Nonlinear forced response of piezoelectric microcantilevers with application to tapping mode atomic force microscopy

摘要

Atomic Force Microscopy (AFM) uses a scanning process performed by a microcantilever beam to create a three dimensional image of a nano-scale physical surface. AFM includes a microcantilever probe with a tip at the end that is controlled in order to keep the force between the tip and the surface constant by changing the distance of the microcantilever from the surface. Some microcantilevers have a layer of piezoelectric material on one side of the microcantilever for actuation purpose. An accurate understanding of the microcantilever motion and tip-sample force is needed to generate accurate imaging. In this paper, the equations of motion for an AFM piezoelectric microcantilever probe are derived for a nonlinear contact force. The analytical expressions for natural frequencies and mode shapes are determined. Then, the analytical frequency response of the piezoelectric probe is found using the method of multiple scales. The effects of nonlinear excitation force on the microcantilever probe's frequency and amplitude have been analytically studied. The force nonlinearities lead to a frequency shift in the response. Accurately modeling this frequency shift during contact mode of the AFM probe is a significant consideration for the generation of more accurate imaging.