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。     

Pull-in voltage study of electrostatically actuated fixed-fixed beams using a VLSI on-chip interconnect capacitance model

A highly accurate computationally efficient closed-form model has been developed to determine the pull-in voltage of an electrostatically actuated fixed-fixed beam. The approach includes the electrostatic spring softening effects due to the fringing field capacitances along with the nonlinear spring hardening effects associated with the load-deflection characteristics of a uniformly loaded fixed-fixed beam. Meijs and Fokkema's highly accurate empirical formula for the capacitance of a VLSI on-chip interconnect has been used to determine the spring softening effects due to the fringing field capacitances. The developed model has been verified by comparing the results with published experimentally verified three-dimensional (3-D) finite element analysis (FEA) results and with those from other published representative closed-form models. The developed model can determine the pull-in voltage with a maximum deviation of 1.27% from the FEA results for small deflections and for large deflections (airgap-beam thickness ratio =12), the deviation from the FEA results is 2.0%. A maximum deviation of 0.5% from the FEA results has been observed for extreme fringing field cases (beamwidth-airgap ratio /spl les/0.5). The model's accuracy range is better compared to the other published models.     

Closed-form empirical relations to predict the static pull-in parameters of electrostatically actuated microcantilevers having linear width variation

We develop novel closed-form empirical relations to estimate the static pull-in parameters of electrostatically actuated tapered width microcantilever beams. A computationally efficient single degree-of-freedom model is employed in the setting of Ritz energy technique to extract the static pull-in parameters of the distributed electromechanical model that takes into account the effects of fringing field capacitance. The accuracy of this single-dof model together with the variable-width equivalent of the Palmer’s fringing model is established through a comparison with 3D finite element simulations. A unique surface fitting model is proposed to characterize the variations of both the pull-in displacement and pull-in voltage, over a realistically wide range of system parameters. Optimum coefficients of the proposed surface fitting model are obtained using nonlinear regression analysis. Empirical estimates of pull-in parameters are validated against finite element simulations, and available experimental and numerical data. An excellent agreement indicates that the proposed relationships are sufficiently accurate to be safely used for the electromechanical design of tapered microcantilever beams.     

Dynamic response of a torsional micromirror to electrostatic force and mechanical shock

In this paper dynamic characteristics of a capacitive torsional micromirror under electrostatic forces and mechanical shocks have been investigated. A 2DOF model considering the torsion and bending stiffness of the micromirror structure has been presented. A set of nonlinear equations have been derived and solved by Runge–Kutta method. The Static pull-in voltage has been calculated by frequency analyzing method, and the dynamic pull-in voltage of the micromirror imposed to a step DC voltage has been derived for different damping ratios. It has been shown that by increasing the damping ratio the dynamic pull-in voltage converges to static one. The effects of linear and torsional shock forces on the mechanical behavior of the electrostatically deflected and undeflected micromirror have been studied. The results have shown that the combined effect of a shock load and an electrostatic actuation makes the instability threshold much lower than the threshold predicted, considering the effect of shock force or electrostatic actuation alone. It has been shown that the torsional shock force has negligible influence on dynamic response of the micromirror in comparison with the linear one. The results have been calculated for linear shocks with different durations, amplitudes, and input times.     

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.     

VM-TEST: Mechanical property measurement using electrostatically actuated vertical MEMS test structures fabricated in thick metal layers

An efficient method is presented to determine the mechanical properties of thick metal layers using the pull-in voltage of electrostatically actuated structures. To fabricate these high aspect ratio beams without severe deformations, additional features were added, which made existing pull-in voltage equations inaccurate and therefore corrections were necessary. ANSYS Multiphysics was used to analyze the differences between ideal beams and the fabricated beams. To demonstrate the proposed approach, both nickel and gold devices were fabricated. To extract the material property values, a sum of least squares fitting scheme was used. A Young’s modulus of 186.2 and 60.8 GPa was obtained for nickel and gold structures respectively. Both values are significantly smaller than values reported for bulk material, but fall within the range of values reported in the literature.     

Mechanical behavior of a circular micro plate subjected to uniform hydrostatic and non-uniform electrostatic pressure

Circular micro plates are used in the many Microelectromechanical devices as micropumps and micro pressure sensors. All such systems exhibit a static instability phenomenon (Divergence) which is known as the “pull-in” instability. In this paper a distributed model was used to investigate the pull-in instability of a circular micro plate subjected to non-uniform electrostatic pressure and uniform hydrostatic pressure. The non-linear governing equation was derived and in order to linearize the obtained governing equations, step by step linearization method was used, then the linear system of equation was solved by finite difference method. The obtained results for only electrostatic actuation were compared with the existing results and good agreement has been achieved. There are exist two method of actuation. The pull-in voltages for these two actuation mechanism were investigated and the obtained results exhibited different effects on each actuation mechanism.     

