Control of the higher eigenmodes of a microcantilever: Applicationsin atomic force microscopy

K.S. Karvinen n, S.O.R. MoheimaniSchool of Electrical Engineering and Computer Science, The University of Newcastle, Callaghan, NSW 2308, Australia

a r t i c l e i n f o

Article history:Received 21 June 2013Received in revised form12 November 2013Accepted 20 November 2013Available online 10 December 2013

Keywords:Atomic force microscopyAmplitude modulationTapping modeContact modeLiquidMultifrequencyMicrocantileverQ controlHigh-frequencyHigher eigenmodesModulated–demodulated control

a b s t r a c t

While conventional techniques in dynamic mode atomic force microscopy typically involve theexcitation of the first flexural mode of a microcantilever, situations arise where the excitation of highermodes may result in image artefacts. Strong nonlinear coupling between the cantilever modes in liquidenvironments may result in image artefacts, limiting the accuracy of the image. Similar observations havebeen made in high-speed contact mode AFM. To address this issue, we propose the application of themodulated–demodulated control technique to attenuate problematic modes to eliminate the imageartefacts. The modulated–demodulated control technique is a high-bandwidth technique, which is wellsuited to the control of next generation of high-speed cantilevers. In addition to potential improvementsin image quality, a high-bandwidth controller may also find application in multifrequency AFMexperiments. To demonstrate the high-bandwidth nature of the control technique, we construct anamplitude modulation AFM experiment in air utilizing low amplitude setpoints, which ensures thatharmonic generation and nonlinear coupling of the modes result in image artefacts. We then utilizefeedback control to highlight the improvement in image quality. Such a control technique appearsextremely promising in high-speed atomic force microscopy and is likely to have direct application inAFM in liquids.

& 2013 Elsevier B.V. All rights reserved.

1. Introduction

The atomic force microscope [1] employs a microcantileverwith an atomically sharp tip on its free end to interrogate thesurface of a sample. Originally operated in contact mode, the mostnotable improvement offered by dynamic mode AFM techniques isthe reduction of tip–sample forces, which enables noninvasiveimaging of sensitive samples. Amplitude modulation AFM, alsoknown as tapping mode AFM [2] when the tip makes intermittentcontact with the sample, is perhaps the most widely used AFMmode; suitable for operation in air and liquid, it can accuratelyestimate the topography of a sample's surface.

Conventional tapping mode AFM involves the excitation of thefirst flexural mode of the microcantilever, disregarding the pre-sence of higher modes. However, a complete understanding of theimaging process necessitates consideration of the multimodalnature of the microcantilever. Nonlinear coupling between canti-lever modes in liquid AFM may result in image artefacts [3–5].High-speed contact mode AFM scans may also exhibit artefactsowing to excitation of higher modes of the cantilever [6–8].

At present, very few solutions exist to address these problems.While the sensor can be calibrated to ignore contributions fromhigher modes [5], this approach is tedious and lacks robustness.Feedback control can be employed with much greater success.

In this contribution, we demonstrate that strong nonlinearcoupling of the tip and sample, combined with the highlyresonant, multimodal nature of the microcantilever, may causeimage artefacts and we utilize modulated–demodulated control intheir mitigation. Such a technique may be particularly relevant forhigh-speed AFM imaging in liquid. Modulated–demodulated con-trol is a vibration control technique that is suitable for the controlof high-frequency resonant dynamics. We have developed a high-bandwidth modulated–demodulated controller [9,10], which canbe configured using low-bandwidth, reconfigurable controllers inthe baseband, thus simplifying the controller implementation.In this work, the controller was designed to offer independentcontrol of a specific mode, without affecting adjacent modes. Weexperimentally verified that the application of modulated–demo-dulated control significantly reduced image distortion.

Another potential application of modulated–demodulated con-trol is in the emerging field of multifrequency AFM. Severalmultifrequency techniques have been developed, including multi-harmonic imaging [11–13], bimodal excitation [13–16], bandexcitation, torsional harmonic AFM and nanomechanical

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n Corresponding author. Tel.: +61 2 4921 7345.E-mail address: kai.karvinen@uon.edu.au (K.S. Karvinen).

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holography [13]. In such applications it may be advantageous toindependently control the characteristics of each mode withoutaffecting the adjacent modes. Such multimodal Q control hasalready been demonstrated in AFM [17].

