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# Preferential growth of intermetallics under temperature gradient at Cu–Sn interface during transient liquid phase bonding: insights from phase field simulation

Shuibao Liang, Anil Kunwar, Changqing Liu, Han Jiang, Zhaoxia Zhou

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Shuibao Liang, Anil Kunwar, Changqing Liu, Han Jiang, Zhaoxia Zhou. Preferential growth of intermetallics under temperature gradient at Cu–Sn interface during transient liquid phase bonding: insights from phase field simulation. Journal of Materials Research and Technology, 2022, 19, pp.345–353. 10.1016/j.jmrt.2022.05.047. hal-04113373

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Original Article

# Preferential growth of intermetallics under temperature gradient at Cu–Sn interface during transient liquid phase bonding: insights from phase field simulation

<figure><img src="image_4.png" /></figure>

Shuibao Liang<sup>a</sup>, Anil Kunwar<sup>b</sup>, Changqing Liu<sup>a,*</sup>, Han Jiang<sup>a</sup>,
Zhaoxia Zhou<sup>c</sup>

ARTICLE INFO

Article history:

Received 19 April 2022

Accepted 8 May 2022

Available online 13 May 2022

Keywords:

Thermomigration

Intermetallic

Orientation

Temperature gradient

Phase field model

ABSTRACT

Transient liquid phase bonding under temperature gradient in electronics in-
terconnections yields intermetallic grains at the bonding interface with different
morphological features compared with conventional soldering process. However, the
interfacial reactions due to the thermal gradient that result in the preferential growth of
intermetallics are yet to be fully understood. In this study, incorporating with the ther-
motransport effect a multiphase field model is developed to elaborate the fundamental
growth mechanism of Cu₆Sn₅ intermetallic in the Sn/Cu solder interconnect under tem-
perature gradient. We particularly account for the effect of orientation and anisotropic
thermal conductivity of Cu₆Sn₅ intermetallic grains in relation to the temperature gradient
during their growth, as observed thermal conductivity with the c-axis of Cu₆Sn₅ interme-
tallic parallel to the gradient can be 1.6 times of those perpendicular to the gradient.
Simulation results show that the temperature gradient can accelerate the growth of the
Cu₆Sn₅ phase, as reported in the experiments. The heat flux is mainly conducted through
the intermetallic grain with c-axis parallel to the temperature gradient, causing faster
growth of the grain than the grain with c-axis perpendicular to the temperature gradient;
the growth rate difference of the two types of grains becomes more pronounced under high
temperature gradient. It is revealed that the faster and preferential growth of this type of
intermetallic grain is attributed to the higher thermomigration induced diffusion flux and
accompanying faster atomic interdiffusion process, especially near the solder/interme-
tallic interface.

© 2022 Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

<footnote><sup>a</sup> Wolfson School of Mechanical, Electrical and Manufacturing Engineering, Loughborough University, Loughborough LE11 3TU, UK</footnote>

<footnote><sup>b</sup> Faculty of Mechanical Engineering, Silesian University of Technology, Konarskiego 18A, 44-100 Gliwice, Poland</footnote>

<footnote><sup>c</sup> Department of Materials, Loughborough University, Loughborough LE11 3TU, UK</footnote>

<footnote>* Corresponding author.
E-mail address: c.liu@lboro.ac.uk (C. Liu).
https://doi.org/10.1016/j.jmrt.2022.05.047
2238-7854/© 2022 Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).</footnote>
# 1. Introduction

