REVIEW OF SCIENTIFIC INSTRUMENTS 78, 103707 共2007兲

Harnessing bifurcations in tapping-mode atomic force microscopy
to calibrate time-varying tip-sample force measurements
Ozgur Sahina兲
The Rowland Institute at Harvard, Harvard University, Cambridge, Massachusetts 02142, USA

共Received 28 August 2007; accepted 1 October 2007; published online 31 October 2007兲
Torsional harmonic cantilevers allow measurement of time-varying tip-sample forces intapping-mode atomic force microscopy. Accuracy of these force measurements is important forquantitative nanomechanical measurements. Here we demonstrate a method to convert the torsional
deflection signals into a calibrated force wave form with the use of nonlinear dynamical response of
the tapping cantilever. Specifically the transitions between steady oscillation regimes are used tocalibrate the torsional deflection signals. © 2007 American Institute of Physics.
关DOI: 10.1063/1.2801009兴
I. INTRODUCTION

Tapping-mode atomic force microscopy 共AFM兲 has enabled practical imaging of materials with nanoscale lateral
resolution.1,2 In this operation mode the force sensing cantilever vibrates at or near its resonance frequency with a large
enough amplitude to avoid sticking of the tip to the sample
and minimize lateral interaction. During the oscillations, attractive and repulsive forces act on the atomically sharp tip.
These forces depend on the material characteristics of thesample and they affect the oscillations of the tapping cantilever. As a result, the cantilever exhibits rich nonlinear dynamical behavior.3–7 Several groups have investigated thecantilever dynamics in the tapping mode in order to extract
information on the material properties. While most of theinitial work was on the use of vibration amplitude andphase,8–12 various authors are now exploring the use of highfrequency vibration harmonics13–18 and higher order resonances on the cantilever.19–21
Recently, torsional harmonic cantilevers are introduced
to measure time-varying tip-sample forces in dynamic force
microscopy.22 These cantilevers have tips that are offset fromtheir longitudinal axis, so that tip-sample forces excite torsional vibrations 共see Fig. 1兲. The sensitivity and high bandwidth of the torsional vibrations enable measurements of theattractive and repulsive forces and their variation with timeor tip-sample separation in the tapping mode. The accuracyof these measurements is crucial for quantitative nanomechanical measurements. Therefore a proper calibration of thetorsional response of the torsional harmonic cantilevers isnecessary. Here we propose a method to calibrate the torsional deflection signals of torsional harmonic cantilevers.
Specifically, we want to convert the electrical signals at theposition sensitive detector to the tip-sample forces.
II. THEORETICAL CONSIDERATIONS

In a tapping-mode experiment performed with a torsional harmonic cantilever, raw torsional deflection signala兲Author to whom correspondence should be addressed. Electronic mail:sahin@rowland.harvard.edu

0034-6748/2007/78共10兲/103707/4/$23.00

wave form is distorted by the torsional resonance.22 Thiseffect is most easily understood and corrected in the frequency domain, where the periodic torsional deflection signals and tip-sample forces are represented with harmonics
and the response of the cantilever is represented with a frequency response function.
Harmonics of the tip-sample forces come at integer multiples of the driving frequency. Each harmonic force component results in a harmonic vibration on the cantilever in proportion to the frequency response of the torsional deflections.
Considering the higher stiffness of higher order torsionalmodes, we approximate the torsional frequency responsewith a simple harmonic oscillator with a resonance frequency and quality factor equal to those of the fundamental
torsional mode. With this assumption, the transfer function
relating the detector signal in volts VT共␻兲 to the tip-sample
forces FTS共␻兲 can be written as follows:VT共␻兲 = coptical

␻T2dFTS共␻兲.
kT ␻T2 − ␻2 + i␻␻T/QT共1兲

Here the torsional resonance frequency and quality factor aredenoted as ␻T and QT. These two parameters are easily andaccurately measured in an AFM system. Torsion constant of
the first torsional mode kT is defined as the angular deflection
for a unit torque around the long axis of the lever. Torque isgenerated by the tip-sample forces acting at an offset distance d from the longitudinal axis of the lever. coptical is thedetector signal for a unit torsional deflection angle. We areneglecting the frequency response of the detector because thecutoff frequency is well above the harmonic frequencies ofinterest in our AFM system. VT共␻兲 is the Fourier transformof the detector signal vT共t兲. Equation 共1兲 can be used to solve
for FTS共␻兲. This calculation requires the constants used inEq. 共1兲 to be measured. An intermediate parameter that isuseful in this calculation and subsequent discussions is thecorrected voltage wave form VTC, which is defined as78, 103707-1© 2007 American Institute of Physics

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Rev. Sci. Instrum. 78, 103707 共2007兲

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TABLE I. Spring constant K1, resonance frequency f 0, torsional resonancefrequency f T, length L, width w, and tip offset distance d of three torsional
harmonic cantilevers.

