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Structural, mechanical and thermodynamic properties of AuIn2 crystal under pressure: A first-principles density functional theory calculation
Accepted Manuscript Structural, Mechanical and Thermodynamic Properties of AuIn2 Crystal under Pressure: A First-Principles Density Functional Theory Calculation Ching-Feng Yu, Hsien-Chie Cheng, Wen-Hwa Chen PII: DOI: Reference:
S0925-8388(14)02165-3 http://dx.doi.org/10.1016/j.jallcom.2014.09.031 JALCOM 32142
To appear in:
Journal of Alloys and Compounds
Received Date: Revised Date: Accepted Date:
27 May 2014 2 September 2014 3 September 2014
Please cite this article as: C-F. Yu, H-C. Cheng, W-H. Chen, Structural, Mechanical and Thermodynamic Properties of AuIn2 Crystal under Pressure: A First-Principles Density Functional Theory Calculation, Journal of Alloys and Compounds (2014), doi: http://dx.doi.org/10.1016/j.jallcom.2014.09.031
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Structural, Mechanical and Thermodynamic Properties of AuIn2 Crystal under Pressure: A First-Principles Density Functional Theory Calculation
Ching-Feng Yua, Hsien-Chie Chengb*, and Wen-Hwa Chena* a
Department of Power Mechanical Engineering, National Tsing Hua University, Hsinchu 30013, Taiwan
b
Department of Aerospace and Systems Engineering, Feng Chia University, Taichung 40724, Taiwan
Abstract The structural, mechanical and thermodynamic properties of cubic AuIn2 crystal in the cubic fluorite structure, and also their temperature, hydrostatic pressure and direction dependences are investigated using first-principles calculations based on density functional theory (DFT) within the generalized gradient approximation (GGA). The optimized lattice constants of AuIn2 single crystal are first evaluated, by which its hydrostatic pressure-dependent elastic constants are also derived. Then, the hydrostatic pressure-dependent mechanical characteristics of the single crystal, including ductile/brittle behavior and elastic anisotropy, are explored according to the characterized angular character of atomic bonding, Zener anisotropy factor and directional Young’s modulus. Moreover, the polycrystalline elastic properties of AuIn2, *
The correspondence: [email protected] (H.C. Cheng), [email protected] (W.H. Chen) 1
such as bulk modulus, shear modulus and Young’s modulus, and its ductile/brittle and microhardness characteristics are assessed versus hydrostatic pressure. Finally, the temperature-dependent Debye temperature and heat capacity of AuIn2 single crystal are investigated by quasi-harmonic Debye modeling. The present results reveal that AuIn2 crystal demonstrates low elastic anisotropy, low hardness and high ductility. Furthermore, its heat capacity strictly follows the Debye T3-law at temperatures below the Debye temperature, and reaches the Dulong-Petit limit at temperatures far above the Debye temperature.
Keywords: Mechanical property, First-principles calculations, Elastic anisotropy, Ductile/brittle Nature, Polycrystalline elastic properties
1. Introduction Recently, evaluation of physical properties of intermetallic compounds (IMC) is an important issue in material science due to their properties have the significant influence on package interconnects in electronic devices. They can also present excellent magnetic, superconducting and chemical properties, due to their strong internal order and metallic bonding. In particular, Au-In system IMC has been the significant topic owing to their applications of microelectronic packaging. 2
The interfacial interactions between Au and In have been extensively investigated by many researches [1-7]. For example, Shohji et al. [1] investigated the growth of IMC in Au and In-48wt%Sn diffusion couples, and found that the AuIn2 layer is formed at the interface between Au and In-48Sn solder during isothermal aging at 70 °C, 90 °C and 110 °C. Sohn et al. [3] conducted the low temperature bonding for optoelectronic packaging by setting up Au-In eutectic system. The eutectic bonding was conducted at 180-210°C at a wafer scale and re-melting temperature measured using thermal-mechanical analysis (TMA) was 470 °C. Their results revealed that several intermetallic compounds, such as AuIn and AuIn2, are found after Au-In bonding by scanning electron microscopy (SEM) and energy dispersive x-ray (EDX) analyses. The interfacial reactions of an In-48wt%Sn solder joint with an electroplated Au/Ni ball grid array substrate at reflow temperature of 150 °C were investigated by Koo et al. [4] using several measurement techniques, such as SEM, transmission electron microscopy (TEM), EDX spectrometry, inductively coupled plasma-atomicemission spectroscopy and x-ray diffractometry. They found that two different IMC layers, AuIn and AuIn2, are created at the solder-substrate interface after one reflow. Chuang et al. [5] demonstrated that continuous AuIn2 and scallop-shaped AgIn2 IMC layers are formed at the interfaces between In-3Ag solder and Au/Ni/Cu pad and between that and Ag/Cu pad, respectively, during the reflow 3
process of In-3Ag solder ball grid array (BGA) packages with electroless nickel immersion gold (ENIG) and immersion silver (ImAg) surface finishes. Lian et al. [6] assessed the feasibility of forming Au-In IMCs, and showed that AuIn2 is developed at a low annealing temperature of 160°C and a short annealing time of 10 minutes. Song and Lu [7] investigated the IMC formation in an Au/In diffusion couple during chemical reaction at an isothermal temperature of 180 °C in vacuum. They indicated that In is fully exhausted and AuIn2 is observed in 2 minutes. Apart from that, the structural, electronic and dynamical properties of AuIn2 have received considerable attention due to its technological importance. This compound is a good metallic conductor, crystallizes in the cubic fluorite structure at ambient conditions [8,9] and holds superconducting properties at lower temperatures. Smeibidl et al. [10] reported that the AuIn2 is a good candidate for nuclear demagnetization as well as for the observation of nuclear magnetic ordering due to its large nuclear moment and its large molar nuclear Curie constant. Because the compound also possesses a small Korringa constant and good thermal conductivity, its applications may include nuclear susceptibility thermometry and nuclear refrigeration [11]. In addition, it is an ideal material for investigating the transition-metal 5d bands [12]. During the past four decades, the structural, electronic and dynamical properties of AuIn2 have also been explored by numerous experimental and theoretical studies 4
[9,12-21]. For instance, charge-transfer and valence-band behavior upon alloying in AuIn2 IMC have been studied by Sham et al. [13] using Au Mössbauer and x-ray photoemission measurements. Nelson et al. [14] studied the (1 0 0) surface of AuIn2 IMC through synchrotron-radiation-excited angle-resolved photoemission, and indicated that the Au 5d band of the compound was relatively flat. Hsu et al. [9] evaluated the structural, electronic, magnetic, and optical properties of AuIn2 using synchrotron-radiation-excited angle-resolved photoemission spectra. The anomalies of resistance, thermoelectric power (TEP) and universal equation of state (UEOS) in AuIn2 was observed by Godwal et al. [16]. They also found that an electronic topological transition (ETT) occurs in the pressure range of 2-4 GPa, and a structural phase transition takes place beyond 8 GPa to a monoclinic phase through imaging plate high-pressure angle-dispersive x-ray diffraction analysis. Garg et al. [18] predicted electronic topological transition in AuIn2 compound at high pressure based on the accurate X-ray diffraction data derived using a diamond anvil cell with the ELETTRA synchrotron source. For the theoretical investigations, Hsu et al. [19] calculated the binding energies of AuIn2 by the first-principles theory [22,23] using WIEN2K code [24]. Gao et al. [20] carried out a numerical procedure for assessing the Seebeck coefficient of AuIn2 based on its electronic band structure. The structural, electronic and dynamical properties of AuIn2 were explored by Uǧur and Soyalp [21] 5
through the plane-wave pseudopotential method within the density functional theory [22,23]. Li et al. [12] studied the electsronic band structure, transport properties, and lattice dynamic in AuIn2 under high pressure with the full potential linearized augmented plane wave and pseudopotential plane wave methods. Although the electronic properties, optical properties and dynamic properties of AuIn 2 have been widely studied, there is a lack of knowledge of its mechanical and thermodynamic properties. In the study, the mechanical and thermodynamic properties of AuIn2 in the cubic fluorite structure are for the first time systematically explored using first-principles density functional theory calculations. This investigation puts the focus on predicting the structural, mechanical, and thermodynamic characteristics, including lattice constants, elastic properties and anisotropy, brittle-ductile behavior, microhardness, Debye temperature and heat capacity, and most importantly, their temperature and pressure
dependences.
