15 December 2010 first published online , doi: 10.1098/rspa.2010.0451 467 2011 Proc. R. Soc. A O. Payton, A. R. Champneys, M. E. Homer, L. Picco and M. J. Miles atomic force microscopy: theory and experiment Feedback-induced instability in tapping mode References lated-urls http://rspa.royalsocietypublishing.org/content/467/2130/1801.full.html#re Article cited in: html#ref-list-1 http://rspa.royalsocietypublishing.org/content/467/2130/1801.full. This article cites 30 articles, 1 of which can be accessed free Subject collections (44 articles) nanotechnology (108 articles) mathematical modelling Articles on similar topics can be found in the following collections Email alerting service here the box at the top right-hand corner of the article or click Receive free email alerts when new articles cite this article - sign up in http://rspa.royalsocietypublishing.org/subscriptions go to: Proc. R. Soc. A To subscribe to on March 12, 2014 rspa.royalsocietypublishing.org Downloaded from on March 12, 2014 rspa.royalsocietypublishing.org Downloaded from
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15 December 2010 first published online, doi: 10.1098/rspa.2010.0451467 2011 Proc. R. Soc. A
O. Payton, A. R. Champneys, M. E. Homer, L. Picco and M. J. Miles atomic force microscopy: theory and experimentFeedback-induced instability in tapping mode
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Proc. R. Soc. A (2011) 467, 1801–1822doi:10.1098/rspa.2010.0451
Published online 15 December 2010
Feedback-induced instability in tapping modeatomic force microscopy: theory and experiment
BY O. PAYTON1,2, A. R. CHAMPNEYS1, M. E. HOMER1,*,L. PICCO2 AND M. J. MILES2
1Department of Engineering Mathematics, University of Bristol,Bristol BS8 1TR, UK
2H. H. Wills Physics Laboratory, University of Bristol, Bristol BS8 1TL, UK
We investigate a mathematical model of tapping mode atomic force microscopy (AFM),which includes surface interaction via both van der Waals and meniscus forces. We alsotake particular care to include a realistic representation of the integral control inherent tothe real microscope. Varying driving amplitude, amplitude setpoint and driving frequencyindependently shows that the model can capture the qualitative features observed in AFMexperiments on a flat sample and a calibration grid. In particular, the model predictsthe onset of an instability, even on a flat sample, in which a large-amplitude beating-type motion is observed. Experimental results confirm this onset and also confirm thequalitative features of the dynamics suggested by the simulations. The simulations alsosuggest the mechanism behind the beating effect; that the control loop over-compensatesfor sufficiently high gains. The mathematical model is also used to offer recommendationson the effective use of AFMs in order to avoid unwanted artefacts.
Keywords: atomic force microscope; tapping mode; feedback; nonlinear; oscillation; control
1. Introduction
Atomic force microscopes (AFMs) are remarkable mechanical devices that arecapable of resolving features on a surface to nanometre precision (e.g. McMasteret al. 1994; Alsteens et al. 2009). While conventional microscopes use lenses tomagnify the surface optically, an AFM works by ‘feeling’ the topography of thesurface using a sharp tip at the end of a small cantilever moving over the area ofinterest (Hansma et al. 1994). As the tip moves over the surface, the position ofthe tip is measured, producing a three-dimensional image with high resolution inthe topography height. As the cantilever has a low stiffness, it is possible to imagedelicate biological samples without damaging them. Another advantage of AFMs,over electron-based imaging methods, is that samples submerged in fluid may beimaged, as the sample does not have to be made from or covered in a conductingmaterial. Along with imaging, AFMs can also be used to measure small forces,for example those required to unravel a strand of DNA (Rief et al. 1999).
There are many modes of operation of AFMs; see Alessandrini & Facci (2005)for a comprehensive review. The simplest is known as contact mode, where thetip is in constant contact with the surface. Although still used for many imagingtasks, such a mode of operation is not advisable if the sample is considered toodelicate (Zhong et al. 1993; Höper et al. 1995). Instead, a form of intermittentcontact called tapping mode (Zhong et al. 1993) is often used. When an AFM isused in the tapping mode, the cantilever is driven using a piezoelectric device atan oscillation frequency close to its resonant frequency. Away from the surface,the tip oscillates freely. When the tip is brought close to the individual atoms onthe sample surface, it is affected by both long-range attractive and short-rangerepulsive van der Waals forces (Ciraci et al. 1992; Dankowicz et al. 2007). Understandard humidity conditions, a layer of water, typically nanometres in thickness,forms over the surface and the tip (Beaglehole & Christenson 1992); as the tipapproaches this layer, a meniscus is formed (Zitzler et al. 2002; Hashemi et al.2008). Both these effects change the amplitude of oscillation of the cantilever,which in turn is carefully measured by the AFM apparatus and is used to buildup an image of the surface.
