# Determination of the Equivalent Thickness of a Taiko Wafer Using ANSYS Finite Element Analysis
## 2 Warpage of an FSM Taiko Wafer According to the Theory of Elasticity in the Spherical Case  
In analogy to what reported in [21], we can develop an approximation of the warpage of a front side metalized taiko wafer. Indeed, according to the theory of elasticity (see Landau-Lifshitz [29]), if the taiko wafer is not subject to the force of gravity, the vertical displacement \(\zeta\) with respect to the neutral surface satisfies the biharmonic equation \(\Delta^{2}\zeta=0\), which is valid in the ring region as well as in the thinned wafer region. For a FSM taiko wafer as well, the biharmonic equation has the Michell's solution [31]. However, in the spherical case, as a first approximation, the warpage can be characterized by one curvature \(\kappa_{sub}\) in the substrate region and one curvature \(\kappa\) in the ring region. In detail, in the ring region, \(R_{int}<r\leq R_{ext}\), the simplest solution does not consider the dependence on the angle \(\theta\) and can be written as:  
\[\zeta_{ext}=\frac{\kappa}{2}r^{2}+b+cr^{2}ln\bigg{(}\frac{r}{R_{ext}}\bigg{)} +dln\bigg{(}\frac{r}{R_{ext}}\bigg{)} \tag{3}\]  
In the internal and thinned substrate region, \(0\leq r\leq R_{int}\), the spherical solution of the biharmonic equation is  
\[\zeta_{int}=\frac{\kappa_{sub}}{2}r^{2}+b_{sub} \tag{4}\]  
The curvatures \(\kappa_{sub}\), \(\kappa\) and the parameters \(b_{sub}\), \(b\), \(c\) and \(d\) can be determined from the boundary conditions \(\zeta_{int}(R_{int})=\zeta_{ext}(R_{int})\) and \(\frac{d\zeta_{int}(R_{int})}{dT}=\frac{d\zeta_{ext}(R_{int})}{dT}\). At the interface between the FSM thin film and the wafer substrate, at a distance \(r\) from the center, the taiko wafer is subject to a radial and a circumferential force, whose values per unit length are set equal to \(F_{r}\) and \(F_{\theta}\), respectively (See Figure 1). The wafer is considered supported at the external ring \(R_{ext}\), and \(N_{r}\) and \(N_{\theta}\) are the reaction forces evaluated per unit length (See Figure 1). Because of the mechanical equilibrium, the resultant forces and moments must be equal to zero. In particular, the equilibrium of forces and moments holds locally for the thin film deposited in the substrate region:  
\[\left\{ \begin{array}{c}F_{r,\theta_{sub}}+F_{r,\theta_{film}}=0\\ \\ M_{r,\theta_{sub}}+M_{r,\theta_{film}}=0\end{array} \right. \tag{5}\]  
Additionally, the reaction moments acting on the ring at \(R_{ext}\) are in equilibrium with those of the substrate. \[M_{r,\theta_{ring}}\Big{|}_{R_{ext}}=0 \tag{6}\] \begin{table}
\begin{tabular}{c c c} \hline
**Symbol** & **Quantity** & **Typical Value** \\ \hline \(h_{s}\) & Substrate Thickness (\(\upmu\)m) & 100 \\ \hline \(H-h_{s}\) & Step height (\(\upmu\)m) & 450 \\ \hline \(h_{f}\) & Front Side Metal thickness (\(\upmu\)m) & 4.5 \\ \hline \(R_{ext}-R_{int}\) & Ring width (mm) & 3.7 \\ \hline \end{tabular}
\end{table} 
Table 1: Look-up table reporting the typical value of the geometrical properties of the investigated FSM-taiko wafer. By similar reasoning to the case of [21], the moments per unit length in the substrate region and ring region can be written as:  
\[\left\{