Thermal Stress Modeling In Microelectronics and Photonic Structures, and the Application of the Probablistic Approach: Review and Extension

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1 Thermal Stress Modeling In Microelectronics and Photonic Structures, and the Application of the Probablistic Approach: Review and Extension E. Suhir Distinguished Member of Technical Staff, Physical Sciences and Engineering Research Division, Bell Laboratories, Lucent Technologies, Inc., 600 Mountain Ave Room 1D-443 Murray Hill, New Jersey 07974, USA Phone: Fax: Abstract Elevated thermal stresses and deformations can result in a mechanical (structural), as well as in a functional (electrical or optical), failure of an electronic or photonic material, structure or package. Predictive modeling, whether analytical ( mathematical ) or numerical (typically, Finite Element), is an effective research/engineering tool for the prediction of the level and the adverse consequences of thermal stresses and deformations. Accordingly, in this review, the author discusses the major attributes of the predictive modeling of the thermal stresses in microelectronics and photonics packaging. In addition, the role that a probabilistic approach might play for understanding the effect of the variability/uncertainty in materials properties, structural geometry, and loading conditions on the thermal stress has been indicated. Based on several practical examples, one can demonstrate that, in some practically important problems of packaging engineering, when such a variability/uncertainty cannot be ignored, the application of a probabilistic approach can be very helpful in the analysis and design of a viable and reliable structure. The review is based primarily on the author s research conducted at Bell Laboratories during the last decade. Key words: Thermal Stress Modeling, Probablistic Approach, Microelectronics, and Photonics. 1. Background: Thermal Stress Failures Thermal ( internal ) loading, and the resulting stresses and deformations can be defined as those that are associated with the change in temperature, and/or as those, which depend on thermomechanical properties of the employed materials. Thermal stresses occur during fabrication, testing, storage, and operation of the microelectronic and photonic equipment. Thermally induced stresses and strains can be due to dissimilar materials that expand/contract at different rates during temperature excursions, and/or to the nonuniform distribution of temperature (temperature gradient). Elevated thermal stresses are the major contributor to the finite service life of microelectronics and photonic structures, packages, and systems. Although the most serious consequences of the elevated thermal stresses are usually associated with mechanical (structural) failures (ductile rupture, brittle fracture, failures due to fatigue, creep, thermal relaxation, thermal shock, stress corrosion, excessive deformation, among other factors), thermal stresses and strains can result also in the functional failures, such as, in the loss of the electrical or optical performance of a compo- International Microelectronics And Packaging Society 215

2 Intl. Journal of Microcircuits and Electronic Packaging nent or device. For instance, transistor junction failure can occur due to of the elevated thermal stress in the integrated circuit, if the heat produced by the chip cannot readily escape 9. Optical performance failure (such as, loss in coupling efficiency) occurs 19, when the lateral thermally induced displacement in the gap between two lightguides exceeds the allowable limit (this limit can be as low as 0.5m). In laser packages, such displacements can be due also to thermal stress relaxation in a laser weld. The ability to understand the sources of thermal stresses and displacements, predict their distribution and the maximum values, and possibly minimize them is of clear practical importance. 2. Thermal Stress Modeling Pioneering work in modeling of thermal stress in bodies comprised of dissimilar materials has been performed by Timoshenko 20, Völkersen 21, and Aleck 1. Timoshenko and Völkersen based their treatment of the problem on a structural analysis (strength-of-materials) approach, while Aleck applied theory-of-elasticity method. Both approaches were then extended and substantially modified in application to modeling of the mechanical behavior of assemblies and structures employed in various fields of engineering, including the area of microelectronics and photonics packaging. The structural analysis approach was used by Chen and Nelson 3, Chang 2, Glascock and Webster 5, Suhir 12,13,14,15, Hall et al. 7, Lau 10, and many other investigators. The theory-of-elasticity approach was used by Zeyfang 23, Eischen 4, Kuo 8, Yamada 22 and other researchers. The structural analysis (engineering) approach enables one to determine, often with sufficient accuracy, the magnitude and the distribution of the interfacial shearing and through-thickness ( peeling ) stresses, as well as the stresses acting in the cross sections of the constituent materials. This approach results in simple and easy-to-use formulas, and can be (and, actually, has been) successfully employed, as a part of a physical design process of a component or device. It can be used to select materials, establish the dimensions of the structural elements, and compare different designs from the standpoint of the stress level. As to the theory-of-elasticity method, it is based on rather general assumptions and equations of the elasticity theory and provides a rigorous mechanical treatment of the problem. This approach is advisable, when there is a need for the most accurate evaluation of the induced stresses and strains. Applied within the framework of linear elasticity, this approach leads, in the majority of cases, to a stress singularity at the assembly edge. Its application has been found particularly useful when there is an intention to further proceed with fracture analysis. The engineering and the theory-of-elasticity approaches should not be viewed, of course, as competitors, but rather as different research tools, which have their merits and shortcomings, and their areas of application. Finite Element modeling has become, since the mid-1950s, the major research tool for theoretical evaluations in mechanical characterize microelectronics and fiber optics structures. There- 216 and structural engineering, including the area of microelectronics and photonics 6. This should be attributed to the availability of powerful and flexible computer programs, which enable one to obtain, within a reasonable time, a solution to almost any stressstrain-related problem. Based on the comparison of the Finite Element and analytical evaluations, it has been found, Mishkevich and Suhir 11, that the approach previously used in application to thermal stress thin film structures, Suhir 17, can be employed to further simplify the interfacial thermal stress calculations in bi-material assemblies with a certain thickness ratios of the constituent materials. This approach suggests that the interfacial shearing stress can be evaluated from a simplified equation obtained under an assumption that the interfacial shearing stress can be evaluated without considering the effect of the peeling stress. Such an assumption is conservative, resulting in a reasonable overestimation of the maximum shearing stress, in comparison with the more accurate value, obtained from the coupled equations for the interfacial stresses 15. The simplified solution suggests that the maximum value of the shearing stress always takes place at the assembly edge, while the solution, based on the coupled equations for the interfacial stresses, indicates that the maximum shearing stress occurs at some distance from the edge. After the shearing stress is determined, the peeling stress can be found from an equation similar to the equation of bending of a beam lying on a continuous elastic foundation. In connection with the wide use of computational methods in thermal stress analyses for microelectronics and photonics packages, it should be pointed out that broad application of computers has, by no means, made analytical solutions unnecessary or even less important, whether exact, approximate, or asymptotic 14. Simple and easy-to-use analytical relationships have invaluable advantages, due to of the clarity and compactness of the obtained information and clear indication of the role of various factors affecting the given phenomenon or the behavior of the given system. These advantages are especially significant when the parameter under investigation depends on more than one variable. As to the asymptotic techniques and formulas, they can be successful in those cases in which there are difficulties in the application of computational methods, such as, in problems containing singularities. But, even when the application of numerical methods encounters no difficulties, it is always advisable to investigate the problem analytically before carrying out computeraided analyzis. Such a preliminary investigation helps to reduce computer time and expense, to develop the most feasible and effective preprocessing model and, in many cases, to avoid fundamental errors. It is noteworthy that the Finite Element method was originally developed for structures with complicated geometry and/or with complicated boundary conditions (such as, avionics structures), when it might be difficult to apply analytical approaches. As a consequence, this method is especially widely used in those areas of engineering in which structures of complex configuration are typical: aerospace, marine, and offshore structures, and some civil engineering structures. In contrast, a relatively simple geometry and simple configurations usually International Microelectronics And Packaging Society

