Effect of Ag Doping on Mechanical Properties of Cu₆Sn₅ Intermetallic Compounds

Abstract

Cu₆Sn₅-xAg alloys (x = 0, 3, 6; %, mass fraction) were synthesized using Ag as a dopant through a high-temperature melting technique. The microstructure of the alloy was analyzed using X-ray diffraction (XRD), scanning electron microscopy (SEM), and other equipment, while the hardness of the alloy was measured to investigate the impact of Ag addition on the structure and microstructure of the Cu₆Sn₅ intermetallic compound. This study explored the influence of varying Ag contents on the properties of Cu₆Sn₅ intermetallic compounds, with calculations based on first principles revealing the mechanical properties and density of states of η′-Cu₆Sn₅ and its Ag-doped systems. The results indicated that Cu₆Sn₅-xAg alloys predominantly existed in three distinct forms, all exhibiting large masses without any impurities or precipitates. First-principle calculations demonstrated that Ag substitution in certain sites suppressed the anisotropy of the Young’s modulus of Cu₆Sn₅, particularly in the Cu1, Cu3, Sn1, and Sn3 positions, while the effect was less significant at the Cu2, Cu4, and Sn2 sites. The introduction of Ag through doping enhanced the covalent bonding within the η′-Cu₆Sn₅ structure, promoting the formation of a stable (Cu, Ag)₆Sn₅ structure.

1. Introduction

The electronics industry is rapidly evolving towards miniaturization and multifunctionality, driving electronic packaging technology to advance in high-density, functionality, and integration aspects [1]. Lead-free solder has emerged as the primary material in electronic packaging due to its excellent electrical conductivity, mechanical properties, and cost-effectiveness [2]. Reflow soldering is predominantly used for connecting electronic components [3]. During the soldering process, a chemical reaction between Sn and the Cu substrate forms an intermetallic compound (IMC) at the solder joint interface. The primary component of the IMC is Cu₆Sn₅, with a melting point of 415 °C and strong thermodynamic and kinetic stability [4,5]. Cu₆Sn₅ exhibits two crystalline structures: a monoclinic crystal structure (η′ phase) below 186 °C and a hexagonal crystal system structure (η-phase) above 186 °C [6]. The phase transformation results in a 2.15% volume change, increasing compressive stress within the solder joint, potentially leading to crack formation or propagation and compromising solder joint reliability [7,8,9].

The mechanical properties of Cu₆Sn₅ alloys have been extensively studied both domestically and internationally. Through nanoindentation experiments, researchers have measured the Young’s modulus (E) and hardness (H) of Cu₆Sn₅, revealing significant anisotropy in both properties when compared to (Cu, Ni)₆Sn₅ [10]. Choudhury [11] et al. further delved into the anisotropic mechanical properties of single-grain Cu₆Sn₅ using nanoindentation and electron backscatter diffraction. However, the experimental methods used to gather information on these anisotropic properties are limited due to challenges in obtaining data from all directions. In microelectronic soldering processes, the Cu₆Sn₅ alloy, a common intermetallic compound, tends to form small interfacial compounds in lead-free solder joint experiments, leading to inaccuracies in measuring its mechanical properties. Doping with trace elements like Ni [12], Co, In [13], Zn, Au, and Pd has been shown to inhibit the phase transition of η-Cu₆Sn₅, thereby enhancing its mechanical properties. Studies have demonstrated that the addition of Zn, Au, and In can stabilize the high-temperature η-Cu₆Sn₅ phase within a specific temperature range, with these atoms occupying specific Cu or Sn sites to create a more thermodynamically stable phase. It was observed that Au and In increased the cell volume of Cu₆Sn₅, while Zn caused a reduction, leading to a balance in strain energy that prevented phase transition [14].

