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Full elastic constants of Cu6Sn5 intermetallic by Resonant Ultrasound Spectroscopy (RUS) and ab initio calculations L. Jiang a, N. Muthegowda b, M.A. Bhatia b, A. Migliori c, K.N. Solanki b, N. Chawla a,b,⇑ a

Materials Science and Engineering, Arizona State University, Tempe, AZ 85287, United States Mechanical and Aerospace Engineering, Arizona State University, Tempe, AZ 85287, United States c Los Alamos National Laboratory, MS E536, Los Alamos, NM 87545, United States b

a r t i c l e

i n f o

Article history: Received 7 May 2015 Accepted 10 May 2015 Available online 21 May 2015 Keywords: Intermetallic Ab initio Resonant Ultrasound Spectroscopy Elastic properties Cu6Sn5

a b s t r a c t Cu6Sn5 intermetallic is an important compound formed during reaction between Sn-rich interconnects and copper metallization. The full elastic constants of Cu6Sn5 were quantiﬁed experimentally by Resonant Ultrasound Spectroscopy (RUS). The single crystal elastic properties were modeled by density functional theory. A mesoscale polycrystalline model, incorporating the single crystal constants was compared to the experimental results, yielding excellent agreement. Ó 2015 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Environmental and health concerns over the use of Pb in electronic packaging (in eutectic Sn–Pb solder) have prompted the development and understanding of the behavior of Sn-rich (Pb-free) solders [1–3]. Solders are typically melted on a metallic substrate, such as Cu or Ni [4]. On a Cu substrate, Sn will react with Cu to form Cu6Sn5 and/or Cu3Sn intermetallic compounds (IMC) [4–7]. Cu6Sn5 may be formed in two phases: g (superlattice based on hexagonal structure) or g0 (monoclinic). A recent, thorough synchrotron X-ray diffraction study by Zeng et al. [8] shows that on cooling from 250 °C or so, g0 structure will be obtained on cooling to room temperature within 100 s. This is typical of most cooling rates in Pb-free solder, and is conﬁrmed by reports that a hexagonal structure is observed on cooling in solder joints [9–12]. The size and morphology of the Cu6Sn5 layer control the mechanical behavior and reliability of solder-Cu joints [13–17]. For example, a relatively thin intermetallic layer is beneﬁcial in achieving a strong mechanical and chemical bond between the solder and Cu substrate [13,14]. At larger thickness, however, the intermetallic often acts as a crack initiation site leading to catastrophic failure and poor toughness of the joint [15–17]. In fact, the thickness of the Cu6Sn5 layer determines whether fracture under mechanical shock conditions is solder or IMC-controlled ⇑ Corresponding author at: Materials Science and Engineering, Arizona State University, Tempe, AZ 85287, United States. Tel.: +1 (480) 965 2402; fax: +1 (480) 727 9321. E-mail address: [email protected] (N. Chawla). http://dx.doi.org/10.1016/j.scriptamat.2015.05.012 1359-6462/Ó 2015 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

[18,19]. Intermetallics also form within the solder, for example, in eutectic Sn–0.7Cu or Sn–Ag–Cu solder. This intermetallic will also affect the mechanical properties of the solder. Thus, a comprehensive knowledge of the mechanical properties of intermetallic phases formed in Sn-rich solders is extremely important. Instrumented indentation has been used to obtain Young’s modulus and hardness of Cu6Sn5 [20–24]. In our previous studies [11,12], we also used micropillar compression as a unique means of probing the mechanical properties of this important intermetallic. Micropillars were milled using focused ion beam (FIB) within single-crystal nodules of Cu6Sn5, and tested in compression using a nanoindenter with a ﬂat tip. We reported the fracture strength, strain-to-failure, and the effects of crystallographic orientation on strength of Cu6Sn5. The strength of the IMC was higher along the c-axis (which happens to be the growth direction in a typical Sn–Cu couple), than along the a-axis. To date, the full elastic constants of Cu6Sn5 have not been measured. There have been a few reports of density functional theory calculations on the elastic constants of this important material [25], but no experimental measurements. Resonant Ultrasound Spectroscopy (RUS) is a technique which combines the resonant frequency response and harmonics with the modes of vibration of a sample. A nearly cubic rectangular parallelepiped of the material of approximately is lightly suspended by opposite body corners between two piezoelectric transducers, which minimize any contribution to the measurement by the transducers. Since a non-unique solution exists for such a system, an iterative solution

