Open AccessArticle

First-Principles Study on the Mechanical Properties of Ni₃Sn₄-Based Intermetallic Compounds with Ce Doping

Ruisheng Zhao

¹,

Yan Cao

²,

Jinhu He

²,

Jianjun Chen

²,

Shiyuan Liu

²,

Zhiqiang Yang

²,

Jinbao Lin

¹ and

Chao Chang

^1,*

School of Applied Science, Taiyuan University of Science and Technology, Taiyuan 030024, China

Shanxi Diesel Engine Industry Co., Ltd., Datong 037036, China

Author to whom correspondence should be addressed.

Coatings 2025, 15(1), 59; https://doi.org/10.3390/coatings15010059

Submission received: 2 December 2024 / Revised: 30 December 2024 / Accepted: 3 January 2025 / Published: 7 January 2025

(This article belongs to the Special Issue Coatings for Advanced Devices)

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Abstract

Ni₃Sn₄ intermetallic compound (IMC) is a critical material in modern electronic packaging and soldering technology. Although Ni₃Sn₄ enhances the strength of solder joints, its brittleness and anisotropy make it prone to crack formation under mechanical stress, such as thermal cycling or vibration. To improve the plasticity of Ni₃Sn₄ and mitigate its anisotropy, this study employs first-principles calculations to investigate the mechanical properties and electronic structure of the doped compounds Ce_x Ni_3−xSn₄ (x = 0, 0.5, 1, 1.5, 2) by adding the rare earth element Ce. The results indicate that the structure Ce_0.5 Ni_2.5Sn₄ has a lower formation enthalpy (

H_{f}

) compared to other doped structures, suggesting enhanced stability. It was found that all structures exhibit improved plasticity with Ce doping, while the Ce_0.5 Ni_2.5Sn₄ structure shows relatively minor changes in hardness (H) and elastic modulus, along with the lowest anisotropy value (

A^{U}

). Analysis of the total density of states (TDOS) and partial density of states (PDOS) reveals that the electronic properties are primarily influenced by the Ni-d and Ce-f orbitals. At the Fermi level, all Ce_x Ni_3−xSn₄ (x = 0, 0.5, 1, 1.5, 2) structures exhibit metallic characteristics and distinct electrical conductivity. Notably, the TDOS value at the Fermi level for Ce_0.5 Ni_2.5Sn₄ lies between those of Ni₃Sn₄ and other doped structures, indicating good metallicity and conductivity, as well as relative stability. Further PDOS analysis suggests that Ce doping enhances the plasticity of Ni₃Sn₄. This study provides valuable insights for the further application of rare earth elements in electronic packaging materials.

Keywords:

intermetallic compounds; first-principles calculations; mechanical properties; electronic structures; rare earth elements

1. Introduction

In modern electronic packaging technology, the study of intermetallic compounds (IMCs) has gained increasing importance, particularly concerning their reliability [1,2]. As solder joints become smaller and electronic devices advance towards higher performance and density integration [3,4], the reliability of IMCs is critical to the mechanical performance and durability of solder joints, directly impacting the overall performance and lifespan of electronic devices [5]. The composition of intermetallic compounds (IMCs) and strain-induced mismatch in the coefficient of thermal expansion (CTE) can lead to mechanical fatigue failures in packaging [6]. Research has shown that changes in the microstructure and mechanical properties of solder joints are often the primary causes of reliability issues [7,8,9]. Sn-based lead-free solder typically refers to lead-free solder with Tin (Sn) as the primary component, widely used in electronic soldering. When Tin-based solder contacts nickel-coated layers, diffusion reactions occur at high temperatures, resulting in the formation of

{N i}_{3} {S n}_{4}

intermetallic compound [10]. This compound is of great interest due to its excellent mechanical properties, corrosion resistance, and wear resistance [11]. However, the brittleness and anisotropy of

{N i}_{3} {S n}_{4}

can lead to cracking under thermal cycling and mechanical stress, thereby impacting the reliability of solder joints [12,13]. Therefore, enhancing the ductility of

{N i}_{3} {S n}_{4}

and reducing its anisotropy are essential for improving the reliability of electronic packaging.

First-principles calculations, based on quantum mechanics, provide high-precision results without relying on empirical parameters or experimental data. This approach is widely utilized in the design and optimization of new materials, particularly for predicting their structures and properties [14,15,16]. Research by Yao et al. indicates that the incorporation of copper atoms reduces the stability and mechanical properties of the

{N i}_{3} {S n}_{4}

phase. As the concentration of copper increases, the bulk modulus, shear modulus, and elastic modulus of

{(N i, C u)}_{3} {S n}_{4}

all decrease, suggesting that copper inclusion may adversely affect mechanical performance and increase the risk of interfacial microcracks [17,18]. Furthermore, studies by Bi and Hu et al. demonstrate that adding small amounts of Co (12.46 at.%) enhances the mechanical properties of

{N i}_{3} {S n}_{4}

intermetallic compound, improving the shear strength of solder joints. First-principles calculations also reveal the primary reasons for this enhancement [19]. Han and Chen et al. investigated

