Cohesive zone criterion for cracking along the Cu/Si interface in nanoscale components

Crack initiation and propagation along the Cu/Si interface in multilayered films (Si/Cu/SiN) with different thicknesses of the Cu layer (20 and 200 nm) are experimentally investigated using a nano-cantilever and millimeter-sized four-point bending specimens. To examine the cohesive zone model (CZM) criterion for interfacial delamination along the Cu/Si interface in nanoscale stress concentration, an exponential type of CZM is utilized to simulate the observed delamination processes using the finite element method. After the CZM parameters for the Cu/Si interface are calibrated by experiment, interface cracking in other experiments is predicted. This indicates that the CZM criterion is universally applicable for describing cracking along the interface regardless of specimen dimensions and film thickness which include the differences in plastic behavior and residual stress. The CZM criterion can also predict interfacial cracking along Cu/Si interfaces with different stress singularities.     

Quantitative evaluation of plasticity of a ductile nano-component

The plasticity of a copper nano-component is experimentally evaluated by a cantilever beam specimen in which the component is sandwiched between hard layers: silicon and silicon nitride. The cantilever is monotonically loaded with a diamond indenter so that a bending moment is applied to the Cu component, and the deflection at the free-end of the cantilever is precisely monitored by transmission electron microscope observation. The load-deflection relationship shows a clear non-linearity, which is due to the plastic deformation of the Cu component. Using the experimental results, the plastic property of the ductile Cu component is inversely analyzed by finite element method assuming that the component obeys the Ramberg-Osgood constitutive law. The plastic parameters, σ₀ (yield stress), n (hardening exponent) and α (yield offset) are optimally fitted to reproduce the experimentally evaluated load-deflection curve. The resultant parameter set is derived as (σ₀, n, α) = (345 MPa, 3.2, 1.25). The Cu nano-component has a much higher yield stress and hardening rate compared to bulk Cu.     

Dominant stress region for crack initiation at interface edge of microdot on a substrate

In order to examine the mechanics of crack initiation at the free interface edge of a microcomponent on a substrate, delamination tests are carried out for two specimen shapes of Cr microdots on a SiO₂ substrate. The microdots of the first specimen are shaped like the frustum of a round cone. The Cr microdots are successfully delaminated from the SiO₂ substrate in a brittle manner and the critical load is measured by atomic force microscopy (AFM) with a lateral loading apparatus. Stress analysis reveals that a singular stress field exists near the interface edge and the strength for the crack initiation is governed by the intensified normal stress field. The critical stress intensity parameter is evaluated as K_σC ≈ 0.24 MPa m^0.39. Similar delamination tests are conducted for microdots shaped like the frustum of an oval cone. The stress distributions at the crack initiation of this specimen shape show a higher normal stress than the first specimen shape in the region near the interface edge of about x < 40 nm, while it is lower in the region of about x > 50 nm (x: distance from the edge). This suggests a limitation of conventional fracture mechanics: namely, the crack initiation in these specimens is not uniquely governed by the intensity of the singular field. It is found that the delamination crack is initiated when the averaged stress σ_ya in the region of 90-130 nm reaches 190-270 MPa, regardless of the specimen shape. This indicates that the dominant stress region of crack initiation is roughly estimated as 90-130 nm and the criterion is given in terms of the averaged stress in the region.     

Effect of residual stress on delamination from interface edge between nano-films

The focus in this study is on the effect of residual stress on the delamination crack initiation from the interface edge between thin films, Cu/TiN, where the stress is intensified by the free edge effect. The delamination tests, where the mechanical stress is applied on the interface, show that the specimen with the thinner Cu film has an apparently higher strength at the interface edge. The residual stress in the films is then evaluated by curvature measurement of film/substrate coupon and the influence on the delamination is analyzed. The residual stress increases with the increase of film thickness and remarkably intensifies the stress near the edge. By superimposing the contributions of the applied load and the residual stress, a good agreement is obtained in the normal stress intensity near the interface edge at the delamination independent of the Cu thickness. This signifies that the combination of intensified stresses due to the applied load and the residual stress governs the crack initiation at the interface edge, and the toughness at the interface edge is evaluated by the stress intensity factor on the basis of the fracture mechanics concept.     

Crack initiation at free edge of interface between thin films in advanced LSI

Since electronic devices are made of multi-layered sub-micron films, delamination along the interface is one of the major failure mechanisms. This paper aims to develop a method for evaluating the mechanical criterion of interface cracking between thin films on a substrate. The focus is put on crack initiation from the free edge of the interface where the stress concentrates due to the mismatch of elastic deformation. In the evaluation, it is important to exclude plastic deformation and fracture of the thin metal film, because they bring about ambiguity on the measured magnitude of interface strength. In this study, an experimental method is proposed on the basis of fracture mechanics concepts, and the validity is examined by tests on Cu (conductor metal)/TaN (barrier metal) interface in a large-scale integrated circuit. The critical stress intensity at delamination crack initiation is successfully analyzed by the boundary element method.     

