ARTICLESAn atomic force microscope tip
designed to measure time-varying
nanomechanical forces
OZGUR SAHIN1*, SERGEI MAGONOV2†, CHANMIN SU2, CALVIN F. QUATE3 AND OLAV SOLGAARD3
1

Rowland Institute at Harvard, Cambridge, Massachusetts 02142, USA
Veeco Instruments, Santa Barbara, California 93117, USA
3
E. L. Ginzton Laboratory, Stanford University, Stanford, California 94305, USA†
Present address: Agilent Technologies, 4330 W. Chandler Boulevard, Chandler, Arizona 85226, USA*e-mail: sahin@rowland.harvard.edu
2

Published online: 29 July 2007; doi:10.1038/nnano.2007.226

Tapping-mode atomic force microscopy (AFM), in which the vibrating tip periodically approaches, interacts and retracts from thesample surface, is the most common AFM imaging method. The tip experiences attractive and repulsive forces that depend on
the chemical and mechanical properties of the sample, yet conventional AFM tips are limited in their ability to resolve thesetime-varying forces. We have created a specially designed cantilever tip that allows these interaction forces to be measured withgood (sub-microsecond) temporal resolution and material properties to be determined and mapped in detail with nanoscalespatial resolution. Mechanical measurements based on these force waveforms are provided at a rate of 4 kHz. The forces andcontact areas encountered in these measurements are orders of magnitude smaller than conventional indentation and AFM-based
indentation techniques that typically provide data rates around 1 Hz. We use this tool to quantify and map nanomechanicalchanges in a binary polymer blend in the vicinity of its glass transition.
Phase changes and chemical compositional variations in materials
on the nanoscale have been studied using various scanning force
microscopy techniques. In these techniques, a force-sensing
cantilever with a sharp tip is placed in continuous contact withthe sample surface. Dynamical properties of the cantilever are
adjusted depending on the material in contact with the tip.
Examples of these techniques include ultrasonic force
microscopy1, force modulation microscopy2, shear modulation
force microscopy and lateral force microscopy3,4. These
techniques typically provide a one-dimensional signal, such as achange in vibration amplitude or deflection, although the tip –sample interaction is affected by multiple parameters. As aresult, extensive modelling and many assumptions are required
to interpret the measurements. A drawback of these techniques
occurs when operating the contact mode under large loading
forces, where tip and sample damage are more likely to occur.
This restricts the range of materials that can be studied andlimits the spatial resolution of the measurements. Force curve
methods in contact-mode and nanoindentation techniques
based on atomic force microscopy (AFM) provide more reliable
stiffness measurements5,6, but low operation speeds (1 s per
measurement) and large loading forces limit their use.
Tapping-mode AFM has become the most widely used imaging
method because the tip’s interaction with the sample surface isrelatively gentle, substantially eliminating plastic deformation andreducing the sample volume involved in the tip–sample
interaction7,8. However, it has been difficult to extract material
properties in the tapping mode. In this article, we present a

specially designed AFM cantilever tip, the torsional harmonic
cantilever (THC), which overcomes this difficulty.
In tapping-mode AFM, the cantilever is excited to vibrate at its
resonant frequency and is brought close to the sample. Intermittent
contact with the surface alters the amplitude and phase of thecantilever vibration. The vibrations are detected with an opticalsystem where a laser beam reflects from the back of the cantilever
and then falls onto a position-sensitive photodetector (Fig. 1).
The laser spot moves up and down on the photodetector as thecantilever vibrates. A feedback mechanism adjusts the height ofthe cantilever base so that the vibration amplitude is maintained
at a set-point value slightly below the free amplitude. As thecantilever is scanned across the surface, the feedback signal isused to map the topography of the sample surface.
The tapping cantilever vibrates approximately in a sinusoidal
trajectory. The amplitude and phase of this trajectory are the two
primary observables. Unfortunately, these quantities only provide
time-averaged values of the tip – sample interaction forces9,10.
Therefore, the information about how the interaction force
varies as the tip undergoes its trajectory is lost to a large extent.
The use of higher harmonic vibrations excited by the tip –sample forces would provide useful information11–16. There are,however, two major difficulties in this approach. First, thesignal-to-noise ratios of the higher harmonic vibrations are notsufficient for practical measurements17. Second, the overallfrequency response of the cantilever, required for translating
harmonic vibration signals into harmonic forces acting on thetip, depends on the shape of the cantilever mode and thenature nanotechnology | VOL 2 | AUGUST 2007 | www.nature.com/naturenanotechnology© 2007 Nature Publishing Group