2. Image artefacts in AFM

2.1. Overview of image artefacts

While the atomic force microscope is capable of obtainingaccurate images detailing nanometer-scaled features, image arte-facts may appear as a result of component limitations.

Many commercial atomic force microscopes utilize piezoelec-tric tube scanners to move the sample in the x, y and z directions.However, piezoelectric ceramics are nonlinear and may suffer fromthermal drift, creep and hysteresis, which produce distortions inthe image [18,19]. The resonant nature of piezoelectric tubescanners and cross-coupling between the axes further limit theachievable performance. Without closed-loop control, it can bedifficult to make accurate measurements [20,21]. The selection ofamplitude setpoint and feedback parameters is also important inthe recovery of accurate surface estimates and even experiencedoperators must often determine good operating conditions by trialand error.

In addition, the probe itself can cause image artefacts. Since theimage is the convolution of the tip and the feature, the quality ofthe tip plays a significant role in the images [18]. For accurateimages, the tip must remain atomically sharp and free fromcontamination. The angle of the microcantilever with respect tothe sample may affect the shape of sharp features.

2.2. Excitation of the higher modes of a microcantilever

The excitation of higher cantilever modes is another cause ofimage artefacts, particularly when there is strong nonlinearcoupling between the modes.

Performing AFM scans in liquid environments can often bechallenging owing to the low Q factors and nonlinear effects of theliquid environment. Experiments have shown that the highereigenmodes can be excited via tip–sample contact and modescan be nonlinearly coupled [3–5]. While the excitation of highermodes may offer more insight into properties of the sample, it isalso likely that such coupling may result in image artefacts. Oneproposed solution to overcome this limitation is to calibrate thesensor to ignore contributions from the higher mode [5], howeverthis solution is tedious to implement and lacks robustness.

The effect of oscillatory flexural modes in contact mode AFMhas been investigated [6,7], highlighting that strong nonlinear

coupling of the tip and the sample results in the excitation ofhigher modes, which adversely affects image quality. Furthermore,the excitation of higher modes was responsible for image distortionin high-speed contact mode scans using spiral scan patterns [8].With high setpoint amplitudes and low tip–sample forces, tappingmode AFM is rarely affected by image artefacts resulting from thenonlinear coupling of the modes. In fact, the higher modes oftencontain useful information, which has resulted in the developmentof dynamic mode multifrequency AFM techniques [13]. However, it ispossible to establish highly nonlinear operation in tapping modeAFM with low setpoints. Such an experiment was set up in order todemonstrate the role of feedback control in the mitigation of imageartefacts caused by the excitation of higher modes and to highlightthe high-bandwidth nature of the proposed solution. Fig. 1 illustratesartefacts in the images of an NT-MDT TGZ1 calibration grating, whichare due to the excitation of the second flexural mode and itsnonlinear coupling to the first mode.

3. Control of the higher modes of a microcantilever

Conditions do occur where the excitation of higher modes canhave an adverse effect on the estimated surface topography. Suchsituations are likely to become even more prevalent in high-speedscanning. Since higher eigenmodes are typically characterized byhigh quality factors, one way to address the issues caused by theexcitation of higher modes is to utilize Q control techniques toattenuate these modes.

Since the introduction of Q control in AFM [22], a plethora ofadvanced techniques have been developed, including adaptivecontrol [23–26], full state feedback control [27], optimal stochasticcontrol [28,29], observer based techniques [30,31], piezoelectricshunt control [32,33] and parametric resonance [34]. Many ofthese techniques are difficult to implement at high frequencies,which complicates the control of higher modes. Therefore, wepropose the application of modulated–demodulated control to thecontrol of higher modes of a microcantilever in AFM.

4. Modulated–demodulated control

Modulated–demodulated control is a vibration control techni-que, applicable to systems possessing high-frequency resonantdynamics. Modulated–demodulated control systems have foundapplication in gyroscopy [35,36], gravity gradiometry [37–39],vibration damping in flexible structures [40], dynamic phase com-pensation [41] and more recently, in atomic force microscopy [9].Advantages of the modulated–demodulated control techniqueinclude the significant reduction of the bandwidth requirements of

Fig. 1. AFM images of an NT-MDT TGZ1 calibration grating with high feedback gain (a) and low feedback gain (b) highlighting artefacts caused by the nonlinear coupling ofthe first and second flexural modes and the profile cross-sections (c). The amplitude setpoint is 25%.