{N i}_{3} {S n}_{4}

with various added elements (Pd, Pt, Ge, Sb, Co, Cu, In, and Pb) and showed that copper and antimony negatively impact ductility, while germanium and lead improve it. The other elements have a minimal effect on ductility and brittleness [20]. Additionally, researchers have explored adding trace amounts of rare earth elements (such as La, Ce, and Sc) as alloying agents in solders. Studies have demonstrated that solder containing rare earth elements exhibits better wettability, creep strength, and tensile strength, while also reducing the thickness of interfacial intermetallic compounds [21,22]. For example, Zhang et al. added 0.03 wt% cerium to SnAgCu solder, and found that the fatigue life of the SnAgCuCe solder joints exceeded that of SnAgCu joints by 30.2%. This indicates that SnAgCuCe solder can replace traditional SnAgCu solder in electronic applications, offering enhanced reliability [23]. Ce was chosen for doping due to its superior ability to improve the ductility and toughness of Ni₃Sn₄, attributed to its unique electronic properties, such as d-f hybridization and increased density of states near the Fermi level. Comparative studies indicate that while other rare earth elements [24], such as La and Sc, enhance strength and hardness, their effects on plasticity and toughness are less pronounced. By introducing rare elements, the grain structure of the solder is effectively refined, which enhances the material’s toughness. Furthermore, these elements improve the wettability between the solder and the substrate, thereby increasing the bonding strength of the welded joints. However, research on the doping of Ni₃Sn₄ with rare earth elements remains relatively scarce. Therefore, exploring the effects of rare earth element doping on the mechanical properties of

{N i}_{3} {S n}_{4}

alloys, particularly concerning plasticity and anisotropy, could help to address this research gap.

This study aims to evaluate the potential of rare earth element Ce doping in Ni₃Sn₄ alloys, particularly in terms of enhancing toughness and influencing the alloy’s anisotropy. By conducting an in-depth analysis of the mechanisms associated with Ce doping, this research seeks to explore new pathways for optimizing material performance, ultimately achieving superior overall mechanical properties. Initially, we examine the structural characteristics and stability of the modified compound, denoted as

{C e}_{x} {N i}_{3 - x} {S n}_{4}

. Subsequently, we investigate how different levels of Ce doping influence the mechanical properties and anisotropy of the material. Finally, we analyze the electronic density of states of the

{C e}_{x} {N i}_{3 - x} {S n}_{4}

alloy. This research aims to provide valuable insights for improving the properties of

{N i}_{3} {S n}_{4}

2. Structural Characteristics and Stability

The

{N i}_{3} {S n}_{4}

crystal is classified within the monoclinic system, belonging to the space group C2/m [25]. Each unit cell comprises six nickel (Ni) atoms and eight tin (Sn) atoms. The Ni atoms occupy two distinct positions: Ni(2a) and Ni(4i). The Ni(2a) atoms are located at the eight vertices of a hexahedron and at the center of the XY plane, while the Ni(4i) atoms reside within the hexahedron and the XZ plane. The initial lattice parameters for the geometric optimization of the

{N i}_{3} {S n}_{4}

structure are as follows: a₀ = 12.21 Å, b₀ = 4.05 Å, c₀ = 5.20 Å, β = 105.03°, and the volume V = 248.434 Å³. A schematic diagram of the

{N i}_{3} {S n}_{4}

crystal structure is presented in Figure 1a. Ce atom doping can substitute Ni atoms at the Wyckoff positions in the crystal, as illustrated in Figure 1. The calculated formation enthalpies for

{C e}_{0.5} {N i}_{2.5} {S n}_{4}

at the 4i and 2a positions are −24.607 kJ/mol and −23.079 kJ/mol, respectively. This indicates a preference for Ce atoms to occupy the 4i position over the 2a position, as evidenced by the lower formation enthalpy of

{C e}_{0.5} {N i}_{2.5} {S n}_{4}

(4i). Therefore, it is assumed in this calculation that all alloying elements occupy the 4i position within the

{N i}_{3} {S n}_{4}

crystal.

To investigate the thermodynamic stability, the following formula is employed to calculate the formation enthalpy

H_{f}

of the

{N i}_{3} {S n}_{4}

-based ternary structure [8]. This enthalpy reflects the alloy’s ability to form: the more negative the

H_{f}

value, the easier it is for the alloy structure to form, indicating a more stable system [25]. the formation enthalpy is calculated using the following formula:

H_{f} = \frac{[E_{U_{p} V_{q} W_{g}}^{T} - (p E_{U} + q E_{V} + g E_{W})]}{p + q + g}

(1)

where

E_{U_{p} V_{q} W_{g}}^{T}

represents the total energy of the intermetallic compound

U_{p} V_{q} W_{g}

(where U, V, and W are the constituent elements, and p, q, and g are their respective atomic numbers);

E_{U}

E_{V}

and

E_{W}

are the per-atom energies in the single crystal structures of U, V, and W, respectively. The calculation results are presented in Table 1. It can be observed that after Ce doping, the value of

{C e}_{0.5} {N i}_{2.5} {S n}_{4}

is relatively low and close to that of the undoped

{N i}_{3} {S n}_{4}

. Compared to other structures with varying Ce doping contents,

{C e}_{0.5} {N i}_{2.5} {S n}_{4}

exhibits greater stability.

Figure 2 illustrates the variation in lattice constants of the alloy

{C e}_{x} {N i}_{3 - x} {S n}_{4}

(x = 0, 0.5, 1, 1.5, 2). Atomic doping can significantly affect the lattice constants. Doping atoms typically either replace or insert themselves into the lattice of the matrix atoms. This alteration can lead to an increase or decrease in the lattice constant, depending on the size and properties of the doping atoms [30]. When the radius of the doping atoms is similar to that of the matrix atoms, the change in lattice constant is usually minimal. Conversely, when the radius of the doping atoms exceeds that of the matrix atoms, the lattice constant tends to increase; if the doping atoms are smaller, the lattice constant decreases. As the amount of Ce replacing Ni increases, the lengths of a (Å), b (Å), and c (Å) exhibit different increases compared to the lattice constant of Ni₃Sn₄, showing distinct directional differences. When the increments of the lattice constants a (Å), b (Å), and c (Å) differ, the crystal typically exhibits noticeable anisotropy in various directions [31,32]. This may weaken the directional dependence of bonding and consequently impact the anisotropic and elastic properties of

{C e}_{x} {N i}_{3 - x} {S n}_{4}

(x = 0, 0.5, 1, 1.5, 2), which will be discussed later.

