# Application of Analytical Modeling in the Design for Reliability of Electronic Packages and Systems

**DOI:**https://doi.org/10.1007/978-3-662-53605-6_370-1

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## Definitions

**Electronic** packaging is a discipline within the field of materials, electrical and mechanical engineering concerned with enclosures and protective features built into the electronic devices. It comprises a wide variety of technologies. It applies both to end products and to components.

**Electronic systems** design deals with the multidisciplinary design issues of complex **electronic** devices and **systems**, such as mobile phones and computers. The subject covers a broad spectrum, from the design and development of an **electronic system** (new product development) to assuring its proper function, service life, and disposal.

## Analytical Classical Mechanics and Applied Physics-Based Modeling, Its Role, and Significance

Mechanical (“physical”) performance of electronic and photonic materials, assemblies, packages, and systems is the bottleneck of the today’s electronic and photonic reliability engineering. Nothing happens to an electron or a photon. But, as we used to say at Bell Labs, “if you put the best chip in the world on a piece of junk called a substrate, you end up with a piece of junk.” Application of analytical modeling enables to reveal and explain the underlying physics associated with often nonobvious, always nontrivial, and sometime even paradoxical problems and situations in the design-for-reliability (DfR) problems in electronics and photonics. Modeling is the major approach of any science, including microelectronics and photonics (see, e.g., Bar-Cohen and Kraus 1988; Chen et al. 2016; Suhir et al. 2008, 2011; Suhir, 1997a, 1999b, 2000a, b, 2011a, b, c, d, 2012; Huang et al. 2015; Fan et al. 2010; Chen et al. 2017). Modeling effort could be experimental, when the testing setup is designed and built depending on what is anticipated to be observed and/or to measured (see, e.g., Bar-Cohen and Kraus 1988; Suhir 1986a, b, 1997a, 1988a, b, c, 1998, 1999a, 2000a, 2001a, b, c, d, e, 2002a, b, 2003, 2006, 2009a, b, c; Deletage et al. 2003; Christiaens et al. 2006; Zhang et al. 2006; Zhou et al. 2009, 2010; Fan and Lee 2010; Shi et al. 2010; Fan 2014; Fan et al. 2015a, b), or theoretical. Experimental models (specimens) are typically of the same physical nature and, in electronics and photonics, on the same scale and made of the same materials, as the actual objects. Theoretical models, on the other hand, use abstract notions. The goal of theoretical models is to reveal nonobvious, latent, even paradoxical, relationships and phenomena hidden in the input information. No theoretical model can provide results, which are not contained in the input data, in the taken assumptions and hypotheses. Theoretical models can be either analytical or numerical (computational). Analytical models employ mathematical methods of analysis. The today’s numerical models are typically computer-aided. Experimental models, on the other hand, can occasionally lead to new results. A famous example is the surprise discovery of radiation by Antoine Henri Becquerel in 1896 that gave birth of nuclear chemistry. But a researcher cannot rely, of course, on such an outcome.

- 1.
Experiments could be often carried out with full autonomy, i.e., without necessarily requiring theoretical support.

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Unlike theory, testing can be used as a final proof of the viability and reliability of a product and is therefore essential requirement, when it comes to making a viable electronic or a photonic device into a reliable product.

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Experiments in the high-tech field, expensive as they might be, are considerably less costly than, e.g., those in naval architecture, or in aerospace engineering, where “specimens” might cost millions of dollars.

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High-tech experimentations are much easier to design, organize, and conduct than in the macro-engineering world.

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Materials, whose properties are not completely known, are nonetheless often and successfully employed in high-tech products.

- 6.
Because every 5 years or so, new generations of electronic or photonic products are developed, there is often simply not enough time to establish all the properties and understand the behavior of materials in these fields of engineering; lack of information about the material properties is certainly an obstacle for carrying out theoretical modeling, but not for implementing the materials themselves.

- 7.
Many leading specialists in high-tech engineering (experimental physicists, materials scientists, chemists, chemical engineers) traditionally use experimental methods as their major research tool; it is not just a coincidence that 11 out of 12 Bell Labs Nobel laureates were experimentalists.

- 1.
Predictive modeling, unlike experimentation, is able to shed light on the role of each particular parameter that affects the behavior and performance of the material or the product of interest.

