Size Effect on The Bending And Tensile Strength of Micromachined Polysilicon Films for MEMS

I. INTRODUCTION Microelectromechanical systems (MEMS) have achieved impressive progress and become a very area of research. But long-term durability of various MEMS devices requires a fundamental understaof the fatigue and fracture characteristics of su…     

Effect of intrinsic stress gradient on the effective mode-I fracture toughness of amorphous diamond-like carbon films for MEMS

The influence of intrinsic stress gradient on the mode-I fracture of thin films with various thicknesses fabricated for Microelectromechanical Systems (MEMS) was investigated. The material system employed in this study was hydrogen-free tetrahedral amorphous diamond-like carbon (ta-C). Uniform gauge microscale specimens with thicknesses 0.5, 1, 2.2, and 3 μm, containing mathematically sharp edge pre-cracks were tested under mode-I loading in fixed grip configuration. The effective opening mode fracture toughness, as calculated from boundary force measurements, was 4.25±0.7 MPa√m for 0.5-μm thick specimens, 4.4±0.4 MPa√m for 1-μm specimens, 3.74±0.3 MPa√m for 2.2-μm specimens, and 3.06±0.17 MPa√m for 3-μm specimens. Thus, the apparent fracture toughness decreased with increasing film thickness. Local elastic property measurements showed no substantial change as a function of film thickness, which provided evidence for the stability of the sp²/sp³ carbon binding stoichiometry in films of different thicknesses. Detailed experiments and finite element analysis pointed out that the dependence of the effective fracture toughness on specimen thickness was due to the intrinsic stress gradient developed during fabrication and post-process annealing. This stress gradient is usually unaccounted for in mode-I fracture experiments with thin films. Thicker films, fabricated from multiple thin layers, underwent annealing for extended times, which resulted in a stress gradient across their thickness. This stress gradient caused an out-of-plane curvature upon film release from its substrate and, thus, combined bending and tensile mode-I loading at the crack tip under in-plane forces. Since the bending component cannot be isolated from the applied boundary force measurements, its contribution, becoming important for thick films, remains unaccounted for in the calculation of the critical stress intensity factor, thus resulting in reduced apparent fracture toughness that varies with film thickness and curvature. It was concluded that in the presence of a stress gradient, accounting only for the average intrinsic stresses could lead in an overestimate of the fracture resistance of a brittle film. Under these considerations the material fracture toughness of ta-C, as determined from specimens with negligible curvature, is K_IC=4.4±0.4 MPa√m.     

Experimental Investigation of Strain Rate Dependence of Nanocrystalline Pt Films

A new microscale uniaxial tension experimental method was developed to investigate the strain rate dependent mechanical behavior of freestanding metallic thin films for MEMS. The method allows for highly repeatable mechanical testing of thin films for over eight orders of magnitude of strain rate. Its repeatability stems from the direct and full-field displacement measurements obtained from optical images with at least 25 nm displacement resolution. The method is demonstrated with micron-scale, 400-nm thick, freestanding nanocrystalline Pt specimens, with 25 nm grain size. The experiments were conducted in situ under an optical microscope, equipped with a digital high-speed camera, in the nominal strain rate range 10⁻⁶–10¹ s⁻¹. Full field displacements were computed by digital image correlation using a random speckle pattern generated onto the freestanding specimens. The elastic modulus of Pt, E = 182 ± 8 GPa, derived from uniaxial stress vs. strain curves, was independent of strain rate, while its Poisson’s ratio was v = 0.41 ± 0.01. Although the nanocrystalline Pt films had the elastic properties of bulk Pt, their inelastic property values were much higher than bulk and were rate-sensitive over the range of loading rates. For example, the elastic limit increased by more than 110% with increasing strain rate, and was 2–5 times higher than bulk Pt reaching 1.37 GPa at 10¹ s⁻¹.     

Effect of Thin Films on Dynamic Performance of Resonating MEMS

Continued advances in microelectromechanical systems (MEMS) technology have led to development of a multitude of new sensors and their corresponding advanced applications. Great many of these sensors (e.g., microgyroscopes, accelerometers, biological, chemical, security, medical, etc.) rely on either sensing elements or elastic suspensions that resonate. Regardless of their applications, sensors are always designed to provide the most exact responses to the signals they are developed to detect and/or monitor. One way to quantify this exactness is to use the Quality factor (Q-factor). MEMS sensors are typically fabricated out of materials that are mechanically sound at the microscale, but can be relatively poor electrical conductors. For this reason, areas of MEMS are coated with various thin metal films to provide electrical pathways. These films, however, adversely alter resonant properties of a device. To facilitate our study, microcantilever configurations were selected to test influence that thin metal films have on resonators. This paper reviews a theoretical analysis of the effect that thermoelastic internal friction has on the Q-factor of microscale resonators and shows that the internal friction relating to TED is a fundamental damping mechanism in determination of quality of high-Q resonators over a range of operating conditions. Using silicon microcantilevers coated with aluminum films from 5 nm to 30 nm thick, on one as well as both sides, Q-factors were experimentally determined using the ring-down method. From the ring-down curve, the Q-factor of each microcantilever was determined. Experimental results show that as thickness of the aluminum film increases, Q-factor of the device decreases. Comparison of analytical and experimental results indicates good correlation, well within the limits based on uncertainty analysis. In addition, preliminary results also show a significant temperature dependence of the Q-factor of aluminum coated microcantilevers.     

