1 Introduction

Cracking of outermost SnPb solder joints is the major failure pattern of the flip-chip device in its early stage of development (Puttlitz Karl and Quinones 1997). The widely accepted reason causing the cracking of SnPb solder joints is that the accumulated shear strain or stress exceeds the fracture strength of SnPb. To decrease the shear strain or stress in the outermost SnPb solder joints and increase the thermal cycling numbers of the flip-chip device, the underfill is filled between the silicon chip and the FR-4 substrate (Rzepka et al. 1998; Fan et al. 2001; Hirohata et al. 2010). Experimental results confirmed that the flip-chip device filled with the underfill has a larger thermal cycling number, because the meshy underfill prevents a larger shear strain or stress from accumulating in the outermost SnPb solder joints. In addition to reducing the shear strain, the filled underfill also provides a stronger coupling between the silicon chip and the FR-4 substrate, this enhanced coupling results in the bending of the flip-chip device (Hirohata et al. 2010). In the flip-chip device filled with the underfill, it is well accepted that the SnPb solder joints have the function of electrical connection, mechanical support and heat transfer, the underfill provides the extra mechanical support, and restricts the increment of the shear strain in the outmost SnPb solder joints.

Although the filled underfill lengthened the thermal cycling number of the flip-chip device, the cracking of SnPb solder joints did not vanish, besides, the edge delamination of the flip-chip device usually emerged in its free edge regions due to the bending deformation of the flip-chip device, and the cracks punching through the silicon chip also appeared due to the enlarged tensile stress (Fan et al. 2001). All these typical failure phenomena could be explained by the elastic fracture mechanics theory. That is, the peeling stress appearing in the free edge of the flip-chip device may cause its edge delamination (the opening mode crack between neighboring layers); the local shear stress near the free edges of the flip-chip device may cause its local delaminaton (the sliding mode crack between neighboring layers); and the enlarged tensile stress in the silicon chip causes the cracks punching through the silicon chip (the opening mode crack in the silicon chip). The former two local delamination patterns are only determined by the interfacial stresses distribution, and the last failure phenomenon stems from the concentrated tensile stress. All these three stress components mentioned above can be calculated with the thermal stresses calculating theory proposed by Hsueh et al. (2002, 2006a, b), which is the simplest method applicable to calculating the thermal stresses in the uniform elastic multilayer system, and depends only on three unknowns tb, r, and c, which indicates the location of the bending axis, the radius of curvature of the system, and the uniform strain component, respectively. These three unknowns are sequentially determined by the following three boundary conditions. That is, the resultant force due to the uniform strain component is zero, the resultant force due to the bending strain component is zero, and the sum of the bending moment with respect to the bending axis is zero.

InSb IRFPAs are firstly fabricated by the flip-chip technology, the underfill is then filled between the InSb chip and the silicon readout integrated circuits (ROIC) to improve the reliability of indium bumps. In the structural model of InSb IRFPAs, the indium bumps possess the viscoplastic feature, the underfill possesses the viscoelastic feature, and the interlayer consisting of the indium bumps and the meshy underfill is not uniform. So the stresses calculation theory proposed by Hsueh C. H. can not be employed to calculate the stress components in InSb IRFPAs (Hu et al. 2014, 2016). If the discrete indium bumps are ignored and substituted with the underfill, the interlayer will reduce to a homogeneous underfill layer. If the curing process of the underfill is furthermore ignored, the thermal stress components in the simplified InSb IRFPAs structure can be calculated with the stresses calculation theory proposed by Hsueh C. H..