Transient liquid phase bonding technique has been widely used in brazing and soldering to achieve effective interconnection of metallic layers [1-3]. However, the transient liquid phase bonding of solder interconnects usually requires relatively long processing durations to achieve full intermetallic interconnects for high temperature applications [4-6], which is one of the important bottlenecks for high-efficiency manufacturing. Through applying various physical loadings, such as temperature gradient [7-9], electric current stressing [10] and ultrasound [11], has been demonstrated to efficiently accelerate the formation of interfacial intermetallic in solder interconnects during transient liquid phase bonding over the last few years. The ease of manipulation of applied temperature gradient during bonding of solder interconnects makes transient liquid phase bonding under temperature gradient one of the most promising interconnect technologies. In general, the presence of a temperature gradient can accelerate the formation and growth of intermetallic at the cold end, and the thickness of the intermetallic phase at the hot end is smaller than that at the cold end, which had been demonstrated in Sn-based/Cu [12], Sn/Ag [13] and Sn/Ni [14] interconnects. Particularly, it was reported that the transient liquid phase bonding process of Sn/Cu system under temperature gradient to form full intermetallic interconnects can be 3-10 times faster than that using the conventional transient liquid phase bonding process [7].

The microstructure of the intermetallic phase and grains of solder interconnects formed by transient liquid phase bonding with temperature gradient shows some distinct features compared with the microstructure formed by conventional transient liquid phase bonding [15,16]. Scalloped intermetallic grains at the Sn/Cu interface are usually observed after conventional transient liquid phase bonding, while the columnar intermetallic grains can be formed during transient liquid phase bonding in the presence of a high temperature gradient. Another significant difference is that the crystallographic orientation of intermetallic grains in the solder interconnects formed by transient liquid phase bonding assisted by temperature gradient exhibits obvious preference [7,17]. Notice that the properties of intermetallic grains own anisotropy in different crystallographic orientations. For instance, the Cu₆Sn₅ intermetallic exhibits a 20% increase in strength and a 58% increase in electric conductivity along the [0001] direction compared to that perpendicular to the [0001] direction [18,19]. Meanwhile, the intermetallic interconnects used in three-dimensional integrated circuits may contain only a few grains [20]. Therefore, the investigation of the orientation of intermetallic grain in solder interconnects has been attracting extensive attentions. Yang et al. [7] analysed the grain orientation of Sn/Cu interconnects after the transient liquid phase bonding with temperature gradient, and found that the Cu₆Sn₅ intermetallics at the cold end have a strong preferred orientation and their (0001) poles tend to be parallel to the applied temperature gradient. Similarly, it was reported that the Cu₆Sn₅ grains at the cold end of the Sn/(111)Cu solder interconnects obtained by the temperature gradient transient

liquid phase bonding have obvious preferential orientation and strong texture [17]. Although it has been pointed out that the aligned intermetallic grains could be similar to that in directional solidification [7], the mechanism of producing the intermetallic grains with preferred orientation still needs to be explored and understood.

Phase field method as a powerful numerical tool for simulating and revealing the microstructure evolution and its influencing factors, it has been deployed for the study of intermetallic growth in Sn/Cu [21-23], Mg/Al [24], and Al/Ni [25,26] systems. In addition to the interdiffusion of atoms, effects of electromigration [27,28], stress [29], and micro-elasticity [30] can be coupled to study the growth characteristics of intermetallics under physical fields. Recently, the intermetallic grain growth of Sn/Cu solder interconnects under temperature gradient has been investigated using the phase field simulation [31,32], providing a good understanding of the growth characteristics of intermetallics. Nevertheless, the thermal properties of intermetallic grains were taken as isotropic in previous studies [31,32], the grain orientation effect and anisotropy of material properties was not considered, which is really critical for grasping the microstructural characteristics of micro- and nano-scale solder interconnects.

In this work, the growth characteristics of intermetallic grains in Sn/Cu solder interconnects under the temperature gradient transient liquid phase bonding is investigated with a developed multiphase field model. In addition to coupling the thermotransport-related free energy density into the multiphase field model, the developed model considers the anisotropy in the thermal properties of intermetallic grains with different orientations. The main objective of this study is to reveal the effect of temperature gradient on the growth of intermetallic grains, focusing on analysing the growth rate and competitive impact of intermetallic grains with different orientations.