- 2.
Although testing can reveal insufficiently robust elements, it is usually incapable to detect superfluously reliable ones; overengineered and superfluously robust products may have excessive weight and be more costly than necessary: in mass production of expensive high-quality devices, superfluous reliability may entail substantial and unnecessary additional costs.

- 3.
Theoretical modeling can often predict the result of an experiment in less time and at considerably lower expense than the actual experiment.

- 4.
In many cases, theory serves to discourage wasting time on useless experiments; the classical example are numerous attempts to build impossible heat engines that have been prevented by a study of the theoretical laws of thermodynamics; while this is, of course, a classical and an outstanding example of the triumph of a theory, there are also numerous, though less famous, examples, when plenty of time and expense were saved because of prior theoretical modeling of a problem.

- 5.
In the majority of research and engineering projects, a preliminary theoretical analysis enables one to obtain valuable information about the phenomenon or the object and gives an experimentalist an opportunity to decide what and how should be tested or measured and in what direction success might be expected.

- 6.
By shedding light on “what affects what,” theoretical predictive modeling often serves to suggest new useful experiments. For example, the theoretical analyses of stresses in bi-material assemblies and in semiconductor thin films (see, e.g., Suhir 1986a and Luryi and Suhir 1986) triggered numerous experimental investigations aimed at the rational physical design of semiconductor crystal grown systems.

- 7.
Theory can be used to interpret empirical results, to bridge the gap between different experiments, and to extend the existing experience on new materials, components, and structures.

- 8.
One cannot do without a good theory when developing rational (optimal) designs; the idea of optimization of structures, materials, and costs has penetrated many areas of modern engineering; no progress in this direction could be achieved, of course, without application of theoretical methods. The best engineering design is the best compromise between reliability, cost-effectiveness, and time to market (to completions).

Analytical modeling (see, e.g., Huang et al. 2015; Luryi and Suhir 1986; Suhir 1989a, 1991, 1997b, c, 2005, 2009a, 2013, 2015a, b; Ou et al. 2014, 2016a, b; Placette et al. 2012; Suhir et al. 2013, 2015a, b, 2016a, b; Suhir and Shakouri 2013; Suhir and Bechou 2013; Suhir and Nicolics 2014) occupies a special place in the predictive modeling effort. Analytical modeling is able not only to come up with simple relationships that clearly indicate “what affects what” but, more importantly, can often explain the physics of phenomena and especially, as it is in this review, some nonobvious and even paradoxical situations, much better than the FEA modeling, or even experimentation, can.

Having said that, we would like to point out that while an experimental approach, unsupported by theory, is blind, theory, not supported by an experiment, is dead. An experiment forms a basis and provides input data for a theoretical model and determines its viability, accuracy, and limits of application. Limitations of a theoretical model are different in different problems and, in the majority of cases, are not known beforehand. It is the experimental modeling that is the “supreme and ultimate judge” of a theoretical model. It is noteworthy that the limitation of a particular theoretical model could be also assessed based on a more general model: limitations of a linear approach could be determined on the basis of a more general nonlinear model, limitations of a deterministic model can be determined on the basis of a probabilistic model, etc. Experiment can often be rationally included into a theoretical solution to an applied problem. Even when some relationships and structural characteristics lend themselves, in principle, to theoretical evaluation, it is sometimes simpler to determine these relationships empirically. For example, the spring constant of an elastic foundation provided by the primary coating of an optical fiber could be evaluated experimentally and then included into the analytical or a numerical predictive model.

Crucial requirements for an effective analytical model are its simplicity and clear physical meaning. A good analytical model should be based on physically meaningful considerations and produce simple and easy-to-use relationships, clearly indicating the role of the major factors affecting a phenomenon or an object of interest. One authority in applied physics remarked, perhaps only partly in jest, that the degree of understanding of a phenomenon is inversely proportional to the number of variables used for its description and that an equation longer than three inches is most likely wrong. It takes a lot of imagination, intuition, appropriate assumptions, and effort to come up with a meaningful analytical solution, while it is merely skills and training that are needed for the application of FEA simulation.

## Analytical vs. FEA Modeling

The most widespread model in the stress-strain evaluations and physical design for reliability of electronic and photonic materials and systems is finite element analysis (FEA) (see, e.g., Deshayes et al. 2003; Dandu et al. 2010; Fan and Suhir 2010; Fan and Ranouta 2012; Fan et al. 2014; Sun et al. 2016; Chen et al. 2017). This method strongly depends on the use of computers and went a long way from its initial application in the 1960s to avionic structures (by J.H. Argyris and his associates at the University of Stuttgart in Germany) to wide applications in materials science and reliability physics in a wide variety of engineering disciplines and efforts (see, e.g., Sun et al. 2016, 2017; Shen et al. 2016; Chen et al. 2016).

