Structural, mechanical, thermodynamic and electronic properties of AgIn₂ and Ag₃In intermetallic compounds: ab initio investigation

Abstract

The structural, mechanical, thermodynamic and electronic properties of two Ag–In phase crystals, i.e., AgIn₂ and Ag₃In intermetallic compounds (IMCs), are explored using ab initio calculations within the generalized gradient approximation. The optimized lattice constants of AgIn₂ and Ag₃In crystals are first investigated in the study. Next, the elastic constants of the two single crystal structures as well as their associated polycrystalline elastic properties, such as bulk modulus, Young's modulus, shear modulus and Poisson's ratio, are predicted through Voigt–Reuss–Hill approximation. The mechanical characteristics of these two crystals, such as ductile–brittle characteristic and elastic anisotropy, are further assessed by way of the calculation of the Cauchy pressures, Zener anisotropy factor and directional Young's modulus. Additionally, the temperature-dependence of Debye temperature and heat capacity are obtained according to a quasi-harmonic Debye model, and their band structures and density of states profiles are evaluated through analysis of electronic characteristics. The calculation results show that these two IMC crystals are not only an elastically anisotropic, low stiff and very ductile material but also a conductor. The elastic anisotropy, mechanical property, Debye temperature and heat capacity of Ag₃In all surpass those of AgIn₂, and also, Ag₃In tends to be much stiffer than AgIn₂. Furthermore, the heat capacity of these two crystals strictly follows with the well-known T³-law at temperature below the Debye temperature and would reach the Dulong–Petit limit at temperature above the Debye temperature.

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1. Introduction

To meet the trend toward green products in the microelectronic packaging industry,¹ numerous studies have been made to develop lead-free solders.^2,3 Nowadays, a substantial number of novel multi-component Pb-free alloys have been proposed as potential alternative to Pb–Sn solders,^4–6 in which some metal elements such as In, Bi, Sb and Zn, in replacement of Pb, are doped into solders. Among these metal elements, In-based alloy solders, as well as pure indium, may be the most promising candidates due to their outstanding wettability, low melting point, high thermal fatigue durability and high ductility.^7–10 Thus, these extraordinary properties make In-based solders attractive for bonding temperature sensitive devices that need low process temperature and high ductility, such as the bond interconnect between central processing unit chips and Cu heat-sinks, light-emitting diodes and thermal inductive sensors.¹¹

Unfortunately, there are many important technical challenges needed to be solved before they can be successfully implemented as an interconnect material. For instance, the Cu pads coated with Ag film are used as the under bump metallurgy (UBM) of chip and substrate metallization for solder bonding in the microelectronics industry.^12,13 During the solder joint assembly process of solder joints on Ag pad, the Ag surface finish dissolves into the liquid In-based solder, leading to the precipitation of Ag–In phase intermetallic compounds (IMCs), such as AgIn₂ and Ag₃In, at the interfaces between the solder and UBM. The IMCs can effectively prompt a remarkable influence on the structural stiffness and material strengths of solder joints, which are significant to the reliability performance of the microelectronic packaging.¹⁴ In literature, extensive investigations have put their focus on the interfacial formation and evolution of AgIn₂ and Ag₃In IMCs.^11,13–20 It is worth noting that prior to the successful application of the Pb-free solder in the advanced interconnect technology, it is very crucial to thoroughly grasp the physical properties of the IMCs for further comprehension of the solder joint reliability. The main goal of the study attempts to provide a more complete and comparative investigation of the structural, mechanical, thermodynamic and electronic properties of the AgIn₂ and Ag₃In IMCs through ab initio calculations by density functional theory (DFT)^21–23 within the generalized gradient approximation (GGA).²²

2. Computational details

In the study, the ab initio DFT calculations are carried out by means of the Cambridge Serial Total Energy Package code.^24,25 The interactions between electrons and core ions are treated using the ultrasoft pseudopotential.²⁶ The exchange-correlation energy is described within GGA of the Perdew–Burke–Ernzerhof formalism,²² which is dependent on both the electron density and its gradient at each space point. After performing careful tests of convergence, it is found that a cutoff energy of 350 eV is sufficient to reduce the error in the total energy less than 0.1 meV per atom.^27,28 For sampling of the Brillouin zone, the Monkhorst–Pack point meshes²⁹ of 8 × 8 × 10 and 12 × 12 × 12 are employed for AgIn₂ and Ag₃In. A quasi-Newton (variable-metric) minimization method using the Broyden–Fletcher–Goldfarb–Shanno update scheme³⁰ is applied to search the ground state. The system would attain the ground state via self-consistent calculation as the total energy is stable less than 5 × 10⁻⁶ eV per atom, the maximum displacement of atoms is 5 × 10⁻⁴ Å, the maximum ionic Hellmann–Feynman force is less than 0.01 eV Å⁻¹ and the maximum stress below 0.02 GPa.

