The Mechanical Properties, Structural Stability and Thermal Conductivities of Y, Sc Doped AuIn₂ by First−Principles Calculations

Abstract

In this paper, based on density functional theory, the structural stability and mechanical properties of AuIn₂ doped with RE (RE = Y, Sc) were investigated. The bulk modulus, shear modulus, Young’s modulus and Poisson’s ratio of the materials were calculated by Viogt−Reuss−Hill approximation. The calculation results show that Sc−S_Au (trace of Au substituted by Sc in AuIn₂), Y−S_Au and Y−S_In have stable structure, and Y−S_Au has obvious effect on the toughness indexes of AuIn₂ alloy. Furthermore, based on Clarke and Cahill modes, the lattice thermal conductivity of the intermetallic compound was calculated and shows the same tendency with the Debye temperature and fast heat transfer rate in the direction of [110].

1. Introduction

AuM₂ (M is Al, Ga, and In), the intermetallic compounds (IMCs) of stoichiometry AB₂ with CaF₂ type (fluorite) crystal structure [1,2], have been widely studied for their applications in wire bonding, chip packaging, aerospace, and medical equipment [3]. However, it is very brittle, fragile and difficult to process. Therefore, in order to obtain a new type of Au−based alloy with high strength and hardness to satisfy the requirements of structural and functional integration, a new type of Au alloy containing intermetallic compounds (IMCs) was developed, because IMCs can hinder the dislocation movement as the second phase in new Au−based alloy. The mechanical properties of IMCs are normally improved by adding trace elements. For example, B is added to Ni₃Al [4] and Fe or Ga to NiAl [5], thus their room temperature plasticity is significantly improved. Meanwhile, the addition of trace elements can also reduce the melting point and increase the fluidity. Yu [6] calculated the mechanical propriety of AuAl₂ by doping Y, Sc, Ga and In elements, and the experimental results show that Y, Sc doping has obvious toughening effect. Sinha [7] studied the phonon dynamics of AuAl₂ and AuGa₂ and found that there is a large deviation in their elastic constants although the bulk modulus of AuGa₂ and AuAl₂ alloys are very close. Rehmann S [8] demonstrated that AuIn₂ doped with Y, Sc, Si displays a striking blue color, nuclear ferromagnetism and type I superconductivity. Godwal [9] found that AuIn₂ starts to amorphize at 24 Gpa under high pressure and proved that the anomalous blue appearing at 2.7–2.8 Gpa is related to the content of Sc. Manjula [10] et al. studied the elastic and bond properties of zirconium and hafnium−doped Rh₃V IMCs. The weaker covalent bond strength resulted in greater ductility, and Rh₃Hf_0.125V_0.875 and Rh₃Hf_0.875V_0.125 were more ductile. Furthermore, Manjula [11] also studied the relationship between valence−electron concentrations, formation energy and elastic mechanical properties of cubic Rh₃AxTi1−x (A is V, Nb, Ta) and the results were further assessed by using charge density plots. Therefore, the doping of RE has crucial effects on the mechanical and thermal properties of the IMCs. Based on the first principle of Density Functional Theory (DFT), the structural stability, elasticity effect and thermal properties of AuIn₂ doped with rare earth elements were studied in order to provide theoretical information for the further application of AuIn₂ and related Au alloy.

2. Calculation Method

In this work, the CASTEP code [12] based on density functional theory [13] is used for the first−principles calculations [14]. According to the principle of selecting low energy, the CA−PZ scheme in the local density approximation (LDA) [15,16] is used to exchange related energy processing. The interaction between ion electronics and valence electrons were described by Ultrasoft pseudo−potentials (USPPs.) in which the valence electron configurations of Au are 4p⁴5d¹⁰6s¹, In 4d¹⁰5s²5p¹, Y 4d¹5s² and Sc 3d¹4s². To obtain the doping sites of RE (RE = Y, Sc), 2×2×2 supercells were built to ensure RE doping concentration of 1%, containing 32 Au atoms and 64 In atoms. The atomic positions and lattice parameters were optimized by the Broyden−Fletcher−Goldfarb−Shannon (BFGS) method. After the convergence test, the k point was set to 2×2×2 and cut−off energy was taken as 360 eV. During the structural optimization, the total energy convergence, the maximum stress and maximum force were set as 2 × 10⁻⁵ eV/atom, 0.1 GPa and 0.05 eV/Å, respectively.