Since the mid-1950s, FEA modeling has become the major simulation tool for theoretical stress-strain evaluations in materials, mechanical, structural, aerospace, maritime, and other areas of engineering and applied science, including electronics and photonics materials science and engineering. This should be attributed, first of all, to the availability of powerful and flexible computer programs, which enable one to obtain, within a reasonable time, a solution to almost any stress-strain-related problem but partially also to the illusion that FEA is the ultimate and indispensable tool for solving any design or stress analysis problem. The truth of the matter is that FEA and broad application of computers has by no means made analytical solutions unnecessary or even less important, whether exact, approximate, or asymptotic. Simple and physically meaningful analytical relationships have invaluable advantages, because of the clarity and compactness of the information and clear indication on the role of various factors affecting the given phenomenon or the material’s behavior or the device performance. These advantages are especially significant when the parameter under investigation depends on more than one variable. As to the asymptotic techniques, they can be successful in many cases, when there are difficulties in the application of computational methods, e.g., in problems containing singularities. Such problems are often encountered in high-tech materials engineering, because of the wide employment of assemblies comprised of dissimilar materials. But even when application of FEA encounters no difficulties, it is always advisable to investigate the problem analytically before carrying out FEA. Such a preliminary investigation helps to reduce computer time and expense, develop the most feasible and effective preprocessing model, and, in many cases, avoid fundamental errors.

It is noteworthy also that FEA has been originally developed for structures with complicated geometry and/or with complicated boundary conditions (such as avionics or some civil engineering structures), when it might be difficult to apply analytical approaches. As a consequence, FEA has been especially widely used in those areas of engineering, in which structures of complex configuration are typical (aerospace, maritime and offshore structures, some civil engineering structures, etc.). In contrast, electronic and photonic structures are usually characterized by simple geometries and can be easily idealized as beams, flexible rods, rectangular or circular plates, composite structures of relatively simple geometry, etc. There is an obvious incentive therefore for a broad application of analytical modeling in electronics and photonics materials science and engineering. Additional incentive is due to the fact that adjacent structural elements in electronics materials engineering often have dimensions that differ by orders of magnitude. Examples are multilayer thin film structures fabricated on thick substrates or adhesively bonded assemblies, in which the bonding layer is typically significantly thinner than the bonded components of the assembly. Since the mesh elements in a FEA model must be compatible, FEA of such structures often becomes a problem of itself, especially in regions of high stress concentration. Such a problem does not occur, however, with an analytical approach.

Another consideration in favor of analytical modeling is associated, as has been mentioned above, with an illusion of simplicity in applying FEA procedures. Many users of FEA programs sincerely believe that the “black box” they deal with will automatically provide the right answer, as long as they push the right key on the keyboard. At times, a hasty, thoughtless, and incompetent application of computers can result in more harm than good by creating an impression that a solution has been obtained, when, in effect, this “solution” might be simply wrong. It is well known that although it is usually easy to obtain * a* FEA solution, especially with the today’s user-friendly software, it might be quite difficult to obtain

*right one. And how would one know that it is indeed the right solution, if there is nothing to compare it with? Clearly, if the FEA data are in good agreement with the results of an analytical modeling (which is typically based on different assumptions: FEA is a numerical continuum mechanics tool, while the available close-form analytical solutions use mostly approximate structural analysis and strength-of-materials methods), then there is a reason to believe that the obtained solution is accurate enough.*

**the**Several practically important examples that illustrate the above general statements are set forth below.

### Interfacial Thermal Stresses Concentrate at the Assembly Ends

A microelectronic package is comprised of dissimilar materials with different coefficients of thermal expansion (CTEs). This, when the package experiences temperature change, results in elevated thermal stresses and, if special measures are not taken (such as the use of metal frame), also in elevated bow. Such bow (warpage) can be an obstacle to the further fabrication process and can affect the reliability of the electronic assembly, and particularly the second level of interconnections (package to the system substrate), the most vulnerable structural element in today’s electronics technologies. The interfacial shearing and peelings stresses in the adhesively bonded or soldered assemblies employed in microelectronics concentrate at the assembly ends and, for large enough assemblies, do not increase with the further increase in the assembly size. The engineering theory of bimetal thermostats of infinite length (Timoshenko 1925) and of finite length (Suhir 1986a, 1989b) is widely used in electronics and photonics modeling effort.

