A Thermomechanical Model for Warpage Prediction of Microelectronic Packages

Transcription

1 A Thermomechanical Model for Warpage Prediction of Microelectronic Packages Eric Egan [a], Gerard Kelly [b], Tom O Donovan [c], Michael Peter Kennedy [d] [a] National Microelectronics Research Centre (NMRC), National University of Ireland, Lee Maltings, Prospect Row, Cork, Ireland. eric.egan@mail.com [b] Cork Institute of Technology, Cork, Ireland. kellyg@cit.ie [c] Dept. of Statistics, National University of Ireland, Cork, Ireland. t.odonovan@ucc.ie [d] Dept. of Microelectronics Engineering, National University of Ireland, Cork, Ireland. peter.kennedy@ucc.ie Abstract The thermomechanical warpage or vertical deflection of microelectronic packages due to temperature change is caused by the mismatch in the coefficients of thermal expansion between vertically asymmetric layers of materials. The structure of many microelectronic packages may be characterized as a multi-layered plate consisting of two regions, that of a area and a mold area. Physical observation of various packages demonstrates that these two regions cause the package to exhibit dual curvatures, which are approximated using closed form sets of algebraic equations. The resulting technique, termed the dual-curvature approach, predicts the thermomechanical warpage of packages composed of temperature-dependent materials and may be extended for application to packages having a diverse range of geometries. The accuracy of the dual-curvature approach is assessed for square, single- packages by means of threedimensional, finite element simulations at nine in-plane locations. It is shown that the dual-curvature approach significantly improves the thermomechanical warpage prediction of microelectronic packages relative to earlier analytical models. Key words Thermomechanical, warpage, deflection, deformation, curvature, bending, multi-layered, plates, PBGA The International Journal of Microcircuits and Electronic Packaging, Volume 5, Number 1, First Quarter, 00 (ISSN ) 100

2 1. Introduction Package warpage resulting from large temperature changes during the manufacturing process is a significant reliability concern [1]. Of the stress-related defects that cause reliability problems in ball grid array microelectronic packages, coplanarity issues, vertical cracking, interfacial delamination, and solder joint fatigue are among the most prominent. Indeed, package warpage is partially attributable to the deformation and tensile stress that are responsible for solder ball coplanarity problems [-5] and vertical cracking [6,7], respectively. Warpage of multilayered structures also increases the risk of delamination between layers [8]. Furthermore, solder joint geometry [, 9] and fatigue [10, 11] are also adversely affected by package warpage. Package warpage is a combination of the chemical shrinkage of the mold encapsulation during the curing process [1] and the thermomechanical deformation of the package due to coefficient of thermal expansion (CTE) mismatch between adjoining materials subjected to a temperature change [13]. Although chemical shrinkage for some packages may minimally affect the package warpage [5], the influence of chemical shrinkage on warpage is highly dependent on both the thermoset polymer chosen for the mold encapsulation and the curing process itself [14]. This paper presents a technique to approximate the package warpage due to thermomechanical stress. Package warpage aspects related to chemical shrinkage, as well as other modeling issues, are discussed briefly in a later section. The dual-curvature approach presented in this paper is a closed form set of algebraic equations for the rapid approximation of the thermomechanical warpage of microelectronic packages. Although any package type could be used to demonstrate the capability of the dual-curvature approach, this research focuses on the plastic ball grid array (PBGA) package due to its vertically asymmetric crosssectional structure that accentuates the warpage due to CTE mismatch. PBGA packages are generally composed of a silicon mounted onto a substrate using attach material. Mold encapsulation is transfer molded around the silicon using a thermoset epoxy [15]. The PBGA is modeled as a two-region, four-layer plate, whose two regions consist of the and mold areas. Figure 1 shows a cross section of the main structural components of a PBGA package, the mold encapsulation, silicon, attach, and substrate. One of the earliest works for thermal stress analytical modeling was provided by Timoshenko in a fundamental paper on beam bending due to CTE mismatch between layers as applied to bi-metal thermostats [16]. Later, this two-layer structure used by Timoshenko was expanded by Cifuentes to one having n layers by developing a thermomechanical set of equations using linear algebra [17]. Suhir was among the first to develop and apply theoretical mechanics routinely for both warpage and thermal stress analysis of plastic encapsulated devices [13]. Although the calculation of curvature due to CTE mismatch is well established, researchers differ in the application of thermoelastic plate mechanics to actual packaging structures. In research discussing the thermal stress of plastic packages in general, Suhir mathematically describes packages having a uniform cross section along its entire length [18]. Other researchers also simplify the structure to that of a uniform cross section and have calculated warpage based on only two layers [4, 7]. This approach of considering the microelectronic package as consisting of only the region will be termed the -only approach. 101

