Introduction

With the rapid development of microelectronic packaging and integration technology in the era of big data and artificial intelligence, a hierarchy of solder joints is needed, and the three-dimensional vertical interconnect structures requires multiple reflow processes during manufacturing. Thus, low-temperature solders have attracted attention and are used for third-level interconnects to ensure that the first and second levels of solder interconnects do not melt during the processing of the third-level solder interconnects.1 Moreover, low-temperature solders are beneficial in alleviating thermal mismatch-induced warpage issues. Due to the many advantages of Sn-Bi based solders,2 such as low melting point, low coefficient of thermal expansion, and low cost, eutectic Sn-58Bi (wt.%) alloy has been recommended as a substitute solder for low-temperature interconnects.

Despite the aforementioned advantages resulting from the low melting point of Sn-Bi solder interconnects, the microstructure of the Sn-58Bi solder matrix is usually prone to coarsening or segregation due to the low activation energy of atomic diffusion and migration,3,4 especially under the loadings of electric current stressing and accompanying thermal gradient.5,6 For instance, the high-density electric current passing through the solder interconnects can induce void formation and polarity growth of intermetallics.7,8,9 The temperature gradient existing in electronic packaging can induce thermomigration issues of solder interconnects,10,11,12 which induces directional diffusion of Bi atoms to the cold end in Sn-58Bi solder interconnects.13 Thus, many studies14,15 have focused on the reliability issues and microstructure characteristics of Sn-58Bi solder interconnects. Generally, under the influence of electromigration or thermomigration, more serious phase migration or segregation is found in Sn-58Bi solder interconnects compared with that observed in Sn-Ag and Sn-Ag-Cu solder interconnects.6,16,17

In recent years, Cu-cored solder has been proved to be one of the effective ways to enhance the performance and reliability of solder interconnects, due to the good electrical and thermal conductivities, and the high melting point. In addition to studies on the performance improvement of Sn-Pb,18 Sn-Ag,19 and Sn-Ag-Cu20,21 solder interconnects by embedding the Cu cores in solder matrix, some researchers22,23 have explored the performance of Cu-cored Sn-58Bi solder interconnects. Mu et al.22 observed the redistribution and accumulation of Sn- and Bi-rich phases around the Cu periphery. Liu et al.23 added graphene nanosheets to Cu-cored Sn-Bi solder interconnects to improve their electromigration resistance. Nevertheless, the quantitative mechanical mechanism and kinetics of phase migration and microstructure evolution affected by the Cu cores remain to be clarified and revealed. Our previous studies24,25 showed that phase field method can be effectively used to simulate and investigate the evolution of the microstructure of Sn-Bi solder interconnects, relying on the advantages of phase interfaces, without explicitly tracking their movement.26,27

In this work, a quantitative phase field model is developed to study the electromigration and phase segregation behavior of Cu-cored Sn-58Bi solder interconnects. The phase migration kinetics and mechanics are investigated, then the current density and temperature distribution characteristics due to the inhomogeneous microstructure redistribution or segregation are discussed in detail. The focus will be placed on revealing the influence of the Cu core on the physical field distribution and the phase migration rate.

Methodology

To study the microstructure evolution and phase migration in Cu-cored Sn-58Bi solder interconnects, a variable concentration c is introduced to represent the mole fraction of Bi atoms, then the total free energy of the system under electric current stressing can be constructed as28

$$\begin{aligned} \begin{aligned} F = \int _V^{} {\left[ {f(c) + \frac{{{\varepsilon ^2}}}{2}{{\left| {\nabla c} \right| }^2} + {f_{em}} + {f_{tm}}} \right] {\hspace{1.0pt}} {\hspace{1.0pt}} } \mathrm{{d}}V, \end{aligned} \end{aligned}$$
(1)

where the first term f(c) is the Gibbs free energy density, \(\varepsilon^2\) is proportional to the surface energy, and \(\varepsilon^2 = 2\gamma {l_w}/{({c_{bct}} - {c_{rhomb}})^2}\),29 where \(\gamma \) denotes the surface energy, \({l_w}\) is the diffusion interface width, and \({c_{bct}}\) and \({c_{rhomb}}\) are the equilibrium concentration of tin-based bct phase and the bismuth-based rhombohedral phase, respectively. The third term \({f_{em}}\) corresponds to the electrostatic free energy density, \({f_{em}} = {N_{\mathrm{{a}}}}e\psi {Z^*}\),30 and \({Z^*} = c{Z_{Bi}} + (1 - c){Z_{Sn}}\), where \({N_a}\) is Avogadro’s number, e is the electronic charge, \(\psi \) is the electric potential, and \({Z_{\text{Bi}}}\) and \({Z_{\text{Sn}}}\) are the effective charge number of Bi and Sn, respectively. The last term \({f_{tm}}\) represents the energy contribution from atomic thermomigration.