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Torsional
FlexuralDetector signal (V)Lock-in amplifier

0.8
0.4
Oscilloscope or
A/D converter

0.0
–0.4
–0.8
0 2 4 6 8 10 12 14 16 18 20
Time (µs)

Laser

Quadrant
photodetector

Piezo element

the framework used by M. Stark and colleagues, which treats thetip –sample interaction as a feedback force on cantileverdisplacement19. This allows the tapping cantilever to be
represented by a linear system (equivalent to the response of a
cantilever fixed at the base and free near the tip). This cantileveris driven by two external forces: tip–sample forces and thedriving force.
In the following analysis, the cantilever is represented as a linear
time-invariant system with a transfer function G that relates thecantilever displacement C(x, y, t) to the total force acting on it,f(x, y, t) ¼ fdrive þ ftip, hence C(x, y, t) ¼ Gf(x, y, t). The opticaldetection system (quadrant photodetector) is represented withthe operator C, which relates the vertical and horizontal detector
signals sx and sy to the cantilever displacement by sx ¼ CxC andsy ¼ CyC. We note that the photodetector signal is directly
proportional to the slope of the cantilever, therefore Cx ¼ @/@x
and Cy ¼ @/@y, evaluated at the location of the laser spot. With
these definitions, the relationship between the detector signals
and the forces on the cantilever can be written ass ¼ CGð fdrive þ ftip ÞSampleFigure 1 Schematic of the tapping-mode atomic force microscope and theexperimental set up. The cantilever is vibrated with a piezo element at theresonance frequency. The up – down (flexural) and twisting (torsional) vibrationsof the cantilever are detected by a quadrant photodetector that is sensitive tothe position of the laser spot reflecting from the back side of the cantilever.
The blue curve in the oscilloscope indicates the flexural oscillation and theorange curve indicates the torsional oscillation.

position of the laser spot18. It is very difficult to determine all ofthese parameters accurately. The THC enhances the signals at
higher harmonics and provides a frequency response that can be
accurately determined.

ð1Þ

where s is a vector with two components representing the verticaland horizontal signals of the detector. In principle, matrices G andC can be separated into flexural and torsional components. This isequivalent to saying that the flexural modes only result in verticalsignals in the detector, and the torsional modes only horizontal
signals. (Note that the actual experimental conditions slightly
deviate from these two conditions due to the crosstalk from large
vertical deflections coupling into lateral signals.) Furthermore, thedriving force will only excite the flexural modes. We assume we
can neglect the effect of in-plane forces between the tip and thesample for the torsional vibrations. In the experiments we scan the
cantilever along the longitudinal axis so that in-plane forces do
not generate torque around that axis. Also, the lateral tip motion
due to torsional oscillations is estimated to be below 1 Å,considerably smaller than the typical sample indentation depths.
Based on these assumptions we can writesX ¼ CX GX ð fdrive þ ftip Þ ¼ HX ð fdrive þ ftip Þ

ð2Þ

sY ¼ CY GY ftip ¼ Hy ftip

ð3Þ

and

THE TORSIONAL HARMONIC CANTILEVER
The THC has a tip that is offset from the long axis of the cantilever(Fig. 2a). When it is vibrated in the tapping mode, tip–sample
interaction forces generate a torque around the long axis of thecantilever and excite the torsional modes (Fig. 2b,d). Torsional
deflections move the laser spot horizontally on the photodetector(Fig. 2c). As in the case of the conventional tapping-mode AFM,
flexural (vertical) vibration signals are used for amplitude
feedback to follow topography. Simultaneously, torsional
vibration signals are used for the calculation of the time-resolved
tip– sample interaction forces. The advantage of the use of
torsional vibrations is explained in the next section, where weanalyse the dynamics of the THC in the tapping mode.