3 Mold Encapsulation Silic on D ie Die Attach Substrate z h 1 h h 4 h 3 x y P diag L mld (a) z y θ P ctr Die Area M old Area (b) L x Reference Plane (c) x Figure 1. Schematic of a PBGA showing: (a) a cross-sectional view, (b) a plan view of the measurement paths, P ctr and P diag, and of the and mold areas, and (c) a three-dimensional model of one-quarter of the package For the warpage of plastic quad flat packages, PQFPs, Suhir considers the warpage to consist of two curvatures: a finite, -area curvature given by the thermomechanical system of equations discussed above and a mold-area curvature having a value of zero [1]. This will be referred to as the -line approach. However, this approach does not take into account the intrinsic, mold-area curvature of PBGAs. The dual-curvature approach modifies the -line approach to include the mold-area curvature and is shown to significantly enhance the predictive accuracy for PBGAs. After showing a physical warpage profile illustrating the need for a dual-curvature analysis, this paper presents the dualcurvature approach along with the thermomechanical system of equations describing multi-layered plate curvature. The dual-curvature approach is then applied to a data set consisting of randomly generated, single- packages as shown in Figure 1, created by 3D simulations employing the finite element method (FEM). This data set is used to assess the accuracy of the dual-curvature approach relative to the -only and -line approaches. 10

4 . Definition of Warpage and Curvature It is convenient to choose the lateral plane intersecting the middle of the as the reference x-y plane as depicted in Figure 1c, because the vertical contraction of materials above and below the reference plane will be of the same order of magnitude. The displacement of the center position in the reference plane is constrained to be zero in all directions. Free vertical contraction of the mold encapsulation and substrate would have to be considered if calculating the warpage profile for either the mold or substrate face. Planes running parallel to the cross section of the PBGA in the vertical, z, direction through the centerlines are the x-z and y-z planes. In this paper, warpage along a given radial path is defined as the normal distance between the two closest parallel lines that encompass the vertical deformation of that path. In the threedimensional simulations to follow, all paths are originally colinear with the reference plane of Figure 1c prior to deformation. The maximum warpage value from all scan paths is the package warpage, δ, whose path direction usually coincides with the diagonal direction, P diag, shown in Figure 1b. An example of positive warpage is depicted in Figure with two distinct curvatures, each positive. If the warpage is concave up, it is termed positive warpage; if concave down, it is termed negative warpage. Curvature, κ, is defined as the inverse of the radius of curvature. To simplify the equations to follow, we assume that the circular arcs due to bending take the form of parabolas [1] because of the small curvature magnitudes that are typical of microelectronic packages. For warpages of constant, uniform curvature, the package warpage, δ, is related to curvature by the relation, Figure. Package warpage definition showing positive (concave-up) warpage δ = 0.5κ s (1) where s is the lateral distance from the package center to the point of interest along any radial path. If the radial path is P ctr, the path length, s, is in the direction of either x or y, as shown in Figure 1b. Because all materials are modeled as isotropic (explained later in the paper) and the package is square and symmetric, reflective symmetry is obtained about the x-axis, y-axis, and the lines y = x and y = -x. Due to these symmetry conditions, only one-eighth of the package needs to be described within the range, 0 < θ < 45 o. As such, the path length, s, can be expressed as x s =. () cosθ 3. Dual Curvature of Single-Die Packages Dual-curvature warpage profiles are easily demonstrable in PBGA packages in particular. A warpage measurement of a 17 mm PBGA with 8 mm is shown below in Figure 3. A laser profiler measures the warpage at room temperature (0 o C) to within an accuracy of 103