Zhou et al. [15] investigated the impact of Ni element doping on the mechanical and electronic characteristics of high-temperature η-Cu₆Sn₅ phases. Their findings revealed that the inclusion of Ni enhanced the mechanical stability, brittleness, modulus, and Debye temperature of high-temperature η-Cu₆Sn₅. It is crucial to ensure the physical and thermal stability of the interfacial layer in solder joints, typically dominated by Cu₆Sn₅ intermetallic compounds, due to the thermal cycling and shock experienced by electronic devices [8,16,17,18,19]. The crystal structures of Cu₆Sn₅ can vary, with η′-Cu₆Sn₅ being the predominant phase in solder joint service. This phase significantly influences the mechanical behavior of microscale solder joints. Elastic anisotropy is linked to microcrack formation, underscoring the importance of a systematic analysis of the mechanical properties and elastic anisotropy of η′-Cu₆Sn₅ to enhance solder joint reliability.

Numerous studies have been conducted on the impact of Ag addition on solder joint reliability and interfacial reactions. While experimental determination of the crystal structure’s anisotropy is relatively consistent, investigating the effects of Ag substitution on properties of Cu₆Sn₅ by simulating the anisotropic distribution of properties like elastic modulus and hardness through mechanical calculations can enhance understanding of the crystal structure’s various mechanical properties. This paper calculates the mechanical properties and density of states of Ag-doped Cu₆Sn₅ intermetallic compounds based on first principles and explores the atomic-scale effect of Ag atomic doping on the anisotropy of Cu₆Sn₅ intermetallic compounds. The theoretical explanation of the enhancement effect resulting from Ag doping is provided, along with the rationale behind the formation of enhancers. These calculations are crucial for studying the diffusion paths of Cu and Sn atoms in η′-Cu₆Sn₅ post-doping, and for designing subsequent polyalloys. The findings of this study are anticipated to be beneficial for the future design and development of Sn-Cu solders.

2. Materials and Methods

Ag powder obtained from Henan Lebleu Metal Materials Technology Co. (Henan, Lebleu, Zhengzhou, China). was utilized as the doping agent. This Ag powder was blended with highpurity Sn powder and Cu powder sourced from Hebei Shengte Metal Materials Co. (Hebei, Shengte, Shijiazhuang, China), both with purities exceeding 99.5%. The Cu-Sn alloy phase diagram indicates that when the mass percentage of Sn is 61 wt% and the mass percentage of Cu is 39 wt%, the Cu₆Sn₅ alloy is formed upon melting. The alloys with varying Ag contents were labeled as S0, S1, and S2 for clarity, and the compositions of the mixed metal powders can be found in

Table 1. Composition of Cu₆Sn₅-xAg alloys.

No.	Composite	Content, ω/%
		Sn	Cu	Ag
S0	Cu6Sn5-0Ag	61	39	0
S1	Cu6Sn5-3Ag	59.17	37.83	3
S2	Cu6Sn5-6Ag	57.34	36.66	6

. To enhance the homogeneity of the mixed powders, ball milling was conducted. A PU-80 high energy ball mill (Cunqiu, Guangzhou, China) was employed with a ball milling duration of 4 h, a rotational speed of 800 r/min, and a mass ratio of large balls, Φ10 mm, to small balls, Φ6 mm, of 1:4. Subsequently, the mixed powders were filled into custom-made moulds, pressed into Φ15 mm × 30 mm press blanks using a WE-30 universal testing machine (Tuopu, Tianjin, China) with a pressing pressure of 100 kN, a loading speed of 2 kN/s, and a duration of 10 min. The press blank was then placed in a crucible and subjected to melting treatment in a muffle furnace at 900 °C for 4 h. Upon completion of melting, the crucible was cooled to room temperature before being removed. A graphite crucible was secured in a fixture and the molten material was poured into a graphite mould for casting, producing a sample.