In the simplified InSb IRFPAs structure described above, the simulated interfacial stresses distribution was uniform in its center region, and the tensile stresses reduced from a larger value in its central region to zero at its free edge, both the shear stress and the peeling stress increased from zero in its central region to larger values close to its peripheral region (Meng et al. 2017a, b). Though these published results have confirmed that the local delamination of InSb IRFPAs is most likely to appear in its negative electrode region, just as the region circled with a yellow rectangle loop shown in Fig. 1, where the local shear stress is largest (Meng et al. 2017a). At the same time, these published results also implied that the larger tensile stress in InSb chip may trigger the cracks punching through InSb chip, just as demonstrated in Fig. 1, where several cracks appear and are distributed parallel to each other. Yet, the following typical failure phenomena appearing in the mass production of InSb IRFPAs can not be clearly explained by those simulated interfacial stress distribution (Meng et al. 2017a, b), such as the cracks in InSb chip are non-uniformly generated, and the widths of the cracks also non-identically presented in the center part of InSb IRFPAs, just as the regions circled with red elliptical loops demonstrated in Fig. 1. So we think that the approach of replacing the discrete indium bumps with the underfill is incompletely correct, and the discrete indium bumps may play a crucial role in triggering off the local failure of InSb IRFPAs.

Fig. 1
figure 1

Typical local failure patterns, the region circled with a yellow rectangle loop locates in the region where the negative electrodes of InSb IRFPAs exists, the regions circled with red elliptical loops stand on indium bumps

Full size image

2 Modeling creation and stress simulation

In the modeling creation of InSb IRFPAs, the InSb chip and the silicon ROIC are both meshed with solid 95 element due to their elastic characteristics, the underfill is a viscoelastic material presenting a certain degree of volume shrinkage during its curing process, the cured underfill can also be meshed with solid 95. The indium is a viscoplastic material and meshed with visco107 element. Its irreversible, temperature and rate dependent inelastic characteristics are described by an Anand’s viscoplastic model (Chang and Patrick 2009), in which nine model parameters are needed and listed in Table 1, where A is the pre-exponential factor, Q is the activation energy, R is the universal gas constant, ξ is the multiplier of stress, m is the strain rate sensitivity, ŝ is the coefficient for deformation resistance saturation value, n is the strain-rate sensitivity for the saturation value of deformation resistance, h0 is the hardening/softening constant, α is the strain rate sensitivity of hardening or softening, and s0 is the initial coefficient for deformation resistance saturation value (Chang and Patrick 2009). For easy focusing on the function of indium bumps in InSb IRFPAs, the negative electrode of InSb IRFPAs and the shallow isolation grooves sandwiched between neighboring mesa junction diodes are both omitted. The employed material parameters, typical sizes, element types are all listed in Table 2, where E is the Young’s modulus, ν is the Poisson’s ratio.

Table 1 Modeling parameters of indium described with Anand’s viscoplastic model
Full size table
Table 2 Material parameters, sizes and elements of InSb IRFPAs at different temperatures
Full size table

If the mesa structure of InSb diode array, the reticulate underfill structure being complementary to the mesa structure of InSb diode array, the indium bump array, and the negative electrode blind holes are all taken into consideration in larger format InSb IRFPAs, an enormous number of elements are generated in its meshing process, such a large number of elements will cost a longer solving time and will need an expensive computing platform. To assess the function of indium bumps in InSb IRFPAs by means of our Dell workstation, Dual CPUs, 72 G memory and 2 T disk, we employed the equivalent structural model of InSb IRFPAs to calculate its stress components. In our equivalent model (Meng et al. 2012; Zhang et al. 2014), the larger 128 × 128 photodiode array is replaced with the smaller 32 × 32 photodiode array, in which the coefficient of thermal expansions (CTE) of the employed materials are enlarged four times in the in-plane of the photodiode array, just as plotted in Fig. 2, and keep unchangeable in the out-of-plane of the photodiode array.

Fig. 2
figure 2

Employed CTEs of InSb, indium, underfill, and silicon ROIC in InSb IRFPAs structural model at different temperatures, these in-plane CTEs are enlarged four times

Full size image

The α in Table 2 is the coefficient of thermal expansion (CTE) of the underfill, can be described by the Eq. 1

$$\alpha = 22.46 \times 10^{ - 6} + 5.04 \times 10^{ - 8} \times ({\text{T }}-273)$$
(1)

where T is in Kelvin (He et al. 2000).

