APPLIED PHYSICS LETTERS 92, 223113 共2008兲

Parametrically coupled multiharmonic force imaging
M. Kurtuluş Abak,1 Ozan Aktas,2 Rashad Mammadov,1 İhsan Gürsel,1,3 andAykutlu Dâna1,a兲
1Institute of Materials Science and Nanotechnology, Bilkent University, 06800 Ankara, Turkey
Department of Physics, Bilkent University, 06800 Ankara, Turkey
3
Department of Molecular Biology and Genetics, Bilkent University, 06800 Ankara, Turkey
2共Received 19 February 2008; accepted 16 May 2008; published online 6 June 2008兲
We report use of nonlinear tip-sample interactions to parametrically convert the frequency
components of periodic tip-sample interaction forces to frequencies where they can be resonantly
detected. One flexural mode of a cantilever is used for tapping-mode imaging and another flexural
mode is used for detection of forces converted in presence of an externally injected mechanicaloscillation at the difference frequency of the detecting mode and a harmonic of the tapping mode.
Material contrast in attractive and repulsive regimes are demonstrated on samples with polymethyl
methacrylate patterns and with deoxyribonucleic acid strands on silicon. © 2008 American Instituteof Physics. 关DOI: 10.1063/1.2940304兴
Measurement and imaging of material dependent nonlinear tip-sample interactions in atomic force microscopy1
共AFM兲 provides information about the structure of thesample on the nanoscale. In the conventional tapping mode,a single mode of the cantilever is used to resonantly detect
the fundamental frequency component of the time varying
interaction force, which is observed in the perturbations of
the amplitude and phase of the oscillation.2 This selective
measurement filters out higher harmonics of the interaction
force and results in loss of information and material contrast
is observed in dissipation related phase changes in theoscillation.3–5 Although excitation of the higher frequency
harmonics can be strongly related to material elasticity they
are not detected in conventional tapping mode.6 The main
reason for this is, for a rectangular cantilever, higher ordermode frequencies are not integer multiples of the fundamental mode frequency.7–11 Several authors have addressed the
problem of measuring the complete tip-sample interaction
during tapping using special modifications to the cantileveror the scanning probe electronics.12,13 Multifrequency excitation of the cantilever simultaneously at the two flexural
resonances has also been demonstrated to give complementary material contrast.14,15
In this article, the highly nonlinear tip-sample interaction
is used to convert the frequency components of the interaction force generated during tapping to a frequency that canbe resonantly measured. Both parametric up-conversion
共PUC兲 and down-conversion 共PDC兲 schemes can be implemented. One of the flexural modes is used in tapping mode
imaging while a sinusoidal mechanical signal is applied at
the difference frequency of the force component to be measured and the mode frequency to be used for detection. The
frequency-converted force components are then resonantly
detected using a lock-in amplifier. Frequency conversion efficiency is also related to the mechanical properties of thesample, therefore the conversion process itself also results inmaterial contrast.
In the experiments different commercial AFM systems
共PSIA XE100 retrofitted with a homemade vibrating sample
holder and an Asylum MFP 3D system fitted with an externala兲Electronic mail: aykutlu@fen.bilkent.edu.tr.

circuit for multifrequency excitation兲 are used. One of theflexural harmonics is used in tapping mode by the AFM controller and the ac deflection of the other flexural mode isdetected using a rf lock-in amplifier 共model SR844兲. Commercial rectangular AFM cantilevers are used. Typical measured mode resonant frequencies are f 0 = 68– 80 KHz, f 1= 410– 500 KHz, and nominal spring constants are k0= 2.7 N / m, k1 = 共f 1 / f 0兲2k0 = 99.3 N / m. Typical quality factorsof the fundamental and second order flexural modes are determined to be Q0 ⯝ 200 and Q1 ⯝ 500. Supplier specified tip
radius is R = 15 nm. Deflection detection noise floor of theAFM systems used are xmin ⬃ 2 ⫻ 10−13 m / 冑Hz and resonant
force measurements are Brownian motion limited. In PUC
experiments, the fundamental mode is driven with an amplitude A0 ⯝ 10 nm at a frequency f 0⬘ = f 0 − ⌬f, and a local oscillator vibration with amplitude ALO ⯝ 1 nm is applied either
by vibrating the tip or the sample at a frequency f LO. Theresult of the mixing is monitored at f 1. Repulsive or attractive branches of the tip-sample interaction can be chosen by
selecting a sufficiently large positive or negative frequency
offset ⌬f.16 The parametrically excited second order mode
amplitudes A1 as a function of the tip-sample separation
zts are plotted in Figs. 1共b兲–1共d兲 for f LO = f 1 − nf 0⬘ wheren = 1 , 3 , 6 共other n not shown for figure clarity兲. The PUC
process is seen to depend on local oscillator amplitude ALO
关Fig. 1共e兲兴. In the down-conversion process, the second orderflexural mode is used in tapping and lever vibration is monitored at a frequency f ⴱ = f 1⬘ − f LO. Since the fundamental mode
spring constant k0 is smaller compared to the higher ordermode spring constant k1, resonant frequency shift of the fundamental mode is more pronounced 关Fig. 1共f兲兴. For resonant
detection of PDC signal, f LO was carefully chosen to coincide the PDC signal with the shifted fundamental mode resonance frequency f 0⬘ 关Fig. 1共g兲兴. However, nonresonant detection of PDC signal is also possible, at the expense ofincreased ALO 关Fig. 1共h兲兴. In all cases, in order to avoid
jumping between attractive and repulsive regimes, ALO must
be kept adequately small. For certain set of parameters, instabilities are observed for very large ALO 共⬎5 nm兲 and insome cases increased stability was observed with increasing
ALO, possibly due to dithering effects.17 The noise spectrum

