A stationary free boundary problem modeling electrostatic MEMS

A free boundary problem describing small deformations in a membrane based model of electrostatically actuated MEMS is investigated. The existence of stationary solutions is established for small voltage values. A justification of the widely studied narrow-gap model is given by showing that steady state solutions of the free boundary problem converge toward stationary solutions of the narrow-gap model when the aspect ratio of the device tends to zero

Delayed pull-in transitions in overdamped MEMS devices

We consider the dynamics of overdamped MEMS devices undergoing the pull-in instability. Numerous previous experiments and numerical simulations have shown a significant increase in the pull-in time under DC voltages close to the pull-in voltage. Here the transient dynamics slow down as the device passes through a meta-stable or bottleneck phase, but this slowing down is not well understood quantitatively. Using a lumped parallel-plate model, we perform a detailed analysis of the pull-in dynamics in this regime. We show that the bottleneck phenomenon is a type of critical slowing down arising from the pull-in transition. This allows us to show that the pull-in time obeys an inverse square-root scaling law as the transition is approached; moreover we determine an analytical expression for this pull-in time. We then compare our prediction to a wide range of pull-in time data reported in the literature, showing that the observed slowing down is well captured by our scaling law, which appears to be generic for overdamped pull-in under DC loads. This realization provides a useful design rule with which to tune dynamic response in applications, including state-of-the-art accelerometers and pressure sensors that use pull-in time as a sensing mechanism. We also propose a method to estimate the pull-in voltage based only on data of the pull-in times.Comment: 17 page

A parabolic free boundary problem modeling electrostatic MEMS

The evolution problem for a membrane based model of an electrostatically actuated microelectromechanical system (MEMS) is studied. The model describes the dynamics of the membrane displacement and the electric potential. The latter is a harmonic function in an angular domain, the deformable membrane being a part of the boundary. The former solves a heat equation with a right hand side that depends on the square of the trace of the gradient of the electric potential on the membrane. The resulting free boundary problem is shown to be well-posed locally in time. Furthermore, solutions corresponding to small voltage values exist globally in time while global existence is shown not to hold for high voltage values. It is also proven that, for small voltage values, there is an asymptotically stable steady-state solution. Finally, the small aspect ratio limit is rigorously justified

On the partial differential equations of electrostatic MEMS devices III: Refined touchdown behavior

AbstractThis paper is a continuation of [N. Ghoussoub, Y. Guo, On the partial differential equations of electrostatic MEMS devices: Stationary case, SIAM J. Math. Anal. 38 (2007) 1423–1449] and [N. Ghoussoub, Y. Guo, On the partial differential equations of electrostatic MEMS devices II: Dynamic case, NoDEA Nonlinear Differential Equations Appl. (2008), in press], where we analyzed nonlinear parabolic problem ut=Δu−λf(x)(1+u)2 on a bounded domain Ω of RN with Dirichlet boundary conditions. This equation models a simple electrostatic Micro-Electromechanical System (MEMS) device consisting of a thin dielectric elastic membrane with boundary supported at 0 above a rigid ground plate located at −1. Here u is modeled to describe dynamic deflection of the elastic membrane. When a voltage—represented here by λ—is applied, the membrane deflects towards the ground plate and a snap-through (touchdown) must occur when it exceeds a certain critical value λ∗ (pull-in voltage), creating a so-called “pull-in instability” which greatly affects the design of many devices. In an effort to achieve better MEMS design, the material properties of the membrane can be technologically fabricated with a spatially varying dielectric permittivity profile f(x). In this work, some a priori estimates of touchdown behavior are established, based on which the refined touchdown profiles are obtained by adapting self-similar method and center manifold analysis. Applying various analytical and numerical techniques, some properties of touchdown set—such as compactness, location and shape—are also discussed for different classes of varying permittivity profiles

Modelling the electrostatic actuation of MEMS: state of the art 2005.

Most of MEMS devices are actuated using electrostatic forces. Parallel or lateral plate actuators are the types commonly used. Nevertheless, electrostatic actuation has some limitations due to its non-linear nature. This work presents a methodic overview of the existing techniques applied to the Micro-Electro-Mechanical Systems (MEMS) electrostatic actuation modeling and their implications to the dynamic behavior of the electromechanical system

Compliant Torsional Micromirrors with Electrostatic Actuation

Due to the existence of fabrication tolerance, property drift and structural stiction in MEMS (Micro Electro Mechanical Systems), characterization of their performances through modeling, simulation and testing is essential in research and development. Due to the microscale dimensions, MEMS are more susceptible and sensitive to even minor external or internal variations. Moreover, due to the current limited capability in micro-assembly, most MEMS devices are fabricated as a single integrated micro-mechanical structure composed of two essential parts, namely, mass and spring, even if it may consist of more than one relatively movable part. And in such a scale of dimensions, low resonant micro-structures or compliant MEMS structures are hard to achieve and difficult to survive. Another problem arises from the limited visibility and accessibility necessary for characterization. Both of these issues are thus attempted in this research work. An investigation on micromirrors with various actuations and suspensions is carried out, with more attention on the micromirrors with compliant suspensions, electrostatic actuation and capable of torsional out-of-plane motion due to their distinct advantages such as the low resonance and the low drive voltage. This investigation presents many feasible modeling methods for prediction and analysis, aiming to avoid the costly microfabrication. Furthermore, both linear and nonlinear methods for structure and electrostatics are all included. Thus, static and dynamic performances of the proposed models are formularized and compared with those from FEA (Finite Element Analysis) simulation. The nonlinear modeling methods included in the thesis are Pseudo Rigid Body Model (PRBM) and hybrid PRBM methods for complex framed microstructures consisting of compliant beam members. The micromachining technologies available for the desired micromirrors are reviewed and an SOI wafer based micromachining process is selected for their fabrication. Though the fabrication was executed outside of the institution at that time, the layout designs of the micro-chips for manufacture have included all related rules or factors, and the results have also demonstrated the successful fabrication. Then investigation on non-contact test methods is presented. Laser Doppler Vibrometer (LDV) is utilized for the measurement of dynamic performances of proposed micromirrors. Two kinds of photo-sensing devices (PSDs), namely, the digitized PSD formed by CCD arrays and the analog PSD composed of a monolithic photosensing cell, are used for static test set-ups. An interferometric method using Mirau objective along with microscope is also employed to perform static tests of the selected micromirrors. Comparison of the tested results and their related theoretical results are presented and discussed, leading to a conclusion that the proposed hybrid PRBM model are appropriate for prediction or analysis of compliantly suspended micromirrors including issues arising from fabrication tolerance, structural or other parametric variations