Gold Microbeams Research Articles

An exact solution on gold microbeam with thermoelastic damping via generalized Green-Naghdi and modified couple stress theories

The vibration of the gold microbeam resonator considering the effects of the material length scale under the temperature effect is investigated in this article. The modified couple stress theory is utilized to show the material length scale effects. The effects of temperature considering the thermoelastic damping are taken into account based on Green-Naghdi thermoelasticity theory. The Green-Naghdi coupled thermoelasticity and modified couple stress governing equations for Euler-Bernoulli beam are derived based on Hamilton’s principle. The Laplace method is utilized to solve the obtained equations. The inverse Laplace method as Talbot method is used to obtain the lateral deflection and temperature response of the microbeam in the time domain. The effects of the various parameters such as the nondimensional material length scale parameter, the thickness of the microbeam, and the initial temperature conditions, on the lateral deflection as well as temperature responses of the microbeam are studied to investigate the results. Raising the material length scale parameter and width to length ratio makes the lateral deflection of the beam decrease in the time domain. Increase in temperature shock and Poisson’s ratio makes the lateral deflection increase, and both have a direct effect on the lateral deflection in the time domain, except that this increase is linear for temperature shock. The temperature of the microbeam decreases by an increase in Poisson’s ratio and progressing in the axial direction. For the final observation, it should be noted that the couple stress has an inverse effect on the lateral deflection.

Correction to: A comparative study of modified strain gradient theory and modified couple stress theory for gold microbeams

This is an erratum to the previously published version due to systematical errors in figures. Only the legends and axes names of the Figs.

Thermoelastic vibrations of a Timoshenko microbeam based on the modified couple stress theory

The dependence of the quality factor of nonlinear microbeam resonators under thermoelastic damping for Timoshenko beams with regard to geometric nonlinearity has been studied. The constructed mathematical model is based on the modified couple stress theory which implies prediction of size-dependent effects in microbeam resonators. The Hamilton principle has yielded coupled nonlinear thermoelastic PDEs governing dynamics of the Timoshenko microbeams for both plane stresses and plane deformations. Nonlinear thermoelastic vibrations are studied analytically and numerically and quality factors of the resonators versus geometric and material microbeam properties are estimated. Results are presented for gold microbeams for different ambient temperatures and different beam thicknesses, and they are compared with results yielded by the classical theory of elasticity in linear/nonlinear cases.

A comparative study of modified strain gradient theory and modified couple stress theory for gold microbeams

Microbeams are common structures encountered in micro- and nano-electromechanical systems. Their mechanical response cannot be modelled by local theories of continuum mechanics due to size effect, which becomes more pronounced as the structural length scale approaches the microstructural length scale. The size effect can be circumvented by higher-order continuum theories. In this study, Euler–Bernoulli microbeams are analysed with modified strain gradient theory (MSGT) and modified couple stress theory (MCST). The weak forms for the numerical implementation are obtained by using variational methods. Then, the set of algebraic equations for the finite element method are derived. As a novel aspect, the performance of MSGT and MCST is compared and the length scale parameters of these theories are identified for gold microbeams from the existing experimental results in the literature. With the help of the identified parameters, the cut-off point for the applicability of the classical beam theories for gold microbeams is assessed. The study suggests use of higher-order theories for the state-of-the-art gold microbeam structures having thickness $$t<30\,\upmu {\hbox {m}}$$ .

Analysis of Gold Microbeams with Higher Order Continuum Theories

AbstractMicrobeams are building blocks for many microstructures as well as microelectromechanical systems (MEMS) and cannot accurately be modelled by classical continuum theories due to size effects based on their micro‐scale. These size effects can be taken into account by the so‐called higher order continuum theories. Modified Strain Gradient Theory (MSGT) and Modified Couple Stress Theory (MCST) are two commonly used theories, which extend the classical local continuum theories of grade one with the introduction of additional length scale parameters. In this contribution, the variational problem governing the elasticity of higher order beam formulation and the finite element implementation based upon, are briefly introduced. To this end, well known Euler‐Bernoulli beam formulation assumptions are used. The size effect for gold‐micro beams is demonstrated and the length scale parameters of gold microbeams for MSGT and MCST are identified form the existing experimental data from literature for the first time. As a novel aspect, significant size effect is demonstrated for the length‐scales associated with the state of the art gold microbeam structures developed for MEMS applications, which reveals the necessity of the use of higher order theories at these length scales. Advantages and drawbacks of these theories are also identified. (© 2017 Wiley‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Size effects on intergranular crack growth mechanisms in ultrathin nanocrystalline gold free-standing films

