A Note on Suhir s Solution of Thermal Stresses for a Die-Substrate Assembly

M. Y. Tsai e-mail: mytsai@mail.cgu.edu.tw C. H. Hsu C. N. Han Department of Mechanical Engineering, Chang Gung University, Kwei-Shan, Tao-Yuan, Taiwan 333, ROC A Note on Suhir s Solution of Thermal Stresses for a Die-Substrate Assembly The well-known, closed-form solution to thermal stresses of a die-substrate assembly is initially provided by Suhir in the mid-1980 s after Timoshenko [1] and Chen and Nelson [2]. It has been revised several times in its die attach (adhesive) peel solution by Suhir [3,4], and Mishkevich and Suhir [5]. However, there still exist some controversies and inconsistencies regarding die stresses, die attach shear and peel stresses, and warpage (deformation) of the assembly. In the study, Suhir s derivation of the solution is closely examined in details, and the corrections to the solution are suggested and verified by comparing with the finite element results. It is shown that, unlike the original Suhir solution, the corrected one gives very good prediction of thermal stresses and deformations of die-substrate assembly. The limitation of the Suhir solution is also discussed in this study. DOI: 10.1115/1.1648056 Introduction In the electronic packaging, die cracking, interface delamination, and package co-planarity are common structural failure phenomena encountered in manufacturing process or thermomechanical reliability evaluation. During the design or testing stage, stress analyses are carried out to understand these package failure behaviors and their causes, and thus to help prevent from these failure occurring so as to come out with an optimized design. To this purpose, a simple and easy-to-use solution of stress analysis is necessary and valuable, not only for providing insight into mechanics and thus identifying influential parameters used in the package designs, but also for helping plan experiments and interpret numerical such as finite element analysis and experimental results. A fundamental solution to thermal stresses of a die-substrate assembly is initially proposed by Suhir 6 8 in the mid-1980 s after Timoshenko 1 and Chen and Nelson 2. Ithas kept being revised in its die attach adhesive peel solution by Suhir himself 3,4, and with his co-author Mishkevich 5. The solution is relatively simple and easy-to-use, compared to Chen and Nelson 2 and Mirman 9, and can be allowed to explicitly and instantly indicate important parameters of affecting the stresses and deformations of the assembly. However, there are still some existing controversies and inconsistencies regarding die stresses, die attach shear and peel stresses and warpage deformation of the assembly. For example, the Suhir solution is inconsistent with the finite element solutions in 10, different versions and wrong formula of the solution was quoted in 11, and recently the misuse of the solution applied to stress analysis of flip-chip assemblies results in inconsistency with the finite element results 12. The original solution with some controversies and inconsistencies is still good enough for parametric identification and trend predictions as seen in the literature. However, for design purposes and concerns, the better and accurate prediction especially in quantity is also desired in engineering applications. The purposes of this study are to summarize the different versions of the Suhir solution available in the literature and identify the inconsistency, and then to propose the corrections to the solution and verify the corrected solution with the finite element results. Contributed by the Electronic and Photonic Packaging Division for publication in the JOURNAL OF ELECTRONIC PACKAGING. Manuscript received September 2003. Associate Editor: Z. Suo. Corrected Suhir s Solution Prior to the correction to the Suhir solution, the relationship between bending moment and curvature in pure bending of plates from Timoshenko 13 can be briefly addressed by M x D 1 1 r x r y D 2 w x 2 2 w y 2 (1) M y D 