MEMS Cantilever Biosensors Driven by Fringe Effect: Voltage-Amplitude Response of Parametric Resonance

Abstract

Abstract This paper deals with voltage-amplitude response of fringing field/effect actuated micro-electromechanical system (MEMS) cantilever beam resonator biosensors at parametric resonance. The system consists of a flexible electrode (cantilever beam) and a fixed electrode (ground plate) in parallel, connected through an AC voltage source. The electrostatic force induced is due to the parallel-plate capacitance, which is directly related to the overlapping area of the two electrodes (Palmer’s Formula). In addition, fringing field electrostatic actuation must be accounted for. This is due to the bending of electric field lines from the ground plate to the top and side walls of the cantilever beam. In this study, the cantilever beam is actuated by the fringing field only. This is modelled by the following approach: the electrostatic force due to the parallel-plate capacitance is neglected due to a hole in the ground plate (removes overlapping area). With the frequency of the AC voltage near the natural frequency of the MEMS cantilever, the resonator will go into parametric resonance. The partial differential equation describing the dynamic behavior of the micro-beam resonator is nondimensionalized and a Reduced Order Model (ROM) is formulated. This is a one-mode of vibration model that is solved analytically using the Method of Multiple Scales (MMS). The voltage-amplitude response (bifurcation diagram) is projected. The force due to fringing actuation is expanded in Taylor Series and the terms up to the 3rd and 5th powers are kept (small terms due to small bookkeeping parameter). The method identified a zero-amplitude solution as well as two branches, one stable and one unstable, with two bifurcation points, subcritical and supercritical. In addition, a two-term reduced order model is developed and solved numerically to obtain steady-state solutions and validate the results. Finally, the equations from the two-term reduced order model are used in AUTO-07p (a continuation and bifurcation software for ODEs) to get a voltage response from the two-term ROM. The response is then compared to those that include the electrostatic force only as well as electrostatic and fringe together. The effect of damping, fringe, and detuning frequency on the voltage-amplitude response are also investigated.