Applicability of the classical curvature-stress relation for thin films on plate substrates

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Abstract
The applicability in various circumstances of the classical equation relating substrate curvature to film stress for the case of a thin film on a plate substrate is examined. Theoretical treatments, based on elementary plate-bending theory and general elasticity theory, are given of the effects of gravity, substrate shape, film nonuniformity, and substrate crystallinity on substrate curvature. Formulas describing the effect of gravity and of film nonuniformity are confirmed experimentally using a laser-beam reflection technique. It is shown that gravity effects can cause significant errors in stress calculations based on the classical curvature versus stress equation, but these effects are largely avoidable or subtractable. Within limits, curvature is independent of substrate shape. For a film of nonuniform thickness, the classical equation does not apply, but in certain cases a simple analog does. Appropriate interpretation of elastic moduli appearing in the classical equation allows the equation to apply when the substrate is elastically anisotropic (e.g., a single crystal) with third-order or higher symmetry about its z axis. These studies augment thin-film stress measurement experimental technique.

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