K.S. Karvinen, S.O.R. Moheimani / Ultramicroscopy 137 (2014) 66–71 67

the baseband controller, enabling the use of low-bandwidth, recon-figurable controllers to configure a high-bandwidth analog controller.It is becoming evident that such an implementation could beextremely useful in high-frequency applications.

The modulated–demodulated controller shown in Fig. 2 consistsof two branches, each containing a demodulator with a low-passfilter FðsÞ, an LTI controller CðsÞ and a modulator [40]. Provided thatthe resonance frequency is significantly higher than the half-powerbandwidth, demodulation extracts the amplitude envelope of themeasured signal, thus reducing the required bandwidth of thecontroller. Modulation of the controller outputs shifts the signalsback to high frequencies. A conceptual overview of this technique ispresented in Fig. 3.

The modulated–demodulated controller in Fig. 2 is linear timeinvariant. Given the baseband system ðFCÞðsÞ the modulated–demodulated controller can be modeled as

CeqðsÞ ¼ e� jϕðFCÞðs� jωcÞþejϕðFCÞðsþ jωcÞ: ð1Þ

Alternatively, if the baseband system is expressed as a minimalstate space representation fA;B;C;Dg_x ¼ AxþBuy¼ CxþDu;one possible state space representation of the resulting modu-lated–demodulated controller is

_x ¼A �ωcIωcI A

" #xþ 2B

0

� �u

y¼ ½C cos ϕ C sin ϕ�xþ2D cos ϕu: ð2Þ

5. Controller design

While it is possible to construct a positive position feedbackcontroller by configuring the baseband with a first order system [9],more complex baseband systems enable more aggressive roll-off,eliminating the excitation of adjacent modes and reducing sensornoise in the closed-loop. We chose to implement a Butterworthfilter in the baseband using the MATLAB function butter(n,wb,’s’). Utilizing fmincon, the optimal order n and cutofffrequency of the filter wb were determined, which minimizedthe H1 norm of the second mode. We simultaneously ensured thatthe controller could not excite adjacent modes appreciably. Afourth order Butterworth filter offered a good compromisebetween complexity and rapid roll-off. Using this design metho-dology, the gain and phase margins were 6 dB and approximately401 respectively. The small gain theorem ensures stability of theclosed-loop system away from resonance; the excitation of adja-cent modes is minimal.

In this application, we only considered the control of the secondmode, as its excitation resulted in image artefacts. However, simula-tion confirmed that controllers can be designed independently toaddress a single mode and then combined to produce an equivalentmultimodal controller. The use of high-order filters in the basebandensures that each controller has a minimal effect on its adjacentmodes. This can clearly be seen in Fig. 7.

6. Controller implementation

Our realization of the modulated–demodulated controller isdepicted in Fig. 5. Analog Devices AD835 analog multipliers wereutilized for the high-bandwidth multiplication, complemented bya host of high-bandwidth amplifiers. An Anadigm AN221E04 fieldprogrammable analog array (FPAA) was employed in the base-band. The use of switched capacitor filters enables the cutofffrequencies to be set with precision; the symmetric implementa-tion of the in-phase and quadrature branches is important toensure linear operation of the controller.

Fig. 2. Modulated–demodulated controller.

Fig. 3. Modulated–demodulated control technique: (a) frequency spectrum of aresonant plant, (b) frequency spectrum after demodulation, (c) synthesis of thecontrol signal in the baseband, and (d) high frequency control signal after modulation.

K.S. Karvinen, S.O.R. Moheimani / Ultramicroscopy 137 (2014) 66–7168

There are several limitations of this control method. DC offsetsin the baseband, owing to physical limitations of the electroniccomponents, enable direct feedthrough of the carrier signal, thusintroducing dynamics at the resonance. DC offsets were minimizedusing DC supplies to counteract the offsets introduced by the FPAAin the in-phase and quadrature branches. The switching frequencyof the FPAA is 4 MHz and this has an impact on high-frequencycontroller implementations; modulation introduces dynamics atf c74 MHz. The effect of these dynamics is small, but observable.Finally, while time delays in the baseband do not affect thebandwidth of the controller, they are undesirable and should beminimized to ensure correct operation.