Fig. 1
figure 1

Gibbs free energy density curves for Sn-rich and Bi-rich phases at 100°C and the fitted free energy density curve.

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The free energy density for each phase in the binary solder alloy system can be obtained by the equation31

$$\begin{aligned} \begin{aligned} f{(c)_i}&= {c_A}f_A^0 + {c_B}f_B^0 + RT\left[ {{c_A}\ln ({c_A}) + {c_B}\ln ({c_B})} \right] \\&\quad + {c_A}{c_B}\left[ {L_0^{A,B} + L_1^{A,B}({c_A} - {c_B})} \right] , \end{aligned} \end{aligned}$$
(2)

where \({c_A}\) and \({c_B}\) are the mole fractions of components Bi and Sn, respectively, and \({c_A} = c \), \({c_A} = 1 - c \). \(f_A^0\) and \(f_B^0\) are the temperature-dependent Gibbs free energies of pure component Bi and Sn, respectively. \(L_0^{A,B}\) and \(L_1^{A,B}\) are the interaction parameters. To facilitate the numerical solution procedure,32 the two curves of the Sn-rich and Bi-rich phases are fitted to a polynomial function to eliminate singularity at the intersection, as show in Fig. 1. Then the free energy density f(c) can be obtained.

Table I Material properties and input parameters for phase field modelling
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When electric current is applied to the solder interconnects, it obeys \(\nabla \cdot \overrightarrow{j} = = 0\), and the electrical potential distribution yields \(\nabla \cdot \left[ {\lambda \cdot \nabla \psi } \right] = 0,\) where \(\lambda \) is the electrical conductivity. The electrical conductivity in the solder matrix is the linear combination of the electrical conductivities of Sn-rich and Bi-rich phases. Moreover, a change in the temperature in the solder interconnects may be induced by Joule heating, and the temperature distribution is governed by \(\rho {C_p}{{\partial T} \big / {\partial t}} = \nabla ({k_t}\nabla T) + \lambda {\left| {\nabla \phi } \right| ^2}\), where \(\rho \), \({C_p}\), and \({k_t}\) are the density, heat capacity, and thermal conductivity, respectively.

Then, the phase separation and coarsening can be described by a Cahn–Hilliard equation as

$$\begin{aligned} \begin{aligned} \frac{{\partial c}}{{\partial t}} = \nabla \cdot \left[ {M\left[ {\nabla \left( {\frac{{\partial f(c)}}{{\partial c}} - {\varepsilon ^2}{{\left| {\nabla c} \right| }^2} + \frac{{\partial {f_{em}}}}{{\partial c}}} \right) + \frac{{{Q^*}\nabla T}}{{{V_m}T}}} \right] } \right] , \end{aligned} \end{aligned}$$
(3)

where M is the atomic mobility, and it is related to the diffusion coefficients as \(M = D/\left( {{{{\partial ^2}f(c)} \big / {\partial {c^2}}}} \right) \), and \( D = \theta {D_{bct}} + (1 - \theta ){D_{rhomb}}\), and the shape function \(\theta \) is defined as \(\theta = ({{{c_{rhomb}} - c)} \big / {({c_{rhomb}} - {c_{bct}})}}\).39 \({D_{bct}}\) and \({D_{rhomb}}\) are the diffusion coefficients in Sn-rich and Bi-rich phases, respectively. The last term includes the contribution from \({f_{tm}}\),25,40 \({V_m}\) is the molar volume, and the effective heat of transport \({Q^*}\) is taken in the form of \({Q^*} = c{Q_{Bi}} + (1 - c){Q_{Sn}}\), where \({Q_{Bi}}\) and \({Q_{Sn}}\) are the heat of transport of Bi and Sn, respectively.