The operators GX and GY represent the combined response offlexural and torsional modes, respectively. Note that equation (3)only deals with ftip, because the torsional vibrations are only
excited by the tip– sample forces. This equation can be used toobtain ftip from sY by estimating the transfer function
corresponding to HY and its inverse. The response of torsionalmodes GY can be estimated with a simple damped harmonic
oscillator that has a resonance frequency of the first torsionalmode and the corresponding quality factor. The operator CY isestimated with a scalar multiplier coptical that corresponds to thebending angle of the cantilever for a unit tip displacement in thetorsional mode. The resulting transfer function HY(v) is given by

DYNAMICS OF THE TORSIONAL HARMONIC CANTILEVER
Analysis of high-frequency torsional and flexural vibrations of thetapping cantilever requires modelling the cantilever as a continuum
mechanical element with multiple vibration modes. We will follow
508

Hy ðvÞ ¼ coptical

v2T =KT;2vT  v2 þ ivvT =QT

ð4Þ

where v is the angular frequency, vT is the torsional resonancefrequency, and QT and KT are the quality factor and effective
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Torque

Q1

Q2

Q3

Q4

Figure 2 Design of the torsional harmonic cantilever with an off-axis tip. a, A scanning electron micrograph image of a torsional harmonic cantilever.
The cantilever is 300 mm long, 3 mm thick and nominally 30 mm wide (50 mm near the free end). The tip is offset 15 mm from the centre of the cantilever.
b, An illustration of the THC interacting with the surface. The offset position of the tip results in a torque around the axis of the cantilever. c, An illustration of
the laser spot on the four-quadrant position-sensitive photodetector. The optical power difference (Q1 þ Q2) 2 (Q3 þ Q4) is proportional to vertical cantileverdeflection, and (Q1 þ Q3) 2 (Q2 þ Q4) is proportional to torsional angle. d, Simulated first torsional mode shape of a THC fixed at the base. This mode isexcited when the tip hits the surface.

spring constant of the torsional resonance. Once this transfer
function is calibrated and sY is sampled with a data acquisition
card, equation (3) can be numerically solved to obtain ftip. Detailsof the numerical calculations and the calibration of the parametersthat determine HY are described in the Methods section.
The two important parameters describing the transfer function
HY are the gain and the bandwidth. The gain is determined bycoptical/KT and the bandwidth is limited by vT. Larger gain
enhances sensitivity, and larger bandwidth provides better
time resolution.
Calculated transfer functions corresponding to HX(v) andHY(v) and experimentally measured vibration spectra of a THC
tapping on a polystyrene sample are plotted in Fig. 3 todemonstrate the advantages of the torsional vibrations. The
flexural frequency response shows multiple resonances
corresponding to the fundamental and higher-order modes atfrequencies up to 20 Hz. (The dips and zeros in the frequency
response arise due to cancellation of the neighbouring modes.) In
the same frequency range, there is only one torsional resonancefrequency. These curves show that the torsional vibrations provide
a larger response (sensitivity) over a wide range of frequencies(bandwidth). The improved sensitivity is due to the larger coptical.
In particular, because the moment arm in the torsional direction ismuch shorter than the length of the cantilever, these vibrations
result in larger angular deflections and have a lower effective
spring constant KT than the higher-order flexural modes.
The vibration spectra measured while tapping on a polystyrene
sample (Fig. 3b,c) show that there is a dominant peak at the drive

frequency (51.2 kHz) in the flexural spectrum. The otherpeaks are the higher harmonics that are excited by the periodic
tip –sample forces, but beyond 300 kHz these harmonics have
low signal-to-noise ratios. This illustrates the drawback of the useof flexural harmonics for the recovery of time-resolved tip–
sample forces. In contrast, the torsional vibration spectrum
shows peaks with good signal levels at the higher harmonics up
to 1 MHz. Note that there is also a good degree of agreement
between the calculated frequency response and the measured
torsional spectrum. These results show that the torsional
vibrations of a THC recover the high-frequency force
components necessary to reconstruct time-resolved tip –sample forces.

