A new type of atomic force microscope based on chaotic motions

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Abstract

Local flow variation (LFV) method of non-linear time series analysis is applied to develop a chaotic motion-based atomic force microscope (AFM). The method is validated by analyzing time series from a simple numerical model of a tapping mode AFM. For both calibration and measurement procedures the simulated motions of the AFM are nominally chaotic. However, the distance between a tip of the AFM and a sample surface is still measured accurately. The LFV approach is independent of any particular model of the system and is expected to be applicable to other micro-electro-mechanical system sensors where chaotic motions are observed or can be introduced.

Introduction

Atomic force microscopy (AFM) is a widely used tool for atom level surface analysis. In an AFM, a microscale cantilever with a sharp tip is used to scan the specimen surface, and the vibration of the cantilever is measured to identify the distance between the tip and the specimen surface. Depending on the tip interaction with the specimen surface, an AFM can work in three modes: contact, no contact, and tapping (where the tip oscillates and touches the surface occasionally). Tapping mode AFM is one of the most potent techniques for topographic imaging of substrates. However, data analysis in this mode is prone to complications. In this mode—due to non-linearities that arise from contact interactions between the tip of a cantilever beam and substrate surface [1]—unwanted dynamic phenomenon such as phase jumps, period doublings, or even chaos are observed [2], [3], [4], [5], [6]. These non-linear effects are difficult to handle using standard data processing and hinder the performance of an AFM system.

To mitigate unwanted dynamical behavior in tapping mode AFM, careful selection of operation parameters and the introduction of various control strategies is the most recommended solution. A large set of parameters, such as driving frequency or stiffness of the cantilever, are available for achieving desired stable periodic motions. However, experimental experience [6], [7] shows that the process of finding parameters which ensure stable motions across the whole sample surface is time consuming. In addition, these solutions do not compensate measurement errors caused by the inherent non-linearity of an AFM.

Recent work shows that chaotic motions can provide valuable information about system's parameters [8], [9], and related non-linear time series analysis (NTSA) methods have been widely applied [10], [11], [12]. Although chaotic motions are observed in different micro-electro-mechanical systems (MEMS) besides AFM [14], few have attempted to apply these NTSA methods to handle chaotic motions in MEMS [13], and the main reasons are listed as follows: (1) Without setting up a one-to-one relationship between the outcome of NTSA and measurement targets, most of the NTSA methods only provide qualitative results. (2) MEMS work in quite high frequency range (at least several KHz range), and most of the NTSA methods are computationally intensive. Thus, these methods cannot process large amounts of data from MEMS in real time.

In this paper, a new NTSA method, local flow variation (LFV) [15], is applied to analyze chaotic motions generated by an AFM system. As a practical implementation of phase space warping (PSW) concept [16], [17], [18], [19], the LFV provides direct, linear, one-to-one relationship between small changes in system's parameters and LFV-based feature vectors for both chaotic and periodic motions. The LFV is also computationally fast, and can be utilized to process large amount of data generated by MEMS in real time and provide quantitative results.

In this paper, the LFV is applied to analyze signals generated by a numerical model of an AFM. Although the response of the AFM is nominally chaotic, the change of the surface features (distance between the surface and the cantilever tip) is still identifiable. The LFV only assumes that AFM is governed by a deterministic model and its parameters (surface features) change slowly in time. Thus, LFV not only provides an alternative approach to mitigate non-linear effects widely observed in MEMS, but shows the potential for various MEMS sensors based on chaotic motions.

In the following section, some basic details of the LFV method are described. Next, a simplified numerical model of and AFM system is introduced, which is used in Section 4, to analyze the chaotic motions generated from the numerical model. The process of AFM sensor calibration and measurement are described next. Finally, the performance of the LFV-based AFM and potential to apply LFV in other MEMS sensors is discussed.

Section snippets

Description of the method

This paper follows a basic framework developed for systems with slowly drifting parameters [20]. Specifically, a hierarchical dynamical system is considered where slowly evolving surface features are causing changes in a fast-time AFM system's parameters. Mathematically this can be described asx˙=f(x,μ(φ),t),φ˙=εg(x,φ)andy=h(x),where overdots denote time (t) differentiation, x∈Rn is a fast-time AFM state variable, φ∈Rm is a slowly changing surface feature that alters a parameter vector μ∈Rp,

Numerical model and its simulation

To illustrate the performance of the LFV-based AFM, a simplified model of an AFM [24]—a non-linear spring mass system with external excitation (Fig. 1)—is studied:mx¨+β(x˙-d˙)+k(x-d)=f(δ),where x and d are the displacement of the tip versus the fixed frame and the controller input, respectively; m, β, and k denote the tip mass, damping coefficient, and stiffness coefficient of the spring, respectively. The interaction between the tip and the sample surface is described by the van der Waals

Calibration and measurement

In simulations, change in the surface topography is simulated by altering z. It is also assumed that the AFM measurement function is given by y=h(x)=x, which is a reasonable assumption, given that laser vibrometer measures AFM beam oscillations. Thus, in this paper, x time series need to be used to infer the surface topography z. As every instrument, the chaotic AFM needs to be calibrated. This process is described in the next section, followed by the validation of the measurement procedure.

Discussion

In practice, x variable is not directly measurable, and the AFM oscillations are recorded using a reflection of a laser beam which is projected on the top surface of the cantilever. However, the phase space reconstruction is still expected to work since it only requires that the measurement function h∈C1, in addition to the fast-time system being deterministic as described in Eq. (1).

In experimental contexts, AFM exhibits chaotic motions in two different tapping modes: (1) “hard tapping” mode,

Conclusion

An AFM based on chaotic motion and utilizing LFV methodology was introduced. This methodology is not dependent on particulars of system model and can be easily applied to other MEMS sensors. The basic concepts and procedures were described and a simple one-degree-of-freedom model of the AFM was used to illustrate the calibration and measurement procedures. The calibration process was used to identify range of parameters for which the chaotic response is possible and to find associated

Acknowledgment

This paper is based on the work supported by the NSF CAREER Grant no. CMS-0237792.

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