In our created model, only one-eighth structure of InSb IRFPAs is created to reduce the number of elements, just as displayed in Fig. 3. Here, the surface symmetric loadings (including the XZ symmetric surface and the diagonal symmetric surface of InSb IRFPAs) are applied, and the center of silicon ROIC bottom surface (labeled as point O) is fixed to avoid the movement of InSb IRFPAs. Besides, the transient analysis of InSb IRFPAs is performed to obtain the interfacial stresses distribution in its cooling process.

Fig. 3
figure 3

Created equivalent structural model of InSb IRFPAs, surface symmetric loadings are applied, and the center of silicon ROIC bottom surface is fixed

Full size image

When InSb IRFPAs are cooled from room temperature to liquid nitrogen temperature, the thermal stresses are generated because of the existed thermal mismatch. According to the relationship between the crack extension and the driving force, we extract the normal tensile stress distribution, which may cause the facture of InSb chip, the interfacial shear stress distribution, which has been confirmed to cause the local delamiantion of InSb IRFPAs in its negative electrode region, and the peeling stress distribution, which may cause the edge delamination of InSb IRFPAs, just as plotted in Figs. 4, 5 and 6, respectively.

Fig. 4
figure 4

Extracted normal stress distribution in different interfaces of InSb IRFPAs along X-axis, a InSb IRFPAs without indium bumps, here, the indium bumps are replaced with the underfill in its meshing process, b InSb IRFPAs with indium bumps, here, the X-axis locates in the XZ plane, and no indium bumps appear in the X-axis, just as the path 1 shown in Fig. 7, the X-axis of (c) and (d) is along the diagonal of InSb IRFPAs with indium bumps, just as the path 2 shown in Fig. 7, where indium bumps alternate with the underfill

Full size image
Fig. 5
figure 5

Extracted shear stress distribution in different interfaces of InSb IRFPAs along X-axis, a InSb IRFPAs without indium bumps, b InSb IRFPAs with indium bumps, the X-axis is along the path 1 in which no indium bumps appear, c and d InSb IRFPAs with indium bumps, here the X-axis is along the path 2

Full size image
Fig. 6
figure 6

Extracted peeling stress distribution in different interfaces of InSb IRFPAs along X-axis, a InSb IRFPAs without indium bumps, b InSb IRFPAs with indium bumps, here the X-axis is along the path 1, c and d InSb IRFPAs with indium bumps, here the X-axis is along the path 2

Full size image

For assessing the function of indium bumps in InSb IRPFAs, the indium bumps are firstly ignored, and the simulated results serve as the reference of stress components, just as demonstrated in Figs. 4a, 5a and 6a, respectively (Meng et al. 2017b). Once the indium bumps are taken into account, two typical paths are selected to uncover the operating distance of indium bumps. The path 1 is in the XZ plane, passes right through the middle of two neighboring rows of indium bumps. The path 2 is along the diagonal of InSb IRFPAs, where the indium bumps alternate with the underfill, just as demonstrated in Fig. 7. In Fig. 7, the InSb chip is discarded, the intermediate layer of InSb IRFPAs is directly demonstrated.

