Touchdown and Pull-In Voltage Behavior of a MEMS Device with Varying Dielectric Properties

Abstract

The pull-in voltage instability associated with a simple MEMS device, consisting of a thin dielectric elastic membrane supported above a rigid conducting ground plate, is analyzed. The upper surface of the membrane is coated with a thin conducting lm. In a certain asymptotic limit representing a thin device, the mathematical model consists of a nonlinear partial dieren tial equation for the deection of the thin dielectric membrane. When a voltage V is applied to the conducting lm, the dielectric membrane deects towards the bottom plate. For a slab, a circular cylindrical, and a square domain, numerical results are given for the saddle-node bifurcation value V , also referred to as the pull-in voltage, for which there is no steady-state membrane deection for V > V. For V > V it is shown numerically that the membrane dynamics are such that the thin dielectric membrane touches the lower plate in nite time. Results are given for both spatially uniform and nonuniform dielectric permittivity proles in the thin dielectric membrane. By allowing for a spatially nonuniform permittivity prole, it is shown that the pull-in voltage instability can be delayed until larger values of V and that greater pull-in distances can be achieved. Analytical bounds are given for the pull-in voltage V for two classes of spatially variable permittivity proles. In particular, a rigorous analytical bound V1, which depends on the class of permittivity prole, is derived that guarantees for the range V > V1 > V that there is no steady-state solution for the membrane deection and that nite-time touchdown occurs. Numerical results for touchdown behavior, both for V > V1 and for V < V < V1, together with an asymptotic construction of the touchdown prole, are given for both a spatially uniform and a spatially nonuniform permittivity prole.