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.
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