Elsevier

Engineering Fracture Mechanics

Volume 75, Issue 14, September 2008, Pages 4217-4233
Engineering Fracture Mechanics

Hybrid spectral/finite element analysis of dynamic delamination of patterned thin films

https://doi.org/10.1016/j.engfracmech.2008.03.006Get rights and content

Abstract

A combined spectral and finite element analysis is performed to investigate the dynamic edge delamination of patterned thin films from a substrate. The analysis is motivated by an emerging experimental technique in which high-amplitude laser-induced stress waves initiate progressive interfacial debonding of thin film interfaces. The numerical method relies on the spectral representation of the elastodynamic solutions for the substrate and the finite element model for the thin film. A cohesive model is introduced along the interface of the bimaterial system to capture the decohesion process. The important role of the film inertia on the crack extension and the appearance of the mixed-mode failure are demonstrated by observing the traction stress evolution at various points along the bond line. Parametric studies on the effect of film thickness, interface fracture toughness, loading pulse shape and amplitude on the debonding process are performed. A semi-analytical investigation on the inertial effect is carried out to predict the final crack length as a function of the film thickness and pulse amplitude.

Introduction

Thin films are crucial elements in a wide range of engineering applications such as integrated circuits, magnetic storage media, thermal sensing elements and micro-electro-mechanical systems. One of the mechanical reliability concerns of such devices is the interfacial adhesion between the film and the substrate. Often considered as the most critical regions in thin film structures, interfaces have been the focus of extensive research to characterize the mechanics of their failure. One of the key issues underlying this research is the development of reliable techniques to measure the adhesive properties to be incorporated directly into the material design and fabrication process selection.

Various experimental techniques have been developed to measure the thin film adhesion including the scratch, peel, pull, blister and indentation tests [1]. In these tests, however, the film/substrate interface is typically subjected to very high stress levels resulting in a large amount of plastic deformation in the film, which makes it difficult to determine the intrinsic interfacial properties [2]. In contrast, laser spallation techniques [3], [4], [5], [6], [7], [8], [9], [10], [11] utilize laser-generated stress waves to load the interface remotely at high strain rates (107/s) for very short times (11ns). At these high strain rates, the material’s yield stress increases drastically, as observed in plate impact experiments [12] and the failure process takes place with minimum plastic deformation.

A schematic of the tensile spallation experiment is shown in Fig. 1. The sample consists of a transparent confining layer, a thin energy-absorbing layer, the substrate and the testing film. An infrared Nd:YAG pulse (λ=1064nm) with a variable energy content between 1 and 150 mJ and a width of about 10 ns is incident on a metallic absorbing layer sandwiched between the confining layer and the substrate. The energy-absorbing layer is chosen to be much thicker (0.4μm) than the critical penetration depth of laser light at this wavelength. A compressive longitudinal stress wave with a Gaussian shape similar to that of the laser pulse is emitted from the absorbing layer. The wave then propagates towards the film–substrate interface and is reflected from the free film surface into a tensile wave, which then loads the testing interface in tension. The laser energy is increased until a longitudinal wave is generated with an amplitude sufficient to fail the film/substrate interface. Interferometric measurements of out-of-plane displacement are made at the surface of the thin film. From displacement measurements at the free surface, the stress history at the interface is inferred using standard wave mechanics and the tensile failure strength of the interface is obtained by monitoring the maximum traction stress acting on the interface at the onset of failure. This technique has been further developed by Wang et al. [2], [13], [14] to measure mixed-mode interfacial strength values.

Another key adhesive property is the fracture toughness, defined as the energy needed to separate a unit area of the bonding surface. Although number of experimental techniques have been proposed to extract the fracture toughness values, many have been challenged in terms of their reliability and repeatability. de Boer and Gerberich [15] have used the nanoindentation technique with the aid of an analytical model provided by Marshall and Evans [16] to measure the adhesive fracture energy. However, this method is limited to brittle, weakly bonded films since ductile, strongly adhered films substantially deform before their delamination from the substrate [17]. To provide additional driving force to delaminate the well bonded interface, Bagchi et al. [18] have proposed a method using a highly stressed superlayer deposited on the top of the test film. The analysis for this test, however, is based on fully elastic assumption and does not take into account the plastic work, resulting in an inaccurate adhesive fracture energy. To exclude the effect of residual stress in the film during the testing process, Charalambides et al. [19] have developed a four-point bending test for the system made of a thin film sandwiched between two rigid substrates. Although the analysis to extract the strain energy release rate from this test is fairly simple and is not dependent on the final crack length, the sample preparation involving high temperature diffusion bonding process might change the original bonding properties. Despite their drawbacks, the three experimental methods described above are still extensively used.