Fig. 2
figure 2

(a) Schematic diagram of the geometric model of Cu-cored solder interconnects. (b) Typical microstructure of Sn-58Bi solder and mapping results to elements.

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In the present work, line-type Cu-cored Sn-58Bi solder interconnects are considered to study their electromigration and phase segregation behavior under electric current stressing at 100°C, as schematized in Fig. 2a, which is similar to the previous experimental studies.22,23 Noting that the Cu-Sn intermetallic layer near the substrates and Cu core is relatively thin,22,41 the intermetallic formation and its influence on the current density and temperature gradient distributions are temporally ignored. An electric current with a density of 1.5 \(\times \) 10\(^4\) A/cm\(^2\) is applied, and it is assumed that the left surface is grounded. For the thermal boundary conditions, similar to previous studies,41,42 it is assumed that the Cu substrates are kept at 100°C, and convection coefficients in the air around all the boundaries are set to be 10 \(\mathrm{{W/}}{\mathrm{{m}}^{\mathrm{{2}}}} \cdot \mathrm{{K}}\),42 and the ambient temperature is taken as 25°C. The interfacial intermetallics in Sn-58Bi/Cu interconnects are relatively thin; thus, the intermetallic effect is temporally ignored.43 To investigate the Cu core effect on the microstructure evolution of Sn-58Bi solder interconnects, the electromigration behavior of the interconnects with different Cu core sizes of \(r_{c}\)= 0 μm, 6.25 μm, and 12.5 μm is examined. Other geometric parameters of the considered solder interconnects are assigned as \(L \)= 90 μm, H= 52 μm, and \(L_{s}\)=31 μm. The material physical properties and values of input parameters for phase field modelling can be seen in Table I. Similar to the previous study,44 typical scanning electron microscopy showing the phase distribution characteristics from the experiment is used for reconstructing a finite element model and describing the initial states of the simulated phases in solder matrix, as shown in Fig. 2b.

Results and Discussion

Microstructural Evolution and Phase Segregation

We first consider the microstructural evolution of the Sn-58Bi solder matrix in the interconnects with different Cu core sizes under a current density of 1.5 \(\times \) 10\(^4\) A/cm\(^2\). The simulated results of microstructure evolution and migration are shown in Fig. 3, where 3a1–c1, a2–c2, and a3–c3 correspond to the cases with Cu core radii \(r_{c}\) = 0 μm, 6.25 μm, and 12.5 μm, respectively. When t = 10 h, the Sn-rich phase and Bi-rich phase are alternately distributed in the solder matrix, and the microstructure characteristics of the solder matrix after reflow or aging (see Fig. 2b) are maintained. As the time proceeds to 30 h and 60 h, it can be seen that the Bi-rich phase gathers on the right anode side to form a Bi-rich layer in the solder matrix, while the Sn atoms migrate in the current density direction, and the Sn-rich phase segregates and accumulates on the left cathode side, which is in good agreement with the experimental observations.45,46 Based on the comparative analysis of the microstructure of solder interconnects with different values of \(r_{c}\), it is exhibited that the existence of Cu cores causes the accumulation of Bi-rich phases in the left half of the cores, as can be seen in 3b2–c2 and b3–c3, which qualitatively agree with the experimental results observed by Mu et al.22 Moreover, the presence of copper nuclei could change the current distribution features, and as a result, the segregation of the Bi-rich phase on the right anode side of the solder matrix is significantly hindered, especially for the case of Cu-cored Sn-58Bi solder interconnects with \(r_{c}\)=12.5 μm. This means that the introduction of Cu core for the Sn-58Bi solder interconnects can reduce the segregation degree of the Bi-rich phase to the anode side, which may reduce the brittle fracture potential47 caused by the Bi-rich layer in the solder interconnects during service.

Fig. 3
figure 3

Simulated results of microstructure evolution of solder matrix in Cu-cored Sn-58Bi solder interconnects with \(r_{c}\)=0 (a1–c1), 6.25 (a2–c2), and 12.5 μm (a3–c3) under electric current stressing with the density of 1.5 \(\times \) 10\(^4\) A/cm\(^2\) for times (a1–a3) t=10 h, (b1–b3) t=30 h, and (c1–c3) t = 60 h. The color bar represents the concentration of Bi (Color figure online).