Furthermore,
the
hydrostatic
pressure-dependent
polycrystalline elastic properties of the crystal are also examined. The validity of the proposed theoretical models is demonstrated through comparison with the published experimental and theoretical data [6,7,9,12,17,25].
6
2. Computational details and theoretical method The Cambridge Serial Total Energy Package code (CASTEP) [26-30] based on the density functional theory (DFT) [22,23,31] is applied in all the present calculations. Furthermore, the ultrasoft pseudopotential formalism of Vanderbilt [32] is adopted to describe the interactions of valence electrons with ion cores, and the electron wave function is expanded in plane waves up to an energy cutoff of 360 eV for all the calculations. The exchange-correlation energy is evaluated using the generalized gradient approximation (GGA) of the Perdew-Burke-Ernzerhof (PBE) formalism [23], which is dependent on both the electron density and its gradient at each space point. To search the ground state, a quasi-Newton (variable-metric) minimization algorithm using the Broyden-Fletcher-Goldfarb-Shanno (BFGS) update scheme [33] is utilized, which provides a very efficient and robust way to explore the optimum crystal structure with a minimum energy. To sample the Brillouin zone [34], the Monkhorst-Pack k-point mesh is employed [35]. It is selected based on the convergence of the k-point mesh, where the change of total energy becomes less than 1 meV/atom [36,37]. In the study, the selected k-point mesh is 161616. The following convergence thresholds are utilized in the geometry optimization: 5×10-6 eV/atom for the total energy variation, the maximum ionic Hellmann-Feynman force within 0.01 eV/Å, the maximum stress within 0.02 GPa and the maximum ionic 7
displacement within 5×10-4 Å. The resulting stress with respect to the optimized crystal structure [38] is calculated by imposing a number of specified homogenous deformations with a finite value. The elastic coefficients can be then determined through a linear fit of the calculated resulting stress as a function of strain, where four strain amplitudes are used with the maximum value of 0.3%. To assess the thermal effects on the thermodynamic properties of the crystal, a quasi-harmonic Debye model is used, and periodic boundary conditions are applied in all the present calculations for predicting the physical properties of the crystal. In principle, the lattice contribution to the heat capacity can be expressed as, C (T ) k
where
( / kT ) 2 exp( / kT ) g ( )d , [exp( / kT ) 1]2
(1)
and k are the reduced Planck constant and the Boltzmann constant,
respectively, T is temperature, and g() denotes the number of states at frequency, and
is the density of states function. As soon as the phonon dispersion relation is derived, the density of states function can be determined. In the study, the phonon density of states and dispersion relation are calculated by using the finite displacement method [39]. This technique is derived based on numerical differentiation of forces on atoms that are computed for a number of unit cells with atomic displacements.
8
3. Results and discussion 3.1 Structural properties Structure optimization is first performed to determine the lattice parameters and the positions of the atoms of AuIn2 monocrystal in the stable ground state, based on the experimental data derived from X-ray diffraction (XRD) analysis [25], as shown in Fig. 1. This figure exhibits that it is a cubic system with Fm-3m space group. The calculated lattice parameters are shown in Table 1, and it is evident that there is a good agreement between the present calculations and the literature data [9,12,17,25]. The maximum difference among them is only about 3.2%, demonstrating the effectiveness of the proposed simulation model.
3.2 Mechanical properties The elastic constants of single-crystalline AuIn2 IMC nanocrystal can be obtained through the following generalized Hooke’s law, 6
{ i } [Cij ]{ j } ,
(2)
j 1
where {σi} and {εj} are the stress and strain vector, respectively. In the cubic system, [Cij] can be described as [40],
9
C11 C12 C12 0 C 12 C11 C12 0 C C12 C11 0 [Cij ] 12 0 0 C44 0 0 Symmetric 0 0 0 0 0 0 0
0 0 0 0 C44 0
0 0 0 . 0 0 C44
(3)
It is clear to see that there are three independent elastic constants in the stiffness matrix, and furthermore, the inverse of the stiffness matrix [Cij] yields the compliance matrix [Sij]. For a cubic system, the requirement of mechanical stability leads to the following restrictions [41] on the elastic constants, C11 2C12 0, C44 0, C11 C12 0 ,
(4)
The calculated elastic constants in Table 1 satisfy the above criteria, indicating that AuIn2 single crystal is a mechanical stable system. In addition, the influences of hydrostatic pressure on the elastic constants are further evaluated, and the calculated results are shown in Fig. 2. A quasi-linear relationship can be detected between the elastic constants and hydrostatic pressure as the applied hydrostatic pressure is in the range of 0-20 GPa. Generally speaking, angular character of atomic bonding is related to material characteristics, such as brittleness or ductility. Pettifor [42] reported that the angular character of atomic bonding in metals and compounds can be depicted through the Cauchy pressure, which is defined as C12–C44. It is noted that as the pressure is positive, the material is expected to be metallic; otherwise, it is a nonmetallic material 10
with directional bonding. Verification of this correlation has been demonstrated on some ductile materials, such as elemental metals and IMCs, including Al, Ni, Ni3Al and Pt3Al [42]. On the contrary, brittle semiconductor, such as Si and Ge, would possess directional bonding characteristics [42]. Table 1 shows that the Cauchy pressure is 30.4, suggesting that AuIn2 crystal possesses ductile nature. Moreover, hydrostatic pressure effect on the Cauchy pressure is also investigated, and the results are shown in Fig. 3. Evidently, an increasing hydrostatic pressure would increase the Cauchy pressure, also indicating the increase of the ductility of AuIn2 single crystal. To quantify the degree of anisotropy of a solid, the Zener anisotropy factor (Az) is usually applied, as shown below [27], Az
2C44 . C11 C12
(5)
As Az is equal to one, the nanomaterial is elastically isotropic; otherwise, it is elastically anisotropic, where a larger deviation from one would imply a higher degree of elastic anisotropy. The calculated Zener anisotropy factor for AuIn2 at zero pressure is listed in Table 1, which is around 0.73, suggesting that the crystal is not an elastic, isotropic material. In addition, Fig. 4 shows the relationship between hydrostatic pressure and Zener anisotropy factor, where the factor would increase linearly with hydrostatic pressure. The Zener anisotropy factor enhances from around 0.73 to 1.05 as hydrostatic pressure varies from 0 to 20 GPa. This implies that AuIn2 single crystal 11
is likely to become elastic and isotropic at a very high hydrostatic pressure. A three-dimensional surface representation of the elastic anisotropy of the cubic AuIn2 single crystal, expressed by the variation of the Young’s modulus of the crystal with crystal direction, is further constructed. The direction dependence of the Young’s modulus of a triclinic crystal system can be described as [43], E 1/ (s11 2(s11 s12 0.5s44 )(l12l22 l22l32 l32l12 ) ,
(6)
where l1, l2, and l3 represent the direction cosines with respect to the a, b and c directions of lattice and Sij (i, j = 1, 2, 3) stands for the compliance coefficients. Using the derived compliance coefficients, the directional Young’s moduli and their projections on the a-b, b-c, and a-c planes of the crystal at five different hydrostatic pressures (i.e., 0, 5, 10, 15 and 20 GPa) can be calculated, as shown in Fig. 5. An enhancement of the directional Young’s modulus of the crystal can be observed with the increase of hydrostatic pressure. The projected configurations of the directional Young’s moduli on the a-b, b-c and a-c planes reveal great similarity, denoting that the strength of AuIn2 would not change with crystallographic plane. Furthermore, it is noticeable that the projected configuration of the directional Young’s moduli would vary from a quasi-square to quasicircle shape as hydrostatic pressure rises from 0 to 20 GPa. This suggests that AuIn2 single crystal tends to become more isotropic at a higher hydrostatic pressure, and the result is consistent with that of the 12
pressure-dependent Zener anisotropy factor shown in Fig. 4. In general, many crystalline solids, such as IMCs, consist of numerous single crystals or grains with different crystal orientations, which are known as polycrystalline materials. Because the single crystals possess an identical property in every direction, they tend to randomly orientate with each other to form a quasi-isotropic material. To facilitate the comparison with the literature experimental data, which are typically polycrystalline results, the elastic properties of the polycrystalline aggregate of randomly orientated anisotropic single-crystalline AuIn2 nanocrystal are predicted through the following Voigt-Reuss scheme [44] together with the calculated single crystal data,
KV
1 C11 C22 C33 2(C12 C13 C23 ) , 9
(7)
K R S11 S22 S33 2(S12 S13 S23 ) , 1
GV
(8)
1 C11 C22 C33 (C12 C13 C23 ) 3(C44 C55 C66 ) , 15
(9) 1
GR 15 4 S11 S22 S33 ) 4( S12 S13 S23 3(S44 S55 S66 ) .