In this study, first-principles calculations based on density functional theory (DFT) were performed using the Cambridge Sequential Total Energy Package (CASTEP) [33]. Traditional nanoindentation allows for rapid and practical assessment of material mechanical properties [34,35,36], while first-principles calculations provide high precision and a deep understanding of material characteristics. During the calculations, the electron exchange-correlation function was modeled using the Perdew–Burke–Ernzerhof (PBE) [37], functional under the generalized gradient approximation (GGA-PBE) [38], and the interaction potential between valence electrons and ions was simulated using the ultrasoft pseudopotential (USPP). A kinetic energy cutoff of 400 eV was selected for the plane wave function expansion, and the Broyden–Fletcher–Goldfarb (BFG) algorithm was employed for structural optimization. The Pulay density mixing method was utilized for electron relaxation in the self-consistent field (SCF) calculations. The convergence criteria for the SCF calculations were set as follows: the total energy must be less than 1.0 ×

10^{- 6}

eV/atom, the force on each atom must be less than 0.05 eV/Å, the tolerance offset must be less than 0.002 Å, and the maximum stress deviation must be less than 0.1 GPa.

To ensure the accuracy of subsequent calculations and obtain the most stable crystal structure, a stringent total energy convergence test was conducted. The elastic constants

C_{i j}

and elastic compliance constants

S_{i j}

for

{C e}_{x} {N i}_{3 - x} {S n}_{4}

were calculated using 2 × 5 × 6 Monkhorst–Pack K points, and the results are presented in Table 2 and Table 3.

3. Mechanical Properties

The mechanical stability of a crystal structure can be assessed using independent elastic constants [39]. The elastic constant matrix can be reduced to 13 variables based on the symmetry of the monoclinic structure. The criteria for mechanical stability in the monoclinic system are expressed by the following equations [40,41,42]:

C_{i j} > 0 (i = 1 - 6)

(2)

[C_{11} + C_{22} + C_{33} + 2 (C_{12} + C_{13} + C_{23})] > 0

(3)

(C_{33} C_{55} - C_{35}) > 0, (C_{44} C_{66} - C_{46}) > 0, (C_{22} + C_{33} - 2 C_{23}) > 0

(4)

Table 2 presents the calculated elastic constants

C_{i j}

for all structures based on

{N i}_{3} {S n}_{4}

. The elastic constants for

{N i}_{3} {S n}_{4}

closely align with other computational results [18,20]. The elastic constants

C_{i j}

for the structures of

{C e}_{x} {N i}_{3 - x} {S n}_{4}

(x = 0, 0.5, 1, 1.5, 2) readily satisfy the corresponding criteria listed in Equations (2)–(4), indicating that the studied

{C e}_{x} {N i}_{3 - x} {S n}_{4}

structures are mechanically stable at 0 K. To further compute the Young’s modulus [43] of

{C e}_{x} {N i}_{3 - x} {S n}_{4}

(x = 0, 0.5, 1, 1.5, 2), the Voigt–Reuss–Hill (VRH) [44] approach can be employed. The Voigt (V) and Reuss (R) values for bulk modulus and shear modulus can be calculated using the following formulas:

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

(5)

\begin{array}{l} G_{V} = (1 / 15) [C_{11} + C_{22} + C_{33} + 3 (C_{44} + C_{55} + C_{66}) \\ - (C_{12} + C_{13} + C_{23})] \end{array}

(6)

\begin{matrix} B_{R} = Ω [a (C_{11} + C_{22} - 2 C_{12}) + b (2 C_{12} - 2 C_{11} - C_{23}) \\ + c (C_{15} - 2 C_{25}) + d (2 C_{12} + 2 C_{23} - C_{13} - 2 C_{22}) \\ + 2 e (C_{25} - C_{15}) + f]^{- 1} \end{matrix}

(7)

\begin{matrix} G_{R} = 15 {4 [a (C_{11} + C_{22} + C_{12}) + b (C_{11} - C_{12} - C_{23}) + c (C_{15} \\ + C_{25}) + d (C_{22} - C_{12} - C_{23} - C_{13}) + e (C_{15} - C_{25}) + f] / Ω \\ + 3 [\frac{g}{Ω} + \frac{C_{44} + C_{66}}{C_{44} C_{66} - C_{46}^{2}}]}^{- 1} \end{matrix}

(8)

a = C_{33} C_{55} - C_{35}^{2}

(9)

b = C_{23} C_{55} - C_{25} C_{35}

(10)

c = C_{13} C_{35} - C_{15} C_{33}

(11)

d = C_{13} C_{55} - C_{15} C_{35}

(12)

e = C_{13} C_{25} - C_{15} C_{23}

(13)

\begin{matrix} f = C_{11} (C_{22} C_{55} - C_{25}^{2}) - C_{12} (C_{12} C_{55} - C_{15} C_{25}) \\ + C_{15} (C_{12} C_{25} - C_{15} C_{22}) + C_{25} (C_{23} C_{35} - C_{25} C_{33}) \end{matrix}

(14)

g = C_{11} C_{22} C_{33} - C_{11} C_{23}^{2} - C_{22} C_{13}^{2} - C_{33} C_{12}^{2} + 2 C_{12} C_{13} C_{23}

(15)

\begin{matrix} Ω = 2 [\begin{matrix} 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}) \end{matrix}] \\ - [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})] + g C_{55} \end{matrix}