# 2. Methodology

## 2.1. The multiphase field model

The schematic diagram of the transient liquid phase bonding of Sn/Cu interconnects under temperature gradient is shown in Fig. 1(a), a high temperature gradient is established by the temperature difference between the heat stage and the heat sink, and the interdiffusion of atoms between the substrate and the solder layer causes the formation of intermetallic compounds. Considering Cu₆Sn₅ as the main intermetallic compound phase in the Sn/Cu system, for simplicity, the Cu₆Sn₅ grains with different orientations are investigated, as sketched in the computational domain shown in Fig. 1(b), and three phases: Cu substrate, Cu₆Sn₅ grains and Sn solder are included. In the multiphase field model [23,29,33], a series of non-conserved order parameters {$\eta_i(x, y, z, t)$} ($i = 1, 2, ..., N$) can be introduced to describe the different phases and grains of the metallic system. The variable $\eta_1$ and $\eta_N$ stand for solder and substrate, respectively. The other N-2 number of order parameters denotes the intermetallic grains with different orientations. Meanwhile, a molar centration field $c$ is introduced to consider atom diffusion and its effect on the phase
<figure><img src="image_5.png" /><figcaption>Fig. 1 – Schematics of (a) thermal gradient bonding, (b) computational domain and temperature gradient distribution. The orientation of intermetallic grains 1 and 2 are identified by the angle between applied temperature gradient and their c-axis.</figcaption></figure>

growth. Then the free energy of the system can be constructed as

$$F = \int_V [f_{ch}(\eta_1, \eta_2, \dots, \eta_N, c) + f_{int}(\eta_1, \eta_2, \dots, \eta_N) + f_{tm}(\eta_1, \eta_2, \dots, \eta_N, c)] dV, \quad (1)$$

where $f_{ch}$ is the chemical energy density and

$$f_{ch} = \sum_{i=1}^{N} h_i \left[ \frac{A_i}{2} (c_i - c_{0,i})^2 + B_i (c_i - c_{0,i}) + C_i \right], \quad (2)$$

where $h_i = \eta_i^2 / \sum_{i=1}^{N} \eta_i^2$ and $A_i, B_i$ and $C_i$ are the coefficients of the parabolically fitted Gibbs energy density [23,34] of the phase i, and $c_{0,i}$ is the equilibrium concentration of phase i. The concentration $c_i$ is the local concentration of Sn in phase i. $f_{int}$ is the interfacial free energy, and it is expressed as

$$f_{int} = m f_0(\eta) + \frac{k}{2} \sum_{i=1}^{N} (\nabla \eta_i)^2, \quad (3)$$

where $m = 6\sigma_{int}/l_{int}$, $\sigma_{int}$ is the interfacial energy grain and $l_{int}$ is the diffuse interface width, the gradient energy coefficient $k_{\eta} = 3\sigma_{int}l_{int}/4$, $f_0$ is a fourth order Landau polynomial function and it is written as [23].

$$f_0(\eta) = \sum_{i=1}^{N} (\eta_i^4 / 4 - \eta_i^2 / 2) + 1.5 \sum_{i=1}^{N} \sum_{j=1, j \neq i}^{N} \eta_i^2 \eta_j^2 + 1/4. \quad (4)$$

The last term of Eq. (1), $f_{tm}$, is the free energy density associated to the thermotransport [32,35,36], it can be calculated by

$$f_{tm} = \sum_{i=1}^{N} h_i \frac{Q_i^* c_i \nabla T}{V_m T}, \quad (5)$$

where $V_m$ is the molar volume, $Q_i^*$ is the effective heat of transport of phase i, and $Q_i^* = c_i Q_i^{Sn} + (1 - c_i) Q_i^{Cu}$, $Q_i^{Sn}$ and

$Q_i^{Cu}$ is respectively the effective heat of transport of Sn and Cu in phase i. The local concentration $c_i$ in each phase i at the coexisting phase region obeys the two equations [37]:

$$c = \sum_{i=1}^{N} h_i c_i, \quad (6)$$

$$\frac{\partial f_{ch,1}}{\partial c_1} = \frac{\partial f_{ch,2}}{\partial c_2} = \dots = \frac{\partial f_{ch,N}}{\partial c_N}, \quad (7)$$