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In order to quantitatively understand the effect of Cu core on the electromigration and segregation behavior of the Sn-58Bi solder interconnects, the average migration displacement of Bi-rich phase in interconnects owning different sized Cu cores is calculated by the following equation:

$$\begin{aligned} \begin{aligned} {S_{em}} = \frac{{\iint \limits _{t = {t_f}} {I(c)xdxdy}}}{{\iint \limits _{t = {t_f}} {I(c)dxdy}}} - \frac{{\iint \limits _{t = 0} {I(c)xdxdy}}}{{\iint \limits _{t = 0} {I(c)dxdy}}}, \end{aligned} \end{aligned}$$
(4)

where \({S_{em}}\) is the displacement at time \({{t_f}}\), and the function I(c) has the value of 1 if \(c>0.5\) and otherwise \(I(c) = 0\) in the solder matrix. As shown in Fig. 4, it can be clearly seen that the migration speed of the Bi-rich phase in the Sn-58Bi solder interconnect without Cu core is the largest, while the migration speed of the Bi-rich phase in the Cu-cored Sn-58Bi solder interconnect with \(r_{c}\)=12.5 μm is the smallest. The migration displacement increases almost linearly with time, which indicates that the Bi-rich layer would grow linearly.46 Based on the analysis of the slope of the curves in Fig. 4, it is found that the migration speeds of the Bi-rich phase corresponding to the cases with \(r_{c}\) = 0 μm, 6.25 μm, and 12.5 μm are 1.09 \(\times \) 10\(^{-11}\), 9.97 \(\times \) 10\(^{-12}\), and 6.39 \(\times \) 10\(^{-12}\) m/s, respectively. The simulated velocity of Bi-rich phase migration corresponding to the Sn-58Bi solder interconnect without Cu core is close to the experimentally obtained migration velocity \(\sim \) 10\(^{-11}\) m/s.46,48

Fig. 4
figure 4

Migration displacement of the Bi-rich phase changes with time for the Cu-cored Sn-58Bi solder interconnects with \(r_{c}\)= 0 μm, 6.25 μm, and 12.5 μm.

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Inhomogeneous Current Density Distribution and Resistance Change

To further reveal the mechanism and reason behind the electromigration and segregation of the Cu-cored Sn-58Bi solder interconnects, the current density distribution directly related to the electromigration driving force needs to be analyzed since \({F_{em}} = {N_{\mathrm{{a}}}}e{Z^*}\nabla \psi \), and \(\overrightarrow{j} = - \lambda \nabla \psi \). It is noted that the electrical conductivity of Sn, Bi, and Cu atoms is significantly different, and the electrical conductivity of Cu atoms is 6.5 and 77.4 times that of Sn and Bi atoms, respectively. Therefore, the current density distribution in the solder matrix is strongly affected by the microstructure characteristics. Figure 5 presents the current density distribution in Sn-rich and Bi-rich phases of the Sn-58Bi solder interconnects. Clearly, the current density in Sn-rich phase and Cu core is much higher than that in the Bi-rich phase, and the electron flow vectors mainly pass through the Sn-rich phase and the Cu core, while the Bi-rich phase hinders the direct migration of the electron flow to the anode side. For the Cu-cored Sn-58Bi solder interconnects, the cores significantly weaken the current crowding effect near the four corners of the solder matrix, and the current crowding phenomenon appears on the contact interfaces between the Cu core and the solder matrix in the direction of the applied electric current, as can be seen in Fig. 5b and c. Although the large current density around the edge of the Cu core could result in a large migration driving force, its effect is mainly reflected in the formation of the Bi-rich phase layer near the left half of the Cu cores or the Sn-rich phase layer near the right half of the cores. Moreover, the increase in the size of the Cu core can reduce the crowding of the current density in the Sn-rich phase and the Bi-rich phase. For the current density of the three cases at t = 30 h shown in Fig. 5, the average current density in Sn-rich and Bi-rich phases is 1.84 \(\times \) 10\(^4\) and 8.82 \(\times \) 10\(^3\) A/cm\(^2\), respectively, when \(r_{c}\) = 0, the average current density in Sn-rich and Bi-rich phases is respectively 1.74 \(\times \) 10\(^4\) and 8.58 \(\times \) 10\(^3\) A/cm\(^2\) corresponding to the case with \(r_{c}\)= 6.25  μm, and the average current density in Sn-rich and Bi-rich phases is 1.32 \(\times \) 10\(^4\) and 7.19 \(\times \) 10\(^3\) A/cm\(^2\), respectively, when \(r_{c}\)= 12.5 μm. It is evident that the larger the Cu core size, the lower current density in the Sn-rich and Bi-rich phases will be. The lower current density induces the decrease in the electromigration driving force applied on atoms; thus, the phase migration and accumulation rate decrease.