Dynamic pull-in of parallel-plate and torsional electrostatic MEMS actuators

An analysis of the dynamic characteristics of pull-in for parallel-plate and torsional electrostatic actuators is presented. Traditionally, the analysis for pull-in has been done using quasi-static assumptions. However, it was recently shown experimentally that a step input can cause a decrease in the voltage required for pull-in to occur. We propose an energy-based solution for the step voltage required for pull-in that predicts the experimentally observed decrease in the pull-in voltage. We then use similar energy techniques to explore pull-in due to an actuation signal that is modulated depending on the sign of the velocity of the plate (i.e., modulated at the instantaneous mechanical resonant frequency). For this type of actuation signal, significant reductions in the pull-in voltage can theoretically be achieved without changing the stiffness of the structure. This analysis is significant to both parallel-plate and torsional electrostatic microelectromechanical systems (MEMS) switching structures where a reduced operating voltage without sacrificing stiffness is desired, as well as electrostatic MEMS oscillators where pull-in due to dynamic effects needs to be avoided.     

Equivalent area nonlinear static and dynamic analysis of electrostatically actuated microstructures

Nonlinear dynamic investigation of electrostatically actuated micro-electro-mechanical-system (MEMS) microcantilever structures is presented. The nonlinear analysis aims to better quantify, than the linear model, the instability threshold associated with electrostatically actuated MEMS structures, where the pull-in voltage of the microcantilever is determined using a phase portrait analysis of the microsystem. The microcantilever is modeled as a lumped mass-spring system. The nonlinear electrostatic force is incorporated into the lumped microsystem through an equivalent area of the microcantilever for a given electrostatic potential. Electro-mechanical force balance plots are obtained for various electrostatic potentials from which the static equilibrium positions of the microcantilever are obtained and the respective conservative energy values are determined. Subsequently, phase portrait plots are obtained for the corresponding energy values from which the pull-in voltage is estimated for the microsystem. This pull-in voltage value is in good agreement with the previously published results for the same geometric and material parameters. The results obtained for linear electrostatic models are also presented for comparison.     

A novel expression obtained by using artificial bee colony algorithm to calculate pull-in voltage of fixed-fixed micro-actuators

Microsystem Technologies - In this paper, a novel, computationally efficient and simple closed-form expression has been derived to accurately calculate the pull-in voltage value of fixed-fixed...     

On the tunability of a MEMS based variable capacitor with a novel structure

In this work a novel MEMS based variable capacitor has been presented. To increase the tunability and decrease the applied voltage, the conventional fixed-fixed beam used in CPW lines has been changed to a fixed-simple supported beam. The proposed structure is a simple cantilever micro-beam in the first step of deflection and is changed to a fixed-simple supported micro-beam in the second step of motion. In the capacitive micro-structures increasing the applied voltage decreases the equivalent stiffness of the structure and leads the system to an unstable condition by undergoing to a saddle node bifurcation. In the proposed structure to avoid pull-in instability and increase the capacitance tuning range, mechanical stiffness of the structure is increased by changing boundary conditions by locating a pedestal in the end of the cantilever beam. The governing nonlinear equation for static deflection of the micro-beam, based on Euler–Bernoulli micro-beam theory has been presented. The results show that the proposed structure increases the capacitance tuning range and decreases the applied voltage. The results also show that the position of the pedestal affects the tunability and the threshold voltage of the structure.     

Analytical approach and numerical /spl alpha/-lines method for pull-in hyper-surface extraction of electrostatic actuators with multiple uncoupled voltage sources

This work presents a systematic analysis of electrostatic actuators driven by multiple uncoupled voltage sources. The use of multiple uncoupled voltage sources has the potential of enriching the electromechanical response of electrostatically actuated deformable elements. This in turn may enable novel MEMS devices with improved and even new capabilities. It is therefore important to develop methods for analyzing this class of actuators. Pull-in is an inherent instability phenomenon that emanates from the nonlinear nature of the electromechanical coupling in electrostatic actuators. The character of pull-in in actuators with multiple uncoupled voltage sources is studied, and new insights regarding pull-in are presented. An analytical method for extracting the pull-in hyper-surface by directly solving the voltage-free K-N pull-in equations derived here, is proposed. Solving simple but interesting example problems illustrate these new insights. In addition, a novel /spl alpha/-lines numerical method for extracting the pull-in hyper-surface of general electrostatic actuators is presented and illustrated. This /spl alpha/-lines method is motivated by new features of pull-in, that are exhibited only in electrostatic actuators with multiple uncoupled voltage sources. This numerical method permits the analysis of electrostatic actuators that could not have been analyzed by using current methods.     