(10)
where KV and KR denote the upper (Voigt) and lower (Reuss) bounds of the bulk modulus (K) of the polycrystalline aggregate, respectively, and GV and GR represent those of the shear modulus (G) of the polycrystalline aggregate. In addition, the effective bulk and shear modulus can be calculated by using the Voigt-Reuss-Hill approximation [45], which is considered as geometric mean, 13
K KV K R ,
(11)
G GV GR .
(12)
By using the calculated polycrystalline bulk and shear modulus, the effective Young’s modulus (E) and Poisson’s ratio (ν) of the AuIn2 polycrystalline aggregate can be calculated as follows,
9 KG , 3K G 3K 2G v . 2(3K G ) E
(13) (14)
The obtained bulk modulus, shear modulus, Young’s modulus and Poisson’s ratio for the polycrystalline AuIn2 are shown in Table 1. It is noted that the present calculations are in good agreement with the literature experimental and theoretical data [6,12,17]. Additionally, their dependences on hydrostatic pressure are also evaluated and the results are presented in Fig. 6. It is shown that the increase of hydrostatic pressure would enlarge the polycrystalline elastic properties. Specifically, there is about 194.9%, 319.6% and 301.9% increase in the bulk modulus, shear modulus and Young’s modulus of polycrystalline AuIn2, as hydrostatic pressure increases from 0 to 20 GPa, revealing that hydrostatic pressure has less influence on the bulk modulus, as compared to the shear and Young’s modulus. Pugh [46] addressed that the ratio of bulk to shear modulus, K/G, can be utilized as an empirical index of ductility/brittleness for polycrystalline materials, and the 14
critical threshold value for distinguishing the physical properties of materials is around 1.75. As the ratio is up to 1.75, the polycrystal would possess a ductile character while it would have a brittle nature as the ratio is below 1.75. The computed K/G ratio for polycrystalline AuIn2 is about 3.84, as shown in Table 1, revealing that the polycrystal would behave like a ductile material. As this ductile compound material appears in solder interconnects, their drop impact reliability [47] should be significantly improved. Figure 7 further illustrates the K/G ratio as a function of hydrostatic pressure, and it is found that the level of ductility of AuIn2 crystal would descend with a decreasing K/G ratio. The stiffness of polycrystalline AuIn2 IMC can be investigated by calculating the microhardness parameter (H) [27], which is the resistance of the physical object against compression of the contacting parts, as described below, H
(1 2v) E . 6(1 v)
(15)
The microhardness of AuIn2 IMC is shown in Table 1. It is found that AuIn2 IMC can be regarded as a low hardness material due to the weak metallic bond. The present microhardness is further compared to the experimental data of Song and Lu [7] using nanoindentation testing, and a good agreement can be seen between them, as exhibited in Table 1. In addition, the pressure-dependent microhardness of AuIn2 IMC is also depicted in Fig. 8. Clearly, hydrostatic pressure would also have a substantial 15
influence on the microhardness, where the microhardness would enhance with hydrostatic pressure.
3.3 Thermodynamic properties The thermodynamic properties, such as Debye temperature and heat capacity, of AuIn2 crystal are explored using a quasi-harmonic Debye model with a periodic boundary condition. It is noted that the phonon effect is considered in this model [48,49]. In the quasi-harmonic Debye model, the non-equilibrium Gibbs function G*(V;P,T) is defined as,
G *(V ; P, T ) E(V ) PV AVib ( (V ); T ) ,
(16)
where E(V) represents the total energy per unit cell, V denotes the volume of the molecular system, T is the temperature of the system, P stands for the constant hydrostatic pressure condition, and AVib indicates the vibrational term, which is described as [50,51],
AVib ( (V ); T ) nKT (
9 3ln(1 e T ) D( )) , 8T T
(17)
where n is the number of atoms in the molecule, D ( ) represents the Debye integral T given as, D( y )
3 y3
y
0
x3 dx , ex 1
(18)
and is expressed as [50], 16
1/3
6 2V 1/2 n k
f (v )
Bs . M
(19)
In Eq. (19), M is the molecular mass per unit cell and Bs denotes the adiabatic bulk modulus, which is approximated by [50],
Bs B(V ) V
d 2 E (V ) . dV 2
(20)
The f (v) function is given in [49,52] and the Poisson’s ratio is determined from the calculated elastic constants. Then, the equilibrium state can be determined by the minimum of the Gibbs function G*(V;P,T) with respect to the volume V, G* (V ; P, T ) 0. V P ,T
(21)
By solving the above equation, the thermal equation of state V(P, T) can be finally derived, by which the Debye temperature can be obtained. The predicted Debye temperature and heat capacity of the AuIn2 single crystal versus temperature and hydrostatic pressure are shown in Figs. 9 and 10. It is observed that the Debye temperature is almost constant in the temperature range of 0-300K, and decreases slightly with the increase of temperature. Moreover, at the fixed temperature, there is a linear relationship between Debye temperature and hydrostatic pressure, and the increase of hydrostatic pressure would raise the Debye temperature. At zero pressure and 300K, the calculated Debye temperature for AuIn2 crystal is around 233.1K. Figure 10 reveals that the heat capacity at temperature below the Debye 17
temperatures (i.e., 233.1K) would remarkably increase with an increasing temperature. In addition, a high law temperature dependence at the low temperature range can be clearly observed, and the predicted power exponent is around 3. The power law relation is the so-called Debye heat capacity theory, i.e., the heat capacity is proportional to the cube of temperature [53]. At zero pressure and 300K, the predicted heat capacity of AuIn2 crystal is 24.40 J mol-1 K-1 and at high temperatures (i.e., far above the Debye temperature), the heat capacity of AuIn2 crystal converges to about 25.01 J mol-1 K-1, i.e., the Dulong-Petit limit [54], irrespective of the magnitude of pressure. The Dulong-Petit limit is widely observed in all solids at high temperatures or more precisely, at temperatures far above the Debye temperature.