The physical structure of the alloys was examined and analyzed by X-ray diffraction (XRD, D/max-RB, Rigaku, Tokyo, Japan), with a Cu target as the X-ray source, a scanning angle from 10° to 90°, and a scanning speed of 10°/min. A scanning electron microscope (SEM, Zeiss, Oberkochen, Germany) was used for the observation of the microstructures in conjunction with the energy spectrometer (EDS) to analyze the elemental distribution of the alloy. The hardness of the alloys was measured by a Vickers hardness tester, model AHVD-1000 (Miange, Suzhou, China), with a load of 200 g and a retention time of 20 s. Ten different positions were selected for each sample.

The crystal structure models of η′-Cu₆Sn₅ and η-Cu₆Sn₅ are depicted in Figure 1. A dopant atom Ag was introduced to replace Cu atoms, forming the (Cu, Ag)₆Sn₅ structural model. These structures underwent geometry optimization using a first-principles approach based on density-functional theory with the CASTEP procedure to calculate elastic constants [20,21,22]. The elastic properties of Cu positions in Ag-substituted Cu₆Sn₅ were then determined. An OTFG ultrasoft pseudopotential was utilized to describe ionic nuclei and valence electron interactions, with the exchange correlation equation following GGA-PBE [23]. Parameters such as k-point (7 × 7 × 2), cut-off energy (408.2 eV), and internal structure calculations using the BFGS algorithm were specified. Convergence criteria included a maximum ion Hellmann–Feynman force of 0.03 eV/Å, total energy tolerance of 5 × 10⁻⁵ eV/atom, maximum ion displacement of 0.001Å, and maximum stress tolerance of 0.05 GPa.

3. Results and Discussion

3.1. Effect of Ag Content on the Microstructure of Alloys

Figure 2 displays the X-ray diffraction (XRD) spectra of Cu₆Sn₅-Ag alloy samples with varying Ag powder content (0%, 3%, and 6%). The physical phase characterization of the Cu₆Sn₅-xAg (x = 0%, 3%, 6%) alloy was conducted, as depicted in Figure 2. The 2θ values of the main peaks in the Cu₆Sn₅-Ag alloy were found to be 30.093° and 42.973°. The lattice constants of Cu₆Sn₅ in the Cu₆Sn₅-xAg alloy were determined using Jade9.0 software and are presented in

Table 2. Effect of Ag addition on the lattice constant of Cu₆Sn₅.

	a (Å)	b (Å)	c (Å)
η′-Cu6Sn5	10.994	7.267	9.862
η′-Cu6Sn5 (GGA)	11.039	7.305	9.903
Doped with 3 wt% Ag	11.036	7.273	9.751
Doped with 6 wt% Ag	11.006	7.284	9.687

. The experimentally obtained XRD data were fitted to the full peaks of the physical phases in Jade software and the magnitude of the lattice constant was calculated. Following geometry optimization, the structural information of η′-Cu₆Sn₅ was compiled. The calculated results, obtained using Generalized Gradient Approximation (GGA), closely matched the experimental values, with lattice constant errors falling within acceptable limits. Error analysis confirmed the reasonableness of the optimization results. Table 2 illustrates that upon Ag atom doping, the lattice constants of η′-Cu₆Sn₅ increased in the a and b directions and decreased in the c direction, resulting in a more uniform volume and reduced anisotropy of the η′-Cu₆Sn₅ structure.

Figure 3 displays the SEM images of Cu₆Sn₅-Ag alloys containing 0%, 3%, and 6% Ag. The micro-morphology of the alloy exhibited three distinct compositions: white zone, light grey zone, and grey zone. Voids appeared on the surface of the alloy, which gradually decreased with the increase in Ag addition. The SEM image reveals the presence of three different substances, with intermetallic compounds appearing as relatively large lumps and no other impurities or precipitates.