3. Results and discussion

3.1. Structural properties

The lattice constants and the atomic positions of AgIn₂ and Ag₃In crystals are first optimized according to the reported experimental crystal structures^31–33 through X-ray diffraction analysis, as shown in Fig. 1. It reveals the AgIn₂ and Ag₃In crystals belong to the lattice system of tetragonal and cubic, respectively. The optimized lattice constants, lattice constant ratios (c/a) and equilibrium volume in the ground state are given in Table 1, together with the previous experimental results for comparison. A good agreement can be found between present results and the previous experimental values.^31–33 However, there is a fair discrepancy in the equilibrium volume between them, which could be mainly due to the well-known overestimation problem using GGA. Fig. 2 shows hydrostatic pressure-induced variation of the relative change of the lattice constants (a/a₀ and c/c₀) and unit cell volume (V/V₀), where a₀, c₀ and V₀ denote the lattice constants and volume at ambient pressure. Apparently, the a/a₀, c/c₀ and V/V₀ considerably decease with the increase of hydrostatic pressure, revealing that the hydrostatic pressure would reduce the size of the unit cell of AgIn₂ and Ag₃In crystals.

Fig. 1 Crystal structures of (a) AgIn₂ and (b) Ag₃In. The big spheres in brown are In atoms and the small spheres in blue are Ag atoms.

Table 1 Calculated lattice constants a, b and c, lattice constant ratio c/a, equilibrium volume V, bulk modulus K, shear modulus G, Young's modulus E, Poisson's ratio v and ratio of bulk modulus to shear modulus K/G of AgIn₂ and Ag₃In single crystals

	Lattice constants (Å)	Lattice constant ratio	V (Å^3)	K (GPa)	G (GPa)	E (GPa)	v	K/G
a Ref. 31.b Ref. 32.c Ref. 33.
AgIn2	a = b = 6.99, c = 5.75, (a = b = 6.87, c = 5.60)^a (a = b = 6.88, c = 5.62)^b	c/a = 0.822 (c/a = 0.817)^a	280.95 (266.12)^a (264.30)^b	50.09	23.93	61.93	0.294	2.09
Ag3In	a = b = c = 4.23 (a = b = c = 4.14)^c	—	75.69 (70.96)^c	83.22	31.86	84.76	0.330	2.61

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Fig. 2 Calculated pressure dependence of a/a₀, c/c₀ and V/V₀.

3.2. Elastic and mechanical properties

Elastic constants can generally give a link between the mechanical and dynamical behaviors of crystals and give important information concerning the nature of chemical bonding operating in solids. For the tetragonal AgIn₂ crystal, there are six independent elastic constants, namely, C₁₁, C₃₃, C₄₄, C₆₆, C₁₂ and C₁₃, in the stiffness matrix. On the other hand, for the stiffness matrix of cubic Ag₃In crystal, it comprises three independent elastic constants, including C₁₁, C₄₄ and C₁₂. The independent elastic constants can be obtained by a linear fit to the stress–strain relationship according to the Hooke's law as applying an appropriate infinitesimal strain pattern to generate stresses. It should be noted that the calculated elastic constants for the crystals should satisfy the following mechanical stability criteria.

For tetragonal crystal:

C₁₁ > 0, C₃₃ > 0, C₄₄ > 0, C₆₆ > 0, (1)

C₁₁ − C₁₂ > 0, C₁₁ + C₃₃ − 2C₁₃ > 0 (2)

and

{2(C₁₁ + C₁₂) + C₃₃ + 4C₁₃} > 0. (3)

For cubic crystal:

C₁₁ + 2C₁₂ > 0, C₄₄ > 0, C₁₁ − C₁₂ > 0. (4)

