*4.2. Stress Intensity Factor Analysis of Solder–IMC Interface Crack under Quasi-Static Load*

4.2.1. Relationship between Stress Intensity Factor at Interface Crack Tip and Crack Length

Figure 7 shows the von Mises equivalent stress distribution at the crack tip when the load is 10 MPa and the crack length is 10 and 20 µm. It can be seen that the equivalent stress of the crack tip increases as the crack length increases, and the high equivalent stress appears both on the solder and IMC. However, it does not mean that failure or

crack propagation will definitely occur in these locations. Under tensile load, the crack between the upper layer and the substrate initiate from the free edge of the actual specimen, especially where defects such as cracking or void brought by the bonding process existed. The initial crack first expands along the interface to a certain depth and then propagates along the interface or is deflected to the solder matrix, which depends on the energy release rate of the two propagation paths. Therefore, the energy release rate will be used to judge whether the crack is initiated and propagated, and the stress intensity factor will be used to determine the crack tip propagation path.

**Figure 7.** The Von Mises equivalent stress distribution in the crack tip with the crack length of: (**a**) 10 µm; (**b**) 20 µm.

The relationship between the stress intensity factor at the crack tip of the IMC–solder interface calculated by the interaction integral and the crack depth is shown in Figure 8. It can be seen that the stress intensity factors of *K*<sup>I</sup> and *K*II of the interfacial crack tip increase with an increase in the crack length under the same load, and the *K*II will increase quickly due to the elastic deformation of the solder, which leads to the increasing tendency of the type II cracking mode and possibly the crack deflection as well.

**Figure 8.** The relationship between the stress intensity factor of the interface crack between the IMC and the solder and crack depth.

Polynomial fitting is performed for the stress intensity factor at the crack tip with different crack lengths in the figure, and the fitting expression is as follows:

$$K = \begin{cases} 3.7 \times 10^3 a^2 + 1.7 \times 10^2 a + 0.013 & \sigma = 10 \text{MPa} \\ 18.6 \times 10^3 a^2 + 8.5 \times 10^2 a + 0.65 & \sigma = 50 \text{MPa} \\ 33.3 \times 10^3 a^2 + 15.4 \times 10^2 a + 1.2 & \sigma = 90 \text{MPa} \end{cases} \tag{1}$$

$$K = \begin{cases} 5.44 \times 10^3 a^2 - 1.27a - 0.054 & \sigma = 10 \text{MPa} \\ 27.2 \times 10^3 a^2 - 5.68a - 0.265 & \sigma = 50 \text{MPa} \\ 48.9 \times 10^3 a^2 - 9.77a - 0.48 & \sigma = 90 \text{MPa} \end{cases} \tag{2}$$

where σ is the peel stress loaded on the upper surface of the solder and a is the crack length. Comparing the stress intensity factors of *K*<sup>I</sup> and *K*II under three loads, it can be seen that *K*<sup>I</sup> and *K*II are proportional to the load, because the material model used in the simulation is a linear elastic model. Therefore, the expressions of *K*<sup>I</sup> and *K*II can be rewritten as follows:

$$\begin{array}{l} \text{K} = 3.7 \times 10^2 a^2 \sigma + 17a\sigma + 0.013\sigma\\ \text{K} = 5.44 \times 10^2 a^2 \sigma - 0.127a\sigma - 0.0054\sigma \end{array} \tag{3}$$

#### 4.2.2. Influence of Solder Thickness on Stress Intensity Factor of Interface Crack

In the existing research on solder joints of several hundred micrometers, due to the much lower elastic modulus of solder versus the rest part of a joint, the solder volume plays an important role in the mechanical properties of the microbumps. If the thickness of the solder layer is too small, the mechanical properties of the microbumps will be adversely affected. The solder thickness in a microbump-based die stacking 3D integration structure is greatly reduced compared to the flip chip interconnection, which necessitates the research on the dependence of SIF on the solder thickness quantitatively. Figure 9 compares the SIF evolution with a progressing crack under different solder thicknesses from 15 to 30 µm. It can be seen that both *K*<sup>I</sup> and *K*II increase with a decrease in solder thickness. This phenomenon is plainly explained by the stress distribution around the crack tip, as shown in Figure 10. The elastic mismatch between IMC and the solder causes stress concentration around the crack tip, which is better alleviated with a thicker solder layer, as can be judged from the more uniform distribution of stress across the cross section of analysis. Therefore, switching from the spherical solder bumps to the Cu pillar-based microbump joints is believed to pose additional failure risk under the drop impact condition.