A Sequential Tensile Method for Rapid Characterization of Extreme-value Behavior in Microfabricated Materials

A high-throughput sequential tensile test method has been developed to characterize the fracture strength distribution of microfabricated polycrystalline silicon, the primary structural material used in microelectromechanical systems (MEMS). The resulting dataset of over 1,000 microtensile tests reveals subtle extreme-value behavior in the tails of the distribution, demonstrating that the common two-parameter Weibull distribution is inferior to a three-parameter Weibull model. The results suggest the existence of a cut-off or threshold stress (1.446 GPa for this particular material) below which tensile failure will not occur. The existence of a cut-off stress suggests that the material’s flaw size distribution and toughness distribution are both also bounded. From an application perspective, the cut-off stress provides a statistically-sound basis for reliable design. While the sequential method is demonstrated here for tensile strength distributions in polycrystalline silicon MEMS, the technique could be extended to a wide range of mechanical tests (bending strength, elastic modulus, fracture toughness, creep, etc.) for both microsystem and conventional materials.     

Experiments and theory in strain gradient elasticity

Conventional strain-based mechanics theory does not account for contributions from strain gradients. Failure to include strain gradient contributions can lead to underestimates of stresses and size-dependent behaviors in small-scale structures. In this paper, a new set of higher-order metrics is developed to characterize strain gradient behaviors. This set enables the application of the higher-order equilibrium conditions to strain gradient elasticity theory and reduces the number of independent elastic length scale parameters from five to three. On the basis of this new strain gradient theory, a strain gradient elastic bending theory for plane-strain beams is developed. Solutions for cantilever bending with a moment and line force applied at the free end are constructed based on the new higher-order bending theory. In classical bending theory, the normalized bending rigidity is independent of the length and thickness of the beam. In the solutions developed from the higher-order bending theory, the normalized higher-order bending rigidity has a new dependence on the thickness of the beam and on a higher-order bending parameter, b_h. To determine the significance of the size dependence, we fabricated micron-sized beams and conducted bending tests using a nanoindenter. We found that the normalized beam rigidity exhibited an inverse squared dependence on the beam's thickness as predicted by the strain gradient elastic bending theory, and that the higher-order bending parameter, b_h, is on the micron-scale. Potential errors from the experiments, model and fabrication were estimated and determined to be small relative to the observed increase in beam's bending rigidity. The present results indicate that the elastic strain gradient effect is significant in elastic deformation of small-scale structures.     

A review of MEMS-based microscale and nanoscale tensile and bending testing

Thin films at the micrometer and submicrometer scales exhibit mechanical properties that are different than those of bulk polycrystals. Industrial application of these materials requires accurate mechanical characterization. Also, a fundamental understanding of the deformation processes at smaller length scales is required to exploit the size and interface effects to develop new and technologically attractive materials. Specimen fabrication, small-scale force and displacement generation, and high resolution in the measurements are generic challenges in microscale and nanoscale mechanical testing. In this paper, we review small-scale materials testing techniques with special focus on the application of microelectromechanical systems (MEMS). Small size and high force and displacement resolution make MEMS suitable for small-scale mechanical testing. We discuss the development of tensile and bending testing techniques using MEMS, along with the experimental results on nanoscale aluminum specimens.     