3. Results and Discussion

3.1. Structural Properties and Formation Energy

As shown in Figure 1a, the crystal structure of AuIn₂ is fluorite (CaF₂) a cubic−type structure and belongs to the Fm3m space group. The AuIn₂ crystal cell (lattice constant a = 6.502 Å) [17] possesses 4 Au atoms and 8 In atoms, where the Au atoms are located in (0,0,0)a and the In atoms in (1/4, 1/4, 1/4) a and (1/4, −1/4, −1/4) a. In atoms are located in the tetrahedral interstice location of face−centered cubic Au. Therefore, there are three possible RE doping sites (RE = Y, Sc) as shown in Figure 1c, namely octahedral interstice location (RE−I): (Au₃₂REIn₆₄), the substitution of Au atoms (RE−S_Au: Au₃₁REIn₆₄) and the substitution of In atoms (RE−S_In: Au₃₂In₆₃RE). The optimized crystal lattice results of the doped and undoped systems are shown in

Table 1. Structural parameters and formation energy of AuIn₂ and doped system.

Phase	a (Å)	V (Å3)	Ef (eV)	Refs
Au4In8	6.479	271.972	−	[18]
Au4In8	6.502	274.879	−	present
AuIn2 (4.2 K)	6.483	272.476	−	[19]
AuIn2 (296 K)	6.508	275.640	−	[19]
AuIn2	6.507	275.513	−	[20]
Au32In64	12.963	2178.29	−	present
Sc−SAu	12.992	2192.95	−1.61	present
Sc−SIn	12.959	2176.28	−0.85	present
Sc−I	13.151	2274.45	4.99	present
Y−SAu	13.014	2204.11	−2.68	present
Y−SIn	12.974	2183.84	−1.15	present
Y−I	13.167	2282.76	6.01	present

. The deviations of the theoretically predicted AuIn₂ lattice parameters from other theoretical values [18] and experimental data [19,20] are within 3%, pointing to the reasonable and feasible setting of the computational parameters in this paper. Compared with AuIn₂, the crystal structure of the doped RE (RE = Y, Sc) systems show lattice distortions due to the difference of atomic radius between RE, Au and In. The lattice distortion caused by the replacement position is larger than that of the gap position, and the lattice distortion caused by Y doping is larger than that of Sc doping.

To understand the energy stability of rare earth element doping in neutral charge AuIn₂, the formation energy (E^f) of three possible RE doping sites (RE = Y, Sc) was calculated by the following relation [21,22,23]:

$$E^f\left(D\right)=E_{total}\left(D\right)-E_{total}\left[A{u}_{32}I{n}_{64},bulk\right]- {\displaystyle \sum }{n}_i{\mu }_i$$

(1)

where E^f (D) denotes the formation energy of doped impurities in the supercell. E_total (D) is the total energy of the supercell with doped impurities, while E_total [Au₃₂In₆₄, bulk] is the total energy of Au₃₂In₆₄ and${\mu }_i$ is the chemical potential of type i atoms. $n_i$states the number of type i atoms removed ($n_i<$0) or added ($n_i>$0) from the supercell during defect formation.

