Multi-frequency atomic force microscopy:A system-theoretic approach

Naveen Shamsudhin ∗,∗∗ Hugo Rothuizen ∗∗

Bradley J. Nelson ∗ Abu Sebastian ∗∗

∗ Institute of Robotics and Intelligent Systems, ETH Zurich,8092 Zurich, Switzerland (e-mail: snaveen@ethz.ch)

∗∗ IBM Research-Zurich, 8803 Rueschlikon, Switzerland

Abstract: Multi-frequency atomic force microscopy (MF-AFM), employing the detectionand/or excitation of multiple cantilever frequencies, has shown great promise in increasing thecompositional sensitivity, and the spatial and temporal resolution of imaging. The multitudeof frequency components generated in MF-AFM encode information about the tip-samplenonlinearity. For quantitative interpretation of the observables in MF-AFM operation, wepropose a two-pronged approach combining special-purpose cantilevers and a system-theoreticmodeling paradigm. This provides an excellent framework to understand and leverage thenonlinear dynamics of the interaction of a multi-eigenmodal cantilever with the nonlinearforce potentials on the sample surface, to develop novel imaging methods. We describeexperimental techniques for accurate in-situ identification of the cantilever (sensor/actuator)transfer functions, which are crucial components to understand the generation of MF-AFMobservables. The modeling framework is verified with experiments and is shown to be able topredict several key features of MF-AFM operation.

Keywords: Atomic Force microscopy, MF-AFM, system identification, cantilever dynamics

1. INTRODUCTION

Since its invention in 1986 [Binnig et al. (1986)], the atomicforce microscope (AFM) has been one of the quintessentialinstruments of nanoscale science and engineering. The tip-sample interaction force is sensed by a micro-cantileverand through this observation local surface properties aremapped with sub-nanometer resolution.

In the most basic form of AFM, the cantilever tip isscanned across the sample surface. Due to tip-sample in-teraction forces, the cantilever deflects and this deflectionis measured. This is known as contact mode AFM. Giventhat the tip traverses over a short region of the tip-sampleinteraction potential, a linear analysis is usually sufficientto analyze this mode of AFM in spite of the highly non-linear nature of the interaction force. However, contact-mode AFM has several drawbacks such as tip and sampledamage owing to the lateral shear forces.

The next significant step in AFM was the introductionof dynamic mode operation [Martin et al. (1987)]. In thismode, the micro-cantilever is typically oscillated at thefirst resonance frequency to intermittently probe the sam-ple. There are two key variants of dynamic-mode AFM.In amplitude modulation (AM-AFM) operation, the ob-servables are the shift in the amplitude and phase of thecantilever oscillation. In frequency modulation (FM-AFM)operation, the cantilever is made to oscillate at the effec-tive resonance frequency of the cantilever-sample systemand the resulting frequency shift is the observable. Dy-namic mode AFM is now an established tool for nanoscaleinvestigation. The dynamics is much more involved given

that the tip traverses a wide range of the interactionpotential. Hence, it took almost 20 years to fully establishthe fundamental theory behind dynamic mode operation[Garcia and Herruzo (2012)].

In the last few years, there is a significant research ef-fort towards multi-frequency AFM (MF-AFM). MF-AFMinvolves techniques where the cantilever motion is mea-sured (and sometimes driven) at multiple resonant fre-quencies [Proksch (2011); Garcia and Herruzo (2012)].These frequencies are usually associated with either thehigher harmonics of the oscillation or the eigenmodes ofthe cantilever. MF-AFM provides several new informationchannels owing to the plethora of spectral componentscreated, many of which can be mapped back into localsurface properties. MF-AFM holds great promise in itsability to significantly enhance the resolution, the imagingthroughput and the ability to discern material properties.The “holy grail” quest of AFM has been compositionalmapping where materials differences are mapped out withthe same nanometer resolution as topographic images. Atpresent the most promising approach to reaching this goalis MF-AFM. However, in spite of the tremendous promiseof MF-AFM techniques, quantitative MF-AFM remainsa significant challenge. Interpreting and controlling thecomplex dynamics of the multi-modal cantilever interact-ing with the highly nonlinear and dissipative tip-sampleinteraction potential is an enormous challenge.