1 1 r y r x D 2 w y 2 2 w x 2 (2) or 2 w x 2 M xm y D1 2 (3) 2 w y 2 M ym x D1 2 (4) where the w is deflection of the plate, the M x and M y are the bending moments, and the r x and r y the radii of bending curvature with respect to the x- and y-axes. The flexural rigidity D is expressed in terms of elastic modulus E, Poisson s ratio and plate thickness t as Et 3 D 121 2 (5) If a spherically bending plate is assumed, then M x M y M (6) The substitution of Eq. 6 into 3 and 4 render 2 w x 2 2 w M y 2 D1 M (7) D where the flexural rigidity of the spherically bending plate D is D Et3 (8) 121 Regarding the solution of thermal stresses for die-substrate assembly, the first completely closed-form solution was proposed by Suhir 6,7, following Timoshenko 1, and Chen and Nelson 2. The schematic of the assembly is shown in Fig. 1, in which E,,. and t represent elastic modulus, Poisson s ratio, the thermal expansion coefficient, and the thickness of each layer with sub- Journal of Electronic Packaging Copyright 2004 by ASME MARCH 2004, Vol. 126 Õ 115

The Suhir solutions and corrections to die attach shear and peel stresses, die stresses and warpage in the assembly subjected to thermal load T are discussed in detail. The thermal load is TT f T i (11) where T f is final temperature, and T i initial one. The difference of thermal expansion coefficient between die and substrate is 3 1 (12) Die Attach Shear Stress. be explicitly expressed by where The die attach shear stress ( 0 ) can 0 x kt sinh kx (13) cosh kl longitudinal compliancek ; (13.1) 1 1 2 1 2 3 t2 (13.2) E 1 t 1 E 3 t 3 4D The Eq. 13.2 has been adopted in Eq. 11 in Suhir 7 and in Eq. 2 in Mishkevich and Suhir 5, but it has been corrected to 1 1 1 3 t2 (13.3) E 1 t 1 E 3 t 3 4D which was described in Eq. 26 in Suhir 8, by an assumption of a spherically bending plate. The interface compliance is in which t 1 3G 1 2t 2 3G 2 t 3 3G 3 (13.4) Fig. 1 Geometrical and material parameters, and free-body diagram for Suhir s model scripts from 1 to 3 for die, die attach adhesive, and substrate, respectively. And the interfacial stresses between die attach and die or substrate are shear stress 1 or 2 ) and peel stress 1 or 2 ). The solution is based on the assumptions as follows: 1. Isothermal loads are assumed. 2. Each layer in the assembly acts as a spherically bending thin plate. 3. Perfectly bonded interfaces exist between the layers in the assembly. 4. The material of die attach adhesive layer is very relatively compliant compared to the die and substrate. 5. The thickness of die attach adhesive layer is very relatively thin compared to the die and substrate. According to the assumptions 4 and 5, the thermal expansion coefficient of die attach 2 is insensitive to the stresses and thus can be neglected or excluded in the formulations, and the interfacial stresses can be approximately written by 1 2 0 (9) and 1 2 0 (10) where 0 and 0 are die attach shear and peel stresses, respectively. and E i G i 21 i ; (13.6) tt 1 t 2 t 3 ; (13.7) DD 1 D 2 D 3 ; (13.8) D i E 3 it i 121 2 (13.9) i It is worth noting that D i in Eq. 13.9 is the flexural rigidities of the plates. However, if the spherically bending plates actually occurring at thermal deformation of a bi-material plate are assumed, the flexural rigidities will become D i E 3 it i (13.10) 121 i The above equation is the one proposed here for the correction in order to be consistent with the spherically bending plate assumption which is used in Eq. 8. Die Attach Peel Stress. The first version of the solution to the die attach peel stress ( 0 ) postulated by Suhir 6,7 is: 0 x T cosh kx cosh kl (14) It has been modified by Suhir 4,5 by considering the satisfaction of free boundary condition at the joint end and obtained as 0 x T s 4 1s 4 cosh kx cosh kl A 0V 0 xa 2 V 2 x (15) 116 Õ Vol. 126, MARCH 2004 Transactions of the ASME

