Effects of Casimir force on pull-in instability in micromembranes

We analyze pull-in instability of electrostatically actuated microelectromechanical systems, and study changes in pull-in parameters due to the Casimir effect. When the size of the device is reduced, the magnitude of the Casimir force is comparable with that of the Coulomb force and it significantly alters pull-in parameters. We model the deformable conductor as an elastic membrane and consider different geometries. Beyond a certain critical size the pull-in instability occurs with zero applied voltage, and the device may collapse during the fabrication process. Copyright c © EPLA, 2007 Introduction. – Electrostatically actuated microelectromechanical systems (MEMS) are becoming increasingly useful in many applications such as switches, micro-mirrors and micro-resonators, see, e.g., [1–3]. At the microscopic scale the electrostatic actuation may dominate over other kinds of actuation. Most of the electrostatically actuated systems [4] are comprised of a conductive deformable plate suspended over a rigid ground plate. An applied electric voltage between the two conductors results in the deflection of the elastic plate, and a consequent change in the system capacitance. The applied electrostatic voltage has an upper limit beyond which the two plates snap together and the device collapses. This phenomenon is called pull-in instability and the corresponding voltage the pull-in voltage; it was simultaneously observed experimentally by Taylor [5] and Nathanson et al. [6]. With the decrease in device dimensions from the micro to the nanoscale additional forces on nanoelectromechanical systems (NEMS), such as the Casimir force [7,8], should be considered. The Casimir force represents the attraction of two uncharged material bodies due to modification of the zero-point energy associated with the electromagnetic modes in the space between them. The existence of the Casimir force poses a severe constraint on the miniaturization of electrostatically actuated devices. At the nanoscale, the Casimir force may overcome elastic restoring actions in the device and lead to the plates’ sticking during the fabrication process. An important feature of the Casimir effect is that even though its nature is quantistic, it predicts a force between macroscopic bodies. van der Waals forces are related to electrostatic interaction among dipoles at the atomic scale [9]. Whereas the Casimir force between semi-infinite parallel plates depends only on the geometry, van der Waals forces depend on material properties of the media. The Casimir force is effective at longer distance than van der Waals forces [9]. van der Waals forces are accounted for in NEMS where interactions occur at the atomic scale, as for example in carbon nanotubes [10]. van der Waals forces are not considered in the work presented below. Here, we analyze the effect of the Casimir and the Coulomb forces on the pull-in parameters of NEMS for a large variety of common two-dimensional (2-D) geometries. We show that beyond a critical size, the pull-in instability occurs at zero voltage. This means that the system collapses during the manufacturing process. We also analyze symmetry-breaking in annular membranes due to the combined effects of the Coulomb and the Casimir forces. Different investigators have studied sticking in MEMS, but they did not consider the combined effect of the Casimir and the Coulomb forces. In [11,12] a rectangular membrane using the 1-D distributed model and considering nonlinear stretching effects has been studied, while in [13] a lumped one degree-of-freedom (d.o.f.) model has been used to analyze the stiction phenomenon