5 approximately ± 3 microns. In Figure 3, the warpage profile along the centerline path, P ctr of Figure 1, shows a distinct dual curvature with a negative curvature in the region, followed by a sharp positive curvature near the edge. The slight tilt of Figure 3a may be due to non-uniform solder ball heights on the underside of the package and is corrected in Figure 3b by an appropriate axis rotation. Several of the extreme points of deviation from the warpage profile in Figure 3a are removed in Figure 3b. The deviations are due to the laser-inscribed markings on the topside of the package. This warpage measurement, along with many other physical observations, suggests the importance of accounting for the mold-area curvature in analytical warpage modeling. z (µm) z (µm) x (mm) 8.5 a) b) 4. PBGA Manufacturing Process PBGA package warpage is a result of the manufacturing process where the package undergoes large temperature variations. After attach and wirebonding, the pre-package structure is heated to approximately 170 o C in a cavity containing the substrate and silicon. After the initial phase of mold encapsulation curing, the PBGA is typically held at approximately170 o C for several hours of postmold curing during which the mold encapsulation reaches full cure. The package then cools down to ambient temperature. Table 1 summarizes the main steps involved in PBGA manufacture. The thermomechanical warpage at the end of the manufacturing process is due to the temperature difference that each material experiences between its final state at room temperature and its stress-free, reference temperature, T ref, i for material, i, at which no thermomechanical stresses yet exist. In the rest of this paper, room temperature is set equal to 5 o C. It is customary to assume that the thermomechanical stress-free state of the x (mm) 8.5 Figure 3. Warpage measurement using a laser profiler of a 17 mm PBGA with 8 mm along the centerline path, P ctr, (a) before and (b) after plot adjustments entire package occurs at the mold curing temperature [1, 4, 5, 19, 0] of step 3. However, in this research, a thermomechanical stress-free state for the, attach, and substrate is assumed at the attach temperature of 150 o C of step 1. Upon cooling, the pre-package structure experiences thermomechanical stresses due to CTE mismatch. On the other hand, the reference temperature for the mold encapsulation is chosen to be the curing temperature of step 3 as suggested by Dunne and Sitaraman [1], since the mold encapsulation is initially a viscous liquid at this stage. Consequently, T ref, 4 is 170 o C. 104

6 5. Development of Dual-Curvature Approach Warpage prediction of PBGA packages by analytical means is desirable for understanding the principles causing the thermomechanical stress state, bending moments, and warpage of microelectronic packages []. Closed form, analytical models offer a one-step, rapid calculation of package warpage and enable the efficient exploration of the design space [3]. Microelectronic packages such as PBGAs may often be characterized as multi-layered, tworegion plates. Due to a temperature change, the CTE mismatch between layers of differing materials causes unequal, free thermal strains. In this paper, the top layer (usually the mold encapsulation) is termed material 1, whereas the bottom layer (usually the substrate) is termed material n, as depicted in Figure 1a. The inherent thermomechanical curvature of each package region may be obtained with a thermomechanical system of n +1 equations, whose detailed theoretical development may be found in the literature [18, 3]. The structure of the region of Figure 1b is approximated as a four-layer plate consisting of isotropic, elastic materials. Its thermomechanical curvature, induced by an isothermal temperature change, T, is given by equations (3) below, where ν i = Poisson s ratio for material i, E i = elastic modulus of material i at the final temperature, h i = thickness of material i, P i = mean axial, in-plane force of material i per unit width of the plate (orthogonal to length and thickness directions), α i = coefficient of thermal expansion (CTE) of material i, calculated as the secant (mean) CTE over the temperature range, T, T = temperature change, T final T ref, where T final = final temperature at which the warpage is to be measured. For predicting PBGA warpage in particular, the dual-curvature approach approximates the PBGA structure as a two-region, four-layer plate, depicted in Figure 1. The multi-layered plate is divided into two regions to account for the and mold areas of the PBGA. The two regions are assumed to be thermomechanically independent from one another. The assumption that the thermomechanical curvature of each region is independent from the other region simplifies the dual-curvature approach but also causes an error due to the mold-area influence on the -area curvature and vice versa. The magnitude of this error is explored with a detailed error assessment later in the paper. 1 ν 1 ν h + h Eh E h 11 1 ν 1 ν h + h ( α α ) E h E h P 1 1 T 33 1 ν ( α α ν + P 1 h h 3 ) T 0 0 P = E h E h 3 ( α 4 α 3 ) T (3) P 4 0 κ h h h h h + h E + h h 3 h 3 i h ν 1 = 11 i i i 105

