Thermal Stress in Assemblies with Identical Adherends

‘‘The practical value of mathematics is, in effect, a possibility to obtain, with its help, results simpler and faster."

—Andrey N. Kolmogorov, Russian mathematician

Introduction

Stresses in adhesively bonded or soldered joints were addressed in numerous publications, including the field of electronics and photonics (see, for example, [l—32]). There is an obvious incentive to employ thermally matched materials in electronic and photonic assemblies. This is true even from the thermally induced stresses point of view, but assemblies with identical silicon adherends are attractive, first of all, from the standpoint of their electrical performance and compact size. In this chapter, we address, as suitable and useful examples and using analytical modeling, several practically important thermal stress problems associated with the employment of assemblies with identical adherends.

Bell Labs Si-on-Si multi-chip flip-chip Packaging Technology

The Bell Labs Si-on-Si multi-chip flip-chip packaging technology [32-35], developed about 30 years ago, is a good example of such a design (Figures 3.1 and 3.2). The design could be viewed as a “giant chip” on a miniaturized board. The technology focused on low cost, high performance, compact size, and high reliability. The bottleneck of the design is reliability and, first of all, the performance of the solder Si-to-Si interconnects. Extensive testing was carried out during the development of the pioneering Bell Labs Si-on-Si design, with an objective to assess the lifetime of solder joint interconnections. A solder bump was modeled, when analytical modeling was carried out, as a short elastic cylinder clamed at its flat ends (Figure 3.3). The results are shown in Figures 3.4 and 3.5. The calculated data indicated that the highest stresses and strains occur in the axial direction, while the loadings are applied in horizontal planes. Figure 3.5 indicates also that there is an incentive for employing solder joints with elevated ratios of their standoffs (heights) to diameter. It is noteworthy that many years later, this finding was implemented in the column-grid array technology as an attractive substitute for

Bell Labs Si-on-Si technology

FIGURE 3.1 Bell Labs Si-on-Si technology: multichip flip-chip package design.

Solder bump in Si-on-Si technology

FIGURE 3.2 Solder bump in Si-on-Si technology: during temperature cycling tests, it is the mismatch between the solder and Si materials that determines solder joint fatigue strength.

Si-on-Si technology

FIGURE 3.3 Si-on-Si technology: solder bump modeled as a short cylinder clamped at the ends and subjected to shearing radial tensile forces applied to its flat ends. (From E. Suhir, “Axisymmetric Elastic Deformations of a Finite Circular Cylinder with Application to Low Temperature Strains and Stresses in Solder Joints,” ASME Journal of Applied Mechanics, vol. 56. No. 2, 1989.)

Calculated stresses and strains in a short cylinder whose plane ends are subjected to radial tension caused by the thermal contraction mismatch of the solder material with Si

FIGURE 3.4 Calculated stresses and strains in a short cylinder whose plane ends are subjected to radial tension caused by the thermal contraction mismatch of the solder material with Si. (From E. Suhir, “Axisymmetric Elastic Deformations of a Finite Circular Cylinder with Application to Low Temperature Strains and Stresses in Solder Joints,” ASME Journal of Applied Mechanics, vol. 56, No. 2, 1989.)

Maximum stresses and strains in the short cylinder shown in Figur

FIGURE 3.5 Maximum stresses and strains in the short cylinder shown in Figure 3.3. (From E. Suhir, “Axisymmetric Elastic Deformations of a Finite Circular Cylinder with Application to Low Temperature Strains and Stresses in Solder Joints,” ASME Journal of Applied Mechanics, vol. 56, No. 2, 1989.)

ball-grid-array technology. The analytical data was confirmed by finite element analysis (FEA). Failure-oriented testing was conducted, and its results are showm in Figure 3.6. Such testing was carried out until half of the tested population failed. The experimental bathtub curve is shown in Figure 3.7. It is noteworthy that its wear-out portion occupies about half of the material’s lifetime. This circumstance

RE 3.6 FEA-predicted configuration of the solder bump at low temperature conditions

FIGU RE 3.6 FEA-predicted configuration of the solder bump at low temperature conditions.

Percentage of failed joints versus number of cycles

FIGURE 3.7 Percentage of failed joints versus number of cycles. (From E. Suhir, “Axisymmetric Elastic Deformations of a Finite Circular Cylinder with Application to Low Temperature Strains and Stresses in Solder Joints,” ASME Journal of Applied Mechanics, vol. 56, No. 2. 1989.)

should be considered when designing solder joint interconnections; their lifetime should not be limited just by their steady-state portion.

