Full-Field Bulge Testing Using Global Digital Image Correlation Paper number 206
Jan Neggers1 (
[email protected]), Johan Hoefnagels1, François Hild2, Stéphane Roux2 and Marc Geers1 1)
Eindhoven University of Technology, Department of Mechanical Engineering, Den Dolech 2, 5612 AZ Eindhoven, Netherlands 2) LMT Cachan, avenue du Président Wilson 61, 94235 Cachan Cedex, France
ABSTRACT The miniature bulge test is an acknowledged method for characterizing freestanding thin films. Nevertheless, some discrepancies in the quantitative results from such tests can be found in literature, explained in part by erroneous assumptions in the analytical description used to compute the global stress and strain from the membrane pressure and deflection. In this research, a new method is presented which renders the analytical description obsolete. A specialized Global Digital Image Correlation technique on high resolution, confocal microscopy, surface height maps of bulged membranes, has been developed. This method is able to capture full-field continuous deformation maps, from which local strain maps are computed. Additionally, local stress maps are derived from full-field curvature maps and the applied pressure. The local stress-strain maps allow the method to be used on inhomogeneous, anisotropic membranes as well as on exotic membrane shapes. INTRODUCTION Current micro-electronic devices or integrated systems often consist of an abundance of different thin films, typically selected on their electronic properties [1-3]. However, due to the inherent large range of thermal expansion coefficients in these materials, large stresses occur during processing or in the lifetime of the product. To improve the reliability of these products the mechanical properties of the individual thin films are required. Since these thin films often have one dimension smaller than the intrinsic micro-structural length scale of the material, they exhibit a so-called size effect which means that the thin film material response is different than their bulk counterparts [4-6]. Therefore, experimental methods that can characterize these thin films in the same form as they are produced and used are invaluable. An important experiment in this class is the bulge test experiment. In a bulge test experiment a thin film or membrane is deflected using a pressure medium. From the pressure and deflection at the apex of the bulge, the stress-strain response is calculated by means of an analytical descriptions of the membrane deformation, called the bulge-equations [7-8]. These bulge equations are very sensitive to the membrane geometry which has been a large source of inaccuracy in the past. Nonetheless, with the currently available (lithographic) Si micro-machining techniques the bulge membranes can be produced with such accuracy that this is no longer a problem. These micro-machining techniques are the same techniques as used to create the micro-electronics, as a result, allowing for the creation of bulge test samples which had a very similar processing history as the real products. The bulge test experiment is recognized as a powerful method to characterize thin-films mechanically, allowing for the measurement of full (isothermal) stress-strain curves, including the plastic regime, of freestanding thin films [9]. Nevertheless, some discrepancies occur in literature when comparing the results quantitatively. The reasons for these discrepancies are most likely due to the fact that the membrane deformation is actually more complex, including (i) localized deformation, mainly close to the membrane boundaries, (ii) inhomogeneous membrane deformation, typically the deformation state ranges from plane strain to plane stress, varying over the membrane, and (iii) anisotropic material behavior. None, of the before are taken into account in the bulge-equations, and are always assumed to be negligible. The membrane deflection is often measured with a single spot, laser interferometric, measurement. In contrast, also full-field surface profilometric measurement techniques are becoming ever more available, e.g. atomic force microscopy or confocal
optical microscopy. Using such a technique to measure the membrane deflection profile allows for the application of Digital Image Correlation (DIC) to obtain the full-field strain, i.e. the strain at every spot on the membrane. Additionally, the applied pressure, underneath the membrane, is constant over the entire membrane, and the membrane is in static equilibrium. This means that from the applied pressure and the measured curvature field the stress-field can be obtained if the membrane thickness is known and the local bending stresses are small. It should be noted that this reasoning did not include any assumptions on the membrane shape, i.e. this method is applicable to any shape bulge membrane without making any assumptions on boundary conditions etc. As a result, a custom bulge test setup has been build that fits underneath the optical confocal microscope (figure 1).
Fig 1 Experimental setup: a miniature plane strain bulge test setup, placed under a confocal optical profilometer.
