First-Principles Study on the Elastic Mechanical Properties and Anisotropies of Gold–Copper Intermetallic Compounds

Abstract

In this study, first-principles calculations were utilized to investigate the lattice constants, elastic constants, and mechanical properties of gold–copper (Au–Cu) intermetallic compounds (IMCs), including AuCu₃, AuCu, and Au₃Cu. We also verified the direction dependence of the Young’s modulus, shear modulus, and Poisson’s ratio of the compounds. The calculated lattice parameters agreed with the experimental data, and the single-crystal elastic constants, elastic modulus E, shear modulus G, bulk modulus B, and Poisson’s ratio ν were calculated. For the Young’s and shear moduli, AuCu₃ showed the highest anisotropy, followed by AuCu and Au₃Cu. The Poisson’s ratios of AuCu₃ and Au₃Cu crystals were isotropic on (100) and (111) crystal planes and anisotropic on the (110) crystal plane. However, the Poisson’s ratio of the AuCu crystal was anisotropic on (100) and (111) crystal planes and isotropic on the (110) crystal plane.

1. Introduction

Gold–copper (Au–Cu) alloy systems are binary mixtures with high electrical and thermal conductivities and excellent mechanical strength and chemical stability, and they are widely used in catalysis, electronics industries, and biological materials [1,2]. There are three Au–Cu intermetallic compounds (IMCs), including AuCu₃, AuCu, and Au₃Cu, in the Au–Cu alloy phase diagram reported by Okamoto et al. [3]. Janczak et al. [4] employed X-ray powder diffraction to study the composition and structure of Au–Cu IMCs during annealing and confirmed the presence of these three compounds. Ravi et al. [5] carried out mutual diffusion experiments of Au–Cu system at different temperatures. Singh et al. [6] examined the alloy behavior of Au–Cu via transmission electron microscopy and high-resolution phase-contrast microscopy. However, to date, the physical and mechanical properties of the three IMCs have not been determined due to the difficulty in obtaining pure samples of sufficient sizes. Several first-principles simulation studies on Au–Cu IMCs have been reported. Mohri et al. [7] conducted a complete phase stability analysis of Au–Cu IMCs. Xie et al. [8] calculated the potential energies, heat of formation, and critical temperatures of order–disorder transitions of AuCu₃, AuCu, and Au₃Cu IMCs and AuCu₃-, AuCu-, and Au₃Cu-type ordered alloys with maximal ordering degrees. Ozolins et al. [9] investigated the phase stability, thermodynamic properties, and bond lengths of Au–Cu alloys. Hu et al. [10] studied the stability and thermal properties of AuCu₃, AuCu, and Au₃Cu and calculated the phonon spectrum and phonon density of states. Kong et al. [11] evaluated the structure, elasticity, and thermodynamic properties of AuCu₃, AuCu, and Au₃Cu.

Although several studies on Au–Cu IMCs have been reported, most of them do not consider the anisotropy of Au–Cu IMCs. However, the abnormal growth of crystal grains, transformation and formation of material structure, and formation of microcracks are closely related to the anisotropy of the material [12,13]. There is, therefore, a need for further research on the elastic mechanical properties and anisotropy to increase the applications of Au–Cu IMC and improve the reliability and structure of designs. In this study, we calculated the elastic constants of monocrystalline AuCu₃, AuCu, and Au₃Cu based on the first-principles method. Next, according to the Voigt–Reuss–Hill approximation, we obtained the Young’s, bulk, and shear moduli and the Poisson’s ratio of polycrystalline IMCs. Finally, we examined the directional dependence and anisotropic degree of IMCs.