Fig. 7
figure 7

Two selected paths in which the interfacial stress distribution is extracted

Full size image

3 Discussion of the role of indium bump in InSb IRFPAs

In the simplified model of InSb IRFPAs, where the indium bumps are replaced with the underfill in its meshing process, the thermal stresses generated in this uniform three-layer system are uniformly distributed in its center region, just as shown in Figs. 4a, 5a, and 6a, respectively. When the indium bumps are taken into consideration, this uniform distribution character of the thermal stresses disappear. For the path 1, which passes right through the middle of neighboring rows of indium bumps, the accumulated tensile stress reduces from 793 to 652 MPa in the top surface of the underfill, and also reduces from 795 to 608 MPa in its bottom surface. The compressive stress in silicon ROIC increases from 311 to 379 MPa. The tensile stress in InSb chip reduce from 1810 to 951 MPa, a local smallest value obtained at those sites farthest to the indium bumps in path 1, and increase from 1810 to 1853 MPa, a local largest value obtained at these sites nearest to the indium bumps in path 1. It is worth pointing out that these stress values are calculated by averaging all the simulated stress values in path 1 with the coordinate ranging from 0 to 0.4 mm. For path 2, where the indium bumps alternate with the underfill, the normal stresses along X-axis fluctuate acutely and distribute periodically. The tensile stress keeps at a lower value of 854 MPa in the InSb chip glued with the underfill, and maintains at a larger value of 4380 MPa in the InSb chip glued with indium bumps. The tensile stress in the center of indium bumps keeps at a lower value of 1704 MPa, and maintains at a larger value of 2339 MPa in its peripheral region, its stress distribution curve is just like the letter U. The silicon ROIC glued with the underfill is in the compressive stress state with a value of − 533 MPa, glued with indium bumps center region is under the tensile stress of 485 MPa, and sharply attenuates from the tensile stress state to the compressive stress state with the coordinate position shifting from the center of indium bump to its free edges. The whole underfill is in the state of tensile stress, which fluctuates between 283 and 736 MPa. In the peripheral region of InSb IRFPAs, where no indium bumps exist, the normal stress distribution along X-axis direction is almost unchangeable, because the free edge effect is dominant in this region.

According to the distribution characteristics of the stresses described above, we think that the indium bumps have a significant effect on the normal stress component along X-axis in the central region of InSb IRFPAs. To be specific, for the InSb chip and the silicon ROIC connected to the two ends of indium bumps, respectively, the simulated tensile stress in InSb chip increases from 1810 MPa (obtained from InSb IRFPAs omitting indium bumps) to 1853 MPa (obtained at the coordinate positions nearest to indium bumps in path 1), further to 4380 MPa (obtained at the sites where indium bumps locate in path 2), the simulated normal stress in the silicon ROIC reduces from − 311 MPa (omitting indium bumps) to − 379 MPa (obtained at the coordinate positions nearest to indium bumps in path 1), after that, increases from the compressive stress state to the tensile stress state with the value of 485 MPa (obtained at the sites where indium bumps locate in path 2). This sharply increased tensile stress in the InSb chip connected with the indium bumps may originate from the larger shrinkage of the indium bumps. The larger shrinkage of the indium bumps along the normal direction of InSb IRFPAs pulls the glued InSb chip to shift downward, at the same time, this downward movement tendency is suppressed by the surrounding underfill, whose shrinkage is smaller than that of the indium bumps. Thus the shrinkage difference between the indium bumps and the underfill lengthens the InSb chip along the normal direction of InSb IRFPAs, furthermore sharply enhances the tensile stress in the InSb chip glued with the indium bumps, because the strain compatibility conditions must be satisfied in the neighboring materials. Due to the same reason, the silicon ROIC glued with the indium bumps is also pulled upward by the shrinkage of the indium bumps, thus the normal stress along X-axis in the silicon ROIC changes from the compressive stress state to the tensile stress state. Besides, when the indium bumps are taken into account, the normal stress distribution in the central region of InSb IRFPAs presents noteworthy cyclical fluctuation, and the fluctuation magnitude in path 1 is smaller than that in path 2. These cyclical distribution characteristics proved that the operating distance of the indium bumps is local and not larger than 20.7 μm, this operating distance is calculated from the two adjacent coordinate points at which the tensile stress of the InSb chip equals to 1810 MPa, a simulated value obtained from the InSb IRFPAs omitting the indium bumps. It is necessary to point out that the operating distance of the indium bump (20.7 μm) is shorter than the diameter of the indium bump (24 μm), this result originates from the strain compatibility conditions in the interface between the indium bumps and the surrounding underfill.

