Statistical analysis and predictions of fracture and fatigue of micro-components

A number of materials typically used in MEMS technology exhibit brittle fracture behaviour which leads to a scatter in strength and a size effect as a consequence. Furthermore, some of these materials, e.g. polycrystalline silicon, show fatigue effects which limit the lifetime under cyclic loading conditions. Probabilistic methods based on the Weibull theory have been established successfully in predicting the strength of micro-components under static loading. However, the consequence of fatigue on reliability predictions has not yet been studied extensively. We present strength as well as lifetime predictions for poly-silicon components with stress concentrations based on experimental data published in the literature. Our results show that while strength predictions for components with stress concentrations based on scaling procedures works well, lifetime prediction is a challenging task associated with large prediction uncertainties. Finally, we relate the crack propagation approach used for our lifetime predictions with micro-mechanical fatigue models that are discussed for poly-silicon.     

Fracture Properties of Silicon Carbide Thin Films by Bulge Test of Long Rectangular Membrane

This paper reports the mechanical properties and fracture behavior of silicon carbide (3C-SiC) thin films grown on silicon substrates. Using bulge testing combined with a refined load-deflection model of long rectangular membranes, which takes into account the bending stiffness and prestress of the membrane material, the Young's modulus, prestress, and fracture strength for the 3C-SiC thin films with thicknesses of 0.40 and 1.42 mum were extracted. The stress distribution in the membranes under a load was calculated analytically. The prestresses for the two films were 322 plusmn 47 and 201 plusmn 34 MPa, respectively. The thinner 3C-SiC film with a strong (111) orientation has a plane-gstrain moduli of 415 plusmn 61 GPa, whereas the thicker film with a mixture of both (111) and (110) orientations exhibited a plane-strain moduli of 329 plusmn 49 GPa. The corresponding fracture strengths for the two kinds of SiC films were 6.49 plusmn 0.88 and 3.16 plusmn 0.38 GPa, respectively. The reference stresses were computed by integrating the local stress of the membrane at the fracture over edge, surface, and volume of the specimens and were fitted with Weibull distribution function. For the 0.40-mum-thick membranes, the surface integration has a better agreement between the data and the model, implying that the surface flaws are the dominant fracture origin. For the 1.42-mum-thick membranes, the surface integration presented only a slightly better fitting quality than the other two, and therefore, it is difficult to rule out unambiguously the effects of the volume and edge flaws. [2007-0191].     

Fracture testing of bulk silicon microcantilever beams subjected toa side load

A custom experimental system was developed to fracture silicon microcantilever beams in side loading (i.e., the load was applied in the noncompliant direction), and the resulting force/deflection (stiffness) characteristics were obtained. A finite element model of these structures was analyzed using ABAQUS, and the resulting model stiffness correlated well with the experimental data. Fracture types were divided into two categories, {111} and {110}, according to the type of silicon crystalline plane along which fracture occurred. The initiation location of each fracture type was identified. The fracture stress (strength) in the beam was obtained from the stress produced in the model at the fracture initiation site for a load equivalent to the experimental fracture force. Numerous beams were tested, and the statistical results were compiled. The distributions and statistical data from each of the fracture types were compared to each other and to previously acquired results from front/back loading (i.e., loading in the compliant direction) of these same structures. Side-loading results indicated that the {110} fracture type had a greater fracture strength than the {111} type. Based on a comparison of the side loading data with the front/back loading data, it was concluded that side wall roughness and especially the edge roughness greatly affected the fracture strength of the silicon micromechanical structures     

A MEMS-Based Evaluation of the Mechanical Properties of Metallic Thin Films

Characterizing the mechanical properties of metal thin films is critical for the design and fabrication of metal microelectromechanical systems and integrated circuit devices. This paper focuses on wafer-level determination of the mechanical behavior of sputtered aluminum and nickel thin films, using a variety of measurement techniques. Elastic moduli have been determined in devices fabricated with standard micromachining techniques using bulge testing of square diaphragms and lateral resonator structures. We find a Young's modulus of ~70 GPa for Al and ~200 GPa for Ni, in agreement with data for the bulk metals. Using pressurize/depressurize cycles, the load-deflection curves of the membranes have also been determined, and in conjunction with finite element simulations, were used to determine the yield strength and fracture strength of these films. Residual stresses in the films have also been investigated using wafer curvature, bulge testing, and X-ray diffraction. The merits of each measurement technique are discussed.     