(a) Imaging with an atomic force microscope in tapping mode
Figure 1 depicts the apparatus used to create tapping mode AFM images, suchas those presented in figure 2. Images are put together using multiple sweepsalong the fast scan direction (the y axis in figure 1); the tip is incrementallystepped in the slow scan direction (the x axis in figure 1) at the completionof each sweep. The fast scan motion has a typical frequency of 2 Hz and theslow scan can take over a minute to complete one period. A tube made frompiezoelectric material moves the tip in both the fast and slow scan directionsas well as the surface displacement (z) direction, in order to try to keep theamplitude of oscillation constant during the scan. The piezoelectric tube changesits dimensions as the voltage placed across it varies. The displacement of thetip in the z direction is measured by reflecting a focused laser off the end ofthe cantilever onto a position-sensitive photodiode. As the tip taps across thesurface, a feedback control loop attempts to keep the amplitude of oscillationof the tip constant by altering the distance between the sample and the tip.Feedback loops and error analysis algorithms are used to deduce the requiredchange in extension of the piezoelectric tube to produce the required tip-sampledistance in order to keep the recorded amplitude constant. Computer softwarerecords the movement of the tip in the z-direction owing to the control and plotsit against the position of the tip in the x and y directions to create a final imagelike the one in figure 2.
(b) Motivation
Tapping mode AFMs are now well-established experimental measurementdevices with numerous celebrated successes (e.g. Round & Miles 2004; Yang et al.2007). Yet, there remain relatively few modelling studies, especially those that aimto correctly capture features observed in commercial AFM devices with feedbackcontrol; although see Melcher et al. (2008) for a description of general purposesoftware Veda for simulations of the dynamics of an AFM cantilever. Cantileversand tips are expensive to produce, making simulation an attractive alternative
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Figure 1. Schematic of the functional elements of a typical AFM. The piezoelectric tube moves thetip in the fast (y) and slow (x) scan directions by altering the voltages on the piezoelectric elementsforming the tube. The head holding the cantilever at the base of the tube oscillates the cantilever(triangular in this diagram). The AFM records the movement of the tip by reflecting the laserbeam onto the dipole photodiode. A feedback control loop alters the extension of the piezoelectrictube (in the z-direction) to move the equilibrium position of the tip closer or further away fromthe sample in order to try to restore the oscillation amplitude to a chosen value. (Online versionin colour.)
200 nm
26 mm26 mm
Figure 2. Experimental tapping mode AFM image of a portion of the calibration grid used. ADigital Instruments, Dimension 3100, Model D31005-1, with a Nano World Arrow-NCR-W, S/N67209l697 Arrow silicon SPM sensor cantilever (length 160 mm, width 45 mm, thickness 4.6 mm, withresonance approx. 285 kHz and force constant 42 N m−1) is used to obtain the image at a scan rateof 0.6242 Hz. (Online version in colour.)
to trial and error experiments. Also, by accurately capturing the behaviour of anAFM in simulations, multiple parameters may be altered and the results recordedautomatically, saving many hours of equipment time and diminishing the risk ofdamaging the piezoelectric tube by crashing the tip into the surface.
The purpose of this paper is not to produce a simulation tool per se,but to examine a low-order mathematical model of a tapping mode AFMthat contains all the important physical effects, including a combination ofnonlinear tip–sample interaction and the effects of the feedback loop controllaws. Such a low-order model is amenable to performing parameter sweeps, toproduce so-called brute-force bifurcation diagrams, and to investigate potential
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instabilities. Nevertheless, as can be seen in figure 2, when imaging with an AFMthe image is rarely perfect. In order to understand the possible systematic sourcesof such artefacts, to distinguish them from thermal noise, it is imperative thatthe fully nonlinear properties of the AFM be modelled as closely as possible.
Previous research has derived models for tapping mode AFMs; see Song &Bhushan (2008) for a review. In particular, studies have included nonlinear surfaceinteraction forces and meniscus effects (Zitzler et al. 2002; Dankowicz et al. 2007;Song & Bhushan 2008), which are highly nonlinear and can lead to chaotic motionif the desired amplitude of oscillation (also known as the setpoint) is chosenincorrectly (Hu & Raman 2006). Such complex motion can be caused by the so-called grazing bifurcations (Jamitzky et al. 2006; Dankowicz et al. 2007; Yagasaki2007; Hashemi et al. 2008), as the amplitude setpoint is decreased. Theoretically,such bifurcations are shown to underlie a functional bistability between twodistinct but nearby equilibrium amplitudes for a given set of parameters.
However, such phenomena have typically been found in open-loop systems. Inthis paper, we shall specifically investigate how the value of gain used in a simplemodel of the control loop affects the stability of the AFM system. We will alsostudy the effects of changing amplitude setpoint in a closed-loop system. At eachstage of the investigation, we take experimental results to support the model.
(c) Outline
The rest of the paper is outlined as follows. Section 2 briefly describes theexperimental apparatus that we have used to compare with the predictionsof our mathematical model. Section 3 then carefully explains the steps takenin developing the model, including how we implement the control law in anattempt to mirror how it operates in practise. In §4 we produce brute-forcebifurcation diagrams through numerical simulation of the model applied to theimaging of a flat surface. A form of instability is found for sufficiently highgains. We also show how experimental bifurcation diagrams contain the samequalitative behaviour. An attempt is made to explain the physical mechanismsthat underlie the observed phenomenon. Section 5 contains a detailed comparisonbetween the simulations and experiments when applied to the calibration grid infigure 2, by variation of several key parameters. Finally, §6 draws conclusions andmakes practical recommendations for effective AFM operation in order to avoidthe effects described here.