The formation energy of the Au atom substituted by RE (RE = Y, Sc) (RE−S_Au: Au₃₁REIn₆₄) and the indium atom substituted by Y (Y−S_In: Au₃₂YIn₆₃) is negative, while the formation energy of RE (RE = Y, Sc) doped at the octahedral interstice location and indium atoms substituted by Sc (Sc−S_In: Au₃₂ScIn₆₃) system is positive. The high formation energy is accompanied by poor thermodynamic stability, so the structure of Y−S_Au (Au₃₁YIn₆₄), as shown in Table 1, is the most stable. Stability was also certified by element electronegativity for the CaF₂ type IMCs, because the electronegativity difference value between Au (2.54) and Y (1.22) is high compared to the difference of other elements; the bigger electronegativity difference value between two elements, the higher stability of the IMCs. The most stable structure is Au substituted by Y from the calculated results presented in Table 1.

In

Table 2. The calculated elastic constants C_ij (in Gpa), the total number of population and bond length of AuIn₂ and doped system.

Phase	C11	C12	C44	Bond Population (Total)	Bond	Length (Å)
AuIn2	126.2	58.3	55.4	138.24	Au−In	2.807
Sc−SAu	111.7	63.0	45.6	135.44	Sc−In	2.875
Y−Su	111.7	66.5	43.7	134.24	Y−In	2.998
Y−SIn	121.1	59.8	45.8	135.36	Y−Au	2.848

, the total bond population of the Au site and In site occupied by the RE atom obviously decrease, while the bond length of the doped systems increase. The higher the value, the stronger the bonding interaction and the greater the interatomic force. Therefore, the order of ductility of the IMCs is AuIn₂ < Y−S_In < Sc−S_Au < Y−S_Au.

3.2. Elastic Constant of Single Crystal

According to Hooke’s law, the elastic constant was obtained from the stress–strain results [24]. In CASTEP code, each strain (ε) imposed on crystal would occur corresponding stress (σ), thus the step size of each strain is 4 and the maximum strain amplitude is 0.003 by the first−principal calculation. For different crystal structures, the number of independent elastic constants are determined by the structural symmetry. AuIn₂ belongs to the cubic crystal system with three independent elastic constants C₁₁, C₁₂ and C₄₄, as shown in Table 2.

To judge the structural stability, satisfaction of the corresponding innate mechanical stability criteria is also needed because different crystal structures have different stability criteria. For the cubic crystal system, the criteria of mechanical stability are as follows [25]:

$$C_{11}>0,C_{44}>0,C_{11}>\left|C_{12}\right|,\left(C_{11}+2C_{12}\right)>0$$

(2)

The values of C_ij in Table 2 are consistent with the above criterion of mechanical stability of cubic crystals, so it indicates that both AuIn₂ and the RE doped are mechanically stable in the ground state. The linear compressive strengths of the cubic crystals along the X, Y, and Z axes are equal, due to C₁₁ = C₂₂ = C₃₃ in the cubic crystals. In Table 2, the C₁₁ of RE−S_Au and Y−S_In are significantly lower than AuIn₂, so the linear compressive strength of the doped system decreases along the X, Y and Z axes. In a cubic crystal, C₁₂ represents the shear stress in the plane (110) along the direction of $\left[110\right]$. According to the values in Table 2, the C₁₂ of RE−S_Au and Y−S_In are greater than AuIn₂, which indicated that the doped system has greater resistance to deformation in the (110) plane for the higher shear stress in the$\left[110\right]$ direction. Meanwhile, C₄₄ represents the shear stress along the [001] direction on the (100) plane and the C₄₄ of RE−S_Au and Y−S_In is smaller than AuIn₂, as shown in Table 2, which indicated that the doped system has less shear stress on the (100) plane in the [001] direction.