When considering the temperature gradient applied to the solder interconnect, the temperature at the bottom surface is set to $T_b$, while the temperature at the top boundary $T_p$ is set depending on the magnitude of the applied temperature gradient. Then the temperature distribution is governed by

$$\nabla \cdot (\mathbf{k}(\eta_1, \eta_2, \dots, \eta_N) \nabla T) = 0, \quad (8)$$

where $\mathbf{k}(\eta_1, \eta_2, \dots, \eta_Q)$ is the thermal conductivity tensor with account for the anisotropy of materials, and $\mathbf{k}(\eta_1, \eta_2, \dots, \eta_N) = \sum_{i=1}^{N} \eta_i^2 \mathbf{k}_i / \sum_{i=1}^{N} \eta_i^2$, and $\mathbf{k}_i$ in the grains with orientation $\eta_i$ is given by $\mathbf{k}_i = M \mathbf{k}_0 M^T$, in which $M$ is the transformation matrix which can be determined by the grain orientations [38].

The evolution of the order parameter follows:

$$\frac{\partial \eta_i}{\partial t} = -L_{\eta} \left( \frac{\delta f_{ch}}{\delta \eta_i} + \frac{\delta f_{int}}{\delta \eta_i} + \frac{\delta f_{tm}}{\delta \eta_i} \right), \quad (9)$$

where mobility $L_{\eta}$ can be determined by the following equation [39]:

$$L_{\eta} = \frac{\sum_{i=1}^{N} \sum_{j=1, j \neq i}^{N} L_{ij} \eta_i^2 \eta_j^2}{\sum_{i=1}^{N} \sum_{j=1, j \neq i}^{N} \eta_i^2 \eta_j^2}, \quad (10)$$

and $L_{ij} = 2m(M_i + M_j) / [3k_{\eta}(c_{0,i}^{eq} - c_{0,j}^{eq})^2]$, where $c_{0,i}^{eq}$ and $c_{0,j}^{eq}$ is respectively the equilibrium concentrations of phase i and j,
$M_i$ and $M_j$ is respectively the diffusion mobility in the two phases.

Moreover, the diffusion of the conserved concentration field c is governed by

$$ \frac{\partial c}{\partial t} = \nabla \cdot \left[ \left( \sum_{i=1}^{N} h_i M_i \right) \nabla \left( \frac{\delta f_{ch}}{\delta c_i} + \frac{\delta f_{tm}}{\delta c_i} \right) \right], \quad (11) $$

where the interdiffusion mobility $M_i$ is associated to the interdiffusion coefficient $D_i$ and grain boundary diffusion coefficient $D_{gb}$. If the grain boundary diffusion is ignored, $M_i$ is calculated by [39].

$$ M_i = \frac{D_i}{V_m \partial^2 f_{ch,i} / \partial c_i^2}. \quad (12) $$

## 2.2. Numerical parameter selection

In the simulation, the coefficients and equilibrium concentration of the polynomial of the free energy density of each phase, i.e., $A_i$, $B_i$, $C_i$, and $c_{0,i}$, at a specific temperature can be determined by interpolation based on the CALPHAD expression provided in Shim et al. [40]. To mimic the transient liquid phase bonding of Sn/Cu solder interconnects, the coefficients for each phase and corresponding equilibrium concentration at 250 °C are listed in Table 1. Noting that the used coefficient $A_i$ of Sn is larger than the obtained value by interpolation to ensure c does not exceed the value 1, and similar assumption was made by Durga et al. [39]. The curves of the Gibbs free energy density of each phase are shown in Fig. 2, the initial concentrations are taken as $c_{i,Cu} = 0.1786$, $c_{i,Cu_6Sn_5} = 0.4981$ and $c_{i,Sn} = 0.999$.