2. Experimental methodology
The mathematical model we shall develop is based on a specific experimental set-up. All experimental data reported here were obtained using a Digital InstrumentsDimension 3100 AFM, Model D31005-1, with the Digital Instrumentspiezoelectric tube and photodetector head model DMLS, and a Nano WorldArrow-NCR-W, S/N 67209l697 Arrow silicon SPM sensor cantilever and tip (oflength 160 mm, width 45 mm and thickness 4.6 mm, with resonance approx. 285 kHzand force constant 42 N m−1). Many of the parameters of the apparatus can beaccurately altered via a data acquisition computer running NanoScope V. 5.3software. Measurements are collected using the AFM in tapping mode, imaging
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a three-dimensional calibration grid (Digital Instruments 498-000-026). This gridhas a 10 mm pitch with holes of depth 200 nm. An AFM image of a 26 × 26 mmarea of the grid can be seen in figure 2.
Using a signal access module (SAM), it was possible to intercept, measure andrecord the raw data from the movement of the laser over the photodiode beforethe signal is fed into the controller. An oscilloscope (Tektronix TDS2012D) witha sampling frequency of 100 MHz was used to capture the data. The oscillationof the cantilever was sampled every 2 × 10−8 s, which corresponds to about 175samples per period of the cantilever’s oscillation at 285 kHz. The differencebetween the maximum and minimum amplitude values within 1000 samples(roughly six periods of oscillation) is recorded as the peak-to-peak amplitude.The advantage of this method is that the control loop can remain active whilethe raw data are collected, allowing the viewing of both the image and the rawdata. Hence aspects of the control loop can be inferred.
3. Model development
(a) Free oscillation
When the cantilever is being driven away from the surface, the position ofthe tip of the probe with time can be faithfully modelled as a driven dampedsimple harmonic oscillator (Rabe et al. 1996; Turner et al. 1997) oscillating inthe z direction only. Applying a periodic piezoelectric force of amplitude g andfrequency u, the equation of motion for the freely oscillating cantilever can thusbe written in the form
z = gu2 cos(ut) − cm
z − km
z , (3.1)
where m is the effective mass of the cantilever together with its tip, c the dampingcoefficient and k its linear stiffness. We can rewrite the effective mass and dampingof the cantilever and tip in terms of its natural frequency u0 and a quality factor,Q via
c = mu0
Qand m = k
u20
. (3.2)
The equation of motion (3.1) then takes the form
z = gu2 cos(ut) − u0
Qz − u2
0z . (3.3)
The values of these modal parameters can be found in table 1.Although many harmonics may be present on the cantilever (Stark et al.
2004), the control loop in our experimental system considers motion in thefirst transverse mode only, as there is only a single laser reflected off the endof the cantilever. Thus, a model that ignores higher cantilever harmonics isbroadly justified. Any features in the images that may be caused indirectly bythe excitation of higher modes will not form part of this paper. See Melcher et al.(2008) for multi-modal simulations of AFMs.
Closer to the surface, we must include an additional term on the right-hand sideof equation (3.1) that captures the forces the tip experiences owing to sample–tip
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Table 1. Parameter values used in numerical simulations of the AFM model.
parameter value explanation
g 1 × 10−8 m driving amplitude (Melcher et al. 2008)Q 200 quality factor (NanoWorld 2009)a 0.42 × 10−9 m intermolecular distance (Bondi 1964)H 1.4 × 10−19 J Hamaker constant (Shokuie et al. 2007)R 2 × 10−9 m radius of tip (NanoWorld 2009)u/(2p) 2.85 × 105 Hz driving frequency (see §2)u0/(2p) 2.85 × 105 Hz natural frequency (see §2)Dt 100 · 2p/u0 sampling interval (see §3)sa 1 × 10−7 m amplitude setpointVt 0.27 Poisson ratio of the tip (Hu et al. 1996)Vs 0.27 Poisson ratio of the sample (Hu et al. 1996)Et 1.65 × 1011 Pa elastic modulus of the tip (Dolbow & Gosz 1996)Es 1.65 × 1011 Pa elastic modulus of the sample (Dolbow & Gosz 1996)se 7.2 × 10−2 J surface energy of water at 25◦C (Dean 1998)don 0.4 × 10−9 m distance at which the meniscus forms (Hashemi et al. 2008)doff 2.32 × 10−9 m distance at which the meniscus breaks (Hashemi et al. 2008)N 40 number of sampling intervals in the integration time within the
control loop (see §3)G 2.12 value of integral gainh 0.2 × 10−9 m thickness of water layer on silicon (Beaglehole & Christenson 1992)
interactions. These forces are owing to two separate effects, van der Waals forcesand capillary action. We shall now treat each in turn.
(b) van der Waals forces
Figure 3 defines the distance d = z + u(t) − z(y) between the sample and thecantilever tip. As d tends to zero, the tip experiences a force due to atomicinteractions, the so called van der Waals force. The tip is not atomically sharp, andeven if it were, it would not interact with a single molecule at a time. Therefore,we model the tip as a spherical surface of radius R coming in contact with alocally flat sample surface. The contact force is modelled as a combination ofsurface adhesion and the restoring forces generated by the elastic spherical probetip pushed against a flat surface. We use an approximation of the contact regionof the van der Waals force called the Derjaguin, Muller and Toporov (DMT)model (Derjaguin et al. 1975) to avoid the singularity as d tends to zero; attip sample distances, d ≤ a, where a is the intermolecular distance, the adhesionand attractive part of the DMT contact force is assumed to be held fixed at thevalue of the van der Waals force at the distance d = a. Thus, we approximate thecontact force F as
F(d) =
⎧⎪⎪⎨⎪⎪⎩
HR6d2
, d > a,
HR6a2
+ 43
√R(a − d)3/2
(1 − V 2t )/Et + (1 − V 2
s )/Es, d ≤ a,
(3.4)
where Vt and Vs are the Poisson’s ratios, and Et and Es are the elastic moduli ofthe tip and sample, respectively.