Mechanical properties of ultrananocrystalline diamond thin films relevant to MEMS/NEMS devices

The mechanical properties of ultrananocrystalline diamond (UNCD) thin films were measured using microcantilever deflection and membrane deflection techniques. Bending tests on several free-standing UNCD cantilevers, 0.5 μm thick, 20 μm wide and 80 μm long, yielded elastic modulus values of 916–959 GPa. The tests showed good reproducibility by repeated testing on the same cantilever and by testing several cantilevers of different lengths. The largest source of error in the method was accurate measurement of film thickness. Elastic modulus measurements performed with the novel membrane deflection experiment (MDE), developed by Espinosa and co-workers, gave results similar to those from the microcantilever-based tests. Tests were performed on UNCD specimens grown by both micro and nano wafer-seeding techniques. The elastic modulus was measured to be between 930–970 GPa for the microseeding and between 945–963 GPa for the nanoseeding technique. The MDE test also provided the fracture strength, which for UNCD was found to vary from 0.89 to 2.42 GPa for the microseeded samples and from 3.95 to 5.03 for the nanoseeded samples. The narrowing of the elastic modulus variation and major increase in fracture strength is believed to result from a reduction in surface roughness, less stress concentration, when employing the nanoseeding technique. Although both methods yielded reliable values of elastic modulus, the MDE was found to be more versatile since it yielded additional information about the structure and material properties, such as strength and initial stress state.     

MEMS-based Universal Fatigue-Test Technique

We have developed a MEMS (micro electro mechanical systems)—based method for fatigue testing of micrometer— millimeter-sized specimens of any material (hence ‘universal’). The miniature, re-usable, stand-alone fatigue test frame is fabricated as a single MEMS chip. Specimens of any material can be manually mounted in the chip and fatigue-tested. We describe the design and construction of the MEMS device and specimens, the test protocol and data analysis procedure, and show stress versus number of cycles to failure (S-N) results for 25 μm thick Al 1145 H19 foil. The S-N results are in accord with expectations, and examination of the fracture surface by scanning electron microscopy shows distinct regions corresponding to slow and fast crack growth.     

横观各向同性弹性半空间地基上正交异性矩形中厚板弯曲解析解

对横观各向同性体通解进行双重傅里叶变换,获得了直角坐标系下横观各向同性弹性半空间地基受任意竖向荷载作用下的位移积分变换解;在此基础上建立了板与地基的变形协调方程,并与三个广义位移变量描述的弹性地基上四边自由正交各向异性矩形中厚板的弯曲控制方程相结合,用三角级数法,得出横观各向同性弹性半空间地基上四边自由正交异性矩形中厚板受任意竖向荷载作用的弯曲解析解。相关算例分析表明,本文方法是有效的。     

Numerical stress-strain analysis of a nonthin elastic plate

A numerical solution to elastic-equilibrium problems for nonthin plates is proposed. The solution is obtained by using the curvilinear-mesh method in combination with Vekua’s method. The efficiency (rapid convergence and accuracy) of this approach is demonstrated by solving test problems for thick plates that can also be solved exactly or approximately by other methods. A numerical solution is obtained to the bending problem for orthotropic nonthin plates of constant and varying thickness __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 3, pp. 119–126, March 2006.     

非线性MEMS微梁的重心有理插值迭代配点法分析

文章利用重心有理插值迭代配点法分析计算非线性MEMS微梁问题。通过处理MEMS微梁的几何通过假设初始函数,将微梁非线性控制方程转换为线性化微分方程,建立逼近非线性微分方程的线性化迭代格式。采用重心有理插值配点法求解线性化微分方程,提出了数值分析MEMS微梁非线性弯曲问题的重心插值迭代配点法。给出了非线性微分方程的直接线性化和Newton线性化计算公式,详细讨论了非线性积分项的计算方法和公式。利用重心有理插值微分矩阵,建立了矩阵-向量化的重心插值迭代配点法的计算公式。数值算例结果表明,重心插值迭代配点法求解微梁非线性弯曲问题,具有计算公式简单、程序实施方便和计算精度高的特点。     

Bending of annular sectorial Mindlin Plates using Kirchhoff results

This study is concerned with the elastic bending problem of a class of annular sectorial plates whose radial edges are simply supported. Exact bending relationships between the Mindlin plate results and the corresponding Kirchhoff plate solutions have been derived based on the concept of load equivalence. These bending relationships facilitate the deduction of thick (Mindlin) plate results from the corresponding classical thin (Kirchhoff) plate solutions, thus bypassing the need to solve the more complicated governing equations of thick plates. The correctness of the relationships is established by solving the bending problem of annular sectorial plates under a uniformly distributed load and comparing the results with existing thick plate solutions.     

Mechanics of shaft-loaded blister test for thin film suspended on compliant substrate

Based on the von Kármán plate theory, the mechanics of a shaft-loaded blister test for thin film/substrate systems is studied by considering elastic substrate deformations and residual stresses in these films. In testing, films are attached to a substrate provided with a circular hole, through which loading is applied to the film by a flat-ended shaft of circular cross-section. The effect of substrate deformation on the deflection of the loaded film is taken into account by using a line spring model. For small deflections, an analytical solution is derived, while for large deflections a numerical solution is obtained using the shooting method. The resulting load-shaft displacement relation, which is essential in blister tests, compares favorably with finite element analysis.     