2. Methods and Computational Details

All first-principles calculations for Au–Cu IMCs were performed using the Cambridge Serial Total Energy Package (CASTEP) code based on the density functional theory (DFT) [14]. The generalized gradient approximation (GGA) with the Perdew–Burke–Ernzerhof (PBE) functional [15] was selected to estimate the exchange-correlation energy for Au and Cu. The kinetic energy cutoff and self-consistent field tolerance for plane waves were set at 440 eV and 1.0 × 10⁻⁶ eV/atom, respectively [11]. For different Au–Cu alloy structures, different Monkhorst–Pack grids [16] were used to sample the Brillouin zone to produce different k-points in each structure. The k-point sampling in the first irreducible zones of AuCu₃, AuCu, and Au₃Cu was 8 × 8 × 8, 9 × 9 × 7, and 8 × 8 × 8, respectively. All k-point settings were convergent with respect to total energy (see Appendix A). The Broyden–Fletcher–Goldfarb–Shannon (BFGS) algorithm [17] was used to optimize the space group and lattice constants of the Au–Cu IMCs. The convergence tolerance of energy and maximum force were set at 4.0 × 10⁻⁶ eV/atom and 0.01 eV/Å. The maximum displacement was set at 4.0 × 10⁻⁴ Å. Figure 1 shows the unit cells of the three Au–Cu IMCs considered herein.

3. Simulation Methods

3.1. Lattice Constants and Elastic Properties

AuCu₃, AuCu, and Au₃Cu were investigated, and

Table 1. Calculated and experimental lattice constants of Au–Cu IMCs.

Phase	Ref.	Crystal System	Space Group	k-Points Mesh	a(Å)	c(Å)
AuCu3	This work Exp. [18] error/%	Cubic	Pm-3m	8 × 8 × 8	3.778589 3.747000 0.84
AuCu	This work Exp. [19] error/%	Tetragonal	P4/mmm	9 × 9 × 7	2.840972 2.806000 1.25	3.708597 3.67000 1.05
Au3Cu	This work Exp. [20] error/%	Cubic	Pm-3m	8 × 8 × 8	4.042688 3.965000 1.96

lists the calculated and experimental lattice constants. The calculated values are consistent with the experimental results with an average deviation of less than 2%.

Elastic constants are vital for crystals. They can correlate the microscopic properties of materials with macroscopic mechanical behaviors. Elastic constants can be calculated using Hooke’s law based on the stress–strain relationship:

$${\sigma }_{ij}=C_{ijkl}{\epsilon }_{ij}$$

(1)

where C_ijkl represents the elastic constant, also known as the stiffness matrix. The stress–strain matrix can be written as follows:

$$\left(\begin{array}{l}{\sigma }_1\\ {\sigma }_2\\ {\sigma }_3\\ {\tau }_1\\ {\tau }_2\\ {\tau }_3\end{array}\right)=\left(\begin{array}{cccccc}{C}_{11}& C_{12}& C_{13}& C_{14}& C_{15}& C_{16}\\ & C_{22}& C_{23}& C_{24}& C_{25}& C_{26}\\ & & C_{33}& C_{34}& C_{35}& C_{36}\\ & & & C_{44}& C_{45}& C_{46}\\ & Sym.& & & C_{55}& C_{56}\\ & & & & & C_{66}\end{array}\right)\left(\begin{array}{l}{\epsilon }_1\\ {\epsilon }_2\\ {\epsilon }_3\\ {\gamma }_1\\ {\gamma }_2\\ {\gamma }_3\end{array}\right)$$

(2)

where σ_i, τ_i, ε_i, and γ_i are the normal stress, shear stress, normal strain, and shear strain, respectively. The elastic constants of AuCu₃, AuCu, and Au₃Cu calculated herein and the reported values are listed in

Table 2. Calculated elastic constants of Au–Cu IMCs (GPa).

