An analytical model for calculating the pull-in voltage of micro cantilever beams subjected to tilted and curled effects

Pull-in is a fundamental phenomenon in electrostatic micro devices. In previous studies on modeling the pull-in voltage of suspended micro cantilever beam subject to residual stress, only curled deformation was considered. This study proposed a modified deformation function, which considered both curled and tilted deformations caused by gradient stress and mean stress, to calculate the pull-in voltage of the suspended cantilever beam with residual deformations. In order to verify the proposed analytical model, suspended poly-silicon cantilever beams with three different lengths, 260 μm, 295 μm and 330 μm, are fabricated through surface micromachining process. It is shown that the residual deformations include both curled and tilted deformations, where the tilted angle and radius of curvature can be identified by white light interferometer (WLI). By comparing the analytical results with measurement results on pull-in voltages, it is found that while only considering curled effect, the average error of calculated pull-in voltage is 10.5%. On the other hand, when both tilted and curled effects are considered, the average error is reduced to 3.2%, which verifies the accuracy improvement of the proposed analytical model.