# Warpage Behavior on Silicon Semiconductor Device:
## 4 Model for Warpage Calculation  
### Literature Analytical Approaches  
Two analytical approaches aimed to predict stress and warpage are described in this section. [PERSON]'s equation [22; 27; 28] are commonly used as reference. This formula is developed for \"membrane-like\" geometries made by two materials, such as the system made by semiconductor wafer and ECD metal, in which one layer (in our case, ECD Cu) is much thinner than other. According to [PERSON], thermal stress \(\sigma\) of thin film can be written as:  
\[\sigma=\frac{E_{s}}{6(1-\upsilon_{s})}\frac{h_{s}^{2}}{h_{f}}\left(\frac{1}{R }-\frac{1}{R_{0}}\right), \tag{4}\]  
in which \(E_{s}\) the Young's modulus, \(\upsilon_{s}\) the Poisson's ratio and \(h_{s}\) the thickness of the substrate, \(h_{f}\) is the thickness of the film, and \(R\) and \(R_{0}\) the curvature radii of the substrate after and before film deposition. Considering that thermal stress is given by \(\sigma=E_{f}(\alpha_{s}-\alpha_{f})\Delta T=E_{f}\Delta\epsilon_{t}\), in which \(\alpha_{s}\) and \(\alpha_{f}\) are the coefficient of thermal expansion, respectively, for substrate and film. Curvature could be calculated from Stoney as follows:  
\[\frac{1}{R}-\frac{1}{R_{0}}=\frac{6E_{f}(1-\upsilon_{s})}{E_{s}}\frac{h_{f}}{ h_{s}^{2}}\Delta\epsilon_{t}. \tag{5}\]  
The application of Stoney approach requires materials must be homogeneous and with linear elastic behavior. Considering the complex non-linear Cu behavior during annealing, Stoney results are not reliable to predict stress and warpage during the entire annealing process. The second considered approach to numerically estimate stress comes from Timoshenko formulation for bi-material thermistors [21] that is extended for three and more materials [29; 30]. This theory permits to translate the thermal input into a set of equivalent forces and momentum; therefore, it is possible to apply the classic theory for an elastic beam. Consider a beam consisting of two layers of different metals, welded together and subjected to a quasi-static temperature change from \(T_{0}\) to \(T\). If the linear thermal expansion coefficients of these two materials are different from each other, heating and cooling will generate the lamina bending. Let be \(\alpha_{s}\) and \(\alpha_{f}\) the linear thermal expansion coefficients of substrate ans film, respectively. Furthermore \(E_{s},E_{f}\) and \(h_{s},h_{f}\) represent the elasticity modules and the thickness of the two materials. For simplicity, the beam has a unit width. Considering that \(I_{f}\) and \(I_{s}\) are, respectively, the inertia momentum for film and substrate, curvature \(1/R\) can be expressed as in Equation (6):  
\[1/R=\frac{(\alpha_{s}-\alpha_{f})\Delta T}{\frac{h_{f}+h_{s}}{2}+\frac{2(E_{f }I_{f}+E_{s}I_{s})}{h_{f}+h_{s}}\left(\frac{1}{E_{f}\alpha_{s}}+\frac{1}{E_{s} \alpha_{s}}\right)}. \tag{6}\]  
Considering a beam of length \(l\), much smaller than the radius of curvature, the maximum deflection delta can be well approximated as:  
\[\delta\approx 1/8\ \frac{l^{2}}{R}. \tag{7}\]  
The stress state within each lamina will be the composition of the stress induced by the bending moment \(M\), obviously dependent on the distance from the neutral axis of the under beam, and from the axial load P, constant on each of the two parts. Bending moment can be obtained according to \(M=\frac{EI}{R}\), in which \(I\) is the moment of geometric inertia of the cross section and \(R\) the radius of curvature of the inflexed beam. Normal load can be expressed from the equation related to flexural moment equilibrium:  
\[Ph/2=E_{s}I_{s}/\rho+E_{f}I_{s}/\rho. \tag{8}\]Considering the cooling case, the normal load expressed in Equation (8) acts as compressive stress in the material with the lower coefficient of thermal expansion (silicon substrate), as tensile stress where the CTE is higher (thick copper film).