Ref.	Phase	C11	C33	C44	C66	C12	C13
This work	AuCu3	173.9		65.0		124.6
Ref. [10]	AuCu3	180.9		65.1		119.3
This work	AuCu	231.7	158.1	46.1	29.2	89.0	117.6
Ref. [11]	AuCu	229.8	159.6	45.8	33.3	90.0	118.4
This work	Au3Cu	165.2		18.8		131.8
Ref. [11]	Au3Cu	165.4		23.9		128.7

. The two sets of data are in good agreement. The elastic flexibility matrix was obtained by inverting the elastic stiffness matrix (i.e., S_ij = [C_ij]⁻¹). According to the symmetry of Au–Cu IMCs, to express the elastic flexibility matrix, there are three S_ij (i.e., S₁₁, S₁₂, and S₄₄) for AuCu₃ and Au₃Cu and six S_ij (i.e, S₁₁, S₁₂, S₁₃, S₃₃, S₄₄, and S₆₆) for AuCu. Herein, the calculated values of S₁₁, S₁₂, and S₄₄ for AuCu₃ were 0.01430, −0.00597, and 0.015380, respectively, and those for Au₃Cu are 0.02075, −0.00921, and 0.05313, respectively. For AuCu, S₁₁, S₁₂, S₁₃, S₃₃, S₄₄, and S₆₆ were 0.00693, −0.00007, −0.00510, 0.01392, 0.02170, and 0.03424, respectively.

Before calculating the Young’s, shear, and bulk moduli and the Poisson’s ratio of a lattice, it is important to examine its mechanical stability. For the cubic crystals, AuCu₃ and Au₃Cu, the mechanical stability criteria are as follows [21]:

$$C_{11}-C_{12}>0, C_{11}>0, C_{44}>0, C_{11}+2C_{12}>0$$

(3)

For the tetragonal crystal AuCu, the criteria are [22]:

$$\begin{array}{l}{C}_{11}>0,{ \mathrm{C}}_{33}>0, C_{44}>0, C_{66}>0, C_{11}-C_{12}>0, \\ C_{11}+C_{33}-2C_{13}>0, 2C_{11}+C_{33}+2C_{12}+4C_{13}>0\end{array}$$

(4)

According to the elastic constants listed in Table 2, the lattices of AuCu₃, AuCu, and Au₃Cu are stable. Moreover, the Young’s, shear, and bulk moduli and the Poisson’s ratio play decisive roles in evaluating the mechanical properties of the materials. Herein, the Voigt–Reuss–Hill (VRH) method [23] was employed to approximate the elastic moduli. The calculation of the elastic moduli based on the VRH approximation depends on the type of crystal. For the cubic crystals AuCu₃ and Au₃Cu, the bulk moduli, B_V and B_R, and shear moduli, G_V and G_R, can be calculated using Equations (5)–(8), respectively [24]:

$$B_V=\frac{1}{3}\left(C_{11}+2C_{12}\right)$$

(5)

$$G_V=\frac{1}{5}\left(C_{11}-C_{12}+3C_{44}\right)$$

(6)

$$B_R=\frac{1}{3S_{11}+6S_{12}}$$

(7)

$$G_R=\frac{15}{4S_{11}-4S_{12}+3S_{44}}$$

(8)

For the tetragonal crystal AuCu, B_V, B_R, G_V, and G_R can be calculated using Equations (9)–(13), respectively [25]:

$$B_V=\frac{1}{9}\left(2C_{11}+2C_{12}+4C_{13}+C_{33}\right)$$

(9)

$$B_R=\frac{C_{33}\left(C_{11}+C_{12}\right)-2C_{13}^2}{C_{11}+C_{12}+2C_{33}-4C_{13}}$$

(10)

$$G_V=\frac{1}{15}\left(2C_{11}-C_{12}-2C_{13}+C_{33}+6C_{44}+2C_{66}\right)$$

(11)

$$G_R=\frac{15}{18B_V/C^2+6/\left(C_{11}-C_{12}\right)+6/S_{44}+3/C_{66}}$$

(12)

$$C^2=\left(C_{11}+C_{12}\right)C_{33}-2C_{13}^2$$

(13)

where B_V, B_R, G_V, and G_R are the upper and lower limits of the polycrystalline bulk modulus B and shear modulus G, respectively. The calculated bulk and shear moduli are the arithmetic average of the two limits [26]. The bulk modulus B and shear modulus G are expressed as follows:

