APPLIED PHYSICS LETTERS 94, 063109 共2009兲

Dynamics of repulsive dual-frequency atomic force microscopy
Robert W. Starka兲
Department of Earth and Environmental Sciences and Center for NanoScience (CeNS),
Ludwig-Maximilians-Universität München, Theresienstr.41, Munich 80333, Germany

共Received 1 October 2008; accepted 21 January 2009; published online 10 February 2009兲
In bimodal atomic force microscopy, two flexural modes are driven at their resonances. The
oscillation of the second eigenmode, which is usually an incommensurate multiple of thefundamental frequency, perturbs the dynamic system. Numerical simulations show that the tip
motion is almost periodic at typical set points and that harmonics and intermodulation frequencies
prevail in the spectrum. The simulations also predict a very small increase in the noise of the firstmode amplitude and phase due to the second mode oscillation. At small average tip sample
separations, however, phases with repulsive and purely attractive forces can occur intermittently.
© 2009 American Institute of Physics. 关DOI: 10.1063/1.3080209兴

Fts共z兲 =再− HR/关6共zs + z兲2兴− HR/6a20 + 34 EⴱD ⱖ a0 ,冑R共a0 − zs − z兲3/2 D ⬍ a0 ,

冎where H is the Hamaker constant, R is the radius of the tip,parameter Eⴱ is the effective contact stiffness, and a0 is aninteratomic distance.
The simulation was implemented with MATLAB 2008aand SIMULINK 7.1 on a Windows PC 共Intel Core 2 Duo兲. The
cantilever model was truncated after three eigenmodes. Cantilever parameters were as follows: resonant frequencies ␻1= 1.0, ␻2 = 6.2669, ␻3 = 17.5475; quality factors Q1 = 200,mode 1

14 decreasing
A1 :A 2124mode 2

3*

1028

#

#

641

2045*0**0#Electronic mail: stark@lmu.de. Tel./FAX: ⫹49 89 2180-4329/-4334.

0003-6951/2009/94共6兲/063109/3/$25.00

-45

共1兲

Matrix A describes the cantilever dynamics 共resonant frequencies and damping兲, vector b the coupling of the forces uto the individual modes. Matrix C accounts for the couplingof the system states x to the output y 共tip position, deflection
angle, and velocity兲.9,10a兲

45

0

#

amplitude (nm)amplitude (nm)16-45y = Cx.

共2兲phase (°)

ẋ = Ax + bu,

The tip sample interaction was modeled as a nonlinear
feedback. The force law was a Derjaguin-Muller-Toporov
model withphase (°)

In a bimodal atomic force microscope 共AFM兲, two flexural modes are simultaneously excited.1–3 The amplitude of
the fundamental mode oscillation is used for topography
feedback, while amplitude and phase of the higher mode
vibration encode the material composition. The excitation ofan additional mode enhances the material specific sensitivity
of dynamic AFM.2–6 Such an enhancement occurs in theattractive1,6 and in the repulsive regime.2,4 The excitation at anoninteger or even nonrational multiple ␻2 of the fundamental frequency ␻1, however, introduces a perturbation to thedynamic system because the external driving force Fd= F1 sin共␻1兲 + F2 sin共␻2兲 is not periodic—or at least the period is very long. In the attractive regime, this perturbation isvery small and high contrast and resolution can be achieved
at low noise.3,6 In the repulsive regime, the situation may bedifferent.
In conventional single frequency amplitude modulation,
AFM gentle imaging conditions are associated with a stable
periodic oscillation state.7,8 Irregular 共chaotic兲 oscillations
can lead to large repulsive tip sample forces.9 Thus, in orderto achieve high resolution and high contrast bimodal AFM in
the repulsive regime, the AFM operator needs a guide for theselection of operation parameters 共amplitude ratio and set
point兲.
Numerical simulations are a convenient way to investigate the influence of operation parameters on the system dynamics because the repulsive tip-sample forces in bimodal
AFM are not accessible experimentally and analytic approximations are not yet available. Here, the cantilever was modeled as a linear and time invariant system. The multiple degree of freedom system was simulated by integrating thestate-space equations-90
6810 12 14 16 6position (nm)810 12 14 16position (nm)-90

FIG. 1. 共Color online兲 Amplitude and phase curves of both modes for theamplitude ratios A1 : A2 = 50, 20, 15, 12, 10, 6, and 4. The curves with
A1 : A2 = 50 共blue兲 and A1 : A2 = 10 共red兲 are marked by * and #, respectively.
The black arrow indicates the direction of decreasing A1 : A2.

94, 063109-1© 2009 American Institute of Physics

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Appl. Phys. Lett. 94, 063109 共2009兲

Robert W. Stark

FFT amplitude

063109-2

100##A 1:A2 = 15

#

*-2

10

#n

#

*

#n #

*

#n #

# n

4

5

#*

#nn-4

100
FFT amplitude*012
A1 :A 2= 50

3A1 :A 2= 10

678

A1 :A 2= 210-210-41001020 30 0 10 20 30 0 10
normalized frequency

20 30FIG. 3. 共Color online兲 FFT spectra at Â1 / A1 = 80% for various amplitude
ratios. For A1 : A2 = 15 the harmonics of the driving frequency 共n兲 andintermodulation products are indicated. Frequencies following the rule
␻ = ⫾ ␻2 ⫾ n␻1 are marked by “#” and those according to ␻ = ⫾ 2␻2 ⫾ n␻1by “*.”