RECONSTRUCTING THE TIP –SAMPLE FORCE WAVEFORM
We recorded the flexural and torsional vibration signals with a dataacquisition card. The vibration signals are quasi-periodic in thetapping mode. Therefore we averaged the data over 12 oscillation
cycles to further reduce the noise. This results in a measurement
bandwidth of 4 kHz. Figure 4a shows a plot of the dataobtained on a highly oriented pyrolytic graphite sample. The
flexural signal is approximately a sinusoid at the drive frequency,with weak higher harmonic vibrations. The torsional vibrations
exhibit a more complicated waveform. When the tip hits the
sample (the lowest point in the flexural waveform), the cantileverbends torsionally in proportion to the torque generated by thetip –sample forces (orange arrow). After this impact, thenature nanotechnology | VOL 2 | AUGUST 2007 | www.nature.com/naturenanotechnology© 2007 Nature Publishing Group

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0.6
Detector signal (V)Response amplitude (a.u.)

0.8
102

Torsional

100

Flexural

10–2

0.4
0.2
0.0
–0.2
–0.4
–0.6

10–4

–0.8
0

2

4

6

8
10
12
14
Frequency (normalized)

16

18

20

0

2

4

6

8

10
12
Time (µs)

14

16

18

20

0

2

4

6

8

10
12
Time (µs)

14

16

18

20

–4

–2

2
4
6
8
Tip–sample distance (nm)

10

40
30

–20

Force (nN)

Detector signal (dBm)

0

–40

20
10
0

–60

-10
–80
0

200

400
600
Vibration frequency (KHz)

800

1,000

–20

20

–40

Force (nN)

Detector signal (dBm)

30

10

–60

0
–10

–80
0

200

400

600

800

1,000

Vibration frequency (KHz)

Figure 3 Frequency response and vibration spectra of a torsional harmonic
cantilever. a, The frequency response of flexural (blue) and torsional (orange)modes of a THC. The frequency axis is normalized so that the first flexural
resonance frequency is one. b,c, Flexural (b) and torsional (c) vibration spectra of a
THC tapping on a polystyrene sample. The first peak of the flexural spectrum in b isat the driving frequency and is the largest component of the cantilever motion. Allother flexural and torsional peaks are the higher harmonics generated by the tip–
sample interaction force. The torsional peaks show much better signal levels at
higher frequencies. Note the similarity between the magnitudes of the peaks in thetorsional spectrum and the corresponding frequency response curve in a.

cantilever exhibits slowly decaying torsional oscillations with afrequency close to the torsional resonance frequency. Thisbehaviour is also reflected in the torsional vibration spectrum
given in Fig. 3c as a pronounced increase in the magnitudes ofthe peaks around the 15th to 17th harmonics.
Figure 4b plots the time-resolved tapping-force waveform
obtained on graphite calculated according to equation (3) using
the torsional deflection signal in Fig. 4a. The waveform is in the
shape of a pulse where the positive forces are the repulsive tip–
sample interactions that arise as the tip indents the sample. The
vertical position of the tip is approximately a sinusoid with a
510

0

12

14

Figure 4 Reconstructing the tip – sample force waveform. a, Oscilloscope
traces of the periodic flexural (blue) and torsional (orange) vibration signals at
the position-sensitive detector, obtained on graphite. b, Time-resolved
tip – sample force measurements calculated on graphite. c, The same data as inb, plotted against tip – sample distance. Negative distances mean that the
sample is indented. Arrows indicate the direction of motion. The solid part of thecurve marks the points between the largest sample indentation and breaking of
the contact on the retraction portion of the curve.

