A solder ball with diameter 𝜑 has volume

When reflown onto two pads without distortion the description of the volume V*bump* reads:

Where K is the bond height for the purely spherical bump and p stands for the pad radius. The surface S*bump* is given as

When weight is added the bump is compressed in the z direction. For modelling purposes we assume an elliptic model.

The volume of this shape is given as

Where a is defined as

The surface is given as:

where:

Upon insertion of all variables into the formula for S*bump* one obtains an expression for the surface energy E*dK* as function of the deviation dK from the equilibrium bond height K:

Where T*0,0* is the surface of the purely spherical bump:

The force constant *Fc* of this spring is deduced from:

*Fc* can be derived as:

Where* j* stands for the ratio *p/K* of the pad radius p and the bond height K.

When a bump is deformed only in the in-plane direction X by a small amount X. The amount of displacement is taken inversely proportional to the cross section of the bump

The excursion *u(z)* of the central axis is given as:

where X is the lateral amplitude. It can be shown that the total surface energy E*x* can described as the sum of the undisturbed part T*0,0* and a contribution due to deflection T*2,0* :

From which the force constant for returning to the central equilibrium position can be derived.

When weight is combined with displacement in the horizontal plane we obtain the following result for the total surface energy:

From this formula it can be seen that that is a coupling between the restoring forces in the vertical and horizontal direction. Compared to the other terms T*2,1* is very small.

**Author**

C. van Veen

Read the pdf file **here**.

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