7. Experimental results

7.1. Microcantilever characterization and control

The measured frequency response of a Bruker DMASP micro-cantilever (Fig. 4) is shown in Fig. 6. The higher modes are clearlysignificant in defining the behavior of the microcantilever. Fig. 7highlights the closed-loop frequency response of the microcanti-lever. The carrier frequency fc was set to 241 kHz and the cutofffrequency of the Butterworth filter fb was 36 kHz.

7.2. AFM scans

To demonstrate the effect of the nonlinearly coupled secondmode on image quality, an NT-MDT NTEGRA atomic force micro-scope was used. An NT-MDT TGZ1 calibration grating, whichfeatures step heights of 21.671.5 nm with a periodicity of370:05 μm, was used. The scan size was set to 10 μm� 10 μm.For setpoints below approximately 30% the effect of the secondflexural mode on the image quality is clearly visible. While suchoperating conditions are not typical in tapping mode experiments,we wish to demonstrate strong nonlinear tip–sample coupling toestablish the importance of feedback control in the mitigation ofsuch artefacts. Fig. 8 shows the improvement of the three-dimensional surface as a result of Q control and Fig. 9 comparescross-sections of the surface profile with and without Q control.Results for each example were obtained successively to eliminatethe possibility of observing variations in other parameters.

The experimental results clearly illustrate the adverse effect of thesecond flexural mode on the image quality. In Fig. 8(a) overshoot isvisible on both the upward and downward steps. With feedbackcontrol, we can eliminate these features entirely, producing a sig-nificantly better image as shown in Fig. 8(b). Reducing the feedbackgain prevents overshoot, however, an artefact remains in the upwardsstep as shown in Fig. 8(c). This profile definitely does not reflect theshape of the rectangular calibration grating. Application of Q controleffectively eliminates this artefact as shown in Fig. 8(d). Increasing thesetpoint to 40% (and above), the nonlinear coupling between themodes is reduced, and there is no appreciable difference between thesurface profiles with and without Q control. These results highlight

application of the modulated–demodulated controller at 240 kHz insuppressing a higher mode. We believe that such a technique will beof great importance in liquid and contact mode AFM experiments,especially with next generation high-frequency microcantilevers.

Fig. 4. Bruker DMASP microcantilever.

Fig. 5. Modulated–demodulated controller implementation.

Fig. 6. Frequency response of a Bruker DMASP probe highlighting the significanceof the higher eigenmodes.

K.S. Karvinen, S.O.R. Moheimani / Ultramicroscopy 137 (2014) 66–71 69

8. Applications in atomic force microscopy

8.1. AFM in liquid

Performing AFM in liquid environments has important applica-tions, particularly in the biological sciences. The excitation ofhigher eigenmodes has a significant effect on the image [5] andwe believe that the modulated–demodulated control techniqueoffers a robust solution for such applications through the directcontrol of higher modes. Such control would most likely requireintegrated actuation, such as the insulated actuation schemeoutlined in [42], as opposed to conventional excitation schemes.

8.2. Multifrequency AFM

Another interesting possibility is multimodal Q control. Thesimultaneous control of multiple modes [17] has already beendemonstrated in tapping mode AFM. In this work, the first andsecond flexural modes were controlled with PPF controllers andhigh-speed scans were performed using the second mode. Thehigh-bandwidth nature of the modulated–demodulated controltechnique makes it suitable for the control of higher modes inmultifrequency AFM experiments. Currently, several multifre-quency techniques are widely used. The open loop bimodaltechnique [43,44] is easy to perform, but does not require controlof the higher modes. Whilst some techniques utilize feedbackloops around the higher modes [16,45], no multifrequency tech-niques currently employ Q control techniques and this maybe of interest in future experiments given the importance of Qcontrol in single frequency experiments. Modulated–demodulated

Fig. 7. Closed-loop control of the second mode of a Bruker DMASP microcantilever: the measured controller frequency response (– –) illustrates that control energy isrestricted to the second mode (left) and the close up view of the second mode illustrates the controller performance and the attenuation of the second mode for increasingcontroller gains (right). The controller parameters were f c ¼ 241 kHz and f b ¼ 36 kHz.

Fig. 8. AFM images highlighting the improvement in image quality using modulated–demodulated control to suppress the second eigenmode with high (a) and (b) and lowfeedback gains (c) and (d) for an amplitude setpoint of 25%. The scan speed was 62:4 μm=s.