The computational domain shown in Fig. 1(b) is discretized into $120 \Delta x \times 180 \Delta z$ grids and $\Delta x = \Delta z = 4.15 \times 10^{-9}$ m, and it is simplified that the two intermetallic grains with the same morphology are initially included in the intermetallic phase layer, while the orientations of intermetallic grains are represented by the angle $\omega$ between the z direction and the c-axis of the grains. The angle $\omega$ for grain 1 and 2 shown in Fig. 1(b) is respectively set to be 0° and 90°. The periodic condition is applied for order parameters $\eta_i$ and concentration c at the left and right edges of the simulation domain, while Neuman type boundary condition is applied at the top and bottom edges. Since that temperatures $T_b$ and $T_p$ are respectively applied to the bottom and top edges, Dirichlet boundary condition is adopted on these two edges, $T_b = 250$ °C and value of $T_p$ is assigned based on the magnitude of the applied temperature gradient. The diffusion coefficients are taken as [32,41]: $D_{Cu} = 5.90 \times 10^{-5} \exp(-138800/RT)$ m$^2$/s, $D_{Cu_6Sn_5} = 2.58 \times 10^{-8} \exp(-85200/RT)$ m$^2$/s and $D_{Sn} = 2.47 \times 10^{-11} \exp(-33020/RT)$ m$^2$/s.

<table><caption>Table 1 - Coefficients for parabolic free energy densities of phases.</caption><thead><tr><th></th><th>Cu</th><th>Cu<sub>6</sub>Sn<sub>5</sub></th><th>Sn</th></tr></thead><tbody><tr><td>A (J/m<sup>3</sup>)</td><td>9.5928×10<sup>9</sup></td><td>2.5517×10<sup>10</sup></td><td>3.5978×10<sup>11</sup></td></tr><tr><td>B (J/m<sup>3</sup>)</td><td>-2.1841×10<sup>9</sup></td><td>1.6914×10<sup>8</sup></td><td>-6.3615×10<sup>9</sup></td></tr><tr><td>C (J/m<sup>3</sup>)</td><td>-1.2184×10<sup>9</sup></td><td>-1.9306×10<sup>9</sup></td><td>-1.4443×10<sup>9</sup></td></tr><tr><td>c<sub>0</sub></td><td>0.0637</td><td>0.4265</td><td>0.9800</td></tr><tr><td>c<sup>tot</sup><sub>0</sub></td><td>0.0453</td><td>0.3266/0.4498</td><td>0.9800</td></tr></tbody></table>

<figure><img src="image_6.png" /><figcaption>Fig. 2 - Gibbs free energy density of each phase at 250 °C.</figcaption></figure>

$10^{-11} \exp(-33020/RT)$ m$^2$/s. The diffuse interface width $l_{gb} = 3.33 \times 10^{-8}$ m, and $\sigma_{gb} = 0.5 J/m^2$ [29], $V_m = 1.629 \times 10^{-5}$ m$^3$/mol. In terms of the thermal conductivities of each phase, thermal conductivity of Sn solder and Cu substate are assumed to be isotropic and $k^{Cu} = 398$ W/(m·K), $k^{Sn} = 67$ W/(m·K) [42]. The Cu$_6$Sn$_5$ intermetallic grains have the hexagonal structure [43], as shown in Fig. 3(a-e), the atom density on different plane is different, possibly induce that the conduct resistance for phonons and electrons is anisotropic, which is similar to that observed in Cu$_3$Sn intermetallics [44]. The thermal conductivity of Cu$_6$Sn$_5$ intermetallic can be connected to its electric conductivity through the Wiedemann-Franz law as $k_{ij} = \sigma_{ij}L_{wf}T$ [45], where $k_{ij}$ and $\sigma_{ij}$ is respectively the elements of the thermal and electric conductivity tensors, $L_{wf}$ is the Lorenz number and $L_{wf} = 1.99 \times 10^{-8}$ W$\Omega$K$^{-2}$ for Cu$_6$Sn$_5$ [46]. Based on the Wiedemann-Franz law and the electric conductivity values of Cu$_6$Sn$_5$ obtained by Liu et al. [19], the crystal orientation dependence of thermal conductivity of Cu$_6$Sn$_5$ can be shown in Fig. 3(f), in which the ratio of the maximum and minimum value is 1.6, thus the obvious anisotropic thermal conductivity of Cu$_6$Sn$_5$ is considered: $k_{11}^{IMC} = k_{22}^{IMC} = 24.78$ W/(m·K), $k_{33}^{IMC} = 39.04$ W/(m·K) [19] [46]. The effective heat of transport of Sn and Cu in each phase are assigned as [31,47-49]: $Q_{Sn}^{Sn} = -11.12$ KJ/mol, $Q_{Sn}^{Cu} = 11.12$ KJ/mol, $Q_{Cu_6Sn_5}^{Sn} = -3.38$ KJ/mol, $Q_{Cu_6Sn_5}^{Cu} = 35.10$ KJ/mol, $Q_{Cu}^{Sn} = -3.38$ KJ/mol and $Q_{Cu}^{Cu} = 3.38$ KJ/mol.