By substituting all these calculated elastic constants into the above equations (i.e., eqn (1)–(4)), it is demonstrated that the present calculations do match all the above criteria, suggesting that these two single crystals are intrinsically stable system. Moreover, the pressure dependence of the elastic constants for AgIn₂ and Ag₃In is assessed, and the results are presented in Fig. 3. Noticeably, there is a quasi-linear dependence between the elastic constants and hydrostatic pressure for the two crystals as the applied hydrostatic pressure is in the range of 0–20 GPa. It can be mainly attributed to the pressure-dependent lattice constants, which would decrease with pressure, as shown in Fig. 2, thereby leading to a greater stiffness. Further by comparing these elastic constants, it is clearly found that C₁₁, C₃₃, C₁₂ and C₁₃ are susceptible to the hydrostatic pressure, while C₄₄ and C₆₆ vary little under the effect of the hydrostatic pressure.

Fig. 3 Elastic constants as function of hydrostatic pressure for AgIn₂ and Ag₃In crystals.

It is generally noted that the angular character of atomic bonding is related to material characteristics, such as brittleness or ductility. Pettifor³⁴ signified that Cauchy pressure can be adopted to capture the nature of the bonding. The materials with a positive Cauchy pressure possess metallic like bonds, thereby holding ductile characteristics. On the other hand, brittle materials would possess a negative Cauchy pressure. For the tetragonal AgIn₂ crystal, the Cauchy pressure can be depicted by two terms, i.e., C₁₂–C₆₆ and C₁₃–C₄₄. On the contrary, there is only one term for the cubic Ag₃In crystal to calculate the Cauchy pressure, denting as C₁₂–C₄₄. The calculated Cauchy pressures for AgIn₂ and Ag₃In single crystals as a function of pressure are shown in Table 2. It is found the Cauchy pressures for AgIn₂ and Ag₃In single crystals are all positive, suggesting that these two crystals would hold a ductile behavior. In addition, an increasing hydrostatic pressure would enhance the Cauchy pressures, which indicates that the ductility of AgIn₂ and Ag₃In single crystals increase with the hydrostatic pressure. By comparing the calculated these Cauchy pressures, one can see that the increasing trend of Cauchy pressure, induced by the hydrostatic pressure, of AgIn₂ is larger than that of Ag₃In.

Table 2 Calculated Zener anisotropic factor A_Z, Cauchy pressure and Vickers hardness H_v of AgIn₂ and Ag₃In crystals

Pressure (GPa)		AZ	C12–C44 (GPa)	C12–C66 (GPa)	C13–C44 (GPa)	Hv (GPa)
0	AgIn2	1.46	—	11.73	10.55	3.76
	Ag3In	2.01	25.68	—	—	5.37
5	AgIn2	1.59	—	33.65	28.07	3.16
	Ag3In	2.33	29.81	—	—	5.10
10	AgIn2	1.68	—	42.07	43.74	2.92
	Ag3In	2.85	41.64	—	—	5.02
15	AgIn2	1.98	—	72.66	68.42	2.56
	Ag3In	3.80	62.69	—	—	4.87
20	AgIn2	2.64	—	104.44	89.43	2.18
	Ag3In	5.47	84.96	—	—	3.58

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The elastic anisotropy of crystals would have a great effect on the physical properties, such as anisotropic plastic deformation, crack behavior and elastic instability. To quantify the degree of anisotropy of a solid, the Zener anisotropy factor (A_Z)^35–37 is usually applied, as shown below.

For tetragonal crystal:

For cubic crystal:

As the Zener anisotropy factors, A_Z, is unity, crystals are considered as an isotropic system; otherwise, they are an elastically anisotropic system. A larger deviation from unity signifies a greater elastic anisotropy. The predicted Zener anisotropy factors for AgIn₂ and Ag₃In single crystals versus pressure are listed in Table 2. It reveals that the Zener anisotropy factor for AgIn₂ and Ag₃In crystals at zero pressure is around 1.46 and 2.01, respectively, which suggests that Ag₃In single crystal is composed of a much higher degree of elastic anisotropy than AgIn₂. Table 2 further shows the Zener anisotropy factor for AgIn₂ and Ag₃In crystals significantly increases with the hydrostatic pressure, implying that the elastic anisotropy of these two single crystals would rapidly rise as the hydrostatic pressure increases from 0 to 20 GPa.

To further evaluate the anisotropic characteristics of AgIn₂ and Ag₃In crystals, a three-dimensional surface representation of the elastic anisotropy of the two single crystals is attempted. The representation demonstrates the variation of the Young's modulus of the crystals with crystal direction. The direction dependence of the Young's modulus of tetragonal and cubic crystal systems can be expressed as,³⁸.