Initiation of interface crack at free edge between thin films with weak stress singularity

Delamination tests using sandwich type specimens are conducted for eight combinations of materials: thin films formed on silicon substrates which are relatively popular in micro-electronic industry, to develop a method for quantitative evaluation and comparison of crack initiation strength at the free edge. The difficulty stems from the difference of stress singularity, K_ij/r^λ (K_ij: stress intensity, r: distance from free edge and λ: order of stress singularity), where λ is depending on the combination of materials. Thus, the critical K_ij has different dimensions, MPa m^λ, in each interface. Using the experimentally observed delamination load, the stress distribution along the interface is analyzed by boundary element method. Since the orders of stress singularity, λ, in the materials are less than 0.07 (weak singularity), the stress field near the interface edge is almost constant in atomic (nanometer) level. Then, the critical strength for the interface cracking is quantitatively represented by the concentrated stress near the edge. The effects of the several factors such as species of thin films, oxidized interlayers and deposition processes of thin films on the interface strength are evaluated on the basis of this critical stress as well.     

Fatigue crack initiation and propagation under cyclic contact loading

A computational model for contact fatigue damage analysis of gear teeth flanks is presented in this paper. The model considers the conditions required for the surface fatigue crack initiation and then allows for proper simulation of the fatigue crack propagation that leads to the appearance of small pits on the contact surface. The fatigue process leading to pitting is divided into crack initiation and a crack propagation period.The model for prediction of identification of critical material areas and the number of loading cycles, required for the initial fatigue crack to appear, is based on Coffin-Manson relations between deformations and loading cycles, and comprises characteristic material fatigue parameters. The computational approach is based on continuum mechanics, where a homogenous and elastic material model is assumed and results of cyclic loading conditions are obtained using the finite element method analysis.The short crack theory together with the finite element method is then used for simulation of the fatigue crack growth. The virtual crack extension (VCE) method, implemented in the finite element method, is used for simulating the fatigue crack growth from the initial crack up to the formation of the surface pit. The relationship between the stress intensity factor K and crack length a, which is needed for determination of the required number of loading cycles N_p for a crack propagation from the initial to the critical length, is shown.     

The use of interlayers in modeling interface crack propagation

Delaminations are a common mode of failure at interfaces between two material layers which have dissimilar elastic constants. There is a well-known oscillatory nature to the singularity in the stress fields at the crack tips in these bimaterial delaminations, which creates a lack of convergence in the modewise energy release rates. This makes constructing fracture criteria somewhat difficult. An approach used to overcome this is to artificially insert a thin, homogeneous, isotropic layer (the interlayer) at the interface. The crack is positioned in the middle of this homogeneous interlayer, thus modifying the original ‘bare’ interface crack problem into a companion ‘interlayer’ crack problem. Individual modes I and II energy release rates are convergent and calculable for the companion problem and can be used in the construction of a fracture criterion or locus. However, the choices of interlayer elastic and geometric properties are not obvious. Moreover, a sound, consistent, and comprehensive methodology does not exist for utilizing interlayers in the construction and application of mixed-mode fracture criteria in interface fracture mechanics. These issues are addressed here. The role of interlayer elastic modulus and thickness is examined in the context of a standard interface fracture test specimen. With the help of a previously published analytical relation that relates the bare interface crack stress intensity factor to the corresponding interlayer crack stress intensity factor, a suitable thickness and elastic modulus are identified for the interlayer in a bimaterial four-point bend test specimen geometry. Interlayer properties are chosen to make the interlayer fracture problem equivalent to the bare interface fracture problem. A suitable mixed-mode phase angle and a form for the fracture criterion for interlayer-based interface fracture are defined. A scheme is outlined for the use of interlayers for predicting interface fracture in bimaterial systems such as laminated composites. Finally, a simple procedure is presented for converting existing bare interface crack fracture loci/criteria into corresponding interlayer crack fracture loci.     

Initiation and arrest of an interfacial crack in a four-point bend test

This paper describes a framework to study the initiation and arrest of an interfacial crack, using a combination of experiment and computation. We consider a test configuration widely used in the microelectronic industry: a sample of two substrates bonded by a stack of thin films, with a pre-crack in one of the substrates, perpendicularly impinging upon the films. When the sample is loaded to a critical level, the pre-crack initiates a new crack on one of the interfaces in the sample. The new crack often runs rapidly on the interface for a considerable length, and then arrests. We introduce a quantity, the initiation energy, to characterize the condition under which the pre-crack initiates the interfacial crack. The initiation energy is independent of the test configuration on the scale of the substrates, but changes greatly with the materials and stacking sequence of the films. We measure the initiation energy experimentally, interpret the data using mechanistic models, and use the initiation energy to predict the arrest crack length.     