# 3. Results and discussion

## 3.1. Effect of temperature gradient on intermetallic grain growth

The growth of the intermetallic phase containing two differently oriented grains under temperature gradients of $\nabla T_a = 0$,
<figure><img src="image_7.png" /><figcaption>Fig. 3 – (a) The unit cell of Cu₆Sn₅. (b) Atom distributions in the 3³ Cu₆Sn₅ unit cells. (c–e) Atom projections in three planes perpendicular to the a-, b-, and c-axes. (f) Surface representation of thermal conductivity change with orientation.</figcaption></figure>

1.5 × 10⁵ and 1.5 × 10⁶ K/m is first investigated. The simulated results show that the existence of the temperature gradient accelerates the growth of the intermetallic phase, as shown in Fig. 4(a), where the contour lines of the grain boundary of intermetallic grains are sketched for t = 3.5 s. Compared the contour lines with the initial grain boundary position line (i.e., dashed line), it is observed that the intermetallic grain size increases, and the intermetallic phase occupy more solder matrix region due to higher interdiffusion coefficient at solder/intermetallic interface than that at solder/substate interface. Moreover, larger temperature gradient drives the faster growth of intermetallic phase (see Fig. 4(b)), which was observed in some previous experiments [50,51]. This is because that the temperature gradient in the top-down direction promotes the interdiffusion of atoms, and the directional thermomigration of atoms induces additional diffusion flux.

Noting that the thermal conductivities of the two intermetallic grains with different orientations are different, and the heat flow through the solder interconnect is affected by the grain morphology, as shown in Fig. 5, in which the cloud map of the nominal heat flux magnitude is displayed. The nominal heat flux magnitude $\bar{\varphi}_q = \varphi_q / \nabla T_a$ is chosen for visualization purpose, where $\varphi_q$ is the heat flux. From the distribution and directions of the heat flux vectors, we note that the heat flux flows from the solder to the bottom substrate. Moreover, the heat flux in grain 1 is greater than that in grain 2 due to their different orientation. For grain 1, its c-axis is parallel to applied temperature gradient direction (i.e., along z-axis), thus its thermal conductivity along the temperature gradient direction is larger than that of grain 2. The conduct resistance for phonons and electrons of grain 1 is smaller, and the heat flux vector is more inclined to be conducted through the grain 1 and bypass grain 2. Theoretically, grain 1 with

<figure><img src="image_8.png" /><figcaption>Fig. 4 – (a) Simulated grain boundary contours under temperature gradients of $\nabla T_a = 0, 1.5 \times 10^5$ and $1.5 \times 10^6$ K/m at 3.5 s, where the dashed contour shows the initial position of intermetallic grains. (b) Increase of the intermetallic layer thickness with time.</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>Fig. 5 – The distribution of heat flux magnitude and flux vector arrows after applying temperature gradient of $\nabla T_a = 1.5 \times 10^5$ (a) and $1.5 \times 10^6$ K/m (b) for 3.5 s.</figcaption></figure>

higher thermal conductivity has a larger heat flux since $\overrightarrow{\phi_q} = -k\nabla T$.