(16)

The bulk modulus (B) represents the resistance of a material to external compression in flexible systems, while the shear modulus (G) describes the material’s ability to resist shear strain. The Young’s modulus (E) and Poisson’s ratio (ν) can be derived from G and B. Additionally, hardness (H) can be obtained from the Young’s modulus (E) and Poisson’s ratio (ν) [45,46], with calculations as follows:

B = \frac{B_{V} + B_{R}}{2}

(17)

G = \frac{G_{V} + G_{R}}{2}

(18)

E = \frac{9 B G}{3 B + G}

(19)

v = \frac{3 B - 2 G}{2 (3 B + G)}

(20)

H = \frac{(1 - 2 v)}{6 (1 + v)} E

(21)

The values of the bulk modulus (B), shear modulus (G), Young’s modulus (E), Poisson’s ratio (ν), hardness (H), and the ratio B/G for

{C e}_{x} {N i}_{3 - x} {S n}_{4}

(x = 0, 0.5, 1, 1.5, 2) can be derived from Equations (17) to (21). The calculated values are presented in Figure 3 and Table 4. Regarding the hardness of pure Ni₃Sn₄, some experimental data are listed in Table 5. A large number of nanoindentation results for Young’s modulus fall within the range of 119.4 to 156 GPa, while the hardness values range from 6.1 to 8.8 GPa. The main variation in these values is due to the influence of the sample’s specific morphology on the nanoindentation measurement process. In this study, for pure Ni₃Sn₄, the calculated values are as follows: B = 99.15 GPa, G = 49.59 GPa, E = 128.31 GPa, ν = 0.284, and B/G = 1.98. These values fall within the range of experimental data obtained via nanoindentation, further confirming the reliability of the computational approach [19,47]. The capacity of a material to withstand deformation is represented by its bulk modulus (B), determined by the strength of the chemical bonds and the compressibility of the material. A higher bulk modulus signifies increased strength, which corresponds to a diminished ability to deform. Table 4 indicates that the bulk modulus (B) decreases with higher Ce doping levels, implying that Ce doping enhances the plasticity of the material.

In general, the ratio of bulk modulus to shear modulus (B/G) is used to describe the ductility or brittleness of a crystal structure [46]. A lower B/G value is associated with brittleness, while a higher B/G value indicates improved ductility. For the

{N i}_{3} {S n}_{4}

structure, the B/G ratio is 1.98. Following the addition of Ce, the B/G value increases with higher Ce content, suggesting an enhancement in ductility. Moreover, a lower Poisson’s ratio signifies that the material experiences less lateral deformation during tensile or compressive loading, making it more susceptible to brittle fracture rather than plastic deformation once a critical stress threshold is reached [50]. In contrast, a higher Poisson’s ratio indicates greater lateral deformation under similar loading conditions, reflecting improved deformability and greater plasticity. Thus, the Poisson’s ratio is directly related to the material’s ability to resist shear stress; a higher Poisson’s ratio corresponds to enhanced plasticity. Consequently, the observed increase in Poisson’s ratio with Ce doping suggests that the plasticity of

{N i}_{3} {S n}_{4}

is enhanced, which is consistent with the findings from the B/G analysis. This indicates that with Ce doping, the material is more likely to undergo plastic deformation rather than brittle fracture under applied stress. Furthermore, as Ce content increases, the hardness (H) value decreases accordingly.

It is generally accepted that the evolution of cracks within solder joints is closely related to the elastic anisotropy of intermetallic compounds (IMCs). In electronic packaging, microcracks induced by elastic anisotropy can adversely affect the reliability of electronic products. During the production and operation of solder joints, the anisotropic plastic deformation of IMCs can lead to the formation of microcracks, thereby reducing the reliability of electronic devices [51]. To investigate the impact of Ce introduction on the elastic anisotropy of IMCs, a universal anisotropy index

A^{U}

was employed. A higher

A^{U}

value indicates greater structural anisotropy of the material. The calculation formula for

A^{U}

is provided below, with the calculated results shown in Figure 4.

A^{U} = 5 \frac{G_{V}}{G_{R}} + \frac{B_{V}}{B_{R}} - 6

(22)

If the value of

A^{U}

is zero, the structure is considered isotropic. A higher

A^{U}

value indicates greater anisotropy in the material. The

A^{U}

value for

{N i}_{3} {S n}_{4}

is 0.66, which is close to values reported in the literature [19]. Calculations suggest that

{C e}_{x} {N i}_{3 - x} {S n}_{4}

is anisotropic, with

{C e}_{0.5} {N i}_{2.5} {S n}_{4}

exhibiting a smaller

A^{U}

value than

{N i}_{3} {S n}_{4}

. This indicates that a small amount of Ce doping can improve the anisotropy of

{N i}_{3} {S n}_{4}

, thereby reducing the tendency for microcrack formation, which helps enhance the reliability of the solder joints.

To further investigate the anisotropy of the

{C e}_{x} {N i}_{3 - x} {S n}_{4}

structure, three-dimensional (3D) surface plots of the bulk modulus and Young’s modulus were created, as shown in Figure 5 and Figure 6. The results indicate that for all

{N i}_{3} {S n}_{4}

based structures, the three-dimensional surface plots display varying degrees of deviation from sphericity, suggesting that both

{N i}_{3} {S n}_{4}

and

{C e}_{x} {N i}_{3 - x} {S n}_{4}

exhibit anisotropy. Furthermore, after Ce doping, the shape of

{C e}_{x} {N i}_{3 - x} {S n}_{4}

deviates even more from sphericity, indicating an enhancement in anisotropy. Notably, only

{C e}_{0.5} {N i}_{2.5} {S n}_{4}

exhibits a shape closer to spherical for its Young’s modulus, suggesting that this structure has relatively lower anisotropy, which is consistent with the computed results.