3.3. Elastic Modulus of Polycrystalline

The elastic modulus of polycrystals are normally obtained from the Voigt−Reuss−Hill approximate calculation model [26,27,28]. For cubic AuIn₂ and its doping systems, the bulk and shear modulus of the Voigt, Reuss and Hill approximations were calculated by the following expressions [29]:

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

(3)

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

(4)

$$G_R={\left[\frac{4}{5}{\left(C_{11}-C_{12}\right)}^{-1}+\frac{3}{5}{C}_{44}{}^{-1}\right]}^{-1}$$

(5)

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

(6)

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

(7)

Young’s modulus E and Poisson’s ratio ν are calculated as follows:

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

(8)

$$\nu =\frac{3B-2G}{6B+2G}$$

(9)

The bulk modulus reflects the resistance of material to the external compression within the elastic range, namely the ability of material to resist deformation. The higher the bulk modulus, the higher the compressibility of the material. The values of the theoretical calculations of the elastic modulus, Poisson’s ratio and Cauchy pressure of AuIn₂ and its doped systems are shown in

Table 3. The calculated bulk modulus B (in Gpa), shear modulus G (in Gpa), Young’s modulus E (in Gpa), Cauchy pressure C₁₂−C₄₄ (in Gpa), G/B ratio, and Poisson’s ratio ν of AuIn₂ and doped system.

Phase	B	GV	GR	GH	E	GH/BH	C12−C44	ν
AuIn2	80.92	46.57	44.06	45.31	117.23	0.56	2.9	0.26
Sc−SAu	79.18	37.07	33.72	35.39	96.19	0.45	17.4	0.31
Y−SAu	80.87	34.95	31.31	33.13	91.66	0.41	22.8	0.32
Y−SIn	80.10	39.90	38.34	39.11	102.63	0.49	14.0	0.29

. AuIn₂ has the highest bulk modulus B 80.92 GPa, while that of the doped system is lower, which illustrates that the doped system has lower compressibility than AuIn₂. The slightly different bulk modulus of the doping systems mean that RE−S_Au (Y−S_Au and Sc−S_Au) have almost similar compression coefficients.

Shear modulus is the ratio of shear stress to shear strain within the range of the material elastic deformation. For cubic crystals, the shear modulus is related to the elastic constant C₄₄ and a larger C₄₄ corresponds to a larger shear modulus. The largest shear modulus of AuIn₂ 45.31 Gpa and the largest C₄₄ 55.4 GPa as shown in Table 3, indicating that the ability of AuIn₂ to resist shear strain is stronger than that of the doped RE.

Young’s modulus is the ratio of stress to deformation within a physical elastic limit according to Hooke’s law. The higher the Young’s modulus, the higher the stiffness of a material, i.e., the material is less likely to deform. As Table 3 shows, the Young’s modulus E of AuIn₂ is 117.23 Gpa, while the Young’s modulus of the doped system is significantly smaller than that of AuIn₂. The order of Young’s modulus of the doping system is Y−S_In > Sc−S_Au > Y−S_Au.

Poisson’s ratio refers to the ratio of the absolute value of the transverse positive strain to the axial positive strain when the material is under unidirectional tension or compression. Poisson’s ratio was used to describe the strength of covalent bonds of the materials [30]. A smaller Poisson’s ratio is favorable to the formation of covalent bonds and a higher hardness. When Poisson’s ratio is in the range of −1.0 to 0.5, it means that the corresponding materials are stable [31]. The Poisson ratios of AuIn₂, Sc−S_Au, Y−S_Au and Y−S_In are 0.264, 0.305, 0.320 and 0.290, respectively, and just within the range of −1.0 to 0.5. Therefore, AuIn₂ and doping systems are stable in the linear elastic range.

The ratio of G/B, Cauchy pressure C₁₂−C₄₄ can also be sed to characterize the toughness and brittleness properties of cubic materials [32,33]. A material is brittle if it satisfies ν < 0.26, G_H/B_H > 0.57, C₁₂−C₄₄ < 0. Otherwise, the material is malleable. As Table 3 shows, G_H/B_H, ν and C₁₂−C₄₄ of AuIn₂ doping systems meets the ductility condition, and the values of ν, C₁₂−C₄₄ for the Y−S_Au system are the largest, while G_H/B_H is the smallest.