The simulated tensile stress in the InSb chip glued with the underfill decreases from 1810 MPa (obtained from InSb IRFPAs omitting indium bumps) to 951 MPa (obtained at the coordinate positions nearest to indium bumps in path 1), further to 854 MPa (obtained at the sites where indium bumps locate in path 2), this reduced tensile stress in the InSb chip also originates the smaller shrinkage of the meshy underfill enclosing the discrete indium bumps. In liquid nitrogen shocking tests of InSb IRFPAs, the in-plane shrinkage of InSb chip is determined by the interaction among the multilayer system, thus, when the lengthened InSb chip in the InSb IRFPAs neglecting the indium bumps is further lengthened by the glued indium bumps, the stretching magnitude of the InSb chip glued with the underfill will reduce, this reduced stretching amplitude further reduces the tensile stress in the InSb chip glued with the underfill. The simulated normal stress in the silicon ROIC glued with the underfill reduces from − 311 MPa (ignoring indium bumps) to − 379 MPa (in path 1), further to − 533 MPa (in path 2). This increased compressive stress originates from the local bending effect of the silicon ROIC, because the silicon ROIC glued with the indium bumps is stretched along the normal direction of InSb IRPFAs, at the same time, the encircled silicon ROIC glued with the underfill is further compressed to satisfy the strain compatibility conditions.

As for the indium bumps, their shrinkage are restrained by the surrounding underfill, the InSb chip and the silicon ROIC connected with its two end surfaces, thus the tensile stresses are generated in the indium bumps, the tensile stress almost keeps at 2330 MPa in its top surface glued with the InSb chip, and almost keeps at 1670 MPa on its bottom surface glued with the silicon ROIC.

Since the stress components can not be directly measured, the surface profile is employed to describe the total surface deformation of InSb IRFPAs. If the deformation amplitude is normalized with the thickness of InSb IRFPAs, we can obtain the Z-components of strain. So we supply the Z-components of strain of InSb IRFPAs in Fig. 8a, c. Apparently, the whole InSb IRFPAs bends upward for the larger shrinkage of the InSb chip. The InSb chip glued with the indium bumps is concave downward, and the InSb chip glued with the underfill is convex upward, this deformation distribution of alternating rises and falls is more evident along the diagonal of InSb IRFPAs. In fact, we notice that the identical surface profile was also published in the InAs/GaSb T2SL dual-band IRFPAs (Plis et al. 2013), just as shown in Fig. 8b. In the InAs/GaSb T2SL dual-band IRFPAs (320 × 256, 30 μm pitch), the pixel area is 24 μm × 24 μm, the diameter of the indium bump is about 10 μm (Huang et al. 2010), thus the surrounding underfill occupies about 89% of InSb IRFPAs surface area. According to the relative occupying areas of the indium bumps and the underfill, we think that the blue region in Fig. 8b is connected with the indium bumps, and the surrounding region painted with red and yellow is connected with the underfill. Comparing the surface profiles of the InAs/GaSb T2SL IRFPAs measured at 77 K with the simulated deformation distribution of InSb IRFPAs obtained at 77 K, we think that our simulation deformation result is correct, because the identical structure of IRFPAs is adopted. Similarly, we also think the extracted stress components of InSb IRFPAs are correct and can be employed to analyze the failure mechanism of InSb IRFPAs.

Fig. 8
figure 8

a simulated Z-components of strain distribution in InSb IRFPAs, b measured surface profile of 320 × 256 InAs/GaSb T2SL dual-band FPAs (Plis et al. 2013), c selected local strain distribution of InSb IRFPAs

Full size image

When the indium bumps are ignored in InSb IRFPAs structural model, the simulated tensile stress in the InSb chip is uniform, this uniform distribution feature of the tensile stress means that the cracks punching through InSb chip may uniformly occur in its center region, but we never observed the uniformly distributed cracks in InSb chip in our mass productions of InSb IRFPAs. On the contrary, we only observed that the local fracture of InSb chip always originates from some certain sites, just as shown in Fig. 1. So we ascertain that the approach of ignoring the indium bumps in InSb IRFPAs structural model is imperfect, can not be employed to interpret the non-uniform distribution of the cracks punching through InSb chip.