3 fore, they can be easily and quite accurate idealized as beams, flexible rods, plates, and various composite structures of relatively simple geometry. Therefore, there is a clear incentive for a broad application of analytical stress modeling for such structures. 3. Probabilistic Approach and contain the corresponding deterministic models as first approximations. It is important to emphasize that the use of probabilistic methods and approaches is due not so much to the fact that the available information is insufficient for a deterministic analysis, but, first of all, to the fact the variability and uncertainty are inherent in the very nature of many physical phenomena, materials characteristics, engineering designs, and application conditions. Probabilistic models enable one to establish the scope and the limits of the application of deterministic theories. These models provide a solid basis for a substantiated and goal-oriented accumulation, and effective use of empirical data. Realizing and emphasizing the fact that the probability of failure of an engineering product is never zero, probabilistic methods enable one to quantitatively assess the degree of uncertainty in various factors, which determine the performance of a product, and to design on this basis a product with a low probability of failure. Probabilistic methods, underlying all the modern methods of forecasting and decision making, allow one to extend the accumulated experience on new products and new designs, which may differ from the existing ones by type, dimensions, materials, and operating conditions. Although probabilistic approaches and models are able to account for a substantially larger number of different factors than the deterministic methods, they should not be viewed as a sort of a panacea that is able to cure all the engineering troubles. These methods cannot perform miracles and have their limitations. For instance, probabilistic methods cannot be applied in situations, where the conditions of an experiment or a trial are not reproducible or when the events are very rare. Quite often a serious obstacle for applying probabilistic methods is the difficulty of obtaining the necessary input information. In such situations, the designer considers the worst case, uses a quasi- deterministic approach, or a more or less consistent combination of probabilistic and deterministic reasoning. However, when the application of probabilistic methods is possible, justified, and is supported by reliable enough input information, these methods provide a powerful, effective, and well-substantiated resource for engineering analyses and designs. Packaging/reliability engineer uses predictive modeling and, particularly, modeling of thermal stresses and other thermal phenomena, at all the stages of the analysis, design, testing, manufacturing, operation, and maintenance of a product or system. The traditional approach in predictive modeling can be referred to as deterministic. Such an approach does not pay sufficient attention to the variability of parameters and criteria used. This approach is acceptable and can be justified in many cases, when the deviations ( fluctuations ) from the mean values are small, when the design parameters are known, or can be predicted with reasonable accuracy, and when the processes and procedures that the engineer deals with are stochastically stable, that is, when small causes result in small effects. There are, however, numerous situations, in which the fluctuations from the mean values are significant and in which the variability, change and uncertainty play a vital role. In the majority of such situations, the product will most likely fail, if these uncertainties are ignored. Therefore, understanding the role and significance of the laws of chance, and the causes and effects of variability in material properties, structural dimensions, tolerances, bearing clearances, loading conditions, stresses and strength, applications, and environments, and a multitude of other design parameters, is critical for the creation and successful operation of a viable and reliable product or structure. In the majority of practical cases, the random nature and various uncertainties in the design characteristics and parameters can be described on the basis of the methods of the theory of probability. Probabilistic methods proceed from the fact that uncertainties are an inevitable and essential feature of the nature of an engineering system or design and provide ways of dealing with quantities that cannot be predicted with absolute certainty. Unlike deterministic methods, probabilistic approaches address more 4. Example: Solder Glass Attachment general and more complicated situations, in which the behavior in a Ceramic Electronic Package of the given characteristic or parameter cannot be determined with certainty in each particular experiment or in a particular situation. However, for products, which are manufactured in large As an example of probabilistic modeling of thermal stresses quantities, and for experiments, which are repeated many times and a probabilistic structural