Ping S. Lee

Prediction of Solder Joint Geometries in Array-Type Interconnects

An approximate mathematical model is developed for predicting the shapes of solder joints in an array-type interconnect (e.g., a ball-grid array or flip-chip interconnect). The model is based on the assumption that the geometry of each joint may be represented by a surface of revolution whose generating meridian is a circular arc. This leads to simple, closed-form expressions relating stand-off height, solder volume, contact pad radii, molten joint reaction force (exerted on the component), meridian curvature, and solder surface tension. The qualitative joint shapes predicted by the model include concave (hourglass-shaped), convex (barrel-shaped, with a truncated sphere as a special case), and truncated-cone geometries. Theoretical results include formulas for determining the maximum and minimum solder volumes that can be supported by a particular pair of contact pads. The model is used to create dimensionless plots which summarize the general solution in the case of a uniform array (i.e., one comprising geometrically identical joints) for which the contact pads on the component and substrate are of the same size. These results relate the values of joint height and width (after reflow) to the solder joint volume and the molten-joint force for arbitrary values of the pad radius and solder surface tension. The graphs may be applied to both upright and inverted reflow, and can be used to control stand-off for higher reliability or to reduce bridging and necking problems causing low yields. A major advantage of the model is that it is numerically efficient (involving only simple, closed-form expressions), yet generates results that are in excellent agreement with experimental data and more complex models. Thus, the model is ideally suited to performing parametric studies, the results of which may be cast in a convenient form for use by practicing engineers. Although in the present paper the array is assumed to be doubly-symmetric, i.e., possess two orthogonal planes of symmetry, the model may be extended to analyze arrays of arbitrary layout. The motivation for predicting joint geometries in array-type interconnects is two-fold: (1) to achieve optimal joint geometries from the standpoint of improved yield and better reliability under thermal cycling and (2) to take full advantage of the flexibility of new methods of dispensing solder, such as solder-jet and solder-injection technologies, which enable the volume of each individual joint to be controlled in a precise manner. Use of dispensing methods of these types permits the solder volumes in the array to be distributed in a non-uniform manner. Results such as those presented here (in combination with appropriate fatigue studies) can be used to determine the optimal arrangement of solder volumes.

Finite Element Method for Predicting Equilibrium Shapes of Solder Joints

This paper discusses the development and application of a finite element method for determining the equilibrium shapes of solder joints which are formed during a surface mount reflow process. The potential energy governing the joint formation problem is developed in the form of integrals over the joint surface, which is discretized with the use of finite elements. The spatial variables which define the shape of the surface are expressed in a parametric form involving products of interpolation (blending) functions and element nodal coordinates. The nodal coordinates are determined by employing the minimum potential energy theorem. The method described in this paper is very general and can be employed for those problems involving the formation of three dimensional joints with complex shapes. It is well suited for problems in which the boundary region is not known a priori (e.g., “infinite tinning” problems). Moreover, it enables the user to determine the shape of the joint in parametric form which facilitates meshing for subsequent finite element stress and thermal analyses.

Effect of Chip and Pad Geometry on Solder Joint Formation in SMT

An analytical model of solder joint formation during a surface mount reflow process is developed for two-dimensional fillets whose flow may be restricted due to “finite” metallizations on a leadless component and the printed circuit board. Although these height and length constraints on the fillet geometry may result in obtuse contact angles, the solution is obtained in the form of an explicit integral, similar to that previously derived by the authors for the case of acute contact angles. This solution may also be recast into the form of elliptic integrals of the first and second kinds, thereby permitting one to evaluate the fillet geometry using mathematical tables or special function software, if desired, rather than resorting to a computer-based numerical quadrature. In addition an approximate zero-gravity solution is given by means of simple closed-form expressions relating the height, length, contact angles, and cross-sectional area of the fillet. Numerical results generated by implementing the “exact” integral solution for the joint profile are given in the form of dimensionless plots, relating fillet geometry to the solder properties (surface tension and density), amount of solder, chip height, and pad length. Also presented in dimensionless form are the approximate results from the zero-gravity model, which are independent of solder properties, yet are of sufficient accuracy for “small” joints. Because of their dimensionless nature, the results of the present paper may be of maximum utility to process engineers aiming to achieve desired joint geometries (e.g., to maximize fatigue life or to eliminate bridging problems), or to board designers responsible for selecting efficient footprint patterns to maximize board density. Models of solder joint formation, such as the one presented here, may be of most value when used in conjunction with stress analysis packages (e.g., finite element programs) and appropriate fatigue models. In this way an integrated approach to the design of solder joints and circuit boards may be taken, resulting in improved product reliability and performance.

