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The IUP Journal of Mechanical Engineering

Lagrange-Kirchhoff Model for the Deformation of a Coated Taiko Silicon Plate

Article Details

Pub. Date	: May, 2023
Product Name	: The IUP Journal of Marketing Management
Product Type	: Article
Product Code	: IJMM040523
Author Name	: A Landi and M Renna
Availability	: YES
Subject/Domain	: Marketing
Download Format	: PDF Format
No. of Pages	: 17

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Abstract

Internal stress in a film deposited on a plate-like substrate causes the film and substrate pair to warp until the mechanical equilibrium is reached. This condition is reached when both the net force and the resulting bending momenta reach a null value. In this paper, the analytical Lagrange-Kirchhoff model for the deformation of a coated 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. These results can be extended to other substrates, such as silicon carbide (SiC).

Introduction

Internal stress in a film deposited on a plate-like substrate causes the film and substrate pair to warp until the mechanical equilibrium is reached. This condition is reached when both the net force and the resulting bending momenta reach a null value. A plate-like or a disc-shaped substrate implies a simplification that the substrate thickness, ts, is constant and small in comparison to its diameter D. From the curvature of elastically deformed coated substrate the average film stress can be calculated. When the thickness of the film, tf, is small comparted to that of the substrate, a simple formula holds, which was first published by Stoney (1909).

Bifurcation is a nonlinear phenomenon which determines the emergence of an abrupt degeneration in the symmetry of large wafers undergoing the large-scale integration process of the semiconductor industry. Because of the stress induced by the thin films manufactured upon the wafer, the warpage can degenerate and increase asymmetrically, resulting in two main principal and different curvatures. The purpose of processing is to contain the warpage and avoid the phenomenon of bifurcation. However, an understanding of the occurrence of this phenomenon can be of help in controlling the warpage of thinned wafers having large diameters. In this paper, the