where t 3D 1 t 1 D 3 ; (15.1) 2D s & k ; (15.2) 4 K 1 1 2 t 1 KD 4D 1 D 3 (15.3) 1 3 2 t 3 E 1 E 3 1 (15.4) which is used in Mishkevich and Suhir 5. However, the above equation should be corrected to K 1 1t 1 1 1 3t 3 E 1 E 3, (15.5) in order to comply with the assumption of the spherical bending plates. The rest of related parameters are A 0 2& V 3ls 3 V 0 ltanh kl s 2 (15.6) sinh 2lsin 2l A 2 2& V 1ls 3 V 2 ltanh kl s 2 sinh 2lsin 2l (15.7) V 0 xcosh x cos x (15.8) V 2 xsinh x sin x (15.9) V 1 x 1 cosh x sin xsinh x cos x (15.10) & Fig. 2 a Three-dimensional finite element model, in which two models with 2-element and 6-element through thickness of die attach layer are used b Two-dimensional axis-symmetrical model Die Stresses. be derived as V 3 x 1 cosh x sin xsinh x cos x (15.11) & along the top line, The die stresses in the longitudinal direction can Top T t 1 and along the bottom line, Bot T t 1 13 td 1 cosh kx (16) t 1 D1 cosh kl 13 td 1 cosh kx (17) t 1 D1 cosh kl Warpage of Assembly. During the derivation of the equations and based on Eqs. 28, 30 and 31 in Suhir 8, the curvature can be written as 1 tt cosh kx x 2D 1 (18) cosh kl where the (x) is the radius of the curvature of the assembly. The relationship between the warpage deflection w(x) and the curvature can be expressed as wx 1 (19) x After the integration, the warpage deflection w(x) can be obtained as wx tt 2D 1 cosh kx1 2 x2 k 2 cosh kl (20) It is worth mentioning that the above formulation has been wrongly printed in Eq. 47 in Suhir 8. To summarize the corrected Suhir solution of the die-substrate assembly, the Eqs. 13 and 15 for die attach shear and peel stresses, respectively, are associated with Eqs. 13.3, 13.10 and 15.5, rather than Eqs. 13.2, 13.9, and 15.4. Eqs. 16 and 17 are for die longitudinal stresses, and Eq. 20 is for the warpage of the assembly. If the problem of a strip assembly is taken, the equations listed above for die attach shear, peel and die stresses, and warpage can be used by setting Poisson s ratio equal zero in Eqs. 13.3, 13.10 and 15.5. That is, the solution for the plate assembly is reduced to the one for a beam assembly. This solution of a strip assembly is very close to a plane stress condition of a two-dimensional state and thus can be used to simulate the die and die attach shear but not peel stresses on the free surface of three-dimensional problem 14 16. Solution Verication and Discussion To verify the solutions, the three-dimensional 3D and twodimensional 2D finite element models were used for justification by comparing their results with original and corrected solutions. The finite element models are shown in Fig. 2 for the assembly size, meshes and boundary condition. For the 3D model in Fig. 2a, only a quarter of the assembly is considered due to the existence of two planes of symmetry. Apart from symmetrical conditions assigned on the planes of symmetry, a point located at the bottom of the line intersected between the two planes of symmetry is constrained in the z thickness direction for stabilizing the assembly. Two configurations of mesh are used: one is with 2 elements through thickness of die attach layer, and the other with 6 elements. Total of 10,800 and 23,400 eight-node solid elements are employed for 2-element and 6-element die attach configurations. The size of the assembly is 6 mm6 mm0.63 mm. For Journal of Electronic Packaging MARCH 2004, Vol. 126 Õ 117

Table 1 Material properties and thickness for the diesubstrate assembly Die i1 Die Attach i2 Substrate i3 E i i GPa i (ppm/ C) t i mm 170 0.28 2.5 0.3 6.3 0.4 36 0.03 26 0.42 15 0.3 Note: 3 mm the 2D model in Fig. 2b, an axis-symmetrical model with extremely fine near and at the free end of the die attach layer is allowed to capture the high stress gradient near the interface corners. Total number of constant-stress elements is 12,000 used in the 2D analysis. Both of the detailed dimensions and material properties are listed in Table 1. The thermal load applied is T 1 C for all analyses. The die stress, warpage of the assembly, and die attach shear and peel stresses determined from the corrected Suhir solution are compared closely with those from