Initial deflection of silicon-on-insulator thin membrane micro-mirror and fabrication of varifocal mirror

Thin membranes fabricated from silicon-on-insulator (SOI) wafer are valuable for deformable mirrors. The mirror is controlled to generate a specific wave-front with precision smaller than a wavelength. Here, we investigate quantitatively the initial deflection of the thin membrane mirrors fabricated from SOI wafer, which are often used in micro-electro-mechanical systems. A 1-μm-thick and 450-μm-diameter mirror fabricated from SOI wafer deflects upward around the circumference at an angle of 0.12°. The maximum deflection of the mirror is 320 nm at the center. The stress conditions of the mirrors are analyzed on the basis of material strength theory. The deflection is explained by the residual stress of the buried oxide layer of SOI wafer. The in-plane stresses of the micro-mirrors of diameters from 450 μm to 860 μm range from compressive stress of 1.2 MPa to tensile stress of 2.1 MPa. Furthermore, based on the above experimental and theoretical analyses, a 1-μm-thick varifocal micro-mirror of the diameter of 400 μm is fabricated. The focus of the mirror is varied from −28 mm to 21 mm with the deviation smaller than 4 nm from parabola in the mirror central region.     

Fabrication of silicon and oxide membranes over cavities usingion-cut layer transfer

Silicon and oxide membranes were fabricated using an ion-cut layer transfer process, which is suitable for sub-micron-thick membrane fabrication with good thickness uniformity and surface micro-roughness. After hydrogen ions were implanted into a silicon wafer, the implanted wafer was bonded to another wafer that has patterned cavities of various shapes and sizes. The bonded pair was then heated until hydrogen-induced silicon layer cleavage occurred along the implanted hydrogen peak concentration, resulting in the transfer of the silicon layer from one wafer to the other. Using this technique, we have been able to form sealed cavities and channels of various shapes and sizes up to 50-μm wide, with a 1.6-μm-thick silicon membrane. As a process variation, we have also fabricated silicon dioxide membranes for optically transparent applications     

Micromechanical force sensors based on SU-8 resist

This paper presents an innovative approach to develop highly sensitive 3D force sensors. In order to increase the probing sensitivity by using a material with low Young’s Modulus, a novel force sensor design based on SU-8 polymer is realized. Therefore, a low cost fabrication process is developed accompanied by design studies and an estimation of mechanical properties of the deforming elements. This paper will present the fabrication process as well as a distinguishing set of test results, analytical results, and simulations on the characterization of the presented SU-8 force sensor. The novel SU-8 design consists of an SU-8 boss-membrane with integrated piezoresistive elements. The SU-8 membranes are structured using photolithography. The SU-8 micromechanical structures are characterized to determine film stresses, bending stiffness, displacement of the stylus, and breaking points. Included in these tests are measurement of the stress behavior at different process steps and simulation of stress distribution in the membrane at different directions of loading. In addition a comparative analytical investigation of the structures is carried out particularly with regard to the displacement of the stylus.     

Influence of the frequency on fatigue of directly wafer-bonded silicon

Fatigue tests on directly wafer-bonded silicon samples were performed using pre-cracked Micro-Chevron samples applying cycling loading frequencies between 0.3 and 40 Hz. The experimental lifetime results were compared with a theoretical prediction using measured subcritical crack growth parameters under static loading conditions. The experimental investigations revealed that the number of cycles required to break the samples increased with frequency. In contrast, the corresponding time-to-failure values did not depend on frequency. Both the qualitative behavior and the quantitative life-time results agreed very well with a prediction based on a fracture mechanical model. Therefore, it could be concluded that fatigue behavior in the considered frequency range is solely controlled by stress corrosion in the bonded interface. Furthermore, the results demonstrate an available approach for life-time prediction of wafer-bonded micro-electro-mechanical systems components stressed by cycling loading.     

Large deflections of circular isotropic membranes subjected to arbitrary axisymmetric loading

Circular membranes with fixed peripheral edges, subjected to arbitrary axisymmetric loading were analyzed. A single governing differential equation in terms of radial stress was used. This nonlinear governing equation was solved using the finite difference method in conjunction with Newton-Raphson method. Three loading cases, namely (1) uniformly loaded membrane, (2) a membrane with uniform load over an inner portion, and (3) a membrane with ring load, were analyzed. Calculated central displacement and the central and edge radial stresses for uniformly loaded membrane, agreed extremely well with the classical solution.     

