# Determination of the Equivalent Thickness of a Taiko Wafer Using ANSYS Finite Element Analysis
## 1 Introduction
Indeed, in 2021, [PERSON] and [PERSON] [21] introduced an analytical method to model the bifurcation in back-side metalized taiko wafers. In the following years, with [PERSON], they reported on their investigation of the bifurcation phenomenon in 8\" metalized wafers [22; 23; 24; 25; 26] by means of FEA methods and extended the investigation to understand the correlation between warpage wafer and warpage die [27]. The aim of this work is to compare the analytical description of the warpage occurring in an FSM-taiko wafer with the results obtained by Finite Element Simulations of the same system achieved by means of the ANSYS(r) mechanical solver. In fact, the Stoney equation is not valid for the case of a taiko wafer since we have to consider the presence of the ring. In the current study we aim to expand the analysis of warpage to the case of a taiko wafer that has been metalized on the front side. Specifically, we aim to investigate the warpage of a front side metal (FSM) taiko wafer in the spherical regime and determine the value of the equivalent thickness for an FSM-taiko wafer. The equivalent thickness concept in the context of a taiko wafer involves making modifications to the mathematical Equation (1), to account for the effects of the wafer's physical properties and determine the influence of the ring region on its warpage. This includes considering the wafer as an idealized structure with a constant curvature, despite the presence of a damaged layer due to back-grinding [28]. A proposal for a modified Stoney formula for a taiko wafer can be written as:  
\[\kappa_{Stoney,\ taiko}=\frac{M_{taiko}}{E_{\text{s}}I_{taiko}} \tag{2}\]  
where \(M_{taiko}\) is the bending moment determined by the FSM and \(E_{s}I_{taiko}\) is the flexural rigidity of the FSM-taiko wafer. A rigorous determination of the values for these quantities involves the use of the theory of elasticity [29; 30]. In this work we considered a novel approach, which consisted of developing an analytical model of the curvature determined by the stress produced by an FSM in a taiko wafer and compared the result with a finite element analysis (FEA) method of the same metalized taiko wafer, which was achieved by using the ANSYS mechanical solver. The novelty of the method lies in the fact that developing the equations which rule the curvature of the taiko wafer will allow an understanding of the physics which determine the mitigation of the warpage in these kinds of wafers. To do so, we define first all the quantities which are involved in the description of an FSM taiko wafer. In Figure 1 the schematic of half of the vertical cross-section of an FSM-taiko wafer is reported. A reference frame is fixed at the center of the taiko wafer, with the origin placed on the neutral plane, such that the back side of the wafer is at \(z=z_{B}\) with respect to the neutral plane. In the schematic, \(h_{\text{s}}\) is the thickness of the substrate, \(H\) the height of the ring region, \(R_{int}\) the internal radius and \(R_{ext}\) the external radius, respectively. On the front side of the wafers a thin film, e.g., metal, having a modulus of elasticity \(E_{f}\) and thickness \(h_{f}\), is indicated in yellow. The neutral axis results from the intersection of the neutral plane and the cross-section plane. Once the reference is fixed, \(h_{\text{s}}+z_{B}\) is the distance between the neutral axis (plane) and the interface substrate/thin film, \(h_{\text{s}}+z_{B}+h_{f}\) is the distance between the neutral axis and the FSM thin film top-surface, whereas \(H-z_{B}-h_{\text{s}}\) is the distance of the ring region surface of the taiko wafer with respect to the neutral axis. The height of the step is given by \(H-h_{\text{s}}\) and usually is a fixed quantity. In Table 1 we report the typical value of the geometrical quantities involved in the investigation of the FSM taiko wafer. Figure 1: Cross section of a front-side metal (FSM) taiko wafer illustrating the quantities determining the warpage of the FSM taiko wafer.