This study investigated the combined effects of thickness (30 vs 100 nm) and average grain size (40 vs 70 nm for the thicker films) on the crack propagation mechanisms in ultrathin nanocrystalline gold microbeams, using a microelectromechanical system device to perform in situ transmission electron microscope (TEM) tensile experiments. Monotonic tensile tests of the two types of microbeams show similar strength levels (∼400 MPa) and ductility (∼2%). However, the thicker specimens exhibit a much more ductile behavior under repeated stress relaxation experiments, which the in situ TEM experiments revealed to be related to differences in intergranular crack propagation mechanisms. The governing crack growth process is in both cases dominated by grain boundary dislocation activities leading to grain boundary sliding. For the thinner specimens, secondary nanocracks are generated (as a result of grain boundary sliding) ahead of the main crack and coalesce together. Instead, secondary nanocracks do not form ahead of the main crack for the thicker specimens; the main crack extends as a result of sustained grain boundary sliding at the crack tip.

ANALYSIS OF GOLD MICRO-BEAMS WITH MODIFIED STRAIN GRADIENT THEORY

Micro-beams are building blocks for many micro- and nano-structures as well as micro-electro-mechanical systems (MEMS) and cannot accurately be modeled by classical continuum theories due to significant size effects at the length scales associated with these structures. Size effects can be taken into account by the so-called higher order continuum theories. In this study, Euler-Bernoulli micro-beams are analyzed with the Modified Strain Gradient Theory (MSGT), which extends the classical local continuum theories of grade one with the introduction of three additional length scale parameters. In this contribution, finite element implementation is briefly demonstrated by using Galerkin discretization techniques for Euler-Bernoulli beams. The size effect for gold-micro beams is demonstrated and the length scale parameters of gold micro-beams for MSGT are identified form the existing experimental data from literature for the first time . As a novel aspect, significant size effect is demonstrated for the length-scales associated with the state of the art gold micro-beam structures developed for NEMS and MEMS applications, which reveals the necessity of the use of higher order theories at these length scales.

Effects of the Length Scale Parameter on the Thermoelastic Damping of a Microbeam Considering the Couple Stress Theory

This paper studies the thermoelastic damping in microbeams considering the couple stress theory with microstructure. This theory includes the microinertia effects, coming from the kinetic energy due to the velocity gradient through the differential macroelements. A Galerkin-based reduced order model and complex frequency approach have been used to determine the quality factor. For a gold microbeam as a case study, the obtained results for different ambient temperatures, beam lengths and thicknesses are compared to those obtained using the classic theory of elasticity. The comparison has been made for different values of the length scale parameter. The effects of the microinertia term on the magnitude of the thermoelastic damping have also been investigated and shown that for which conditions these effects are significant.

Effect of the speed, the rotation and the magnetic field on the Q-factor of an axially clamped gold micro-beam

In this paper, a study of the thermoelastic damping of a gold microbeam resonator in the context of the generalized thermoelasticity with one relaxation time has been improved. An explicit formula of thermoelastic damping has been derived in general form includes the speed of the beam in the direction of its principal axis and without any external body force. Influences of the speed, the rotation, and the magnetic field have been studied. We discussed the effect of the speed of the beam, the rotation, and the magnetic field on the thermoelastic damping (Q-factor). The moving and the rotation of the microbeam have significant effects on the thermoelastic damping and the energy dissipation. The external magnetic field with tiny induced magnetic field acts on the microbeam has a significant effect on the thermoelastic damping and the energy dissipation. The energy dissipation of the microbeam decreases when the speed of the microbeam, the rotation angular speed, and the external magnetic field acts on the microbeam decrease.