For tetragonal crystal:

E = 1/((l₁⁴ + l₂⁴)s₁₁ + l₃⁴s₃₃ + l₁²l₂²(2s₁₂ + s₆₆) + l₃²(1 − l₃²)(2s₁₃ + s₄₄)). (7)

For cubic crystal:

E = 1/(s₁₁ + 2(s₁₁ − s₁₂ − 0.5s₄₄)(l₁²l₂² + l₂²l₃² + l₃²l₁²)). (8)

where l₁, l₂, and l₃ represent the direction cosines with respect to the a, b and c axis. Using the calculated compliance constants (s_ij), the directional Young's modulus at zero pressure is obtained and shown in Fig. 4. In addition, Fig. 4(a) shows that the geometry of the projected directional Young's modulus for AgIn₂ in the a–b crystal plane is a quasi-square shape while it slightly changes to an ellipse shape in the b–c and a–c crystal planes, indicating that the anisotropic feature in the b–c and a–c crystal planes is much less significant than a–b plane. On the other hand, as shown in Fig. 4(b), the projected directional Young's moduli for Ag₃In in the three planes are identical and present in the four-pointed star shape, which is far away from the circle shape, i.e., Ag₃In holds the highly anisotropic characteristic. These figures further suggest that the degree of elastic anisotropy for Ag₃In single crystal is much more significant than AgIn₂, which is consistent with the calculated Zener anisotropy.

Fig. 4 Direction dependence of Young's modulus and their projections onto the a–b, b–c and a–c planes for (a) AgIn₂ and (b) Ag₃In crystals (unit: GPa).

Generally speaking, the bulk modulus (K) is the measure of resistance to volume change by applied hydrostatic pressure, and shear modulus (G) is the measure of resistance to reversible deformation upon shear stress. Young's modulus (E) reveals the resistance of materials against uniaxial tensions. The Poisson's ratio (v) is normally used to quantify the stability of the crystal against shear deformation, which is in the range of −0.5–0.5, and the larger the Poisson's ratio, the better the plasticity. These polycrystalline mechanical properties are very important for industrial applications, and they can be determined based on the following Voigt–Reuss scheme³⁹ in terms of independent elastic constants,

K_R = [S₁₁ + S₂₂ + S₃₃ + 2(S₁₂ + S₁₃ + S₂₃)]⁻¹, (10)

G_R = 15[4(S₁₁ + S₂₂ + S₃₃) − 4(S₁₂ + S₁₃ + S₂₃) + 3(S₄₄ + S₅₅ + S₆₆)]⁻¹. (12)

The above formulate yield the upper (Voigt) and lower (Reuss) bounds of the polycrystalline elastic properties. Furthermore, the Voigt–Reuss–Hill approximation⁴⁰ is utilized to give the effective values of the bulk and shear modulus,

Then, the Young's modulus and the Poisson's ratio of the AgIn₂ and Ag₃In IMCs can be calculated by using the bulk and shear modulus,

The obtained bulk modulus, shear modulus, Young's modulus and Poisson's ratio for the polycrystalline AgIn₂ and Ag₃In are shown in Table 1. Evidently, it is found that these four results (i.e., K, G, E and v) of Ag₃In are larger than those of AgIn₂. In addition, the hydrostatic pressure dependence of the polycrystalline mechanical properties of the AgIn₂ and Ag₃In IMCs are also examined, as can be seen in Fig. 5. By further comparing the results in the figure, it is found that the pressure has a much larger impact on the bulk modulus, as compared to the shear modulus and Young's modulus. The degree of variation in the bulk modulus with the pressure range of 0–20 GPa can be as much as about 203% and 127% for AgIn₂ and Ag₃In IMCs, respectively.

Fig. 5 The calculated bulk modulus, shear modulus and Young's modulus as a function of hydrostatic pressure of the polycrystalline AgIn₂ and Ag₃In IMCs.