7 Table 1. Manufacturing Process Stages for PBGA Step Manufacturing Stages Temperature ( o C) 1 Die attach curing 150 Wirebonding 40 3 Mold encapsulation 170 curing 4 Postmold curing 170 Due to assumptions in the dual-curvature approach, the package curvature is assumed to take the form of a step function, changing by an amount of κ - κ at the interface of mld the and mold areas, as shown in Figure 4. The geometric and material parameters of the PBGA simulated for Figure 4 are given in Table A1 of the Appendix. The step-function representation of the complicated curvature profile shown in the three-dimensional FEM simulations greatly simplifies the equations to follow. The approximation of the package curvature as a step function is given by equation (4). Curvature can be defined as the second derivative of the vertical deformation as given in equations (5) and (6). The inherent thermomechanical curvatures of each region are calculated using equations (3). and where κ, for s S κ = κ mld, for s S (4) w s κ =, for s S (5) mld w s κ =, for s S, (6) κ, κ mld = the curvatures of the and mold areas, respectively, w = vertical height in the z-direction of the deformed (warped) reference plane relative to its undeformed position, s = radial distance from the package center to any point in the reference plane, S = radial distance from the package center to any point along the edge. Equation (7) provides the dual-curvature prediction of the warpage profile of microelectronic packages with structures similar to that of Figure 1. The dual-curvature approach assumes the package warpage to take the form of two parabolas joined together at the -mold interface with the same vertical displacement and slope. The dual-curvature equation results from twice integrating the second derivatives in equations (5) and (6) and setting the deflection and slope of the two curves equal at the -mold interface, κs w(s) = κmlds, for s S + ( κ κ ) mld S S s, for s S. (7) Equation (7) gives the warpage in the z direction along a given radial path of length s from the package center in any lateral direction. 106

8 0.0 k (1/mm) D FEM Results Dual-Curvature Approach s (mm) Figure 4. Curvature of a 1 mm PBGA with 8 mm along P diag from 3D FEM simulations and the dual-curvature approach 6. Modeling of Material Properties It has been customary in analytical modeling of package warpage to use isotropic properties [1, 4, 5, 13, 17-19]. In addition, isotropic modeling of all four materials is considered reasonable due to two observations. Firstly, warpage measurements of several PBGA packages using a laser profiler have not generally shown significant warpage differences between the x and y directions. Secondly, in a previous study involving threedimensional FEM simulations, anisotropic modeling of the substrate results in a change of package warpage of under % relative to isotropic modeling using the x-y, in-plane properties for all materials [4]. Similar results have been reported upon investigation of the possible anisotropy of the silicon where warpage values between the anisotropic and isotropic cases differed by approximately 5% [5]. In general, only material properties in the x and y directions have a considerable impact on the warpage, with the z-direction properties having negligible impact. Correct modeling of the stress-free temperatures is another important concern. Because the reference temperatures of the constituent materials are usually not the same, each reference temperature must be included in the thermomechanical system of equations in (3) by altering T to the relation given below, T = T T (8) i final ref,i In addition, the CTE s in equations (3) are expressed as the secant CTE s, not the tangent, instantaneous CTE s. The secant 107