4. Conclusions The structural, mechanical and thermodynamic properties of AuIn2 crystal in the cubic fluorite structure under hydrostatic pressure have been thoroughly investigated by the first-principles density functional calculations incorporated with the quasi-harmonic Debye model. The calculated lattice constants, bulk modulus, Young’s modulus and microhardness of AuIn2 crystal at ambient pressure are in good agreement with the previous experimental and theoretical data, demonstrating the effectiveness of the proposed theoretical models. Furthermore, the predicted elastic 18
constants of AuIn2 single crystal at zero pressure do satisfy the mechanical stability criteria, implying that this cubic single crystal IMC is a mechanically stable system. The calculated Zener anisotropy factor and directional Young’s modulus values show that the crystal is somewhat elastically anisotropic; however, the degree of anisotropy would be lessened as the increase of hydrostatic pressure. In other words, the anisotropic AuIn2 single crystal would turn into an elastically isotropic material with an increasing hydrostatic pressure. From the calculated Cauchy pressure and K/G ratio, it is demonstrated that both single-crystalline and polycrystalline AuIn2 compounds possess considerable ductility. Furthermore, based on the calculated microhardness value, it is detected that AuIn2 polycrystal is not a hard material, which could be due to the weak metal bond. The polycrystalline elastic properties of the crystal, such as bulk modulus, Young’s modulus and shear modulus, and even its ductility and microhardness would increase with hydrostatic pressure. The analysis also results show that the Debye temperature of AuIn2 single crystal remains constant in the temperature range of 0-300K, and at larger temperatures, it would reduce slightly with temperature. Besides, it turns out that AuIn2 single crystal holds the Debye heat capacity universal T3-dependence at temperature below the Debye temperature, and the heat capacity of the single crystal reaches the Dulong-Petit limit at temperature far above the Debye temperature. Apart from that, 19
the Debye temperature at constant temperature would increase nearly linearly and significantly with hydrostatic pressure, while the heat capacity would slightly decrease with it.
Acknowledgement The work is partially supported by National Science Council, Taiwan, ROC, under grants NSC 101-2221-E-007-009-MY3, MOST 103-2221-E-007-010- and MOST 103-2221-E-035-024-MY3. The authors also thank the National Center for High-Performance Computing (NCHC) for computational resources support.
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[30] Z. G. Mei, M. Stan, J. Yang, J. Alloys Comp. 603 (2014) 282. [31] Y.C. Shih, C.S. Chen, K.C. Wu, J. Mech. 30 (2014) 241. [32] D. Vanderbilt, Phys. Rev. B 41 (1990) 7892-7895. [33] J.D. Head, M.C. Zerner, Chem. Phys. Lett. 122 (1985) 264. [34] L. Brillouin, Comptes Rendus Hebdomadaires des Seances de l’Academie des Sciences 191 (1930) 292. [35] H.J. Monkhorst, J.D. Pack, Phys. Rev. B 13 (1976) 5188. [36] J.D. Tucker, Ab Initio-Based Modeling of Radiation Effects in the Ni-Fe-Cr System, Ph.D. Thesis, University of Wisconsin, Madison, US, 2008. [37] D.W. Boukhvalov, Y.W. Son, Nanoscale 4 (2012) 417. [38] V. Milman, M.C. Warren, J. Phys.: Condens. Matter 13 (2001) 241. [39] B. Montanari, N.M. Harrison, Chem. Phy. Lett. 364 (2002) 528. [40] J.F. Nye, Physical Properties of Crystals, Oxford University Press, Oxford, 1985. [41] D.C. Wallace, Thermodynamics of Crystals, Wiley, New York, 1972. [42] D.G. Pettifor, Mater. Sci. Technol. 8 (1992) 345. [43] A. Authier, A. Zarembowitch, International Tables for Crystallography D (2006) 72. [44] J.K.D. Verma, B.D. Nag, J. Phys. Soc. Jpn. 20 (1965) 635. [45] R. Hill, Proceed. Phys. Soc. 65 (1952) 349. 23
[46] S.F. Pugh, Philos. Mag. 45 (1954) 823. [47] H.C. Cheng, H.K. Cheng, S.T. Lu, W.H. Chen, IEEE T. Device Mat. Re. (2014) 499. [48] M.A. Blanco, E. Francisco, V. Luana, Comput. Phys. Commun. 158 (2004) 57. [49] E. Francisco, J.M. Recio, M.A. Blanco, A. Martin Pendás, A. Costales, J. Phys. Chem. A 102 (1998) 1595. [50] M.A. Blanco, A. Martin Pendás, E. Francisco, J.M. Recio, R. Franco, J. Molec. Struct. Theochem. 368 (1996) 245. [51] M. Flórez, J.M. Recio, E. Francisco, M.A. Blanco, A.M. Pendás, Phys. Rev. B 66 (2002) 144112. [52] E. Francisco, M.A. Blanco, G. Sanjurjo, Phys. Rev. B 63 (2001) 094107. [53] P. Debye, Annalen der Physik (Leipzig) 39 (1912) 789. [54] A.T. Petit, P.L. Dulong, Ann. Chim. Phys. 10 (1819) 395.
24
Table and Figure Captions Table 1: Calculated lattice constants a, b, and c, elastic constants C11, C12, and C44, Cauchy pressure C12–C44, bulk modulus K, shear modulus G, Young’s modulus E, Poisson’s ratio v, Zener anisotropy factor Az, ratio of bulk modulus to shear modulus K/G, and microhardness parameter H of single-crystalline and polycrystalline AuIn2 IMC
Fig. 1 Crystal structure of cubic AuIn2. The spheres in brown represent In atoms and the spheres in yellow denote Au atoms Fig. 2 Elastic constants as a function of hydrostatic pressure for AuIn2 single crystal Fig. 3 Cauchy pressure versus hydrostatic pressure for AuIn2 single crystal Fig. 4 Pressure-dependent Zener anisotropy factor for AuIn2 single crystal Fig. 5 Direction dependence of Young’s modulus for AuIn2 single crystal (unit: GPa) and their projections onto the a-b, b-c and a-c planes associated with five different hydrostatic pressures: (a) 0 GPa, (b) 5 GPa, (c) 10 GPa, (d) 15 GPa and (e) 20 GPa Fig. 6 Pressure-dependent bulk modulus (K), shear modulus (G) and Young’s modulus (E) for polycrystalline AuIn2 Fig. 7 Pressure-dependent K/G ratio for polycrystalline AuIn2 Fig. 8 Pressure-dependent microhardness parameter for polycrystalline AuIn2 25
Fig. 9 Temperature-dependent Debye temperature for AuIn2 single crystal Fig. 10 Temperature-dependent heat capacity Cv for AuIn2 single crystal
26
Tables and figures Table 1: Calculated lattice constants a, b, and c, elastic constants C11, C12, and C44, Cauchy pressure C12–C44, bulk modulus K, shear modulus G, Young’s modulus E, Poisson’s ratio v, Zener anisotropy factor Az, ratio of bulk modulus to shear modulus K/G, and microhardness parameter H of single-crystalline and polycrystalline AuIn2 IMC C11
C12
C44
C12–C44
(GPa)
(GPa)
(GPa)
(GPa)
a=b=c=6.72
75.9
42.6
12.2
-
-
-
-
-
a=b=c=6.66
Lattice constants (Å) This study
K (GPa)
G (GPa)
E (GPa)
v
Az
K/G
H (GPa)
30.4
53.8
14.0
38.6
0.38
0.73
3.84
1.11
-
-
-
-
-
-
-
-
1.37
-
-
-
-
-
35.4
-
-
-
-
-
-
-
-
62.0
-
-
-
-
-
-
a=b=c=6.68
-
-
-
-
54.0
-
-
-
-
-
-
a=b=c=6.51
-
-
-
-
-
-
-
-
-
-
-
a=b=c=6.51
-
-
-
-
-
-
-
-
-
-
-
Song and Lu (2014) (Exp.) [7] Lian et al. (2009) (Exp.) [6] Li et al. (2007) (GGA) [12] Hsu et al. (2005) (GGA) [17] Hsu et al. (1994) (Exp.) [9] Goral and Eyring (1986) (Exp.) [25]
27
In Au
c b
a
Fig. 1 Crystal structure of cubic AuIn2. The spheres in brown represent In atoms and the spheres in yellow denote Au atoms
28
Fig. 2 Elastic constants as a function of hydrostatic pressure for AuIn2 single crystal
29
Fig. 3 Cauchy pressure versus hydrostatic pressure for AuIn2 single crystal
30
Fig. 4 Pressure-dependent Zener anisotropy factor for AuIn2 single crystal
31
(a) 0 GP
(b) 5 GPa
32
(c) 10 GPa
(d) 15 GPa
33
(e) 20 GPa Fig. 5 Direction dependence of Young’s modulus for AuIn2 single crystal (unit: GPa) and their projections onto the a-b, b-c and a-c planes associated with five different hydrostatic pressures: (a) 0 GPa, (b) 5 GPa, (c) 10 GPa, (d) 15 GPa and (e) 20 GPa
34
Fig. 6 Pressure-dependent bulk modulus (K), shear modulus (G) and Young’s modulus (E) for polycrystalline AuIn2
35
Fig. 7 Pressure-dependent K/G ratio for polycrystalline AuIn2
36
Fig. 8 Pressure-dependent microhardness parameter for polycrystalline AuIn2
37
Fig. 9 Temperature-dependent Debye temperature for AuIn2 single crystal
38
Fig. 10 Temperature-dependent heat capacity Cv for AuIn2 single crystal
39
▼ The mechanical and thermodynamic properties of AuIn2 are reported for the first time using first-principles density functional theory calculation. ▼ The calculated lattice constants and elastic properties of AuIn2 are consistent with the published theoretical and experimental data. ▼ The results reveal that AuIn2 demonstrates low elastic anisotropy, low hardness and high ductility. ▼ It is worth to note that the anisotropic AuIn2 tends to become elastically isotropic as hydrostatic pressure increases.