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

(14)

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

(15)

Next, Young’s modulus E and Poisson’s ratio v can be calculated from B and G as Equations (16) and (17), respectively:

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

(16)

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

(17)

The calculated B, G, E, and v are listed in

Table 3. Calculated bulk modulus B, shear modulus G, Young’s modulus E, B/G ratio, Poisson ratio ν, and Vickers hardness H_V of Au–Cu IMCs.

Phase	B (GPa)			G (GPa)			B/G	E (GPa)	ν	HV (GPa)
	BV	BR	B	GV	GR	G
AuCu3	141.03	141.03	141.03	48.88	39.31	44.09	3.20	119.80	0.358	2.919
AuCu	141.08	138.37	139.73	44.10	38.94	41.52	3.37	113.34	0.365	2.611
Au3Cu	142.93	142.93	142.93	17.97	17.91	17.94	7.97	51.65	0.440	0.462

. The bulk modulus reflects the resistance of a material to external uniform compression in an elastic system and is related to the elasticity of the chemical bond. Herein, the bulk moduli of the three IMCs were similar. The magnitude of the Young’s modulus indicates the stiffness of the material. The higher the Young’s modulus of a material, the less likely it is to deform. Table 3 shows that the Young’s modulus of Au–Cu IMCs decreased with increasing Au content (from AuCu₃ to Au₃Cu), and the shear modulus showed a similar trend. Poisson’s ratio is an elastic constant that reflects the lateral deformation of a material. The calculated results show that Poisson’s ratio increased with an increase in Au content in the three IMCs (Table 3). Vickers hardness H_V is calculated using Equation (18) [27]:

$$H_V=0.92{\left(G/B\right)}^{1.3137}{G}^{0.708}$$

(18)

The calculated H_V is listed in Table 3. The Vickers hardness Hv and Young’s modulus E of Au–Cu IMCs showed the same trend (i.e., AuCu₃ > AuCu > Au₃Cu). Pugh [28] established a ductility index (B/G ratio) to evaluate the ductility of materials. High values of B/G indicate high ductility, and vice versa. If the ratio is higher than 1.75, the material is ductile; otherwise, it is brittle. Herein, the B/G ratios of the three Au–Cu IMCs were higher than 1.75, implying that they are ductile materials. The ductility increased with the Au content. The Poisson’s ratio ν is also related to B/G. A material with ν greater than 0.26 is considered ductile [29]. For the Au–Cu IMCs, B/G was greater than 1.75, and ν was greater than 0.26, confirming that they are ductile.

3.2. Elastic Anisotropy

Elastic anisotropy determines many basic properties of materials and is important for predicting the fracture toughness of materials. The universal anisotropy index A^U and percent anisotropy indices of compression and shear (A_B and A_G) are used to evaluate the elastic anisotropy of a material, and they are expressed as follows [11]:

$$A^U=5\frac{G_V}{G_B}+\frac{B_V}{B_R}-6$$

(19)

$$A_B=\frac{B_V-B_R}{B_V+B_R}\times 100\%$$

(20)

$$A_G=\frac{G_V-G_R}{G_V+G_R}\times 100\%$$

(21)

For A^U, A_B, and A_G, if the value is 0, the crystal is isotropic. The greater their deviation from 0, the higher the degree of anisotropy. Herein, A^U of AuCu₃ was 1.217, indicating that AuCu₃ is anisotropic. The A_B and A_G were 0 and 10.85, respectively, indicating that AuCu₃ has no compression but shear anisotropy. The A^U, A_B, and A_G for AuCu were 0.682, 0.97, and 5.27, respectively, indicating that AuCu has lower universal, compression, and shear anisotropy. The A^U for Au₃Cu was 0.017, indicating that the degree of universal anisotropy of Au₃Cu is low. Also, the A_B and A_G for Au₃Cu were 0 and 0.17, respectively, indicating that the compound has no compressive anisotropy and weakest shear anisotropy.