Fig. 5
figure 5

Current density distribution in Sn-rich and Bi-rich phases of the solder interconnects with \(r_{c}\)= 0 μm (a), 6.25 μm (b), and 12.5  μm (c) at t = 30 h. The microstructure characteristics corresponding to Fig. 3b1–b3, and the left plot in (a–c) shows the current density distribution in the Sn-rich phase and Cu core, and the right plot displays the current density distribution in the Bi-rich phase.

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Due to the difference in the electrical conductivities of Bi, Sn, and Cu atoms, the electromigration and segregation of Bi-rich phase can influence the electrical resistance of the solder interconnects, so the voltage across the solder interconnects changes with time, as shown in Fig. 6. Generally, the voltage increases linearly with time due to the Bi-rich phase segregation and accumulation to the anode side, which is consistent with previous experimental studies.48,49 Despite the lower rate of Bi-rich phase migration for the interconnects with a larger Cu core, the current crowding effect near the core periphery drives the Bi segregation to the anode end of the cores, and thus the rate of voltage increase is almost the same as that of the solder interconnect without Cu cores. The magnitude of voltage across the Cu-cored Sn-58Bi solder interconnects is lower than the solder interconnect without the Cu core, and the increase in voltage is smaller for the larger Cu-cored Sn-58Bi solder interconnect due to the hindered phase migration and segregation. The lower voltage of the Cu Sn-58Bi solder interconnects means that Joule heating will be less, and the higher thermal conductivity of Cu is conducive to the heat release. Therefore, the existence of the Cu core will be beneficial to the thermal management of the solder interconnects in integrated circuits.

Fig. 6
figure 6

Variation in the voltage across the Cu-cored Sn-58Bi solder interconnects with \(r_{c}\)= 0 μm, 6.25 μm, and 12.5  μm.

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Inhomogeneous Temperature Gradient Distribution

When electric current flows through the solder interconnects, the generation of Joule heat may change the temperature distribution of the solder interconnects. Although the temperature difference between Bi-rich and Sn-rich phase is small, the temperature gradient is large. The high temperature gradient may promote the directional migration of the Bi-rich phase in the solder matrix. The thermomigration driving force is usually considered to be proportional to the magnitude of the temperature gradient, \({F_{tm}} = {{{Q_{}}^{\mathrm{{*}}}\nabla T} \big / {{V_m}T}}\), and the temperature change is small. Figure 7 shows the temperature gradient distribution in Sn-rich and Bi-rich phases of the Sn-58Bi solder interconnects. We can see that the temperature gradient in Sn-rich phase and the Cu core is lower than that in the Bi-rich phase. The heat flow is transferred from the central region to the cathode and anode sides, and mainly through the Sn-rich phase and the Cu core, due to the better thermal conductivity of the Sn-rich phase and Cu compared to Bi-rich phase. Another feature is that as the size of the Cu core increases, the average temperature gradient in the solder matrix decreases due to the high thermal conductivity of Cu. For the temperature gradient of the three cases at t = 30 h shown in Fig. 7, the average temperature gradient in Sn-rich and Bi-rich phases is 1.05 \(\times \) 10\(^3\) and 2.57 \(\times \) 10\(^3\)°C/m, respectively, when \(r_{c}\) = 0; those values change to 9.00 \(\times \) 10\(^2\) and 2.31 \(\times \) 10\(^3\)°C/m for the interconnect with \(r_{c}\)= 6.25 μm. The average temperature gradient in Sn-rich and Bi-rich phases is 4.71 \(\times \) 10\(^2\) and 1.31 \(\times \) 10\(^3\)°C/m, respectively, when \(r_{c}\)= 12.5 μm. Moreover, the maximum temperature gradient in Sn-rich and Bi-rich phases is 9.16 \(\times \) 10\(^3\) and 5.28 \(\times \) 10\(^4\)°C/m, respectively, when \(r_{c}\)=0; those values change to 7.99 \(\times \) 10\(^3\) and 4.51 \(\times \) 10\(^4\)°C/m for the interconnect with \(r_{c}\)= 6.25 μm. The maximum temperature gradient in Sn-rich and Bi-rich phases is 4.24 \(\times \) 10\(^3\) and 1.95 \(\times \) 10\(^4\)°C/m, respectively, when \(r_{c}\)= 12.5 μm. It should be noted that the formation of the Bi-rich phase layer induces the high temperature gradient near the anode side. The temperature gradient int the Bi-rich phase layer near the anode side is larger for the interconnect with \(r_{c}\)= 12.5 μm due to the thermal conductivity difference in phases and shorter path length for the heat transfer from the Cu core to the Bi-rich phase close to the anode side.

