# Determination of the Equivalent Thickness of a Taiko Wafer Using ANSYS Finite Element Analysis
## 1 Introduction  
Vertical power MOSFETs based on silicon [1] and silicon carbide [2] devices, intended for the automotive sector, need a considerable cutdown of the switching losses and Joule heating effect [3, 4]. A method to achieve this consists of reducing the thickness of the semiconductor wafer. However, the combined effect of a thinned substrate along with, e.g., a front side metallization (FSM) produces a severe warpage of the whole wafer, which can hinder the performance and subsequent manufacturing processes needed to complete the devices. Containing the warpage of the wafer determined by the FSM residual stress is crucial in the semiconductor industry. Indeed, the handling of the wafer as the size increases can be an issue; also in the case of SiC substrates [5, 6]. A method to solve the issue involves the use of taiko wafers [7, 8, 9, 10, 11, 12], which consists of back-ground wafers having a ring region of the wafer which is preserved. The result is that, because of the intrinsic structure of the taiko wafer, for a given metal stress and metal film thickness, the resulting warpage of aFSM-thinned wafer is mitigated with respect to the case of a canonical or flat metallized thinned wafer. To gain an analytical description of the system, we start with the simplification of a disk-shaped wafer. A thin FSM metal film, deposited on the front side of a semiconductor wafer, warps the whole metalized wafer until the net forces and the resulting bending moment's zero-condition is reached. From this it is possible to gain an estimate of the resulting curvature. If the substrate thickness \(h_{s}\) is constant and negligible with respect to the diameter \(D\), of the wafer, by considering a thickness \(h_{f}\) of the film, uniform and small compared to that of the substrate, an average film stress, \(\sigma_{f}\), can be determined from the curvature of the elastically deformed coated substrate, according to the equation proposed by [PERSON] [13], which, corrected with the biaxial modulus [14] can be written as:  
\[\sigma_{f}=\frac{E_{s}h_{s}^{2}}{6h_{f}(1-v_{s})}\kappa \tag{1}\]  
Equation (1) provides a direct linear relationship between the measured average curvature \(\kappa\) and the average film stress \(\sigma_{f}\). In the equation \(E_{s}\) and \(v_{s}\) are the Young's modulus and Poisson ratio of the substrate, respectively. In general, the Stoney equation is appropriate in the linear regime where both the residual stress \(\sigma_{f}\) and the resulting curvature \(\kappa\) are low. However, as the stress increases, deviations from linearity are observed and the curvature \(\kappa\) for a given stress \(\sigma_{f}\) of the film becomes lower than what is predicted by Equation (1). Corrections to the Stoney equation have been considered in the past. For example, a cubic polynomial correction to the linear dependence, has been proposed [15]. Additionally, an effort to extend Stoney's formula for non-uniform wafer thickness has been made in [16] for the case of wafers with a slight thickness change in the peripheral region. However, this work only considers the step as a small disturbance compared to the overall wafer thickness, thus it does not offer a comprehensive solution in the case of a taiko wafer. Approaches based on Finite Element Analysis (FEA) have also been considered in the case of flat substrates using standard structural analysis tools such as ANSYS [17; 18; 19] and ABAQUS [20]. These works highlight that it is challenging to accurately predict the final shape of the warped flat substrate. Indeed, a numerical and stable solution is often not the physical one, unless a small perturbation which would result in a bifurcation, is applied. As far as we are aware, there have been few comprehensive studies on general cases of taiko wafers. The way the warpage changes for these wafers is therefore unclear. In this study, we offer an analytical explanation derived from the theory of elasticity to comprehend the resulting warpage in an FSM-taiko wafer.