To further investigate the tangential anisotropy of Au–Cu IMCs, we employed the anisotropy factors A₁, A₂, and A₃. The index of A₁ represented the shear anisotropy factor between [011] and [010] crystal orientations on the (100) crystal plane, Similarly, A₂ was the shear anisotropy factor between [101] and [001] orientations on the (010) crystal plane, and A₃ was that between [110] and [010] orientations on the (001) crystal plane. For cubic crystals, A₁, A₂, and A₃ were expressed as follows [11]:

$$A_1=A_2=A_3=\frac{4C_{44}}{C_{22}+C_{33}-2C_{13}}$$

(22)

For tetragonal crystal [11]

$$A_1=A_2=\frac{4C_{44}}{C_{11}+C_{33}-2C_{13}}, A_3=\frac{4C_{66}}{2C_{11}-2C_{12}}$$

(23)

The greater the difference between the anisotropy factors and 1, the higher the anisotropy of the crystal was. The calculated values are shown in

Table 4. Calculated anisotropic index of Au–Cu IMCs.

Phase	AU	AB	AG	A1	A2	A3
AuCu3	1.217	0	10.85	2.63	2.63	2.63
AuCu	0.682	0.97	5.27	1.19	1.19	0.41
Au3Cu	0.017	0	0.17	1.13	1.13	1.13

. The A₁, A₂, and A₃ for AuCu₃ were 2.63, which is the highest deviation from 1, indicating that AuCu₃ exhibits the highest shear anisotropy. For Au₃Cu, the A₁, A₂, and A₃ were 1.13, indicating negligible shear anisotropy. For AuCu, the A₁ and A₂ were 1.19, and A₃ was 0.41, indicating that AuCu has mild shear anisotropy, between that of AuCu₃ and Au₃Cu.

To further evaluate the anisotropy of Au–Cu IMCs, the Young’s modulus in three dimensions was calculated. For the cubic crystals AuCu₃ and Au₃Cu, the three-dimensional (3D) expression of E is given by Equation (24) [24]:

$$\frac{1}{E}=S_{11}-2\left(S_{11}-S_{12}-0.5S_{44}\right)\left(l_1^2{l}_2^2+l_2^2{l}_3^2+l_1^2{l}_3^2\right)$$

(24)

For the tetragonal crystal, AuCu [30]

$$\frac{1}{E}=S_{11}\left(l_1^4+l_2^4\right)+\left(2S_{13}+S_{44}\right)\left(l_1^2{l}_3^2+l_2^2{l}_3^2\right)+S_{33}{l}_3^4+\left(2S_{12}+S_{66}\right)l_1^2{l}_2^2$$

(25)

where l₁, l₂, and l₃ are the direction cosines of the a-, b-, and c-axes, respectively. Figure 2 and Figure 3 show E for the 3D surface and cross-section of the IMCs. The degree of anisotropy depends on the deviation of a geometrical body from the spherical shape. If a geometrical body is a sphere, it exhibits isotropy. Thus, the Young’s modulus of AuCu₃ showed the highest anisotropy, and that of Au₃Cu showed the lowest anisotropy, as shown in Figure 2. The Young’s modulus of AuCu₃ had a maximum value of 169.05 GPa in the <111> crystal directions and a minimum value of 69.94 GPa in the <100> directions. In comparison, Au₃Cu had a maximum Young’s modulus of 54.099 GPa in the <111> crystal orientation directions and a lowest Young’s modulus of 48.19 GPa in the <100> crystallographic directions. The tetragonal AuCu had a maximum Young’s modulus of 147.25 GPa in the [201], [021], [201], [021], [201], [021], [201], and [021] directions and a minimum Young’s modulus of 71.85 GPa in the [001] and [00] directions. In directions perpendicular to the normal direction to the (001) plane, the bulk modulus had a maximum value of 568.86 GPa, whereas it was minimum in the [001] and [001] crystal directions (269.50 GPa). The maximum and minimum Young’s moduli are listed in

Table 5. Anisotropy ratios and maximum and minimum Young’s moduli E for AuCu₃, AuCu, and Au₃Cu.