As Figure 2 shows, Cauchy pressure C₁₂−C₄₄ has the same trend as ν but the G_H/B_H shows an opposite trend. The Y−S_Au system has the highest Cauchy pressure and ν but the lowest G_H/B_H which are especially different from AuIn₂. This indicates that the earth elements doping (RE = Sc, Y) in AuIn₂, especially the Y−S_Au, improve the plasticity of the alloy.

3.4. Anisotropy of Elastic Modulus

The elastic anisotropy is one of the important mechanical properties of crystalline materials and is related to the generation of microcracks [32,34]. The value of elastic anisotropy A^U = A_shear = 0 and A₁ = 1 when the material is isotropic. Once the value of A deviates from 0, it shows that the crystal exhibits elastic anisotropy. The universal elastic anisotropy index (A^U) [34], the anisotropy shear percentage (A_shear) [35] and the (100) plane shear factor (A₁) [36] are expressed as follows:

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

(10)

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

(11)

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

(12)

$$A_1=\frac{4C_{44}}{2C_{11}-2C_{12}}$$

(13)

The calculated values of anisotropy indices A^U, A_shear and A₁ of AuIn₂ and its doping system are shown in

Table 4. Calculated elastic anisotropic indexes (A^U, A_comp, A_shear, and A₁) of AuIn₂ and doped systems.

Phase	AU	Acomp	Ashear	A1
AuIn2	0.286	0	2.77	1.632
Sc−SAu	0.496	0	4.73	1.873
Y−SAu	0.582	0	5.49	1.934
Y−SIn	0.202	0	1.99	1.494

. The change trend of the anisotropy indices are shown in Figure 3.

The A^U value 0.582 of Y−S_Au and 0.202 of Y−S_In in Table 4 show that Y−S_Au reached the maximum elastic anisotropy, and Y−S_In the minimum. The results can also be verified by A_shear which shows the same trend as A^U of Y−S_Au from Figure 3a, while Y−S_In has the smallest A^U and A_shear values. The order of elastic anisotropy is Y−S_Au > Sc−S_Au > Y−S_In. This is consistent with the discussion of the bond length and formation energy in Table 1 and Table 2.

The values of shear anisotropy factor (A₁) and the (|1−A₁|) are used to characterize the shear anisotropy. The maximum value (0.934) of |1−A₁| in Y−S_Au indicates the highest shear anisotropy in (100), (010) and (001) planes, and the minimum (0.494) in Y−S_In the lowest in the same planes.

For cubic crystals, the anisotropy of bulk modulus B, shear modulus G and Young’s modulus E are calculated according to the following expressions (s_ij is elastic compliance constant) [37,38]:

$$\frac{1}{B}=\left(s_{11}+2s_{12}\right)\left(l_1^2+l_2^2+l_3^2\right)$$

(14)

$$\frac{1}{G}=s_{44}+4\left(s_{11}-s_{12}-\frac{s_{44}}{2}\right)\left(l_1^2{l}_2^2+l_2^2{l}_3^2+l_1^2{l}_3^2\right)$$

(15)

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

(16)

The elastic anisotropy is related to the different atoms’ arrangement in the crystal structure. In order to observe the elastic anisotropy more intuitively, the elastic anisotropy is calculated according to the above expressions and directly characterized by three−dimensional (3D) surface structure, and the anisotropy increases with the increase of spherical deviation as shown in Figure 4, Figure 5 and Figure 6.

The bulk modulus B of AuIn₂ and doping systems are isotropic because their 3D diagrams are almost spherical, as Figure 4 shows. This is consistent with the conclusion that the A_comp values of AuIn₂ and the doping systems are zero. Au atoms replaced by RE (RE = Y, Sc) elements and In atoms by Y do not change the anisotropy of the bulk modulus, as Figure 4b–d shows.