the original ones and the finite element results. Note that the corrected solutions are associated with Eqs. 13.10 and 15.5, while the original ones with Eqs. 13.9 and 15.4. The results from the finite element analysis, used in comparison, are those on the x-z plane of symmetry, since the assumption of a spherically bending plate is taken in the Suhir solution. And the 2-element model used is for neglecting the variation of die attach shear and peel stresses across the thickness and would be equivalent to the approximation of the Eqs. 9 and 10 used in Suhir s formulations. The die stress distributions along the top and bottom lines of the die are shown in Fig. 3 for these three different results. It can be seen that the die stress from the corrected Suhir solution is very consistent with one from the finite element analysis, while the die stress from the original one is about 15% lower than the other two solutions. The results of the warpage of the assembly, as shown in Fig. 4, also indicate that the results from the corrected Suhir solution and the finite element coincide very well, while the warpage from the original one is about 10% higher than the ones from the others. It is worth mentioning that the die stresses and warpage on the x-z plane of symmetry in the 3D model are exactly the same as those from 2D axis-symmetrical model. Regarding the die attach shear and peel stresses, the results obtained from these three analyses are shown Fig. 4 Warpage deflection of the assembly under T ÄÀ1 C, from the corrected and original Suhir s solutions and finite element results in Figs. 5 and 6, respectively, and indicate that the corrected solution is closer to the finite element solution than the original one. And the original solutions give about 8% lower than the corrected solutions on the maximum values of the die attach shear and peel stresses. Fig. 5 Die attach shear stress distributions for the assembly under TÄÀ1 C, from the corrected and original Suhir s solutions and finite element results Fig. 3 Die stress x distributions along the top and bottom lines of the die for the assembly under TÄÀ1 C, from the corrected and original Suhir s solutions and finite element results Fig. 6 Die attach peel stress distributions for the assembly under TÄÀ1 C, from the corrected and original Suhir s solutions and finite element results 118 Õ Vol. 126, MARCH 2004 Transactions of the ASME

Fig. 7 Die attach shear stress distributions for the assembly under TÄÀ1 C, from the finite element results with 2-element and 6-element through thickness of die attach layer and 2D axis-symmetrical models Fig. 8 Die attach peel stress distributions for the assembly under TÄÀ1 C, from the finite element results with 2-element and 6-element through thickness of die attach layer and 2D axis-symmetrical models However, based on the theoretical elasticity assumption, there exist two stress singularities occurring at wedge corners 17,18: one is at an interface corner between the die and die attach, and the other between the substrate and die attach. The stress singularity means the stress values at the singular point become unbound. As a result, the finer the mesh gets around the singular area in the finite element analysis, the larger the stress value becomes. The 3D 6-element and 2D axis-symmetrical models show this phenomenon in Figs. 7 and 8, respectively, for die attach shear and peel stress distributions near the die/die attach top interface, and the die attach/substrate bottom interface. By comparison of the 2-element model with the other two models, it is shown that the approximation of the Eqs. 9 and 10 used in the theory is no longer valid as the region approaches to x/l0.98 and further. That is, the theory close to one-dimensional formulation in the die attach layer cannot work well in this small and strongly twodimensional effect region with x involved. Of course, the size of this small two-dimensional region is approximately the same order as the thickness of die attach layer. It is