Elastic flexure of bilayered beams subject to strain differentials
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S. Lee Department of Materials Science and Engineering, National Tsing Hua University, Hsinchu, Taiwan 30043 (Received 24 November 1999; accepted 7 September 2000)
The residual stresses present in a thin film and the curvature formed at its substrate during deposition have been a great concern to electrochemists and process engineers. Here a new hybrid analytical method is presented to reanalyze the flexural problem subjected to a strain differential in the general case. It was shown that the present solutions for ultrathin films agree with Stoney’s equation. Moreover, single or dual neutral axes resulted, depending on materials and thickness ratios between the film and the substrate. Quantitative differences with others in the solutions of deformed curvature and residual stress are discussed in a representative case of GaAs top coat/Si substrate wafers.
I. INTRODUCTION
The deposition of a coating on a substrate seems to have broad applications in science and technology. In the case of surface engineering, for instance, a thin film or multilayered film/coating is deposited on the surface to protect the substrate from environmental attack. The nature of the attack may be mechanical, chemical, or thermal, depending on the application. In the semiconductor industry, a thin film of GaAs is deposited onto a semiconductor substrate such as single-crystal Si to form a chip. To design an integrated circuit (IC), a polymer film is deposited on a metallic substrate to form a base for the IC. In general, these bilayered plates have one thing in common: they are composites consisting of two phases with distinct physical and mechanical properties. There are many means to “glue” the film to the substrate, solgel, physical and chemical vapor deposition, plasma spray, and molecular beam epitaxy, to name a few.1 During the processing, differential strains across the interface are unavoidably introduced owing to mismatch of lattice parameters or thermal expansion coefficients (TEC) between the two distinctive phases. As a result, residual stresses are generated in the bimaterial media even in the absence of externally applied stress. If those stresses build up to a significant level, the plate may fracture or severely warp, thereby rendering the part useless. This problem has been a concern particularly in the semiconductor industry. In 1909, Stoney derived a simple relationship between the radius of curvature of the deformed shape and the thickness ratio for a given amount of differential strain in 2780
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J. Mater. Res., Vol. 15, No. 12, Dec 2000 Downloaded: 11 Mar 2015
a two-layer composite plate when the film is ultrathin.2 The equation has been used widely in a variety of applications. However, disagreements and conflicting solutions with regard to the stress and deformation fields still appear in the open literature.3–7 More recently, Chu revisited the problem and presented a physics-based solution for the general case.7 His result for deformed curvature, when evaluated in the limiting case, di
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