In this work, the analytical Lagrange-Kirchhoff model for the deformation for a taiko plate silicon is proposed. Under the conditions of validity of the linear theory of thin plates, the model is in agreement with the experimental data.

The successful handling of large semiconductor wafers is crucial for scaling up their production. Early-stage warpage control allows to prevent undesirable asymmetric warpage, known as wafer bifurcation or buckling. Indeed, even in a gravity-free environment, thinning an 8" or 12" semiconductor wafer can result in warpage and bifurcation. To mitigate this issue, the taiko method, which involves creating a thicker ring region around the rim of the wafer, has been widely used. Previous research has focused on the theoretical factors affecting the warpage of a backside metalized taiko wafer. This work extends the case to a front-side metalized taiko wafer and introduces the concept of the equivalent thickness of a taiko wafer. The equivalent thickness of a taiko wafer, influenced by the ring region, lies in between the thickness of the central region and that of the annular region. Because of the limited number of taiko wafers that can be produced on a production line, modelling can be beneficial. In this work we compared the results of a developed analytical model with those obtained from a finite element analysis (FEA) approach with ANSYSY® software to model the equivalent thickness of a taiko wafer. We investigated the curvature as a function of the stress of the metal layer, considering key design factors such as the substrate region thickness, the thickness of the thin metal film, the step height, and the width of the ring region. By systematically varying the thickness of the central region of the taiko wafer, we explored the curvature as a function of stress induced by thermal load in the linear regime and determined the slopes in the linear region of the curvature vs. stress curves. The aim of this study is to identify regularities and similarities with the Stoney equation and investigate the validity of the analytical approach for the case of a taiko substrate. It results that there is a good agreement between the analytical model of a taiko wafer and the numerical analysis gained by the FEA methods.

A connection between the phenomenon of wafer bifurcation and that of spontaneous symmetry breaking (SSB) has been established. Indeed, by developing an analytical approximation of the elastic energy of a wafer coated with a thin layer (e.g. a metal), it is shown as the elastic potential energy, interpreted as a quantity contributing to the thermodynamic free energy, can be investigated within the framework of the Landau theory of the second order phase transitions. The elastic energy of a bifurcated wafer is a complex function of the stress and the curvatures in the two perpendicular directions. In this work, it is shown as a translation of the coordinate system of the curvatures allows to gain a potential which has a “Mexican hat” shape. This is a distinctive trait of the phenomenon of spontaneous symmetry breaking (SSB). Moreover, it is shown as the values of the coordinates at the minimum of the SSB potential agrees with those provided by the theory. Bifurcation is hence a phenomenon that can also be interpreted as a spontaneous symmetry breaking where the rotational symmetry of a disc shaped wafer is broken. It occurs because of a lowering of the wafer energy in the SSB broken symmetry configuration.

An extension of the Stoney formula for the case of a back side metallized 8” silicon taiko wafer has been developed, in the elastic regime, within the frame of the theory of elasticity. A good correlation between the calculated warpage, determined by the stress released by a given back side metallization (BSM), and the corresponding experimental warpages of the same thick metal layers deposited on an 8” silicon taiko wafer provides evidences of the correctness of the developed theory. This development suggests the possibility to extend this approach to the case of 8” taiko wafers based on a wide band gap semiconductor such as silicon carbide (SiC).