12
3

K1
共N/m兲

f0
共kHz兲

fT共kHz兲

L
共␮m兲

w
共␮m兲d
共␮m兲

2.26
6.18
7.97

47.3
74.7
59.1

809.0
1127.01115.7

300
275
38030
3030

152522III. EXPERIMENTAL RESULTS
FIG. 1. Schematic 共left兲 and SEM picture 共bottom right兲 of a torsional
harmonic cantilever. Torsional vibrations excited due to tapping allow us tomeasure time-varying tip-sample forces 共top right, curve measured ongraphite兲.

VTC共␻兲 =␻T2 − ␻2 + i␻␻T/QT
␻T2VT共␻兲 = copticaldFTS共␻兲.
kT共2兲

Corrected voltage wave form VTC共␻兲 can be directly calculated from the measured detector voltages with the knowledge of ␻T and QT. In time domain VTC has the same wave
form, within a scalar factor, as FTS. However, it remains inthe electrical units 共volts兲. Evaluation of this scalar factor issufficient to calibrate the torsional response of torsional harmonic cantilevers.
The method of calibration presented in this letter is enabled by the fact that both torsional and vertical mechanicaldetection channels respond to the same time-average 共dc兲forces in proportion to their effective spring constants. Oncethe vertical spring constant is calibrated with establishedtechniques and the time-average force FTS共␻ = 0兲 is quantitatively measured by the vertical deflections, the corresponding average torsional deflection signal VTC共␻ = 0兲 will beused to determine the scalar factor in Eq. 共2兲.
In practice, dc deflection measurements of a cantilever
are affected by mechanical and thermal drifts in both thecantilever and detector positions. This fact complicates theuse of dc measurements in both vertical and lateral channels.
However, nonlinear dynamical response of the cantilever vibrations in tapping mode provides an opportunity to eliminate drift and perform reliable calibration, as we explainnext.
The tapping cantilever exhibits multiple steady oscillation states depending on the drive amplitude and frequency.23During the transitions between different steady oscillation
regimes 共such as attractive and repulsive regimes兲, timeaverage forces quickly jump from a negative 共attractive兲
value to a positive 共repulsive兲 value. The switching time islimited by the resonance frequency and quality factor of thecantilever and it is in the millisecond time scale. Therefore,
the difference between the dc deflection signals before andafter a transition will be a drift-free quantity. Instead of comparing the absolute values of the dc signals in the verticaland lateral channels, we are going to compare the changes inthe dc values before and after an oscillatory state transition.

To demonstrate this calibration scheme we worked withthree torsional harmonic cantilevers. Their vertical spring
constants, vertical and torsional resonance frequencies,
nominal width, length, and tip offset distances are given in
Table I. Among these parameters, we are going to use torsional resonance frequencies and vertical spring constants in
our calibration method. Torsional resonance frequencies areused to calculate VTC by Eq. 共2兲, and vertical spring constants are used to measure time-averaged tip-sample forcesfrom vertical deflections of each cantilever. Torsional resonance frequencies can be measured accurately by obtaining
frequency tuning curves with the lateral photodetector signal
共note that vertical piezodrive can actuate torsional resonances due to minor asymmetries兲; however, vertical spring
constants require careful calibration. Various methods forcalibrating the vertical spring constants are reported in theliterature.24–26 We have used the method based on Ref. 25that calibrates the cantilever spring constant against thermomechanical noise, which is provided by the commercial software running the AFM. This calibration method requires theknowledge of cantilever deflection sensitivity 共V/nm兲 andthat depends on the laser spot position along the length of thecantilever. Deflection sensitivities for each cantilever are obtained from the slopes of vibration amplitude versus verticaldistance curves 共commonly referred as tapping-mode force
curves兲. Then, the thermal noise spectrum around the fundamental vertical resonance frequency of each cantilever isused to determine the vertical spring constant.
For the calibration of time-varying tip-sample forces, wehave obtained amplitude versus distance and phase versus
distance curves with each cantilever on graphite. The cantilevers are driven slightly above their resonance frequencies,
where the free amplitudes drop to approximately 60% of
the resonant value. This condition favors the existence of
two oscillation states.23 Multiple cycles of amplitude versus
distance and phase versus distance curves are plotted inFigs. 2共a兲 and 2共b兲 with results from each cantilever on adifferent column. At each data point vertical dc deflectionsof the cantilevers 共generally referred as tapping mode deflection signals兲 are recorded and converted into force units
关Fig. 2共c兲兴 by multiplying with the corresponding verticalspring constants. Note that there is an offset in the force
values due to detector misalignment. The corresponding torsional deflection signals at each point are recorded and thecorrected voltage wave form VTC is obtained. Time-averagevalues of VTC are plotted in Fig. 2共d兲.
Jumps in the phase-distance curves show the points

