# Warpage Behavior on Silicon Semiconductor Device:
## 4 Model for Warpage Calculation
### Finite Element Model for Warpage Calculation  
A Finite Element Model is developed with Comsol Multiphysics to calculate the warpage variation, curvature and mechanical stress due to temperature variation, inside the range between 50-90 \({}^{\circ}\)C, in which warpage variation is found linear. It is created a 3D model, considering a simplified stack. Modeled structure is a 50 \(\times\) 10 \(\times\) 0.75 mm beam, made by silicon substrate (thickness 730 \(\upmu\)m) and ECD copper layer (thickness 20 \(\upmu\)m), neglecting TiW (0.3 \(\upmu\)m) and seed Cu (0.2 \(\upmu\)m) layers. These are neglected due to their lower thickness. Geometry is discretized with linear hexa mesh, not-coherent between silicon and copper due to the huge thickness difference. Considering that aim of analysis is to estimate stress and deformation inside a well-defined temperature range, materials are imposed \"stress and strain free\" at 50 \({}^{\circ}\)C and then they are heated up to 90 \({}^{\circ}\)C. Cu and Si are joined together using a \"continuity\" contact formulation, which fixes to 0 the relative displacements between Si and Cu nodes in contact. Displacement boundary conditions are imposed on structure vertices in order to allow the beam free expansion along the three directions.  
Material properties, reported in Table 1, are taken from internal material database based on material supplier data and nanoindentation tests to determin the Young modulus [31]. It is assumed material behavior as elastic and neglecting the temperature dependence. Linear elastic warpage and stress behavior is calculated with the customized FE model and two different literature approaches. One is derived from the Timoshenko theory for bi-metal thermostats [24], and the other is the Stoney equation [22], to calculate thermal warpage in thin film. Warpage results are benchmarked with experimental warpage observed in the second cycle during the first heating phase (50-90 \({}^{\circ}\)C). Stress along x direction and warpage variation in temperature range 50-90 \({}^{\circ}\)C are calculated with FEM, considering the system stress and strain free at 50 \({}^{\circ}\)C. The results, shown in Figure 9, highlight stress is uniformly distributed on the strip, compressive on the copper side due to positive temperature variation. Warpage is correlated by means of a parabolic law with strip length.  
The comparison between warpage and stress variation coming from considered methods is resumed in Table 2. FEM warpage is the closest with measurements and underestimates the experimental behavior of about 5%. Comparing between them the calculation methods, FEM predicts the highest warpage and highest stress, and it is the closest to the experimental measured warpage. The quite low difference between the [PERSON] and [PERSON] analytical approaches could be justified due to the fact that [PERSON] formula could be reduced assuming copper thickness much lower than silicon thickness. The difference observed in stress value can be addressed due to the fact that both the Timoshenko  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Material** & **Young’s Modulus** & **CTE** & **Poisson Ratio** \\  & **GPa** & **ppm/K** & **Poisson Ratio** \\ \hline Silicon & 120 & 2.5 & 0.28 \\ Copper & 77.5 & 16.4 & 0.364 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Considered material data for simulation.  
and Stoney approaches neglect the contribution of copper Poisson ratio, as explained in Section 4.1, while finite element model accounts this factor.