Improved yield and performance of ball-grid array packages: design and processing guidelines for uniform and non-uniform arrays

Two models are presented for analyzing ball-grid array (BGA) solder interconnects: (a) a model for predicting solder joint geometries after an upright or inverted reflow process and (b) a reliability model for estimating maximum shear strain and fatigue life under a global thermal mismatch load condition. The reliability model includes a newly derived closed-form expression relating the maximum shear strain in a solder joint to the load, material, and joint shape parameters. The models are used to generate design/processing guidelines that may be used to improve the yield of a BGA soldering operation and the in-service reliability of the interconnect. Potential benefits of using a nonuniform array (i.e. one including joints of different size and shape) are explored.

Improved Analytical Estimate of Global CTE Mismatch Displacement in Areal-Array Solder Joints

An analytical expression is derived for determining the maximum solder joint shearing displacement occurring in an a real-array interconnect under global CTE mismatch loading. The result may be viewed as a “load correction factor” to be applied to the commonly used estimate which is based on the free thermal expansion of component and substrate. The new expression for the correction factor includes the following parameters: (a) dimensions and material properties of component and substrate; (b) array size and population; (c) material properties of solder; and (d) geometric parameters of the individual joints. The theoretical result is based on modeling the assembly as two circular elastic disks connected by a shear-type “elastic foundation” whose distributed shear stiffness is related to the joint/array characteristics. The analytical expression and the graphical aids presented herein may provide convenient alternatives to performing time-consuming and expensive finite element “macro-analyses” on the assembly for the purpose of specifying boundary conditions for a subsequent “micro-analysis” on a single joint.

Shearing Deformation in Partial Areal Arrays: Analytical Results

Simple, closed-form expressions, based on elasticity theory, are derived for determining the location and magnitude of the maximum shearing displacement in a partial areal array of solder joints. Both uniform and nonuniform thermal loadings are considered, as is the heterogeneity of the component, which often arises due to different values of coefficient of thermal expansion (CTE) and elastic properties among the module’s constituent materials. The model is based on the following assumptions: (a) the square geometry of the array and component may be replaced with an equivalent axisymmetric geometry; and (b) the stiffness of the solder joints is negligible with respect to that of the component and substrate. The “soft joint” assumption corresponds to low-modulus solders or to thermal excursions occurring at high temperatures or low frequencies, for which significant stress relaxation occurs in the solder. For arrays exhibiting higher stiffness characteristics, the model yields conservative estimates of shearing displacement. Results indicate that, unlike homogeneous-component models under uniform temperature changes, the critical joints are not necessarily at the outer corners of the array. Other candidate locations predicted by the model (and observed in experimental and numerical studies) include the inner corner joints and any joints positioned beneath the die corners. The analytical results, also presented graphically, are found to depend on only three dimensionless parameters: the ratio of inner to outer array dimension, the ratio of die size to outer array dimension, and a “mismatch parameter,” which depends on the material, geometry, and loading characteristics of the problem. The results can be used to quickly determine the location and magnitude of peak shearing displacement in the array, possibly minimizing or eliminating the need to perform expensive and time-consuming finite element macroanalyses on entire assemblies involving hundreds of joints. Thus, the analyst may proceed directly to a detailed finite element microanalysis of the critical joint for fatigue life estimation, using the calculated shearing displacement as a required boundary condition in the finite element model.