Simplest Elongated Assembly with Identical Adherends

Such an assembly can be viewed as a special case of the assembly addressed in Chapter 2. Let us consider an assembly comprised of two identical elongated rectangular adherends and a low-modulus adhesive or solder bond. Based on equation (2.5), we conclude that the effective coefficient of thermal expansion (CTE) ae of the assembly is not different of the CTE a, of the bonded components (components #1) of the assembly, w'hose effective Young’s modulus and thickness are substantially greater than Young’s modulus and thickness of the bonding material. Then, using formula (2.6), we conclude that the force (per unit assembly width) acting in the cross sections of the adherend materials in the midportion of the assembly is practically zero, and the force acting in the cross sections of the bonding layer (zero component) in its midportion is

Here, E0 is Young’s modulus of the bonding material, v0 is its Poisson’s ratio, h0 is its thickness, is its CTE, a, is the CTE of the bonded components material, and At is the change in temperature from the manufacturing to the low (testing, operation) temperature. The corresponding normal stress is

The axial compliance X of the assembly is due to the adherends only and, for two identical adherends, is expressed, using formula (2.9), as

The interfacial compliance of the assembly can be determined, using equation (2.16), as follows:

Then, the interfacial shearing stress factor is, in accordance with formula (2.15), The maximum interfacial shearing stress can be found as

Therefore, the assembly of interest is an elongated one and is characterized by the following parameters:

Component

CTE, a, 1/°C

Young's Modulus, E, kg/mm2.

Poisson's Ratio, v

Thickness, h, mm

0

7.4E-6

12.500

0.38

0.25

#1

2.4E-6

36,000

0.33

2.00

Temperature change is A? = 275°C. Then we have:

Assembly with Identical Adherends Subjected to Different Temperatures: Thermal Stresses in a Multileg Thermoelectric Module Design

Motivation

There is an obvious incentive to determine what could possibly be done to reduce the thermal stresses in a thermoelectric module (TEM)/power-generator design. These designs (Figures 3.8-3.10) can be modeled as assemblies comprised of two identical components subjected to different temperatures. The bonding system is a plurality of identical column-like supports (legs) spaced at equal distances from each other. It has been shown [l l—13] that compliant bonds could bring down the thermally induced interfacial stresses in them, so could thinner (dimension in the horizontal direction) and longer (dimension in the vertical direction) ТЕМ legs result in a appreciable stress relief, and, perhaps, since the device’s size is always important, such a relief could be achieved even if shorter legs are employed, as long as they are thin and the spacing between them is significant? It is imperative, of course, that if thin ТЕМ legs are employed for lower stresses, there is still enough interfacial real estate, so that the adhesive strength of the ТЕМ assembly is not compromised. On the other hand, owing to a lower stress level in an assembly with thin legs, assurance of its interfacial strength might be less of a challenge than for a conventional assembly with stiff, thick, and closely positioned legs.

Calculated failure rate during accelerated testing of solder joint interconnections

FIGURE 3.8 Calculated failure rate during accelerated testing of solder joint interconnections.

Experimental bathtub curve and probability of nonfailure versus failure rate and the number of cycles

FIGURE 3.9 Experimental bathtub curve and probability of nonfailure versus failure rate and the number of cycles. (From E. Suhir, “Axisymmetric Elastic Deformations of a Finite Circular Cylinder with Application to Low Temperature Strains and Stresses in Solder Joints,” ASME Journal of Applied Mechanics, vol. 56, No. 2, 1989.)

Thermoelectric module (TEM)/power generator. (From K. Yazawa, G

FIGURE 3.10 Thermoelectric module (TEM)/power generator. (From K. Yazawa, G.

Solbrekken, A. Bar-Cohen, Thermoelectric-Powered Convective Cooling of Microprocessors,

IEEE Transactions on Advanced Packaging, vol. 28, No. 2, 2005.)

Background

The thermoelectric cooling technology uses the Peltier effect to create heat flux between two assembly components. A cooler based on the Peltier effect (Peltier device or a solid-state refrigerator or a thermoelectric cooler or module) transfers heat from the “hot plate” to the “cold plate,” thereby consuming energy.

Although the device can be used either for heating or for cooling (refrigeration), the main application in practice is cooling, and the major advantages of a Peltier cooler (compared to, say, a vapor-compression refrigerator) are its lack of moving parts or circulating liquid, as well as its small size and flexible shape. But could it also have high power efficiency, low cost, and high reliability? During the last decade, TEMs have received increased attention of the research and engineering communities.

Various aspects of the ТЕМ technology has been addressed by a number of investigators (see, for example, [36-50]). Although the majority of the research work has been naturally focused on the thermoelectric properties of the ТЕМ materials and modules (See Beck coefficient, electrical resistivity, thermal conductivity) and their functional (thermoelectric) performance, some investigators have pointed out the importance of the ТЕМ short- and long-term mechanical (physical) robustness and addressed, using primarily FEA, the mechanical behavior of the ТЕМ materials and structures. An analytical approach has been used first for the evaluation of thermal stresses in a simplified two-leg ТЕМ design [50]. The emphasis was on the assessment of the effect of the size of the bonded regions on the maximum stress. It has been shown that the employment of thinner and longer legs could indeed result in a substantial stress relief, thereby leading to a more mechanically robust ТЕМ. In the analysis that follows, the taken approach is generalized for the case of а ТЕМ with any number of legs. The analysis is carried out considering that flexible (long and thin) legs provide a certain level of the mechanical compliance between the ceramic components, but do not experience axial thermal loading.

 
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