S0925-8388(14)02165-3 http://dx.doi.org/10.1016/j.jallcom.2014.09.031 JALCOM 32142
To appear in:
Journal of Alloys and Compounds
Received Date: Revised Date: Accepted Date:
27 May 2014 2 September 2014 3 September 2014
Please cite this article as: C-F. Yu, H-C. Cheng, W-H. Chen, Structural, Mechanical and Thermodynamic Properties of AuIn2 Crystal under Pressure: A First-Principles Density Functional Theory Calculation, Journal of Alloys and Compounds (2014), doi: http://dx.doi.org/10.1016/j.jallcom.2014.09.031
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Structural, Mechanical and Thermodynamic Properties of AuIn2 Crystal under Pressure: A First-Principles Density Functional Theory Calculation
Ching-Feng Yua, Hsien-Chie Chengb*, and Wen-Hwa Chena* a
Department of Power Mechanical Engineering, National Tsing Hua University, Hsinchu 30013, Taiwan
b
Department of Aerospace and Systems Engineering, Feng Chia University, Taichung 40724, Taiwan
Abstract The structural, mechanical and thermodynamic properties of cubic AuIn2 crystal in the cubic fluorite structure, and also their temperature, hydrostatic pressure and direction dependences are investigated using first-principles calculations based on density functional theory (DFT) within the generalized gradient approximation (GGA). The optimized lattice constants of AuIn2 single crystal are first evaluated, by which its hydrostatic pressure-dependent elastic constants are also derived. Then, the hydrostatic pressure-dependent mechanical characteristics of the single crystal, including ductile/brittle behavior and elastic anisotropy, are explored according to the characterized angular character of atomic bonding, Zener anisotropy factor and directional Young’s modulus. Moreover, the polycrystalline elastic properties of AuIn2, *
The correspondence: [email protected] (H.C. Cheng), [email protected] (W.H. Chen) 1
such as bulk modulus, shear modulus and Young’s modulus, and its ductile/brittle and microhardness characteristics are assessed versus hydrostatic pressure. Finally, the temperature-dependent Debye temperature and heat capacity of AuIn2 single crystal are investigated by quasi-harmonic Debye modeling. The present results reveal that AuIn2 crystal demonstrates low elastic anisotropy, low hardness and high ductility. Furthermore, its heat capacity strictly follows the Debye T3-law at temperatures below the Debye temperature, and reaches the Dulong-Petit limit at temperatures far above the Debye temperature.
Keywords: Mechanical property, First-principles calculations, Elastic anisotropy, Ductile/brittle Nature, Polycrystalline elastic properties
1. Introduction Recently, evaluation of physical properties of intermetallic compounds (IMC) is an important issue in material science due to their properties have the significant influence on package interconnects in electronic devices. They can also present excellent magnetic, superconducting and chemical properties, due to their strong internal order and metallic bonding. In particular, Au-In system IMC has been the significant topic owing to their applications of microelectronic packaging. 2
The interfacial interactions between Au and In have been extensively investigated by many researches [1-7]. For example, Shohji et al. [1] investigated the growth of IMC in Au and In-48wt%Sn diffusion couples, and found that the AuIn2 layer is formed at the interface between Au and In-48Sn solder during isothermal aging at 70 °C, 90 °C and 110 °C. Sohn et al. [3] conducted the low temperature bonding for optoelectronic packaging by setting up Au-In eutectic system. The eutectic bonding was conducted at 180-210°C at a wafer scale and re-melting temperature measured using thermal-mechanical analysis (TMA) was 470 °C. Their results revealed that several intermetallic compounds, such as AuIn and AuIn2, are found after Au-In bonding by scanning electron microscopy (SEM) and energy dispersive x-ray (EDX) analyses. The interfacial reactions of an In-48wt%Sn solder joint with an electroplated Au/Ni ball grid array substrate at reflow temperature of 150 °C were investigated by Koo et al. [4] using several measurement techniques, such as SEM, transmission electron microscopy (TEM), EDX spectrometry, inductively coupled plasma-atomicemission spectroscopy and x-ray diffractometry. They found that two different IMC layers, AuIn and AuIn2, are created at the solder-substrate interface after one reflow. Chuang et al. [5] demonstrated that continuous AuIn2 and scallop-shaped AgIn2 IMC layers are formed at the interfaces between In-3Ag solder and Au/Ni/Cu pad and between that and Ag/Cu pad, respectively, during the reflow 3
process of In-3Ag solder ball grid array (BGA) packages with electroless nickel immersion gold (ENIG) and immersion silver (ImAg) surface finishes. Lian et al. [6] assessed the feasibility of forming Au-In IMCs, and showed that AuIn2 is developed at a low annealing temperature of 160°C and a short annealing time of 10 minutes. Song and Lu [7] investigated the IMC formation in an Au/In diffusion couple during chemical reaction at an isothermal temperature of 180 °C in vacuum. They indicated that In is fully exhausted and AuIn2 is observed in 2 minutes. Apart from that, the structural, electronic and dynamical properties of AuIn2 have received considerable attention due to its technological importance. This compound is a good metallic conductor, crystallizes in the cubic fluorite structure at ambient conditions [8,9] and holds superconducting properties at lower temperatures. Smeibidl et al. [10] reported that the AuIn2 is a good candidate for nuclear demagnetization as well as for the observation of nuclear magnetic ordering due to its large nuclear moment and its large molar nuclear Curie constant. Because the compound also possesses a small Korringa constant and good thermal conductivity, its applications may include nuclear susceptibility thermometry and nuclear refrigeration [11]. In addition, it is an ideal material for investigating the transition-metal 5d bands [12]. During the past four decades, the structural, electronic and dynamical properties of AuIn2 have also been explored by numerous experimental and theoretical studies 4
[9,12-21]. For instance, charge-transfer and valence-band behavior upon alloying in AuIn2 IMC have been studied by Sham et al. [13] using Au Mössbauer and x-ray photoemission measurements. Nelson et al. [14] studied the (1 0 0) surface of AuIn2 IMC through synchrotron-radiation-excited angle-resolved photoemission, and indicated that the Au 5d band of the compound was relatively flat. Hsu et al. [9] evaluated the structural, electronic, magnetic, and optical properties of AuIn2 using synchrotron-radiation-excited angle-resolved photoemission spectra. The anomalies of resistance, thermoelectric power (TEP) and universal equation of state (UEOS) in AuIn2 was observed by Godwal et al. [16]. They also found that an electronic topological transition (ETT) occurs in the pressure range of 2-4 GPa, and a structural phase transition takes place beyond 8 GPa to a monoclinic phase through imaging plate high-pressure angle-dispersive x-ray diffraction analysis. Garg et al. [18] predicted electronic topological transition in AuIn2 compound at high pressure based on the accurate X-ray diffraction data derived using a diamond anvil cell with the ELETTRA synchrotron source. For the theoretical investigations, Hsu et al. [19] calculated the binding energies of AuIn2 by the first-principles theory [22,23] using WIEN2K code [24]. Gao et al. [20] carried out a numerical procedure for assessing the Seebeck coefficient of AuIn2 based on its electronic band structure. The structural, electronic and dynamical properties of AuIn2 were explored by Uǧur and Soyalp [21] 5
through the plane-wave pseudopotential method within the density functional theory [22,23]. Li et al. [12] studied the electsronic band structure, transport properties, and lattice dynamic in AuIn2 under high pressure with the full potential linearized augmented plane wave and pseudopotential plane wave methods. Although the electronic properties, optical properties and dynamic properties of AuIn 2 have been widely studied, there is a lack of knowledge of its mechanical and thermodynamic properties. In the study, the mechanical and thermodynamic properties of AuIn2 in the cubic fluorite structure are for the first time systematically explored using first-principles density functional theory calculations. This investigation puts the focus on predicting the structural, mechanical, and thermodynamic characteristics, including lattice constants, elastic properties and anisotropy, brittle-ductile behavior, microhardness, Debye temperature and heat capacity, and most importantly, their temperature and pressure
dependences.