In Figure 5, the deviation from sphere of the three−dimensional shear modulus G graph shows that the shear modulus G of AuIn₂ and its doping system are anisotropic. The deviation is consistent with the maximum A_shear value of Y−S_Au (5.49%). In addition, the larger deformation of Sc−S_Au, but smaller than Y−S_Au, is related to the order of anisotropy index Y−S_Au > Sc−S_Au > AuIn₂. In contrast, the anisotropy index A_shear of Y−S_In is the smallest (1.99%) and the deviation of the three−dimensional graph from the sphere is small. Therefore, the order of shear modulus anisotropy is Y−S_Au > Sc−S_Au > AuIn₂ > Y−S_In.

As Figure 6 shows, the section protruding along the vertex angle of the Y−S_Au spatial distribution map displays a larger degree of deformation and higher degree of elasticity. In addition, the three−dimensional plots of Young’s modulus of Y−S_Au and Sc−S_Au are almost identical, which is related to their similar anisotropy index A^U (Y−S_Au: 0.582, Sc−S_Au: 0.496), the same as AuIn₂ and Y−S_In (AuIn₂: 0.286, Y−S_In: 0.202). In contrast, the anisotropy index A^U of Y−S_In is the smallest (0.202) and the almost round three−dimensional graph indicates that the elastic anisotropy of Y−S_In is the smallest. Therefore, the order of Young’s modulus anisotropy is Y−S_Au > Sc−S_Au > AuIn₂ > Y−S_In, which is consistent with the anisotropic order from A^U and A_shear.

To illustrate the elastic anisotropy of AuIn₂ and its doped system in the (001) plane, we plotted the projections of AuIn₂ and its doping system in the (001) plane with two major elastic modulus (bulk modulus and Young’s modulus), as shown in Figure 7. Meanwhile, the bulk modulus and Young’s modulus of AuIn₂ doped with RE in each direction are listed in

Table 5. The bulk modulus and Young’s modulus of AuIn₂ and doped systems in [100] and [110] directions (GPa) are calculated.

Phase	B		E
	[100]	[110]	[100]	[110]
AuIn2	80.87	80.87	89.32	94.89
Sc−SAu	80.32	80.32	66.16	73.4
Y−SAu	110.12	110.12	62.11	73.3
Y−SIn	80.45	80.45	81.60	86.56

. For the cubic system, the atoms have the same arrangement in [100], [010] and [001] due to lattice symmetry, so they have the same elastic constants (B and E) along [100], [010] and [001]. In Figure 7a, the projections of bulk modulus of AuIn₂ and the doping system are circular, thus the bulk modulus of AuIn₂ and the doped systems are isotropic. In Figure 7b, the Young’s modulus of the model of Y−S_Au displays the largest anisotropy. Moreover, in Table 5, the ratios of E[110]/E[100] of AuIn₂, Sc−S_Au, Y−S_Au and Y−S_In are 1.06, 1.11, 1.18 and 1.02, respectively. The deviation order of E[110]/E[100] of the above models is Y−S_Au > Sc−S_Au > AuIn₂ > Y−S_In.

3.5. Debye Temperature and Thermal Conductivity

The Debye temperature of the alloy were calculated according to the sound velocity (${\upsilon }_m$). The formula is as follows [39,40,41]:

$${\theta }_D=\frac{h}{k}{\left[\frac{3n}{4\pi }\left(\frac{N_A\rho }{M}\right)\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}{\upsilon }_m$$

(17)

$${\upsilon }_m={\left[\frac{1}{3}\left(\frac{2}{v_t^3}+\frac{1}{v_l^3}\right)\right]}^{\raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}, v_t={\left(\frac{G}{\rho }\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}, v_l={\left(\frac{3B+4G}{3\rho }\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$$

(18)

where$v_t$and $v_l$ are transverse and longitudinal sound velocities in the materials. h is Planck constant, approximately 6.626 × 10⁻³⁴ J∙s; k is Boltzmann constant 1.38 × 10⁻²³ J/K; n is the number of atoms; N_A is Avogadro constant, 1.38 × 10⁻²³ mol⁻¹. $\rho$ is the density and M is the molar mass of the material.