9 CTE is the mean of the tangent CTE over the given temperature range. If a given material is not temperature dependent, the tangent, instantaneous CTE at T final will equal the secant CTE from the molding temperature to T final. The secant CTEs for use in equations (3) for temperature-dependent materials exhibiting elastic properties are defined as, 1 T = Tfinal εth,i α sec, i = αtan,i(t) dt = T Ti, (9) i T = Tref,i where α sec,i = secant CTE of material i from T ref,i to T final, α tan,i = temperature-dependent tangent CTE of material i, ε th,i = free thermal strain of material i from T ref, i to T final. If one or more of the elastic moduli are temperature dependent, the package will exhibit nonlinear warpage behavior relative to temperature. However, assuming all materials to behave elastically from the molding temperature to T final, the thermomechanical warpage of temperature-dependent materials is characterized by using the values of the elastic moduli at T = T final in equations (3). Intermediate values of the elastic moduli do not affect the thermomechanical warpage at T final for temperature-dependent elastic materials. As noted by Kelly et al [19, 6], another concern that may affect package warpage is the chemical shrinkage of the mold encapsulation during the polymerization process prior to full cure. The mold encapsulation typically consists of a thermoset polymer heavily filled with silica to reduce its effective CTE and temperature dependency [15]. The polymerization process of thermoset polymers from their near-liquid state to hardened, fully-cured materials is due to the cross-linking of the polymeric chains which causes the materials to experience a volumetric shrinkage [1]. The linear compressive strains due to chemical shrinkage that occurs after the mold encapsulation bonds to the surrounding materials will eventually contribute to the package warpage and may be considered by increasing the effective value of mold encapsulation CTE. Equation (10) increases α 1 of the mold encapsulation in equations (3) to that given below, where ε cs α 1 = αsec, 1 + T, (10) 1 α sec, 1 = secant CTE of the mold encapsulation due to thermomechanical contraction from T ref,1 to T final, ε cs = linear compressive strain (ε cs < 0) due to chemical shrinkage during polymerization. It should be noted that relation (10) implicitly assumes that the mold encapsulation behaves as a temperature-dependent elastic material from step 3 of Table 1 to the time of warpage measurement. Research undertaken by Oota found this assumption can be reasonable for materials having a high glass transition temperature [7]. A more accurate, albeit more complicated, description of the chemical shrinkage process is possible through a viscoelastic analysis of the mold encapsulation [8, 9]. 7. Error Metric for Accuracy Assessment of Predictive Techniques In assessing the predictive accuracy of the dual-curvature approach for single- PBGAs, 108

10 the values of certain design variables are randomly varied within specified limits in a set of 50 three-dimensional FEM simulations. As such, 50 random and independent packages are created. Comparing the simulation warpage profile with that of the dual-curvature prediction provides a set of errors from which to characterize the predictive accuracy. Instead of using an error metric such as mean absolute percentage error, where dividing by warpages near zero provides unrealistically high percentage errors, we propose to use another error metric termed the scaled RMSE. This error metric is the root mean squared error, RMSE, divided by the standard deviation of the data set as shown below, scaled RMSE = where N 1 N i= 1 N 1 N 1 i= 1 (w w i (w w i i,pred) mean) 1/, (11) N = number of data in independent data set, w i = actual value of warpage at a given point for package i from 3D simulation, w i, pred = predicted value of warpage at a given point for package i, w mean = mean value of actual warpages from simulation data set for a given point. While the RMSE provides a type of averaged error for the entire data set, the scaled RMSE scales this averaged error relative to the magnitude of the scatter of the data. For data having a small scatter of values, the scaled RMSE places a premium on accuracy by dividing by a small standard deviation. By scaling the RMSE to the size of the standard deviation of the data set, the accuracy of different predictive models can be compared for data sets with different amounts of scatter. 8. Accuracy Assessment of Dual- Curvature Approach for Single-Die Packages To test the accuracy of the dual-curvature approach for predicting the warpage of packages as depicted in Figure 1, an error assessment data set is constructed of 50 FEM simulations of 3D packages. All design parameters involving the mold encapsulation are randomly varied within prescribed limits, whereas the parameters of the other materials remain constant as shown in Table. The material properties of the, attach, and substrate of the multichip BGA examined by Moore and Jarvis [0] are selected for the single- packages of the error assessment data set. By varying the material properties and geometry of the mold encapsulation, virtual PBGAs having diverse geometric aspect ratios and curvature characteristics are constructed. The three-dimensional FEM simulations employ 8-noded, rectangular parallelepiped elements with a maximum inter-nodal distance of 300 µm. Using a bi-material plate having geometric and curvature characteristics that are typical for PBGAs as a benchmark case, the FEM warpage results are found to be within 0.1% of the theoretical solution for the bimaterial plate. Therefore, the accuracy of the FEM solution is considered to reliably predict the 3D vertical deformation of a multi-layered, plate-like structure with elastic properties. The simulations assume an isothermal temperature drop from 170 o C to 5 o C for the randomly generated PBGA packages. The stress-free reference temperature for the, attach, and substrate materials is taken to be 150 o C. The reference temperature for the mold encapsulation is 170 o C. 109

