MEMS薄膜平均应力梯度及等效弹性模量在线提取方法

薄膜沿厚度方向的平均应力梯度及薄膜的弹性模量对器件性能有重要影响.提出了一种利用静电作用下悬臂梁的吸合电压提取薄膜沿厚度方向的平均应力梯度及等效弹性模量的方法,该方法的关键在于实现悬臂梁吸合电压的快速精确计算.考虑了悬臂梁由应力梯度引起的沿宽度方向的弯曲及实现其固定端接近理想固支的方法,提高了吸合电压的计算精度.实际模拟表明该测量方法计算速度快、精度高,能够应用于实际工艺过程中材料参数的在线测量.     

Stability analysis of an electrostatically actuated out of plane MEMS structure

In the present research, stability and static analyses of microelectromechanical systems microstructure were investigated by presenting an out-of-plane structure for a lumped mass. The presented model consists of two stationary electrodes in the same plane along with a flexible electrode above and in the middle of the two electrodes. The nonlinear electrostatic force was valuated via numerical methods implemented in COMSOL software where three-dimensional simulations were performed for different gaps. The obtained numerical results were compared to those of previous research works, indicating a good agreement. Continuing with the research, curves of electrostatic and spring forces were demonstrated for different scenarios, with the intersection points (i.e., equilibrium points) further plotted. Also drawn were plots of deflection versus voltage for different cases and phase and time history curves for different values of applied voltage followed by introducing and explaining pull-in and pull-out snap-through voltages in the system for a specific design. It is worth noting that, at voltages between the pull-in and pull-out snap-through voltages, the system was in bi-stable state. Based on the obtained results, it was observed that the gap between the two electrodes and the applied voltage play significant roles in the number and type of the equilibrium points of the system.

     

Nonlocal and strain gradient based model for electrostatically actuated silicon nano-beams

Conventional continuum theory does not account for contributions from length scale effects which are important in modeling of nano-beams. Failure to include size-dependent contributions can lead to underestimates of deflection, stresses, and pull-in voltage of electrostatic actuated micro and nano-beams. This research aims to use nonlocal and strain gradient elasticity theories to study the static behavior of electrically actuated micro- and nano-beams. To solve the boundary value nonlinear differential equations, analogue equation and Gauss–Seidel iteration methods are used. Both clamped-free and clamped–clamped micro- and nano-beams under electrostatical actuation are considered where mid-plane stretching, axial residual stress and fringing field effect are taken into account for clamped–clamped cases. The accuracy of solution is evaluated by comparison of the pull-in voltages of different micro-electro-mechanical systems with those results previously published in the literature. A main drawback of the previous theoretical researches using nonlocal or strain gradient methods was that they don’t account for effects of the size on the Young modulus of the beam and merely they adjust the length scale parameters for small sizes to fit data with experimental results. In the present study, the experimental voltages for static pull-in of different micro- and nano-beams are used to estimate the silicon Young’s modulus, nonlocal and length scale parameters. Using the estimated parameters, pull-in voltages of silicon micro- and nano-beams based on strain gradient and nonlocal theories are compared with respect to each other and also with the experimental and previous theoretical results. The conducted results demonstrate that the predicted pull-in voltages using proposed strain gradient theory will give the best fit with experimental observations.     

MEMS薄膜弹性模量及残余应力提取的数值算法

材料弹性模量、残余应力的在线提取,已成为MEMS领域中日益迫切的需要。文中首先简要介绍现有的一些主要的材料参数提取方法,然后提出了一种无需静电作用下两端固支梁吸合电压的显式解析表达式,而是利用决定静电作用固支梁弯曲挠度的微分方程直接计算吸合电压,再提取材料参数的算法。避免了在推导吸合电压的显式表达式过程中可能引入的误差,有利于保证材料参数测量的精度。模拟结果表明这种算法速度快、精度高,对实际应用有较好的参考价值。     

