# Determination of the Equivalent Thickness of a Taiko Wafer Using ANSYS Finite Element Analysis
## 6 ANSYS Results
Figure 5: Directional deformation along the z-direction for each pair of paths in the x and y directions, respectively, at the top of the Al metal layer, at the Si/Al interface and at the bottom of the central Si substrate of the taiko wafer, reported as a function of the coordinate x (y) along the path, for the case of FSM-taiko wafer that is 200 \(\upmu\)m thick. All the paths provide the same deformation. The curvature has been determined from a parabolic approximation and the trend according to the theory reported for comparison. Figure 6: Curvature as a function of the stress on a taiko wafer having \(h_{\text{s}}=200\)\(\upmu\)m and the other quantities fixed to the typical value reported in Table 1. In the graph, the line determined by the slope of the Stoney equation and the tangent line to the curvature vs. stress curve are shown. The slope and hence the curvature of the taiko wafer is lower with respect to the Stoney case. In Figure 7 we report the curvature as a function of the stress for a set of seven taiko wafers having \(h_{s}\) ranging across the interval 100-400 \(\upmu\)m and the other quantities fixed to the typical value reported in Table 1. In particular, the height \(H-h_{s}\) was set to a constant value of 450 \(\upmu\)m. For each \(h_{s}\), we calculated the slope of the tangent line to the origin of the curve. In Figure 8, the values of these slopes (\(\kappa/\sigma\)) have been reported as a function of the thickness \(h_{s}\) of the substrate in the central region of the taiko wafer. For comparison, we also reported the slope determined by the Stoney equation for a flat wafer. As can be observed, the slope, and hence the curvature as a function of the stress, though increasing with the decrease in the thickness of the wafer, is lower with respect to the case of a flat substrate or a Stoney case, because of the mitigation effect determined by the taiko wafer. To complete our investigation on the FSM-taiko wafer we calculated the slope as a function of the thickness as provided by the analytical developments reported in the first part of this work, by Equations (26) and (34). The results have been compared with those provided by ANSYS and reported in the same Figure 8. The values of the slopes resulting from the analytical calculations are practically the same; and hence there is one value for the \(z_{B}\) value. Moreover, though the calculated values are well below the value provided by ANSYS, it is possible to gain the order of magnitude of the slope and the trend. Finally, it is worth observing that by adding the two contributions of the slopes \(\frac{\kappa_{H}}{\sigma_{f}}\) and \(\frac{\kappa_{B}}{\sigma_{f}}\), determined by the developed theory, a good agreement with the ANSYS simulation experiment was found. This indicated that the theory explains well the physics which is behind the mitigation of the warpage of a taiko wafer. Lastly, in Figure 9, we report the equivalent thickness calculated according to Equation (28), of the wafer as calculated from the theory and from the ANSYS results as a function of \(h_{s}\). These results show that an FSM taiko wafer warps as an equivalent standard wafer whose substrate thickness is higher. Figure 7: Curvature as a function of the stress for a taiko wafer with \(h_{s}\) ranging across the interval of 100–400 \(\upmu\)m and the other quantities fixed to the typical value reported in Table 1.