Stochastic noise in atomic force microscopy

Abstract

Having reached the quantum and thermodynamic limits of detection, atomic force microscopy (AFM) experiments are routinely being performed at the fundamental limit of signal to noise. A critical understanding of the statistical properties of noise leads to more accurate interpretation of data, optimization of experimental protocols, advancements in instrumentation, and new measurement techniques. Furthermore, accurate simulation of cantilever dynamics requires knowledge of stochastic behavior of the system, as stochastic noise may exceed the deterministic signals of interest, and even dominate the outcome of an experiment. In this article, the power spectral density (PSD), used to quantify stationary stochastic processes, is introduced in the context of a thorough noise analysis of the light source used to detect cantilever deflections. The statistical properties of PSDs are then outlined for various stationary, nonstationary, and deterministic noise sources in the context of AFM experiments. Following these developments, a method for integrating PSDs to provide an accurate standard deviation of linear measurements is described. Lastly, a method for simulating stochastic Gaussian noise from any arbitrary power spectral density is presented. The result demonstrates that mechanical vibrations of the AFM can cause a logarithmic velocity dependence of friction and induce multiple slip events in the atomic stick-slip process, as well as predicts an artifactual temperature dependence of friction measured by AFM.

Multiscaling behavior of atomic-scale friction

“…Labuda et al [20][21][22] divided noises of a typical AFM experiment into three categories: (1) detection noise which comes from AFM optical device, (2) force noise that is mainly caused by thermal fluctuations, and (3) displacement noise which is related to the design of AFM system. Their study was based on analyzing the noises of nanofriction data using the method of power spectral density and simulating the effect of each of these noises on nanofriction.…”

Dynamics of Atomic Stick-Slip Friction Examined with Atomic Force Microscopy and Atomistic Simulations at Overlapping Speeds

“…In the PTT model, only those thermal vibrations of the tip apex are considered. However, other thermal noise sources, such as thermally induced vibrations of the cantilever, or athermal instrument noise, such as mechanical vibrations of the AFM apparatus and electronic 60 Hz noise, are not included despite the fact that they too can lower the activation barrier to slip by adding energy into the contact [42,43]. Both athermal and thermal noise sources are inherent in every experiment, but not fully captured in simulations.…”

“…Physically, the plateau represents attaining a high enough scanning speed that available vibrations of the atoms at the end of the tip apex no longer have enough time assist in overcoming the local interfacial potential energy barriers [42]. In the PTT model, only those thermal vibrations of the tip apex are considered.…”

Estimating Damping in Microresonators by Measuring Thermomechanical Noise Using Laser Doppler Vibrometry

“…The PSD of finite time signals can be estimated by segmenting the time series using an appropriate window function, computing the Fourier transform of each segment, and then appropriately reassembling the transforms. Such approaches (which include the widely used Welch periodogram technique [37]) are computationally efficient but suffer from spectral leakage and a bias in the estimation of the quality factor [38], [39]. Therefore, we employed a direct method (also called the Daniell method) that eliminates spectral leakage by avoiding segmenting.…”

“…The underlying statistical concepts and numerical implementation of the Daniell method are discussed in detail in Ref. [38].…”