**Figure 9.** The relationship between the stress intensity factor *K*<sup>I</sup> of the interfacial crack tip and solder thickness under different solder thicknesses.

**Figure 10.** The von Mises equivalent stress distribution in the crack tip with different solder thicknesses: (**a**) 20 µm; (**b**) 30 µm.

#### *4.3. Investigation on Crack Growth Behavior*

The analyses above have revealed the increase in the stress intensity factors *K*<sup>I</sup> and *K*II with increasing crack length. Further investigation of the crack propagation behavior, especially the propagation path, needs the quantitative analyses on the crack tip energy release rates *J*<sup>1</sup> and *J*2. Hu [15] found the propagation behavior of a semi-infinite plane crack at the interface of a two-phase material in 1989 and revealed that theoretically the crack deviated from the original main crack propagation path by a minimum length. They further deduced the relationship between the stress intensity factor after crack deflection and along the original path. The maximum energy release rate can be used to determine the crack deflection angle. The criterion of deflection of quasi-static interface crack propagation behavior is as follows:

$$\frac{G\_S}{G} > \frac{\Gamma}{\Gamma\_i} \tag{4}$$

Among them: *G<sup>s</sup>* = *J* = q *J* 2 <sup>1</sup> + *J* 2 2 , *G* = *K* 2 *<sup>E</sup>*<sup>∗</sup> , Γ is the fracture toughness of the solder, and Γ<sup>i</sup> is the fracture toughness of Ni3Sn<sup>4</sup> IMC. In this paper, the maximum fracture toughness of solder SAC305 is set to be 295 N/m, which is measured by Loo [16]. To be able to directly compare the fracture toughness values of the Ni-Sn–IMC interface from the various existing research, the fracture toughness is converted into a critical stress intensity factor. For the Ni3Sn<sup>4</sup> layer, a critical stress intensity factor of 4.22 ± 0.45 MPa m1/2 measured by Ghosh [14] was adopted, which equals 165.5 N/m; thus, we obtain Γ Γi = 1.78. It can be seen from the expression of the crack tip energy release rate that when the material is of linear elastic property, the ratio *<sup>G</sup><sup>S</sup> G* is irrelevant to load. For the convenience of calculation, the ERR is calculated with the uniaxial load of 50 MPa, and the IMC and solder thicknesses are set as 1 and 20 µm, respectively. The energy release rate at the interface crack tip under different crack lengths is calculated as follows:

For the homogeneous two-material interface:

$$J\_1 = \frac{K\overline{K}}{E^\* \cosh^2(\pi\varepsilon)}\tag{5}$$

$$J\_2 = -\frac{\text{Re}[Kr^{i\varepsilon}]\text{Im}[Kr^{i\varepsilon}]}{\pi\varepsilon\cosh^2(\pi\varepsilon)} \times \left[\frac{1-\nu\_1}{4\mu\_1}(1-e^{-2\pi\varepsilon}) + \frac{1-\nu\_2}{4\mu\_2}(e^{2\pi\varepsilon}-1)\right] \tag{6}$$

where ε is the oscillatory index

$$
\varepsilon = \frac{1}{2\pi} \ln(\frac{1-\beta}{1+\beta}) \tag{7}
$$

β is the second Dundurs' constant

$$\beta = \frac{\mu\_1(\kappa\_2 - 1) - \mu\_2(\kappa\_1 - 1)}{\mu\_1(\kappa\_2 + 1) + \mu\_2(\kappa\_1 + 1)}\tag{8}$$

and *κ* is Kolosov's constant

$$\kappa = \begin{cases} \frac{3-\nu\_p}{1+\nu\_p} \text{ plane stress} \\ 3-4\nu\_p \text{ plane strain} \end{cases} \tag{9}$$

where

$$\frac{1}{E^\*} = [\frac{1-\nu\_1}{4\mu\_1} + \frac{1-\nu\_2}{4\mu\_2}] \tag{10}$$

The results of the relevant parameters of the dual-material SAC305–IMC interface in the above formula are shown in Table 2. The trend of *<sup>G</sup><sup>S</sup> <sup>G</sup>* with crack length is calculated, as shown in Figure 11.