design in microelectronics, one can in identical conditions, this behavior can be described by probabilistic/statistical relationships. These relationships manifest Cerquad ) package of an integrated circuit (IC) device. The examine a solder ( seal ) glass attachment in a ceramic ( Cerdip/ themselves as trends in a large number of random events. mechanical performance of solder glass in the package, subjected Probabilistic models reflect the physics of phenomena and the to the temperature change during its fabrication, testing, or storage, is affected primarily by the stresses occurring due to of the variability of the behavior and performance of an engineering product much better than the deterministic ones. It would not be thermal expansion (contraction) mismatch of the seal glass and an exaggeration to say that all the fundamental theories and approaches of modern physics and engineering are probabilistic, break when stretched or bent. the body of the package. Solder glass is a brittle material and can International Microelectronics And Packaging Society 217

4 Intl. Journal of Microcircuits and Electronic Packaging The thermally induced stresses arising in the solder glass can be minimized, if all the ceramic components of the package (the window frame, the base, and the lid) have the same coefficients of expansion. In such a case, the shearing stresses, which concentrate at the end portions of the glass seal, are expected to be low, and the normal stresses in the glass layer will be due to the thermal expansion (contraction) mismatch of the seal glass with the ceramics. If, for some reason, ceramics with different coefficients of expansion are used, it is imperative that these coefficients, being temperature dependent, match well at least at lowtemperature conditions, when the expected thermally induced stresses are the highest. Being brittle materials, solder glasses are able to withstand much higher stresses in compression than in tension. Therefore, it is desirable, for low compressive stresses in the glass layer, that the glass has a somewhat smaller coefficient of expansion than the ceramics. The maximum interfacial shearing stress in a thin solder glass layer can be computed by the formula 13, where h 1 is the layer thickness, and K is the parameter of the assembly compliance. K is given as follows, and λ is the longitudinal compliance of the assembly, h 0 is the total thickness of the ceramics parts, E 0 and E 1 are Young s moduli of the ceramics and the glass, respectively, ν 0 and ν 1 are Poisson s ratios of these materials, (1) (2) (3) (4) are these coefficients for the given temperature t, t 0 is the annealing ( zero stress, setting up ) temperature, and α 0 (t) and α 1 (t) are the instantaneous ( actual ) values of the coefficients of expansion. In an approximate analysis, one can assume that the axial compliance, λ, of the assembly is due to the seal glass only, that is, and therefore, the maximum normal stress, σ max, in the glass can be computed by a simplified formula, While the geometrical characteristics of the assembly, the temperature change, and the elastic constants of the ceramics and the glass can be determined with quite high accuracy, the reliability in the prediction of the difference in the coefficients of thermal expansion of the glass and the ceramics is not as good. This is due, first of all, to the fact that these coefficients exhibit strong temperature dependence and are very sensitive to variations in the glass composition, and thermal treatment. However, what is even more important, is that, due to the clear incentive to minimize the thermal expansion (contraction) mismatch of the glass and the ceramics, such a mismatch is characterized by a small difference of close and not necessarily very small numbers. This leads to an elevated uncertainty in the evaluation of this difference, thereby, justifying the application of a probabilistic approach. Treating the coefficients of thermal expansion as random variables, one can evaluate the probability, P, that the stress in the seal glass is compressive and does not exceed the allowable value, σ *, as the probability that the difference, falls within the interval ψ = α 0 - α 1 (9) (7) (8) κ is the interfacial compliance of the assembly, σ max is the maximum normal stress in the midportion of the glass layer, t is the change in temperature, α = α0 α1 is the difference in the effective coefficients of thermal expansion for the ceramics and the glass, (5) (10) Let us assume that the random variables α 0 and α 1 follow the normal law of distribution, such that their probability density functions are expressed by the formulas,. (11) Then, one can conclude that the function, y, of nonfailure (safety (6) margin) is also normally distributed, 218 International Microelectronics And Packaging Society