Nonlinear behavior of capacitive micro-beams based on strain gradient theory

This paper studies the size dependent behavior of materials in MEMS structures. This behavior becomes noticeable for a structure when the characteristic size such as thickness or diameter is close to its internal length-scale parameter and is insignificant for the high ratio of the characteristic size to the length-scale parameter, which is the case of the silicon base micro-beams. However, in some types of micro-beams like gold or nickel bases, the size dependent effect cannot be overlooked. In such cases, ignoring this behavior in modeling will lead to incorrect results. Some previous researchers have applied classic beam theory on their models and imposed a considerable hypothetical value of residual stress to match their theoretical results with the experimental ones. The equilibrium positions or fixed points of the gold and nickel micro-beams are obtained and shown that for a given DC voltage, there is a considerable difference between the obtained fixed points using classic beam theory, modified couple stress theory, and modified strain gradient theory. In addition, it is shown that the calculated static and dynamic pull-in voltages using higher order theories are much closer to the experimental results and are higher several times than those obtained by classic beam theory.

Experimental methods for the characterization of fatigue in microstructures

The mechanical fatigue behavior of gold microbeams is analyzed. Dedicated devices have been designed and built able to produce alternate loading on gold specimens; the electrostatic actuation is used as driving force. Gold beams are tested under both bending and tensile alternate loadings. Results were used to plot S-N curves and fatigue Goodman-Smith diagram in order to estimate the fatigue limit of the material in presence of mean and alternate stress conditions. The surface topography evolution is studied and failure modes are discussed.

Modeling of gold microbeams as strain and pressure sensors for characterizing MEMS packages

This work presents the modeling of gold microbeams for characterizing Micro-electro-mechanical systems (MEMS) packages in terms of both strains induced to the MEMS devices and hermetic sealing capability. The proposed test structures are based on arrays of rectangular-shaped clamped-free and clamped–clamped beams, to be realized with a film of electroplated gold by surface micromachining technology. The resonant frequency of the microbeams is modeled by FEM simulations as a function of substrate deformations, which could be induced by the package. Clamped–clamped bridges show a linear change of the square of the resonant frequency in case of in-plane deformations, in fairly good agreement with an approximate analytical model. Cantilever beams are modeled as variable capacitors to detect out-of-plane deformations. Finally, an analytical model to study cantilever beams as resonators for detecting pressure changes is discussed and compared with preliminary experimental results, showing an impact on the quality factor in a range from 10−2 mbar to 1 bar.

MEMS Mechanical Fatigue: Effect of Mean Stress on Gold Microbeams

In this paper, the reliability of gold components for microelectromechanical systems applications is studied by evaluating the mechanical fatigue behavior. The lifetime of microspecimens under alternate loads with different mean stress conditions is tested with the introduction of an original experimental procedure. Dedicated fatigue test devices are designed and fabricated with the aim of producing an alternate load on the embedded specimen. Electrostatic actuation is used as a driving force. Gold microbeams are tested under tensile variable load with different levels of mean and alternate stresses. FEM models are exploited to estimate the stress level produced in the material under the variable actuation conditions. Experimental results are analyzed to investigate the effects of different mean stress levels on lifetime under mechanical fatigue. The fatigue results are represented by a Goodman-Smith diagram scheme. Surface topography degradation is monitored through scanning electron microscope images. The modes of failure are in agreement with the literature, which is a consequence of fatigue degradation.

Residual stress measurement method in MEMS microbeams using frequency shift data

The dynamical behaviour of a set of gold microbeams affected by residual stress has been studied. Experimental frequency shift curves were obtained by increasing the dc voltage applied to the specimens. Comparison with different analytical and numerical models has been carried out in order to identify both analytical and finite element models in the presence of residual stress. Residual strain and stress, due to the fabrication process, have been widely reported in the literature in both out-of-plane microcantilevers and clamped–clamped microbeams by using mainly the value of pull-in voltage and static deflection data. In the case of a microcantilever, an accurate modelling includes the effect of the initial curvature due to microfabrication. In double-clamped microbeams, a pre-load applied by tensile stress is considered. A good correspondence is pointed out between measurements and numerical models so that the residual stress effect can be evaluated for different geometrical configurations.