In summary, the plasticity of the

{C e}_{x} {N i}_{3 - x} {S n}_{4}

structures is enhanced compared to the undoped version. Notably, the

{C e}_{0.5} {N i}_{2.5} {S n}_{4}

structure not only demonstrates improved plasticity but also exhibits the lowest degree of anisotropy, which reduces the tendency for microcrack formation. This enhancement contributes to the reliability of the solder joints.

4. Electronic Structure

It is widely recognized that the mechanical properties and energy characteristics of materials are largely determined by their electronic structure. To better understand the bonding mechanisms of intermetallic compounds with different alloying elements, we analyzed the total density of states (TDOS) and partial density of states (PDOS) for

{C e}_{x} {N i}_{3 - x} {S n}_{4}

(x = 0, 0.5, 1, 1.5, 2). The results are illustrated in Figure 7 and Figure 8, where the Fermi level (

E_{F}

) is positioned at zero energy. For all studied structures, the alloy density of states curves are continuous at the Fermi energy (0 eV), and the density of states (DOS) values at the Fermi level (

N_{E F}

) are all greater than zero, indicating that all structures exhibit metallic characteristics and good electrical conductivity [52]. A smaller density of states at the Fermi level correlates with greater structural stability; conversely, a larger density can lead to bonding instability within the system [53]. Figure 7 shows that the alloy density of states at the Fermi level increases with higher Ce content. Therefore, as the Ce content increases, the structural stability decreases gradually. This result is consistent with the conclusion from the previous formation energy calculations, which show that as the Ce content increases, the system’s energy increases, leading to a gradual decline in structural stability. Among the

{C e}_{x} {N i}_{3 - x} {S n}_{4}

structures,

{C e}_{0.5} {N i}_{2.5} {S n}_{4}

has the lowest alloy density of states at the Fermi level, consistent with previous structural stability analyses. Furthermore, it was found that the density of states at the Fermi level (0 eV) for the doped structures is higher than that of

{N i}_{3} {S n}_{4}

, suggesting an enhancement in conductivity following doping. A greater electronic density of states within the alloy system facilitates the formation of metallic bonds, which contributes to improved ductility and reduced brittleness [53]. The continued addition of Ce is expected to enhance the alloy’s ductility, aligning with the results derived from B/G, Poisson’s ratio ν, and Cauchy pressure

C_{i j}

assessments regarding the impact of Ce content on the alloy’s ductility-brittleness balance.

In metal compound

{N i}_{3} {S n}_{4}

, two primary bonding peaks are observed at −2.31 eV and −1.78 eV. The main factor influencing the bonding in

{N i}_{3} {S n}_{4}

is the hybridization between Ni-d and Sn-p states, with the DOS values contributed by Ni-d states spanning from 0 to −5.5 eV. Below −6 eV, the Sn-s states predominantly influence the DOS, consistent with previous findings [54]. In the Ce-doped intermetallic compound

{C e}_{x} {N i}_{3 - x} {S n}_{4}

, these bonding peaks are weakened, and a new bonding peak appears around 0.8–1.4 eV, primarily dominated by Ce-f electrons. This indicates that the hybridization between Ni-d and Sn-p electrons is diminished after Ce replaces Ni in

{N i}_{3} {S n}_{4}

. The introduction of Ce also reveals a contribution from Ce-f states to the DOS and suggests that there may be covalent bonding between Ni and Ce.

Furthermore, the formation of covalent interactions aligns with the trends observed in the Poisson’s ratio. In the Ce-doped Ni₃Sn₄ system studied here, we observed a significant increase in the overlap between Ce-f and Sn-p orbitals with increasing Ce content. This indicates that upon Ce substitution of Ni₃Sn₄, the hybridization between Ni-d and Sn-p electrons is weakened. Additionally, a new peak appears in the 0 eV to 4 eV range, primarily dominated by Ce-d electrons. The strength of atomic interactions can be inferred from the overlap of f- and p-hybrid orbitals. According to Gschneidner et al. [55], a clear relationship between electronic structure and the ductile or brittle behavior of crystal structures has been established. Therefore, we hypothesize that Ce doping enhances the ductility of Ni₃Sn₄, which is consistent with the increased ductility observed in our mechanical property calculations.

In summary, the electronic structure analysis reveals that the total density of states (TDOS) and partial density of states (PDOS) of

{C e}_{x} {N i}_{3 - x} {S n}_{4}

(x = 0, 0.5, 1, 1.5, 2) are primarily influenced by Ni-d and Ce-f states. The addition of Ce increases the toughness of the

{N i}_{3} {S n}_{4}

alloy, aligning with the trends observed in the previous calculations.