Microfabrication of palladium-silver alloy membranes for hydrogen separation

In this paper, a process for the microfabrication of a wafer-scale palladium-silver alloy membrane (Pd-Ag) is presented. Pd-Ag alloy films containing 23 wt% Ag were prepared by co-sputtering from pure Pd and Ag targets. The films were deposited on the unetched side of a <110>-oriented silicon wafer in which deep grooves were etched in a concentrated KOH solution, leaving silicon membranes with a thickness of ca. 50 /spl mu/m. After alloy deposition, the silicon membranes were removed by etching, leaving Pd-Ag membranes. Anodic bonding of thick glass plates (containing powder blasted flow channels) to both sides of the silicon substrate was used to package the membranes and create a robust module. The hydrogen permeability of the Pd-Ag membranes was determined to be typically 0.5 mol H/sub 2//m/sup 2//spl middot/s with a minimal selectivity of 550 for H/sub 2/ with respect to He. The mechanical strength of the membrane was found to be adequate, pressures of up to 4 bars at room temperature did not break the membrane. The results indicate that the membranes are suitable for application in hydrogen purification or in dehydrogenation reactors. The presented fabrication method allows the development of a module for industrial applications that consists of a stack of a large number of glass/membrane plates.     

FEM analysis of concrete structures subjected to mode-I and mixed-mode loading conditions

In this study, the finite element method (FEM) for a body containing displacement discontinuity is used for the investigation of tensile fracture behavior under mode-I and mixed-mode loading conditions in concrete structures. A mechanical model for the tensile fracture behavior is reduced to a mathematical problem, and the analysis method is proposed. With the aid of this method, several factors which govern tensile fracture are examined, such as the unloading path in the tension-softening behavior and the transmission of shear stresses across crack surfaces. A plain concrete beam without a notch is analyzed by first neglecting and then taking into account the unloading path in the tension-softening behavior to demonstrate the phenomenon of cracking localization in mode-I crack growth. Pullout test specimens of practical significance are analyzed in order to study the crack growth phenomenon under mixed-mode loading conditions. Cases with and without lateral confinement are considered and the results obtained from the present analysis are compared with those obtained from available experimental data. A simple model for shear transfer across crack surfaces is established. By incorporating this model in the program, a pullout test specimen with lateral confinement is analyzed to examine the influence of shear transfer across crack surfaces on cracking localization.     

Microrivets for MEMS packaging: concept, fabrication, and strengthtesting

Microriveting is introduced as a novel and alternative joining technique to package MEMS devices. In contrast to the existing methods, mostly surface bonding, the reported technique joins two wafer pieces together by riveting, a mechanical joining means. Advantages include wafer joining at room temperature and low voltage, and relaxed requirements for surface preparation. The microrivets, which hold a cap-base wafer pair together, are formed by filling rivet holes through electroplating. The cap wafer has a recess to house the MEMS devices and also has through-holes to serve as rivet molds. The seed layer on the base wafer becomes the base of the rivet. The process requires only simple mechanical clamping of the wafer pair during riveting, compared with the more involved procedures needed for wafer bonding. Directionality of electroplating in an electric field is what makes this process simple and robust. Strength testing is carried out to evaluate the joining with microrivets. Different modes of rivet failure under different loading conditions are identified and investigated. Effective strength between 7 and 11 MPa was measured under normal loading with nickel microrivets. Joining strengths comparable to conventional wafer bonding processes, ease of fabrication with repeatability, and compatibility with batch fabrication show that microriveting is a feasible technique to join wafers for MEMS packaging, especially when hermetic sealing is not essential     

The effect of atmospheric moisture on crack propagation in the interface between directly bonded silicon wafers

Infra-red video sequences were taken of directly bonded silicon wafer pairs undergoing the razor blade crack length bond strength measurement in a specially designed jig. A series of tests were carried out under controlled atmospheres of nitrogen at various relative humidities. Analysis of the video images showed that the crack continues to propagate rapidly for several minutes after the blade has stopped moving, and that the presence of moisture has a strong positive influence on the rate of crack propagation under static loading. A new Maszara protocol is suggested based on modelling crack growth using our experimentally derived constants.     

Optimization of axisymmetric elastic modulus distributions around a hole for increased strength

Holes in engineering structures cause stress concentrations that often lead to failure. In nature, however, blood vessel holes (foramina) in load-bearing bones are not normally involved in structural failures. It has been found that this behavior is linked to the material distribution near the hole. In the present paper, we have investigated the effectiveness of optimizing the radial distribution of the isotropic elastic modulus around a circular hole to increase load-carrying capacity. Bezier curves were used to describe the radial distribution of the elastic modulus. Since changing the elastic modulus usually affects the strength, the ratio of maximum principal stress to strength was chosen as the objective function for optimization. Using non-dimensional analysis of the 2-D elasticity equations, we identified three parameters that govern the optimum design and are applicable to a wide range of materials, loading, and geometries. The first is a material parameter that describes the relationship between the strength and elastic modulus, the second is the geometric parameter given by the ratio of the optimized field to the hole radius, and the third is the biaxial load ratio. The effect of failure criterion choice on the optimum elastic modulus distribution is also investigated. Optimum elastic modulus distributions for materials whose strength increases faster than the stiffness, as density and/or composition is varied, completely eliminated the effect of the hole by locally stiffening areas that experience high stresses. When the strength lagged behind the stiffness, optimum designs were similar to those found in bones, and relied on modulus distributions that direct the loads away from the hole.     