5 (12) Here, the mean, <ψ>, and the variance, D ψ, of the random variable ψ are <ψ> = <α 0 > - <α 1 > and D ψ = D 0 + D 1, respectively. The probability that the variable y falls within the interval (0, ψ * ) can be computed as follows, where (13) (14) and the reliability (safety) indices, γ and γ *, are expressed by the formulas,. (15) If the probability P of nonfailure is close to unity, then, one can be confident that the normal stresses in the glass layer are compressive and do not exceed the allowable level σ *, while the shearing stresses (which concentrate at the ends of the glass layer) are small and do not exceed the τ * = kh 1 σ * level. Let, for instance, the elastic constants of the solder glass be E 1 = 0.66 x 10 6 kg/cm 2 and ν 1 = 0.27, the seal temperature be 485 C, the lowest (testing) temperature be -65 C (so that the change in temperature is t = 550 C), the calculated effective coefficients of 6 1 expansion at this temperature be α 1 = / C and 6 1 α 0 = / C, the standard deviations of these coefficients be D = D1 = / C, and the (experimentally obtained) compressive strength of the glass be σ u = 5,500 kg/cm 2. With the safety factor of, say, η = 4, we have σ * = σ u /4 = 1,375 kg/ cm 2. The required level of the function ψ of nonfailure, as predicted by equation (10), is as follows, the mean value of this function is of the form,, = and γ * = , and the probability of nonfailure predicted on the basis of the formula (13), is as follows, P = φ 1 (6.5475) + φ 1 (1.2726) - 1 = If the standard deviations D 0 and D of the coefficients of 1 thermal expansion were only / C, then the reliability indices would become γ = and γ * = , respectively, and the probability of nonfailure would increase to P = Hence, the standard deviations, which reflect the degree of uncertainty in the prediction of the coefficients of expansion, have a strong effect on the probability criterion, and must be kept sufficiently small for higher reliability of the package. In this connection, it is important to point out that it is advisable that materials vendors provide information not only about the mean values of the coefficients of expansion, but about their variances (standard deviations) as well. This would enable a designer to establish the adequate allowable stress level, so as not to compromise the package reliability. Let us examine now several additional problems to illustrate the effectiveness of the application of the probabilistic approach in the thermal stress. 5. Some Other Problems Problem # 1. (The use of Bayes statistics). A bi-material assembly is subjected to temperature change and, hence, experiences thermally induced loading. It has been predicted (for instance, on the basis of accelerated testing) that the probabilities that the constituent materials, #1 and #2, will not fail during the lifetime of the assembly are p 1 and p 2, respectively. The field report indicated that the assembly failed. The details were not reported, however, and are not available. What is the probability that it was the material #1 that failed? The following four hypotheses were possible prior to the obtained field report: H 0 = {none of the materials will fail}; H 1 = {only the first material will fail}; H 2 = {only the second material will fail}; H 3 = {both materials will fail}. The probabilities of these hypotheses are as follows, P 0 = p 1 p 2 ; P 1 = (1-p 1 )p 2 ; P 2 = p 1 (1-p 2 ); P 3 = (1-p 1 )(1-p 2 ). (1.1) The conditional probabilities, associated with the observed event A, the assembly failed, are as follows, P(A/H 0 ) = 0, P(A/H 1 ) = P(A/H 2 ) = P(A/H 3 ) = 1 (1.2) The Bayes formula,, and the variance is of the form, (1.3) D ψ = D 0 + D 1 = ( 1 / C) 2. Then, the reliability indices, calculated by the formulas (15), are γ The formula indicates, how the posterior probability, P(H i /A), of International Microelectronics And Packaging Society 219

6 Intl. Journal of Microcircuits and Electronic Packaging the event A can be determined from the known prior probabilities, P(H i ), and, hence, makes it possible to revise the probabilities of the initial hypotheses on the basis of the new information. Applying the formula (1.3), one can obtain the following expression for the probability that only the first material failed, (1.4) If, for instance, p 1 = 2p 2 = p = 0.99 (the probability on nonfailure of the first material is twice as large as the probability of nonfailure of the second material), then, If the probability of nonfailure is the same for both materials (p 1 = p 2 = p), then, with p = 0.99, one can have, (1.5) For very reliable materials (p = ~ 1 ), the probability that only the first material failed is P(H 1 /A) = 2 1. On the other hand, for very unreliable materials (p = ~ 0), this probability is zero, quite likely that both materials failed. Problem # 2. (Another example of the use of Bayes statistics). The estimated probability that the thermally induced bow of a Printed Circuit Board (PCB), manufactured at the given factory, meets the specification requirement is p. A series of control tests was carried out to determine whether the given batch of PCB s meets the specification. However, due