Wavelet cross-correlation and phase analysis of a free cantilever subjected to band excitation

Francesco Banfi,Gabriele Ferrini

doi:10.3762/bjnano.3.33

Francesco Banfi, Gabriele Ferrini

Open Access

https://doi.org/10.3762/bjnano.3.33

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Abstract

This work introduces the concept of time–frequency map of the phase difference between the cantilever response signal and the driving signal, calculated with a wavelet cross-correlation technique. The wavelet cross-correlation quantifies the common power and the relative phase between the response of the cantilever and the exciting driver, yielding “instantaneous” information on the driver-response phase delay as a function of frequency. These concepts are introduced through the calculation of the response of a free cantilever subjected to continuous and impulsive excitation over a frequency band.

Highlights

Atomic force microscopy (AFM) has made important progresses towards the characterization of material properties at the nanoscale by means of dynamic techniques that extended the microscope capabilities well beyond simple topographic measurements [1,2]
Among the techniques developed in dynamic AFM, multimode excitation and the so called band-excitation methods have been put forward recently [3,4,5]
A fundamental feature of wavelet phase analysis consists of measuring the phase response of the cantilever with respect to complex excitation signals, and displaying the results in the time–frequency plane, with a resolution set by the Heisenberg principle, as shown in the simulations reported in Figure 2, Figure 4, and Figure 6 for a damped harmonic oscillator

Summary

Introduction

Atomic force microscopy (AFM) has made important progresses towards the characterization of material properties at the nanoscale (elastic constants, force interactions, friction, molecular interactions, to name only a few) by means of dynamic techniques that extended the microscope capabilities well beyond simple topographic measurements [1,2]. Among the techniques developed in dynamic AFM, multimode excitation and the so called band-excitation methods have been put forward recently [3,4,5] All of these techniques are based on the frequency, amplitude and phase response around one or more cantilever oscillation modes when the tip interacts with the sample surface. Thermal noise analysis has been performed, with the aid of wavelet transforms, to characterize the time–frequency response of a thermally excited cantilever in dynamic force spectroscopy [10,11,12]. In these previous works, the focus was on the time

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Discussion

Conclusion