Furthermore,
the
hydrostatic
pressure-dependent
polycrystalline elastic properties of the crystal are also examined. The validity of the proposed theoretical models is demonstrated through comparison with the published experimental and theoretical data [6,7,9,12,17,25].
6
2. Computational details and theoretical method The Cambridge Serial Total Energy Package code (CASTEP) [26-30] based on the density functional theory (DFT) [22,23,31] is applied in all the present calculations. Furthermore, the ultrasoft pseudopotential formalism of Vanderbilt [32] is adopted to describe the interactions of valence electrons with ion cores, and the electron wave function is expanded in plane waves up to an energy cutoff of 360 eV for all the calculations. The exchange-correlation energy is evaluated using the generalized gradient approximation (GGA) of the Perdew-Burke-Ernzerhof (PBE) formalism [23], which is dependent on both the electron density and its gradient at each space point. To search the ground state, a quasi-Newton (variable-metric) minimization algorithm using the Broyden-Fletcher-Goldfarb-Shanno (BFGS) update scheme [33] is utilized, which provides a very efficient and robust way to explore the optimum crystal structure with a minimum energy. To sample the Brillouin zone [34], the Monkhorst-Pack k-point mesh is employed [35]. It is selected based on the convergence of the k-point mesh, where the change of total energy becomes less than 1 meV/atom [36,37]. In the study, the selected k-point mesh is 161616. The following convergence thresholds are utilized in the geometry optimization: 5×10-6 eV/atom for the total energy variation, the maximum ionic Hellmann-Feynman force within 0.01 eV/Å, the maximum stress within 0.02 GPa and the maximum ionic 7
displacement within 5×10-4 Å. The resulting stress with respect to the optimized crystal structure [38] is calculated by imposing a number of specified homogenous deformations with a finite value. The elastic coefficients can be then determined through a linear fit of the calculated resulting stress as a function of strain, where four strain amplitudes are used with the maximum value of 0.3%. To assess the thermal effects on the thermodynamic properties of the crystal, a quasi-harmonic Debye model is used, and periodic boundary conditions are applied in all the present calculations for predicting the physical properties of the crystal. In principle, the lattice contribution to the heat capacity can be expressed as, C (T ) k
where
( / kT ) 2 exp( / kT ) g ( )d , [exp( / kT ) 1]2
(1)
and k are the reduced Planck constant and the Boltzmann constant,
respectively, T is temperature, and g() denotes the number of states at frequency, and
is the density of states function. As soon as the phonon dispersion relation is derived, the density of states function can be determined. In the study, the phonon density of states and dispersion relation are calculated by using the finite displacement method [39]. This technique is derived based on numerical differentiation of forces on atoms that are computed for a number of unit cells with atomic displacements.
8
3. Results and discussion 3.1 Structural properties Structure optimization is first performed to determine the lattice parameters and the positions of the atoms of AuIn2 monocrystal in the stable ground state, based on the experimental data derived from X-ray diffraction (XRD) analysis [25], as shown in Fig. 1. This figure exhibits that it is a cubic system with Fm-3m space group. The calculated lattice parameters are shown in Table 1, and it is evident that there is a good agreement between the present calculations and the literature data [9,12,17,25]. The maximum difference among them is only about 3.2%, demonstrating the effectiveness of the proposed simulation model.
3.2 Mechanical properties The elastic constants of single-crystalline AuIn2 IMC nanocrystal can be obtained through the following generalized Hooke’s law, 6
{ i } [Cij ]{ j } ,
(2)
j 1
where {σi} and {εj} are the stress and strain vector, respectively. In the cubic system, [Cij] can be described as [40],
9
C11 C12 C12 0 C 12 C11 C12 0 C C12 C11 0 [Cij ] 12 0 0 C44 0 0 Symmetric 0 0 0 0 0 0 0
0 0 0 0 C44 0
0 0 0 . 0 0 C44
(3)
It is clear to see that there are three independent elastic constants in the stiffness matrix, and furthermore, the inverse of the stiffness matrix [Cij] yields the compliance matrix [Sij]. For a cubic system, the requirement of mechanical stability leads to the following restrictions [41] on the elastic constants, C11 2C12 0, C44 0, C11 C12 0 ,
(4)
The calculated elastic constants in Table 1 satisfy the above criteria, indicating that AuIn2 single crystal is a mechanical stable system. In addition, the influences of hydrostatic pressure on the elastic constants are further evaluated, and the calculated results are shown in Fig. 2. A quasi-linear relationship can be detected between the elastic constants and hydrostatic pressure as the applied hydrostatic pressure is in the range of 0-20 GPa. Generally speaking, angular character of atomic bonding is related to material characteristics, such as brittleness or ductility. Pettifor [42] reported that the angular character of atomic bonding in metals and compounds can be depicted through the Cauchy pressure, which is defined as C12–C44. It is noted that as the pressure is positive, the material is expected to be metallic; otherwise, it is a nonmetallic material 10
with directional bonding. Verification of this correlation has been demonstrated on some ductile materials, such as elemental metals and IMCs, including Al, Ni, Ni3Al and Pt3Al [42]. On the contrary, brittle semiconductor, such as Si and Ge, would possess directional bonding characteristics [42]. Table 1 shows that the Cauchy pressure is 30.4, suggesting that AuIn2 crystal possesses ductile nature. Moreover, hydrostatic pressure effect on the Cauchy pressure is also investigated, and the results are shown in Fig. 3. Evidently, an increasing hydrostatic pressure would increase the Cauchy pressure, also indicating the increase of the ductility of AuIn2 single crystal. To quantify the degree of anisotropy of a solid, the Zener anisotropy factor (Az) is usually applied, as shown below [27], Az
2C44 . C11 C12
(5)
As Az is equal to one, the nanomaterial is elastically isotropic; otherwise, it is elastically anisotropic, where a larger deviation from one would imply a higher degree of elastic anisotropy. The calculated Zener anisotropy factor for AuIn2 at zero pressure is listed in Table 1, which is around 0.73, suggesting that the crystal is not an elastic, isotropic material. In addition, Fig. 4 shows the relationship between hydrostatic pressure and Zener anisotropy factor, where the factor would increase linearly with hydrostatic pressure. The Zener anisotropy factor enhances from around 0.73 to 1.05 as hydrostatic pressure varies from 0 to 20 GPa. This implies that AuIn2 single crystal 11
is likely to become elastic and isotropic at a very high hydrostatic pressure. A three-dimensional surface representation of the elastic anisotropy of the cubic AuIn2 single crystal, expressed by the variation of the Young’s modulus of the crystal with crystal direction, is further constructed. The direction dependence of the Young’s modulus of a triclinic crystal system can be described as [43], E 1/ (s11 2(s11 s12 0.5s44 )(l12l22 l22l32 l32l12 ) ,
(6)
