Abstract
Based on the transform of reference plane of bending from substrate mid-plane to an optimum site, and synchronously involved with the hypothesis of a coupled lattice shared by atoms of two crystals on the both sides of interface between film and substrate, a new closed-form solution with symmetric form in mathematics for residual stress in a film-substrate system is developed. Examples are discussed for the thermal and epitaxial stresses. As the ratio of film thickness to substrate thickness becomes larger, stress given by the new solution differs obviously from those predicted by the previous solutions [For instances: G. G. Stoney, Proc. R. Soc. (London) A82, 172 (1909); L. B. Freund, J. A. Floro, and E. Chason, Appl. Phys. Lett. 74, 1987 (1999); and Q. S. Mu, AIP Adv. 8, 065224 (2018)], which may be considered as an extended modification of this paper to the existing studies.
Highlights
Number of applications1,2 of films is enormous: photovoltaic cells, protective coatings, integrated electronic circuits and microelectro-mechanical systems, to name a few
Where κ is substrate curvature, hs is substrate thickness, and Ms is the biaxial modulus of substrate
The definition of film force f is f = σfhf, where σf is the average stress in film and hf is film thickness
Summary
Number of applications of films is enormous: photovoltaic cells, protective coatings, integrated electronic circuits and microelectro-mechanical systems, to name a few. What needs to be underlined here is that the forms possessed by them are not symmetric in mathematics for the two layers of substrate and film, one main reason for this lies in their derivations where the reference plane around which the bending of a film-substrate system occurring is always assumed as the mid-plane of substrate This assumption will become inappropriate when hf is not obviously less than hs, so hf
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