To recognize the brittle and ductile behavior of materials, Pugh⁴¹ has presented a relationship, which empirically identifies the plastic properties of materials by the ratio of the bulk modulus to shear modulus. The critical value for separating ductile and brittle materials is around 1.75. If K/G ratio is over 1.75, the material behaves in a ductile way; otherwise the material behaves in a brittle way. The calculated K/G ratios for AgIn₂ and Ag₃In at zero pressure are exhibited in Table 1. It is found that the K/G ratios are larger than 1.75, revealing the AgIn₂ and Ag₃In IMCs are completely ductile materials. Table 1 further demonstrates that the K/G ratio of Ag₃In is larger than that of AgIn₂, suggesting that the Ag₃In crystal is much more ductile than AgIn₂.

The hardness of a material generally plays an essential role in its applications, especially using as an abrasive resistant phase. Teter⁴² found a linear correlation of Vickers hardness H_v with shear modulus G. Nevertheless, the hardness is not only associated with the shear modulus G, but also the bulk modulus K for many materials. In the quest for a better formula, an amendatory non-linear correlation formula associated with K and G has been proposed by Chen et al.,⁴³

H_v = 2(k²K)^0.585 − 3, k = G/K. (17)

However, Tian et al.⁴⁴ reported that eqn (17) may make the hardness negative for some low hardness materials due to the “−3” term, so they modified it which then can always obtain positive values. In the study, the correction by Tian et al.⁴⁴ is adopted to calculate the Vickers hardness of the AgIn₂ and Ag₃In, as described below,

H_v = 0.92k^1.137G^0.708, k = G/K. (18)

The calculated Vickers hardness H_v associated with these two polycrystalline IMC crystals as a function of hydrostatic pressure are listed in Table 2. It turns out that they are all a low stiff material, and the low Vickers hardness of the two polycrystalline IMC crystals may be attributed to the weak metal bond. The results also reveal that the Vickers hardness of Ag₃In at various pressures are all lager than those of AgIn₂, suggesting that the Ag₃In is much stiffer than AgIn₂. Additionally, the hydrostatic pressure would also have a substantial influence on the Vickers hardness, where the Vickers hardness would decrease with the hydrostatic pressure.

3.3. Thermodynamic properties

In order to investigate thermodynamic properties of AgIn₂ and Ag₃In IMCs, the quasi-harmonic Debye model is applied, in which the phonon effect is considered. This model is described in detail in ref. 45 and 46. In the quasi-harmonic Debye model, the non-equilibrium Gibbs function G*(V;P,T) is taken in the form of,

G*(V;P,T) = E(V) + PV + A_Vib(θ(V);T), (19)

where E(V) is the total energy per unit cell, P corresponds to the constant hydrostatic pressure condition, and A_Vib is the vibrational term, which can be expressed as,^47,48where n is the number of atoms in the molecule, is the Debye integral given as,and θ can be expressed as,⁴⁸where M is the molecular mass per unit cell and K_s is the adiabatic bulk modulus, which is approximated given by,⁴⁷

The f(v) function is given in^48,49 and the Poisson's ratio v is determined through calculated elastic constants. Then, the nonequilibrium Gibbs function G*(V;P,T) can be minimized with respect volume V as follow,

By solving equation, the thermal equation of state V(P,T) can be obtained. The heat capacity is expressed as,⁵⁰

The calculated Debye temperature and heat capacity as function of hydrostatic pressure and temperature of AgIn₂ and Ag₃In are shown in Fig. 6 and 7. The analysis results show that the Debye temperature is nearly independent of the temperature at the temperature range of 0–1000 K. Moreover, under a fixed temperature, the Debye temperature increases with the hydrostatic pressure. As the pressure increases from 0 to 20 GPa, the Debye temperature for AgIn₂ and Ag₃In at 300 K would increase by about 47% and 40%, respectively. At zero pressure and 300 K, the calculated Debye temperature for AgIn₂ and Ag₃In are about 179.8 and 199.9 K. Fig. 7 reveals the temperature dependence of the heat capacity of AgIn₂ and Ag₃In. It is found that the heat capacity at low temperatures shows high-temperature dependence, i.e., the third power of temperature, which is a consequence of the Debye T³ law.⁵¹ However, at temperature above the Debye temperature, the heat capacity tends to reach the well-known Dulong–Petit limit,⁵² i.e., about 12.3 and 24.4 J mol⁻¹ K for AgIn₂ and Ag₃In, which is common to all solids at high temperature or a temperature far above the Debye temperature. Additionally, it can be also observed from the figure that the heat capacity increases with temperature at a constant pressure while decreasing with pressure at a constant temperature. By further looking at Fig. 7, it is clear to see that temperature tends to have a greater impact on the heat capacity of the crystal than pressure within the temperature and pressure range studied.