known amplitude and phase. We can therefore eliminate the time
variable and plot all the force measurements in Fig. 4b against
tip –sample distance, as shown in Fig. 4c. Hereafter, these curves
will be referred to as time-resolved tapping-force curves. There
are two force values recorded for a given tip– sample distance:
one for the approaching tip and one for the retracting tip. Thesetime-resolved tapping force curves reveal the richness of the tip–
sample interaction in the tapping mode. The rate of increase in
the repulsive forces during the indentation is proportional to thestiffness of the sample. The magnitudes of the negative forces
depend on the van der Waals parameters of the tip– sample
ensemble, as well as the capillary forces due to a liquid meniscus
acting on the tip20. The measurements also reveal a hysteresis in
the attractive forces on approach and retraction of the tip, with anature nanotechnology | VOL 2 | AUGUST 2007 | www.nature.com/naturenanotechnology© 2007 Nature Publishing GroupARTICLES
Phase

Topography

10th harmonic

10 nm

10°

10 V

10 nm

10°

10 V

10 nm

10°

10 V

10 nm

10°

10 V

10 nm

20°

10 V

50 nm

20°

10 V

50 nm

90°

10 V

85 °C

115 °C

145 °C

160 °C

175 °C

190 °C

215 °C

Figure 5 Changes in the mechanical properties of a polymer blend near the glass transition. Topography, phase and tenth-harmonic images of a thin polymer
film composed of PS and PMMA recorded at different temperatures. The circular features are PMMA, and the matrix is PS. Brighter colour (more yellow) represents
larger height, phase or harmonic amplitude. The scan area is 2.5  5 mm. The colour bar represents different height and phase ranges at each temperature (therange is given in the top left corner of each panel). For the harmonic images, the colour bar represents a 10 V lock-in output signal at all temperatures. Note thatheight and phase contrast increases with temperature, whereas the harmonic contrast is first increasing and then decreasing.

much larger attractive force associated with the unloading portionof the indentation forces (solid line in Fig. 4c).
The variation in the repulsive force with sample indentation
depends on the material properties of the tip and the sample, as
well as the shape of the tip. In stiffer materials, a larger loading
force is required to produce a given depth of indentation. An
approximate model for macroscopic bodies describing theindentation of a flat surface with a spherical probe predicts thatthe forces vary with the following relationship21:
4  pﬃﬃﬃ 3=2
F tip ¼ E R d þ Fadh ;3ð5Þwhere Ftip is the tip – sample force, E * is the reduced Young’s
modulus, R is the tip radius, d is the indentation depth and Fadh

is the tip–sample adhesion force. This formula is valid forcontacts where adhesion is either constant during the indentation(Derjaguin-Muller-Toporov model) or negligible (Hertz model)21.
In general, adhesion plays an important role in contact
mechanics and the formula relating forces to indentations takes
on a more complicated form. We worked with a THC that has arelatively sharp tip to decrease the role of adhesive forces. (Note,the second THC was fabricated with a process similar to thatused for commercial AFM cantilevers that results in a 7-nm tip
radius, characterized by blind reconstruction from the AFM
image of a sample with sharp edges.) For a given tip, thedifferences between materials arise only in the reduced Young’s
modulus and the adhesion force. Although the validity ofmacroscopic models for nanoscale contacts is yet to be
determined22, the results presented hereafter are in agreement

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Indentation force curves on PS

Indentation force curves on PMMA
60

Indentation force (nN)

80 °C
115 °C
25 °C
145 °C60

80 °C
115 °C

50

50

25 °C
145 °C40

40
160 °C30
160 °C20
175 °C10

190 °C
0
–10

–8
–6
–4
–2
Indentation depth (nm)030

175 °C20

190 °C10

215 °C–10 –8–6–4–2
Indentation depth (nm)

0
0

10 G
Effective modulus (Pa)

PMMA
PS

1G

100 M

10 M
0

50

100
150
Temperature (°C)

200

250

Figure 6 Indentation forces as a function of temperature in a polymer
blend. a,b, Time-varying force measurements plotted against sample
indentation for polystyrene (a) and PMMA (b). c, Corresponding effective Young’s
modulus values are calculated by fitting the force curves to equation (5).

with the above relationship, at least for indentations comparable toor less than the tip radius.