An investigation into the electrostatic force representation for electrically actuated microelectromechanical systems

The ever-increasing demand for microelectromechanical systems (MEMS) in modern electronics has reinforced the need for extremely accurate analytical and reduced-order models to aid in the design of MEMS devices. Many MEMS designs consist of cantilever beams with a tip mass attached at the free end to act as a courter electrode for electrical actuation. One critical modeling aspect of electrically actuated MEMS is the electrostatic force that drives these systems. The two most used representations in the literature approximate the electrostatic force between two electrodes as a point force. In this work, the effects of the representation of the electrostatic force for electrically actuated microelectromechanical systems are investigated. The system under investigation is composed of a beam with an electrode attached to its end. The distributed force, rigid body, and point mass electrostatic force representations are modeled, studied, and their output results are compared qualitatively. Static and frequency analyses are carried out to investigate the influences of the electrostatic force representation on the static pull-in, fundamental natural frequency, and mode shape of the system. A nonlinear distributed-parameter model is then developed in order to determine and characterize the response of electrically actuated systems when considering various representation of the electrostatic forces. The results show that the size of the electrode may strongly affect the natural frequencies and static pull-in when the point mass, rigid body, and plate representations are considered. From nonlinear analysis, it is also proven that the representation may affect the hardening behavior of the system and its dynamic pull-in. This modeling and analysis give guidelines about the usefulness of the electrostatic force representations and possible erroneous assumptions that can be made which may result in inaccurate design and optimal performance detection for electrostatically actuated systems.

     

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.     

Tilt-angle stabilization of electrostatically actuated micromechanical mirrors beyond the pull-in point

Recently proposed optical subsystems utilizing microelectromechanical system (MEMS) components are being developed for use in optical crossconnects, add-drop multiplexers, and spectral equalizers. Common elements to these subsystems are electrostatically actuated micromechanical mirrors that steer optical beams to implement the subsystem functions. In the past, feedback control methods were used to obtain precise mirror orientations to minimize loss through optical switch fabrics or to stabilize attenuation through spectral equalizers. However, the mirror tilt angle range is limited because of inherent instability beyond a critical tilt angle (pull-in angle), and the usual feedback schemes do not counteract this effect. This work presents a feedback control method to enable operation of electrostatic micromirrors beyond the pull-in angle, yielding advantages including greater scalability of switch arrays and increased dynamic range of optical attenuators. Both static and dynamic tilting behaviors of electrostatic micromirrors under the feedback control are studied. In addition, a practical implementation of the feedback control system by using linear voltage control law is developed. A voltage slightly larger than the pull-in voltage is first applied when the mirror is at small angle positions, and the voltage is then linearly reduced as the mirror approaches the desired position. Experimental measurements, showing that tilt angles beyond the pull-in point can be achieved, are in good agreement with theoretical analysis.     

A comprehensive model to study nonlinear behavior of multilayered micro beam switches

A novel model to study the pull-in behavior of nonlinear electromechanically coupled systems has been developed. The proposed model is based on the multilayered cantilever and fixed–fixed micro beam type MEMS switches. Due to the complexity of the nonlinear beam mechanics, exact analytical solutions are not generally available; therefore, the derived nonlinear equation has been numerically solved fully using the nonlinear finite difference method. Furthermore, the results obtained are summarized and compared with the other existing empirical and analytical models. These results can be useful in the optimization of MEMS switch designs or other actuators. In addition, the method developed in this paper has a good potential for analyzing other types of complex MEMS devices. An erratum to this article can be found at     

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.     

Using dynamic voltage drive in a parallel-plate electrostatic actuator for full-gap travel range and positioning

The nonlinear dynamics of the parallel-plate electrostatically driven microstructure have been investigated with the objective of finding a dynamic voltage drive suitable for full-gap operation. Nonlinear dynamic modeling with phase-portrait presentation of both position and velocity of a realistic microstructure demonstrate that instability is avoided by a timely and sufficient reduction of the drive voltage. The simulation results are confirmed by experiments on devices fabricated in an epi-poly process. A 5.5-V peak harmonic drive voltage with frequency higher than 300 Hz allows repetitive microstructure motion up to 70% of gap without position feedback. The results of the analysis have been applied to the design of a new concept for positioning beyond the static pull-in limitation that does include position feedback. The measured instantaneous actuator displacement is compared with the desired displacement setting and, unlike traditional feedback, the voltage applied to the actuator is changed according to the comparison result between two values. The "low" level is below the static pull-in voltage and opposes the motion, thus bringing the structure back into a stable regime, while the "high" level is larger than the static pull-in voltage and will push the structure beyond the static pull-in displacement. Operation is limited only by the position jitter due to the time delay introduced by the readout circuits. Measurements confirm flexible operation up to a mechanical stopper positioned at 2 /spl mu/m of the 2.25 /spl mu/m wide gap with a 30 nm ripple.     

Design of high stroke electrostatic micropumps: a charge control approach with ring electrodes

A novel design strategy to avoid pull-in occurrence in electrostatic micro electro mechanical systems is proposed. It combines charge control with ring electrodes, on a circular geometry. This idea is introduced here for the design of efficient and reliable high stroke electrostatic diaphragm micropumps, while it has a broad potential applicability. A minimal lumped one degree-of-freedom model is derived and used to introduce and demonstrate the proposed approach for a circular plate geometry. Finite element models are subsequently adopted for a more detailed device modelling. As expected, charge control exhibits a stabilizing effect with respect to voltage drive, but not sufficient to achieve a full-range stability for the considered geometry. When the electrode area is properly defined, stability range can be extended up to gap closure in the central part of the membrane. In this configuration, the increase in voltage required for full-range device drive would be relevant, while in charge control the penalty is considerably lower. Finally, loading conditions and geometrical parameters for an optimized actuation are suggested.