Few-cycle Regime Atomic Force Microscopy

Abstract

Traditionally, dynamic atomic force microscopy (AFM) techniques are based on the analysis of the quasi-steady state response of the cantilever deflection in terms of Fourier analysis. Here we describe a technique that instead exploits the often disregarded transient response of the cantilever through a relatively modern mathematical tool, which has caused important developments in several scientific fields but that is still quite unknown in the AFM context: the wavelet analysis. This tool allows us to localize the time-varying spectral composition of the initial oscillations of the cantilever deflection when an impulsive excitation is given (as in the band excitation method), a mode that we call the few-cycle regime. We show that this regime encodes very meaningful information about the tip-sample interaction in a unique and extremely sensitive manner. We exploit this high sensitivity to gain detailed insight into multiple physical parameters that perturb the dynamics of the AFM probe, such as the tip radius, Hamaker constant, sample's elastic modulus and height of an adsorbed water layer. We validate these findings with experimental evidence and computational simulations and show a feasible path towards the simultaneous retrieval of multiple physical parameters.

Figure 1

Scheme of the implementation of the few-cycle regime atomic force microscopy. Panel (a) shows the shape of the cantilever excitation signal in the time domain (a modulated sinc excitation) and in the frequency domain (band excitation). Panel (b) shows the tip-sample deflection time dynamics (as a result of the excitation shown in panel (a)) as a function of varying excitation intensity (from v = 1 to v = 6 in increasing order). The black traces are the response of the free cantilever, the red traces are the response of the interacting cantilever, with a tip rest position at 7.5 nm from the surface. The gray horizontal line marks the 7.5 nm deflection amplitude to help visualize when the cantilever interacts with the surface.

Figure 2

Wavelet transform of the cantilever’s deflection (the signal) that is interacting with a highly oriented pyrolytic graphite (HOPG) sample when the band excitation described in Fig. 1(a) is applied to it. The ambient humidity is controlled and reduced to a nominal zero. The cantilever rest position is at 7.5 nm from the surface. The colors in the spectrogram represent the squared modulus (magnitude) of the wavelet transform. The frequency scale, the Fourier amplitude scale and the color scale are logarithmic base 2 (octaves) while the time scale is linear. The Fourier transform (white line, left and top axis) is superposed on the wavelet transform. The gray line under the arrows is the instantaneous frequency associated with the first flexural mode during the cantilever excitation; the arrows’ rotation shows the instantaneous phase shift between signal and reference. The reference is taken to be the cantilever’s deflection when subject to the same band excitation while located far from the sample (in the absence of tip-sample interactions). Arrows pointing right: signal in phase with reference, phase shift = 0; left: signal in anti-phase with reference, phase shift = π; up: signal leading reference by π/2, phase shift = π/2; down: signal lagging reference by π/2, phase shift = −π/2.

Figure 3

Representation of the steps involved in extracting the metric trajectories that are localized in the instantaneous frequency-amplitude-phase 3D space (the ifap 3D space). (a) Instantaneous time-varying values of frequency, phase shifts and amplitude, top to bottom, comprising the dynamical trajectories. In all plots the red traces correspond to the trajectories describing the evolution of amplitude and frequency for the free cantilever (no tip-sample interaction), the blue traces for the interacting cantilever. Since in this context the absolute phases do not have a physical meaning, only the phase shifts of the interacting cantilever with respect to the free cantilever are shown. The external band excitation amplitude increases linearly from left to right, from v = 1 to v = 6. These trajectories are measured along the instantaneous frequency trajectory in the time-frequency plane of the wavelet transform (see gray line in Fig. 2). (b) Metric trajectory in the ifap 3D space. We have color-coded and numbered each distinct interacting case in the upper plots and associated it with a unique point in the lower plot. To do so, we define a metric as the difference between the dynamical variables of the interacting cantilever and those corresponding to the free cantilever by observing their values at the time when the maximum free amplitude occurs (vertical light gray line). By repeating this process for a family of experiments performed at the same equilibrium position (7.5 nm in this case) and varying excitation amplitude, we generate a metric trajectory that is evidenced by the black line connecting the metric points. This line expands from a place near the origin (lowest excitation amplitude and quasi-free cantilever) to points further from the origin as the excitation intensity increases (interacting cantilever). All the points lay almost exactly on a plane, represented in shaded gray with its normal pointing to the observer.

Figure 4

Illustration of the sensitivity of the few cycle regime in discriminating values of relevant physical parameters: Hamaker constant (H), tip radius (R), liquid height above the surface (h) and sample’s Young’s modulus (E). To illustrate the sensitivity of the method we show the experimental trajectory in black (the same as in Fig. 3) and simulated trajectories in colors (details of simulations in Methods Section) where we systematically vary one fitting parameters at a time (H, R, h or E). An optimal value for each fitting parameter minimizes the distance between the simulated trajectories and the experimental trajectory. Warm colors (yellow and green) are used for simulated metric trajectories with fitting parameters lower than optimal. Cool colors (fuchsia, blue and cyan) are used for the simulated metric trajectories with fitting parameters higher than optimal. For simplicity, all traces that are relatively close to the experimental metric trajectory are shown in red. The specific values of the physical parameters being varied are shown in the lower plots, along with the calculated error. The calculated error is the sum of the Euclidean distances between the points of the simulated metric trajectory and the corresponding points in the experimental one. The color code in these error plots is consistent with the one used in the upper plot. From these error plots we can estimate a value of Hamaker constant of 100 zJ, a tip radius of 40 nm, a water layer height of 0.15 nm and a value of sample Young’s modulus of 14 GPa. All simulations and experiments were performed at a cantilever rest position of 7.5 nm. The sinc excitation amplitudes for the points along an experimental (and simulated) metric trajectory were linearly increased as in Fig. 3.

Figure 5

Schematics of the iterative process performed to retrieve multiple physical parameters from the experimental results. Briefly, in a first step the experimental cantilever traces with varying excitation intensities are recorded for the cases of interacting cantilever and free cantilever, followed by the corresponding calculation of the wavelet transform. Then, a metric trajectory is generated as described in Fig. 3. In a parallel manner a simultaneous process generating simulated metric trajectories is performed by systematically varying different physical parameters (e.g., water layer height, sample elastic modulus, tip radius, Hamaker’s constant) until a good fit is obtained between the experimental and simulated metric trajectories based on a minimization of error, as described in Fig. 4. As a result of this iterative process, multi-physical parameters pertaining the tip-sample contact interaction are retrieved with high accuracy.