Deflection of a cantilever rectangular plate induced by surface stress with applications to surface stress measurement

Abstract

Surface stress plays important roles in the fabrication and applications of thin-film substrate systems. Bending test of cantilever microbeams has been commonly applied to characterize the surface stress. Stoney’s equation, ideally valid for completely unconstrained plates, is typically used to convert the measured deflection to a surface stress. To assess the validity of Stoney’s equation for the more complicated case of a plate with a clamped end, an analytical solution has been obtained in this study for the deflection of a cantilever rectangular plate due to surface stresses at its upper and lower surfaces. The analytical solution is given by the summation of single Fourier cosine series in the length and the width directions of the plate and a lower order polynomial. Numerical results for the deflection, slope, and curvature for the midpoint of the free end are presented for cantilever plates with aspect ratios ranging from 0.1 to 10 and for different Poisson’s ratios. In most practical measurements of surface stress, the aspect ratio is greater than one and the maximum percentage errors of Stoney’s equation for the deflection, slope, and curvature for the midpoint of the free end are 16%, 16%, and 10%, respectively. The present analytical solution based on Fourier cosine series with the first two leading terms can provide a significant improvement over Stoney’s equation. The maximum percentage errors for the deflection, slope, and curvature for the midpoint of the free end are reduced to 3%, 2%, and 3%, respectively.