The distribution of elements was analyzed using energy dispersive X-ray spectroscopy (EDS) point scans attached to the scanning electron microscope (SEM). As shown in Figure 4a and Figure 4b for SEM image of Cu₆Sn₅-3%Ag alloy and Cu₆Sn₅-6%Ag alloy respectively. The analysis revealed the main components to be elemental Sn, Cu₃Sn intermetallic compounds, Cu₆Sn₅ intermetallic compounds, and Ag₃Sn intermetallic compounds, as depicted in Figure 4.

Table 3. Cu₆Sn₅-3%Ag alloy spot scan elemental results (at%).

Elemental	1	2	3	4
Sn	25.99	45.19	23.69	94.82
Cu	73.55	54.59	2.300	5.14
Ag	0.44	0.21	74.00	0.04
Phase name	Cu3Sn	Cu6Sn5	Ag3Sn	Sn

and

Table 4. Cu₆Sn₅-6%Ag alloy spot scan elemental results (at%).

Elemental	1	2	3	4
Sn	81.19	24.40	42.60	24.78
Cu	1.140	3.090	56.83	74.19
Ag	17.67	72.51	0.57	1.03
Phase name	Sn	Ag3Sn	Cu6Sn5	Cu3Sn

present the results of the EDS point scan specifically for the Cu₆Sn₅-Ag alloy. 1–4 in Table 3 corresponds to 1–4 in Figure 4a, indicating the EDS results for each point, 1–4 in Table 4 corresponds to 1–4 in Figure 4b, indicating the EDS results for each point. The addition of Ag had a notable impact on the microstructure and morphology of the alloy. Compared to the alloy without Ag, the surface grain became more uniform the presence of Ag₃Sn intermetallic compounds helped to refine the structure of the alloy and enhance its mechanical properties. The elemental distribution analysis conducted through EDS surface scanning is depicted in Figure 5. EDS analysis reflects that the green component represents the Sn element, the red component represents the Cu element, and the blue component represents the Ag element.

3.2. Microhardness of Alloys

The Vickers hardness test results of Cu₆Sn₅-xAg alloys are presented in Figure 6. In Figure 6a, the Vickers hardness of the Cu₆Sn₅ alloy ranges from approximately 200 HV to 510 HV. Figure 6b illustrates that the Vickers hardness of the alloy doped with 3% Ag falls within the range of 240 HV to 360 HV, while in Figure 6c, the Vickers hardness of the alloy doped with 6% Ag ranges from 180 HV to 300 HV. The addition of Ag to the alloy narrows the range between the maximum and minimum values compared to the Cu₆Sn₅ alloy. This phenomenon is attributed to the formation of new phases in the alloy with the addition of Ag, as evidenced by the SEM test results shown in Figure 4.

3.3. Mechanical Property Analysis

The stiffness coefficients of the crystal structures for both the low-temperature phase η′-Cu₆Sn₅ and the high-temperature phase η-Cu₆Sn₅ were determined through simulation. The stiffness coefficient represents the amount of deformation generated by a unit stress when an object is subjected to force. A higher stiffness coefficient suggests that the object can deform more easily under force and is more resistant to external forces [24,25,26]. This coefficient is closely linked to performance parameters like the material’s modulus of elasticity [27]. The elastic constant of the crystalline solid is calculated by the CASTEP code according to the stress–strain method. In this method, three normal strains (ε1, ε2, and ε3) and three shear strains (γ4, γ5, and γ6) act on the crystal. The corresponding stress (σ1, σ2, σ3 and τ4, τ5, τ6) is obtained after deformation of the crystal; then, the independent elastic constants can be obtained from Hooker’s law.