A modified finite element method for determining equilibrium capillary surfaces of liquids with specified volumes

This paper describes a modified finite element method (MFEM) for determining the static equilibrium shape of the capillary surface of a liquid with a prescribed volume constrained by rigid boundaries with arbitrary shapes. It is assumed that the liquid is in static equilibrium under the influence of surface tension, adhesion, and gravity forces. This problem can be solved by employing the conventional FEM; however, a major difficulty arises due to the presence of the volume (integral) constraint and usually requires the use of the Lagrange multiplier method, the sequential unconstrained minimization technique, or the augmented Lagrange multiplier method. With the MFEM, the space variables defining the equilibrium surfaces (or curves) are expanded in terms of parametric interpolation functions, which are designed such that the boundary conditions and the integral constraint equation are automatically satisfied during each iteration of a direct numerical search process. Hence, there is no need to include Lagrange multipliers and/or penalty factors and the problem can be treated more simply as one involving unconstrained optimization. This investigation indicates that the MFEM is more efficient and reliable than the other methods. Results are presented for several case study problems involving liquid solder drops. Copyright

An Improved Analytical Model for Time-Dependent Shearing Deformation in Area-Array Interconnects

An analytical model is developed for predicting the time-dependent shearing displacement in area-array solder interconnects due to global CTE mismatch under thermal cycling. As a first step toward incorporating the creep deformation of the solder, the material is modeled as viscoelastic and temperature-independent. This permits one to invoke the correspondence principle of viscoelasticity to map the authors’ previously derived, closed-form solution for an elastic nonprismatic (concave, convex, or cylindrical) Timoshenko beam under shear loading into the associated viscoelastic solution. This leads to general analytical results for the frequency-dependent shear displacement amplitude in the critical joint. The results are expressed conveniently in terms of a ‘‘fullcreep correction factor’’ and a ‘‘frequency correction factor,’’ which explicitly show the effects of the following parameters on the joint deformation: joint shape; array population; array, component, and substrate dimensions; viscoelastic material properties of the interconnect material; elastic properties of the component and substrate materials; and loading frequency. To demonstrate the technique for a particular viscoelastic constitutive law, the solder is assumed to behave elastically under hydrostatic loads and as a viscoelastic Kelvin solid under deviatoric conditions. For this special case the creep portion of the deformation is shown to be dependent upon only two dimensionless parameters: a dimensionless loading frequency and a materialand shape-dependent joint parameter. The results of the study may be useful in identifying design and process modifications that may improve the thermal fatigue life of area arrays. @S1043-7398~00!00404-7#

Prediction of Solder Geometry for an Axisymmetric Through-Hole Joint

Solutions for axisymmetric profiles of solder joints formed between a cylindrical pin and a printed circuit board (PCB) are presented. The dimensionless differential equation governing the formation of the solder joints is developed and then solved numerically for the cases of single upright joints, single inverted joints, and through-hole joints. Results are presented in terms of the following dimensionless parameters: bond number, solder volume, board thickness, and tinning radius. The dimensionless approach makes the results from this study suitable for use in a broader range of applications.

AN APPROXIMATE MODEL FOR PREDICTING THE SIZE AND SHAPE OF WAVE-SOLDERED SURFACE-MOUNT JOINTS

An approximate method is developed to determine the effect of pad dimensions, component dimensions, conveyor angle and solder properties (surface tension and density) on the size and shape of solder joints formed during a wave soldering process. The model uses a dipping analogy to simulate the wave soldering operation, and is thus also applicable to dipping processes for soldering, pre-tinning or bumping. The study represents an extension of previous work by the authors in which the effect of conveyor angle was not included and the exact solution for the molten solder capillary surface was utilized. In the present investigation a constant curvature approximation is introduced in the dipping sequence as well as in the final joint configuration. This reduces the problem to one of a geometrical nature only. The results of the model are presented in dimensionless form, compared with the exact theoretical results for a horizontal conveyor, and compared with data from actual production joints. The fundamental concept upon which the model is based may also be extended to through-hole and leaded surface-mount components.