Pull-in behavior of electrostatically actuated multistable microstructures

We present an analysis of the electromechanical behavior and stability of a capacitive-based Micro Electro Mechanical Systems (MEMS) device with non-monotonous stiffness-deflection dependence. As an example, we consider a flexible initially curved double clamped micro beam actuated by a distributed electrostatic force. Since the system exhibits both mechanical snap-through buckling and electrostatic pull-in instability, the equilibrium curve has two bifurcation points implying the existence of multiple equilibrium configurations. The multistability phenomenon described in the present work is a result of interaction between mechanical and electrostatic nonlinearities of the system and differs from the electrostatic pull-in based bistability and mechanical bistability associated with the snap-through buckling. The governing equations of the geometrically nonlinear curved Euler-Bernoulli beam are formulated in the framework of the shallow arch approximation. Actuating force is calculated using second order perturbation solution of the Laplace equation for an electric potential. A coupled electro mechanical model is built by the Rayleigh-Ritz method with linear undamped eigenmodes of a straight beam as base functions. After verification of the model results, we analyze the influence of initial geometry of the beam on the location (in terms of actuation voltage and deflections) of the critical points on the bifurcation diagram. It was found that for snap-through to take place, the initial elevation of the beam should be larger than a certain value whereas the existence of electrical pull-in instability is unconditional. In addition, the stable relative deflection of a curved beam is larger than of initially straight beam. Based on the model results, we present an example of a multistable actuator design. Devices of various configurations were fabricated of single crystal silicon using deep reactive ion etching and the existence of multiple stable states and multiple instability points was demonstrated experimentally.