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103707-3

Rev. Sci. Instrum. 78, 103707 共2007兲

Calibrating time-varying forces in AFM

FIG. 2. 共Color online兲 Amplitude vs distance 共a兲, phase vs distance 共b兲, average vertical force vs distance 共c兲, and average lateral deflection signal vs. distance共d兲 curves in a tapping-mode experiment performed with three cantilevers. The horizontal axis is arbitrarily referenced.

where cantilever oscillations switch from one steady state tothe other. Note that both the averages of vertical forces in2共c兲 and the corrected voltage wave forms in 2共d兲 exhibit
jumps at the corresponding data points. A computer programis used to analyze the data and locate the jumps and record
their magnitudes in newtons for 2共c兲 and in volts for 2共d兲during multiple ramp cycles. The ratio of the jump magnitudes in 2共c兲 and 2共d兲 correspond to the inverse of the scalar
factor in the second part of Eq. 共2兲.
Table II gives the average values and standard deviationsof the scaling factors for each cantilever obtained from thedata in Fig. 2. After calculating VTC from the detector signal
and taking its inverse Fourier transform, multiplication withthe scaling factor will give the calibrated time-varying tipsample force wave form.
IV. DISCUSSION

Several calibration methods for the torsional deflection
signals of AFM cantilevers have been proposed and used forquantifying lateral force microscopy measurements.27–32
Those methods that involve lateral tip-sample interactions inTABLE II. Scaling factors for the calibration of the three torsional harmonic
cantilevers. Standard deviations of the measurements on each cantilever are
given in parentheses. Scaling factors calibrated against thermomechanicalnoise are given on the right column.

123

Scaling factor 共nN/mV兲
and standard deviation

Scaling factor
共thermomechanical兲
共nN/mV兲

2.84 共5.0%兲
3.61 共3.3%兲
10.8 共4.5%兲

3.38
5.64
11.8

contact mode cannot be used directly for the calibration oftorsional harmonic cantilevers because vertical forces arealso generating torque due to the offset tip. More importantly, the primary quantity that is being measured by thetorsional harmonic cantilevers is the vertical tip sampleforce, not the lateral force. On the other hand, the torsion
constant kT can be calibrated against the thermal noise. Then,according to Eq. 共2兲, the scaling factor can be obtained byestimating coptical and d. Following this approach we calibrated kT of each cantilever against thermal noise, usedcoptical value calculated from the vertical mode force curves
共we assume photodetector gains are the same for both lateraland vertical channels兲, and use d values in Table I to estimate
the scaling factor in Eq. 共2兲. The resulting scaling factors aregiven in Table II.
The values of the scaling factor directly calculated withthe use of oscillatory state jumps and thermal noise based
estimates differ as much as 50%. We believe the calibration
method presented in this letter is more accurate compared toa calibration against thermal noise. First of all, it does not
require the measurements of intermediate variables kT,
coptical, and d. coptical and d contain uncertainties 共coptical depends on the laser spot position and d is subject to misalignments in the manufacturing process兲. Misalignments in thetip positioning can be as much as 2 ␮m in the conventional
cantilever manufacturing process. Considering a nominal dof 20 ␮m, this will introduce considerable uncertainty into
the thermal noise based calibration. Furthermore, determination of coptical is not as easy as it is for the vertical deflections,where the slopes of tapping-mode force curves provide
coptical values. Yet, there is a possibility to use an analogues
method involving lateral force versus lateral displacement
curves.28 Second, thermal noise based measurement of kT

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Rev. Sci. Instrum. 78, 103707 共2007兲

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values is less accurate for stiff torsional modes of tappingmode cantilevers because the rms deviation in cantilever position is only slightly above the detector noise. In ourmethod, the force jumps used for calibration are typicallyaround 5 nN, which leads to detectible changes in the torsional deflections. The uncertainty of the calibration method
presented here mainly comes from the measurement of thevertical spring constant of the torsional harmonic cantilever.
This value can be obtained more accurately 共typically within
10%兲 with the existing calibration methods.24–26
In summary we have presented a method to calibratetime-varying tip sample forces measured by torsional harmonic cantilevers operated in the tapping mode. The method
uses the fact that both vertical and lateral deflections of
the torsional harmonic cantilevers respond to the same vertical tip-sample force. After calibrating the vertical deflection
signals with established techniques, we calibrate lateraldeflection signals by comparing the time-average 共dc兲 deflection signals in both channels. We eliminate drift related
errors by using differential dc measurements near thetransitions between steady oscillation states of the tapping
cantilever.
ACKNOWLEDGMENTS

This work is supported by the Rowland Junior Fellows
program. The author thanks to Bede Pittenger and David M.
Rhodes from Veeco Instruments for kindly providing thecantilevers used in this study.
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