Table 5. Calculated elastic constants Cij (in GPa) of η-Cu6Sn5.

i	Ci1	Ci2	Ci3	Ci4	Ci5	Ci6
1	124.6	55.6	62.5	0.0	−7.9	0.0
2	55.6	115.7	45.0	0.0	10.4	0.0
3	62.5	45.0	112.3	0.0	−1.0	0.0
4	0.0	0.0	0.0	29.4	0.0	3.5
5	−7.9	10.4	−1.0	0.0	38.2	0.0
6	0.0	0.0	0.0	3.5	0.0	30.5

and

Table 6. Calculated elastic constants Cij (in GPa) of η′-Cu6Sn5.

i	Ci1	Ci2	Ci3	Ci4	Ci5	Ci6
1	145.0	52.8	51.9	0.0	0.0	0.0
2	52.8	145.0	51.9	0.0	0.0	0.0
3	51.9	51.9	160.5	0.0	0.0	0.0
4	0.0	0.0	0.0	43.8	0.0	0.0
5	0.0	0.0	0.0	0.0	43.8	0.0
6	0.0	0.0	0.0	0.0	0.0	46.1

shows the elastic constants of η-Cu₆Sn₅ and η′-Cu₆Sn₅. This coefficient is closely linked to the material’s modulus of elasticity and other performance parameters, allowing for the calculation of related properties.

Table 7. Mechanical properties of the low-temperature phase η′-Cu₆Sn₅.

	Voigt	Reuss	Hill
Bulk Modulus (B, GPa)	75.417	74.541	74.979
Shear Modulus (G, GPa)	32.256	30.722	31.489
Young’s Modulus (E, GPa)	84.695	81.032	82.867
Hardness (H, GPa)	4.007	3.693	3.849
Poisson ratio (v)	0.313	0.319	0.316
B/G	2.338	2.426	2.381

and

Table 8. Mechanical properties of the high-temperature phase η-Cu₆Sn₅.

	Voigt	Reuss	Hill
Bulk Modulus (B, GPa)	84.855	84.701	84.778
Shear Modulus (G, GPa)	46.326	46.134	46.230
Young’s Modulus (E, GPa)	117.580	117.135	117.358
Hardness (H, GPa)	7.008	6.967	6.987
Poisson ratio (v)	0.269	0.270	0.269
B/G	1.832	1.836	1.834

present the mechanical properties of both crystal structures, calculated using the first-nature principle.

The mechanical stability of the monoclinic system can be assessed using the following criteria [28]: the system is stable when C_{ii >} 0 (i = 1 − 6), [C₁₁ + C₂₂ + C₃₃ + 2 (C₁₂ + C₁₃ + C₂₃)] > 0, C₃₃C₅₅-C²₃₅ > 0, (C₄₄C₆₆-C²₄₆) > 0, and (C₃₃C₂₂-2C₂₃) > 0. It is evident that η′_-Cu₆Sn₅ and its variants with different levels of Ag doping meet the stability requirements based on elastic constants, indicating mechanical stability. In practical scenarios, the polycrystalline elastic modulus is often more relevant than single-crystal constants. The elastic properties of η′-Cu₆Sn₅ in its low-temperature phase, high-temperature phase η-Cu₆Sn₅, and various Ag-doped systems, including bulk modulus (B), shear modulus (G), Young’s modulus (E), and Poisson’s ratio (v), were determined using the Voigt Reuss Hill (VRH) approximation. The calculations for these properties are conducted as follows [29,30,31,32]:

$$B_H=\frac{1}{2}\left(B_R+B_V\right)$$

(1)

$$G_H=\frac{1}{2}\left(G_R+G_V\right)$$

(2)

$$E=\frac{9G_H{B}_H}{G_H+3B_H}$$

(3)

$$\nu =\frac{3B_H-E}{6B_H}$$

(4)

The Vickers hardness of polycrystalline materials is typically determined using a theoretical model that relies on the material’s bulk modulus and shear modulus, as expressed in the following equation [33]:

$$H_V=0.92{(\frac{G}{B})}^{1.137}{G}^{0.708}$$

(5)

Figure 7 and Figure 8 display three-dimensional graphs illustrating the bulk modulus (B), Young’s modulus (E), and hardness (H) of the crystal structures of the low-temperature phase η′-Cu₆Sn₅ and the high-temperature phase η-Cu₆Sn₅.