## 3.2. The preferential growth of intermetallic grains

Another observation from Fig. 4(a) is size difference between grain 1 and grain 2, as further confirmed in Fig. 6(a). It is found that size of grain 1 becomes larger than that of grain 2, in particular under the condition of high temperature gradient. From the grain boundary contours shown in Fig. 4(a), it is noticed that under high temperature gradient grain 1 not only grows faster towards the solder matrix, but also slightly expands laterally and occupies more spaces than grain 2. It is indicated that the grain 1 more preferentially exists than grain 2 in the intermetallic phase layer, which is similar to the preferential growth and existence of intermetallic grains with high electric conductivity under high current density [52,53]. This phenomenon was also found in the experimental studies of transient liquid phase bonding under temperature gradient [7], that is, the Cu₆Sn₅ intermetallic grains with c-axis along/ near the applied temperature gradient occupy most of the intermetallic phase region. Although the experimental phenomena had been pointed out to be similar to the directional solidification [5], the mechanism of the preferential growth of intermetallic grains remains unclear. Our simulation results have demonstrated the preferential growth of Cu₆Sn₅ intermetallic grains under temperature gradient, and have found that the anisotropy in thermal conductivity may be responsible for the faster and preferential growth of the intermetallic grains under temperature gradient.

To further reveal the temperature gradient effect on the grain size difference, the simulation cases for other different various temperature gradients are investigated, it is found that the grain size difference increases gradually with increasing the temperature gradient, as shown in Fig. 6(b), and the preferential growth of the grains 1 will be more significant under higher temperature gradient.

Although the anisotropy of interfacial energy and diffusivity can affect the formation and growth of intermetallic grains [30,54], the grain orientation of Cu₆Sn₅ intermetallics

generally does not exhibit obvious preference after transient liquid phase bonding without the temperature gradient [7]. Therefore, it can be inferred that the preferential growth of intermetallic grains under temperature gradient is mainly determined by their thermal properties, which is analogous to the preferential growth of intermetallic grains with high electric conductivity under electric current stressing [55].

## 3.3. Thermomigration induced flux and enhanced atom diffusion

To further understand the mechanism behind the preferential growth of grain 1, we look back at the additional diffusion flux due to the atomic thermomigration, which is expressed as $J_{tm} = - \sum_{i=1}^{N} h_i M^* Q_i^* c_i \nabla T / TV_m$ [15,36]. By investigating the additional diffusion fluxes respectively across grain 1 and grain 2, as shown in Fig. 7, the diffusion flux of grain 1 is higher than that of grain 2, specifically in the vicinity of the solder/ intermetallic interface. Besides, the difference in diffusion flux between grain 1 and grain 2 is larger under higher temperature gradient, see the comparison of Fig. 7(a) and (b). The higher diffusion flux makes it easier for the interdiffusion of atoms to form intermetallics, which leads to a faster growth of intermetallic grain 1. Furthermore, the variation in Sn fraction at t = 0, 1.75 and 3.5 s is depicted in Fig. 8, the Sn concentration difference across grain 1 and grain 2 increases with time. The presence of high temperature gradient drives the faster diffusion of Sn atoms to the intermetallic phase and the substrate, and more significant difference of Sn concentration across grain 1 and grain 2 is observed in Fig. 8(b), the space occupied by grain 1 is larger than grain 2. It is understandable because thermal conductivity is proportional to electric conductivity at the same temperature based on the Wiedemann-Franz law. Liu et al. [19] found that the mobility of the electron carriers is larger along the c-axis of the Cu₆Sn₅ crystals, and the carrier mobility can also play a dominant role in the thermal conductivity of metals, implying that the scattering effect on carrier is smaller, thus the additional diffusion flux caused by thermomigration is larger and resulting in faster growth of grain 1. Once grain 1 grows faster than grain 2,
<figure><img src="image_10.png" /><figcaption>Fig. 6 – (a) Grain size changes with time for different temperature gradients. (b) Grain size difference between grain 1 and grain 2 vs temperature gradient.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Fig. 7 – Thermomigration induced flux along the two paths $x = 0$ and $60\Delta x$ under $\nabla T_a = 1.5 \times 10^5$ (a) and $1.5 \times 10^6$ K/m (b).</figcaption></figure>

the grain boundary between grain 1 and grain 2 will no longer be parallel to the applied temperature gradient due to constraints of Young's equation, then the induced temperature gradient concentration at the grain boundaries could further promote the growth of grain 1 in the lateral direction (see Fig. 4(a)).