**Table 2.** Parameters of two-material SAC305–IMC interface.


**Figure 11.** The variation trend of *<sup>G</sup><sup>S</sup> <sup>G</sup>* with crack length.

Hu found that the interface cracks start from the free edge of the sample, propagate at one to two times the thickness of the film along the interface, and then deflect into the matrix, expanding to a depth of four to five times the thickness of the film and finally parallel to the interface. From Figure 11, it can be seen that the ratio of the crack tip ERR after deflection to that propagating along the interface increases with the increase in the main crack length. When the main crack expands to a length of about 16 µm, the ratio will be greater than the ratio of the fracture toughness of the solder matrix to the fracture toughness of the interface. At this time, the crack will deviate from the original interface path and deflect into the matrix. The deflection angle is calculated by *ω* = arctan J2 J1 , and we can find *w* = 42◦ . It can also be seen from the above figure that if the ratio of the fracture toughness of the solder matrix to the interface fracture toughness is greater than the ratio between two paths, then the crack will always expand along the interface without deflecting to the solder matrix.

Figure 12 compares the influence of solder thickness on the interfacial crack growth behavior. It can be seen from the figure that when the solder thickness decreases, the critical main crack length for crack deflection will decrease. When the solder thickness is 15, 20, 25 and 30 um, the critical crack deflection length is 16, 23, 27 and 29 um, respectively, due to the reason related in Section 4.2.2, i.e., the decrease in solder cushioning causes an increase in stress concentration in the solder matrix, thus increasing the advantage of deflected cracking path.

**Figure 12.** The influence of different solder thicknesses on interfacial crack growth behavior.

Figure 13 shows the variation of arctan |J2/J1|, or in other words, the virtual crack deflection angle, whether or not deflection actually takes place. With the main crack length under different solder thickness conditions, the angle increases rapidly at first, and then closes to a constant value. The crack deflection angle trend is consistent with the research of HH YU et al. on the interfacial cracking behavior of chromium films on silica substrates [12]. The asymptotic value of the crack deflection angle is about 42◦ .

**Figure 13.** Relationship between the crack deflection angle and the main crack length under different solder thicknesses.

#### **5. Experimental Validation and Discussion**

The cross section of the solder joint in the case of drop failure with different solder thickness is shown in Figure 14. According to the SEM analysis, when the solder thickness is 20 µm, the crack length of 8 µm deflects, and the deflection angle is 32.8◦ . When the solder thickness is 30 µm, the crack length of 28 µm deflects, and the deflection angle is 37◦ . When the solder height is 37 µm, the crack length of 32 µm deflects, and the deflection angle is 30.4◦ . The measured deflection angle of the interfacial crack is 30◦ to 40◦ , which is larger than the asymptotic value of deflection angle at the moment of deflection initiation, calculated by numerical simulation. This is owed to the microbump not only

being subjected to normal stress, but it is also subjected to a shear force parallel to the interface during the drop experiment, while the load used in the numerical calculation is only the normal stress. In practice, when the crack propagates to a certain length, the portion of type II cracking produced by the shear stress cannot be negligible. The change of initial deflection angle versus the solder thickness is in good agreement with the numerical calculation based on ERR and fracture toughness. Therefore, in general, the numerical methods adopted in this paper can be used as an effective way to predict the cracking behavior in an actual microbump joint.

**Figure 14.** Cross section of the solder joint with different solder thicknesses, listed as: (**a**) 20 µm; (**b**) 30 µm; (**c**) 37 µm.

The actual crack propagation behavior is affected by many factors, such as interfacial defects or inhomogeneity of microstructure. There is a clear competition between interfacial propagation and solder matrix propagation; for example, it was found in the test vehicles of inferior interfacial strength, e.g., the bonding was carried out at lower than optimal temperatures, and the crack would not deflect due to the increased value. In addition, the competition of the crack path in a well-bonded test vehicle is often observed as minute crack branching, as shown in Figure 15. These small-scale branched cracks often terminated within 1 µm. As the fracture progresses, the deflected path gradually gains favor.

**Figure 15.** Crack branching during the initial stage of the drop test: (**a**) The minute crack branching increase as the fracture progresses; (**b**) The first crack branching.