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Abak et al.

FIG. 1. 共Color online兲 Excitation of the second order mode through parametric mode coupling as a function of tip-sample separation during tappingwith the fundamental mode in attractive regime. The sample is 100 nm thick
PMMA on silicon. 共a兲 Typical force-distance and amplitude-distance curves
acquired simultaneously with the PUC measurements. Second order mode
amplitude A1 when 共b兲 f LO = f 1 − f d, 共c兲 f LO = f 1 − 3f d, and 共d兲 f LO = f 1 − 6f d. 共e兲Parametrically up-converted amplitude A1 increases with increasing LO amplitude ALO. 共f兲 Perturbation of the dynamic properties of the fundamental
mode is observed in the noise spectrum of the cantilever deflection signal
when the second mode is tapping in attractive 共left arrow兲 and repulsive
modes 共right arrow兲 and away from the surface 共middle arrow兲. 共g兲 Parametrically down-converted signal at the shifted fundamental mode resonance frequency during tapping with higher order flexural mode. As ALO isincreased parametrically down-converted signal amplitude increases. 共h兲Parametrically down-converted signal to 30 KHz 共nonresonant兲.

of the phase signal was observed before measurements toensure stability.
In order to understand the parametric conversion of force
harmonics, we use numerical analysis. The cantilever dynamics can be modeled using lumped independent harmonic
oscillators for different flexural modes. The resonant frequency shift of the cantilever mode is given by18

␦ f共zts兲 =f2
kA冕

1/f

Fts关zts + Acos共2␲ ft兲兴cos共2␲ ft兲dt,共1兲0where Fts is the tip-sample interaction force, A is the amplitude of oscillation, k is the lumped spring constant, and f is
the resonant frequency. When the cantilever is driven by afixed amplitude sinusoidal force Fd共t兲 = Fdcos共2␲ f dt兲 at frequency f d, the complex oscillation amplitude Ã can be calculated throughÃ共␻兲 = h共␻兲Fd = ⌺n

1Fd,2
kn 共1 − ␻2/␻r,n兲 + i␻/␻r,nQn

共2兲

where ␻ = ␻d = 2␲ f d is the drive frequency, Fd is the driving
force amplitude, Qn is the effective quality factor, and␻r,n = 2␲共f n + ␦ f n兲 is the resonance frequency of the mode n
perturbed by the tip-sample interaction. The nonlinear tipsample interaction can be modeled using a piecewise
description.16 In this approach the interaction force F共zts兲 canbe described in terms of tip and sample elastic moduli and tip
radius as a function of tip-sample separation zts. For the sakeof simplicity the nonconservative effects are ignored. ForA0 ⯝ 10 nm and ALO ⬍ 1 nm, and assuming that the resulting
oscillations at frequencies other than f LO and f d ⬇ f 0 are also
smaller than 0.1 nm, the time-dependent tip-sample separation zts共t兲 can be approximated as

FIG. 2. 共Color online兲 Simulation results for PUC process in the repulsive
regime while fundamental mode is used for tapping and second order modeis used for detection. 共a兲 Amplitude vs zts for si and PS samples for Q0= 200 and 150. 共b兲 Phase vs A0 shows dissipation contrast. 共c兲 Parametrically
up-converted force amplitude calculated for ALO = 0.2 nm and f LO = f 1 − f ⬘0 vsA0 shows that contrast is dominated by elasticity of sample.