Extension of the Tanaka-Mura model for fatigue crack initiation in thermally cut martensitic steels

A multi scale numerical approach for evaluation of crack initiation and propagation in thermally cut structural elements made of martensitic steel is presented. A numerical simulation of micro-crack initiation is based on the Tanaka-Mura micro-crack nucleation model, where individual grains of synthetic microstructure are simulated using the Voronoi tessellation. Three improvements are added to this model (multiple slip bands, micro-crack coalescence and segmented micro-crack generation). Crack propagation is then solved on a macro scale model using linear elastic fracture mechanics approach. Some experimental tests have also been performed to check the accuracy of the numerical model. The results of the proposed computational model show a reasonable correlation with the experimental results.     

Disappearance of stress singularity at interface edge due to nanostructured thin film

The purpose of this study is to examine the stress distribution near the interface between a nanostructured thin film and a solid body. We focus on a nanostructured thin film that consists of Ta₂O₅ helical nanosprings fabricated on a Si substrate by dynamic oblique deposition. The mechanical properties of the thin film are obtained by vertical and lateral loading tests using a diamond tip built into an atomic force microscope. The apparent shear and Young’s moduli, G′ and E′, of the thin film are 2-3 orders of magnitude lower than those of a conventional solid Ta₂O₅ film. Moreover, the thin film shows strong anisotropy. A finite element analysis for two types of components with different interface edges between the thin film and an elastic solid body is conducted under uniform displacement. One has a free edge where the surface-interface angle is 90°-90°, and the other has a short interface crack. These analyses indicate the absence of not only stress singularity but also high stress concentration near the free edge and the interface crack tip. The characteristic stress distributions near the interface are due to the nanoscopically discrete structure of the thin film.     

Crack initiation and crack propagation under thermal cyclic loading

Theoretical and experimental investigations of crack initiation and crack propagation under thermal cyclic loading are presented. For the experimental investigation a special thermal fatigue test rig has been constructed in which a small circular cylindrical specimen is heated up to a homogeneous temperature and cyclically cooled down under well defined thermal and mechanical boundary conditions by a jet of cold water. At the end of the cooling phase the specimen is reheated to the initial temperature and the following cycle begins. The experiments are performed with uncracked and mechanically precracked specimens of the German austenitic stainless steel X6CrNi 1811.

In the crack initiation part of the investigation the number of load cycles to initiate cracks under thermal cyclic load is compared to the number of load cycles to initiate cracks under uniaxial mechanical fatigue loading at the same strain range as in the cyclic thermal experiment. The development of initiated cracks under thermal cyclic load is compared with the development of cracks under uniaxial mechanical cyclic load.

In the crack propagation part of the investigation crack growth rates of semi-elliptical surface cracks under thermal cyclic loading are determined and compared to suitable mechanical fatigue tests made on compact-tension and four-point bending specimens with semi-elliptical surface cracks. The effect of environment, frequency, load shape and temperature on the crack growth rate is determined for the material in mechanical fatigue tests.

The theoretical investigations are based on the temperature distribution in the specimen, which is calculated using finite element programs and compared to experimental results. From the temperature distribution, elastic and elastic-plastic stress distributions are determined taking into account the temperature dependence of the material properties. The prediction of crack propagation relies on linear-elastic fracture mechanics. Stress intensity factors are calculated with the weight function method and crack propagation is determined using the Paris relation.

To demonstrate the quality of the crack growth analysis the experimental results are compared to the prediction of crack propagation under thermal cyclic load.     

Influence of the interface ply orientation on the fatigue behaviour of bonded joints in composite materials

The paper deals with the study of the fatigue behaviour of bonded joints in composite materials. The influence of the orientation of the composite layer at the adhesive–adherend interface is investigated on single lap joints prepared by carbon fabric/epoxy laminates bonded together with a two-part epoxy adhesive. Different laminate lay-ups ([45/0₂]_s and [45₂/0]_s), overlap lengths (20 and 40 mm) and corner geometry of bonded area (square edge and fillet, respectively) were investigated under tension–tension fatigue. Particular attention was devoted to the analysis of the fatigue damage evolution to identify initiation and subsequent growth of cracks. A previous model developed by the authors, for the prediction of the fatigue life of bonded joints as the sum of an initiation and propagation phase, was successfully applied to summarise the new data.     

Influence of interfacial reaction layer morphologies on crack initiation and propagation in Ti/Al joint by laser welding–brazing

Influence of interfacial reaction on crack initiation and propagation in Ti/Al joint by laser welding–brazing (LWB) was investigated to obtain strong joints. According to in situ observation of crack initiation and propagation and analysis of fracture surface, the effects of different reaction layer including interfacial layer with insufficient interfacial reaction, lamella-shaped morphology, cellular/serration-shaped morphology, club-shaped morphology, and thick continuous morphology on crack initiation and propagation was discussed. It was found that the joint with lamella-shaped, cellular or serration-shaped, club-shaped interfacial layer has the highest mechanical property in current study.     