Elastic–plastic modeling of heat-treated bimorph micro-cantilevers

This paper models the residual stress distributions within micro-fabricated bimorph cantilevers of varying thickness. A contact model is introduced to calculate the influence of contact on the residual stress following a heat treatment process. An analytical modeling approach is adopted to characterize bimorph cantilevers composed of thin Au films deposited on thick poly-silicon or silicon-dioxide beams. A thermal elastic–plastic finite element model (FEM) is utilized to calculate the residual stress distribution across the cantilever cross-section and to determine the beam tip deflection following heat treatment. The influences of the beam material and thickness on the residual stress distribution and tip deflections are thoroughly investigated. The numerical results indicate that a larger beam thickness leads to a greater residual stress difference at the interface between the beam and the film. The residual stress established in the poly-silicon cantilever is greater than that induced in the silicon-dioxide cantilever. The results confirm the ability of the developed thermal elastic–plastic finite element contact model to predict the residual stress distributions within micro-fabricated cantilever structures with high accuracy. As such, the proposed model makes a valuable contribution to the development of micro-cantilevers for sensor and actuator applications.

     

Analysis of a novel RF MEMS switch using different meander techniques

In this paper, two types of RF MEMS switches namely step structure and Normal beam structure are designed and analyzed using different meander techniques. Three techniques namely plus, zigzag and three-square meander were used to lower the pull-in voltage. The actuating beam is designed with the rectangular perforations affects the performance of a switch by lowering the pull-in voltage, switching speed and results in better isolation. In this paper a comparative analysis is done for all three meander techniques with and without perforations on the beam. Total six structures have been designed with the combination three meanders and two different beam structures. The proposed stepdown structure exhibits high performance characteristics with a very low pull-in voltage of 1.2 V having an airgap of 0.8 µm between the actuation electrodes. The gold is used as beam material and HfO₂ as the dielectric material such that the upstate and downstate capacitance is seen as 1.02 fF and 49 fF. The FEM analysis is done to calculate the spring constant and thereby the pull-in voltage and behavior of the switch is studied with various parameters. The switch with a step structure and three-square meander configuration has shown best performance of all by requiring a pull-in voltage of 1.2 V and lower switching time of 0.2 µs. The proposed switch also exhibits good RF performance characteristics with an insertion loss below − 0.07 dB and return loss below − 60 dB over the frequency range of 1–40 GHz. At 28 GHz a high isolation of − 68 dB is exhibited.

     

Elastic–plastic modeling of heat-treated bimorph micro-cantilevers

This paper models the residual stress distributions within micro-fabricated bimorph cantilevers of varying thickness. A contact model is introduced to calculate the influence of contact on the residual stress following a heat treatment process. An analytical modeling approach is adopted to characterize bimorph cantilevers composed of thin Au films deposited on thick poly-silicon or silicon-dioxide beams. A thermal elastic–plastic finite element model (FEM) is utilized to calculate the residual stress distribution across the cantilever cross-section and to determine the beam tip deflection following heat treatment. The influences of the beam material and thickness on the residual stress distribution and tip deflections are thoroughly investigated. The numerical results indicate that a larger beam thickness leads to a greater residual stress difference at the interface between the beam and the film. The residual stress established in the poly-silicon cantilever is greater than that induced in the silicon-dioxide cantilever. The results confirm the ability of the developed thermal elastic–plastic finite element contact model to predict the residual stress distributions within micro-fabricated cantilever structures with high accuracy. As such, the proposed model makes a valuable contribution to the development of micro-cantilevers for sensor and actuator applications.     

Pull-in and freestanding instability of actuated functionally graded nanobeams including surface and stiffening effects

