Phase noise induced due to amplitude fluctuations in dynamic force microscopy

Abstract

In dynamic force microscopy, the force sensor is driven on its resonance frequency and the amplitude of the cantilever is sustained at a constant value. The amplitude typically ranges between 0.1 and 30 nm. If a large amplitude is set, the cantilever tip senses both long-range and short-range interaction forces provided that the tip is close to the sample surface. The short-range interactions are decisive for the atomic contrast in atomic force microscopy (AFM) images. They can be separated from the long-range interactions by setting an amplitude which encompasses the typical range of the interaction force, i.e., the subangstrom regime for van der Waals contribution. It is distinctive for cantilevers operated at small driving amplitudes that the cantilever deflection can be considered as a sinusoidal signal superimposed with a quasimonochromatic random signal originating from fluctuations. If one measures experimentally the standard deviation of the phase ${\ensuremath{\sigma}}_{\ensuremath{\varphi}}$ of the signal with respect to a monochromatic reference signal, a universal relationship between the standard deviation of the phase ${\ensuremath{\sigma}}_{\ensuremath{\varphi}}$ and the cantilever amplitude ${x}_{0}$ is found. The smaller the ratio of rms amplitude of the sinusoidal signal and the rms value of random signal is, the larger the phase fluctuations are. Phase fluctuations are of importance for measurements at small amplitudes, since they determine the limit of phase-sensitive measurements or the lateral imaging resolution in the so-called pendulum mode of AFM operation. In this paper we develop a heuristic model, which provides an analytical formula for the probability density of phase noise of a sinusoidal signal superimposed by a quasimonochromatic one with respect to a reference oscillator. The variance of the phase noise can be deduced from the distribution functions. The suggested model is verified experimentally and is compared with theoretical predictions. The amplitude-dependent phase fluctuations are a powerful tool to determine the spring constant of the beam or to calibrate its oscillation amplitude.