zts共t兲 = z0 + ALOcos共2␲ f LOt兲 + A0 cos共2␲ f dt兲,共3兲

where z0 is the average tip-sample separation. In order tocalculate A0 at a given tip-sample separation, Eqs. 共1兲 and 共2兲
are solved iteratively for Ã for a given set of parameters andthe proper solution is selected. In the analysis, we make theapproximation that ALO, A1 Ⰶ A0, and higher order parametric
processes are ignored 共i.e., parametric mixing happens onlyonce兲. Using Eq. 共3兲 and the interaction model, a timedependent interaction force F共t兲 can be calculated. The harmonic component of F共t兲 at frequency ␻ can be calculatedby F␻ = 兰F共t兲exp共−i␻t兲dt. The amplitude of oscillation of thecantilever can be then calculated through Eq. 共2兲. Using thedescribed approach, A0 versus z0 关Fig. 2共a兲兴 and parametrically converted force amplitude F␻1 versus A0 关Fig. 2共c兲兴 arecalculated where curves are plotted for silicon 共Si兲, and polystyrene 共PS兲 surfaces. PS is chosen as an example soft material. Since dissipative interactions were not explicitly included in the simulations, in order to observe the effect ofdissipation, the calculations are repeated with Q factors of200 and 150. In tapping mode, feedback fixes A0. Therefore,
to compare material contrast observed in different channels
for a given A0 set point, we label the equilibrium points as
1 ⬟ 共PS, Q = 200兲, 2 ⬟ 共PS, Q = 150兲, 3 ⬟ 共Si, Q = 200兲, and4 ⬟ 共Si, Q = 150兲, respectively. Conventional phase contrastof the tapping mode is seen to depend only on dissipation, as
seen in Fig. 2共b兲. However, parametrically converted forces关Fig. 2共c兲兴 show elasticity dependent contrast.
In order to demonstrate that the material dependent contrast can be imaged using the PUC/PDC processes, images
were obtained on polymethyl methacrylate 共PMMA兲 patterns

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Abak et al.

FIG. 3. 共Color online兲 Material contrast observed through PUC in the attractive regime on PMMA patterns on silicon. 共a兲 Phase contrast image. 共b兲
Parametrically up-converted force detected at the second order mode frequency when sample is oscillated at f LO ⬇ f 1 − f 0, 共c兲 f LO ⬇ f 1 − 2f 0, and 共d兲f LO ⬇ f 1 − 3f 0. Images acquired with 0.5 Hz per line, 5 ␮m image width. 共e兲
Material contrast can observed through PDC in the repulsive regime withplasmid DNA strands on silicon: Conventional phase 共full scale 30°兲 and 共f兲
PDC signal at 90 KHz. Double strand 共arrow兲, curled particles, and bundlesof DNA can be discerned. Softer material gives darker signature. Images
acquired with 1.5 Hz per line, 0.25 ␮m image width.

on silicon 关Fig. 3共a兲–3共d兲兴 and on plasmid DNA strands on
silicon 关Figs. 3共e兲 and 3共f兲兴. On large PMMA patterns on
silicon, in the attractive regime where there is little dissipation, PUC contrast is observed for the conversion of fundamental and third harmonics 关Figs. 3共b兲 and 3共d兲兴 even when
there is no significant phase contrast 关Fig. 3共a兲兴. The absenceof dissipation is also directly observed in the low contrast ofFig. 3共c兲, where the second harmonic of the interaction force
is monitored. In the attractive regime, simulations show reduced parametric generation. Therefore, in order to observe
the contrast, considerably larger 共about a nanometer兲 ALO
was used. In order to demonstrate applicability of the technique to biological molecules, a sample with DNA strands on
silicon was imaged using the second order flexural mode fortapping and using PDC. In this case, repulsive mode is easier
to obtain and elasticity dependent PDC contrast can be observed. The observed contrast is parallel with theoretical predictions of Fig. 2共c兲, where increasing sample modulus results in increased parametric conversion. It was alsoobserved that 共data not shown兲 increasing ALO or using amuch lower amplitude set point results in destruction of isolated DNA strands due to increased peak interaction forces.
Simulations show that for standard cantilevers, the interaction pressures are large enough to disturb the molecule’sstructural integrity. Therefore the technique is applicable formolecular imaging but care must be given to keep the peakforces low.
The method presented here can be applied using a standard rectangular cantilever and a standard AFM system witha vibrating sample holder or an AFM system that allows
access to the lever dither piezo by external electronics. In
multifrequency excitation through the cantilever dither piezo,
the oscillation amplitudes are determined by the lever transfer function, and imaging the fifth harmonic of the force wasseen to be most convenient using the PUC process due toenhanced intermediate frequency excitation. Elasticity dependent contrast on molecular level can be observed using astandard cantilever. Further optimization of the cantileverproperties can be done to operate the technique with lower
interaction forces. The technique presented here can be used
to study tip-sample interactions and nanoscale contact mechanics in greater detail than possible with a standard tapping mode AFM. The technique can also potentially be applied for label-free biomolecular detection using integrated
micro and nanomechanical devices.
This work was partially funded by Tübitak Grant No.
107T547 and by funding from the State Planning Agency ofthe Turkish Republic. Aykutlu Dâna thanks Özgür Şahin forenlightening discussions on tip-sample interactions.
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