For quantitative MF-AFM, we need to utilize the fullpower of the theory of dynamical systems. Note thatthe MF-AFM dynamics is substantially more compli-cated than the regular dynamic mode AFM. The system-

Preprints of the 19th World CongressThe International Federation of Automatic ControlCape Town, South Africa. August 24-29, 2014

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theoretic approach to modelling AFM dynamics was firstintroduced for the analysis of AM-AFM dynamics [Sebas-tian et al. (2001), Sebastian et al. (2007)]. The systemsviewpoint was also utilized in an elegant manner to obtaintime-resolved tip-sample interaction forces [Stark et al.(2002)]. In this paper we extend this powerful modellingapproach to investigate MF-AFM operation. We also pro-pose accurate techniques to experimentally identify thesemulti-modal cantilever dynamics, correcting for the ob-servation problem that arises from the commonly usedoptical beam deflection method. Using the systems model,we demonstrate an algorithm to recover the instantaneoustip-sample forces under any generic multi-tone cantileverexcitation encompassing all common methods of MF-AFMoperation, not limited to bi-modal, n-modal, inter-modalor band-excitation. We demonstrate the validity of theMF-AFM model through experiments using a tunableelectrostatic force potential.

2. MICRO-CANTILEVERS FOR MF-AFM

Fig. 1. Scanning electron micrograph of the fabri-cated MF-AFM cantilever. The cantilever possessesa stepped rectangular geometry with two segments.

The micro-cantilever is the transducer through whichwe probe tip-sample interaction forces. In contact-modeAFM, the only cantilever parameter of interest is itsstatic stiffness. In traditional dynamic mode AFM, theparameters of interest are limited to the cantilever stiff-ness, resonant frequency and quality factor associated withthe first eigenmode. Hence the dynamic response of themicro-cantilever over a wide frequency range was not ofgreat interest. Moreover, these cantilevers are generallyactuated by electrically exciting an externally mountedpiezo-electric actuator. Spurious vibrational modes areintroduced by the imperfect coupling between the externalactuator and the micro-cantilever. Since the cantileversare typically actuated and sensed at the first resonancefrequency, these spurious modes did not pose much ofa problem in traditional dynamic-mode AFM. However,for quantitative MF-AFM, it is essential to have a well-defined dynamic behaviour for the actuator-cantilever sys-tem. This could be achieved by integrating piezo-electricactuators onto the cantilever. In conventional rectangularcantilevers, there is a wide separation between the resonantfrequencies corresponding to the different eigenmodes. ForMF-AFM it is probably advantageous to reduce the spac-ing between the resonant frequencies to ensure significantcoupling between the different modes.

Based on this, the authors recently introduced specialMF-AFM cantilevers which have a stepped rectangular

geometry with two rectangular segments and integratedaluminum nitride actuators [Sebastian et al. (2012); Sham-sudhin et al. (2012)]. They are also equipped with con-ductive platinum silicide tips [Bhaskaran et al. (2009)] fornanoscale electrical sensing. A scanning electron micro-graph of one such cantilever is shown in Fig. 1. The new ge-ometry spectrally close-packs the eigenmodes Mi, with thefourth eigenfrequency ωM4 ≈ 13.2 · ωM1 compared to therectangular case where ωM4 ≈ 34.4·ωM1 . Furthermore, theintegrated actuation allows for linear cantilever-actuatordynamics and an excellent phase response, making themwell suited for quantitative MF-AFM.

