#### 6.5.3 Stress Estimate

The grain size determines the final intrinsic stress magnitude (in materials with low adatom
mobility) as can be seen from Fig. 6.4. Bigger grains lead to smaller stress generation and
smaller grains lead to higher stress generation [99]. The resulting grain size and distribution
is affected by several processing and material parameters. However, a process engineer can
control – to some extent – the grain size distribution only by the growth rate and the
nucleation distribution (rate). Thus, the intrinsic stress can be engineered, in principle,
solely by those two parameters. All processing steps after deposition have been neglected for
this study, which may also alter the microstructure of the material (e.g. annealing). A
summary of the models which discuss those processes can be found in a previous work
[99].

The results obtained from the simulation were compared to the stress measurements of
tungsten films (low adatom mobility material)[85] as depicted in Fig. 6.13. The mean of the
distribution is the estimate for the residual stress and the distribution spreading is the
expected deviation of the estimate.

There is a tendency for the experimental data to lie above the estimated average stress. This
fact can be explained by a possibly undervalued nucleation distribution of the tungsten,
which could lead to overestimated averages. Nevertheless, more measurements are
needed to confirm this tendency. In summary, given a set of growth parameters, the
presented methodology provides the most probable interval, where the average residual
stress due to coalescence can lie, and the most likely value for the residual stress in
low adatom mobility metal thin films. In the case of high mobility materials the
compressive component after coalescence must be introduced, but the procedure remains
unchanged.

The problem of residual stress estimation requires a statistical approach due to the random
character of the coalescence process. Additionally, the obtained confidence interval provides
to the engineer a safeguard for the design of the TSVs. Naturally, further improvements are
needed such as 3D simulations (VW growth and FEM), non-instantaneous island nucleation,
and variable deposition rate.