# Determination of the Equivalent Thickness of a Taiko Wafer Using ANSYS Finite Element Analysis
## 4 Evaluation of \(z_{B}\)  
To gain the mechanical stress \(\sigma_{f}\), of the thin metal film, from measurement of warpage we need to evaluate \(z_{B}\). This quantity can be calculated from the equilibrium of the forces. In the substrate region by following [33], we can write the total force acting in the substrate region in the radial direction as:  
\[\frac{E_{f}}{1-v_{f}}\int_{h_{1}-t_{f}}^{h_{1}}zdz+\frac{E_{s}}{1-v_{s}}\int_{ -h_{2}}^{h_{1}-t_{f}}zdz=0 \tag{14}\]  
where \(h_{1}=h_{s}+z_{B}+h_{f}\) and \(-h_{2}=z_{B}\). \[\frac{M_{sub}^{T}}{\kappa}=E_{f}\int_{h_{s}+z_{B}}^{h_{s}+z_{B}+h_{f}}z^{2}dz+E _{s}\int_{z_{B}}^{h_{s}+z_{B}}z^{2}dz=\frac{E_{f}}{3}\bigg{[}\left(h_{s}+z_{B} +h_{f}\right)^{3}-\left(h_{s}+z_{B}\right)^{3}\bigg{]}+\frac{E_{s}}{3}\Big{[} \left(h_{s}+z_{B}\right)^{3}-z_{B}{}^{3}\Big{]} \tag{15}\]  
At a given distance \(r\) we can write,  
\[\frac{\partial^{2}\zeta_{int}}{\partial r^{2}}\Bigg{[}\frac{E_{f}}{1-v_{f}} \int_{h_{s}+z_{B}}^{h_{s}+z_{B}+h_{f}}zdz+\frac{E_{s}}{1-v_{s}}\int_{z_{B}}^{h_ {s}+z_{B}}zdz\Bigg{]}dw \tag{16}\]  
and the total force acting on the substrate region is  
\[d\theta\frac{\partial^{2}\zeta_{int}}{\partial r^{2}}\Bigg{[}\frac{E_{f}}{1-v_ {f}}\int_{h_{s}+z_{B}}^{h_{s}+z_{B}+h_{f}}zdz+\frac{E_{s}}{1-v_{s}}\int_{z_{B}}^ {h_{s}+z_{B}}zdz\Bigg{]}\int_{0}^{R_{int}}rdr \tag{17}\]  
being \(dw=rdrd\theta\). In the ring region we need to evaluate the quantity:  
\[d\theta\frac{E_{s}}{(1-v_{s}^{2})}\int_{-H+z_{B}+h_{s}}^{z_{B}+h_{s}}zdz\int_{ R_{int}}^{R_{ext}}\bigg{(}\frac{\partial^{2}\zeta_{ext}}{\partial r^{2}}+\frac{v_{s}}{r} \frac{\partial\zeta_{ext}}{\partial r}\bigg{)}rdr \tag{18}\]  
Hence, in the case of the taiko wafer, the total force acting on the taiko wafer, in the radial direction, equals zero, being in equilibrium:\[\frac{\partial^{2}\zeta_{int}}{\partial r^{2}}\Big{|}_{R_{int}}\left[\frac{E_{f}}{ 1-v_{f}}\int_{h_{s}+z_{B}}^{h_{s}+z_{B}+h_{f}}zdz+\frac{E_{s}}{1-v_{s}}\int_{z_{ B}}^{h_{s}+z_{B}}zdz\right]\int_{0}^{R_{int}}rdr+\frac{E_{s}}{\left(1-v_{s}^{2} \right)}\int_{-H+z_{B}+h_{s}}^{z_{B}+h_{s}}zdz\int_{R_{int}}^{R_{int}}\left( \frac{\partial^{2}\zeta_{ext}}{\partial r^{2}}+\frac{v_{s}}{r}\frac{\partial \zeta_{ext}}{\partial r}\right)rdr=0 \tag{19}\]  
In the limit \(H\to h_{s}\), \(R_{ext}=R_{int}\), Equation (19) reduces to the canonical case of a flat disk wafer:  
\[\left[\frac{E_{f}}{1-v_{f}}\int_{h_{s}+z_{B}}^{h_{s}+z_{B}+h_{f}}zdz+\frac{E_{ s}}{1-v_{s}}\int_{z_{B}}^{h_{s}+z_{B}}zdz\right]\int_{0}^{R_{int}}rdr=0 \tag{20}\]  
where \(z_{B}\) is equal to  
\[z_{B}=-\frac{\frac{E_{f}h_{f}\left(2h_{s}+h_{f}\right)}{2\left(1-v_{f}\right) }+\frac{E_{s}}{2\left(1-v_{s}\right)}h_{s}^{2}}{\frac{E_{f}}{\left(1-v_{f} \right)}h_{f}+\frac{E_{s}}{\left(1-v_{s}\right)}h_{s}} \tag{21}\]  
which is the value of the limit \(\underset{H\to h_{s}}{lim}z_{B}\) if \(H\to h_{s}\). Moreover, if \(h_{f}=0\), \(\underset{H\to h_{s}}{lim}z_{B}=-\frac{h_{s}}{2}\). To evaluate \(z_{B}\) in the general case, it is necessary to combine, the modified Stoney formula for the taiko wafer (Equation (9)) and calculate the value of the integral (see Appendix B of [21] according to the function \(\zeta_{ext}(r)\) evaluated in Appendix A in [21]. By doing so, a linear equation is obtained, whose solution provides the value of the neutral axis \(z_{B}\)  
\[\frac{\partial^{2}\zeta_{int}}{\partial r^{2}}\Big{|}_{R_{int}}\left[\frac{E_{ f}}{1-v_{f}}\int_{h_{s}+z_{B}}^{h_{s}+z_{B}}zdz+\frac{E_{s}}{1-v_{s}}\int_{z_{B}}^{h_{s}+ z_{B}}zdz\right]\int_{0}^{R_{int}}rdr+\frac{E_{s}}{\left(1-v_{s}^{2}\right)}\int_{-H+ z_{B}+h_{s}}^{z_{B}+h_{s}}zdz\int_{R_{int}}^{R_{ext}}\left(\frac{\partial^{2} \zeta_{ext}}{\partial r^{2}}+\frac{v_{s}}{r}\frac{\partial\zeta_{ext}}{ \partial r}\right)rdr=0 \tag{22}\]  
where \(\zeta_{ext,\ Norn}\) is the normalized function reported in Equation (A1) of Appendix A of ref. [21].