A Highly Accurate Pull-in Voltage Model for an Atomic Force Microscope Probe

A readily usable closed-form model has been developed to determine the pull-in voltage of an atomic force microscopy (AFM) cantilever probe under electrostatic actuation. The analytical model is derived based on the Euler-Bernoulli beam theory, Taylor series expansion of the nonlinear electrostatic force, and deflection function of the first natural mode of a cantilever beam. The model takes account of the electrostatic force associated with the fringing field capacitances between the cantilever probe and the substrate to predict a more accurate pull-in voltage. The developed closed-form model has been verified by comparing the model predicted values with published experimental results with a maximum deviation of 3.36%. The model has also been compared with a published closed-form model and 3-D electromechanical finite element analysis (FEA) carried out by the authors. The results are found to be in excellent agreement.