Fig. 9. Comparison of surface profiles with Q control (–) and without Q control(– –) for amplitude setpoints of 25% (high gain) (top), 25% (low gain) (middle) and40% (bottom).

Fig. 10. Q factor enhancement of the second mode using modulated–demodulatedcontrol.

K.S. Karvinen, S.O.R. Moheimani / Ultramicroscopy 137 (2014) 66–7170

controllers can be designed to offer near independent control ofadjacent resonances, which could enhance/suppress certainmodes of a microcantilever in a desirable way. As an example,Fig. 10 highlights that Q enhancement can significantly increasethe response of the second mode, making it almost as significantas the fundamental mode.

9. Conclusions

Recent work suggests that nonlinear coupling between themodes of a microcantilever may result in image artefacts. Currentlyno robust solutions exist to address this issue. The introduction ofultrahigh-frequency microcantilevers for next generation high-speedimaging will only further complicate the task as high-bandwidthcontrol is characteristically difficult to implement.

We have outlined the application of the modulated–demodulatedcontrol technique to the control of higher modes of a microcantileverin atomic force microscopy. We have shown that modulated–demodulated control, a vibration control technique, which appearsuseful in the control of high-frequency resonant dynamics, cansignificantly improve the image quality by suppressing highermodes, which may be nonlinearly coupled with the drive mode.Such findings may have important implications in liquid and/orcontact mode AFM, where nonlinear coupling between the modes isoften significant and does cause image artefacts. Furthermore, owingto the resonant nature of the control architecture, the modulated–demodulated control technique enables the control of individualmodes of the microcantilever, even at very high frequencies. It istherefore possible to perform simultaneous control of multiplemodes, which may find application in multifrequency AFM techni-ques, where Q control is yet to find significant application.

References

[1] G. Binnig, C.F. Quate, C. Gerber, Phys. Rev. Lett. 56 (1986) 930–933.[2] Q. Zhong, D. Inniss, K. Kjoller, V.B. Elings, Surf. Sci. 290 (1993) L688–L692.[3] S. Basak, A. Raman, Appl. Phys. Lett. 91 (2007) 064107.[4] X. Xu, J. Melcher, S. Basak, R. Reifenberger, A. Raman, Phys. Rev. Lett. 102

(2009) 060801.[5] X. Xu, J. Melcher, A. Raman, Phys. Rev. B—Condens. Matter Mater. Phys. 81

(2010) 035407.[6] O.D. Payton, L. Picco, D. Robert, A. Raman, M.E. Homer, A.R. Champneys, M.J. Miles,

Nanotechnology 23 (2012) 205704.[7] O.D. Payton, L. Picco, M.J. Miles, M.E. Homer, A.R. Champneys, Nanotechnology

23 (2012) 265702.[8] I.A. Mahmood, S.O.R. Moheimani, Nanotechnology 20 (2009) 365503.[9] K.S. Karvinen, S.O.R. Moheimani, in: Proceedings of the 6th IFAC Symposium

on Mechatronic Systems, 2013, pp. 399–405.

[10] K.S. Karvinen, S.O.R. Moheimani, in: Proceedings of the 2013 AustralianControl Conference, 2013, pp. 461–466.

[11] R.W. Stark, W.M. Heckl, Rev. Sci. Instrum. 74 (2003) 5111–5114.[12] R.W. Stark, Nanotechnology 15 (2004) 347–351.[13] R. Garcia, E.T. Herruzo, Nat. Nanotechnol. 7 (2012) 217–226.[14] R. Proksch, Appl. Phys. Lett. 89 (2006) 113121.[15] J.R. Lozano, R. Garcia, Phys. Rev. Lett. 100 (2008) 076102.[16] D. Ebeling, S.D. Solares, Beilstein J. Nanotechnol. 4 (2013) 198–207.[17] M.G. Ruppert, M.W. Fairbairn, S.O.R. Moheimani, in: Proceedings of IEEE/ASME

International Conference on Advanced Intelligent Mechatronics, 2013, pp. 77–82.

[18] D. Ricci, P.C. Braga, Atomic force microscopy, in: Methods in Molecular Biology,vol. 242, Humana Press, 2004, pp. 25–37.