to the test equipment employed, the test results are not absolutely certain. It has been established that the probability that the tests give the correct answer is p 1, and the probability of an erroneous answer is p 2. The given PCB passed the control tests. What is the probability that the PCB meets the specification? The probability of the hypothesis H 1 = {the given PCB meets the specification}, is P(H 1 ) = p. The probability of the hypothesis H 2 = {the given PCB does not meet the specification}, is P(H 2 ) = 1-p. If, in reality, the hypothesis H 1 takes place, than the probability that the PCB passes the tests is, clearly, P(A/H 1 ) = p 1. If the hypothesis H 2 is fulfilled, then the probability that the PCB passes the tests is P(A/H 2 ) = p 2. The posterior ( actual ) probability of the event A, the PCB meets the specification, as predicted by the Bayes formula (1.3), is follows, might be still faulty, but only in two cases out of a thousand. Problem # 3. (This problem shows, how the probability distribution function for a quantity of interest can be obtained from the known probability distribution functions for the quantities it depends upon). A solder joint in a Flip Chip design of an integrated circuit (IC) package experiences thermally induced shear strain due to the thermal expansion (contraction) mismatch of the chip and the substrate materials. This strain can be assessed by the following simplified formula, (3.1) Here, ε = α t is the thermal mismatch strain, α is the difference in the coefficients of thermal expansion (CTE) of the soldered components (the chip and the substrate), t is the change in temperature, h is the solder bump s height (stand-off), and l is the distance of the bump from the center of the chip. The strain, ε, and the stand-off, h, are normally distributed random variables with the probability density functions,, (3.2) respectively. Find the distribution of the shear strain, g (this strain is deemed to be responsible for the reliability of the joint). The distribution of the shear strain, g, can be found, based on the formula (3.1), as follows,, (3.3) where z = ε/h is the ratio of the random variables e and h. Using the formula for the probability density function of the ratio of two random variables 18, one can obtain, where the following notation is used,. (3.4) 220. (2.1) If, for instance, p = P(H 1 ) = 0.96, P(A/H 1 ) = p 1 = 0.98, and P(A/H 2 ) = p 2 = 0.05, then P(A) = Thus, if the PCB passes the tests, it The integral in the formula (3.4) can be represented as follows, International Microelectronics And Packaging Society

7 , (3.10) where (3.5) (3.11) where the parameter β is expressed by the formula, is the mean value of the shear (angular) strain and, and (3.6) is the Laplace function (probability integral). The variance, D h, of the solder joint standoff, h, is typically much smaller than the variance, D ε, of the thermal mismatch strain, and therefore the formula (3.6) for the β value can be simplified as follows, (3.12) is its variance. Note that the above distribution for the shear strain, g, could have been obtained also directly from the probability density function, fε(ε), and the relationship ε = (h/l)γ. Indeed, (3.7) The β value is rather large, since the mean value of the standoff is, as a rule, substantially larger than its standard deviation. Then, the following approximate formula can be used for the evaluation of the function Φ(β),, (3.8) (3.13) The expression (3.13) is similar to the formula (12) and the probability P that the strain γ can be found within the interval (0, γ * ) can be evaluated by the formula, P = P{0 γ γ * } = Φ 1 (δ* - δ) + Φ 1 (δ) - 1, where the function Φ 1 (t) = erf t is expressed by the formula (14) and the parameters δ and δ *, and the expression (3.3) can be written as follows, Problem # 4. (The application of the extreme value statistics). An electronic component is operated in thermal cycling conditions for n cycles. Assuming that the amplitude R(t) of the maximum thermally induced stress, when a single cycle is applied, is distributed in accordance with the Rayleigh law,., (4.1) (3.9) determine the most likely extreme stress value for a large n number. If only the randomness in the thermal expansion mismatch were considered, then, substituting in the obtained expression D h The extreme response, Y n, expected during a certain number, = 0 and <h> = h, one would obtain, n, of observations is a random variable. Its probability density function, g(y n ), and the probability distribution function, G(y n ), can be obtained from the basic distributions, f(x) and F(x), for International Microelectronics And Packaging Society 221