3. MF-AFM MODELING FRAMEWORK

fts

bp

Gbp

Gtp

pb

Gtp Φ

p(t)∑b(t) Gbp

pts

fts

CollimatorPolarizer

Cube Beam Splitter

λ / 4 Filter

Focusing Lens

QuadrantPhotodiode

Vo

Fig. 2. Schematic illustration of the experimental setupand a system-theoretic model that captures the MF-AFM operation.

To interpret the complex dynamics of the above mentionedmulti-modal cantilever interacting with a highly nonlin-ear tip-sample interaction force, we resort to a system-theoretic modeling framework, where the cantilever-sampleinteraction is modeled as the interaction between two dis-tinct linear time-invariant cantilever models Gbp and Gtpinteracting with the sample nonlinearity Φ (Fig. 2). Thecantilever dynamics have been decoupled into two separatemodels, where the transfer function Gbp relates the tipdisplacement to the voltage signal applied to the piezo-electric actuator. This voltage signal is hitherto referredto as the base forcing signal, b(t). Gtp relates the displace-ment to the force experienced by the tip. The tip-sampleforces fts enter the feedback loop through Gtp, results ina displacement pts, which adds onto the displacement pbgenerated byGbp in response to b(t). There is no restrictingassumption on the nonlinearity Φ which may comprise ofelectric, magnetic or mechanical origin.

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4. MODEL IDENTIFICATION

Next we present the in-situ experimental identificationof the Gbp and Gtp transfer functions associated withthe systems-model. A key challenge arises from the factthat optical beam deflection systems are widely employedto measure the cantilever deflection signal (see Fig. 2)necessitated by the requirements on bandwidth and res-olution. However, the measured photodiode signal (Vo) isproportional to the relative angle generated at the end ofthe cantilever (i.e at the point of laser focus) and not tothe absolute cantilever tip displacement [Marti (1999)].

It is common practice to identify the dynamics of the firsteigenmode for regular dynamic mode AFM and only a fewgroups have tried to identify the multi-modal dynamics ofthe cantilever [Scherer et al. (2000), Stark et al. (2005),Pini et al. (2010)]. Multi-modal identification via thermalnoise forcing is also demonstrated [Salapaka et al. (1997)],but it is not possible to obtain any useful phase informa-tion. But in all these approaches a methodology to identifythe multi-modal displacement dynamics from the angularmeasurement is clearly lacking.

4.1 Base-Forcing Transfer Functions Gba and Gbp

−150

−100

−50

0

Ma

gn

itu

de

(d

B)

104

105

106

−400

−200

0

Frequency (Hz)Ph

as

e (

de

g)

Fig. 3. Identification of base-forcing transfer functions;The measured angular response to a base excitationis shown in blue, the pole-zero fit Gba in red, and thebi-quad expansion of the pole-zero model is shown incyan, and finally Gbp is shown in green.

To identify the base-forcing transfer functions, the inte-grated actuator is driven with a single tone voltage signal,b(t) and the input frequency is varied while simultaneouslymonitoring the photodiode signal, Vo(t) using a lock-inamplifier. To obtain the angular information, a(t) fromthe photodiode signal, Vo, it needs to be scaled by theinverse of a constant parameter, KE . KE is defined asKE = α0 · D−1 where α0 [rad/meter] is the ratio of thetip angle to the tip displacement generated by a static tipforcing and D [volts/meter] is the displacement sensitivityobtained from a standard approach curve. For a divingboard cantilever, α0 = (3/2) ·L , where L is the cantileverlength, but since ours has a dual-beam structure withintegrated actuators and electrodes, we used finite-elementanalysis to estimate the ratio α0. The complete frequencyresponse is recorded upto 2 MHz (Fig. 3). The drive signalis chosen to have a fixed peak-to-peak amplitude of 300 mVto ensure adequate signal-to-noise ratio and at the sametime preserving the linearity of cantilever motion.

