A Novel Semianalytical Approach for Finding Pull-In Voltages of Micro Cantilever Beams Subjected to Electrostatic Loads and Residual Stress Gradients

Beam structures are widely used in microelectromechanical systems (MEMS) sensors and actuators, and modeling of pull-in behavior of beams subjected to electrostatic force is essential for MEMS actuators. However, from a fabrication perspective, MEMS microbeams are usually curled due to residual stress gradients, and this causes difficulties to accurately estimate the pull-in voltages. As a result, the characteristics of microbeams subjected to both residual stress gradients and electrostatic forces must be investigated to provide accurate information for the design of sensors and actuators. In this paper, a novel semianalytical formulation for computing the pull-in voltage of a curled cantilever beam due to residual stress gradients is proposed. By assuming an admissible deformation shape and using the energy method to determine the coefficients of the shape functions, it is possible to find the pull-in characteristics of the curled cantilevers. Detailed parametric studies are subsequently performed to quantify the influence of various geometry and processing parameters on the pull-in characteristics of those microbeams. Finally, we present a fitted formula for MEMS engineers to estimate pull-in voltages for beams with residual stress gradients for design optimization. The proposed method can also be extended for handling bilayered curled cantilever beams due to thermomechanical mismatches. Therefore, the method and results presented in this paper should be useful in micro sensor and actuator design.