Timoshenko has indicated in his classical 1926 paper that while the stresses acing in the cross sections of the thermostat strips can be predicted based on the strength-of-material (structural analysis) approach, the interfacial shearing and peeling stresses can be predicted only on the basis of the elasticity theory. Many attempts of mechanical engineers to do that led, however, to cumbersome analytical expressions. Suhir managed to generalize Timoshenko’s theory for the case of finite size assemblies, typical in electronics and photonics. His theory (Suhir 1986a, 1989b) is based on the concept of interfacial compliance. Unlike Timoshenko’s model, this theory enables to evaluate the interfacial shearing and peeling stresses using relatively simple closed form expressions and to demonstrate that the predicted maximum values of these stresses increase with an increase in the assembly size for short assemblies with compliant bonds but remain unchanged for large assemblies with stiff interfaces.

Particularly, a simple formula \( \tau (x)= kT\frac{\sinh kx}{\cosh kl} \) has been obtained for the interfacial shearing stress, assuming that the attachment is significantly thinner than the bonded components and its effective Young’s modulus is considerably lower than the moduli of the bonded materials. In the above formula, \( T=\frac{\Delta \alpha \Delta t}{\lambda } \) is the thermally induced force in the midportion of a long enough assembly; Δ*α* is the CTE difference of the assembly component materials; Δ*t* is the change in temperature (from the fabrication temperature, at which the induced thermal stress is considered zero); *λ* = *λ*_{1} + *λ*_{2} is the total axial compliance of the assembly; \( {\lambda}_1=\frac{1-{\nu}_1}{E_1{h}_1} \) and \( {\lambda}_2=\frac{1-{\nu}_2}{E_2{h}_2} \) are the axial compliances of its components; *h*_{1} and *h*_{2} are the component thicknesses; *E*_{1} and *E*_{2} are the effective Young’s moduli of the component materials; *ν*_{1} and *ν*_{2} are their Poisson’s ratios; \( k=\sqrt{\frac{\lambda }{\kappa }} \) is the parameter of the interfacial shearing stress; *κ* = *κ*_{0} + *κ*_{1} + *κ*_{3} is the total interfacial compliance of the assembly; \( {\kappa}_0=\frac{h_0}{G_0} \) is the interfacial compliance of the bonding layer; \( {\kappa}_1=\frac{h_1}{3{G}_1} \) and \( {\kappa}_2=\frac{h_2}{3{G}_2} \) are the interfacial compliances of the assembly components; \( {G}_0=\frac{E_0}{2\left(1+{\nu}_0\right)} \), \( {G}_1=\frac{E_1}{2\left(1+{\nu}_1\right)} \), and \( {G}_2=\frac{E_2}{2\left(1+{\nu}_2\right)} \) are shear moduli of the materials; and *l* is half the assembly length. The origin of the longitudinal coordinate *x* is at the mid-cross section of the assembly. The maximum shearing stress takes place at the end of the bonded assembly and is *τ*(*l*) = *τ*_{max} = *kT* tanh (*kl*). For assemblies characterized by the *kl* values exceeding, say, 2.5, this formula yields: *τ*(*l*) = *τ*_{max} = *kT*, so that the maximum shearing stress becomes assembly size independent. It has been assumed in this model that the longitudinal interfacial displacements are somewhat greater than the displacements of the inner points of the given cross section and that these displacements can be evaluated as a product of the interfacial position-independent compliance of the assembly and the interfacial shearing stress in the given cross section. This stress changes from zero at the mid-cross section of the assembly to its maximum value *τ*(*l*) = *τ*_{max} = *kT* tanh (*kl*) at the assembly end.