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Figure 3. Schematic of the cantilever at three positions, equilibrium, contact and general,illustrating how the origin ze of the oscillating z displacement moves up or down depending onthe amplitude control input u(t) and the topography z(y) as given in equation (3.6). d is thedistance between the tip and the sample surface and y is the position on the surface of the sample.
(c) Capillary action
It has been established that under standard atmospheric conditions, a thinwater layer covers both the surface of the tip of the AFM and the sample (Zitzleret al. 2002). This layer acts to attract the tip while the tip’s fluid layer is joinedwith the surface’s fluid layer by means of a fluid meniscus bridge. The attractiveforce G is proportional to the volume of water contained in the meniscus (Zitzleret al. 2002; Hashemi et al. 2008). Thus, we assume that the total meniscus forceas a function of the distance from the sample is
G(d) =⎧⎨⎩
− 4pseR1 + d/h
, meniscus formed,
0, meniscus not formed,(3.5)
where h is the thickness of the fluid layer on the sample (assumed uniform andto be equal to that on the tip) and se is the surface energy of the fluid. Alsothe distance don of the tip from the surface at which the meniscus forms is notequal to the distance doff at which the meniscus breaks (Hashemi et al. 2008). Themeniscus is formed when the tip is travelling down towards the surface (d < 0)and d < don and is broken only when the tip is moving away from the surface(d > 0) and d > doff . This inequality leads to a hysteresis in the overall force thatthe tip experiences.
(d) Control
The equilibrium displacement of the tip above the surface is a function of thesurface topography, z(y), measured relative to the height of initial contact withthe surface, and the movement in the z direction owing to the control loop u(t)(figure 3), so
ze = u(t) − z(y). (3.6)
The control system in the experimental AFM used in this study (DigitalInstruments dimension controller, model D3100HP-2) is effectively a blackbox system; however, it is known that a form of proportional, integral and
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derivative (PID) controller is used. As mentioned above, the role of the controlin a tapping mode AFM is to maintain a constant amplitude of oscillation byshifting the origin of the z oscillation relative to the surface. The rest of thissection will describe how a control law was implemented in the model, whichappears to mimic that of the experimental apparatus.
Preliminary simulations and experiments with the AFM using different valuesof PID gains indicated that for the relatively large features on the calibrationgrid (or lack of features on a stationary flat silicon surface), the integral gainhad by far the greatest influence on the results. Furthermore, a faithful fit to theexperimental results could be obtained by setting the proportional and derivativegains to zero and allowing the integral gain to vary.
In order for the AFM to function properly, ze must always be less than halfthe peak-to-peak amplitude ge of the oscillating cantilever; else the tip will beoscillating away from the surface and not imaging. This is achieved by selectinga peak-to-peak amplitude setpoint, sa = ge(1 − ra), in which 0 < ra � 1.
As the cantilever travels across the surface, the simulated control measuresthe peak-to-peak amplitude gm of the oscillating cantilever, by recording themaximum and minimum z-values in a small time interval Dt, where Dt �2p/u, so that many periods of motion are contained in this window (table 1).The controller compares gm with the setpoint amplitude sa over a period oftime t.
We allow the cantilever to oscillate for 2000 time-sampling intervals Dt awayfrom the surface, then measure the equilibrium peak-to-peak amplitude reached;we call this amplitude ge. We then reduce ge by an amount rage to give the peak-to-peak amplitude setpoint sa. As sa decreases, the control will try and keep thetip oscillating closer to the surface and vice versa. Thus
u(t) = gint
t
∫ t
t−t
(sa − gm(s)) ds. (3.7)
We approximate this integral with a summation (which also mimics the behaviourof the control scheme in the experimental apparatus):
u(t) = gint
N
N∑n=1
(sa − gm(t − nDt)). (3.8)
Effectively, this sum calculates a mean error over the N previous samples. Thiserror is then multiplied by a gain value gint to give the final control input u(t).
We found that the size of N plays a significant role in the quality of the imageproduced by the simulated AFM: too small, and the tip either crashes into thesurface or is lifted fully away; too large, and any features on the surface aresmeared out. The value of t is unknown in the actual AFM, our value of Nwas chosen so that the simulated calibrated images closely match those from theexperiments (§5).
We attempted weighting the amplitudes of previous sampling intervals to givemore recent amplitudes a larger effect on the control u(t) in either an exponentialor linear fashion. The results were, however, discouraging. In fact, the mostacceptable fit between the simulation and the experimental data was obtainedusing equation (3.8) with N = 40.
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In order to fit parameters accurately, only a single parameter was varied ineach physical or simulated run. The baseline values used are given in table 1.The physical parameter values have been chosen owing to their success in theexisting simulations in the literature or from the equipment manufacturer in thecase of the cantilever properties. In the case of the control parameters, theirvalues were obtained by careful comparison with the experimental data, whichwe explain as follows.
Experimentally, the amplitude setpoint is given as a voltage, althoughphysically it affects the equilibrium distance between the tip and the surface(sa in the simulations) through the microscope control. The voltage is passedto the piezoelectric device that controls the movement of the sample in the zaxis, and is proportional to the extension of this device, although the constant ofproportionality is not known.
