropies of η′-Cu₆Sn₅ and Cu₃Sn were investigated using the first-principles calculations method. The conclusions of the present work are as follows.

• For η′-Cu₆Sn₅ and Cu₃Sn, the elastic constants of single crystals and polycrystalline aggregates were obtained using first-principles calculations. Accordingly, E, G, B, and ν of the two IMCs all exhibited the following relationship: Cu₃Sn > η′-Cu₆Sn₅. Moreover, the values of G/B and ν showed that the two IMCs were ductile materials.
• The directional dependence analysis of E had anisotropy ratios of 1.39 and 1.33 for monocrystalline η′-Cu₆Sn₅ and Cu₃Sn, respectively. The E anisotropy reflected the following relationship: η′-Cu₆Sn₅ > Cu₃Sn.
• The two monocrystalline IMCs exhibited evident anisotropy of G; the anisotropy ratios were 1.32 and 1.37 for η′-Cu₆Sn₅, and Cu₃Sn, respectively. Accordingly, G anisotropy abided by the following relationship: Cu₃Sn > η′-Cu₆Sn₅. In addition, the anisotropic degree of E and G was similar for η′-Cu₆Sn₅ and Cu₃Sn.
• For B of the two monocrystalline IMCs, the anisotropy ratio of η′-Cu₆Sn₅ was higher, reaching 1.72. For Cu₃Sn, the anisotropy ratios were 1.40. The relationship for B anisotropy was η′-Cu₆Sn₅ > Cu₃Sn. Furthermore, the anisotropy of B was higher compared with G and E for η′-Cu₆Sn₅; however, in the case of Cu₃Sn, the anisotropic degree of B, G, and E was similar.