[19] S.O.R. Moheimani, Rev. Sci. Instrum. 79 (2008) 071101.[20] M. Ratnam, B. Bhikkaji, A.J. Fleming, S.O.R. Moheimani, in: Proceedings of the

44th IEEE Conference on Decision and Control, and the European ControlConference, CDC-ECC '05, 2005, pp. 1168–1173.

[21] I.A. Mahmood, S.O.R. Moheimani, Rev. Sci. Instrum. 80 (2009) 063705.[22] J. Mertz, O. Marti, J. Mlynek, Appl. Phys. Lett. 62 (1993) 2344–2346.[23] I. Gunev, A. Varol, S. Karaman, C. Basdogan, Rev. Sci. Instrum. 78 (2007)

043707.[24] A. Varol, I. Gunev, B. Orun, C. Basdogan, Nanotechnology 19 (2008) 075503.[25] P. Agarwal, T. De, M.V. Salapaka, Rev. Sci. Instrum. 80 (2009) 103701.[26] M.W. Fairbairn, S.O.R. Moheimani, IEEE Trans. Nanotechnol. 11 (2012)

1126–1134.[27] B. Orun, S. Necipoglu, C. Basdogan, L. Guvenc, Rev. Sci. Instrum. 80 (2009)

063701.[28] J.L. Garbini, K.J. Bruland, W.M. Dougherty, J.A. Sidles, J. Appl. Phys. 80 (1996)

1951–1958.[29] K.J. Bruland, J.L. Garbini, W.M. Dougherty, J.A. Sidles, J. Appl. Phys. 83 (1998)

3972–3977.[30] D.R. Sahoo, A. Sebastian, M.V. Salapaka, Appl. Phys. Lett. 83 (2003) 5521–5523.[31] D.R. Sahoo, T. De, M.V. Salapaka, in: Proceedings of the 44th IEEE Conference

on Decision and Control, and the European Control Conference, CDC-ECC '05,vol. 2005, 2005, pp. 1185–1190.

[32] M.W. Fairbairn, S.O.R. Moheimani, A.J. Fleming, J. Microelectromech. Syst. 20(2011) 1372–1381.

[33] M. Fairbairn, S.O.R. Moheimani, Rev. Sci. Instrum. 84 (2013) 053706.[34] G. Prakash, A. Raman, J. Rhoads, R.G. Reifenberger, Rev. Sci. Instrum. 83 (2012)

065109.[35] R.P. Leland, in: Proceedings of the 40th IEEE Conference on Decision and

Control, vol. 4, 2001, pp. 3447–3452.[36] Y.C. Chen, R.T. M'Closkey, T.A. Tran, B. Blaes, IEEE Trans. Control Syst. Technol.

13 (2005) 286–300.[37] M.A. Gerber, Astronaut. Aeronaut. 16 (1978) 18–26.[38] C.A. Affleck, A. Jircitano, in: Position Location and Navigation Symposium,

Record, The 1990's—A Decade of Excellence in the Navigation Sciences, 1990,pp. 60–66.

[39] R.E. Bell, R. Anderson, L. Pratson, Leading Edge 16 (1997) 55–59.[40] K. Lau, D.E. Quevedo, B.J.G. Vautier, G.C. Goodwin, S.O.R. Moheimani, Control

Eng. Pract. 15 (2007) 377–388.[41] C. Hendrickson, R.T. M'Closkey, J. Dyn. Syst. Meas. Control, Trans. ASME 134

(2012) 011024.[42] B. Rogers, T. Sulchek, K. Murray, D. York, M. Jones, L. Manning, S. Malekos,

B. Beneschott, J.D. Adams, H. Cavazos, S.C. Minne, Rev. Sci. Instrum. 74 (2003)4683–4686.

[43] T.R. Rodríguez, R. García, Appl. Phys. Lett. 84 (2004) 449–451.[44] N.F. Martinez, S. Patil, J.R. Lozano, R. García, Appl. Phys. Lett. 89 (2006) 153115.[45] S.D. Solares, G. Chawla, J. Appl. Phys. 108 (2010) 054901.