Fig. 7
figure 7

Temperature gradient distribution in Sn-rich and Bi-rich phases of the solder interconnects with \(r_{c}\)= 0 μm (a), 6.25 μm (b), and 12.5  μm (c) at t = 30 h. The microstructure corresponding to Fig. 3b1–b3, and the left plot in (a–c) shows the temperature gradient distribution in the Sn-rich phase and Cu core, and the right plot displays the temperature gradient distribution in the Bi-rich phase.

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The temperature gradient direction in the solder matrix on the cathode side and anode side is different, but the influence of temperature gradient direction on the microstructure evolution cannot be captured through the simulated results shown in Fig. 3. To unravel the mechanism, the distributions of the electromigration driving force and thermomigration driving force along the applied electric current direction are calculated, as can be seen in Fig. 8. In order to reflect the correlation between the driving force and the phase distribution in the solder matrix, the driving force for the solder interconnects with \(r_{c}\)= 6.25 μm is investigated. It is clear that both the electromigration driving force and the thermomigration driving force are unevenly distributed, but the direction of the thermomigration driving force near the cathode side is opposite to that near the anode side. Therefore, in the anode region, the two driving forces are in the same direction, while the thermomigration driving force direction is opposite to that of the electromigration driving force in the cathode region.

Fig. 8
figure 8

Distribution analysis of the electromigration driving force and thermomigration driving force along the applied electric current direction through the coordinate origin for the Cu-cored Sn-58Bi solder interconnects at t = 30 h.

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Although the thermomigration driving force has an additive or counteracting effect on the electromigration driving force, the magnitude of the thermomigration driving force is about three orders of magnitude smaller than the electromigration driving force, so that the thermomigration driving force only has a minimal impact on the phase electromigration and segregation behavior of Cu-cored Sn-58Bi solder interconnects.

Conclusion

In this work, a quantitative phase field model coupling electromigration and thermomigration driving forces is developed to study the electromigration and phase segregation behavior of Cu-cored Sn-58Bi solder interconnects under electric current stressing. The simulated microstructure characteristics are highly consistent with the previous experimental observations, and the Bi-rich phase and Sn-rich phase are alternately distributed in the solder matrix. With the increase of time, the Bi-rich and Sn-rich phases segregate and accumulate on the anode and the cathode sides, respectively. The presence of Cu core induces severe phase segregation near the Cu core periphery, but it weakens the current crowding effect in the solder matrix. The migration and segregation of the Bi-rich phase to the anode side are hindered, and the Bi-rich phase migration speed decreases with increasing Cu core size. Moreover, the electron flow or heat flow mainly passes through the Sn-rich phase, and the current density and temperature gradient are not homogeneously distributed in the solder. The voltage across the solder interconnects increases due to the phase segregation and accumulation on the cathode and anode sides. Furthermore, the temperature gradient in the Sn-rich phase and Cu core is smaller than that in the Bi-rich phase. The ratio of electromigration driving force to thermomigration driving force can be up to 1000; thus the atomic thermomigration could have minor effects on the microstructure evolution of Cu-cored Sn-58Bi solder interconnects under electric current stressing.