The longitudinal and transverse sound velocity, average sound velocity and Debye temperature ${\theta }_D$ of AuIn₂ including Y, Sc doped systems are listed in

Table 6. The density $\rho$ (g/cm³), sound velocity (longitudinal $v_l$ (m/s), transverse $v_t$ (m/s) and mean $v_m$ (m/s), acoustic Grüneisen constant ${\gamma }_a$ and Debye temperature ${\theta }_D$ (K) of AuIn₂ and doping system.

Phase	AuIn2	Sc−SAu	Y−SAu	Y−SIn
$\rho$	6.196	6.127	6.147	6.185
$v_l$	4776	4541	4510	4624
$v_t$	2704	2403	2321	2515
$v_m$	1941	1794	1753	1849
${\gamma }_a$	1.570	1.807	1.902	1.713
${\theta }_D$	243	216	209	227

. It can be seen from Table 6 that the higher the density of the material, the higher the sound speed and Debye temperature. In contrast to AuIn₂ with the highest density (6.196 g/cm³), Y−S_Au shows the smallest Debye temperature (209 K), longitudinal speed of sound (4510 m/s), transverse speed of sound (2321 m/s), and mean speed of sound (1753 m/s).

The anharmonic of atomic interactions in crystalline solid materials is usually characterized by the acoustic Grüneisen constant ${\gamma }_a$ which is related to the heat transfer and expansion of the material and can be evaluated in terms of $v_l$ and$v_t$ [41,42]:

$${\gamma }_a=\frac{3}{2}\left(\frac{3v_l^2-4v_t^2}{v_l^2+2v_t^2}\right)$$

(19)

The highest ${\gamma }_a$ (1.902) of Y−S_Au and the lowest ${\gamma }_a$ (1.570) of AuIn₂ in Table 6 indicate that the influence of external temperature and pressure on the lattice dynamics of the doping system are much greater than that of AuIn₂.

The thermal conductivity will decrease to the limit as the temperature increases, thus the minimum thermal conductivity of materials can be used to evaluate the thermal conductivity ability in terms of crystalline solid materials [43]. The Clarke model [44,45] and Cahill model [46] were used to study the minimum lattice thermal conductivity $k_{min}$ of the IMCs. The Clarke model is described as follow:

$$k_{min}=0.87k_B{M}_a^{-\frac{2}{3}}{E}^{\frac{1}{2}}{\rho }^{\frac{1}{6}}$$

(20)

$$M_a=\left[M/\left(m·N_A\right)\right]$$

(21)

where $M_{a },\rho$, $M$ and $m$ represent the average atomic mass, density, molar mass and total number of atoms of the unit cell, respectively.

Cahill’s model is described as follows:

$$k_{min}=\frac{k_B}{2.48}{n}^{\frac{2}{3}}\left(v_l+2v_t\right)$$

(22)

where n is the number of atoms per unit volume.

In

Table 7. Calculated thermal conductivities $k_{min}$ ($\mathrm{W}·{\mathrm{m}}^{-1}·{\mathrm{K}}^{-1}$ ) of AuIn₂ and doping systems.

Phase	Clarke Model				Cahill Model
	$${\mathit{M}}_{\mathit{a}} \left(10{-}^{22}\right)$$	$${\mathit{k}}_{\mathit{m}\mathit{i}\mathit{n}}$$	$${\mathit{k}}_{\mathit{m}\mathit{i}\mathit{n}\left[100\right]}$$	$${\mathit{k}}_{\mathit{m}\mathit{i}\mathit{n}\left[001\right]}$$	n (1028)	$${\mathit{k}}_{\mathit{m}\mathit{i}\mathit{n}}$$	$${\mathit{k}}_{\mathit{m}\mathit{i}\mathit{n}\left[100\right]}$$	$${\mathit{k}}_{\mathit{m}\mathit{i}\mathit{n}\left[110\right]}$$	$${\mathit{k}}_{\mathit{m}\mathit{i}\mathit{n}\left[111\right]}$$
AuIn2	2.362	0.461	0.403	0.415	4.359	0.702	0.723	0.901	0.700
Sc−SAu	2.336	0.420	0.348	0.367	4.305	0.639	0.665	0.830	0.642
Y−SAu	2.343	0.409	0.337	0.366	4.288	0.624	0.654	0.819	0.632
Y−SIn	2.358	0.432	0.385	0.390	4.324	0.662	0.677	0.860	0.661

