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

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 film. In a certain asymptotic limit representing a thin device, the mathematical model consists of a nonlinear partial differential equation for the deflection of the thin dielectric membrane. When a voltage V is applied to the conducting film, the dielectric membrane deflects 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 deflection 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 finite time. Results are given for both spatially uniform and nonuniform dielectric permittivity profiles in the thin dielectric membrane. By allowing for a spatially nonuniform permittivity profile, 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 profiles. In particular, a rigorous analytical bound V1, which depends on the class of permittivity profile, is derived that guarantees for the range V > V1 > V∗ that there is no steady-state solution for the membrane deflection and that finite-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 profile, are given for both a spatially uniform and a spatially nonuniform permittivity profile.