Dynamics and Touchdown in Electrostatic MEMS

Perhaps the most widely known nonlinear phenomena in nanoand microelectromechanical systems is the “pull-in” or “jump-to-contact” instability. In this instability, when applied voltages are increased beyond a certain critical voltage there is no longer a steady-state configuration of the device where mechanical members remain separate. This instability affects the design of many devices. It may present a limitation on the stable range of operation, as with a micropump, or may be utilized to create contact, as with a microvalve. Here, a mathematical model of an idealized electrostatically actuated MEMS or NEMS device is constructed for the purpose of studying the dynamics and touchdown behavior of systems operated in the pull-in regime. The model is analyzed in the viscosity dominated limit. This gives rise to a non-linear parabolic equation of reaction-diffusion type. The model is studied using a combination of analytical and numerical techniques. INTRODUCTION The advent of microelectromechanical systems (MEMS) has revolutionized numerous branches of science and industry. The rapidly developing field of nanoelectromechanical systems (NEMS) promises even more radical changes. In both MEMS and NEMS electrostatics plays a key role. Already employed in devices such as accelerometers (Bao, 1996), micropumps (Saif, 1999), optical switches (Camon, 1999), microgrippers (Chu, 1994), micro force gauges (Tilmans, 1992), and transducers (An∗Address all correspondence to this author. derson, 1995), researchers are continually exploring new uses for the Coulomb force. In the electrostatic approach voltage differences are applied between mechanical components of the system. The induced Coulomb force is then varied in strength by changing the applied voltage. The simplicity and importance of this technique has inspired numerous researchers to develop and study mathematical models of electrostatic-elastic interactions. The earliest such study may be found in the pioneering work of H.C. Nathanson and his coworkers (Nathanson, 1967). In their study of a resonant gate transistor they constructed and analyzed a mass-spring model of electrostatic actuation. They predicted and offered the first theoretical explanation of the pullin instability. In an interesting historical coincidence, the prolific British fluid dynamicist, G.I. Taylor, performed a similar study at roughly the same time as Nathanson (Taylor, 1968). While Taylor was interested in the electrostatic deflection of soap films, the small-aspect ratio model introduced in his work is the basis of many modern studies of electrostatic deflections in MEMS and NEMS. Since Nathanson and Taylor’s seminal work numerous investigators have analyzed mathematical models of electrostatic actuation in attempts to further understanding and control of the pull-in instability. The reader is referred to (Pelesko and Bernstein, 2002) for an overview of the field. Despite the roughly three decades of work in this area, the dynamics of electrostaticelastic systems remain relatively unexplored. In this paper we begin such a study by analyzing the dynamics of an idealized electrostatic-elastic system operated in the viscous dominated regime. That is, in our mathematical model we allow the system to vary in time, but we ignore inertial effects. This simplifies 1 Copyright  2003 by ASME