Non-uniform, axisymmetric misfit strain: in thin films bonded on plate substrates/substrate systems: the relation between non-uniform film stresses and system curvatures

Abstract

Current methodologies used for the inference of thin film stress through curvature measurements are strictly restricted to stress and curvature states which are assumed to remain uniform over the entire film/substrate system. By considering a circular thin film/substrate system subject to non-uniform, but axisymmetric misfit strain distributions in the thin film, we derived relations between the film stresses and the misfit strain, and between the plate system's curvatures and the misfit strain. These relations feature a "local" part which involves a direct dependence of the stress or curvature components on the misfit strain at the same point, and a "non-local" part which reflects the effect of misfit strain of other points on the location of scrutiny. Most notably, we also derived relations between the polar components of the film stress and those of system curvatures which allow for the experimental inference of such stresses from full-field curvature measurements in the presence of arbitrary radial non-uniformities. These relations also feature a "non-local" dependence on curvatures making a full-field measurement a necessity. Finally, it is shown that the interfacial shear tractions between the film and the substrate are proportional to the radial gradients of the first curvature invariant and can also be inferred experimentally.

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“…This relation is independent of the thin film and substrate radii, and is identical to its counterpart for the thin film and substrate with the same radii subjected to nonuniform misfit strain [Huang et al 2005]. The above relation clearly shows that the radial and circumferential stress components will be equal only if the equivalent curvature components are also equal.…”

“…The above equation is once again independent of the thin film and substrate radii, and is identical to its counterpart for the thin film and substrate with the same radii subjected to nonuniform misfit strain [Huang et al 2005]. Equations (19) and (20) provide direct relations between thin-film stresses and substrate curvatures.…”

Surface evolution and stability transition of silicon wafer subjected to nano-diamond grinding

“…When the amorphous layer/silicon substrate system is under a non-spherical status (κ x κ y ), the stress of the ultrathin amorphous layer is non-uniform, 22,25 the stresses in the amorphous layer can be obtain from the linear elastic constitutive model as…”

The effect of thin film/substrate radii on the Stoney formula for thin film/substrate subjected to nonuniform axisymmetric misfit strain and temperature

“…This provides a means to determine the interface shear stress that is responsible for the delamination of thin film/substrate systems. The results of [Huang et al 2005] and [Huang and Rosakis 2005], however, are limited to thin film and substrate of the same radius. They are extended to thin film and substrate of different radii in the present paper.…”

“…The above equation depends on the thin film and substrate radii, and is different from its counterpart for the thin film and substrate with the same radii subjected to nonuniform temperature change [Huang and Rosakis 2005].…”