Experimental Study of Bending Behaviour of Reinforcements

In composite reinforcement shaping, textile preform undergo biaxial tensile deformation, in plane shear deformation, transverse compaction and out-of-plane bending deformations. Bending deformations have been neglected in some simulation codes up to now, but taking into account them would give more accurate simulations of forming especially for stiff and thick textiles. Bending behaviour is specific because the reinforcements are structural parts and out of plane properties cannot be directly deduced from in-plane properties, like for continuous material. Because the standard tests are not adapted for stiff reinforcements with non linear behaviour a new flexometer using optical measurements has been developed to test such reinforcements. This new device enables to carry out a set of cantilever tests with different histories of load. A series of tests has been performed to validate the test method and to show the capacities of the new flexometer to identify non linear non elastic behaviour.     

On elastocapillarity: A review

Elastocapillary phenomena involving elastic deformation of solid structures coupled with capillary effects of liquid droplets/films can be observed in a diversity of fields,e.g.,biology and microelectromechanical systems(MEMS).Understanding the physical mechanisms underlying these phenomena is of great interest for the design of new materials and devices by utilizing the effects of surface tension at micro and nano scales.In this paper,some recent developments in the investigations on elastocapillary phenomena are briefly reviewed.Especially,we consider the deformation,adhesion,self-assembly,buckling and wrinkling of materials and devices induced by surface tensions or capillary forces.The main attention is paid to the experimental results of these phenomena and the theoretical analysis methods based on continuum mechanics.Additionally,the applications of these studies in the fields of MEMS,micro/nanometrology,and biomimetic design of advanced materials and devices are discussed.     

Deformation problem for an elastically fixed plate modeling a coating partially delaminated from the substrate (Plane Strain)

An asymptotic solution of the problemindicated in the title is obtained at distances large compared with the plate width and some promising methods for its use, in particular, for calculating the coefficients in the boundary conditions of the plate elastic fixation which models a coating partially delaminated from the substrate, are outlined. The possibility of considering the delamination in the approximation of the plate weak bending (the plate approximation) and the possibility of neglecting the tangential stress action along the contact boundary are implemented. The substrate is considered as a half-infinite elastic solid. This solution was obtained by using the Fourier transform and the solution of the resulting equation by the Wiener-Hopf method. The obtained asymptotic solution can be used to study problems related to coating delamination, especially on soft thick substrata.     

Determination of material properties using nanoindentation and multiple indenter tips

Nanoindentation is a useful method to probe the material properties of a solid. Its effective use lies in interpreting the data collected from a nanoindentation experiment with an associated analytical/numerical solution of the corresponding problem configuration. In this paper, a parametric finite element study has been performed to develop a new procedure for extracting elastic–plastic properties of a material through nanoindentation experiments with a substantially improved accuracy for the elastic properties of a elastic–plastic solid. The procedure involves data collected through the use of two, different, nanoindenter tips. Non-dimensional functions were constructed for two different indenter geometries to show that test results from multiple indenters, when appropriately manipulated, deliver superior results, compared to using one indenter. The material was assumed to be an isotropic elastic–plastic solid with power law hardening. Friction between the indenter and the material was included in the cases studied. The ratio of yield strength to elastic modulus was assumed to be in the range 0.0005–0.02 and the hardening coefficient was assumed to be between 0 and 0.4. Poisson’s ratio was fixed at 0.3.     

A methodology for determining mechanical properties of freestanding thin films and MEMS materials

We have developed a novel chip-level membrane deflection experiment particularly suited for the investigation of sub-micron thin films and microelectro-mechanical systems. The experiment consists of loading a fixed-fixed membrane with a line load applied at the middle of the span using a nanoindenter. A Mirau microscope interferometer is positioned below the membrane to observe its response in real time. This is accomplished through a micromachined wafer containing a window that exposes the bottom surface of the specimen. A combined atomic force microscope/nanoindenter incorporates the interferometer to allow continuous monitoring of the membrane deflection during both loading and unloading. As the nanoindenter engages and deflects the sample downward, fringes are formed and acquired by means of a CCD camera. Digital monochromatic images are obtained and stored at periodic intervals of time to map the strain field. Stresses and strains are computed independently without recourse to mathematical assumptions or numerical calibrations. Additionally, no restrictions on the material behavior are imposed in the interpretation of the data. In fact, inelastic mechanisms including strain gradient plasticity, piezo and shape memory effects can be characterized by this technique.The test methodology, data acquisition and reduction are illustrated by investigating the response of 1-μm thick gold membranes. A Young's modulus of , a yield stress of and a residual stress of are consistently measured. The post-yield behavior leading to fracture exhibits typical statistical variations associated to plasticity and microcrack initiation.