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Ultramicroscopy 110 (2010) 618–621

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Ultramicroscopy

0304-39

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/ultramic

Intermittency in amplitude modulated dynamic atomic force microscopy

Ferdinand Jamitzky a, Robert W. Stark b,n

a Leibniz Supercomputing Centre, Garching, Germanyb Center for Nanoscience and Department of Earth and Environmental Sciences, Ludwig-Maximilians-Universitat Munchen, Theresienstr. 41, 80333 Munchen, Germany

a r t i c l e i n f o

Keywords:

Atomic force microscope

Chaos

Intermittency

Dynamic mode

91/$ - see front matter & 2010 Elsevier B.V. A

016/j.ultramic.2010.02.021

esponding author. Tel.: +49 89 2180 4329; fa

ail address: stark@lmu.de (R.W. Stark).

a b s t r a c t

From a mathematical point of view, the atomic force microscope (AFM) belongs to a special class of

continuous time dynamical systems with intermittent impact collisions. Discontinuities of the velocity

result from the collisions of the tip with the surface. Transition to chaos in non-linear systems can occur

via the following four routes: bifurcation cascade, crisis, quasi-periodicity, and intermittency. For the

AFM period doubling and period-adding cascades are well established. Other routes into chaos,

however, also may play an important role. Time series data of a dynamic AFM experiment indicates a

chaotic mode that is related to the intermittency route into chaos. The observed intermittency is

characterized as a type III intermittency. Understanding the dynamics of the system will help improve

the overall system performance by keeping the operation parameters of dynamic AFM in a range, where

chaos can be avoided or at least controlled.

& 2010 Elsevier B.V. All rights reserved.

1. Introduction

Dynamic atomic force microscopy (AFM) is widely used forhigh-resolution imaging. One of the most commonly usedimaging modes is based on feedback regulation employing anamplitude modulation scheme (tapping mode) [1]. In this mode ofoperation, the tip oscillates with an amplitude of a few tens ofnanometers. Far away from the specimen surface, the tiposcillation corresponds to that of a harmonic oscillator [2,3]. Thisoscillation is perturbed when the oscillating tip impacts on thesurface. This hard interaction with the specimen, however,introduces a strong non-linearity in the dynamic system, whicheven may cause a chaotic system response. To fully understandthe various operation states of dynamic AFM it is essential toidentify and to classify possible states of system response.

Several consequences of the non-linearity have been identifiedso far. They include the coexistence of several oscillatory states[4,5], generation of higher harmonics [6–13], energy transferbetween two modes in multifrequency AFM in air [14,15] ortransfer caused by a viscous liquid environment [16,17]. So fartwo routes into chaos have been identified. The grazing impact atthe transition from non-contact to the tapping regime leads to abifurcative dynamics from a periodic to a chaotic response[18,19]. At very small average separations between tip andsample, i.e. if the cantilever is soft as compared to the surfacepotential, period doubling can occur [20–23]. In a dynamicsystem, the transition to chaos can occur via (i) period doubling

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[24], (ii) intermittency [25,26], (iii) crisis, or (iv) quasi-periodicity[27]. A period-doubling cascade has been observed experimen-tally and in numerical simulations in dynamic AFM [20–23].Numerical simulations showed that also quasi-periodicity mayoccur in multifrequency AFM [15].

There is a promising candidate to substantiate the experimentalrelevance of an intermittency route into chaos. Intermittencymight occur due to coexisting oscillatory states where the systemswitches between the different states [15]. Experimentally, such aswitching behavior can be realized by a sticky sample such as apolymer surface or adhesion due to capillary condensation. In thiscase, the cantilever then can oscillate or it might get stuck tothe sample resulting in a pinned configuration [28]. A choice ofparameters, where the energy of the oscillator corresponds tothe adhesion energy are on the same order of magnitude is ofspecial interest. We have experimentally realized such a system toinvestigate time series data with tools of non-linear dynamics tocharacterize the complex dynamics of a cantilever in the vicinity ofa surface. The case study provides experimental evidence forintermittency as an additional route to chaos in dynamic AFM.

2. Materials and methods

An atomic force microscope in a tip-scanning configurationwas operated in tapping mode (Dimension 3100, Veeco, SantaBarbara, CA). Time series data of the cantilever deflection wererecorded with a flexible resolution analog to digital converter(NI PCI-5911, National Instruments, Austin, TX) at a sample rate of200 kSamples/s at 18.5 bit.