Because of fasttechnological development, electrostatic nanoactuator devices like nanosensors, nanoswitches, and nanoresonators are highly considered by scientific community. Thus, this article presents a new solution technique in solving highly nonlinear integro-differential equation governing electrically actuated nanobeams made of functionally graded material. The modified couple stress theory and Gurtin–Murdoch surface elasticity theory are coupled together to capture the size effects of the nanoscale thin beam in the context of Euler–Bernoulli beam theory. For accurate modelling, all the material properties of the bulk and surface continuums of the FG nanoactuator are varied continuously in thickness direction according to power law. The nonlinearity arising from the electrostatic actuation, fringing field, mid-plane stretching effect, axial residual stress, Casimir dispersion, and van der Waals forces are considered in mathematical formulation. The nonlinear nonclassical equilibrium equation of FG nanobeam-based actuators and associated boundary conditions are exactly derived using Hamilton principle. The new solution methodology is combined from three phases. The first phase applies Galerkin method to get an integro-algebraic equation. The second one employs particle swarm optimization method to approximate the integral terms (i.e. electrostatic force, fringing field, and intermolecular forces) to non-integral cubic algebraic equation. Then, solved the system easily in last phase. The resulting algebraic model provides means for obtaining critical deflection, pull-in voltage, detachment length, minimum gap, and freestanding effects. A reasonable agreement is found between the results obtained from the present method and those in the available literature. A parametric study is performed to investigate the effects of the gradient index, material length scale parameter, surface energy, intermolecular forces, initial gap, and beam length on the pull-in response and freestanding phenomena of fully clamped and cantilever FG nanoactuators.

     

Application of piezoelectric layers in electrostatic MEM actuators: controlling of pull-in voltage

In this paper, a novel method has been developed to control the pull-in voltage of the fixed-fixed and cantilever MEM actuators and measure the residual stress in the fixed-fixed model using of the piezoelectric layers that have been located on the upper and lower surfaces of actuator. In the developed model, the tensile or compressive residual stresses, fringing-field and axial stress effects in the fixed-fixed end type micro-electro-mechanical systems actuator have been considered. The non-linear governing differential equations of the MEM actuators have been derived by considering the piezoelectric layers and mentioned effects. The results show that due to different applied voltage to the piezoelectric layers, the pull-in voltage can be controlled and in the fixed-fixed type the unknown value of the residual stress can be obtained.     

Full characterisation of pull-in in single-sided clamped beams

The most striking characteristic of the voltage-to-deflection curve of an electrostatically actuated beam is pull-in. The actual value of the pull-in voltage depends on: drive mode, temperature dependence and dielectric charging related drift. These aspects have been analysed using structures designed for a 9 V nominal pull-in voltage and fabricated in a commercially available epipoly process. Single-sided clamped beams have been used to avoid any influence of residual stress in the beam on pull-in. Typical results are: less than 5% variation of the pull-in voltage over a wafer, 0.17–1.9 V hysteresis depending on drive mode, a −1 mV/K TC and −12 mV drift during the first 2 weeks of operation.     

Design,modeling, and mechanical characterizations of micromachined InP-based actuator for tunable MOEMS applications

A novel InP-based microactuator, which is actuated by electrostatic means, has been proposed, designed, fabricated, and characterized for tuning applications in the 1.5 μm wavelength domains. Its structural design is based on the global optimization method. The tunable device is a big square membrane, which is supported by four identical cantilever beams. The three alternating layers Si₃N₄/SiO₂ as a distributed Bragg reflector (DBR) mirror, which were previously reported, have been formed on the top of the membrane. Based on the optical interferometric measurements, the proposed Fabry–Perot filter has demonstrated a maximum deflection of ∼321 nm with an applied voltage up to 12 V, an average sensitivity of ∼27 nm/V, a pull-in voltage of 12.7 V, and a release voltage of 10.7 V. It is also observed that its natural frequency is 88.4 kHz. This measured frequency implies that the tuning speed of our device is fast for optical operations within 0.01 ms. In addition, our device’s mirror remains so flat with a good planarity of 0.07°, which is strictly required for the filter’s optical performance. This optical performance can be achieved, when the micromachined structure has a tuning displacement up to ∼38 nm with a low tuning voltage up to 5 V. When compared with the finite element models (FEM), which were generated by the commercialized software, Coventor™, our experimental results agree well in terms of the natural frequency, pull-in voltage and deflections. Thus, our tunable filter, which is based on the optimized design, enables better performances including reduced actuation voltages, large pull-in voltage, improved device reliability, and fast switching times. Our device can also quickly snap back to the original position. In addition, the undesired spring-softening effect has been reduced.     

