# Determination of the Equivalent Thickness of a Taiko Wafer Using ANSYS Finite Element Analysis
## 5 ANSYS Finite Element Model of an FSM-Taiko Wafer in the Spherical Case  
Along with the analytical approach reported in the previous section, an investigation of the FSM-taiko wafer has been carried out by realizing the finite element method (FEM) simulations with the software tool ANSYS(r) Mechanical Enterprise 2022/R2. The geometry of the system is the same as the one reported in Figure 1. The mechanical behavior of a system consisting of a 200 mm (8\({}^{\prime\prime}\)) taiko silicon wafer, coated with a 4.5 um Al metal layer on the front side, has been simulated in the spherical case. The simulations conditions were the following. Mesh preferences were set as nonlinear mechanical. A regular radial mesh was optimized by dividing the geometry model of the silicon wafer and Al layer into 12 parts resulting from the intersections of three concentric rings with four quadrants, for a total of 28 parts, by considering both the silicon substrate and the aluminum layer. Moreover, the taiko wafer was laid on four supports, symmetrically positioned at the edge. Each support consists of an elastic facet having dimensions of 4 mm \(\times\) 3 mm, whose foundation stiffness was set equal to 1 N/mm\({}^{3}\). The face-to-face contacts between the silicon and aluminum layers were set as bonded with a tolerance of 10\({}^{-3}\) mm. In the FEM analysis the physics chosen was the Static Structural with physics type structural and solver target Mechanical APDL. A direct solver type was used by setting the large deflection to on, and the weak springs condition has been set to on. In Figure 2a,b a snapshot of the top plan-view and rear plan-view of the simulated taiko wafer film, with the ring region and the four supports highlighted, is shown. In Figure 2c a magnification of the pin region is reported. Figure 2: (**a**) Top plan-view of the simulated taiko wafer, with the 12 regions dividing the metal layer; (**b**) Rear plan-view of the simulated taiko wafer, with the 16 regions dividing the substrate layer and the ring region. In the picture the four pins are highlighted in green; (**c**) Magnification of a pin region; (**d**) Mesh convergence investigation as a function of the size of the planar cell element. In the insight, top plan-view of the simulated taiko wafer, with planar mesh elements; (**e**) Mesh convergence investigation as a function of the number of divisions “slicing” the substrate and ring regions, respectively. In the insight, the section of the substrate and ring region with eight divisions each, is reported; (**f**) Mesh convergence investigation as a function of the number of divisions “slicing” the metal layer. In the insight, the section of the metal layer with four divisions is reported. The silicon substrate (001) has been simulated by taking into account of its elastic constants opportunely chosen according to the face orientation to reproduce a [PERSON]'s modulus of \(E_{\mathrm{Si}}\) = 130 GPa and [PERSON]'s ratio and \(\
u_{\mathrm{Si}}\) = 0.27, respectively. The Al layer has been considered as an isotropic material, with a Young's modulus of \(E_{\mathrm{Al}}\) = 68 GPa and \(\
u_{\mathrm{Al}}\) = 0.33, respectively. The coefficients of thermal expansion (CTEs) of silicon and aluminum have been gained from a set of values tabulated at specific temperatures and then interpolated for intermediate values. For each simulation a thermal load was applied to the whole system, by considering the stress-free condition at the higher nominal temperature T of the environment. The FSM-taiko wafer was cooled down from the nominal temperature T in \({}^{\circ}\)C to the temperature of 25 \({}^{\circ}\)C. Temperatures parametrized from 30 \({}^{\circ}\)C up to 170 \({}^{\circ}\)C have been simulated. A mesh converge study has been carried out and reported in Figure 2d-f. All the 28 regions were divided into a set of cells characterized by a planar size in the order of millimeters and by two edge divisions determining a \"slicing\" of the metal layer and the set formed by the ring region and the substrate region.