MEMS Mechanical Fatigue: Experimental Results on Gold Microbeams

This paper proposes a new strategy for detecting material strength loss under mechanical fatigue on the basis of the pull-in voltage of the test device. Gold microbeam specimens are tested for mechanical fatigue using an electrostatically actuated dedicated device. The design of the fatigue device is discussed, providing the analysis of stress distribution inside the specimen. The finite-element method is used to simulate the electromechanical coupling. The fatigue limit is estimated through the ldquostaircaserdquo method, and a Wohler curve is obtained from experiments. The surface topography evolution is monitored by scanning electron microscope images; specimen failure modes and material degradation are discussed, revealing the local yield of the material on the upper surface of the beam. The type of degradation appears to be in agreement with the established literature as a consequence of fatigue.

Mechanical fatigue analysis of gold microbeams for RF-MEMS applications by pull-in voltage monitoring

This work is focused on the reliability of gold microcantilevers under the effect of mechanical fatigue. A dedicated device for testing the material is designed and built; the material degradation is monitored during the tests by means of a novel technique based on the control of the pull-in voltage of the device, which was demonstrated to be related to the loss of mechanical strength. The fatigue effect is produced through the excitation of the device at a frequency near the resonance; the excitation frequency and the time of actuation are used as a counter for the number of cycles. The lifetime of the device is measured under variable levels of vibration amplitudes; the number of cycles to failure is estimated within a specific range of actuation voltages by means of the Wohler diagram obtained by experiments. The fatigue limit is also estimated following the stair-case method.

The effects of annealing prior to dealloying on the mechanical properties of nanoporous gold microbeams

Freestanding nanoporous gold beams fracture during dealloying owing to volume shrinkage. This study illustrates that annealing after release, yet prior to dealloying, prevents failure during the selective dissolution step. Experiments in which annealing was performed at temperatures ⩽400 °C illustrate that permanent buckling of the freestanding solid alloy beams is required to prevent failure of the nanoporous beams during dealloying. In contrast, annealing of beams prior to release, or annealing temperatures ⩽200 °C (which do not cause permanent buckling deformation), do not mitigate dealloying failures. This work illustrates that the fabrication yield and the residual stresses (in the final nanoporous beams) correlate with the permanent buckling deflection of the alloy due to annealing, with the maximum yield (>95%) and the minimum residual stress (∼4–7 MPa) occurring for annealing at ∼300 °C. The present experimental results are used with elementary elastic buckling models to explain the critical temperature that produces permanent deflection, as well as discuss the mechanisms underlying the above phenomena.

All-optical micromechanical chemical sensors

The authors describe experimental results from micromechanical resonators coated with a chemoselective polymer that detect chemical vapors from volatile organic compounds using all-optical interrogation. The shift in the resonant frequency of the gold microbeam is read out using photothermal actuation and microcavity interferometry. Response times of less than 5s are achieved for vapor concentrations as low as 60ppm using optical powers of a few megawatts.

Dependence of the resonant frequency of micromachined gold microbeams on polarisation voltage

We demonstrate the feasibility of micromachined tuneable gold cantilever beams. The test structures, composed of a deflectable cantilever microbeam, two lateral driving electrodes and a working part, are fabricated using a micromachining process with a single mask. The microbeams are typically 1 μm wide and 2 μm high, with variable lengths ranging from 40 to 60 /gmm to achieve natural resonant frequencies ( f 0) greater than 100 kHz. These resonators are electrostatically excited with a signal: v( t) = V p + v sin ωt. Varying the polarisation voltage ( V p) produced a shift in the observed resonant frequency ( f obs) of a few percent around f 0, which makes this system tuneable. We describe the fabrication process, the frequency response and the frequency response and the frequency tuning of this device.