“Pull-in” of a pre-stressed thin film by an electrostatic potential: A 1-D rectangular bridge and a 2-D circular diaphragm

A one-dimensional (1-D) rectangular pre-stressed thin film clamped at two opposite ends is actuated by an electrostatic voltage applied to a pad directly underneath. The pre-stress is allowed to be either tensile or compressive in nature. At a critical applied voltage, the film becomes unstable and makes direct contact with the pad, leading to “pull-in”. An elastic model is constructed to account for the phenomenon over ranges of film–pad gap and residual stress. The results compare favorably with finite element analysis (FEA) in literature and experimental data, and possess distinct advantages over other available closed-form solutions. The model is further extended to a two-dimensional (2-D) axisymmetric diaphragm clamped at the periphery. Interrelationship between bridge–pad gap, residual membrane stress, critical voltage, and pull-in central deflection is derived such that new design criteria can be derived for micro-/nano-electromechanical devices. Implications for nano-structures are also discussed.