expected that at the joint end (x/l1), the die attach shear and peel increase dramatically near the die/die attach top interface, but decrease abruptly to the negative value near the die attach/substrate bottom interface due to the existence of the singularity. Therefore, the application of the Suhir solution inherently with the assumptions of Eqs. 9 and 10 to the region from x/l0.98 to 1 is inadequate, so that the Suhir solution at this region can only be used as reference value instead of actual values. Conclusions The Suhir closed-form solution to thermal stresses for diesubstrate assembly was examined in detail in this study. The corrections to the solution were proposed and evaluated by comparing with the finite element solutions. It is shown that, unlike the original Suhir solution, the corrected one give very good predictions of thermal stresses as well as deformations of the diesubstrate assembly. As a result, the existing controversies and inconsistencies in the literature, regarding die stresses, die attach shear and peel stresses and warpage deformation of the assembly were resolved. The limitation of the Suhir solution has been also discussed in this study. Acknowledgment The authors would like to acknowledge the financial support to this research by the National Science Council, ROC, under Grant No. NSC89-2212-E-182-010. References 1 Timoshenko, S. P., 1925, Analysis of Bi-Metal Thermostats, J. Opt. Soc. Am., 11, pp. 233 255. 2 Chen, W. T., and Nelson, C. W., 1979, Thermal Stresses in Bonded Joints, IBM J. Res. Dev., 232, pp. 179 187. 3 Suhir, E., 1989, Interfacial Stresses in Bimetal Thermostats, ASME J. Appl. Mech., 56, pp. 595 600. 4 Suhir, E., 1991, Approximate Evaluation of the Elastic Interfacial Stresses in the Thin Films with Application to High-T c Superconducting Ceramics, Int. J. Solids Struct., 278, pp. 1025 1034. 5 Mishkevich, V., and Suhir, E., 1993, Simplified Engineering Approach for the evaluation of Thermally Induced Stresses in Bi-Material Microelectronic Structures, ASME Structural Analysis in Microelectronics and Fiber Optics, EEP., Vol. 7, pp. 127 133. 6 Suhir, E., 1986, Stresses in Adhesively Bonded Bi-material Assemblies Used in Electronic Packaging, Mater. Res. Soc. Symp. Proc., 72. 7 Suhir, E., 1986, Stresses in Bi-Mettal Thermostats, ASME J. Appl. Mech., 53, pp. 657 660. 8 Suhir, E., 1987, Die Attachment Design and Its Influence on Thermal Stresses in the Die and the Attachment, Proc. 37th Electronics Components Conference, IEEE/EIA, pp. 508 517. 9 Mirman, B. A., 1990, Creep Strains in an Elongated Bond Layer, IEEE Trans. Compon., Hybrids, Manuf. Technol., 134, pp. 914 928. 10 Glaser, J. C., 1990, Thermal Stresses in Compliantly Joined Materials, ASME J. Electron. Packag., 112, pp. 24 29. 11 Matijasevic, G. S., Wang, C. Y., and Lee, C. C., 1993, Thermal Stress Considerations in Die-Attachment, Thermal Stresses and Strain in Microelectronics Packaging, John H. Lau, ed., Van Nostrand Reinhold, pp. 194 220. 12 Michaelides, S., and Sitaraman, S. K., 1999, Die Cracking and Reliable Die Design for Flip-Chip Assemblies, IEEE Trans. Adv. Packag., 224, pp. 602 613. 13 Timoshenko, S., and Woinowsky-Krieger, S., 1956, Theory of Plates and Shells, 2nd Edition. 14 Tsai, M. Y., and Morton, J., 1991, The Stresses in a Thermally Loaded Bimaterial Interface, Int. J. Solids Struct., 288, pp. 1053 1075. 15 Tsai, M. Y., and Morton, J., 1992, A Stress Analysis of a Thermally Loaded Bimaterial Interface: A Localized Hybrid Analysis, Mech. Mater., 13, pp. 117 130. 16 Hsu, G. H., 2002, Investigation of Thermo-Mechanical Behaviors of Flip Chip BGA in IC Packaging, Master Thesis, Chang Gung University, Taiwan. 17 Bogy, D. B., and Wang, K. C., 1971, Stress Singularities at Interface Corners in Bonded Dissimilar Isotropic Elastic Materials, Int. J. Solids Struct., 7, pp. 993 1005. 18 Hein, V. L., and Erdogan, F., 1971, Stress Singularities in a Two-Material Wedge, Int. J. Fract., 73, pp. 317 330. Journal of Electronic Packaging MARCH 2004, Vol. 126 Õ 119