where l1, l2, and l3 represent the direction cosines with respect to the a, b and c directions of lattice and Sij (i, j = 1, 2, 3) stands for the compliance coefficients. Using the derived compliance coefficients, the directional Young’s moduli and their projections on the a-b, b-c, and a-c planes of the crystal at five different hydrostatic pressures (i.e., 0, 5, 10, 15 and 20 GPa) can be calculated, as shown in Fig. 5. An enhancement of the directional Young’s modulus of the crystal can be observed with the increase of hydrostatic pressure. The projected configurations of the directional Young’s moduli on the a-b, b-c and a-c planes reveal great similarity, denoting that the strength of AuIn2 would not change with crystallographic plane. Furthermore, it is noticeable that the projected configuration of the directional Young’s moduli would vary from a quasi-square to quasicircle shape as hydrostatic pressure rises from 0 to 20 GPa. This suggests that AuIn2 single crystal tends to become more isotropic at a higher hydrostatic pressure, and the result is consistent with that of the 12
pressure-dependent Zener anisotropy factor shown in Fig. 4. In general, many crystalline solids, such as IMCs, consist of numerous single crystals or grains with different crystal orientations, which are known as polycrystalline materials. Because the single crystals possess an identical property in every direction, they tend to randomly orientate with each other to form a quasi-isotropic material. To facilitate the comparison with the literature experimental data, which are typically polycrystalline results, the elastic properties of the polycrystalline aggregate of randomly orientated anisotropic single-crystalline AuIn2 nanocrystal are predicted through the following Voigt-Reuss scheme [44] together with the calculated single crystal data,
KV
1 C11 C22 C33 2(C12 C13 C23 ) , 9
(7)
K R S11 S22 S33 2(S12 S13 S23 ) , 1
GV
(8)
1 C11 C22 C33 (C12 C13 C23 ) 3(C44 C55 C66 ) , 15
(9) 1
GR 15 4 S11 S22 S33 ) 4( S12 S13 S23 3(S44 S55 S66 ) .
(10)
where KV and KR denote the upper (Voigt) and lower (Reuss) bounds of the bulk modulus (K) of the polycrystalline aggregate, respectively, and GV and GR represent those of the shear modulus (G) of the polycrystalline aggregate. In addition, the effective bulk and shear modulus can be calculated by using the Voigt-Reuss-Hill approximation [45], which is considered as geometric mean, 13
K KV K R ,
(11)
G GV GR .
(12)
By using the calculated polycrystalline bulk and shear modulus, the effective Young’s modulus (E) and Poisson’s ratio (ν) of the AuIn2 polycrystalline aggregate can be calculated as follows,
9 KG , 3K G 3K 2G v . 2(3K G ) E
(13) (14)
The obtained bulk modulus, shear modulus, Young’s modulus and Poisson’s ratio for the polycrystalline AuIn2 are shown in Table 1. It is noted that the present calculations are in good agreement with the literature experimental and theoretical data [6,12,17]. Additionally, their dependences on hydrostatic pressure are also evaluated and the results are presented in Fig. 6. It is shown that the increase of hydrostatic pressure would enlarge the polycrystalline elastic properties. Specifically, there is about 194.9%, 319.6% and 301.9% increase in the bulk modulus, shear modulus and Young’s modulus of polycrystalline AuIn2, as hydrostatic pressure increases from 0 to 20 GPa, revealing that hydrostatic pressure has less influence on the bulk modulus, as compared to the shear and Young’s modulus. Pugh [46] addressed that the ratio of bulk to shear modulus, K/G, can be utilized as an empirical index of ductility/brittleness for polycrystalline materials, and the 14
critical threshold value for distinguishing the physical properties of materials is around 1.75. As the ratio is up to 1.75, the polycrystal would possess a ductile character while it would have a brittle nature as the ratio is below 1.75. The computed K/G ratio for polycrystalline AuIn2 is about 3.84, as shown in Table 1, revealing that the polycrystal would behave like a ductile material. As this ductile compound material appears in solder interconnects, their drop impact reliability [47] should be significantly improved. Figure 7 further illustrates the K/G ratio as a function of hydrostatic pressure, and it is found that the level of ductility of AuIn2 crystal would descend with a decreasing K/G ratio. The stiffness of polycrystalline AuIn2 IMC can be investigated by calculating the microhardness parameter (H) [27], which is the resistance of the physical object against compression of the contacting parts, as described below, H
(1 2v) E . 6(1 v)
(15)
The microhardness of AuIn2 IMC is shown in Table 1. It is found that AuIn2 IMC can be regarded as a low hardness material due to the weak metallic bond. The present microhardness is further compared to the experimental data of Song and Lu [7] using nanoindentation testing, and a good agreement can be seen between them, as exhibited in Table 1. In addition, the pressure-dependent microhardness of AuIn2 IMC is also depicted in Fig. 8. Clearly, hydrostatic pressure would also have a substantial 15
influence on the microhardness, where the microhardness would enhance with hydrostatic pressure.
3.3 Thermodynamic properties The thermodynamic properties, such as Debye temperature and heat capacity, of AuIn2 crystal are explored using a quasi-harmonic Debye model with a periodic boundary condition. It is noted that the phonon effect is considered in this model [48,49]. In the quasi-harmonic Debye model, the non-equilibrium Gibbs function G*(V;P,T) is defined as,
G *(V ; P, T ) E(V ) PV AVib ( (V ); T ) ,
(16)
where E(V) represents the total energy per unit cell, V denotes the volume of the molecular system, T is the temperature of the system, P stands for the constant hydrostatic pressure condition, and AVib indicates the vibrational term, which is described as [50,51],
AVib ( (V ); T ) nKT (
9 3ln(1 e T ) D( )) , 8T T
(17)
where n is the number of atoms in the molecule, D ( ) represents the Debye integral T given as, D( y )
3 y3
y
0
x3 dx , ex 1
(18)
and is expressed as [50], 16
1/3
6 2V 1/2 n k
f (v )
Bs . M
(19)
In Eq. (19), M is the molecular mass per unit cell and Bs denotes the adiabatic bulk modulus, which is approximated by [50],
Bs B(V ) V
d 2 E (V ) . dV 2
(20)
The f (v) function is given in [49,52] and the Poisson’s ratio is determined from the calculated elastic constants. Then, the equilibrium state can be determined by the minimum of the Gibbs function G*(V;P,T) with respect to the volume V, G* (V ; P, T ) 0. V P ,T
(21)
By solving the above equation, the thermal equation of state V(P, T) can be finally derived, by which the Debye temperature can be obtained. The predicted Debye temperature and heat capacity of the AuIn2 single crystal versus temperature and hydrostatic pressure are shown in Figs. 9 and 10. It is observed that the Debye temperature is almost constant in the temperature range of 0-300K, and decreases slightly with the increase of temperature. Moreover, at the fixed temperature, there is a linear relationship between Debye temperature and hydrostatic pressure, and the increase of hydrostatic pressure would raise the Debye temperature. At zero pressure and 300K, the calculated Debye temperature for AuIn2 crystal is around 233.1K. Figure 10 reveals that the heat capacity at temperature below the Debye 17
temperatures (i.e., 233.1K) would remarkably increase with an increasing temperature. In addition, a high law temperature dependence at the low temperature range can be clearly observed, and the predicted power exponent is around 3. The power law relation is the so-called Debye heat capacity theory, i.e., the heat capacity is proportional to the cube of temperature [53]. At zero pressure and 300K, the predicted heat capacity of AuIn2 crystal is 24.40 J mol-1 K-1 and at high temperatures (i.e., far above the Debye temperature), the heat capacity of AuIn2 crystal converges to about 25.01 J mol-1 K-1, i.e., the Dulong-Petit limit [54], irrespective of the magnitude of pressure. The Dulong-Petit limit is widely observed in all solids at high temperatures or more precisely, at temperatures far above the Debye temperature.