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For the experiment a soft rectangular cantilever was used (CSC38/Cr-Au, Type B, 0.03 N/m Micromash, Talinn, Estonia). Wemeasured a resonance frequency (frequency of maximumresponse) of 12.5472 kHz and a quality factor of 103. Thecantilever was excited at a frequency of 12.5259 kHz to a freeamplitude of about 40 nm. The force curve size and rate were setto 300 nm and 0.01 Hz, respectively. These values correspond toan approach-retract speed of 6 nm s�1. A silicon (1 0 0) waferwith a natural oxide layer served as specimen. The wafer wascleaned with ethanol and ultra-pure water. Measurements werecarried out at 90% relative humidity.

3. Results and discussion

A dynamic approach retract experiment was carried out, i.e.the oscillating tip was approached and after snap-in to the samplesurface it was retracted again. The raw time series data of thedeflection signal is shown in Fig. 1. Data were decimated (reducedby a factor of 10) for a better graphical representation. In the lefthalf of the plot, the tip approaches towards the sample until thesignal breaks down at about sample number 15,000. This sharptransition was caused by a snap-in of the tip onto the sample. The

-4

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Fig. 1. (color online) Time trace of the deflection signal of the tip during approach

and retract to the sample surface. The red rectangle indicates the position of the

details shown in Fig. 2.

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Fig. 2. (color online) Deflection signal right after the snap-out of the tip starting at

time 25,300.

sharp turning point between approach and retract is visible in theraw data at about 16,000 samples. The tip was then retracted fromthe surface again but still adhered to the surface until samplenumber 25,000 when the tip started to temporally detach fromthe surface. This transition can be seen as an abrupt widening ofthe deflection signal. A close-up of the deflection signal is shownin Fig. 2, where the intermittent behavior can be clearly observed.As the tip is further retracted the quasi-free oscillations prevailand after sample number 30,000 the tip is no longer attached tothe surface and shows a sinusoidal motion.

In the following discussion we will focus on the retractionphase of the tip, which shows an irregular response that deservesfurther attention. Comparing approach and retraction phase ahysteresis is observed between both, as the tip was much longerattached to the surface in contrast to the approach phase.However, the snap-out phase shows a very complex behavior(Fig. 2). Short bursts of a quasi-free motion were alternating withphases where the tip was almost attached to the surface. The tipsuddenly snapped away from the surface and started to oscillatewith increasing amplitude until the oscillations became so strongthat the tip was caught again by the surface and stayed very closeto the surface. The change of the quasi-free and the confinedmotion of the tip occurred randomly. However, a clear trend canbe observed due to the retracting motion of the height actuator.The further the tip was retracted from the surface, the longer thequasi-regular phases prevailed until at about 320,000 the tipfinally detached from the surface. Later, the tip trajectory onlyshowed a slight harmonic distortion.

The deflection signal in the retraction phase showed a strongintermittent behavior due to the sticking or not sticking to thesurface. Phases of sticktion of the cantilever to the surfacealternate with phases of quasi-regular motion of the tip. But eventhough the changes between the snap-on and free-motion phasesseem to alternate erratically the embedded trajectory clearlyshows the existence of an attractor that governs the overalldynamic behavior of the system. The attractor can be made visibleby embedding the time series of the deflection signal into anartificial phase space. For that purpose delay coordinates are used[29,30], where the signal time series St is delayed for differenttimes t and these time series are collected together in anm-dimensional vector

Dn ¼ ðSn�ðm�1Þt; Sn�ðm�2Þt; . . .; SnÞ: ð1Þ

Thus, from N scalar signal values N�(m�1)t embeddingvectors are obtained. In the following, we start with anembedding dimension of m=3. Here, t is defined as one sampleof the DAC card and thus t=16 corresponds to a delay time ofapproximately one period (precise value t=15.97) samples forone period. For short delay times of less than a third of the period(t=5) the structure of the attractor is shown in Fig. 3. Theattractor consists of a spiral structure corresponding to the quasi-free motion of the tip and a point-like attractor, whichcorresponds to the cantilever sticking to the surface. The spiralstructure is very pronounced and corresponds to a limit circle ofthe system, which is given by the free motion of the cantilever.