Once the indium bumps are included in the InSb IRFPAs structural model, the simulated tensile stress in the InSb chip is non-uniform, is locally concentrated and enlarged. This non-uniform distribution of the local maximum tensile stress in the InSb chip indicates that the fracture of InSb chip most likely emerges at those sites where the local maximal tensile stress exceeds the tensile strength of InSb chip, just as shown in Fig. 4c. Thus, the non-uniform distribution of the cracks punching through InSb chip shown in Fig. 1 can be explained by the non-uniform distribution of the local maximal tensile stress in InSb chip shown in Fig. 4c. Up to now, we believe that the indium bumps should not be ignored in the structural modeling process of InSb IRFPAs.

In addition to the tensile stress in the InSb chip, the interfacial shear stress and the interfacial peeling stress in the interface between the InSb chip and the underlying intermediate layer are extracted and plotted in Figs. 5 and 6, respectively. Obviously, the interfacial shear stress distribution and the interfacial peeling stress distribution are unchangeable in the peripheral region of InSb IRFPAs, because no indium bumps appear in this region. In the central region of InSb IRFPAs, the periodical rising and falling of the interfacial shear stress and the peeling stress reappear, especially in path 2. The interfacial shear stresses and the interfacial peeling stresses between the InSb chip and the indium bumps are largest. Once these two interfacial stresses exceed the gluing strength between InSb chip and indium bumps, the local delamination will emerge in their interfaces, just as demonstrated in Fig. 1, where several local delamination phenomena appear and are circled with red elliptical loops. In these local delamination regions, the local delamination gap width is widest in the interface between the InSb chip and the indium bumps, and gradually disappears in the interface between the InSb chip and the underfill. Judging from the distribution of the crack width, we think that the InSb chip does not separate from the underfill because of the smaller shear stress or peeling stress exists in this interface. The parallel longer cracks are mainly along the cleavage planes of the InSb chip. In addition to the local delamination standing on the indium bumps in the central region of InSb IRFPAs, we have also observed the local delamination appearing near the free edge of InSb IRFPAs, covers 4–5 indium arrays, and is circled with a yellow rectangular loop, just as demonstrated in Fig. 1. This local delamination derives from the interfacial shear stress appearing in the Fig. 5c, and ranging from X = 1.1 to 1.2 mm in the position along X-axis (Meng et al. 2017a). If the indium bumps are ignored, both the interfacial shear stress and the interfacial peeling stress will disappear in the center region of the InSb IRFPAs, just as shown in both Figs. 5a and 6a (Meng et al. 2017b). Under this condition, the small scope delamination existing in the interface between the indium bumps and the InSb chip will not be explained by the fracture mechanics theory. Up to now, we ascertain that the indium bumps play a pivotal role in giving rise to the local delamination of InSb IRFPAs in its central region.

4 Conclusion

Employing the created InSb IRFPAs structural model, drawing upon the stresses distribution driving the crack extension in the elastic fracture mechanism, we systematically discuss the effect from the indium bumps on the typical failure phenomena of InSb IRFPAs. We find that the discrete indium bumps greatly enhance the tensile stress in the InSb chip, which easily incurs the cracks punching through the InSb chip, also greatly enhance the local shear stress and the local peeling stress in its top surface glued with the InSb chip, which easily incur the local delamination between the InSb chip and the indium bumps. Besides, the distribution of the interfacial shear stress, which easily incurs the delamination of InSb IRFPAs in its negative electrode region, remains unchangeable. Thus, once the indium bumps are included in the InSb IRFPAs structural model, all the recorded failure phenomena, such as the cracks punching through the InSb chip, the local delamination between the InSb chip and the indium bumps, the local delamination strip of InSb IRFPAs in its negative electrode region, all can be directly explained by the driving force compelling the crack extension. Furthermore, the simulated Z-components strain of InSb IRFPAs at 77 K also was confirmed by the measured surface profile of the InAs/GaSb T2SL IRFPAs at 77 K. Up to now, we ascertain that the function of the indium bumps in aspect of causing the local delamination of InSb IRFPAs should be emphasized and supplemented, once ignoring the indium bumps in the structural modeling of InSb IRFPAs, some local failure phenomena, such as the local delamination standing on the indium bumps, the non-uniform distribution of the cracks punching through the InSb chip, can not be explained in the framework of the elastic fracture mechanism theory.