8 Intl. Journal of Microcircuits and Electronic Packaging the random response, X, in the case of n=1, by the formulas 18, g(y n ) = n{f(x)[f(x)} n-1 } x=yn, (4.2) G(y n ) = {[F(x)] ṇ } x=yn. (4.3) From the initial ( basic ) distribution (4.1), using the formula (4.2), one can obtain, Application of the probabilistic approach enables one to quantitatively assess the role of various uncertainties in the materials properties, geometrical characteristics and loading conditions, and, owing to that, to design and manufacture a viable and reliable product. Future work should include research aimed at the accumulation of the information about the probabilistic characteristics of materials, technologies and anticipated loading conditions., (4.4) where References 222 (4.5) and D x is the variance of the ordinates X(t) of the induced stress. The condition, yields the following, g (y n ) = 0 (4.6) (4.7) In the case of large n, the second term in this equation is significantly smaller than the first term, and the most likely value y n * can be found from the equation as. (4.8) As evident from the obtained formula, the most likely extreme stress, y n*, is by the factor of 2 n n larger than the most likely stress D in the case of single loading. If, for instance, n = x 1800 (the electronic component experiences one cycle per day and is operated for 5 years), then, 2 n n = Hence, the most likely thermally induced stress that is expected to occur during 5 years of operation exceeds the 2ln most n likely stress during one day of operation by a factor of only Conclusion 2ln n 1. B. J. Aleck, Thermal Stresses in a Rectangular Plate Clamped Along an Edge, ASME Journal of Applied Mechanics, Vol. 16, pp , F.-V. Chang, Thermal Contact Stresses of Bi-Metal Strip Thermostat, Applied Mathematics and Mechanics, Vol. 4, No. 3, pp , Tsing-hua University, Beijing, China, W. T. Chen and C. W. Nelson, Thermal Stresses in Bonded Joints, IBM Journal, Research and Development, Vol. 23, No. 2, pp , J. W. Eischen, C. Chung, and J. H. Kim, Realistic Modeling of the Edge Effect Stresses in Bimaterial Elements, ASME Journal of Electronic Packaging, Vol. 112, No. 1, H. H. Glascock and H. J. Webster, Structural Copper: a Pliable High Conductance Material for Bonding to Silicon Power Devices, IEEE Transactions on Components, Hybrids, and Manufacturing Technology, Vol. 6, No. 4, pp , J. C. Glaser, Thermal Stresses in Compliantly Joined Materials, ASME Journal of Electronic Packaging, Vol. 112, No. 1, P. M. Hall, Strains in Aluminum-Adhesive-Ceramic Trilayers, ASME Journal of Electronic Packaging, Vol. 112, No. 4, A. Y. Kuo, Thermal Stress at the Edge of a Bi-Metallic Thermostat, ASME Journal of Applied Mechanics, Vol. 57, G. A. Lang, Thermal Fatigue in Silicon Power Devices, IEEE Transactions on Electron Devices, Vol. 17, pp , J. H. Lau, Thermoelastic Solutions for a Semi-Infinite Substrate with an Electronic Device, ASME Journal of Electronic Packaging, Vol. 114, No. 3, V. Mishkevich and E. Suhir, Simplified Engineering Approach for the Evaluation of Thermally Induced Stresses in Bi-Material Microelectronic Structures, International Journal of Microelectronic Packaging, Vol. 1, E. Suhir, Calculated Thermally Induced Stresses in Adhesively Bonded and Soldered Assemblies, Proceedings of the International Symposium on Microelectronics, ISHM 86, Atlanta, Georgia, pp , October E. Suhir, Stresses in Bi-Metal Thermostats, ASME Journal The following conclusions can be drawn from the above study, Predictive modeling is an effective tool for the prediction and prevention of mechanical and functional failures in microelectronics and photonics materials, structures, packages and systems. of Applied Mechanics, Vol. 53, No. 3, pp , September International Microelectronics And Packaging Society

9 E. Suhir, Analytical Modeling in Structural Analysis for Electronic Packaging: Its Merits, Shortcomings and Interaction with Experimental and Numerical Techniques, ASME Journal of Electronic Packaging, Vol. 111, No. 2, pp , June E. Suhir, Interfacial Stresses in Bi-Metal Thermostats, ASME Journal of Applied Mechanics, Vol. 56, No. 3, pp , E. Suhir and B. Poborets, Solder Glass Attachment in Cerdip/ Cerquad Packages: Thermally Induced Stresses and Mechanical Reliability, ASME Journal of Electronic Packaging, Vol. 112, No. 2, pp , E. Suhir, Approximate Evaluation of the Elastic Interfacial Stresses in Thin Films with Application to High-Tc Superconducting Ceramics, International Journal of Solids and Structures, Vol. 27, No. 8, E. Suhir, Applied Probability for Engineers and Scientists, McGraw-Hill, New York, E. Suhir, Thermal Stress Failures in Microelectronics and Photonics: Prediction and Prevention, Future Circuits International, issue 5, S. Timoshenko, Analysis of Bi-Metal Thermostats, Journal of the Optical Society of America, Vol. 11, pp , O. Völkersen, Die Nietkraftverteilung in Zubeanspruchten Nietverbindungen mit konstaten Laschenquerschnitten, Luftfahrtforschung, Vol. 15, pg. 41, 1938 (in German). 22. S. E. Yamada, A Bonded Joint Analysis for Surface Mount Components, ASME Journal of Electronic Packaging, Vol. 114, No. 1, R. Zeyfang, Stresses and Strains in a Plate Bonded to a Substrate: Semiconductor Devices, Solid State Electronics, Vol. 14, pp , International Microelectronics And Packaging Society 223