TORSIONAL HARMONIC IMAGES
The multidimensional signal in Fig. 3 contains information about
many properties of the sample surface. A simple method to
produce qualitative maps of stiffness is to record the amplitudeof a single higher harmonic13,23,24. Theoretical studies of thehigher harmonic response indicate that the amplitude of a higher
harmonic depends on the stiffness of the sample; however, aproper choice of the higher harmonic is important25. The THC
has good signal levels at torsional harmonics up to the torsional
resonance frequency, so it provides the flexibility to image withany of these harmonics to produce the corresponding torsional
harmonic image of the surface. We generate torsional harmonic
images by using a lock-in amplifier that is tuned to the frequency
of a particular higher harmonic.
With topography, phase and torsional harmonic images, wehave investigated the structural properties of an approximately
50-nm-thick binary polymer film on a silicon substrate near theglass transition temperature of the two components, polystyrene(PS) and polymethylmethacrylate (PMMA). The PS and PMMA
have distinct glass transition temperatures (around 100 8C and
130 8C, respectively) and form sub-micrometre domains within
the film. Each polymer component transitions from a stiff glassy
phase to a rubbery phase at its respective glass transition
temperature. In the temperature range between the glasstransitions of PS and PMMA, the regions of rubbery PS phasewill be more compliant than the glassy PMMA regions. Using
512

torsional harmonic images, we can observe the glass transitionsof individual components of the composite polymer film.
Figure 5 shows the topography, phase and tenth torsional
harmonic images obtained at temperatures between 85 and 215 8C.
Images at different temperatures correspond to approximately thesame position on the sample, though some shift results fromthermal drift in the system. Lateral resolution, however, is notaffected by thermal drift because the imaging time is in the orderof seconds, and the drift is 0.5 nm min – 1. The tenth harmonic
was chosen because it is close to the torsional resonancefrequency. At low temperatures, the two material components are
distinguishable in all three images. PMMA forms the roundfeatures and PS forms the matrix.
The topography of the surface shows height variations between
the two components at low temperatures. However, dramaticchanges in height are observed at high temperatures. Above 190 8C,
the boundaries between the different polymer components become
unclear. At elevated temperatures, the rubbery material gains
liquid-like mobility, and flows and rearranges so that it forms
round (droplet-like) features on the surface. This leads to larger but
smoother topographical variations.
Phase images also show dramatic changes with temperature. The
phase of the cantilever oscillations depends on the energy dissipation
during the tip–sample contact9. Formation and disruption of aliquid neck between the tip and the sample, hysteresis in adhesion
forces, and viscosity of the sample are the major energy-dissipation
mechanisms. In particular, on a compliant sample, attractive forces
between the tip and the sample are strong enough to pull thesample locally and raise it above its equilibrium level. Once thecontact is broken, the mechanical energy stored in the raised
sample is dissipated. Because of this, the polymer components
dissipate energy more at elevated temperatures where they are
more compliant. As well as this mechanism, increased viscosity inthe rubbery phase also leads to increased energy dissipation. The
increase in the contrast in the phase images near the firsttransition at 160 8C is primarily due to these effects; however, thecontrast becomes larger even when both materials are in therubbery phase. This illustrates one of the limitations of phaseimaging. In general, phase images identify different material
components; however, interpreting the nature of the contrast isdifficult owing to the multiple mechanisms involved26.
Below 160 8C, the domains of PS and PMMA are
distinguishable in the torsional harmonic images, but thecontrast is small. This is because the two materials have similar
mechanical properties in their glassy phases. At 160 8C and 175 8C,
the contrast between the PS and PMMA domains increases in thetorsional harmonic image. A reduction in the amplitude of theharmonic signal is recorded on the PS matrix. According to ref. 25,
a reduction in higher harmonic amplitudes results from anincreased mechanical compliance of a material. This indicates theglass transition of PS. The contrast is reduced at 190 8C and almost
completely disappears at 215 8C, as the PMMA graduallytransitions to the rubbery phase. At 215 8C, both materials are in
the rubbery phase, the harmonic amplitude near the PMMA and
PS is small, and there is little contrast between the two materials.
The torsional harmonic images provide a qualitative picture of themechanical changes in the sample that is not directly accessible
from the phase or topography images; however, a better
understanding of the changes in the samples can be obtained byanalysing the time-varying tip–sample forces.