The 3D graphs illustrate the anisotropy in the structure of the low-temperature phase η′-Cu₆Sn₅, showing variations in the Young’s modulus and hardness across different directions compared to the high-temperature phase η-Cu₆Sn₅. Pugh [34] introduced the ductility index (B/G) to quantify a material’s ductility, where higher values indicate greater ductility. Both phases of Cu₆Sn₅ have B/G values above 1.75, signifying ductility. Moreover, Poisson’s ratio (v) is associated with B/G, with v values above 0.26 indicating ductility. In both phases, v exceeds 0.26, confirming their ductile nature.

To visually compare the differences between the maximum and minimum values, face projections were conducted in three directions. Figure 8, Figure 9 and Figure 10 illustrate the projections of the bulk modulus (B), Young’s modulus (E), and hardness (H) in these directions.

The mechanical properties of the high-temperature phase η-Cu₆Sn₅ structure are typically isotropic, while those of the low-temperature phase η′-Cu₆Sn₅ structure tend to be anisotropic, as evident from projection diagrams. It has been established that the mechanical properties of the intermetallic compound Cu₆Sn₅ undergo changes during the structural transformation from high-temperature phase η-Cu₆Sn₅ to low-temperature phase η′-Cu₆Sn₅. This alteration is attributed to the anisotropic transformation of the elastic modulus and hardness of Cu₆Sn₅, which also contributes to the formation of microcracks during the structural shift from η′-Cu₆Sn₅ to η-Cu₆Sn₅.

The low-temperature phase η′-Cu₆Sn₅ crystals exhibit a monoclinic crystal structure and are classified under the C2/C space group. Each η′-Cu₆Sn₅ crystal contains seven atoms at distinct positions: Cu1 (4a), Cu2 (4e), Cu3 (8f1), Cu4 (8f2), Sn1 (4e), Sn2 (8f1), and Sn3 (8f2). Doping calculations were conducted for seven potential doping positions. The bulk modulus (B) is a common measure of a material’s resistance to compressive deformation. When subjected to external pressure, a material’s volume is compressed, making it compressible. The bulk modulus is the reciprocal of compressibility, indicating that materials with higher B values have greater compression resistance and are harder to compress. The bulk modulus (B) values were determined based on the single-crystal elastic constants.

Table 9. Mechanical properties of η′-Cu₆Sn₅ and Ag-element substitute-doped η′-Cu₆Sn₅ (Cu1, Cu2, Cu3).

	η′-Cu6Sn5			Ag-Cu1			Ag-Cu2			Ag-Cu3
	Voigt	Reuss	Hill	Voigt	Reuss	Hill	Voigt	Reuss	Hill	Voigt	Reuss	Hill
B	75.4	74.5	75.0	74.2	74.0	74.1	112.9	111.7	112.3	113.7	112.2	112.9
E	84.7	81.0	82.9	98.0	94.7	96.3	125.7	114.2	120.0	132.2	119.7	126.0
H	4.0	3.7	3.8	5.7	5.3	5.5	5.3	4.3	4.8	5.9	4.8	5.3

and

Table 10. Mechanical properties of η′-Cu₆Sn₅ and Ag-element substitute-doped η′-Cu₆Sn₅ (Cu4, Sn1, Sn2, Sn3).