The explanation for the crack branching is that the grain boundary is the low strength region and alternative crack propagation path. Here, a test vehicle with Cu–SnAg–Cu microbump structure was used to enhance the interfacial reaction, and the joint was ionmilled cross-sectionally before SEM observation to exhibit grain contrast, as shown in

Figure 16. The second phase was identified as Cu6Sn<sup>5</sup> IMC. IMC particles can be seen clearly in the junction of Sn grains, which is formed by Cu atoms diffusing along the grain boundary and precipitating in the junction in the form of Cu6Sn<sup>5</sup> during the solidification process. These Cu6Sn<sup>5</sup> particles play a significant role in the arresting and deflection of cracks. As can be seen in Figure 16a,b, crack tips meet the second phase and stop propagating. A higher driving force is required to either propagate around the second phase by deflection, or to continue through the second phase, the latter being less probable from an energetic point of view. Therefore, once arrested by the boundary junction, cracks would further proceed along the boundary of the IMC particle and Sn grain, while the preferred direction of all possible ones is related to the deflection angle, finally forming fracture patterns that differ from one sample to another in shape. The Ag3Sn IMC grains were believed to not have a significant impact on the crack propagation path since they were present in the form of a primary eutectic component located inside each Sn grain [17,18]. It has been previously reported that under thermal cycling or coupled thermomechanical– electrical load, the fatigue crack preferred an intergranular propagation path [19,20], in which case, the reconstructed grain structure and recrystallization might contribute to the weakening of grain boundary strength. This inclination seems to apply well to the highly dynamic and purely mechanical drop impact scenario. We can also reasonably suspect that if the interfacial IMC grows to a certain thickness that leaves visible voids due to the volume shrinkage effect, the bonding interface will be much weakened in that the crack will only propagate along the voided interface.

**Figure 16.** Influence of second-phase particles on crack propagation path: (**a**) crack tips meet the second phase and stop propagating; (**b**) The crack propagate around the second phase by deflection; (**c**) EDX analysis diagram.

Combining the results in Figure 5, it can also be further deduced that the stage III of resistance change plays a significant role in determining the joint lifetime under drop impact, and one possible way to enhance the durability is to eliminate the grain boundaries; thus, the deflected path would cost higher energy than in a joint of the multi-grain solder layer. The research of controlling the grain number of the solder layer in a microbump joint is currently ongoing among various researchers [21,22].

#### **6. Conclusions**

In this paper, we report for the first time the cracking failure characteristics in microbump joints for chip-on-chip stacked interconnections. Experimental tests were carried out using a JEDEC standard test board to reveal the joint resistance change and the crack morphology. To elucidate the crack deflection during the joint degradation process, a local finite-element model was established to calculate the stress intensity factor at the crack tip, and the numerical results were further incorporated into a fracture mechanics model to obtain the crack deflection criteria. The main conclusions are summarized below:

(1) The main failure mode of microbump interconnections for 3D CoC packaging is that cracks were first initiated at the edge of the IMC–solder interface. After propagating along the interface for a distance, they deflected into the solder matrix, eventually penetrating the entire joint. The electrical resistance change is closely linked to the cracking progress.

(2) Stress intensity factor of a zero-thickness crack tip at the interface of the solder and IMC is calculated under quasi-static load by the method of interaction integral method. Both *K*<sup>I</sup> and *K*II increase with the increase in the crack length under the same load, and reducing the solder thickness causes higher SIF due to less alleviated mechanical mismatch.

(3) The crack propagation path is studied using a criterion based on energy release rate and fracture toughness. The calculation results show that the cracks on the interface between the solder and IMC will deflect into the solder matrix after extending to a certain depth along the interface. The deflection angle for crack initiation converges to 40◦ with the increase in crack length. The critical length of the main crack for crack deflection increases with the increase in solder thickness, which is experimentally confirmed by an actual drop test on samples with different solder heights.

(4) The crack propagation path in actual drop test samples was influenced by factors, including the actual strength of the bonding interface and the grain structure of the solder layer. Grain boundaries are the favored path for the deflected cracks.

**Author Contributions:** Conceptualization, W.Z. and Z.C.; methodology, Z.L.; software, L.S. and Z.L.; validation, M.F. and Z.L.; investigation, Z.C.; writing—original draft preparation, Z.L.; writing review and editing, Z.C., Y.G. and Z.L.; supervision, W.Z.; project administration, W.Z.; funding acquisition, W.Z. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by "National Natural Science Foundation of China, grant number U20A6004" and "Natural Science Foundation of Hunan Province, grant number 2021JJ40734", and "State Key Laboratory of High Performance Complex Manufacturing, grant number ZZYJKT2020-08", and "Natural science research project of Hunan Provincial Department of Education, grant number 19B228".

**Institutional Review Board Statement:** The study did not require ethical approval.

**Data Availability Statement:** Data sharing not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