QUANTITATIVE NANOMECHANICAL MEASUREMENTS
The time-resolved tapping-force measurements are capable ofproviding a more detailed and potentially quantitative
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understanding of the changes observed on the polymer film bygenerating indentation force versus indentation depth curves(solid part of the curve in Fig. 4c). In Fig. 6, we plot these curves
recorded while the cantilever is scanning over the PS and PMMA
regions at each temperature (see Supplementary Information,text and Fig. S3, for complete time-resolved tapping-force
curves). The slopes of the indentation forces gradually decrease
with increasing temperature. This indicates that both materials
are becoming increasingly compliant at elevated temperatures. In
order to quantify these changes we use equation (5) to calculate
the corresponding E* values based on curve fitting with theassumption of a spherical tip with 7-nm radius. These values areplotted against temperature in Fig. 6c. At room temperature, theYoung’s modulus of PS and PMMA is measured as 2.3 GPa and
3.7 GPa, respectively, both of which are in the range of typicalbulk values for these materials. The two polymers exhibit only asmall reduction in stiffness at temperatures up to 115 8C.
However, at higher temperatures, stiffness of both materials
decreases by almost two orders of magnitude owing to the glasstransition. According to these results, the apparent glasstransition temperatures of the PS and PMMA are approximately
160 8C and 180 8C, respectively. The glass transition of PS occurs
at lower temperatures compared with PMMA, as expected. At
temperatures beyond the glass transition, these materials are
viscoelastic, and indentation forces contain rate-dependent
components that are not accounted for in equation (2), resulting
in deviations from the model.
The relatively high transition temperatures are expectedconsequences of the frequency dependence of the material
response around the glass transition27,28. With an increase in
the measurement frequency, the glass transition temperature
shifts towards higher temperatures at a rate of 5 –10 8C per
frequency decade. We note that our technique measures themechanical properties at the harmonic frequencies (50–1,000 kHz),
whereas the glass transitions of bulk materials are typicallydetermined through thermal measurements or dynamic
mechanical measurements at frequencies below 100 Hz (ref. 27).

CONCLUSIONSThe THC allows detailed mechanical measurements in the tapping
mode, thereby enabling the study of a wide range of materials. The
torsional harmonic images allow qualitative mapping of
mechanical property variations, and the time-resolved tappingforce curves provide a detailed and quantifiable understandingof the changes observed on the surface. The time-resolved
tapping-force measurements are performed within 250 ms, with a
few nanometres of sample indentation and peak loading forces
around 10 nN. Our new method has the inherent nanometrescale lateral resolution of the tapping-mode AFM. These results
represent orders of magnitude improvements in temporal
resolution, spatial resolution, indentation and mechanical loading
compared to conventional tools. With high operating speed,
increased force sensitivity and excellent lateral resolution, ourtool facilitates practical mapping of nanomechanical properties.