FIG. 2. 共Color online兲 共a兲 Interaction forces during approach for A1 + A2
= 15 nm 共red curve # in Fig. 1兲. The frames for both time series are indicated. 共b兲 Time series data in a quasiperiodic regime. A period of 15 oscillations prevails for Â1 / A1 = 80%. 共c兲 Time series obtained in the intermittent
regime. Phases of noncontact and repulsive contact occur intermittently. Theinsets in 共b兲 and 共c兲 show typical force pulses.

Q2 = 400, Q3 = 400, tip radius R = 20 nm, spring constant k= 10 N / m. Material parameters were Eⴱ = 156 GPa and a0= 0.16 nm. The system was integrated with the variable stepsolvers ode15s and ode45 in MATLAB/SIMULINK in the “rapidacceleration” mode. Both solvers yielded similar results. Themaximum integration step width was set to t = 2␲ / 1280.
“Zero crossing” detection was enabled to precisely calculate
the transition at a0 in Eq. 共2兲.
The simulations focused on the parameter range used for
imaging in bimodal AFM.2,4,6,11 The excitation frequencies
were set to ␻dr共1兲 = ␻1, ␻dr共2兲 = ␻2. In order to compare simulations for different amplitude ratios, the composite free amplitude was set to A0 = A1 + A2 = 15 nm. The second mode
amplitude was kept smaller than the first one 共A1 ⬎ A2兲. In
the following, the hat denotes values under tip sample interaction. Thus, Âi is the average oscillation amplitude with tip
sample interaction and Ai the free amplitude without. Amplitudes and phases were calculated by a sliding fast Fouriertransform 共FFT兲 with a rectangular window 共128 cycles at ␻1
width and 50% overlap兲.
Figure 1 shows approach curves for different amplitude
ratios A1 : A2. The shape of the amplitude curve 共first mode兲was only weakly affected by the second mode excitation.
The slope of the amplitude response of the second eigen-mode, however, varied during approach. Without secondmode excitation, the system remained in the noncontact regime, also referred to as attractive regime or low amplitude
solution 共data not shown兲.7,8 For A1 : A2 ⬎ 50: 1 共blue, ⴱ in
Fig. 1兲 the response changed at 13.4 nm. The system transited from a regime with purely attractive to a regime withrepulsive interaction forces. The transition was induced by
the perturbation caused by the higher mode excitation. This
result confirms the experimental observation of a stabilized
repulsive regime in bimodal AFM.12
The amplitude of the second mode dropped significantly
during the transition and the second mode phase became
insensitive to the approach. For decreasing ratios the amplitude Â2 was less affected. The amplitude and phase curves
for A1 : A2 = 10 共red, # in Fig. 1兲 responded continuously tothe variation in the tip sample contact. At separations smaller
than z = 6 nm the amplitude and phase curves became very
noisy.
The general shape of the amplitude and phase response
can be understood by comparing the energy stored in bothmodes. For large amplitude ratios 共e.g., A1 : A2 = 50: 1兲, theenergy ratio is E1 / E2 = ␻12A12 / ␻22A22 Ⰷ 1. Much more energy is stored in the first than in the second mode. Thus, theresponse of the first mode dominates. For ratios E1 / E2 ⬍ 1共e.g., A1 : A2 = 4 : 1兲, however, the second mode contains mostof the energy. At intermediate amplitude ratio of A1 : A2⬇ 10: 1, the energy stored in both modes is of the same orderof magnitude leading to a similar phase response at bothmodes.
Figure 2共a兲 displays the interaction forces during approach for A1 : A2 = 10. The peak forces were in the order of
100 nN in the regime dominated by the first mode. At aposition of z ⬍ 7 nm and less, the repulsive peak forces increased to a maximum of 500 nN. At z = 7 nm, the amplitudes also dropped to Â1 = 6.4 nm and Â2 = 0.95 nm. At thispoint, the energy stored in both modes was comparable,
since Ê1 / Ê2 = ␻12Â12 / ␻22Â22 ⬇ 1. Figure 2共b兲 shows indi-Downloaded 19 Mar 2009 to 133.28.47.30. Redistribution subject to AIP license or copyright; see http://apl.aip.org/apl/copyright.jsp

063109-3

Appl. Phys. Lett. 94, 063109 共2009兲

Robert W. StarkTABLE I. Mean values and standard deviation 共brackets兲 of amplitude and phase values. Last line: A1= 15 nm, A2 = 7.5 nm.
A1 : A2A1 = 15 nm
50:1
10:1
5:1
2:1
2:1a