The resulting angular frequency response is fit with a pole-zero model (curve in red in Fig. 3) to the obtain Gba (curvein blue in Fig. 3). The excellent fit further confirmingthe linearity of the frequency response. Remarkably thephase response shows a collocated dynamic behavior overa Megahertz bandwidth. We restrict our identification tothe first four eigenmodes of the cantilever. Performinga biquad expansion on Gba, we can rewrite it as as asummation, which essentially represents an orthogonalsummation of the first four eigenmodes of the cantilever.

Gba(s) =

4∑i=1

βi ω2i

s2 + 2ζbiωis+ ω2i

=

4∑i=1

Hb,i(s) (1)

Now, to obtain Gbp (curve in green Fig. 3), we scale eachof the eigenmodes Hb,i by modal correction factors αi,

Gbp(s) =

4∑i=1

(1

αi

)Hb,i(s) (2)

αi captures the angle generated by a unit displacement atthe ith eigenmode and are obtained from finite elementanalysis. Thus we have identified both the transfer func-tions Gba and Gbp.

4.2 Tip-Forcing Transfer Functions Gta and Gtp

To identify the tip-forcing transfer functions, we need amechanism to ensure forcing at the cantilever tip andto excite its multiple eigenmodes. We propose an iden-tification strategy utilizing electrostatic tip forcing. Thecantilever is made of n-doped silicon, this means that itcan form a conductive plate of a capacitor. To create aconfined electrostatic-forcing at a narrow region aroundthe tip, we microfabricated pillars of dimensions 50 µm inheight and 8 µm in diameter (see Fig. 4). There pillars werecreated on an n-doped silicon wafer using deep reactive ionetching process. The cantilever tip was then positionedapproximately 200 nm above one of these pillars.

Fig. 4. Si pillars fabricated by DRIE positioned at the tipof the micro-cantilever. Scale bar: 100µm

The net electrostatic force acting on the cantilever for agiven pillar bias voltage Ve can be decomposed into thesum of three forces, essentially the tip-apex force, the tip-cone force, and the body force as proposed by Colcheroet al. (2001).

fts = Φe (Ve, p) = Φapex + Φcone + Φbody (3)

Due to the confinement of the electrostatic field to a nar-row region of tip, and for small cantilever deflections, we

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can estimate a frequency-independent Φe linearly depen-dent on the square of the electrostatic voltage, fts ≈ A0·V 2

e

The silicon pillar was biased with a band-limited noisesignal and the resultant photodiode signal, Vo was recordedat a sampling frequency of 10 MHz and the frequency re-sponse was estimated using Welch’s averaged periodogrammethod. The linearity of the output response to the inputelectrostatic forcing is verified by the high coherence valueof the identified frequency response and further confirmedby the excellent pole-zero fit as shown in Fig. 5. Followinga similar calibration procedure as for identifying the baseexcitation dynamics Gbp, we scale each harmonic oscillatorwith the corresponding modal correction factors αi.

−150

−100

−50

0

Ma

gn

itu

de

(d

B)

104

105

106

−400

−200

0

200

Frequency (Hz)Ph

as

e (

de

gre

es

)

Fig. 5. Identification of tip-forcing transfer functions; Themeasured angular response to a tip excitation is shownin blue, the pole-zero fit Gta in red, and the bi-quadexpansion of the pole-zero model is shown in cyan,and Gtp0 is shown in green.

Gtp0(s) =

4∑i=1

(1

αi

)γi ω

2i

s2 + 2ζtiωis+ ω2i

(4)

The dc gain of the tip-forcing transfer function Gtp mustequal the inverse of the cantilever static stiffness ks.The static stiffness is typically determined by using thecantilever’s thermal noise response or by Sader’s method.Since both techniques use theoretical calibration factorsbased on rectangular cantilever geometries, we employa calibrated FemtoTools MEMS force sensor to directlyquantify the stiffness of the cantilever. The force sensor isused to measure the restoration force when the cantilevertip is pressed against it.