### Incentive for Using Low-Modulus and/or Low-Yield-Stress Bonding Materials

Since the interfacial stresses concentrate at the assembly ends and because compliant bonds act as effective strain buffers between dissimilar materials in assemblies subjected to the change in temperature, there is an obvious incentive for using low-modulus bonding materials and thick bonding layers, especially at the assembly ends, where the interfacial stresses concentrate. But how low is “low,” how thick is “thick,” and how large is the expected size of the inelastic deformations, if any, at the peripheral portions of the assembly? It has been shown (Suhir 2003) that this size, which leads to low cycle fatigue condition during temperature cycling tests, can be established from the condition that the forces at the ends of the assembly’s elastic midportion are equal to the forces applied from the inelastic peripheral portions and that the latter forces can be evaluated as products of the yield stress in shear of the bonding material and the length of the peripheral inelastic region. The calculated distribution of the interfacial shearing and peeling stresses, when a low-yield-stress material is employed as a suitable bond to attach a highly vulnerable low-expansion photonic chip to a high-expansion copper sub-mount (heat sink). The situation is similar, if it is not a bond, but a thin coating layer (Suhir 2001d) is considered, or when a low-yield-stress solder is used as the bonding material (Suhir 2006). In the latter case, the bonding material relieves the stresses at the assembly ends to the level of the yield stress. This might not be favorable for the bonding material, but might be nevertheless advisable for a vulnerable chip. It has been shown also (Suhir 2001e) that one effective way to bring down the thermally induced interfacial shearing stress at the ends of an adhesively bonded or soldered assembly is to employ a design, in which a bonding material with a high modulus is used in the midportion of the assembly, while a low-modulus bond is used at its peripheral portions. No significant stress relief can be expected, if Young’s modulus of the peripheral low-modulus bond is too low (in the extreme case of a zero modulus, this material will not play any role at all) or too high (say, equal to the Young’s modulus of the high modulus adhesive material in the midportion). The most effective compromise, as far as the maximum stress is concerned, should consider the lengths and the moduli of the bonding materials in the midportion and at the peripheral portions of the assembly. What is even less obvious is that the CTE of the bonding material plays no role, as long as the bonding layer is significantly thinner than the bonded components and/or has a considerably lower modulus. This is indeed the case in a typical adhesively bonded or soldered assembly in electronics and optics. This is not true in the case of a polymer-coated fiber with a low-modulus coating at the ends, when both the fiber and its coating are equal “players.”

The case of an assembly bonded at its ends could be regarded in a way as a sort of an opposite situation in comparison with an assembly with a low-modulus bond at its ends. In an assembly bonded at the ends, the Young’s modulus is finite, of course, at the peripheral regions, and is zero in the midportion of the assembly. It has been shown (Suhir 2002a) that if the peripheral bonding layers are long enough and comprised of high modulus materials, the induced stresses will not be different of those in assemblies with continuous bonding layers. This conclusion is particularly important when manufacturers try to necessarily bring by all means an encapsulation (underfill) material for the entire under-chip space. Such an effort makes sense, if good thermal management is important, but is not necessary from the standpoint of the level of the induced stresses.

### Transverse Grooves in Bonded Components Could Relieve Interfacial Stresses

Jorma Kivilahti and Tomi Reinikainen (Espoo, Finland) have observed, using FEA modeling, significant decrease in the magnitude and the nonuniformity of the interfacial shearing stresses in test samples with transverse grooves in the bonded components (“pins”). This paradoxical phenomenon has been explained using analytical stress modeling (Reinikainen and Suhir 2009; Suhir and Reinikainen 2008, 2009a, 2010): the detected stress relief is due to the increase in the interfacial compliance owing to the “conversion” of pin inner portions into additional layers of the bond. If the grooves are deep enough, the additional interfacial compliance of the bonding structure has a favorable effect on the magnitude and the distribution of the interfacial stresses. On the other hand, the grooves isolate the inner portions of the pins from the direct action of the external tensile forces, thereby increasing the axial compliance of the joint. While the increase in the axial compliance has a negative effect on the interfacial stresses, the carried out numerical example has indicated that the favorable effect of the increased interfacial compliance suppresses the adverse effect of the increase in the axial compliance of the pins, so that the cumulative effect of the grooves is highly favorable. This effect, as determined by the analytical model, is, however, not as strong as the FEA predicts, but is still appreciable. This is because the FEA overestimates the stresses at the edge regions: at the very ends of the assembly, even singularities (infinitely high stresses) might occur. The observed and explained effect of the reduction in the level and uniformity of the interfacial stresses in grooved joints can be effectively used in the next-generation shear-off test methodologies for solder joint interconnections: such methodologies would enable obtaining more consistent and more stable test data.

## Conclusion

Analytical modeling is a powerful tool that enables one to explain paradoxical situations in the behavior and performance of electronic materials and products and make a viable device into a reliable product.

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