Strongly buckled square micromachined membranes

The buckling of compressively prestressed square membranes with built-in edges is investigated experimentally and analyzed theoretically. The buckling depends weakly on Poisson's ratio and essentially is a function of the reduced prestrain, ε¯₀ =ε₀a²/h², where ε₀ is the physical prestrain, a is the width, and h is the thickness of the membrane. As ε¯₀ becomes increasingly negative, the membrane undergoes two symmetry breaking buckling transitions. Beyond the first transition occurring at ε¯_cr1, the buckling profile has all the reflection and rotation symmetries of a square. The reflection symmetries are lost through a second instability transition at ε¯_cr2. The bifurcation points, ε¯_cr1 and ε¯_cr2, and buckling profiles were calculated using analytical energy minimization and nonlinear finite-element simulation. Both methods agree. The buckling of micromachined plasma-enhanced chemical vapor deposition silicon nitride membranes on a silicon wafer is interpreted in terms of the theoretical results. Good matching between measured and calculated buckling profiles is found. The extracted strain values are consistent irrespective of the size and buckling mode of the membranes. From the average strain across the wafer ε₀=-3.50×10^-4 and complementary wafer curvature measurements, a Young's modulus of 130 GPa is deduced. Methods for the straightforward extraction of ε₀ from experimental center deflections of buckled square membranes are described     

Fracture and fatigue behavior of single crystal silicon microelements and nanoscopic AFM damage evaluation

Quasi-static bending and fatigue tests of single-crystal silicon microelements fabricated by photoetching were performed. The microelements were subjected to simple bending and three-point bending with two-support roll length of 1.5 mm. The tests were conducted by using a specially designed electromagnetic actuator based testing machine (load range: 0.1 mN–5 N, accuracy: 0.02 mN), which enables mechanical testing including fatigue of microelements. Mechanical testing including fatigue of microelements could be performed with sufficient precision. Single-crystal silicon microelements deformed elastically until final catastrophic failure, showing a brittle nature. The influence of specimen size on quasi-static fracture behavior was investigated: fracture strength increased with a decrease in sample width, and the maximum fracture strength reached 7.7 GPa. The influence of water on fatigue strength was discussed. The fracture surface and sample surface were examined using an atomic force microscope. Nanoscopic damage during testing was evaluated, and the fracture mechanisms were discussed. Received 20 October 1997/Accepted 5 January 1998     

Geometrically nonlinear analysis of structural membranes

A finite element analysis technique for the solution of geometrically nonlinear problems of flat arbitrary structural membranes subject to general loading is presented. The discrete model employed is a general isoparametric quadratic element with three degrees of freedom per node. Results for membrane deformations and internal stresses are shown to be consistent with other available solutions.     

Stress analysis of the concrete housing of a water intake structure

The stress analysis of the concrete housing of a water intake structure is studied to locate destressed areas and areas of high stress concentration on the structure, and to predict the distribution of the displacements. The analysis is performed in the loading cases of end-of-construction, backfilling and reservoir water loading. The distribution of principal stresses indicates that the walls of the structure are subject to axial load and bending and that in certain areas pure tension or compression is present. Maximum tensile and compressive stresses are within the allowable strength limits for concrete.     

A new technique for measuring the mechanical properties of thinfilms

Accurate measurement of mechanical properties is very difficult for films that are only a few microns thick. Previously, these properties have been determined by indirect methods such as cantilever beam and diaphragm bulge tests. This paper presents a new technique to measure the Young's modulus of thin films in a direct manner consistent with its definition. Strain is measured by a laser-based technique that enables direct and accurate recording of strain on a thin-film specimen. Load is recorded with a 1-lb load cell, and an air bearing is used to eliminate friction in the loading system. The specimen is phosphorus-doped polysilicon that has a gage cross section of 3.5 μm thick by 600 μm wide. All 29 uniaxial tensile tests show brittle behavior, and the average values of Young's modulus and fracture strength are measured to be 170±6.7 GPa and 1.21±0.16 GPa, respectively. One fatigue test is also reported in this paper