Finally the transfer function, Gtp is obtained as,

Gtp(s) =1

ks

Gtp0(s)

|Gtp0(j0)|=

4∑i=1

Htp,i (5)

4.3 Optical Filter Sb and St

From the identified transfer functions Gba, Gbp, Gta, Gtp,we can derive two filters Sb and St given by

Sb(ωn) =

∣∣∣∣GbaGbp

∣∣∣∣ and St(ωn) =

∣∣∣∣GtaGtp

∣∣∣∣ (6)

and the magnitude response of the filters is shown in Fig.6.

These zero-phase filters capture the effect of the opticalbeam deflection system, namely the translation from tip

104

105

106

50

60

70

80

90

100

110

120

130

Frequency (Hz)

Mag

nit

ud

e (

rad

/m)

ωM

1

ωM

2

ωM

3

ωM

4

Sb

St

Fig. 6. Magnitude response of the optical filters Sb and St.

St Votsfts Gtp pts

Gta

KEa

Sb Vobb Gbp pb

Gba

KEa

Fig. 7. Block diagram description of the measurable pho-todiode signal and the forcing signals.

displacement signal to the angular signal. For example,the measurable photodiode signal arising from forcing thecantilever with a base forcing signal, b(t) or a tip forcingsignal, fts(t) is schematically shown in Fig. 7. Thesefilters are essential in de-convolving the tip displacementfrom the measured photodiode signal during MF-AFMoperation as illustrated in the next section.

5. SIGNAL FLOW IN MF-AFM

Based on the identified Gbp, Gtp, KE and the optical filtersSb and St, we can now create an accurate description ofthe tip displacement as well as the tip-sample interactionforces.

5.1 Tip Displacement in MF-AFM

If the cantilever is actuated with an M -tone base forcingsignal b(t), in the absence of tip-sample interaction forces,the linearity of Gbp ensures that the tip response pb(t) isalso M -toned given by,

pb(t) =

M∑ω=1

pf (ω) = Gbp

{M∑ω=1

b(ω)

}(7)

where ω = [ω1, ..., ωM1, ..., ωM ] ∈ RM

However, in the presence of a sample nonlinearity, Φ, theinteraction of the M -tone driven-cantilever with Φ gener-ates spectral components in the tip motion which are dif-ferent from the actuation frequencies, including harmonicsand intermodulation products. The net displacement ofthe cantilever tip p(t) can be written as the summationof the displacements pb(t) and pt(t) induced by the baseforcing and the tip forcing from the sample respectively(Fig. 2).

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p(t) = Gbp b(t) +Gtp fts(t) (8)

p(t) = pb(t) + pts(t) (9)

The corresponding photodiode signal can also be writtenas a summation,

Vo(t) = Vob(t) + Vots(t) (10)

By taking the discrete fourier transform DFT of thesampled photodiode signal, we can express Eq. (10) as

Vo(ωn) =

N−1∑i=0

Vo(ti)e−j2πin/N (11)

Vo(ωn) = Vob(ωn) + Vots(ωn) (12)

The contribution of the tip-forcing displacement pts com-ing from the sample can be estimated by removing thebase actuation component from the measured photodiodesignal Vo.

Vots(ωn) = Vo(ωn)−KE · Sb(ωn) · pb(ωn) (13)

pts(ωn) =1

KE· S−1

t · Vots(ωn) (14)

Taking the inverse DFT of Eq (14), we arrive at thedisplacement pts generated by the tip-sample forces fts,

pts(ti) =1

N

N−1∑n=0

pts(ωn)ej2πin/N (15)

One of the perennial challenges of AFM is the accuratedescription of the instantaneous tip-sample interactionforces. Using the experimentally identified model com-ponents along with the system-theoretic framework, thetip-sample forces fts in MF-AFM experiments can bedirectly estimated without any prior assumptions of theparametrization of the forcing nonlinearity Φ. We canreformulate Eq.(8), and solve for the tip-sample force asfollows,

fts(t) = G−1tp {p(t)−Gbp b(t)} (16)

5.2 Experimental demonstration

To verify the systems-model in a multi-frequency AFMscenario, electrostatic forcing was again chosen as thecandidate for generating the tip forcing signal. A blockdiagram representation is shown in Figure 8. The elec-trostatic forcing operator, denoted by Φe will generate aforce that is a function of the voltage signal, Ve and thetip displacement, p(t).