具Â1典 共nm兲

具Â2典 共nm兲

具␾1典 共deg兲

具␾2典 共deg兲

12.0共9.1e − 8兲
11.8共2.2e − 5兲
10.9共2.4e − 4兲
10.0共4.8e − 4兲
8.0共9.7e − 4兲
12.0共9.7e − 4兲¯
0.05共7.2e − 6兲
1.8共2.2e − 4兲
3.4共5.1e − 4兲
7.0共8.0e − 4兲
6.9共4.0e − 4兲

+36.2共9e − 12兲⫺36.7共0.001兲
⫺35.6共0.006兲
⫺35.8共0.009兲
⫺36.0共0.016兲
⫺36.1共0.010兲

¯⫺83.6共0.013兲
⫺19.27共0.009兲
⫺9.24共0.007兲
⫺3.7共0.005兲
⫺5.4共0.005兲

aTotal amplitude A1 + A2 = 15.

vidual force pulses at a set-point of 80%. After 15 cycles theforce pattern is almost repeated, which indicates a quasiperiodic regime. This means that the oscillation is almost periodic microscopically, but may be not periodic macroscopically. Within a period, the maximum peak force was highlyvariable.
At smaller positions z, the second mode dominates thetip sample interaction since Ê2 ⬎ Ê1. No periodic response
could be identified and the interaction force followed an irregular pattern 关Fig. 2共c兲兴. Time periods with repulsive as
well as purely attractive forces occurred intermittently. The
inset shows individual force pulses 共repulsive and attractive兲.
The intermittent dynamics encountered here also explains theinstable phase signal shown in Fig. 1. Note that the simulations neglect energy dissipation in the tip sample contact,which may play an important role in the stabilization of thequasiperiodic regime.
In Fig. 3, FFT spectra of the tip displacement for an
amplitude set-point of Â1 / A1 = 80% are given for various amplitude ratios. Harmonics and intermodulation products prevail in the spectrum 关Fig. 3共a兲兴. Such intermodulation frequencies occur due to the nonlinear tip sample contact.13,14
The harmonics of the fundamental frequency ␻ = n␻1 are indicated by the letter n. Since the harmonics were very prominent in the spectrum; also intermodulation products of very
high order were present. Figure 3共b兲 illustrates the effect ofincreasing second mode excitation. At A1 : A2 = 50, the spectrum shows well-defined harmonics and intermodulation
products. By increasing the amplitude ratio, their peaks
broaden. This broadening is due the quasiperiodic oscillation, where the instantaneous partial frequencies are only
nearly harmonic frequencies. With the amplitude ratio increasing to A1 : A2 = 2 the instantaneous frequencies of theharmonics peaks were severely detuned from their periodic
values and the distinct harmonic and intermodulation peaks
vanished. This effect implies that the trajectory was nolonger periodic macroscopically.
Experimentally, the excitation of an additional mode also
might imply additional noise. In order to evaluate the noisecaused by the second mode, a line scan on a flat surface wassimulated for the set-point Â1 / A1 = 80%. A sliding window
FFT 共window size: 256 cycles, without overlap兲 was used tocalculate amplitudes and phase 共Table I兲. The standard deviation of Â1 increased by several orders of magnitude 共from
9 ⫻ 10−8 to 2 ⫻ 10−5 nm兲 due to the second mode excitation
共A1 : A2 = 50兲. The standard deviation increased further to1 ⫻ 10−3 nm for A1 : A2 = 2. Such a subpicometer noise, however, will be irrelevant in practical measurements since typical measurement noise is larger. Nevertheless, these amplitude fluctuations 共process noise兲 are responsible for thestabilization of the repulsive regime. The second mode amplitude is less sensitive. At large A2 共A1 : A2 = 2, A1 = 15 nm,
A2 = 7.5 nm兲 the amplitude noise of the second mode is reduced. At this amplitude ratio 共A1 : A2 = 2, E1 ⬍ E2兲 the firstmode responded to the decreasing distance 共Â1 / A1 = 80%兲,while Â2 only slightly varied 共Â2 / A2 = 92%兲.
In summary, the simulations predict that quasiperiodic
and intermittent regimes can exist in bimodal AFM depending on the operating conditions. In the intermittent regime,
oscillation cycles with repulsive and purely attractive interaction forces occur intermittently. This type of dynamics ispredicted for small amplitude ratios A1 : A2 or small setpoints. The simulations also show that in the quasiperiodic
regime stable imaging is possible because the additional amplitude noise caused by the second mode excitation is verysmall. Although amplitude fluctuations are small, the peakforces can vary considerably. The amplitude ratio is a key
parameter for the amplitude and phase response. In the caseof A1 Ⰷ A2, the dynamics is dominated by the first mode response. For imaging in the noncontact regime, the amplitude
ratio thus should be adjusted to A1 Ⰷ A2 in order to obtain
sensitive and stable imaging. In the repulsive regime, thesensitivity of the second eigenmode phase can be increased
by increasing A2. The operator will have to balance sensitivity and strength of interaction forces.
Financial support by the DFG excellence cluster “Nanosystems Initiative Munich 共NIM兲” and the European Commission 共Grant No. NMP4-CT-2004-013684兲 is gratefully
acknowledged.
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