International Journal of Non-Linear Mechanics 43 (2008) 521–526www.elsevier.com/locate/nlm

A new type of atomic force microscope based on chaotic motions

Ming Liu1, David Chelidze∗

Department of Mechanical Engineering and Applied Mechanics, University of Rhode Island, Kingston, RI, USA

Received 23 July 2007; received in revised form 5 February 2008; accepted 4 March 2008

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 andmeasurement procedures the simulated motions of the AFM are nominally chaotic. However, the distance between a tip of the AFM and asample surface is still measured accurately. The LFV approach is independent of any particular model of the system and is expected to beapplicable to other micro-electro-mechanical system sensors where chaotic motions are observed or can be introduced.� 2008 Elsevier Ltd. All rights reserved.

Keywords: AFM: Atomic force microscope; Chaotic motion; Phase space warping; Smooth orthogonal decomposition; Local flow variation

1. Introduction

Atomic force microscopy (AFM) is a widely used tool foratom level surface analysis. In an AFM, a microscale can-tilever with a sharp tip is used to scan the specimen surface,and the vibration of the cantilever is measured to identify thedistance between the tip and the specimen surface. Dependingon the tip interaction with the specimen surface, an AFM canwork in three modes: contact, no contact, and tapping (wherethe tip oscillates and touches the surface occasionally). Tap-ping mode AFM is one of the most potent techniques for topo-graphic imaging of substrates. However, data analysis in thismode is prone to complications. In this mode—due to non-linearities that arise from contact interactions between the tipof a cantilever beam and substrate surface [1]—unwanted dy-namic phenomenon such as phase jumps, period doublings, oreven chaos are observed [2–6]. These non-linear effects are dif-ficult to handle using standard data processing and hinder theperformance of an AFM system.

∗ Corresponding author.E-mail addresses: liumingmech@gmail.com (M. Liu),

chelidze@egr.uri.edu (D. Chelidze).URL: http://www.mce.uri.edu/chelidze/ (D. Chelidze).

1 Present address: Migma Systems, Inc., 1600 Providence Highway,Walpole, MA 02081, USA.

0020-7462/$ - see front matter � 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.ijnonlinmec.2008.03.001

To mitigate unwanted dynamical behavior in tapping modeAFM, careful selection of operation parameters and the intro-duction of various control strategies is the most recommendedsolution. A large set of parameters, such as driving frequencyor stiffness of the cantilever, are available for achieving de-sired stable periodic motions. However, experimental experi-ence [6,7] shows that the process of finding parameters whichensure stable motions across the whole sample surface is timeconsuming. In addition, these solutions do not compensate mea-surement errors caused by the inherent non-linearity of an AFM.

Recent work shows that chaotic motions can provide valu-able information about system’s parameters [8,9], and relatednon-linear time series analysis (NTSA) methods have beenwidely applied [10–12]. Although chaotic motions are observedin different micro-electro-mechanical systems (MEMS) besidesAFM [14], few have attempted to apply these NTSA methodsto handle chaotic motions in MEMS [13], and the main rea-sons are listed as follows: (1) Without setting up a one-to-onerelationship between the outcome of NTSA and measurementtargets, most of the NTSA methods only provide qualitative re-sults. (2) MEMS work in quite high frequency range (at leastseveral KHz range), and most of the NTSA methods are compu-tationally intensive. Thus, these methods cannot process largeamounts 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

522 M. Liu, D. Chelidze / International Journal of Non-Linear Mechanics 43 (2008) 521–526

an AFM system. As a practical implementation of phase spacewarping (PSW) concept [16–19], the LFV provides direct, lin-ear, one-to-one relationship between small changes in system’sparameters and LFV-based feature vectors for both chaotic andperiodic motions. The LFV is also computationally fast, andcan be utilized to process large amount of data generated byMEMS in real time and provide quantitative results.

In this paper, the LFV is applied to analyze signals gener-ated by a numerical model of an AFM. Although the responseof the AFM is nominally chaotic, the change of the surfacefeatures (distance between the surface and the cantilever tip) isstill identifiable. The LFV only assumes that AFM is governedby a deterministic model and its parameters (surface features)change slowly in time. Thus, LFV not only provides an alter-native approach to mitigate non-linear effects widely observedin MEMS, but shows the potential for various MEMS sensorsbased on chaotic motions.