	AuCu3				AuCu				Au3Cu
	Whole	yz	xz	xy	Whole	yz	xz	xy	Whole	yz	xz	xy
Emax (GPa)	169.05	124.86	124.86	124.86	147.25	143.82	143.82	144.2	54.09	52.48	52.48	52.48
Emin (GPa)	69.94	69.94	69.94	69.94	71.85	71.85	71.85	83.41	48.19	48.19	48.19	48.19
Anisotropy ratios	2.42	1.79	1.79	1.79	2.05	2.00	2.00	1.73	1.12	1.09	1.09	1.09

. The anisotropy ratio E_max/E_min was employed to quantify the degree of anisotropy of the Young’s modulus, and the larger the anisotropy ratio, the higher the anisotropy was [31]. AuCu₃ had the highest E_max/E_min of 2.42, and Au₃Cu had the minimum value of 1.12.

Shear modulus G is related to l and n, which are mutually perpendicular vectors (Figure 4). A crystal shears at the plane perpendicular to the vector n. In a particular direction l, G changes with n, and it can be determined using Equation (26) [32].

$$\begin{array}{ll}\frac{1}{G}\left(\mathit{l},\mathit{n}\right)& =4\left[2S_{12}-\left(S_{11}+S_{22}-S_{66}\right)\right]l_1{n}_1{l}_2{n}_2+S_{66}{\left(l_1{n}_2-l_2{n}_1\right)}^2\\ & +4\left(l_1{n}_2+l_2{n}_1\right)\left[\left(S_{16}-S_{36}\right)l_1{n}_1+\left(S_{26}-S_{36}\right)l_2{n}_2\right]+4\left[2S_{23}-\left(S_{22}+S_{33}-S_{44}\right)\right]l_2{n}_2{l}_3{n}_3\\ & +4\left(l_2{n}_3+l_3{n}_2\right)\left[\left(S_{24}-S_{14}\right)l_2{n}_2+\left(S_{34}-S_{14}\right)l_3{n}_3\right]+4\left[2S_{31}-\left(S_{33}+S_{11}-S_{55}\right)\right]l_3{n}_3{l}_1{n}_1\\ & +4\left(l_3{n}_1+l_1{n}_3\right)\left[\left(S_{35}-S_{25}\right)l_2{n}_3+\left(S_{15}-S_{25}\right)l_1{n}_1\right]+S_{44}{\left(l_2{n}_3-l_3{n}_2\right)}^2+S_{55}{\left(l_3{n}_1-l_1{n}_3\right)}^2\\ & +2S_{45}\left(l_2{n}_3+l_3{n}_2\right)\left(l_3{n}_1+l_1{n}_3\right)+2S_{56}\left(l_3{n}_1+l_1{n}_3\right)\left(l_1{n}_2+l_2{n}_1\right)+2S_{64}\left(l_1{n}_2+l_2{n}_1\right)\left(l_2{n}_3+l_3{n}_2\right)\end{array}$$

(26)

l and n are related as follows:

$$n_1{l}_1+n_2{l}_2+n_3{l}_3=0$$

(27)