Effect of residual stress on RF MEMS switch

Studies have been carried out on a RF MEMS shunt switch to analyze the effect of residual stress on its electromechanical characteristics. This paper presents the simulated results as well as theoretically calculated results of a shunt switch due to the presence of residual stress gradient in respect of resonant frequency, pull down voltage and switching characteristics. The effect of introduction of holes in the beam is also studied. The calculated results, corresponding to the switch (without holes) at zero residual stress, of resonant frequency, pull-down voltage and switch on and off time are 28.14 kHz, 28.2 V, 16.35 μsec and 8.6 μsec respectively. Modal analysis of the both the structures (with and without holes) are carried out for different values of residual stress gradients. Modal analysis predicted that higher values of tensile stress gradient are not favorable for switching action. The pull-down voltages and switch on and off times are simulated at different stress gradients. With the increase in compressive stress gradient, the pull-down voltage is found to increase, whereas, switch on and off times is decreased. Corresponding to −20 MPa/μm residual stress gradient, the resonant frequency, pull-down voltage and switch on and off times are found to be 74.5 kHz, 63.5 V, 7.5 μsec and 3.36 μsec respectively. Introduction holes in the structure modified these values to 63.77 kHz, 53.1 V, 8.7 μsec, 3.92 μsec respectively.     

Analysis and analytical modeling of static pull-In with application to MEMS-based voltage reference and process monitoring

The pull-in voltage of one- and two-degrees-of-freedom (DOF) structures has been symbolically and numerically analyzed with respect to drive mode dependence and hysteresis. Moreover, the time and temperature stability has been investigated and tested. Modeling results have been applied in the design of both folded-spring-suspended 1-DOF structures and single-side-clamped 2-DOF beams with a nominal pull-in voltage in the 5-10 V range and fabricated in an epi-poly process. Asymmetrically driven structures reveal pull-in close to the value predicted by the model (V/sub pi/ 1-DOF is 4.65 V analytically simulated and 4.56 V measured; V/sub pi/ 2-DOF is 9.24 V analytically simulated, 9.30 V in FEM and 9.34 V measured). Also the hysteresis is in close agreement (release voltage, V/sub r/, 1-DOF is 1.41 V analytically simulated and 1.45 V measured; V/sub r/ 2-DOF is 9.17 V analytically simulated, 9.15 V in FEM and 9.27 V measured). In symmetrically operated devices the differences between the computed and measured V/sub pi/ and V/sub r/ are much larger and are due to process dependencies, which make these devices very suitable for process monitoring. The 2-DOF asymmetrically operated device is the most suitable for MEMS-based voltage reference. The stability in time is limited by charge build-up and calls for a 100-hour initial burn-in. Temperature dependence is -100 /spl mu/V/K at V/sub pi//spl ap/5 V, however, is calculable and thus can be corrected or compensated.     

A sticking model of suspended polysilicon microstructure includingresidual stress gradient and postrelease temperature

A sticking (stiction) model for a cantilevered beam is derived. This model includes the effect of the bending moment, which stems from stress gradient along the vertical direction of structural polysilicon, and the temperature during the release process. The bending moment due to the stress gradient will play an important role in evaluating antisticking efficiency since liquid tension and surface energy of microstructures tend to become smaller by newly developed antisticking techniques. The effects of stress gradient and temperature were analyzed and verified with surface-micromachined polysilicon cantilevers. By modifying the substrate polysilicon with grain-hole formation technique, the effects of residual stress gradient in polysilicon on stiction could be observed in the condition of low work of adhesion     

Design considerations of rectangular electrostatic torsionactuators based on new analytical pull-in expressions

An important design issue of electrostatic torsion actuator is the relative locations of the actuating electrodes, where the bias voltage is applied. These geometrical design parameters affect both the pull-in angle as well as the pull-in voltage. In this paper, a new approximated analytical solution for the pull-in equation of an electrostatic torsion actuator with rectangular plates is derived. The analytical expression is shown to be within 0.1% of the one degree of freedom (1DOF) lumped-element model numerical simulations. Moreover, the analytical expressions are compared with the full coupled-domain finite-elements/boundary-elements (FEM/BEM) simulations provided by MEMCAD4.8 Co-solve tool, showing excellent agreement. The approach presented here provides better physical insight, more rapid simulations and an improved design optimization tool for the actuator     