5. Conclusions

In this study, we employed first-principles calculations to simulate the ternary intermetallic compound based on

{N i}_{3} {S n}_{4}

and investigate the structural, elastic, and electronic properties following the substitution of Ni with cerium (Ce). The results can be summarized as follows:

(1): In the doped structure of ${C e}_{0.5} {N i}_{2.5} {S n}_{4}$ , the formation enthalpies for Ce occupying the 4i and 2a sites are −24.607 kJ/mol and −23.079 kJ/mol, respectively. This indicates that Ce atoms preferentially occupy the 4i site over the 2a site. Additionally, the formation enthalpy $H_{f}$ value for the structure with x = 0.5 is the lowest among the ${C e}_{x} {N i}_{3 - x} {S n}_{4}$ (x = 0, 0.5, 1, 1.5, 2) structures, suggesting that Ce doping favors the stability of the ${C e}_{0.5} {N i}_{2.5} {S n}_{4}$ (4i) structure relative to the others.
(2): After Ce doping, the elastic modulus of the ${C e}_{x} {N i}_{3 - x} {S n}_{4}$ structure decreased, while its plasticity improved across the board. Additionally, both the Poisson’s ratio (ν) and the ratio of bulk modulus to shear modulus (B/G) increased with higher Ce doping levels. Notably, the ${C e}_{0.5} {N i}_{2.5} {S n}_{4}$ structure exhibited minimal changes in hardness (H) and elastic modulus, yet its anisotropy value ( $A^{U}$ ) was the lowest, indicating that Ce doping effectively enhances the ductility and reduces the anisotropic characteristics of ${N i}_{3} {S n}_{4}$ .
(3): The electronic structure analysis of ${C e}_{x} {N i}_{3 - x} {S n}_{4}$ (x = 0, 0.5, 1, 1.5, 2) reveals that the total density of states (TDOS) and partial density of states (PDOS) are primarily influenced by Ni-d and Ce-f orbitals. All studied structures exhibit metallic characteristics and good electrical conductivity, as evidenced by nonzero TDOS at the Fermi level. Increasing Ce content leads to higher TDOS at the Fermi level, indicating reduced structural stability, consistent with formation energy calculations. The introduction of Ce also weakens the hybridization between Ni-d and Sn-p states, while generating a new bonding peak at 0.8~1.4 eV dominated by Ce-f orbitals. This suggests the formation of covalent bonds between Ni and Ce, and an enhanced overlap between Ce-f and Sn-p orbitals with higher Ce content. These electronic changes contribute to stronger metallic bonding and improved ductility, aligning with mechanical property calculations, such as B/G ratios, Poisson’s ratio, and Cauchy pressure. Overall, the analysis confirms that Ce doping enhances the toughness of ${N i}_{3} {S n}_{4}$ , providing insights into the role of electronic structure in tailoring the mechanical properties of intermetallic compounds.

To date, there have been no reported experimental or theoretical values. Therefore, the results of this study can provide a reference for the further application of this type of material and serve as a validation for future research efforts.

Author Contributions

R.Z. (First Author): Conceptualization, methodology, investigation, simulation, writing—original draft; J.C.: investigation; J.H.: investigation; Y.C.: investigation; Z.Y.: methodology, investigation; J.L.: methodology; S.L.: conceptualization; C.C. (corresponding author): conceptualization, funding acquisition, resources, supervision, writing—review and editing. All authors have read and agreed to the published version of the manuscript.

Funding

This work was supported by The Regional Collaboration Project of Shanxi Province (No. 202204041101044).

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

All data generated or analyzed during this study are included in this published article.

Conflicts of Interest

Yan Cao, Jinhu He, Jianjun Chen, Shiyuan Liu and Zhiqiang Yang were employed by the company Shanxi Diesel Engine Industry Co., Ltd. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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Figure 1. Crystal structures of (a)

{N i}_{3} {S n}_{4}

, (b)

{C e}_{0.5} {N i}_{2.5} {S n}_{4} (4 i),

(c)

{C e}_{0.5} {N i}_{2.5} {S n}_{4}

(2a), (d)

{C e}_{1} {N i}_{2} {S n}_{4}

, (e)

{C e}_{1.5} {N i}_{1.5} {S n}_{4}

, and (f)

{C e}_{2} {N i}_{1} {S n}_{4}

Figure 1. Crystal structures of (a)

{N i}_{3} {S n}_{4}

, (b)

{C e}_{0.5} {N i}_{2.5} {S n}_{4} (4 i),

(c)

{C e}_{0.5} {N i}_{2.5} {S n}_{4}

(2a), (d)

{C e}_{1} {N i}_{2} {S n}_{4}

, (e)

{C e}_{1.5} {N i}_{1.5} {S n}_{4}

, and (f)

{C e}_{2} {N i}_{1} {S n}_{4}

Figure 2. Changes in lattice constants a (Å), b (Å), and c (Å) with varying Ce content.

Figure 3. The variation in bulk modulus, shear modulus, and Young modulus with the Ce atom.

Figure 4. The variation of

A^{U}

with the Ce atom fraction.

Figure 4. The variation of

A^{U}

with the Ce atom fraction.

Figure 5. The surface constructions of bulk modulus for (a)

{N i}_{3} {S n}_{4}

, (b)

{C e}_{0.5} {N i}_{2.5} {S n}_{4}

, (c)

{C e}_{1} {N i}_{2} {S n}_{4}

, (d)

{C e}_{1.5} {N i}_{1.5} {S n}_{4}

, and (e)

{C e}_{2} {N i}_{1} {S n}_{4}

Figure 5. The surface constructions of bulk modulus for (a)

{N i}_{3} {S n}_{4}

, (b)

{C e}_{0.5} {N i}_{2.5} {S n}_{4}

, (c)

{C e}_{1} {N i}_{2} {S n}_{4}

, (d)

{C e}_{1.5} {N i}_{1.5} {S n}_{4}

, and (e)

{C e}_{2} {N i}_{1} {S n}_{4}

Figure 6. The 3D surface plots of Young’s modulus for (a)

{N i}_{3} {S n}_{4}

, (b)

{C e}_{0.5} {N i}_{2.5} {S n}_{4}

, (c)

{C e}_{1} {N i}_{2} {S n}_{4}

, (d)

{C e}_{1.5} {N i}_{1.5} {S n}_{4}

, and (e)

{C e}_{2} {N i}_{1} {S n}_{4}

Figure 6. The 3D surface plots of Young’s modulus for (a)

{N i}_{3} {S n}_{4}

, (b)

{C e}_{0.5} {N i}_{2.5} {S n}_{4}

, (c)

{C e}_{1} {N i}_{2} {S n}_{4}

, (d)

{C e}_{1.5} {N i}_{1.5} {S n}_{4}

, and (e)

{C e}_{2} {N i}_{1} {S n}_{4}

Figure 7. Comparison of total density of states (TDOS) in

{C e}_{x} {N i}_{3 - x} {S n}_{4}

(x = 0, 0.5, 1, 1.5, 2).