Usually, the maximum and minimum values in each direction are used to evaluate the shear modulus G. Figure 5 and Figure 6 show the 3D shape and cross-section of the shear modulus for the three Au–Cu IMCs. The shear anisotropy of AuCu₃ was the highest, followed by that of AuCu and Au₃Cu, which was consistent with A_G in Table 4. Furthermore, the maximum shear modulus of AuCu₃ was maximum in directions perpendicular to the (001), (010), and (100) crystal planes and minimum in the <111> direction. On the other hand, the minimum shear modulus was maximum in the <110> direction and maximum in the <100> direction. Similar to AuCu₃, the maximum shear modulus of Au₃Cu was minimum in <111> and maximum in directions perpendicular to the (001), (010), and (100) crystal planes. Additionally, the minimum shear modulus of Au₃Cu was maximum in the <100> direction and minimum in the <110> direction. As shown in Figure 5, the maximum shear modulus of AuCu was maximum in the [110], [110], [110], and [110] crystal directions and minimum in the [011], [101], [011], [101], [011], [101], [011], and [101] directions. On the other hand, the minimum shear modulus of AuCu was maximum in the [110], [110], [110], [110], [001], and [001] directions and minimum in the <111> and [100], [100], [010], and [010] crystal directions. The maximum and minimum values of the shear modulus G and the anisotropy ratio G_max/G_min for the compounds are listed in

Table 6. Anisotropy ratios and maximum and minimum share moduli G for AuCu₃, AuCu, and Au₃Cu.

	AuCu3				AuCu				Au3Cu
	Whole	yz	xz	xy	Whole	yz	xz	xy	Whole	yz	xz	xy
Gmax (GPa)	65.02	65.02	65.02	65.02	71.31	46.10	46.10	71.31	18.82	18.82	18.82	18.82
Gmin (GPa)	24.68	24.68	24.68	24.68	27.69	29.24	29.24	29.24	16.69	16.69	16.69	16.69
Anisotropy ratios	2.63	2.63	2.63	2.63	2.58	1.58	1.58	2.44	1.13	1.13	1.13	1.13

. AuCu₃ had the largest anisotropy ratio (2.63), and Au₃Cu had the lowest (1.13). For AuCu, the anisotropy ratio was 2.58 in all planes, and the anisotropy ratios of the yz and xz planes were the same (1.58), whereas that on the xy plane was 2.44.

Poisson’s ratio is an important index of elastic constant reflecting the transverse deformation of materials. For the cubic crystals AuCu₃ and Au₃Cu, the ratio is expressed as follows [33]:

$$\begin{array}{ll}v\left(hkl,\theta \right)& =\{S_{12}+\frac{S_0}{h^2+k^2+l^2}[{\left(\frac{h^{2}l}{\sqrt{h^2+k^2}\sqrt{h^2+k^2+l^2}}\cos \theta -\frac{hk}{\sqrt{h^2+k^2}}\sin \theta \right)}^2\\ & +{\left(\frac{k^{2}l}{\sqrt{h^2+k^2}\sqrt{h^2+k^2+l^2}}\cos \theta +\frac{hk}{\sqrt{h^2+k^2}}\sin \theta \right)}^2+{\left(\frac{l\sqrt{h^2+k^2}}{\sqrt{h^2+k^2+l^2}}\cos \theta \right)}^2]\}\\ & /\left[-S_{11}+2S_0\frac{{\left(hk\right)}^2+{\left(hl\right)}^2+{\left(lk\right)}^2}{{\left(h^2+k^2+l^2\right)}^2}\right]\end{array}$$

(28)

$$S_0=S_{11}-S_{12}-\frac{1}{2}{S}_{44}$$

(29)