11 For assessing the accuracy of dual-curvature warpage prediction, the entire warpage profile is analyzed and not only the corner warpage. The warpage values at nine points are selected as shown in Figure 5. The mean warpage value at points A through C is called the package-edge warpage. Likewise, the mean warpage value at points D through F is termed the -edge warpage, and the mean warpage value at points G through I is called the inner warpage. Points G through I are placed mid-way between the edge and the package center, while the path containing points B, E, and H is mid-way between paths P ctr and P diag. Table 3 gives the error assessment results in terms of the RMSE and scaled RMSE, given by equation (11). The standard deviation of the package-edge warpage values from the error assessment data set is 31 µm. The dualcurvature approach is observed to be less accurate even in relative terms for points along the package edge as seen by comparing the scaled RMSE values. The scaled RMSE of 5% for the package-edge warpage illustrates that, in general, the package-edge warpage has an error that is approximately 0.5 times the standard deviation of the warpages from the data set. The RMSE itself shows that the absolute value of the warpage prediction error along the outer edge of the PBGAs is approximately 7.9 µm. Table 4 compares the dual-curvature approach with the earlier techniques termed the -only and -line approaches. Only errors collected for the package-edge warpage are compared because warpage values for points D through I are the same for all methods since all methods assume the same -area curvature. The -only results show that extrapolating the curvature to the end of the package will typically cause very poor predictive quality. Although the -line technique is a much better approximation than the -only approach, the dual-curvature approach is a significant improvement upon the -line approach, reducing the error of the -line approach by approximately one-half. For a visual representation of the type of predictive accuracy afforded by the dualcurvature approach, the warpage profiles from the worst and best cases of the error assessment data set are provided in Figure 6. The criterion for deciding the quality of the warpage estimate for each package is the RMSE of the dual-curvature predictions from all nine points divided by the package warpage, δ, as defined from Figure. As such, an estimate of the error across the entire lateral area is scaled to the overall warpage magnitude. Using this criterion, the packages from runs 17 and 45 of the data set are chosen as the representative worst and best cases, respectively. The parametric values for these two packages are given in Table A of the Appendix. To systematically analyze the effect of main package characteristics on the accuracy of the dual-curvature approach, the variables: κ κ mld, h tot, and L mld /L tot are linearly regressed against the error criterion for quality warpage prediction which is calculated for each of the 50 packages in the error assessment data set. Here, κ and κ mld are determined from appropriate use of equations (3), h tot is the total thickness of the package, and L tot is the sum of L and L mld. Using a statistical t-test, only the factors κ κ mld and h tot are found to be statistically relevant. The following equation is an estimate of the quality of prediction error metric discussed above using multivariate linear regression, 110

12 Table. Material Properties and Geometry of PBGA Simulations for Error Assessment Data Set. Parameter Mold Encapsulant Die Die Attach Substrate Min Max L j (mm)* h i (mm) E i (GPa) α i (x10-6 / o C) ν i * Length values are in terms of the half-lengths, L and L mld, along the center-line path, P ctr, from Figure 1 A B C D E F G H I Figure 5. Quarter-model of a PBGA depicting 9 points for error assessment of the dual-curvature warpage profile 111