In §4 we find that, upon increasing the gain gint, an instability sets in at aspecific integral gain value, in both the simulation and the experimental results.We use the onset of this instability to calibrate the integral gain value gint againstthe experimental gain value G. Our findings suggest that the relationship betweenthem does not follow a linear scale. Instead, the relationship
G = 3√
gint (3.9)provides a satisfactory fit for most values of experimental gain used.
(f ) Running simulations
The model was solved using the Matlab differential equation solver ode45,using a relative error tolerance of 1 × 10−3, which implies an absolute toleranceof O(10−10). Matlab’s event handling was used to locate the boundaries d = don/offand d = a, so that the correct forces G(d) and F(d) are applied.
At each sampling interval, t = nDt, the value of z(y) progresses to the nextvalue by stepping in y(t). Over a flat sample surface, this remains constant butto simulate the effect of the calibration grid, a square waveform is fed with atime period of 600Dt and a peak-to-peak amplitude the same as the depth of thetroughs in the physical calibration grid (200 nm). The value of u(t) calculatedfor a given sampling point t = nDt is then used in the following sampling pointt = (n + 1)Dt to calculate the value of ze using the value of z(y) at the new time. AMatlab event file is then used to update the new value of d given by equation (3.6).The delay of one sampling time-step in the control is justified by the time takenfor the amplifiers in the real AFM to extend or retract the piezoelectric tube.Nevertheless, this time is small when compared with the controller’s lock-in time,and so the effect on the overall dynamics of the system is negligible.
4. Nonlinear effects for a flat sample
(a) Variation of setpoint and gain
Figure 4 shows the results of running quasi-static simulations for a range ofsetpoint amplitudes ra, for different integral gain values gint. For each value ofsa, we plot the peak-to-peak amplitudes of the oscillation of the tip calculated
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Figure 4. Numerical bifurcation diagrams showing peak-to-peak amplitude of oscillation of the tipas a function of amplitude setpoint, sa, for different values of the integral gain: (a) G = 1.34, (b)G = 1.64, (c) G = 2.12, (d) G = 3. Here the imaged surface was flat, all other parameters werefixed at their values in table 1. (Online version in colour.)
for the final 200 sample intervals Dt, after allowing transients to decay over2000Dt. As the setpoint is reduced (by increasing ra), note the various differentregions of either periodic motion (represented by a single amplitude value) ornon-periodic motion (a sample of different amplitude values). These effects aremore pronounced when the gain is largest. For gains lower than the lowest valuesused in figure 5, there is no evidence at all of non-periodic motion.
The effects seen in figure 4 can also be seen in similar simulated bifurcationdiagrams, where the setpoint sa is held fixed and gain is varied quasi-statically.The results are shown in figure 5a–c for the three separate gain values indicatedby the numbers 1–3 in figure 4c. The left-hand panels of figure 5a–c show thecorresponding experimental data. Gaps in the experimental plots are owing todata having not been collected for the particular gain and setpoint combination.Nevertheless, qualitative and quantitative agreement can be seen between theexperiments and the simulations. In particular, in each case an instability sets inat a specific value as the gain is increased, and the amplitude variation of thenon-periodic response becomes more pronounced as the gain is further increased.
Figure 5d–f reproduces the phase portraits at the correspondingly labelledpoints in figure 5b. The plots display the simulation data from the final 200sampling intervals after 2000 intervals of transient behaviour. Note that the phaseportrait (in figure 5d) of stable operation corresponds to a pure periodic orbit,
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Figure 5. Experimental (left) and numerical (right) bifurcation diagrams of the peak-to-peakamplitude as the gain is increased. (a–c) Peak-to-peak amplitude of the tip after an initial 2000sample intervals while increasing the integral gain at a constant setpoint amplitude, sa. (a–c) Correspond to the regions labelled 1–3 in figure 4 (sa = 1.439 × 10−7 m, 1.296 × 10−7 m and1 × 10−7 m, respectively). The ringed regions on the simulation plots show the areas investigatedexperimentally on the left. (d–f ) Simulated phase portraits taken at the corresponding integralgains marked in (b). Error bars have not been plotted so that the increase in amplitude in the non-periodic region is clearer to see. The size of the error is comparable to the spread in the amplitudewithin the periodic region of the experimental results and is primarily caused by noise from thephotodiode. (Online version in colour.)
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Figure 6. Numerical bifurcation diagram of the relative phase difference between the drivingamplitude and the motion of the tip of the cantilever as the amplitude setpoint is decreased.The inset shows the finer structure of the diagram. This plot corresponds to figure 4c. (Onlineversion in colour.)
whereas those in the instability regions (figure 5e,f ) correspond to non-periodicmotion. It is hard to say from the phase portraits whether this motion is trulychaotic. Rather, the shape of the phase portraits seems to be more consistent withthe complicated quasi-periodic motion in which several independent frequenciesare present.
(b) Cause of the instability
One possible explanation of the instability seen in figures 5 and 6 might be somekind of dithering between the two distinct stable equilibria in the amplitude ofopen-loop tapping mode AFMs, as reported by Dankowicz et al. (2007). Accordingto their results, those two different setpoints are best characterized by their verydifferent values of phase difference between the driving and the response of thecantilever. Hence we replot in figure 6 the bifurcation diagram shown in figure 4c,plotting now the phase difference on the y-axis.
Note from the figure that there is a clear correlation between the amplitudeand the phase, as the amplitude setpoint is reduced. The regions of instabilityhave corresponding large ranges of phase difference. Note that we do not seeany evidence of passage close to two distinct equilibrium amplitudes. Rather, theregions of instability visit a wide range of different phase differences.