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

2. Description of the method

This paper follows a basic framework developed for systemswith slowly drifting parameters [20]. Specifically, a hierarchi-cal dynamical system is considered where slowly evolving sur-face features are causing changes in a fast-time AFM system’sparameters. Mathematically this can be described as

x = f(x, μ(�), t), � = �g(x, �) and y = h(x), (1)

where overdots denote time (t) differentiation, x ∈ Rn is afast-time AFM state variable, � ∈ Rm is a slowly changingsurface feature that alters a parameter vector μ ∈ Rp, and y ∈R is a measurement taken from the micro-cantilever (usuallylaser-based measurement of the AFM cantilever oscillations).Here, the fast-time is characterized by the time scales associatedwith the oscillations of the AFM cantilever, and slow-time ischaracterized by the scan rate of the AFM. In this paper, onlytopological features of the surface are considered (i.e., � ∈R). However, in general, the same methodology can be usedto identify other surface features, if they affect the fast-timedynamics of the AFM system.

The non-stationary drifts in parameters μ cause geometricalternations in the phase space of the fast-time AFM system,which are described by the PSW concept [20]. Specifically, thePSW describes the deformation or warping of the phase spacetrajectories caused by parameter drifts. Recent studies demon-strate that the parameter drifts also influence the probabilitydensity or distribution of measured trajectory points in the phasespace of the fast-time subsystem [21,22]. So the changes incorresponding statistical moments also reflect parameter vari-

ations and can be used as a PSW metrics, which is the basicidea behind the LFV.

In AFM, there is no direct access to the state variable x, onlythe scalar time series of measurements {yi}Ni=1 are available.Therefore, instead of working in the true phase space of theAFM system, reconstructed phase space data are considered{y(i)}N−D+1

i=1 :

y(i) = [yi, yi+�, . . . , yi+(D−1)�]T ∈ Rd , (2)

where � is a delay time and D is an embedding dimension.These delay coordinate embedding (DCE) parameters can bedetermined by standard techniques [23] assuming AFM is adeterministic system. The reconstructed phase space snapshotsare obtained from time series recorded over intermediate timeperiods, over which the drift in parameters can be neglected(the AFM system is assumed to be quasi-stationary over inter-mediate time scales).

2.1. Local flow variation

The LFV characterizes the change in the probability distribu-tion of trajectories in fast-time subsystem’s phase space (or thecorresponding reconstructed phase space), which is caused bydrifts in the system parameters μ. To quantify the LFV, the firstmoment of the reconstructed points (FB(�) = EB[y(n; �)]) ina small volume B of the reconstructed phase space is estimatedusing

FB(�)�‖B‖−1∑

y(n;�)∈By(n; �), (3)

where y(n; �) ∈ RD is a sample of the reconstructed fast-timetrajectory for the current surface parameter �. It is reasonableto hypothesize that FB(�) is a function of drifting parameter� for a given B. Thus, an LFV tracking function (FLVTF) isdefined as

lB(�) = FB(�) − FB(�0), (4)

where � and �0 are the current value and reference value ofthe surface parameters, respectively. If FB(�) is continuousand differentiable, we can expect a linear observability of thedrifting parameter:

lB(�) = dFB

d�

∣∣∣∣�=�0

(� − �0) + O(|� − �0|2). (5)

The linear observability is generally expected to hold in theabsence of bifurcations in the AFM fast-time dynamics.

Given a large amount of data, appropriate choice of the hyper-volume B, and assuming m < D, Eq. (5) can be used directly toreconstruct surface features. However, in practice only a limitedamount of data is available and the optimal choice of hyper-volume is not obvious. In addition, the measurements are usu-ally contaminated by some noise. Therefore, to mitigate all theabovementioned difficulties, the reference reconstructed phasespace is partitioned into Ns hyper-cuboids so that each con-tains approximately the same amount of points. Then, Eq. (5)

M. Liu, D. Chelidze / International Journal of Non-Linear Mechanics 43 (2008) 521–526 523

is evaluated in each of these hyper-cuboids and surface charac-teristics (topography, in this case) are inferred by multivariateanalysis of the resulting feature vectors. In particular smoothorthogonal decomposition (SOD) [20] is used for this purpose.

The LFVTFs are calculated for each small volume Bk (k =1, . . . , Ns) reconstructed points from data recorded over inter-mediate time interval i, and are assembled together into a fea-ture vector as

vi = [li1; li2; . . . ; liNs], (6)

where lik = lBkfor the data from i-th record, vi ∈ RDNs , and

“;” indicates columnwise concatenation of each lik . For furtheranalysis, the feature vectors are assembled in time sequenceinto a tracking matrix Y ∈ Ru×DNs as row vectors, where u isthe total number of data sets recorded over intermediate disjointtime intervals.

2.2. Smooth orthogonal decomposition

Given the matrix Y, the SOD is used to identify its linearprojection � = Yq that maximizes both its smoothness in timeand overall variation. The SOD can be described by the fol-lowing generalized eigenvalue problem:

[YT Y]q = �[(DY)T DY]q, (7)

where D is a differential operator. This type of generalizedeigenvalue problem is usually solved by generalized singularvalue decomposition of the matrix pair [Y, DY] [20].

