Experimental Characterization of the static behaviour of microcatntilevers electrostatically actuated

This paper concerns the experimental validation of some mathematical models previously developed by the authors, to predict the static behaviour of microelectrostatic actuators, basically free-clamped microbeams. This layout is currently used in RF-MEMS design operation or even in material testing at microscale. The analysis investigates preliminarily the static behaviour of a set of microcantilevers bending in-plane. This investigation is aimed to distinguish the geometrical linear behaviour, exhibited under small displacement assumption, from the geometrical nonlinearity, caused by large deflection. The applied electromechanical force, which nonlinearly depends on displacement, charge and voltage, is predicted by a coupled-field approach, based on numerical methods and herewith experimentally validated, by means of a Fogale Zoomsurf 3D. Model performance is evaluated on pull-in prediction and on the curve displacement vs. voltage. In fact, FEM nonlinear solution performed by a coupled-field approach, available on commercial codes, and by a FEM non-incremental approach are compared with linear solution, for different values of the design parameters.

💡 Research Summary

This paper presents an experimental validation of several mathematical and numerical models that predict the static behavior of electrostatically actuated free‑clamped microcantilevers, a configuration widely used in RF‑MEMS and microscale material testing. Eight in‑plane bending cantilevers were fabricated from doped polysilicon (E ≈ 166 GPa, ν = 0.23) with varying lengths (100–800 µm), widths (15 µm), thicknesses (2 µm) and gaps (0.05–0.5 µm), thereby covering a broad range of aspect ratios that influence both mechanical stiffness and electrostatic field distribution.

The experimental setup employed a Fogale Zoomsurf 3D optical interferometer, providing sub‑nanometer vertical resolution and 0.6 µm lateral resolution. Each specimen was incrementally biased from 0 V up to the pull‑in event (maximum 200 V supplied by the instrument), while tip displacement was recorded in real time. Measurements were repeated multiple times to ensure repeatability and averaged for analysis.

Three modeling approaches were compared: (1) a conventional linear beam theory assuming small deflections; (2) a non‑incremental nonlinear finite‑element method (FEM) developed by the authors, using a Special Beam Element (SFET) capable of handling large displacements; and (3) a coupled‑field FEM available in commercial ANSYS, which combines PLANE121 electrostatic elements with PLANE183 structural elements and employs mesh morphing to accommodate geometric changes.

Results show that the linear model matches experimental data only for low voltages where tip deflection remains below roughly 5 % of the beam length. As voltage increases, geometric nonlinearity becomes dominant, especially for specimens with large length‑to‑gap (R₂) and length‑to‑thickness (R₃) ratios. The SFET‑based nonlinear FEM reproduces the full displacement‑versus‑voltage curve with high fidelity, predicting pull‑in voltages within 10–15 % of the measured values. The ANSYS coupled‑field solution, while conceptually similar, suffers from mesh‑distortion issues and tends to over‑predict pull‑in for large gaps.

Parametric studies on Young’s modulus (150–166 GPa) and beam thickness reveal that thickness has a far greater impact on pull‑in voltage than the modulus, confirming the sensitivity of electrostatic actuation to geometric parameters. Comparison with the analytical Senturia‑Osterberg pull‑in formula shows reasonable agreement but systematic deviations attributable to three‑dimensional fringe fields not captured in the simplified model.

In conclusion, accurate static prediction of electrostatically actuated microcantilevers requires a nonlinear structural analysis that incorporates coupled electro‑mechanical effects. The authors’ SFET‑based non‑incremental FEM offers superior accuracy and computational efficiency compared to standard linear theory and commercial coupled‑field solvers, making it a valuable tool for MEMS designers seeking reliable pull‑in predictions and insight into the interplay of geometry, material properties, and electrostatic forces.