Another feature of the bistability found by Dankowicz et al. (2007) is hysteresisupon decrease and subsequent increase in the setpoint amplitude. To test forthis, we ran simulations similar to those in figure 4c by first quasi-staticallydecreasing then increasing sa to check for hysteresis in the locations of the regionsof instability. We found limited evidence of hysteresis in the points of onset ofthe instability. The forward and backward sweeps did indicate the presence ofmultiple attractors at some parameter values inside the instability regions.
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Figure 7. (a–d) Experimental and (e–h) simulated time traces for the amplitude z with variation ofgint. The simulation values of gint used in (e–h) were 0.3, 0.5, 5 and 9, respectively, which correspondto G-values of 1.64, 2.12, 6.71 and 3.0 according to equation (3.9). (a–d) The correspondingexperimental results taken at G = 1, 2, 4 and 8, respectively. Note that the experimental datarecorded is the voltage V recorded by the photodiode, which is linearly proportional to z (see textfor details). (Online version in colour.)
To further test the accuracy of our model, equivalent simulations were carriedout using Veda (Melcher et al. 2008) without feedback control, but with otherparameters set as in table 1. These simulations showed results that are remarkablysimilar to those using our model, but showed none of the instabilities we havefound here for sufficiently high gains. This gives some confidence in our assertionthat the instability seen in the experiments is indeed owing to the action of thefeedback control.
To gain further insight into the cause of the instability, it is useful to considerthe dynamics at unstable parameter values in more detail, in both the experimentsand the simulations. Figure 7 shows the z movement of the cantilever againsttime, from experimental data (figure 7a–d) and the corresponding numericalsimulation (figure 7e–h). In experiments, we keep the surface stationary in thex and the y directions and record the actual z displacement by interceptingthe signals sent to the photodiode via the SAM box, and sampling them at100 MHz. The experimental results are thus only proportional to the displacementin the z direction.
Note the close qualitative agreement between experiment and numerics. Forhigh enough gains both theory and experiments show a repeating significantfeature of growing amplitude followed by arch-like slow excursions as the controlpushes the tip into the surface. For low gains, the arches are not present;
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instead it appears that an envelope wave exists with a ‘beating’ frequencyof roughly 500 Hz (from the experimental data using a driving frequencyof 285 kHz).
Beating motion has previously been observed in tapping mode AFMs, owingto the tip hitting the surface and bouncing off too high (Stark & Heckl 2000)when the impulse from the surface coincides constructively with the oscillationpiezoelectric device in the head of the AFM. This still might be playing a part inthe formation of the structures but our bifurcation studies make it clear that thestructures are most strongly dependent on the value of the integral gain, ratherthan any nonlinear detuning causing constructive interference.
We are left with the inevitable conclusion that the instability is not primarilydue to the non-smooth nature of the cantilever dynamics, but is a delay-inducedinstability (e.g. Stepan 1989; Gopalsamy 1992) caused by choosing too great avalue for the integral gain. Qualitatively, the extreme form of the instability whenthe arch-like features are present comes about because, if the gain is increased, theresulting control movement for a given error will increase. In the situation wherethe cantilever is oscillating at an amplitude lower than its amplitude setpoint, thecontrol will therefore move the oscillation midpoint up away from the surface. Ifthe gain is too high, the tip may become free of the surface leading to a large gainin the peak-to-peak z amplitude. Owing to the integral nature of the control, it isnot until a number of sample intervals later that the control once again pushes thecantilever towards the surface causing an arch-like feature. There is a period ofO(NDt) before the cantilever makes contact with the surface again, whereuponthe high gain again causes an over-compensation, which results eventually inanother lift-off event.
5. Measurement of test surfaces
We now investigate the effect of altering the driving amplitude (g), drivingfrequency (u) and setpoint (sa) on the image of the calibration grid. In simulatingthe calibration grid, z(y) is no longer kept constant with time; instead, z(y)is a square wave with a peak-to-peak amplitude the same as the depth of thecalibration grid troughs (200 nm).
(a) Variation of driving amplitude
With all other parameters remaining constant (as given in table 2), theamplitude of oscillation was varied. A selection of experimental results withincreasing amplitude can be seen in figure 8a–c. Below 100 mV the tip fails toestablish contact with the surface.
As the amplitude of oscillation increases, two features in the height trace canbe seen to vary. First, as the tip moves over the grid into a trough, the decay rateof the recorded height increases, which decreases the ‘shadow’ effect. Second, theamount of detail that the AFM picks up from the surface changes with amplitude.At low amplitudes, the tip seems to ‘bounce’ over the surface (Stark & Heckl2000); at intermediate amplitudes, the AFM works as it was designed to andproduces good images. If the amplitude is increased further then the tip seems to
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Figure 8. (a–c) The experimental traces and (d–f ) corresponding simulated results as the drivingamplitude is changed over a calibration grid. (a–c) Traces made by the experimental AFM atdriving amplitudes of 117, 118 and 125 mV, respectively. (d–f ) Simulated control movementsover the modelled step function at driving amplitudes of 5 × 10−9, 5 × 10−8 and 5 × 10−7 m,respectively. (Online version in colour.)
Table 2. Parameter values used in AFM experiments (unless stated otherwise in the text).
have trouble imaging the higher parts of the image. However, the troughs of thegrid are resolved well, with the exception of a dip being formed as the tip dropsoff the edge of the grid and hits the bottom of the trough.
