# Determination of the Equivalent Thickness of a Taiko Wafer Using ANSYS Finite Element Analysis
## 6 ANSYS Results  
For a given nominal temperature T, we measured the equivalent von Mises stress of the FSM layer and the warpage of the taiko wafer. In particular, the warpage was probed by determining the directional deformation along the z direction of the whole system, as well as on a set of three pairs of paths extending along the diameter of the wafer. Each pair of these pathswas oriented according to two perpendicular directions, the x and y direction of the coordinate system. Of the three pairs, the first pair of paths was set at the top of the Al layer, the second pair of paths at the interface between the Si substrate and the Al layer and the third pairs at the bottom of the central region of the taiko wafer. The value of the warpage was recorded for each nominal temperature and the curvature calculated. In Figure 3a we report the distribution of the directional deformation along the z direction for a taiko wafer 200 \(\upmu\)m thin, simulated at a nominal temperature of T = 50 \({}^{\circ}\)C. All the other quantities have been set to the typical values. In Figure 3b we report the projection of the distribution of the directional deformation in the z direction for the same 200 \(\upmu\)m thin taiko wafer, simulated at a nominal temperature of T = 50 \({}^{\circ}\)C. The magnification is 100\(\times\). In Figure 4 from a to f we report the indication of the three pairs of x- and y-paths for this specific case. In Figure 5 we report the directional deformation in the z direction for each pair of paths 1x, 1y, on the top of the metal layer, 2x, 2y, at the interface between the metal layer and the semiconductor, 3x and 3y, at the bottom of the central taiko semiconductor region, respectively, reported as a function of the coordinate x (y) along the path itself, for the case of the FSM-taiko wafer that is 200 \(\upmu\)m thick. In general, we observed that for each pair of paths there is no difference in the curvatures in the x- and y-directions, confirming that the wafer has warped symmetrically. Moreover, all the paths provide the same deformation. The curvature at the center of the wafer has been determined from a parabolic approximation and the trend according to the theory reported for a comparison. We can observe as the parabolic trend fits well with the directional deformation of the taiko wafer. Figure 4: (**a**) Snapshot of the first path used to probe the z-deformation along the x-direction, on the top of the Al layer; (**b**) Snapshot of the first path used to probe the z-deformation in the y direction, on the top of the Al layer; (**c**) Snapshot of the 2 nd path used to probe the z-deformation in the x direction, at the interface of the Al/Si layers; (**d**) Snapshot of the 2 nd path used to probe the z-deformation in the y direction, at the interface of the Al/Si layers; (**e**) Snapshot of the 3 rd path used to probe the z-deformation in the x direction, at the bottom of the central substrate region of the taiko wafer; (**f**) Snapshot of the 3 rd path used to probe the z-deformation in the y direction, at the bottom of the central substrate region of the taiko wafer. Figure 3: (**a**) Distribution of the directional deformation in the z direction for a taiko wafer 200 \(\upmu\)m thin, simulated at a nominal temperature of T = 50 \({}^{\circ}\)C; (**b**) Projection of the distribution of the directional deformation in the z direction for a taiko wafer 200 \(\upmu\)m thin, simulated at a nominal temperature of T = 50 \({}^{\circ}\)C, the magnification is 100\(\times\). In Figure 6 the curvature as a function of the stress (von Mises) for a taiko wafer having \(h_{\text{s}}=200\)\(\upmu\)m and the other quantities fixed to the typical value reported in Table 1 has been reported. 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, defined by the ratio \(\kappa/\sigma\) and hence the curvature of the taiko wafer, is lower with respect to the Stoney case.