	Ag-Cu4			Ag-Sn1			Ag-Sn2			Ag-Sn3
	Voigt	Reuss	Hill	Voigt	Reuss	Hill	Voigt	Reuss	Hill	Voigt	Reuss	Hill
B	111.4	110.5	111.0	97.9	95.9	96.9	97.6	89.8	93.7	97.2	97.1	97.2
E	121.7	108.7	115.2	118.4	110.2	114.3	125.4	95.7	110.7	119.7	118.9	119.3
H	5.0	3.9	4.5	5.7	5.0	5.4	6.5	4.0	5.3	5.9	5.9	5.9

illustrate the stress deformation resistance magnitude of η′-Cu₆Sn₅ and Ag-elemental substitution doped η′-Cu₆Sn₅, with the ranking as follows: Ag-Cu3 > Ag-Cu2 > Ag-Cu4 > Ag-Sn3 > Ag-Sn1 > Ag-Sn2 > η′-Cu₆Sn₅ > Ag-Cu1. Specifically, Ag-Cu3 has a bulk modulus of 112.9 GPa, while Ag-Cu1 has a bulk modulus of 74.1 GPa, indicating that Ag exhibits the highest resistance to volume change when replacing Cu3 in η′-Cu₆Sn₅ and the lowest resistance when replacing Cu1 in η′-Cu₆Sn₅. Young’s modulus (E) is a common indicator of elasticity that reflects mechanical properties, with higher values corresponding to greater stiffness. The stiffness ranking for η′-Cu₆Sn₅ and Ag-element substitution doped η′-Cu₆Sn₅ is Ag-Cu3 > Ag-Cu2 > Ag-Sn3 > Ag-Cu4 > Ag-Sn1 > Ag-Sn2 > Ag-Cu1 > η′-Cu₆Sn₅. The modulus of elasticity for Ag-Cu3 is 126.0 GPa, which is 43.1 GPa higher than that of η′-Cu₆Sn₅, indicating that Ag replacement at the Cu3 position results in the greatest hardness.

Hardness (H) refers to a material’s ability to resist localized surface pressure from a hard object. The hardness ranking in Table 7 and Table 8 is as follows: Ag-Sn3 > Ag-Cu1 > Ag-Sn1 > Ag-Cu3 > Ag-Sn2 > Ag-Cu2 > Ag-Cu4 > η′-Cu₆Sn₅, showing that replacing Ag at the Sn3 position results in the highest hardness. Figure 11, Figure 12 and Figure 13 illustrate that when Ag replaces doped Cu and Sn atoms at the Cu1, Cu3, Cu4, Sn1, and Sn3 sites, it reduces the anisotropy of the bulk modulus. However, replacing the Cu2 and Sn2 sites does not have a significant effect. Similarly, replacing Ag at doped Cu1, Cu3, Sn1, and Sn3 sites reduces the anisotropy of Young’s modulus, while replacing the Cu2, Cu4, and Sn2 sites does not show a noticeable impact.

The distribution of the bulk modulus bottom projection in the [010] direction becomes more homogeneous when the Ag element is substituted at the Cu1, Cu3, Cu4, Sn1, and Sn3 sites, as illustrated in Figure 14, Figure 15 and Figure 16. This demonstrates a reduction in the anisotropy of Cu₆Sn₅ following Ag element doping. Similarly, the anisotropy of Young’s modulus was also suppressed with the introduction of Ag element through substitution at the Cu1, Cu3, Sn1, and Sn3 sites.

The anisotropy ratio Emax/Emin was utilized to quantify the degree of anisotropy, where a larger ratio indicates higher anisotropy. Specifically, for η′-Cu₆Sn₅, Emax = 84.7 Gpa and Emin = 81.0 GPa, resulting in an anisotropy ratio of 1.045. In the case of Ag-Cu1, Emax = 98.0 Gpa and Emin = 94.7 GPa, leading to an anisotropy ratio of 1.034. Lastly, for Ag-Sn3, Emax = 119.7 Gpa and Emin = 118.9 GPa, yielding an anisotropy ratio of 1.006. This suggests that the substitution of Ag for Cu1 and Sn3 in η′-Cu₆Sn₅ results in a lower elastic modulus of anisotropy.