K.S. Karvinen, S.O.R. Moheimani / Ultramicroscopy 137 (2014) 66–71 71

http://refhub.elsevier.com/S0304-3991(13)00311-2/sbref1http://refhub.elsevier.com/S0304-3991(13)00311-2/sbref2http://refhub.elsevier.com/S0304-3991(13)00311-2/sbref3http://refhub.elsevier.com/S0304-3991(13)00311-2/sbref4http://refhub.elsevier.com/S0304-3991(13)00311-2/sbref4http://refhub.elsevier.com/S0304-3991(13)00311-2/sbref5http://refhub.elsevier.com/S0304-3991(13)00311-2/sbref5http://refhub.elsevier.com/S0304-3991(13)00311-2/sbref6http://refhub.elsevier.com/S0304-3991(13)00311-2/sbref6http://refhub.elsevier.com/S0304-3991(13)00311-2/sbref7http://refhub.elsevier.com/S0304-3991(13)00311-2/sbref7http://refhub.elsevier.com/S0304-3991(13)00311-2/sbref8http://refhub.elsevier.com/S0304-3991(13)00311-2/sbref11http://refhub.elsevier.com/S0304-3991(13)00311-2/sbref12http://refhub.elsevier.com/S0304-3991(13)00311-2/sbref13http://refhub.elsevier.com/S0304-3991(13)00311-2/sbref14http://refhub.elsevier.com/S0304-3991(13)00311-2/sbref15http://refhub.elsevier.com/S0304-3991(13)00311-2/sbref16http://refhub.elsevier.com/S0304-3991(13)00311-2/sbref19http://refhub.elsevier.com/S0304-3991(13)00311-2/sbref21http://refhub.elsevier.com/S0304-3991(13)00311-2/sbref22http://refhub.elsevier.com/S0304-3991(13)00311-2/sbref23http://refhub.elsevier.com/S0304-3991(13)00311-2/sbref23http://refhub.elsevier.com/S0304-3991(13)00311-2/sbref24http://refhub.elsevier.com/S0304-3991(13)00311-2/sbref25http://refhub.elsevier.com/S0304-3991(13)00311-2/sbref26http://refhub.elsevier.com/S0304-3991(13)00311-2/sbref26http://refhub.elsevier.com/S0304-3991(13)00311-2/sbref27http://refhub.elsevier.com/S0304-3991(13)00311-2/sbref27http://refhub.elsevier.com/S0304-3991(13)00311-2/sbref28http://refhub.elsevier.com/S0304-3991(13)00311-2/sbref28http://refhub.elsevier.com/S0304-3991(13)00311-2/sbref29http://refhub.elsevier.com/S0304-3991(13)00311-2/sbref29http://refhub.elsevier.com/S0304-3991(13)00311-2/sbref30http://refhub.elsevier.com/S0304-3991(13)00311-2/sbref32http://refhub.elsevier.com/S0304-3991(13)00311-2/sbref32http://refhub.elsevier.com/S0304-3991(13)00311-2/sbref33http://refhub.elsevier.com/S0304-3991(13)00311-2/sbref34http://refhub.elsevier.com/S0304-3991(13)00311-2/sbref34http://refhub.elsevier.com/S0304-3991(13)00311-2/sbref36http://refhub.elsevier.com/S0304-3991(13)00311-2/sbref36http://refhub.elsevier.com/S0304-3991(13)00311-2/sbref37http://refhub.elsevier.com/S0304-3991(13)00311-2/sbref39http://refhub.elsevier.com/S0304-3991(13)00311-2/sbref40http://refhub.elsevier.com/S0304-3991(13)00311-2/sbref40http://refhub.elsevier.com/S0304-3991(13)00311-2/sbref41http://refhub.elsevier.com/S0304-3991(13)00311-2/sbref41http://refhub.elsevier.com/S0304-3991(13)00311-2/sbref42http://refhub.elsevier.com/S0304-3991(13)00311-2/sbref42http://refhub.elsevier.com/S0304-3991(13)00311-2/sbref42http://refhub.elsevier.com/S0304-3991(13)00311-2/sbref43http://refhub.elsevier.com/S0304-3991(13)00311-2/sbref44http://refhub.elsevier.com/S0304-3991(13)00311-2/sbref45

Control of the higher eigenmodes of a microcantilever: Applications �in atomic force microscopyIntroductionImage artefacts in AFMOverview of image artefactsExcitation of the higher modes of a microcantilever

Control of the higher modes of a microcantileverModulated–demodulated controlController designController implementationExperimental resultsMicrocantilever characterization and controlAFM scans

Applications in atomic force microscopyAFM in liquidMultifrequency AFM

ConclusionsReferences