**I. Introduction**

In a soldering process the assemblies seeks to minimize the amount of free energy. This phenomenon , which takes place when the solder is in the liquid state and can be used to accurately place components is called self-alignment. Selfalignment can be achieved through the use of well defined solder pads and a well defined amount of solder. Next to that lateral dimensions and number of interconnects play a role.

*From a distorted situation to a situation with minimum distortion.*

**II. Physics**

It can be shown that the free energy of a solder bump can be describes as:

A fraction of the energy surface for a realistic set of parameters is given the figure below.

Where γ is the surface energy. Considering the fact that the coupling terms between the Z and XY direction are very weak it is handy to simplify the equation into:

where

An assembled component has 5 degrees of freedom: X, Y, Z, φ, θ. The origin is taken in center of the component. X and Y are the main axes of the component. Z is the vertical direction. φ is the azimuth and θ is the out of plane angle.

**III. Calculation**

We will assume to have a component with N x N bumps at pitch P. Upon summation over all bumps of the component we obtain estimators for the uncertainties in the different degrees of freedom:

These outcomes show that three uncertainties σX, σY and σZ determine the outcome. σX and σY , which are both located in-plane, are determined by the accuracy of the lateral position of the joint. This translates to the difference of the XY coordinates of the location of the center of gravity of the solder pads for both the substrate and the component. For the sake of ease σX and σY are taken identical. σZ is the variation in the Z direction. It arises form inaccuracies in the pad radii and from the variation in solder quantities in the joint. It can be estimated as follows:

The relation between joint height and solder volume Vbump and solder pad radii, p and q, for the lower and upper pad

respectively, is given as:

where

where η is the diameter of the volume equivalent solder ball. The variation σZ in the bump height Zbump as function of the volume is estimated by the following expression:

where

where

**IV. Validation**

σX σY depend on lateral location errors, whereas σZ depends on dimensional variations in the artwork and solder ball volume consistency. This can be seen in the table below:

Factors that contribute to a better result are: enhancing the number of bumps, a “best-in-class” solder ball supplier and a very good lithography process for making substrate and component artwork.

**V. Author**

C. van Veen

Download the pdf file **here**.

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