A ‘shadow’ is caused when ze is larger than the half peak-to-peak amplitudeof oscillation of the cantilever for more than one time step. Until the control loopcan lower the value of ze enough for ze to be less than the half peak-to-peakamplitude, no information is collected from the surface topography as the tip isoscillating free from the surface, and so is not being affected by G(d) or F(d).As seen in figure 8, the region of shadow can be minimized using a larger drivingamplitude. In doing so, the amplitude increases at a quicker rate when the tippasses over a falling edge to meet the surface below.
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To simulate the above amplitude experiment, the driving amplitude wasincreased from 5 × 10−9 to 1 × 10−7, while all other parameters were heldconstant (as given in table 1). We show the results in figure 8d–f. Figure 8ashows that when the drive amplitude leads to a half oscillation amplitude ofthe tip which is less than the setpoint, the AFM is not capable of imagingthe surface. Put simply, the tip is not making contact with the surface but isjust oscillating above it. When the driving amplitude is just large enough forthe tip to make contact with the surface, a situation similar to that shown infigure 8b can be seen where the AFM image has a ‘shadow’ after any sharpdrop in the topography height. The shadow is caused by the tip not beingin contact with the surface. The oscillation amplitude of the tip needs toincrease sufficiently to make contact with the surface, while the control movesthe surface up to meet the tip. The effect of this is that the fall of the inputstep function is smeared into a slope. When the driving amplitude is increasedfurther, as shown in figure 8c, the shadow is all but removed owing to thereduced time required for the tip’s oscillation amplitude to become large enoughfor the tip to make contact with the surface again after falling off the topof a step.
(b) Variation of driving frequency
In order for the AFM to function properly, the piezoelectric device in thehead of the microscope must drive the cantilever at a frequency at or justbelow its resonant frequency. Figure 9 shows the results of altering the drivingfrequency about the resonant frequency of the tip used (285 kHz experimentallyand the scaled 100 Hz in the simulation), which correspond to traces in figure 9b,g.The quality of the trace degrades rapidly either side of the corrected resonantfrequency. Experimentally, the AFM control loop can image when the drivingfrequency is reduced to 0.2 per cent below the resonant frequency and increasedto 0.07 per cent above. Even when the driving frequency is at the boundary ofthe usable frequencies, the rising edge can be recognized, but flat surfaces are notrecorded well in the tip traces. Neither are the falling edge nor the trough of thegrid recorded well at these extreme frequencies. At the correct driving frequency(figure 10b,g), the plots give a very good indication of the position of any changein topography.
The experimental and simulation results (figure 9a–e and 9f –j, respectively)taken as the driving frequency was altered can be understood by realizing thatmoving the driving frequency away from the natural frequency has the effect ofreducing the driving oscillation amplitude. When the resulting amplitude fallsbelow the setpoint, then the AFM is no longer able to image. In particular,the same key features are seen when the frequency is either too low or toohigh. However, in terms of quantitative comparison, the results in figure 9 aresomewhat less convincing than the corresponding results in figures 7, 8 and 10.This could be owing to the possibility that higher modes of vibration becomingincreasingly important as the difference between u and u0 increases. It is alsobelieved that as u moves away from u0, some of the physical parameters of thesystem would change. These changes were seen as being beyond the scope of this
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time (ms) time (ms)264 20 40 60 80 100 120 140 160 180 200
Figure 9. (a–e) Experimental and (f –j) simulated plots showing the results of altering the drivingfrequency away from the natural frequency of the cantilever. (b,g) Results when driving frequencyis equal to the natural frequency of the experimental cantilever. (a,f ) The results of imaging acalibration grid at a driving frequency of 0.2% below the natural frequency. (c,h), (d,i) and (e,j)have driving frequencies of 0.04%, 0.06% and 0.07% above the natural frequency away from thesurface, respectively. (Online version in colour.)
paper and so not included in the model. A more complicated spatially extendedapproach to modelling the cantilever would be required to look into this further(e.g. Song & Bhushan 2008).
(c) Variation of the setpoint
We now proceed to investigate the effect of changes in setpoint; the resultsare shown in figure 10. For the experimental results, figure 10a–f, the parametersare held constant at the values stated in table 2, while the driving frequencyis altered. The driving amplitude is also increased from 119 to 200 mV so thatstructures can be seen in more detail near the edges of the range of frequenciesused. At a driving amplitude of 200 mV, the tip makes contact with the surface ata setpoint of 0.9614 V (figure 10a), corresponding to the AFM beginning to recordjust the tops of the calibration grid. As the setpoint is decreased further more
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Figure 10. (a–f ) Experimental and (g–l) simulated plots showing the effect of lowering the setpointvalue. The experimental value of ge is 0.9361 V which corresponds to (b). The correspondingsimulation value of ge used is 1.96 × 10−6 m, shown in (b,h). (a,g) Increase in the setpoint valueusing ra = −0.02. (c,i) Decrease in the setpoint using ra = 0.05, (d,j) a decrease using ra = 0.1, (e,k)a decrease using ra = 0.4 and (f,l) a decrease with ra = 0.6. (Online version in colour.)
detail is picked up from the surface. When the tip starts to get very close to thesurface (figure 10f ), the amplitude traces pick out the changes in topographyvery clearly. This could be owing to the tip having only the opportunity toincrease its amplitude of oscillation while the control loop and piezoelectric tubeare temporarily increasing the setpoint during a change in surface topography. Itis also very clear that the cross section becomes much less noisy as we lower thesetpoint. This supports the results found in §4, especially those in figures 4 and7, where the amplitude variance owing to the beating effect generally decreasesonce the setpoint is below half the value at which the tip starts to make contactwith the surface (ra < 0.5).