Figure 7. Comparison of total density of states (TDOS) in

{C e}_{x} {N i}_{3 - x} {S n}_{4}

(x = 0, 0.5, 1, 1.5, 2).

Figure 8. Total and partial density of states (TPDOS).

Table 1. Through experiments and calculations, the lattice parameters (Å), cell volume (

Å^{3}

), and formation enthalpy (Hf).

Table 1. Through experiments and calculations, the lattice parameters (Å), cell volume (

Å^{3}

), and formation enthalpy (Hf).

Phase	Reference	a (Å)	b (Å)	c (Å)	β (◦)	Volume ( $Å^{3}$ )	Hf (kJ·mol⁻¹)
${N i}_{3} {S n}_{4}$	This work	12.210	4.054	5.196	105.03	248.434	−24.744
	Calculation [26,27]	12.220	4.060	5.270	104.970	253.46	−24.000
	Calculation [20]	12.299	4.084	5.288	105.190	-	−26.000
	Calculation [18]	12.418	4.111	4.315	105.480	-	−26.700
	Calculation [19]	12.334	4.100	5.325	105.010	-	−26.900
	Experiment [28]	12.210	4.060	5.22	105.50	258.83	-
	Experiment [29]	12.199	4.061	5.22	105.25	249.58	-
${C e}_{0.5} {N i}_{2.5} {S n}_{4} (4 i)$	This work	12.283	4.093	5.322	105.41	257.941	−24.607
${C e}_{0.5} {N i}_{2.5} {S n}_{4}$ (2a)	This work	12.278	4.062	5.210	105.08	250.892	−23.079
${C e}_{1} {N i}_{2} {S n}_{4}$	This work	12.493	4.114	5.231	105.32	259.300	−24.472
${C e}_{1.5} {N i}_{1.5} {S n}_{4}$	This work	12.496	4.658	5.365	105.31	301.195	−24.341
${C e}_{2} {N i}_{1} {S n}_{4} i$	This work	12.783	4.891	5.423	105.73	326.357	−24.214

Table 2. Elastic constants (GPa) of

{C e}_{x} {N i}_{3 - x} {S n}_{4}

(x = 0, 0.5, 1, 1.5, 2).

Table 2. Elastic constants (GPa) of

{C e}_{x} {N i}_{3 - x} {S n}_{4}

(x = 0, 0.5, 1, 1.5, 2).

Phase	Reference	$C_{11}$	$C_{12}$	$C_{13}$	$C_{22}$	$C_{23}$	$C_{33}$	$C_{44}$	$C_{55}$	$C_{66}$	$C_{15}$	$C_{25}$	$C_{35}$	$C_{46}$
${N i}_{3} {S n}_{4}$	This work	181.06	68.70	60.12	146.08	75.24	162.32	62.74	60.61	47.03	−24.46	11.23	−11.34	14.92
	Calculation [20]	176.60	64.30	57.40	146.00	76.10	169.70	60.80	49.50	46.70	−22.90	7.00	−9.6	6.90
	Calculation [18]	155.48	70.68	69.34	164.33	68.26	149.86	62.74	59.99	59.95	−21.97	13.99	−8.73	4.90
${C e}_{0.5} {N i}_{2.5} {S n}_{4}$	This work	168.12	65.37	60.08	143.16	65.21	157.03	55.11	53.51	43.32	−17.6	12.42	−9.71	11.21
${C e}_{1} {N i}_{2} {S n}_{4}$	This work	152.87	63.28	59.21	122.79	64.52	140.27	52.66	52.91	43.67	−16.88	11.23	−11.34	9.73
${C e}_{1.5} {N i}_{1.5} {S n}_{4}$	This work	140.13	55.79	50.12	119.17	73.30	137.71	51.64	50.49	43.21	−16.02	10.07	−15.12	9.27
${C e}_{2} {N i}_{1} {S n}_{4}$	This work	125.31	45.61	40.27	93.22	63.57	104.28	51.13	40.53	32.01	−6.11	7.01	−11.09	4.38

Table 3. Elastic compliance constants (GPa) of

{C e}_{x} {N i}_{3 - x} {S n}_{4}

(x = 0, 0.5, 1, 1.5, 2).

Table 3. Elastic compliance constants (GPa) of

{C e}_{x} {N i}_{3 - x} {S n}_{4}

(x = 0, 0.5, 1, 1.5, 2).