The simulation results have demonstrated that the grain 1 whose c-axis along the applied temperature gradient grow

preferentially, and the thermal conductivity along the temperature gradient of this type of grain is larger than that of grain 2. Predictably, if the nucleation stage of intermetallic grain is considered in follow-up model and studies, the grain difference under the temperature gradient will be greater. The findings acquired from this study would provide some theoretical understandings of the microstructure evolution and control of solder interconnects through transient liquid phase

<figure><img src="image_12.png" /><figcaption>Fig. 8 – Sn fraction distributions along the two paths $x = 0$ and $60\Delta x$ under $\nabla T_a = 1.5 \times 10^5$ (a) and $1.5 \times 10^6$ K/m (b).</figcaption></figure>
bonding under temperature gradient, and may have certain
inspirations for the microstructure optimization by high
temperature gradients in emerging processing techniques
such as additive manufacturing.

4. Conclusion

In this study, the preferential growth of Cu₆Sn₅ intermetallics
in Cu/Sn interconnects during the transient liquid phase
bonding under temperature gradient is investigated using a
multiphase field model, which is primarily attributed to the
anisotropic characteristics of thermal conductivity and orien-
tation of intermetallic grains at the Cu–Sn interface. It is found
that the higher temperature gradient drives the intermetallic
phase grow faster due to the additional diffusion flux induced
by thermomigration. The intermetallic grains with c-axis par-
allel to temperature gradient exhibit higher thermal conduc-
tivity compared to the grains with c-axis perpendicular to the
temperature gradient, hence the heat flux in the grains with c-
axis parallel to the temperature gradient is greater; this causes
the significant preferential growth of these intermetallic grains
given the accelerated diffusion for atomic transports.
Comparative analysis of growth rate of two different oriented
intermetallic grains under temperature gradient shows that
the high temperature gradient induces the faster and prefer-
ential growth of the grain with c-axis in parallel with the tem-
perature gradient, resulting in a significant increase in the size
of these intermetallic grains. It is revealed that the preferential
grain growth is closely correlated with the higher thermomi-
gration induced diffusion flux across the grains with c-axis
parallel to the temperature gradient, which could potentially
suppress the growth of the grains with c-axis perpendicular to
the temperature gradient. The main contribution of the pre-
sent study is to elaborate the anisotropic effect of thermal
conductivity on the growth of Cu₆Sn₅ intermetallic grains in
Cu/Sn systems under temperature gradient, in particular, we
found the preferential growth of Cu₆Sn₅ intermetallic grains
due to the better thermal conductivity along the temperature
gradient. Moreover, the developed model could be further
applied for investigations of the growth of other intermetallic
grains under temperature gradient. It is anticipated that pres-
ently obtained findings enable a step forward for the under-
standing of the orientation and microstructure characteristics
of intermetallic grains through transient liquid phase bonding
under temperature gradient, and further studies could incor-
porate the competitive effects of multiple intermetallic grains
under thermomigration.

Declaration of Competing Interest

The authors declare that they have no known competing
financial interests or personal relationships that could have
appeared to influence the work reported in this paper.

Acknowledgments

The authors wish to acknowledge the support from EPSRC
(UK) research grants (No. EP/R004501/1 and No. EP/R032203/1)

and the National Science Centre, Poland (DEC-2021/42/E/ST5/
00339). The author H. Jiang wishes to acknowledge the finan-
cial support from China Scholarship Council (201806150013).
S. Liang would like to thank Dr. J. Hektor (Department of Ma-
terials Science and Applied Mathematics, Faculty of Tech-
nology & Society, Malmö University) for valuable discussions
and kind help.

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