4. Conclusions The structural, mechanical and thermodynamic properties of AuIn2 crystal in the cubic fluorite structure under hydrostatic pressure have been thoroughly investigated by the first-principles density functional calculations incorporated with the quasi-harmonic Debye model. The calculated lattice constants, bulk modulus, Young’s modulus and microhardness of AuIn2 crystal at ambient pressure are in good agreement with the previous experimental and theoretical data, demonstrating the effectiveness of the proposed theoretical models. Furthermore, the predicted elastic 18
constants of AuIn2 single crystal at zero pressure do satisfy the mechanical stability criteria, implying that this cubic single crystal IMC is a mechanically stable system. The calculated Zener anisotropy factor and directional Young’s modulus values show that the crystal is somewhat elastically anisotropic; however, the degree of anisotropy would be lessened as the increase of hydrostatic pressure. In other words, the anisotropic AuIn2 single crystal would turn into an elastically isotropic material with an increasing hydrostatic pressure. From the calculated Cauchy pressure and K/G ratio, it is demonstrated that both single-crystalline and polycrystalline AuIn2 compounds possess considerable ductility. Furthermore, based on the calculated microhardness value, it is detected that AuIn2 polycrystal is not a hard material, which could be due to the weak metal bond. The polycrystalline elastic properties of the crystal, such as bulk modulus, Young’s modulus and shear modulus, and even its ductility and microhardness would increase with hydrostatic pressure. The analysis also results show that the Debye temperature of AuIn2 single crystal remains constant in the temperature range of 0-300K, and at larger temperatures, it would reduce slightly with temperature. Besides, it turns out that AuIn2 single crystal holds the Debye heat capacity universal T3-dependence at temperature below the Debye temperature, and the heat capacity of the single crystal reaches the Dulong-Petit limit at temperature far above the Debye temperature. Apart from that, 19
the Debye temperature at constant temperature would increase nearly linearly and significantly with hydrostatic pressure, while the heat capacity would slightly decrease with it.
Acknowledgement The work is partially supported by National Science Council, Taiwan, ROC, under grants NSC 101-2221-E-007-009-MY3, MOST 103-2221-E-007-010- and MOST 103-2221-E-035-024-MY3. The authors also thank the National Center for High-Performance Computing (NCHC) for computational resources support.
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24
Table and Figure Captions Table 1: Calculated lattice constants a, b, and c, elastic constants C11, C12, and C44, Cauchy pressure C12–C44, bulk modulus K, shear modulus G, Young’s modulus E, Poisson’s ratio v, Zener anisotropy factor Az, ratio of bulk modulus to shear modulus K/G, and microhardness parameter H of single-crystalline and polycrystalline AuIn2 IMC
Fig. 1 Crystal structure of cubic AuIn2. The spheres in brown represent In atoms and the spheres in yellow denote Au atoms Fig. 2 Elastic constants as a function of hydrostatic pressure for AuIn2 single crystal Fig. 3 Cauchy pressure versus hydrostatic pressure for AuIn2 single crystal Fig. 4 Pressure-dependent Zener anisotropy factor for AuIn2 single crystal Fig. 5 Direction dependence of Young’s modulus for AuIn2 single crystal (unit: GPa) and their projections onto the a-b, b-c and a-c planes associated with five different hydrostatic pressures: (a) 0 GPa, (b) 5 GPa, (c) 10 GPa, (d) 15 GPa and (e) 20 GPa Fig. 6 Pressure-dependent bulk modulus (K), shear modulus (G) and Young’s modulus (E) for polycrystalline AuIn2 Fig. 7 Pressure-dependent K/G ratio for polycrystalline AuIn2 Fig. 8 Pressure-dependent microhardness parameter for polycrystalline AuIn2 25
Fig. 9 Temperature-dependent Debye temperature for AuIn2 single crystal Fig. 10 Temperature-dependent heat capacity Cv for AuIn2 single crystal
26
Tables and figures Table 1: Calculated lattice constants a, b, and c, elastic constants C11, C12, and C44, Cauchy pressure C12–C44, bulk modulus K, shear modulus G, Young’s modulus E, Poisson’s ratio v, Zener anisotropy factor Az, ratio of bulk modulus to shear modulus K/G, and microhardness parameter H of single-crystalline and polycrystalline AuIn2 IMC C11
C12
C44
C12–C44
(GPa)
(GPa)
(GPa)
(GPa)
a=b=c=6.72
75.9
42.6
12.2
-
-
-
-
-
a=b=c=6.66
Lattice constants (Å) This study
K (GPa)
G (GPa)
E (GPa)
v
Az
K/G
H (GPa)
30.4
53.8
14.0
38.6
0.38
0.73
3.84
1.11
-
-
-
-
-
-
-
-
1.37
-
-
-
-
-
35.4
-
-
-
-
-
-
-
-
62.0
-
-
-
-
-
-
a=b=c=6.68
-
-
-
-
54.0
-
-
-
-
-
-
a=b=c=6.51
-
-
-
-
-
-
-
-
-
-
-
a=b=c=6.51
-
-
-
-
-
-
-
-
-
-
-
Song and Lu (2014) (Exp.) [7] Lian et al. (2009) (Exp.) [6] Li et al. (2007) (GGA) [12] Hsu et al. (2005) (GGA) [17] Hsu et al. (1994) (Exp.) [9] Goral and Eyring (1986) (Exp.) [25]
27
In Au
c b
a
Fig. 1 Crystal structure of cubic AuIn2. The spheres in brown represent In atoms and the spheres in yellow denote Au atoms
28
Fig. 2 Elastic constants as a function of hydrostatic pressure for AuIn2 single crystal
29
Fig. 3 Cauchy pressure versus hydrostatic pressure for AuIn2 single crystal
30
Fig. 4 Pressure-dependent Zener anisotropy factor for AuIn2 single crystal
31
(a) 0 GP
(b) 5 GPa
32
(c) 10 GPa
(d) 15 GPa
33
(e) 20 GPa Fig. 5 Direction dependence of Young’s modulus for AuIn2 single crystal (unit: GPa) and their projections onto the a-b, b-c and a-c planes associated with five different hydrostatic pressures: (a) 0 GPa, (b) 5 GPa, (c) 10 GPa, (d) 15 GPa and (e) 20 GPa
34
Fig. 6 Pressure-dependent bulk modulus (K), shear modulus (G) and Young’s modulus (E) for polycrystalline AuIn2
35
Fig. 7 Pressure-dependent K/G ratio for polycrystalline AuIn2
36
Fig. 8 Pressure-dependent microhardness parameter for polycrystalline AuIn2
37
Fig. 9 Temperature-dependent Debye temperature for AuIn2 single crystal
38
Fig. 10 Temperature-dependent heat capacity Cv for AuIn2 single crystal
39
▼ The mechanical and thermodynamic properties of AuIn2 are reported for the first time using first-principles density functional theory calculation. ▼ The calculated lattice constants and elastic properties of AuIn2 are consistent with the published theoretical and experimental data. ▼ The results reveal that AuIn2 demonstrates low elastic anisotropy, low hardness and high ductility. ▼ It is worth to note that the anisotropic AuIn2 tends to become elastically isotropic as hydrostatic pressure increases.