Current applications of the laser spallation techniques do not enable direct measurement of interfacial toughness. Pronin and Gupta [20] have proposed to use a compressively strained niobium layer to buckle the film and create a sharp pre-crack prior to loading by laser-generated stress waves. The analysis is, however, limited to the time of crack initiation at which the value of the strain energy release rate is computed with the aid of a transient finite element analysis. The repeatability of the test is dependent on the reproducibility of the flaw geometry. Design of spallation experiments to initiate stable interface debonding and directly measure the interfacial toughness is indeed a significant challenge. Recent experiments [21] have demonstrated the feasibility of using laser-generated stress waves to initiate a mixed-mode edge delamination from the corner of a pattered film (Fig. 2a). Successful edge delamination of a patterned aluminum (Al) film on a silicon (Si) substrate is shown in Fig. 2b. Edge delamination initiated at a much lower stress level (laser fluence) than in the center of the film, far from the stress concentration at the corner, clearly pointing to an edge-driven delamination process.

To support the experiment described above and provide some insight on the mechanics of patterned film delamination, we develop in this paper a numerical scheme to capture the dynamic initiation, propagation and arrest of the crack. Although the fracture problem depicted in Fig. 2 is three-dimensional (3D), we adopt in this work a simpler two-dimensional (2D) plane strain model to capture the role of the corner stress concentration on the initiation and extent of delamination. One of the methods adopted in this work is the spectral scheme, which has been used successfully in the simulation of various fundamental 2D and 3D dynamics fracture problems [22], [23], [24], [25]. Hendrickx et al. [26] have extended the application of this numerical method to study the thin film delamination problems by developing a special spectral formulation for a domain of finite thickness, perpendicular to the fracture plane, subjected to an anti-plane shear loading. However, the extension of this method to study the 2D in-plane loaded film system has not been derived due to its complexity. To capture the dynamic failure mechanism of a thin film on a substrate subjected to a 2D in-plane laser-generated load pulse and to allow the development of the tensile and mixed-mode interfacial stresses [2], we develop a numerical method which relies on the combination of a spectral scheme for the substrate and an explicit finite element model for the film. The interface is characterized by a rate-independent extrinsic bilinear cohesive model used by Geubelle and Baylor [27] to simulate the decohesion process.

Cohesive modeling has been successfully used to solve a variety of failure processes in thin film and multilayer systems. Examples include microcracking and decohesion of films under tension [28], delamination of weakly bonded coatings in indentation tests [29], crack growth in sandwich beams [30], decohesion of thin film segments [31] and laser-induced blistering of thin films [32]. Mohammed and Liechti [33] also used cohesive zone modeling to capture the nucleation of a crack at a bimaterial corner.

In a recent publication, Liang and co-workers [34] have presented a numerical study of laser-induced thin film delamination for the case of “blanket” (non-patterned) films. In this study, a volumetric damage model was used to model the onset and propagation of delamination, while the finite element method was adopted to capture the dynamic response of the substrate and the film.

In this paper, the proposed cohesive spectral/finite element scheme is applied to investigate the dynamic failure problem of the patterned film system subjected to a time-dependent in-plane tensile loading. Several significant issues are addressed in this work including the spontaneous initiation, propagation and arrest of an interfacial crack, the role of the inertial effect on the crack extension and the energetics of the delamination process. The paper is organized as follows: in the next section, we focus on the details of the numerical implementation including the problem description, the spectral scheme formulation, the cohesive finite element model and the coupling algorithm. In Section 3, we present some numerical results and a parametric study of the effect of the loading pulse and the film thickness. Finally, we reveal in Section 4 the relation between energy transition and the failure process as the foundation of a semi-analytical model to predict the final crack length.

Section snippets

Numerical implementation

As described in the introduction, the numerical modeling used in this work relies on the combination of three key components: (i) an explicit spectral scheme to capture the elastodynamic response of the substrate, (ii) an explicit finite element scheme to simulate the response of the thin film, and (iii) a cohesive model to capture the spontaneous initiation, propagation and arrest of interfacial cracks. Details on the formulation and implementation of these three components are provided below

Spatial convergence study

We start the discussion of the numerical results by focusing on the spatial convergence of the proposed hybrid spectral/finite element solution. The spatial discretization of the interface must be sufficiently fine to capture (i) the stress concentration in the vicinity of the corner of the patterned film and (ii) the cohesive failure process taking place ahead of the dynamically propagating crack front. The dynamic failure problem investigated hereafter involves an Al/Si system whose

Semi-analytical model of final crack length

The role of film inertia on the extent of delamination can also be understood through the evolution of various energy components entering the dynamic fracture problem (Fig. 14): the kinetic energy K associated with the debonded portion of the film, the fracture energy EF dissipated in the delamination event, and the strain energy U stored in the film domain. As soon as the crack starts to propagate from the corner, K increases rapidly, reaches a maximum value before gradually decreasing as the

Conclusion

We have performed a numerical analysis of laser-induced delamination of thin patterned films. The underlying numerical method combines a spectral scheme for the substrate and a finite element model for the thin film. The transient delamination of the film subjected to a Gaussian compressive stress pulse coming from the substrate is simulated using a rate-independent bilinear cohesive model. We have demonstrated the ability of the coupling scheme to capture the initiation, propagation and arrest

Acknowledgements

The authors would like to acknowledge the support from the National Science Foundation through grant CMS 04-08487 and 07-26742, and from the VEF fellowship program.

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