Fig. 6 Variation of Debye temperature as a function of temperature for (a) AgIn₂ and (b) Ag₃In IMCs.

Fig. 7 Variation of heat capacity C_v as a function of temperature for (a) AgIn₂ and (b) Ag₃In IMCs.

3.4. Electronic properties

The calculated energy band structures of AgIn₂ and Ag₃In IMCs along the high symmetry directions in the first Brillouin zone are shown in Fig. 8. It is evident that the valence and conduction bands overlap with each other and there is no band gap at the Fermi level, indicating that AgIn₂ and Ag₃In IMCs are not insulator or semiconductor. In order to elucidate the major contribution of orbit in the band structure and bonding characteristics, the total density of state (TDOS) and partial density of state (PDOS) of AgIn₂ and Ag₃In IMCs are investigated, as can be seen in Fig. 9. There is no band gap for AgIn₂ and Ag₃In IMCs, and TDOS of AgIn₂ and Ag₃In IMCs at the Fermi level are N(E_F) = 2.583 and 1.326 electrons per eV, respectively. Furthermore, TDOS indicates that there is an energy gap from −13.40 eV to −10.00 eV for AgIn₂ and −13.70 eV to −9.72 eV for Ag₃In. The large peaks in TDOS from −15.62 to −13.40 eV for AgIn₂ and −15.66 to −13.70 eV for Ag₃In mainly drive from In-4d state. The low-wide peaks in TDOS between −9.99 and 11.12 eV for AgIn₂ has contributions substantially from Ag-4d and In-5p states as well as In-5s and Ag-5s states have a little contribution. For Ag₃In, the low-wide peaks in TDOS between −9.70 and 16.63 eV is mainly contributed by the Ag-4d and Ag-4p states, and the Ag-5s, In-5s and In-5p states would have scarcely any contribution to the TDOS. In addition, the PDOS of both AgIn₂ and Ag₃In IMCs indicate that In-5s and In-5p states are in low-lying states, and a strong hybridization can be observed in the energy range between −6 eV to −2 eV due to an overlap with Ag-4d state.

Fig. 8 Calculated band structure of (a) AgIn₂ and (b) Ag₃In IMCs (the Fermi level is set to zero).

Fig. 9 Calculated total and partial density of states (TDOS and PDOS) for (a) AgIn₂ and (b) Ag₃In (the Fermi level is set to zero).

4. Conclusions

In the study, the structural, mechanical, thermodynamics and electronic properties of the AgIn₂ and Ag₃In crystals are investigated for the first time by the first-principles density functional calculation, quasi-harmonic Debye model and the Voigt–Reuss–Hill approximation. Their dependences on temperature, pressure and also crystal direction are also examined. Some brief essential concluding remarks are drawn below.

(1) The calculated lattice constants of both AgIn₂ and Ag₃In crystals are found to be in good agreement with the experimental data, implying the effectiveness of the proposed theoretical models.

(2) These two single crystals are demonstrated to be mechanically stable. Among the calculated independent elastic constants, C₁₁, C₃₃, C₁₂ and C₁₃ are sensitive to the hydrostatic pressure, while C₄₄ and C₆₆ vary little under the effect of the hydrostatic pressure.

(3) The predicted Cauchy pressures show that the AgIn₂ and Ag₃In crystals are categorized as the ductile material. Moreover, the Cauchy pressures would increase with the increase of hydrostatic pressure, and the increasing trend of Cauchy pressure of AgIn₂ is larger than that of Ag₃In.

(4) The Zener anisotropy factor shows that Ag₃In single crystal would hold a much higher degree of elastic anisotropy than AgIn₂. The result is also confirmed by the direction dependence of the Young's modulus.

(5) The mechanical properties, including bulk modulus, shear modulus and Young's modulus, of the polycrystalline AgIn₂ and Ag₃In crystals would also increase with the hydrostatic pressure, and the pressure has a much larger influence on the bulk modulus rather than the shear and Young's modulus. Additionally, according to the calculated the K/G ratio, it is found that the Ag₃In crystal is much more ductile than AgIn₂.

(6) The polycrystalline IMC crystals turn out to be a low stiff material according to the predicted Vickers hardness, mainly due to the weak metal bond. The Vickers hardness would also decrease with an increasing hydrostatic pressure.

