Reduction of frequency noise and frequency shift by phase shifting elements in frequency modulation atomic force microscopy.

We recently reported the analysis of the frequency noise in the frequency modulation atomic force microscopy (FM-AFM) both in high-Q and low-Q environments [Rev. Sci. Instrum. 80, 043708 (2009)]. We showed in the paper that the oscillator noise, the frequency fluctuation of the oscillator, becomes prominent in the modulation frequency lower than f(0)∕2Q, where f(0) and Q are the resonance frequency and Q-factor. The magnitude of the oscillator noise is determined by the slope of the phase versus frequency curve of the cantilever at f(0). However, in actual FM-AFM in liquids, the phase versus frequency curve may not be always ideal because of the existence of various phase shifting elements (PSEs). For example, the spurious resonance peaks caused by the acoustic excitation and a band-pass filter in the self-oscillation loop increase the slope of the phase versus frequency curve. Due to those PSEs, the effective Q-factor is often increased from the intrinsic Q-factor of the cantilever. In this article, the frequency noise in the FM-AFM system with the PSEs in the self-oscillation loop is analyzed to show that the oscillator noise is reduced by the increase of the effective Q-factor. It is also shown that the oscillation frequency deviates from the resonance frequency due to the increase of the effective Q-factor, thereby causing the reduction in the frequency shift signal with the same factor. Therefore the increase of the effective Q-factor does not affect the signal-to-noise ratio in the frequency shift measurement, but it does affect the quantitativeness of the measured force in the FM-AFM. Furthermore, the reduction of the frequency noise and frequency shift by the increase of the effective Q-factor were confirmed by the experiments.