The idea behind the SOD is that it will find the projectionswhere the local fluctuations in LFVTF due to noise and bifurca-tions are minimized. The largest eigenvalue or smooth orthog-onal value (SOV) �1 of Eq. (7) is associated with the smoothestprojection or smooth orthogonal coordinate (SOC) �1 spannedby the corresponding smooth orthogonal mode (SOM) q1.

In previous studies [19], it was shown that the dominantSOCs can be used to reconstruct slow-time trajectory. In fact,in most of the cases it was shown that there was linear, one-to-one connection between the SOCs � and slow-time variable�. In this paper, it is expected that the dominating SOC can beused to reconstruct surface topography.

3. Numerical model and its simulation

To illustrate the performance of the LFV-based AFM, a sim-plified model of an AFM [24]—a non-linear spring mass sys-tem with external excitation (Fig. 1)—is studied:

mx + �(x − d) + k(x − d) = f (�), (8)

where x and d are the displacement of the tip versus the fixedframe and the controller input, respectively; m, �, and k de-note the tip mass, damping coefficient, and stiffness coefficientof the spring, respectively. The interaction between the tip andthe sample surface is described by the van der Waals attrac-tion/repulsion force f (�):

f (�) = T

�2− 6T

30�8, (9)

m

R

f(δ)

k

d

xz

Fig. 1. Simplified model of the AFM system.

−20 −15 −10 −5 0 5 10 15−1.5

−1

−0.5

0

0.5

1

1.5x 105

Displacement (nm)

Vel

ocity

(nm

/s)

Fig. 2. Phase portrait of the simulated AFM tip motion (x versus x).

where � = z − x, z is the distance between the fixed frameand the sample surface, is the molecular diameter, and T isdefined as T = (AHR)/6, AH is the Hamaker constant, and Rdenotes the radius of the tip.

The system is driven harmonically d = A sin(t), whereA is the driving amplitude and is the excitation frequency.The parameters used in the simulation are: m = 3.1 × 10−7 g,k = 0.0167 N/m, R = 150 nm, = 8.03 × 103 rad/s, � = 0.1,AH = 10−19 J, = 3.16 × 10−9 m, and A = 2 nm. Scanningprocesses of the AFM are simulated by changing z. When the tipis close to the sample surface at about 10 nm distance, chaoticmotions are observed (see Fig. 2 for an example of the chaotictrajectory of the AFM tip motion). More details about this

524 M. Liu, D. Chelidze / International Journal of Non-Linear Mechanics 43 (2008) 521–526

model and the conditions for chaotic responses are available in[4,25].

4. Calibration and measurement

In simulations, change in the surface topography is simu-lated by altering z. It is also assumed that the AFM measure-ment function is given by y = h(x) = x, which is a reasonableassumption, given that laser vibrometer measures AFM beamoscillations. Thus, in this paper, x time series need to be used toinfer the surface topography z. As every instrument, the chaoticAFM needs to be calibrated. This process is described in thenext section, followed by the validation of the measurementprocedure.

4.1. Calibration

Since z can be controlled using piezoelectric ceramic trans-ducers in real AFM systems, the calibration approach is straightforward. The distance z is slowly varied in time from 12.23to 12.54 nm as shown in Fig. 3(b). During this process, about3.4 million data points are collected and are separated into420 disjoint data sets of 213 points in each. A phase space isreconstructed from each data set using time delay � = 7 andembedding dimension D = 5. The phase space reconstructedfor a fixed value of z = 12.23 nm is used as a reference phasespace. Each of these reconstructions is partitioned into 64 smalldisjoint hyper-volumes using intervals defined for the three

2 4 6 8 10

101

102

Number20 40 60 80

12.3

12.4

12.5

Time (s)

20 40 60 80

−0.05

0

0.05

Time (s)20 40 60 80

12.2

12.3

12.4

12.5

12.6

Time (s)

z (n

m)

z, z

(nm

)

SO

V�

1

Fig. 3. Calibration using SOD: 10 largest SOVs (a); distance between the cantilever tip and the sample surface z versus time (b); identified dominant SOC�1 corresponding to the largest SOV �1 (c); surface variation z (solid line) and calibrated measurement z (dotted line) (d).

middle coordinates. The partitions are based on a referencephase space data, where the number of reconstructed trajectorypoints in each hyper-volume is approximately the same. Moredetails about the partitioning algorithm are available in [18].

The LFV procedure outlined above is used to build a 320 ×420 tracking matrix Yc that characterizes the effects of chang-ing surface topography z. Results of the SOD analysis are de-picted in Fig. 3. The largest SOV is much larger than the rest,which means that there is only one smoothly drifting parameterin the system, refer to Fig. 3(a). The corresponding SOC (�1)

shows a linearly increasing trend similar to z trend as depicted inFig. 3((b) and (c)). The best linear fit in the least-squares sense isdetermined to get a calibration function z=a�1+b=aYq1+b.From Fig. 3(d), it is clear that z tracks z quite well. The result-ing calibration parameters (q1, a, and b) are recorded for futuremeasurements.