(7) The Debye temperature of these two single crystals is independent of the temperature. Under a constant temperature, the Debye temperature would increase almost linearly with hydrostatic pressure. Besides, the Debye temperature of Ag₃In is higher than that of AgIn₂by as much as about 20 K.

(8) The heat capacity of these two crystals strictly follows with the well-known T³-law at temperature below the Debye temperature, where it at a solid at constant volume and pressure is proportional to the cube of temperature, and at temperature above the Debye temperature, tends to reach the Dulong–Petit limit. By contrast, it would decrease with pressure at a constant temperature. Most importantly, the heat capacity of Ag₃In crystal is approximately twice that of AgIn₂.

(9) The evaluated energy band structure shows that the AgIn₂ and Ag₃In are conductors.

(10) The analysis of the PDOS reveals that a strong hybridization can be observed in the energy range between −6 eV to −2 eV due to an overlap with Ag-4d state.

In summary, the investigation can help better explore the material properties of these two IMC crystals, and perhaps even their dependences on the thermal-mechanical reliability of the In-based lead-free solder joints.

Acknowledgements

The work is partially supported by National Science Council, Taiwan, ROC, NSC 101-2221-E-007-009-MY3, MOST 103-2221-E-007-010- and MOST 103-2221-E-035-024-MY3. The authors also thank the National Center for High-Performance Computing (NCHC) for computational resources support.