Phase	$S_{11}$	$S_{12}$	$S_{13}$	$S_{22}$	$S_{23}$	$S_{33}$	$S_{44}$	$S_{55}$	$S_{66}$	$S_{15}$	$S_{25}$	$S_{35}$	$S_{46}$
${N i}_{3} {S n}_{4}$	0.0076	−0.0033	−0.0010	0.0108	−0.0041	0.0086	0.0172	0.0190	0.0232	0.0035	−0.0041	0.0019	−0.0055
${C e}_{0.5} {N i}_{2.5} {S n}_{4}$	0.0081	−0.0033	−0.0016	0.0103	−0.0033	0.0084	0.0192	0.0210	0.0244	0.0031	−0.0041	0.0018	−0.0050
${C e}_{1} {N i}_{2} {S n}_{4}$	0.0093	−0.0040	−0.0018	0.0126	−0.0045	0.0102	0.0198	0.0216	0.0239	0.0034	−0.0049	0.0026	−0.0044
${C e}_{1.5} {N i}_{1.5} {S n}_{4}$	0.0095	−0.0043	−0.0008	0.0156	−0.0075	0.0121	0.0201	0.0238	0.0241	0.0036	−0.0067	0.0049	−0.0043
${C e}_{2} {N i}_{1} {S n}_{4}$	0.0100	−0.0044	−0.0010	0.0220	−0.0126	0.0184	0.0198	0.0283	0.0316	0.0020	−0.0079	0.0071	−0.0027

Table 4. Calculated elastic modulus and relevant values of

{C e}_{x} {N i}_{3 - x} {S n}_{4}

alloys (unit in GPa).

Table 4. Calculated elastic modulus and relevant values of

{C e}_{x} {N i}_{3 - x} {S n}_{4}

alloys (unit in GPa).

Phase	Reference	$B_{V}$	$G_{V}$	$B_{R}$	$G_{R}$	$B$	$G$	$E$	$v$	H	B/G
${N i}_{3} {S n}_{4}$	This work	99.15	53.03	98.58	46.87	99.15	49.95	128.31	0.284	7.18	1.98
	Calculation [48]	-	-	-	-	100.86	49.89	128.48	0.287	-	2.02
	Calculation [47]	-	-	-	-	100.19	51.62	132.15	0.280	-	1.94
	Calculation [18]	98.47	53.96	97.56	49.42	98.01	53.96	136.78	0.270	-	1.82
${C e}_{0.5} {N i}_{2.5} {S n}_{4}$	This work	94.50	48.87	93.94	44.98	94.22	46.92	120.72	0.286	6.68	2.00
${C e}_{1} {N i}_{2} {S n}_{4}$	This work	88.21	45.38	87.49	41.02	87.85	43.19	111.34	0.289	6.08	2.03
${C e}_{1.5} {N i}_{1.5} {S n}_{4}$	This work	83.89	43.56	82.75	37.22	83.32	40.39	104.32	0.291	5.61	2.06
${C e}_{2} {N i}_{1} {S n}_{4}$	This work	69.07	36.29	68.77	29.27	68.92	32.78	84.88	0.295	4.49	2.10

Table 5. The Young’s modulus (E) and hardness of

{N i}_{3} {S n}_{4}

were obtained through experiments and calculations. (unit in GPa).

Table 5. The Young’s modulus (E) and hardness of

{N i}_{3} {S n}_{4}

were obtained through experiments and calculations. (unit in GPa).

Phase	Reference	$E$	H
${N i}_{3} {S n}_{4}$	This work	128.31	7.18
${N i}_{3} {S n}_{4}$	Experiment [19]	124.6	5.81
${N i}_{3} {S n}_{4}$	Experiment [49]	123.4~134.0	6.1~7.0
${N i}_{3} {S n}_{4}$	Experiment [18]	135.3 ± 8.4	5.0 ± 0.63
${(C o, N i)}_{3} {S n}_{4}$	Experiment [19]	132.3	5.98
${(C u, N i)}_{3} {S n}_{4}$	Experiment [18]	126.3 ± 7.6	4.7 ± 0.72

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Zhao, R.; Cao, Y.; He, J.; Chen, J.; Liu, S.; Yang, Z.; Lin, J.; Chang, C. First-Principles Study on the Mechanical Properties of Ni₃Sn₄-Based Intermetallic Compounds with Ce Doping. Coatings 2025, 15, 59. https://doi.org/10.3390/coatings15010059

AMA Style

Zhao R, Cao Y, He J, Chen J, Liu S, Yang Z, Lin J, Chang C. First-Principles Study on the Mechanical Properties of Ni₃Sn₄-Based Intermetallic Compounds with Ce Doping. Coatings. 2025; 15(1):59. https://doi.org/10.3390/coatings15010059

Chicago/Turabian Style

Zhao, Ruisheng, Yan Cao, Jinhu He, Jianjun Chen, Shiyuan Liu, Zhiqiang Yang, Jinbao Lin, and Chao Chang. 2025. "First-Principles Study on the Mechanical Properties of Ni₃Sn₄-Based Intermetallic Compounds with Ce Doping" Coatings 15, no. 1: 59. https://doi.org/10.3390/coatings15010059

APA Style

Zhao, R., Cao, Y., He, J., Chen, J., Liu, S., Yang, Z., Lin, J., & Chang, C. (2025). First-Principles Study on the Mechanical Properties of Ni₃Sn₄-Based Intermetallic Compounds with Ce Doping. Coatings, 15(1), 59. https://doi.org/10.3390/coatings15010059

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First-Principles Study on the Mechanical Properties of Ni₃Sn₄-Based Intermetallic Compounds with Ce Doping

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1. Introduction

2. Structural Characteristics and Stability

3. Mechanical Properties

4. Electronic Structure

5. Conclusions

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First-Principles Study on the Mechanical Properties of Ni3Sn4-Based Intermetallic Compounds with Ce Doping

Abstract

1. Introduction

2. Structural Characteristics and Stability

3. Mechanical Properties

4. Electronic Structure

5. Conclusions

Author Contributions

Funding

Institutional Review Board Statement

Informed Consent Statement

Data Availability Statement

Conflicts of Interest

References

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First-Principles Study on the Mechanical Properties of Ni₃Sn₄-Based Intermetallic Compounds with Ce Doping