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Figure 10g–l shows the results of numerical simulations where the amplitudesetpoint, sa, is changed; sa varies between 102 and 60 per cent of half theequilibrium peak-to-peak amplitude (1.96 × 10−7 m), while all other parametersremain as given in table 1. Even at setpoints greater than half the equilibriumpeak-to-peak amplitude, the tip is still weakly able to pick up the surface(figure 10g). This is because of the control loop over-compensating for a correctionand bringing the cantilever closer to the surface than sa by a few microns andso the tip makes contact with the surface. The simulated results agree wellwith those collected experimentally. In both simulation and experimental results,the AFM does not image the surface correctly until the setpoint is reduced to90 per cent of the half peak-to-peak amplitude of the freely oscillating cantilever.Another common feature is the decrease in the noise with a decrease in setpoint.Presumably this noise is because of the instability we have investigated in §4.While it is not always possible to use lower amplitude setpoints for fear ofdamaging a delicate sample surface (the lower the setpoint the more force thetip will exert on the surface), it is clear from the results above that a setpointbelow 50 per cent of the half peak-to-peak amplitude of the freely oscillatingcantilever leads to much better image quality.
6. Conclusion
In this paper we have investigated a model of an AFM in tapping mode.Specifically, a damped, driven oscillating mass model was extended using surfaceinteractions (van der Waals and fluid meniscus forces) to simulate the cantileverin an AFM in tapping mode. A simple model of a control with integral gain, aspresent in the experimental apparatus within a commercial black box controller,was included in the model. The predictions of the mathematical model correctlydescribe the experimental results from the AFM, under variation of the keyparameters that may be varied in an experiment: amplitude setpoint, drivingamplitude and frequency.
The key finding of our study has been that the choice of too large a value for thegain parameter in the control loop can cause an instability. For very high gains,the instability can lead to the arch-like structures in the oscillation amplitudeseen when imaging a flat surface in figure 8c,d. In reality, the AFM is not likelyto be used with such high gains, but typical operating conditions could easilybe high enough to cause the envelope waves seen in figure 8a,b to be produced,degrading the final image as a result.
It would be interesting in future work to probe further the cause of thisinstability. The presence of beating-type behaviour is strongly suggestive ofa delay-induced oscillatory bifurcation (see Stepan 1989; Gopalsamy 1992). Aprecise mathematical analysis of such an instability is beyond the scope of thispaper, and is likely to be challenging, because the underlying state is a large-amplitude non-smooth periodic orbit. It would be interesting to see thoughwhether such a form of instability can be captured in a simplified ‘toy’ modelof the combined cantilever/controller dynamics. We note similarities with otherwork particularly (Dombovari et al. 2008) with respect to the delay and lossof contact.
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We have also not probed in detail whether the dynamics are chaotic or notinside the regions of instability. Clearly, the dynamics have a lot of structure,which suggests that the dominant part of the motion is a periodic modulation ofthe periodic tapping motion. Nevertheless, the fine structure we observe in thebrute force bifurcation diagrams might also be consistent with a (weak) chaoticmodulation of the underlying quasi-periodic motion. The mechanism seems verydifferent to that which appears to underlie the previous observation of chaoticdynamics in an AFM (Hu & Raman 2006). In a sense, whether or not chaos ispresent in the motion we observe is not as serious an issue in practice as how toavoid the instability in the first place.
One motivation of this work has been to help understand how cleaner AFMimages may be produced in tapping mode. Specifically, we have provided atheoretical framework for understanding how appropriate machine parametersshould be selected. The results for the flat sample show the importance of selectingvalues for the amplitude setpoint and integral gain within the regions of stabilityin the bifurcation diagrams shown in figures 5 and 6. For example, the resultsshow that stability can always be achieved by lowering the amplitude setpoint,or by lowering the integral gain. Nevertheless, there is a trade off; too low a valueof the setpoint can cause the cantilever to crash into the surface, and too smalla value of gain will stop the AFM from functioning effectively, as the transienteffects will be too large.
The results we obtained from experiments and simulations on a calibrationgrid suggest further optimal conditions for imaging such a surface. The bestresults were obtained from a peak-to-peak amplitude setpoint of about 60 percent that of free oscillation, with as large a driving amplitude as possible andas slow a scan speed. However, it is also important to put these conclusionsin context. Although these parameters may provide the best images for a hardsurface, the force at which the tip hits the surface, and any damage caused bythese forces, has not been taken into account. For softer surfaces another forcecurve such as the so-called Johnson, Kendall and Roberts (JKR) approximation(Johnson et al. 1971) may yield better results than the DMT model, althoughthe models of the control loop and fluid layer would still apply, as would thegeneral simulation methodology. The results from this investigation, however,demonstrate that models of the AFM, which include the control loop and fluidlayer allow the tip dynamics to be recreated, understood and explained.
O.P. gratefully acknowledges the financial support of the Engineering and Physical SciencesResearch Council (grant no. EP/E032249/1). The authors also acknowledge helpful conversationswith Arvind Raman and Harry Dankowicz.
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