13 Table 3. Error Summary of Dual-Curvature Approach from 50-Run Error Assessment Data Set Warpage Locations RMSE (um) S. Dev. (um) Scaled RMSE Package edge Die edge Inner Table 4. Comparison of Warpage Prediction Techniques Using Package-Edge Warpage Prediction Technique RMSE (um) S. Dev. (um) Scaled RMSE Die-Only Die-Line Dual Curvature RMSE δ A I ( κ ) ( ) κmld 0.135htot + 586( κ κmld )h tot κ κmld, (1) where RMSE A-I / δ = quality metric of predictive accuracy for the warpage profile of a given package, RMSE A-I = RMSE of points A through I (see Figure 5) of the dual-curvature approach for a given package, δ = package warpage (as defined in Figure ) from the three-dimensional FEM simulation. Equation (1) shows that the most important parameter causing error in the dual-curvature approach is the difference in the inherent thermomechanical curvatures between the and mold areas. In general, as a package becomes thicker and as the inherent curvatures of the two regions become more significantly different, the assumption of thermomechanical independence between the two regions becomes compromised. It is recommended not to use the dual-curvature approach if equation (1) results in a value of RMSE A-I / δ > 0.5. This condition occurs in 9 of the 50 runs of the error assessment data set. Of these nine runs, the mean absolute error at the package edge is 11.7 µm. For packages where equation (1) calculates RMSE A-I / δ to be less than 0.5, the mean absolute package-edge error is only 3.8 µm. Equation (1) is useful for ascertaining the projected accuracy of the dual-curvature approach on a case-by-case basis. 11

14 z (µm) D-C approach 3D Simulation 0 Run Run s (mm) Figure 6. Diagonal warpage profiles representing the worst (run 17) and best (run 45) cases for the dual-curvature approach from the error assessment data set 9. Conclusions Microelectronic packages such as PBGAs typically have high residual stress arising from the manufacturing process. The thermomechanical warpage of microelectronic packages is caused by thermal stress due to the CTE mismatch of the vertically asymmetric structure. Package warpage can be considerable and may cause numerous reliability concerns. Physical observations of different packages illustrate the need to consider the curvature of the mold area outside of the area in any theoretical modeling of package warpage. Therefore, an analytical warpage prediction technique called the dual-curvature approach is described which individually calculates the inherent curvatures of the and mold regions and then creates a warpage profile by twice integrating the curvature function. The dualcurvature warpage profile for square, single- 113

15 packages is a series of two parabolas that are joined at the -mold interface with the same slope and vertical deflection. This paper shows how the thermomechanical warpage of packages consisting of temperaturedependent elastic materials with differing stress-free temperatures can be predicted using the dual-curvature approach. A detailed assessment of the dual-curvature accuracy for warpage prediction of single- PBGAs is ascertained relative to two earlier techniques. The dual-curvature approach is tested against a data set constructed of 50 randomly generated, three-dimensional, FEM simulations of PBGAs, providing a range of packages having diverse material properties and structures. The dual-curvature approach is significantly more accurate than the previous approaches and is able to reduce the warpage prediction error by almost one-half relative to the best of the previous techniques. After a statistical error analysis, it is shown that the dual-curvature approach is most accurate for predicting the warpage of packages that are thin and whose and mold-area curvatures are not significantly different. A quality of prediction error metric estimate is provided as a projection of the accuracy of the dualcurvature approach for use on a case-by-case basis. In summary, the dual-curvature approach is shown to provide fairly accurate predictions of the thermomechanical warpage of square, single- packages with two symmetric, lateral regions such as PBGAs consisting of temperature-dependent elastic materials. In addition, the simplicity of the dual-curvature methodology enables its extension to packages having diverse geometries, allowing a rapid, approximate prediction of the warpage profile of almost any package in any lateral direction. 114

16 Appendix Table A1. Parametric Values for Curvature Plot of Figure 4 Parameter Mold Die Die Substrate Enc. Attach L j (mm)* N/A N/A h i (mm) E i (GPa) α i (x10-6 / o C) ν i *Note: the PBGA is cooled to 5 o C with the reference temperature of the, attach, and substrate being 150 o C and the reference temperature of the mold encapsulation being 170 o C. Table A. Parametric Values for Error Assessment Cases of Figure 6 Parameter Run-17 Package Run-45 Package L mld (mm) h 1 (mm) E 1 (GPa) α 1 (x10-6 / o C)

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