4.2. Measurement

In practice, changes in surface topography described by zare not expected to be always smooth. However, the calibra-tion parameters allow to track arbitrarily small changes in z.For validation purpose, an artificial surface profile z shown inFig. 4(a) is used for further analysis. One thousand two hun-dred and forty eight data sets (214 points in each) are collectedwhile varying z according to Fig. 4(a), and a 320 × 1248tracking matrix Ym is constructed based on LFV procedure.The phase space reconstruction parameters and partitions are

M. Liu, D. Chelidze / International Journal of Non-Linear Mechanics 43 (2008) 521–526 525

12

12.2

12.4

12.6

z (n

m)

−0.2

0

0.2

�1

0 50 100 150 200 250 300 350 40012

12.5

13

t (s)

z, z

(nm

)

Fig. 4. (a) z(t), distance between the tip and the sample surface; (b) linearprojection of the feature matrix along the SOM from the calibration; (c) z(t)

(solid line) and estimated measurement (dotted line).

exactly the same as for the reference phase space used in thecalibration.

Removing DC components from the columns of Ym andprojecting it along the q1 SOM identified during the calibrationyields the trend �1 depicted in Fig. 4(b). Using the calibrationfunction determined a priori, z=a�1 +b is plotted in Fig. 4(c),from which it is clear that the z tracks the variation in z quitewell, and can be used as its measurement.

5. Discussion

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

In experimental contexts, AFM exhibits chaotic motions intwo different tapping modes: (1) “hard tapping” mode, whenthe equilibrium position of the tip is quite close to the samplesurface and (2) “light tapping” mode, when the tip only touchthe sample surface occasionally [26]. In this paper, the simu-lated AFM system operates in the “hard tapping” mode. Thismode is rarely used in practice, since frequent surface touchingdamages the cantilever tip quickly. However, the LFV method-ology does not depend on any specific system model. Thus,similar results are also expected for the “light tapping” mode.

The numerical model considered in this paper is a very sim-ple, one-degree-of-freedom model, where only the first vibra-tion mode of the cantilever beam is considered. In addition, theinteraction between the cantilever tip and the sample surfaceis greatly simplified. Although more sophisticated models are

available in literatures [25,27], the use of this simplified modelis justified since: (1) the model is able to generate chaotic re-sponses observed in practice and (2) the LFV methodologydoes not depend on the specific model. This same generalitycan also be exploited in other MEMS sensors where chaoticmotions are possible (e.g., a mass sensor, which is chaoticallyexcited).

The performance of the LFV-based AFM is influenced bythe size of data sets collected for a given intermediate-size timeinterval. Generally speaking, the more data are available, themore accurate the tracking result can be. Based on simulationresults presented here, it takes 400 s to collect 15 million datapoints. Since actual AFM systems work in frequencies thatare much higher than what is used in the simulation (about3 KHz), availability of data should not hinder the performancein practical context.

Nominally chaotic motions dominated the data used in thispaper. However, due to the bifurcations caused by the parame-ter drift, periodic and quasi-periodic motion windows were alsoobserved. The SOD is usually capable to filter out bifurcationnoise caused by small periodic windows. However, if the pe-riodic window persists over a long time period the quality ofmeasurement may suffer. In the simulation used for this paper,one of the largest periodic bands happened around 210 s, whichcaused unexpected peak in the measurement (see Fig. 4(c)).Thus, it is very important during calibration to identify thesuitable range of d and z for which mostly chaotic motionsare expected. In addition the simulated time series were noisefree, which is impossible in real applications. However, sincethe LFV relies on the estimated moments of the reconstructedphase space points’ probability distribution instead of preciseposition of individual points, the LFV is still expected to pro-vide reasonable results for noisy signals. The influence of thenoise on the performance of the PSW-based algorithms is stillunder investigation.

6. Conclusion

An AFM based on chaotic motion and utilizing LFVmethodology was introduced. This methodology is not de-pendent on particulars of system model and can be easilyapplied to other MEMS sensors. The basic concepts and pro-cedures were described and a simple one-degree-of-freedommodel of the AFM was used to illustrate the calibration andmeasurement procedures. The calibration process was used toidentify range of parameters for which the chaotic responseis possible and to find associated calibration parameters. Itwas demonstrated that chaotic motions can be used to recon-struct the changing topography of a sample surface. Possiblecomplications and corresponding countermeasures were alsodiscussed.

Acknowledgment

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

526 M. Liu, D. Chelidze / International Journal of Non-Linear Mechanics 43 (2008) 521–526

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