, the highest $k_{min}$of AuIn₂ and the lowest value of Y−S_Au are consistent with the highest Debye temperature of AuIn₂ (243 K) and the lowest Debye temperature of Y−S_Au (209 K) correspondingly. Thus, the minimum thermal conductivity coefficient $k_{min}$ shows the same tendency as the Debye temperature ${\theta }_D$ because lower Debye temperatures correspond to lower lattice thermal conductivity [47].

In Clarke’s model, the $k_{min}$ of AuIn₂ doped with RE is lower than that of AuIn₂ because the atoms with heavy atomic mass are replaced by the lighter one. The order of atomic mass is Au > In > Sc > Y.

In Cahill’s model, the $k_{min}$ of AuIn₂ is larger than the RE doped systems because of the phonon scattering effect caused by the atom replacement. Furthermore, the $k_{min}$ in Cahill’s model is higher than that in the Clarke model because of the neglect of phonons in the latter.

Based on the anisotropic of sound speed ($v_l$, $v_{t1}$ and $v_{t2}$), $k_{min}$ was used to characterize the anisotropy of the minimum thermal conductivity in the Cahill model:

$$k_{min}=\frac{k_B}{2.48}{n}^{\frac{2}{3}} \left(v_l+v_{t1}+v_{t2}\right)$$

(23)

As shown in Table 7, the $k_{min\left[111\right]}$ calculated by the Cahill model is extremely different in the directions of [100], [100] and [111]. The maximum value (0.901) of $k_{\min \left[110\right]}$ and the minimum value (0.770) of $k_{min\left[111\right]}$ of AuIn₂ indicate the faster heat transfer rates in the direction of [110], which are due to the closest atom alignment along the [110] direction in the cubic crystal, and are the same for the doping systems.

4. Conclusions

In this paper, we have explored the structure stability, elasticity and thermal conductivity of AuIn₂ doped RE (RE = Sc, Y) by the first–principal calculations. According to the formation energies, together with the comparison to the element electronegativity difference value for the CaF₂ type IMCs, the most stable doping of RE (RE = Sc, Y) in AuIn₂ was Y−S_Au (traces of Au atom were replaced by Y), but the octahedral interstice location doping was unstable.

The value of elastic constants of both AuIn₂ and RE doped systems meet the criterion of mechanical stability and the doped systems have greater resistance to deformation along the $[110$] direction than along the [001] direction. Moreover, the higher Cauchy pressure and Poisson’s ratio but lower G_H/B_H of RE doping systems show that the RE elements in doping made the CaF₂ type IMCs show ductile characteristics, of which the Y−S_Au system is the most obvious. Furthermore, the increasing bond length of AuIn₂ after RE doping is consist with the mechanical calculation results.

The anisotropic indexes of elastic anisotropy (A^U, A_comp, A_sheer, and A₁), the 3D surface structure of the elastic modulus and the planar projection indicate that the anisotropic order of the elastic modulus is Y−S_Au > Sc−S_Au > AuIn₂ > Y−S_In. Furthermore, the anisotropy of minimum thermal conductivity $k_{\min }$ was calculated according to Clark and Cahill, and the results by Clarke models are in the same order as the corresponding Debye temperature, but opposite in terms of the elastic modulus. The maximum value of $k_{\min \left[110\right]}$ and the minimum value of $k_{\min \left[111\right]}$ are related to the different atomic mass from the Clarke model and to the different atom alignment, phonon scattering effect from the Cahill model.