The point-like structure corresponds to a fixed-point of thesystem, which is given by the rest position of the tip on thesurface. The system can thus oscillate between these two statesand does this on the strange attractor connecting both. When thetip is retracted further, the point-like structure is dissolved untilfinally the limit circle of the quasi-free motion dominates asshown in Fig. 4. The temporal coherence can be analyzed by usinga large delay time. The motion shows a high coherence as can beobserved in Fig. 5. Even for four times the period of the drivingforce of the system it shows a very pronounced structure and theattractor decays into several parts.

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-3

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Fig. 3. Delay coordinate embedding of the deflection signal time series into an

artificial phase space. The delay time was less than a third of the period, starting at

time 25,500 (t=5).

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Fig. 4. Delay coordinate embedding of the deflection signal a large delay time at

time 28,800 (t=20).

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Fig. 5. Delay coordinate embedding of the deflection signal into an artificial phase

space using a large delay time of approximately three times the period of the

driving force at time 25,500 (t=45).

1e-11

1e-09

1e-07

1e-05

0.001

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1e-04 0.001 0.01 0.1 1

frequency (sample-1)

pow

er s

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ral d

ensi

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.u.)

Fig. 6. Power spectral density obtained during the intermittent phase during

retraction. The lines indicate the exponents �1/2 (solid line) and �1 (dashed).

Note that the frequency scale is in cycles per sample, the driving frequency of

12.5 kHz thus corresponds to f=0.06 sample�1.

F. Jamitzky, R.W. Stark / Ultramicroscopy 110 (2010) 618–621620

Phenomenologically intermittency can be classified into threetypes depending on the exponent of the power spectrum [31].Fig. 6 shows the power spectrum for the intermittent phase of thedeflection signal. Also shown are lines with exponent �1 and �2.An exponent of �1 corresponds to type II or type III intermittencywhile type I intermittency would correspond to an exponent of�1/2. For example, intermittent behavior has also been observedin the Duffing system [31], which is very similar to dynamic AFMoperating with small oscillation amplitudes. For the Duffing

system, a type-I intermittency was identified. Nevertheless, wedid not observe an exponent of �1/2 in our experiment (Fig. 6).Thus, type I intermittency can be ruled out. Another possiblecandidate might be the on–off intermittency, which is a type IIIintermittency and shows an exponent of �1 in the powerspectrum. This type III of intermittency is an universalphenomenon and has been observed amongst others byTownsend [32] for example in the structure of turbulent flows.

A physical interpretation of the underlaying physics isstraight-forward. The cantilever intermittently oscillates in twodistinct states characterized by a large and a very small oscillationamplitude. The small amplitude regime corresponds to the tipsticking to the surface and oscillating in a pinned configuration.The large amplitude state corresponds to the tip oscillating freely.Both states act as attractors. A steady state, however, was notreached. The transition between both states was controlledphysically by the failure of the adhesive connection between tipand sample due to capillary condensation. This failure, however, isa statistical process and the exact duration or rupture force cannotbe predicted. Thus, the transition between both states due toadhesive capture or adhesive rupture gives rise to a chaoticdynamics.

In practice, such a situation is relevant for imaging very stickysurfaces, such as polymers, with tapping mode AFM. If theadhesive forces are large enough to capture the oscillating tip,statistical capture and release events lead to intermittentdynamics. In such a situation, the oscillation amplitude asmeasured with a lock-in amplifier can be ill-defined and thetopography feedback may fail to track the surface. In such a casethe chaotic motion of the tip can show up as additional noise inthe images.

4. Conclusions

The dynamic atomic force microscope shows a rich variety ofchaotic modes. While period bifurcations are well established, aroute into chaos via intermittency has not been described. In ourexperiments we found an intermittency of either of type II or typeIII. The difference between both is the occurrence in one-dimensional maps such as the logistic map [33,34]. In a casestudy we show the occurrence of an intermittent oscillatorybehavior on a sticky sample. Our observations are relevant for

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dynamic AFM imaging of sticky specimen. If sticktion forces arelarge enough, intermittent dynamics can occur. This intermit-tency route represents another route into chaos in addition to thewell documented period doubling or grazing bifurcations.

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