Effect of Initial Substrate Curvature on Nonlinear Bending Measurements of Thin-Film Stress
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EFFECT OF INITIAL SUBSTRATE CURVATURE ON NONLINEAR BENDING MEASUREMENTS OF THIN-FILM STRESS. D. E. Fahnline The Pennsylvania State University, Materials Research Laboratory, University Park, PA 16802
ABSTRACT A recently reported nonlinear model of the bending of a thin-film/substrate bilayer provides a means for determining stress in thin films even for large deflection and ellipsoidal bending. This model replaces the usual Stoney's equation, which is valid only for small deflections. However, the model omits consideration of the commonly observed initial curvature of the substrate before deposition. In the small deflection regime the principle of superposition justifies simply subtracting the initial curvature from the final curvature after deposition, but for large deflections this is inappropriate, because the principle of superposition is no longer valid. The present paper presents a modified form of the nonlinear model incorporating initial substrate bending. The resulting equations show that initial substrate curvature causes magnified nonlinear effects and provide a means for determining film and substrate elastic properties in addition to thin-film stress.
INTRODUCTION Fahnline, Masters, and Salamon [1] have reported a partially successful test of a nonlinear bending model for thin films deposited on wafer substrates proposed by Harper and Wu [2] as well as by Masters, Salamon, and Fahnline [3]. This model serves as a large deflection replacement for the commonly employed Stoney's equation valid only for small deflections. The results verified the usefulness of the model for measuring thin-film stress using ellipsoidally as well as spherically bent samples, for distinguishing ellipsoidal bending due to nonlinear effects from that due to other causes, and also for measuring the substrate Poisson ratio, but there were two qualitative discrepancies. First, the ellipsoidal sample did not exhibit the two stable states predicted by the model, and second, the sample appeared to develop an unpredicted anticlastic shape as a result of thin-film deposition. The present paper attempts to explain these discrepancies by including initial substrate curvatures in the model ab initio, rather than by subtracting them out at the end on the basis of the principle of superposition. An added benefit of this analysis and of the method described for mounting substrates during deposition is that it provides a method for measuring film and substrate elastic properties related to one proposed by Witvrouw and Spaepen [4].
Mat. Res. Soc. Symp. Proc. Vol. 239. @1992 Materials Research Society
252
THE ORIGINAL NONLINEAR
THEORY
For purposes of comparison, Fig. 1 illustrates the theoretical predictions of the original nonlinear bending model with no initial curvatures for a tf = 0.046-micron thick Tantalum film on a ts = 88.9-micron (5-mil) thick square polycarbonate wafer. The substrate elastic modulus and Poisson ratio are E. = 2.25 GPa and vs = 0.37, respectively, and the corresponding film values are Ef = 186 GPa, and vf = 0.3
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