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L. Jiang et al. / Scripta Materialia 107 (2015) 26–29

is reached, using ‘‘seed’’ values for elastic constants of an isotropic case, as well as measurements of the mass and dimensions of the sample. Details of this method are available elsewhere [26,27]. In this paper we report on the full elastic constants of polycrystalline Cu6Sn5, measured experimentally by RUS. First principles calculations by density functional theory were conducted, assuming a single crystal Cu6Sn5 (for both hexagonal and monoclinic crystal structures, i.e., g and g0 ). The single crystal values were then incorporated into a mesoscale model to compute the polycrystalline aggregate’s properties. Reuss averages for elastic constants, such as Young’s, Shear, and Bulk modulus were computed and compared to the experimental results, yielding good agreement. Vacuum-melted, polycrystalline specimens of Cu6Sn5 were prepared. High purity Sn pieces were mixed with Cu shot (ESPI, Ashland, OR, 99.995% pure, packed under argon). To minimize any oxides, the components were mixed in a helium glovebox with a partial pressure of O2 less than 20 ppm. Inside the glovebox, the mixture was placed in a quartz ampoule (12 mm in diameter). With a closed stopcock, the quartz ampoule was taken out of the glove box, evacuated to 105 Torr and hermetically sealed with a blow torch. The sealed ampoules were heated to 1000 °C for 4 h, and periodically mixed by rotating the ampoule, in order to homogenize the liquid metal. The ampoules were then water quenched, removed from the ampoule, and annealed at 210 °C for 12 days, followed by at 300 °C for 16 days, to obtain a homogeneous microstructure of Cu6Sn5 intermetallic. Annealing at 210 °C for 12 days resulted in a microstructure of primarily Cu6Sn5 but some residual Cu3Sn. We decide to raise the temperature then to 300 °C to obtain some Sn liquid phase. This resulted in a 100% Cu6Sn5 microstructure with some residual porosity due to the Sn liquid phase, Figure 1. The residual porosity was about 4.9 ± 0.1%, as was measured by image segmentation of polished microstructures. RUS measurements were conducted on three specimens using between 19 and 39 resonance peaks to determine, in a highly redundant way, two independent isotropic elastic moduli. RUS sweeps frequency, drives a broadband transducer in weak contact with the specimen, and detects the macroscopic-vibration frequencies or normal modes [28,29]. The elastic moduli Cij are determined from the resonances using an inverse calculation with input specimen mass and dimensions, known crystallographic symmetry (in this case polycrystal or isotropic), and the resonance frequencies. Density was computed from the measured dimensions and the mass (masses ranged from about 0.1 g to 0.2 g, dimensions from about 2 mm to 3 mm for a rectangular parallelepiped specimen with shape errors of order 0.002 mm). Errors in moduli arise mainly from errors in the accuracy of the specimen shape, limiting absolute accuracy to about 0.5–1%. In this case, isotropic crystallographic symmetry was appropriate for the texture-free specimen. (RUS analysis codes detect errors in isotropy by yielding much larger errors than above if texture is present and an isotropic model is

used.) The measurement cell was inside a conventional 4He-ﬂow cryostat for temperature control, holding the temperature at 295 K within 1 K using Si-diode thermometry. The ab initio calculations were performed using the Vienna ab initio simulation package (VASP) [30,31]. The electron–ion interaction was described using the projector augmented wave (PAW) method [32] with a plane wave energy cutoff of 304 eV. The exchange–correlation energy was described by the spin-polarized generalized gradient approximation (GGA) with the Perdew–Burk e–Ernzerhof (PBE) functional [33]. Brillouin-zone integration was performed using the gamma centered Monkhoest-Pack scheme [34] with K-point meshes of 2 5 4 and 3 4 4 for the hexagonal (g-Cu6Sn5) and monoclinic (g0 -Cu6Sn5) phases, respectively. The ionic relaxation was carried out using a conjugate gradient algorithm [35] with a 0.1 meV energy convergence criterion. The structure optimization of g and g0 phases is shown in Figure 2. In the case of the g phase, the experimentally reported structure of a hexagonal NiAs-type cell (space group P63/mmc) was used with one missing Sn atom to maintain the initial stoichiometry of 6:5 (with a supercell comprised of 55 atoms), see Gangulee et al. [36]. Similarly, for the g0 phase, which belongs to the NiAs-Ni2In a supercell comprised of 44 atoms was used, see Larsson et al. [10]. For both g and g0 phases, we used experimentally measured unit cell parameters as an initial input to calculated the lattice parameters using ab initio approach. Finally, strain based approaches, as described in Fast et al. [37] and Karki et al. [38], were used with the above mentioned convergence criteria to calculate the elastic properties of g and g0 phases. To obtain the equilibrium lattice parameters for g and g0 phase structures, a series of total energy calculations were performed. The cell external and internal parameters were optimized by minimizing the stresses and inter-atomic forces at ﬁxed volume. The calculated zero temperature total energy as a function of the cell volume for the g and g0 phases was plotted as shown in Figure 3, which predicts that the g0 phase is much more stable than the g phase at 0 K. The equilibrium lattice parameters were then found by minimizing the total energy with respect to the volume. Table 1 lists the unit cell parameters of the g and g0 phases