Determination of residual stresses in a single crystalline 3C-SiC micro-fabricated structure using FE model and measured resonance frequencies

Residual stresses can significantly affect the performance of micro-electromechanical devices. In this work, a set of single crystalline 3C-SiC microbeams (cantilevers and bridges) were fabricated for the purpose of determining residual stresses in the bridges. The cantilevers and bridges were actuated mechanically using a piezoelectric disc. The cantilevers exhibit no residual stresses. Residual stress in the bridges was estimated using an FE ANSYS developed model together with the measurement of resonance frequencies. Residual stress in the bridges was estimated to be 426 ± 29 MPa. The estimated residual stress lies between the upper (string model) and lower (stress-free mode shape model) stress values which were calculated to be 491 and 394 MPa respectively.     

Modeling, Design, Fabrication, and Demonstration of a Digital Micromirror With Interdigitated Cantilevers

We present the modeling, design, fabrication, and measurement results of a novel digital micromirror based on a new actuator called interdigitated cantilevers. In contrast to conventional micromirrors that rotate through the twisting actuation of a hinge, this micromirror has a symmetric bidirectional rotation through a bending actuation of interdigitated cantilevers hidden under a mirror plate. For the static and dynamic characteristics of the proposed micromirror, analytical models were developed first on the basis of the Euler–Bernoulli beam equation, as well as both distributed- and lumped-parameter models. The results of the developed analytical models are in good agreement with those of a finite-element-method (FEM) simulation, having just a 10% deviation. On the basis of these analytical models, we successfully designed, fabricated, and evaluated a micromirror with a mirror size of $16 muhbox{m} times 16 muhbox{m}$. The fabricated micromirror has a mechanical rotation angle of $pm 10^{circ}$, a pull-in voltage of 54 V, a resonant frequency of 350 kHz, and a switching response time of 17 $muhbox{s}$. The measurement results compare favorably with those of analytical models and FEM simulations, with deviations of less than 15% and 10%, respectively. $hfill$[2009-0139]      

Experimental characterization of electrostatically actuated in-plane bending of microcantilevers

Experimental validation of numerical models developed by the authors to predict the static behaviour of microelectrostatic actuators is described. Cantilever microbeams, currently used in connection with RF-MEMS and micro-scale material testing were analysed. A set of microcantilevers, bending in the plane of the wafer, i.e. in the same plane as the profiling system’s target, was tested. This differs from the popular case of out-of-plane microbeams, usually studied in the literature. Geometry nonlinearity caused by large deflection of the microbeam was investigated and nonlinear coupled formulation of electromechanical equilibrium was performed. Coupled-field analysis was implemented using the Finite Element Method (FEM), to predict displacements and pull-in voltage measured by Fogale Zoomsurf 3D, subsequently plotting the displacement-versus-voltage curve to complete model validation. FEM nonlinear analysis, based on iterative approach with mesh morphing, and FEM non-incremental approach, including a special element proposed by the authors, are compared to the linear solution and to experimental results. Geometry nonlinearity appears relevant in microbeam modelling and requires a nonlinear solution of the coupled problem. Investigative work, which compared the results of 2D and 3D models to experimental data, revealed that some three dimensional effects are significant in model validation, but the 2D approach may be effective in predicting static behaviour provided that at least a microbeam thickness equivalent is adopted.     

On the dynamic pull-in of electrostatic actuators with multiple degrees of freedom and multiple voltage sources

This study considers the dynamic response of electrostatic actuators with multiple degrees of freedom that are driven by multiple voltage sources. The critical values of the applied voltages beyond which the dynamic response becomes unstable are investigated. A methodology for extracting a lower bound for this dynamic pull-in voltage is proposed. This lower bound is based on the stable and unstable static response of the system, and can be rapidly extracted because it does not require time integration of momentum equations. As example problems, the dynamic pull-in of two prevalent electrostatic actuators is analyzed.