Measuring the Mechanical Properties of Thin Metal Films by Means of Bulge Testing of Micromachined Windows
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bulge test has long been used for measuring the mechanical properties of thin films [1 4]. It is, at least in principle, a very simple test: A uniform pressure is applied to a circular membrane. Measurement of the membrane deflection as a function of the applied pressure, allows one to determine such mechanical properties as the elastic modulus, yield stress, the residual stress in the membrane, etc. All too often, however, sample preparation has been neglected. In order to obtain accurate and reproducible results, bulge test samples have to be flat, taut membranes with a well-defined size and geometry [4-8]. We have used silicon micromachining techniques to fabricate free-standing membranes of gold and aluminum that meet these requirements. These membranes are square or rectangular in shape and have precisely controlled dimensions. The sample preparation technique involves a minimum of sample handling and is applicable to a wide range of materials.
THEORY The elastic deflection of square membranes by a uniform pressure p has been analyzed by several authors [3, 4, 91 and is approximately described by the following expression: 3 p `=c,(v (1 Etv4 Wo + C2 °'Gt --•-- Wo, (1v) a a
579 Mat. Res. Soc. Symp. Proc. Vol. 356 0 1995 Materials Research Society
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where c 1(v) = 1/(0.792+0.085v) 3 and c2 = 3.393. In this expression, wo is the deflection of the center of the membrane, t is the film thickness and 2a is the width of the membrane. Thus, fitting Eq. (1) to experimental results allows one to determine the biaxial modulus Y=E/(1-v) and the residual stress arres. The constant c I is a weak function of the Poisson's ratio of the film and a rough estimate of this quantity is usually sufficient to accurately calculate the biaxial modulus. The stress distribution in square membranes is very non-uniform, precluding the use of square membranes for measuring the plastic properties of thin films. The deflection of rectangular membranes with sufficiently large aspect ratios, e.g. exceeding 4, is also given by Eq. (1), where now c l(v) = 8/6(l+v) and c2 = 2 [4]. Experimental pressure-deflection data allow one to measure the residual stress and the plane-strain modulus M=E/(1-v 2 ) of the film. Membranes with large aspect ratios quickly approach a simple state of plane strain [8]. Rectangular membranes are therefore very useful for the study of thin film plasticity. The stress and strain in a long membrane are given by = p(a 2 +w2) (2) (2) 2Wot SG =a + aArc sin 2 - 1, (3) E
2a 2Wot w)
so that the stress-strain curve of the film can be readily calculated from the pressure-deflection curve of the membrane. If both square and rectangular membranes are tested, Poisson's ratio can be calculated from the ratio of the biaxial modulus and the plane-strain modulus. Since cl is a function of Poisson's ratio, an iterative procedure starting with a first estimate for v has to be used. SAMPLE PREPARATION AND CHARACTERIZATION Silicon nitride films with a residual tensile stress were deposited by means of low pressure chemical vapor deposi
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