An experiment is performed whereby the cantilever tipis positioned above the Si pillar (as shown in Fig. 4).The cantilever is actuated bi-modally at its first twoeigenfrequencies with b(t) = 0.6sin(ωM1

t)+0.05sin(ωM2t).

Based on the systems-model, by suitable selection of Ve(t),each of the eigenmodal vibrations are cancelled selectively

pf

Gtp Φe

p(t)∑b(t) Gbp

pts

fts Ve

Fig. 8. Systems-model for electrostatic cantilever forcing.

Fig. 9. Experimental validation of the systems-model (a)Electrostatic forcing signal Ve(t) (b) Correspondingphotodiode deflection signal Vo(t); No cancellation inblue, eigenmode M1 cancellation in green, and bi-modal (M1 and M2) cancellation in red.

as shown in Fig. 9. Fig. 9(b) shows the photodiode signalwhereas 9(a) shows the corresponding Ve(t) signal. This isa strong validation of not just the systems-model, but alsoof the various steps involved in the model identificationprocess.

Next we present imaging experiments where the cantilevertip was scanned in non-contact mode above the surface ofthe Si pillar which had been milled by a focused ion beam(FIB) to produce narrow stripes (see Fig. 10(a)). As thecantilever tip is scanned above this sample, the topographyvariation of the stripes creates a varying Φe consequentlymodulating the force fts felt by the tip.

In the first experiment, the pillar was biased with avoltage signal at half the second eigenmodal frequencyVe(t) = Ve0sin

(2π

ωM2

2 t)

while b(t) = 0. Fig. 10(b) showsthe demodulated ωM2 amplitude. In this scenario, theresulting amplitude signal is enhanced due to the forcingat the tip apex caused by the protruding stripes, whilethe background signal from cantilever body is low. In thesecond experiment, we additionally apply a base excitationb(t) = sin (2πωM2t+ φ) which in accordance with thesystems-model should cancel the M2 oscillation when thecantilever is above the stripes. The resulting image isshown in Fig. 10(c) where this cancellation is clearlyvisible. This experiment serves as a further validation ofthe modeling approach as well as provides insight intodevelopment of directed MF-AFM operation modes in thefuture.

6. CONCLUSIONS

Quantitative multi-frequency AFM is a challenging prob-lem where system-theoretic approaches can play a signifi-cant role especially in combination with experimental toolssuch as micro-cantilevers with well defined dynamic be-haviour over MHz bandwidth. In this article we illustrate

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(a) Scale Bar S= 4µm

(b) Ve(t) = Ve0sin(2π

ωM22t)and b(t) = 0

(c) Ve(t) = Ve0sin(2π

ωM22t)and b(t) = sin

(2πωM2

t+ φ)

Fig. 10. (a) Scanning electron micrograph of the structurescreated using the FIB. (b,c) Images obtained based onthe ωM2 amplitude signal.

this by unravelling the complex dynamics arising from sucha cantilever interacting with nonlinear tip-sample interac-tion forces using a systems-model. The various elementsof the model are experimentally identified. In particular,an experimental methodology is proposed to overcome thechallenge posed by optical beam deflection systems thatare used to sense the cantilever deflection signal. Accuratedescription of the tip displacement as well as the instan-taneous tip-sample interaction force is derived from theobservable photodiode signal. Finally, experimental resultsare presented to validate the modeling framework wherethe tip-sample interaction forces are generated throughelectro-static means.

ACKNOWLEDGEMENTS

We would like to thank M. Despont for contributions tothe cantilever design and P. Jonnalagadda for clean-roomsupport. We thank the Swiss National Foundation fortheir financial support through the SystemsX.ch projectMecanX.

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