GLOBAL DIC To obtain the curvature-field typically the position field, i.e. surface map, has to be differentiated twice, which is rather noise sensitive and in-accurate, especially when the curvatures are small, which is the case for strains relevant for typical microelectronic materials. To solve this, a highly specialized DIC method is developed based on the currently quickly developing Global DIC method [10-11]. With this method an infinitely differentiable deformation field is obtained which is extremely insensitive to micro-fluctuations from which high accuracy curvature fields can be computed. In DIC, the goal is typically to find the displacement field u ( x ) , described by the degrees of freedom u , between two images, the references image f ( x ) and the deformed image g ( x ) . The main characteristic of global DIC or GDIC is that u is being solved for globally, as opposed to locally. In this way GDIC typically enforces a continuity of u ( x ) . The degrees of freedom can be similar to the nodal degrees of freedom in a Finite Element Method (FEM) model, and if Quadrilateral Four node (linear) elements are used in GDIC, the method is typically named FEM-DIC or Q4-DIC [11]. However, in GDIC it is not necessary to use a FEM basis. To capture an infinitely differentiable deformation field, the deformation field is discretized using a set of polynomial functions n x y , a
b
a , b [0 , 1, 2 , ]
A GDIC approach consists of the minimization of the mean squared difference
(0.1)
2
ROI
f ( x ) g x u ( x ) d x 2
(0.2)
where the displacement field is expressed in a chosen basis u (x)
u
n
φ n (x)
(0.3)
n
where φ n are chosen vector functions, and u n are associated degrees of freedom, which if combined in a column are denoted as φ and u respectively. The minimization of
2
with respect to the unknowns u is solved iteratively using an ex-
plicit procedure, assuming u i 1 u i d u i . Linearizing this system of equations allows the minimization to be rewritten as a matrix-vector product as
2
ui
M idu b i 0
(0.4)
where M can be considered as a mass matrix with vector components Mi
ROI
φ · f ( x )· f ( x )·φ t d x
(0.5)
and b as a column with vector components bi
ROI
f ( x ) g x u i ( x ) · f ( x )·φ d x
(0.6)
Solving this system iteratively until convergence results in the discretized deformation field from which the strain field and curvature fields can easily be determined. The development of the method is still in progress, nevertheless, there are no foreseen reasons why it would not be possible to capture the full-field local stress and strain maps. RESULTS As an early result the method was applied to 100 nm thick SiN free standing bulge membranes which are created by backetching a (1 6 mm) rectangular opening in the Si below the SiN layer. On top of this membrane a pattern is applied using spherical Ag particles with a distributed size between 80-500 nm. The membrane is then stretched to 5% and 10% strain yielding to correlation images f and g respectively (Figure 2). The method was able to directly correlate between the two strain steps without using intermittent steps. The resulting displacement fields in the x, y, and z directions are shown in figure 3.
Fig. 2 Two measured bulge test profiles f and g at 5% and 10% strain respectively. A pattern is applied on the surface using 80-500 nm Ag particles. In global DIC, the image g is mapped onto f by fitting its deformation u ( x ) .
Fig. 3 The displacement fields in x, y and z directions (subfigures a, b, c) obtained from the global DIC method, using basis functions of the type n x y , where a [0 ,1, 2 , 3, 4 ] and b [ 0 ,1] . a
b
CONCLUSIONS A specialized global DIC method is developed which can cope with the semi-3D data obtained from surface profilometry measurement techniques. The method has proven very robust and able to correlate directly between large strain steps, yielding continuous full-field displacement fields with sub-pixel displacement resolution. The displacement fields are insensitive to micro-fluctuations, e.g. measurement noise, and are therefore especially capable in capturing small curvature variations. From the full-field curvature fields and the applied bulge test pressure the local full-field stress can be determined. The proposed method allows the capturing of full-field stress-strain information of the entire bulge membrane. More importantly it allows for measuring only a part of the membrane, i.e. away from the boundary. As a result the measurement can be performed in a region where the deformation state is well understood allowing for easier extraction of important constitutive material parameters. An elaborate description of the method including proof of principle experiment is currently under development and will be published within respectable time.
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