(a)

Cu Sn (b) Figure 1. Cu6Sn5 microstructure of samples used for RUS analysis.

Figure 2. (a) Hexagonal crystal of Cu6Sn5 with 30 Cu atoms and 25 Sn atoms, and (b) monoclinic structure of Cu6Sn5 with 24 Cu atoms and 20 Sn atoms.

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L. Jiang et al. / Scripta Materialia 107 (2015) 26–29

Figure 3. Calculated zero temperature energy as a function of the unit cell volume for the g-Cu6Sn5 (hexagonal) and g0 -Cu6Sn5 (monoclinic).

Table 1 Unit cell parameters of the g and g0 phases obtained by ﬁrst principles calculations (at 0 K) and the comparison of these parameters with available experimental data (at 300 K). Structure

g-Cu6Sn5 g0 -Cu6Sn5

Lattice constants

DFT Experiment [7] DFT Experiment [8]

a (Å)

c (Å)

a/c

b (Å)

b (°)

4.17 4.19 11.37 11.02

5.01 5.04 10.02 9.83

1.20 1.21 – –

– – 7.51 7.28

– – 98.58 98.84

obtained by ﬁrst principles calculations (at 0 K) and the comparison of these parameters with available experimental data (at 300 K) [36,10]. The calculated lattice parameters are in good agreement with the reported experimental values, as shown in Table 1. The elastic constants of g and g0 phases can be determined by Taylor expansion of the total energy with respect to a small lattice strain, see Fast et al. [37] and Karki et al. [38]. Here, the total energy of the strained system can be expressed as,

EðV; aÞ ¼ EðV 0 ; 0Þ þ V 0

X

Cab

a;b

1X aab þ C abcd aab acd 2 a;b;c;d

!

where EðV; aÞ is the total energy of the strained system, V0 is the volume of unstrained system, EðV 0 ; 0Þ is the corresponding total energy of the system, aab is the symmetric distortion matrix (small lattice strain tensor), and C abcd is the adiabatic elastic constant matrix. The minimized structures were then used to calculate the elastic constants of g and g0 phases by applying strains along different orientations [38]. The energy was minimized with respect to the internal atomic coordinates to determine the changes in the total energy (EðV; aÞ) and deduce the elastic properties. The total energy as a function of the applied strains for the g and g0 phases was calculated at zero temperature. The second-order polynomial ﬁts of the calculated total energy (as shown in Figure 3) were then used to compute elastic constants. Finite element simulation with ABAQUS was used to quantify the elastic properties of Cu6Sn5 polycrystalline, to compare with the experimental results. Elastic constants for each crystal were obtained from DFT calculations and random orientations were assigned to each crystal in a 50-grain sample as a representative volume element (RVE), as shown in Figure 4. Polycrystalline was meshed with four-node tetrahedral elements (C3D4). The samples were strained in simple shear and tension boundary conditions to calculate the averaged elastic constants of the polycrystalline aggregate.

Figure 4. Finite element mesh of polycrystalline model used to simulate the elastic properties of Cu6Sn5.

The experimentally measured and calculated elastic constants for the g and g0 phases and the comparisons of these constants with available literature DFT data are listed in Table 2. A Voigt average approach was used to calculate Young’s and Shear modulus of Cu6Sn5. The Hill criterion [39] was used based on the following relations (see Table 3):

Km ¼

1 2 ðC 11 þ C 22 þ C 33 Þ þ ðC 12 þ C 23 þ C 31 Þ 9 9

Gv ¼

1 2 1 ðC 11 þ C 22 þ C 33 Þ ðC 12 þ C 23 þ C 31 Þ þ ðC 44 þ C 55 þ C 66 Þ 15 15 5

where C11, C22, C33, etc., are the stiffness components, Kv is the bulk modulus and Gv is the shear modulus. Young’s modulus can then be related to the shear and bulk moduli by the following equation:

1 1 1 ¼ þ E 3G 9K To account for the porosity in the experimental sample, the equation by Mackenzie was used [40]:

E ¼ E0 1 f 1 p þ f 2 p2 where E is Young’s modulus of a material with porosity fraction p. E0 is the modulus of the fully-dense material. f1 and f2 are constants, of 1.9 and 0.9, respectively, and m is the Poisson ratio (taken here as being equal to 0.3). The single crystal data do not agree very well with the experiments, naturally, because the experimental data are for a polycrystalline aggregate. However, when the single crystal data are included in a mesoscale simulation, to simulate polycrystalline behavior, the agreement with the experiment is quite good. This is particularly true when the porosity in the experimental sample is accounted for. For example, the calculated average Young’s and bulk moduli of the g phase are 89.7 GPa and 60.5 GPa, which are in very good agreement with the experimental values of 88.2 ± 5.4 GPa and 59.1 ± 0.3 GPa respectively. In the case of the monoclinic g0 phase (not measured experimentally), the calculated average mechanical properties are in good agreement with the DFT values of Lee et al. [25]. In addition, our calculations show that the g0 phase has higher elastic modulus than the g phase. In summary, the calculated lattice constants and elastic properties of the hexagonal g phase agree with the experimental results. Finally, we provide some discussion on how the measured values fall relative to values reported in the literature. Fields and Low

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L. Jiang et al. / Scripta Materialia 107 (2015) 26–29 Table 2 Elastic constants of g and g0 compared with experimental and literature values. C11

C22

C33

C12

C13

C23

C44

C55

C66

Experiment – Resonant Ultrasound Spectroscopy (RUS), GPa g-Cu6Sn5 96.5 ± 3.4 96.5 ± 3.4 96.5 ± 3.4

32.4 ± 1.3

32.4 ± 1.3

32.4 ± 1.3

32.1 ± 2.3

32.1 ± 2.3

32.1 ± 2.3

Polycrystalline simulation – Mesoscale Model, GPa g-Cu6Sn5 108.3 108.3

36.4

36.8

36.4

35.4

36.2

35.8

40.5 – 62.2

35.6 – 69.4

35.6 – 60.6

40.0 41.2 42.3

40.0 – 51.9

40.0 – 48.0

108.3

Single Crystal Simulation – Density Functional Theory (DFT), GPa g-Cu6Sn5 101.1 101.1 121.2 g0 -Cu6Sn5 153.5 166.4 149.4 g0 -Cu6Sn5 [11] 156.4 165.2 155.8

Table 3 Young’s moduli calculated from Hill’s bounds for experimentally measured, DFT single crystal values, and mesoscale averages. E (GPa)

G (GPa)

K (GPa)

Experiment – Resonant Ultrasound Spectroscopy (RUS) g-Cu6Sn5 80.2 ± 4.9 32.1 ± 2.3 g-Cu6Sn5 (corrected for porosity) 88.2 ± 5.4 35.3 ± 2.6

53.7 ± 0.3 59.1 ± 0.3

Polycrystalline Simulation – Mesoscale Model g-Cu6Sn5 89.7

35.8

60.5

Single Crystal Simulation – Density Functional Theory g-Cu6Sn5 94.5 38.1 g0 -Cu6Sn5 122.1 47.4 g0 -Cu6Sn5 [1] 116.7 45.0

60.8 95.8 95.5

[41] reported Young’s moduli of 85.6 GPa for Cu6Sn5. The samples were produced by hot-pressing of stoichiometric powder produced by gas atomization, which resulted in a polycrystalline intermetallic with relatively ﬁne grain size. The reported modulus is similar to the values reported here, although, of course, Fields and Low did not measure all the elastic constants as reported here. The difference between modulus measurements from the bulk, such as those reported by us and Fields and Low [41], and that of nanoindentation on an actual soldered sample, may be due to texture in intermetallics found in solder/Cu joints. Prakash and Sritharan [42] observed a <1 0 2> and <1 0 1> texture in Cu6Sn5 (ordered hexagonal structure) and <1 0 2> and <0 3 1> texture in Cu3Sn (ordered orthorhombic) grown from reaction between Pb–Sn solder on a Cu substrate. This texture may have contributed to the higher modulus values reported for nanoindentation, compared to polycrystalline bulk samples. Other authors have reported values for Cu6Sn5 ranging between 85 and 102 GPa [43]. References [1] [2] [3] [4] [5]

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