For the tetragonal crystal, AuCu [34]:

$$\begin{array}{ll}v\left(hkl,\theta \right)& =-\{\frac{S_{11}}{h^2+k^2}\left[{\left(\frac{h^{2}L}{\sqrt{h^2+k^2+L^2}}\cos \theta -hk\sin \theta \right)}^2+{\left(\frac{k^{2}L}{\sqrt{h^2+k^2+L^2}}\cos \theta +hk\sin \theta \right)}^2\right]\\ & +\frac{S_{12}}{h^2+k^2}\left[{\left(\frac{hkl}{\sqrt{h^2+k^2+L^2}}\cos \theta -k^2\sin \theta \right)}^2+{\left(\frac{hkL}{\sqrt{h^2+k^2+L^2}}\cos \theta +h^2\sin \theta \right)}^2\right]\\ & +S_{13}\left[\left(h^2+k^2\right){\cos }^2\theta +L^2\right]\\ & +\frac{S_{66}}{h^2+k^2}\left(\frac{hkl}{\sqrt{h^2+k^2+L^2}}\cos \theta -k^2\sin \theta \right)\left(\frac{hkL}{\sqrt{h^2+k^2+L^2}}\cos \theta +h^2\sin \theta \right)\\ & +\left(S_{33}-2S_{13}-S_{44}\right)\frac{\left(h^2+k^2\right)L^2}{h^2+k^2+L^2}{\cos }^2\theta \}\\ & \times \frac{h^2+k^2+L^2}{S_{11}\left(h^4+k^4\right)+\left(2S_{12}+S_{66}\right)h^2{k}^2+\left(2S_{13}+S_{44}\right)\left(h^2+k^2\right)L^2+S_{33}{L}^4}\end{array}$$

(30)

where h, k, and l are the Miller indices. L = (a/c)l, where a and c are the lattice constants of the tetragonal crystal. The Poisson’s ratios of the three IMCs on three low-index crystal planes ((100), (110), and (111)) along different directions are shown in Figure 7, Figure 8 and Figure 9. The Poisson’s ratios of AuCu₃ on the (100) and (111) crystal planes were 0.417 and 0.300, respectively, and the shape was circular, indicating that it is isotropic. However, it showed high anisotropy on the (110) crystal plane, and the minimum and maximum values were 0.040 and 0.745 in [110] and [001] directions, respectively. For AuCu, the Poisson’s ratio was circular on the (110) crystal plane with a value of 0.428, indicating no anisotropy. AuCu exhibited anisotropy on the (100) plane with Poisson’s ratios of 0.019 and 0.736 for the [010] and [001] directions, respectively. AuCu also showed high anisotropy on the (111) plane with Poisson’s ratios of 0.518 and 0.260 in [110] and [112] crystal directions, respectively. Further, the Poisson’s ratio of Au₃Cu on (100) and (111) crystal planes was 0.444 and 0.437, respectively, indicating no significant anisotropy on the planes. However, it exhibited anisotropy on the (110) crystal plane, where the minimum and maximum values were 0.39 and 0.48 in the [110] and [001] directions, respectively. In summary, for Poisson’s ratio, Au₃Cu and AuCu₃ crystals exhibited isotropy on the (100) and (111) crystal planes and anisotropy on the (110) crystal plane, whereas AuCu was anisotropic on the (100) and (111) crystal planes and isotropic on the (110) crystal plane.

4. Conclusions

In this study, we employed the first-principles method to extensively explore the elastic mechanical properties and anisotropy of Au–Cu IMCs, and the following conclusions were drawn from the results:

• The Young’s modulus, shear modulus, and Poisson’s ratio increased with Au content (i.e., from AuCu₃ through AuCu to Au₃Cu). However, the bulk moduli of the compounds were similar.
• The Au–Cu IMCs exhibited excellent ductility in this order: Au₃Cu > AuCu > AuCu₃.
• For the Young’s modulus and shear modulus, the three Au–Cu IMCs were anisotropic, and among them, AuCu₃ showed the highest anisotropy.
• The Poisson’s ratios of Au₃Cu and AuCu₃ were isotropic on the (100) and (111) crystal planes and anisotropic on the (110) crystal plane. However, the Poisson’s ratio of the AuCu crystal was anisotropic on the (100) and (111) crystal planes and isotropic on the (110) crystal plane.