Computer simulation study of the atomistic mechanism of deformation and fracture initiation in thin fcc metal films

A molecular dynamics computer simulation was performed using a potential of embedded atom method (EAM) of copper so as to study the atomistic mechanism of plastic deformation in thin metal films. When a thin film of Cu crystal is elongated by more than 8%, small islands of surplus atoms start to nucleate between (111) planes. The formation of new (111) islands occurs by the movement of atoms along the 111 direction from two successive (111) planes. Multiplication of these (111) planes serves as the mechanism of elongation, releasing the accumulated elastic stress. Vacancies and small clusters, thereof, are left behind at the positions from which atoms have migrated and also at the part of newly nucleated (111) planes in which atoms were not filled completely. Stacking fault tetrahedra (sft) are nucleated directly in the deformed region as a result of movement of atoms along 111 on the tetrahedral (111) cap.     

On a moving dielectric crack in a piezoelectric interface with spatially varying properties

This paper provides a comprehensive theoretical analysis of a finite crack propagating with constant speed along an interface between two dissimilar piezoelectric media under inplane electromechanical loading. The interface is modeled as a graded piezoelectric layer with spatially varying properties (functionally graded piezoelectric materials, i.e., FGPMs). The analytical formulations are developed using Fourier transforms and the resulting singular integral equations are solved with Chebyshev polynomials. Using a dielectric crack model with deformation-dependent electric boundary condition, the dynamic stress intensity factors, electric displacement intensity factor, crack opening displacement (COD) intensity factor, and energy release rate are derived to fully understand this inherent mixed mode dynamic fracture problem. Numerical simulations are made to show the effects of the material mismatch, the thickness of the interfacial layer, the crack position, and the crack speed upon the dynamic fracture behavior. A critical state for the electromechanical loading applied to the medium is identified, which determines whether the traditional impermeable (or permeable) crack model serves as the upper or lower bound for the dielectric model considering the effect of dielectric medium crack filling.     

Fatigue strength of the Cu/Si interface in nano-components

The fracture behavior of the Cu/Si interface in a nano-cantilever specimen with a 200 nm-thick Cu film (Specimen-200), which possesses a nanometer-scale strain-concentrated region, is examined under a cyclic bending load. The fatigue strength is around GPa level owing to the high yield stress of the Cu nano-film and the deformation constraint associated with the neighboring hard materials. The S-N curve shows clear dependence of fatigue life on the applied stress in the high-stress range, Δσ. Specimens with a 20 nm-thick Cu film (Specimen-20) are also investigated for comparison. The stress range in the fatigue fracture of Specimen-20 is higher than that of Specimen-200 for the same fatigue life. However, there is good coincidence in the Δσ/σ_s (σ_s: strength in monotonic load) vs. N_f (number of cycles to fracture) at high Δσ. The S-N curves suggest the existence of a fatigue threshold (Δσ_w) at low Δσ. The ratio of fatigue limit to the fracture stress in a monotonic loading, Δσ_w/σ_s, is large compared with the magnitude of bulk metal, which suggests the brittle behavior of the interface. Moreover, the fatigue limits have good coincidence with their yield stresses.     

Effect of thermomechanical loads on the propagation of crack near the interface brittle/ductile

The purpose of this paper is to understand the combined effect of thermal and mechanical loading on the initiation and behaviour of sub-interface crack in the ceramic. In this study a 2D finite element model has been used to simulated mixed mode crack propagation near the bimaterial interface. The assembly ceramometalic is subjected simultaneously to thermomechanical stress field. The extent of a plastic zone deformation in the vicinity of the crack-tip has a significant influence on the rate of its propagation. The crack growth at the joint specimen under four-point bending (4PB) loading and the influence of residual stresses was also evaluated by the maximum tensile stress criterion. The J-integral at the crack tip is generally expressed by the thermomechanical local stresses. The results obtained show the effect of the temperature gradient ΔT, the size of the crack and the applied stresses on the J-integral.     

A cumulative damage theory for fatigue crack initiation and propagation

A method for evaluating the cumulative damage resulting from the application of cyclic stress (or strain) sequences of varying amplitude is presented. Both the crack initiation and propagation stages of the fatigue failure process are included. The development is based on the concept of plastic strain energy dissipation as a function of cyclic life. The damage accumulated at any stage is evaluated from a knowledge of the fatigue limit in the initiation phase and an ‘apparent’ limit obtained through fracture mechanics for the propagation phase. The proposed damage theory is compared with two-level strain cycle test data of thin-walled specimens, and is found to be in fairly good agreement.