Dynamic mulitmode analysis of non-linear piezoelectric microcantilever probe in bistable region of tapping mode atomic force microscopy

• The governing equations for non-linear contact force AFM are derived.
• Results show that non-linear excitation creates a shift in the frequency response.
• This frequency shift is important for generation of accurate images.
• The frequency shift leads to a bistable region.
• One mode is insufficient, two modes are sufficient for most applications.

Atomic Force Microscopy (AFM) uses a scanning process performed by a microcantilever probe to create a three dimensional image of a nano-scale physical surface. The dynamics of the AFM microcantilever motion and tip–sample force is needed to generate accurate images. In this paper, the non-linear dynamics of a piezoelectric microcantilever probe in tapping AFM will be investigated. The equations of motion are derived for a non-linear contact force, the analytical expressions for natural frequencies and mode shapes are obtained, and the method of multiple scales is used to find the analytical frequency response of the microcantilever. Results show that the non-linear excitation force creates a shift in the frequency response curve during contact mode. This frequency shift is an important consideration in the modeling of AFM dynamics for generation of accurate images as well as for accurate readings when using the AFM microcantilever for other applications. The frequency shift also leads to a bistable region, in which a high and a low amplitude solution coexist. The response of the microcantilever at a single input frequency and voltage are analyzed for both high and low amplitude solutions with the main difference being that the high amplitude solution makes contact with the sample while the low amplitude solution does not. This contact results in higher harmonics of the microcantilever being excited. The response in the bistable region is compared to the response in the monostable region. Additionally, a convergence analysis is used to determine the number of modes necessary to describe the motion of the microcantilever in tapping mode. It is determined that one mode is insufficient, two modes are sufficient for most applications, and five modes is the most that would be necessary for applications that require very precise readings.