References

1. F. Su, W. J. Li, T. B. Lan and W. Shang, J. Mech., 2014, 30, 625–630.
2. D. M. Jacobson and G. Humpston, Gold Bull., 1989, 22, 9–18.
3. J. H. Lau, C. P. Wong, N. C. Lee and S. W. Ricky Lee, Electronics Manufacturing with Lead-Free, Halogen-Free Conductive-Adhesive Materials, McGraw-Hill Handbooks, New York, 2003.
4. M. N. Islam, Y. C. Chan, M. J. Rizvi and W. Jillek, J. Alloys Compd., 2005, 400, 136–144.
5. F. Wang, M. O'Keefe and B. Brinkmeyer, J. Alloys Compd., 2009, 477, 267–273.
6. A. A. El-Daly and A. E. Hammand, J. Alloys Compd., 2011, 509, 8554–8560.
7. L. S. Goldmann, R. D. Herdzik, N. G. Koopman and V. C. Marcotte, IEEE Trans. Parts, Hybrids, Packag., 1977, 3, 194–198.
8. K. Shimizu, T. Nakanishi, K. Karasawa, K. Hashimoto and K. Niwa, J. Electron. Mater., 1995, 24, 39–45.
9. W. K. Jones, Y. Liu, M. Shah and R. Clarke, Soldering Surf. Mount Technol., 1998, 10, 37–41.
10. S. Wakiyama, H. Ozaki, Y. Nabe, T. Kume, T. Ezaki and T. Ogawa, in: Electronic Components and Technology Conference, Reno, Nevada, USA, 2007, pp. 610–615.
11. P. J. Wang, J. S. Kim and C. C. Lee, in: Electronic Components and Technology Conference, San Diego, California, USA, 2009, pp. 1317–1321.
12. C. T. Peng, C. M. Liu, J. C. Lin, H. C. Cheng and K. N. Chiang, IEEE Trans. Compon. Packag. Technol., 2004, 27, 684–693.
13. T. H. Chuang, C. C. Jain and S. S. Wang, J. Mater. Eng. Perform., 2009, 18, 1133–1139.
14. F. Tai and F. Guo, Sci. China, Ser. E: Technol. Sci., 2009, 52, 243–251.
15. R. I. Made, C. L. Gan, L. L. Yan, A. Yu, S. W. Yoon, J. H. Lau and C. K. Lee, J. Electron. Mater., 2009, 38, 365–371.
16. C. Kind, R. Popescu, R. Schneider, E. Müller, D. Gerthsen and C. Feldmann, RSC Adv., 2012, 2, 9473–9487.
17. M. A. Haque and W. Gierlotka, International Journal of Mining, Metallurgy and Mechanical Engineering, 2013, 1, 173–177.
18. C. S. Yoo, H. Wei, J. Y. Chen, G. Shen, P. Chow and Y. Xiao, Rev. Sci. Instrum., 2011, 82, 113901.
19. S. Sarkar, L. Balisetty, P. P. Shanbogh and S. C. Peter, J. Catal., 2014, 318, 143–150.
20. N. Arora, B. R. Jagirdar and K. J. Klabunde, J. Alloys Compd., 2014, 610, 35–44.
21. P. Hohenberg and W. Kohn, Phys. Rev., 1964, 136, 864–871.
22. J. P. Perdew, K. Burke and M. Ernzerhof, Phys. Rev. Lett., 1996, 77, 3865–3868.
23. Y. C. Shih, C. S. Chen and K. C. Wu, J. Mech., 2014, 30, 241–246.
24. R. Hao, X. Zhang, J. Qin, S. Zhang, J. Ning, N. Sun, M. Ma and R. Liu, RSC Adv., 2015, 5, 36779–36786.
25. R. Yu, X. Chong, Y. Jiang, R. Zhou, W. Yuan, W. Jiang and J. Feng, RSC Adv., 2015, 5, 1620–1627.
26. D. Vanderbilt, Phys. Rev. B: Condens. Matter Mater. Phys., 1990, 41, 7892–7895.
27. D. W. Boukhvalov and Y. W. Son, Nanoscale, 2012, 4, 417–420.
28. J. D. Tucker, Ab Initio-Based Modeling of Radiation Effects in the Ni–Fe–Cr System, Ph.D. thesis, University of Wisconsin, Madison, US, 2008.
29. H. J. Monkhorst and J. D. Pack, Phys. Rev. B: Solid State, 1976, 13, 5188–5192.
30. J. D. Head and M. C. Zerner, Chem. Phys. Lett., 1985, 122, 264–270.
31. E. Hellner, Z. Metallkd., 1951, 42, 17–19.
32. E. E. Havinga, H. Damsma and P. Hokkeling, J. Less-Common Met., 1972, 27, 169–186.
33. A. N. Campbell, R. Wagemann and R. B. Ferguson, Can. J. Chem., 1970, 48, 1703–1715.
34. D. G. Pettifor, Mater. Sci. Technol., 1992, 8, 345–349.
35. K. Lau and A. K. McCurdy, Phys. Rev. B: Condens. Matter Mater. Phys., 1998, 58, 8980–8984.
36. W. H. Chen, C. F. Yu, K. N. Chiang and H. C. Cheng, Intermetallics, 2015, 62, 60–68.
37. C. F. Yu, H. C. Cheng and W. H. Chen, J. Alloys Compd., 2015, 619, 576–584.
38. A. Authier and A. Zarembowitch, International Tables for Crystallography D, New York, Springer Verlag, 2006, pp. 72–98.
39. J. K. D. Verma and B. D. Nag, J. Phys. Soc. Jpn., 1965, 20, 635–636.
40. R. Hill, Proc. Phys. Soc., London, Sect. A, 1952, 65, 349–350.
41. S. F. Pugh, Philos. Mag., 1954, 45, 823–843.
42. D. M. Teter, MRS Bull., 1998, 23, 22–27.
43. X. Q. Chen, H. Niu, D. Li and Y. Li, Intermetallics, 2011, 19, 1275–1281.
44. Y. J. Tian, B. Xu and Z. S. Zhao, Int. J. Refract. Met. Hard Mater., 2012, 33, 93–106.
45. M. A. Blanco, E. Francisco and V. Luana, Comput. Phys. Commun., 2004, 158, 57–72.
46. E. Francisco, J. M. Recio, M. A. Blanco and A. Martin Pendás, J. Phys. Chem. A, 1998, 102, 1595–1601.
47. M. A. Blanco, A. Martin Pendás, E. Francisco, J. M. Recio and R. Franco, J. Mol. Struct.: THEOCHEM, 1996, 368, 245–255.
48. M. Flórez, J. M. Recio, E. Francisco, M. A. Blanco and A. M. Pendás, Phys. Rev. B: Condens. Matter Mater. Phys., 2002, 66, 144112.
49. E. Francisco, M. A. Blanco and G. Sanjurjo, Phys. Rev. B: Condens. Matter Mater. Phys., 2001, 63, 094107.
50. N. W. Ashcroft and N. D. Mermin, Solid State Physics, Saunders College, Philadelphia, 1976.
51. P. Debye, Ann. Phys., 1912, 39, 789–839.
52. A. T. Petit and P. L. Dulong, Ann. Chim. Phys., 1819, 10, 395–413.

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