3.4. Calculation of the Density of States of Alloys

The total and fractional density of states of η′-Cu₆Sn₅ and its doped systems were analyzed using first principles. Figure 17 presents the TDOS and PDOS diagrams of η′- Cu₆Sn₅ and its doped system. The TDOS in the valence band of η′- Cu₆Sn₅ exhibits two main peaks at energy levels of −3.24 eV and −7.55 eV, contributed by Cu-d and Sn-s states, respectively. Furthermore, a strong covalent bond structure is formed by Cu-d and Sn-p hybridisation at −3.24 eV. The main peaks in the doped systems shift to −3.53 eV and −7.85 eV with Ag doping, indicating an effect on the density of states. Ag doping enhances the metallic properties of η′- Cu₆Sn₅, consistent with previous studies [35,36]. Additionally, a new main peak at 2.47 eV in the doped system is attributed to the interaction of Ag-p with Sn-p and Cu-d, leading to increased stability of η′- Cu₆Sn₅. For η-Cu₆Sn₅, there are two main energy peaks below the Fermi energy level. The strong peak at −3.11 eV coincides with the partial density of states of the Cu-d orbitals; it is formed by the hybridisation of Cu-d orbital electrons at different positions, and at the same time, in this region, the cu-d orbital electrons are weakly hybridised with Sn-p orbital electrons. In the same region, the electrons in the Cu-d orbitals are weakly hybridised with the electrons in the Sn-p orbitals.

This newly introduced hybridisation reduces the formation energy of the intermetallic compound system and strengthens the chemical bond due to higher electron density distribution in the bonding orbitals. Consequently, (Cu, Ag)₆Sn₅ exhibits improved bonding ability and force, contributing to a stable structure and enhanced physical and mechanical properties.

4. Conclusions

In this study, the physical and morphological characterization of the Cu₆Sn₅-xAg (x = 0, 3, 6; %, mass fraction) alloy was conducted by adjusting the addition of Ag elemental content, followed by testing the hardness of the alloy. First-principle calculations were utilized to simulate the crystal structure and mechanical properties of Cu₆Sn₅ intermetallic compounds. Additionally, the mechanical properties of Ag were examined when replacing Cu and Sn in Cu₆Sn₅. The resulting alloys with Ag addition exhibited three main phases: Sn, Cu₃Sn, Cu₆Sn₅ intermetallic compounds, and Ag₃Sn intermetallic compounds. The introduction of Ag led to an increase in the Cu₆Sn₅ lattice constants in the a and b directions, while decreasing in the c direction, resulting in a more uniform bulk and reducing the anisotropy of the η′-Cu₆Sn₅ structure. The low-temperature phase η′-Cu₆Sn₅ structure displayed noticeable anisotropy, with the Young’s modulus and hardness varying significantly in different directions compared to the high-temperature phase η- Cu₆Sn₅. Through Ag substitution doping, it was found that Ag element substitution at the Cu1, Cu3, Cu4, Sn1, and Sn3 sites helped mitigate the anisotropy of the bulk modulus, while substitution at the Cu2 and Sn2 sites had minimal impact. Similarly, Ag element substitution at the Cu1, Cu3, Sn1, and Sn3 sites reduced the anisotropy of the Young’s modulus, with substitution at the Cu2, Cu4, and Sn2 sites showing little effect. For Ag-Cu1, Emax = 98.0 Gpa and Emin = 94.7 GPa, with an anisotropy ratio of 1.034, and for Ag-Sn3, Emax = 119.7 Gpa and Emin = 118.9 GPa, with an anisotropy ratio of 1.006. This suggests that Ag substitution at Cu1 and Sn3 in η′- Cu₆Sn₅ results in less elastic modulus anisotropy. Additionally, a new main peak at an energy value of 2.47 eV was observed in the doped system, indicating that the stability of η′- Cu₆Sn₅ is enhanced by Ag doping due to the strong interaction of Ag-p with Sn-p and Cu-d. Overall, this study offers valuable insights and theoretical guidance for future applications of the Cu₆Sn₅ alloy.