TAPPING-MODE OPERATION PARAMETERS

The set-point and free vibration amplitudes for the measurements in Fig. 3 were
40 nm and 60 nm (first THC, K ¼ 2.4 N m – 1, f0 ¼ 47.4 kHz, Q ¼ 55). The setpoint and free vibration amplitudes for the measurements in Figs 4 and 5 were
70 nm and 100 nm (second THC, K ¼ 6 N m – 1, f0 ¼ 50.8 kHz, Q ¼ 91). The
drive frequency (50.5 kHz) for the imaging was chosen to be slightly below
resonance frequency. This favours repulsive-mode operation. Scan rate during
the imaging was chosen to be 1 Hz. The bandwidth used for lock-in detection of
torsional harmonics was 3 kHz.
CALIBRATION OF THE TRANSFER FUNCTION H Y(v)The torsional resonance frequency vT and the corresponding quality factor QT
were determined by the frequency-tuning software that comes with the AFM
system (QT ¼ 800 and vT ¼ 809.1 kHz for the first THC, and QT ¼ 1,050
and vT ¼ 528.0 kHz for the second THC). The effective torsional spring
constants of the cantilevers near the offset tip were measured by a vibrometer(KT ¼ 550 N m – 1 for the first THC and KT ¼ 480 N m – 1). The optical gain
coptical for torsional deflections was determined in two steps. First, the optical
gain for the fundamental vertical mode (detector voltage for a unit tip
displacement in this mode) was measured by obtaining force– distance curves inthe tapping mode. Second, we multiplied this value by a geometrical factor that
corresponds to the ratio of the slopes on the cantilever for a unit tip displacementin the torsional mode and flexural mode. The geometrical factor is estimatedfrom the mode shapes of a rectangular cantilever. We neglect the frequency
dependence of the optical gain because the cutoff frequency of the photodetector
is around 3 MHz.
SAMPLING AND PROCESSING OF DETECTOR SIGNALS

The flexural and torsional (vertical and lateral) signals generated by the quadrant
photodetector were low-pass-filtered with a cutoff frequency of 2 MHz, and were
continuously downloaded to a computer with the data acquisition card. A
Labview program was used to perform digital signal processing in order toobtain time-resolved forces. In this program, data were divided into packets
where each packet has a number of samples that correspond to 12 oscillation
cycles (first THC) or 120 oscillation cycles (second THC). For each data packet,
discrete Fourier transform was calculated, and equation (3) was solved for ftip in
the frequency domain with the following equation:
ftip ð jÞ ¼

N1X
H 1 ðvk ÞSY ðvk Þeivk ð j1ÞN k¼1 Y

ð6Þ

Here, SY(vk) is the discrete Fourier transform of the sampled torsional deflection
signal, H21
Y is the inverse of the transfer function given in equation (4), N is thenumber of data points per period and ftip(j) is the discrete form of tip – sample
forces. Note that H21
Y diverges at frequencies much larger than the torsional
resonance frequency. We introduce a cutoff frequency in the calculations that isslightly above the torsional resonance frequency to solve this issue (we use up to
20 harmonics for the first THC and 13 harmonics for the second THC). In orderto represent ftip(j) with a larger number of points (resampling), we extend SY(vk)by zero padding beyond the cutoff frequency of the calculations. In that case,equation (6) is used with the final N value. The force waveforms presented in thiswork are resampled at 512 points per period.
The biggest source of error in these calculations is the crosstalk between
the detector signals. The vertical channel has a large signal component at thedrive frequency, which appears at the lateral channel as a crosstalk. We use acurve-fitting procedure to estimate and subtract the crosstalk signal from the
waveform calculated by equation (6) (see Supplementary Information). The
deviation in the final effective modulus estimates is approximately 5% for thenominal values around 1 GPa, and it gradually increases to 20% on samples
around 50 MPa.
SAMPLE PREPARATION

MATERIALS AND METHODSGraphite Freshly cleaved highly oriented pyrolytic graphite was used.

EXPERIMENTAL SETUP

We used a commercial AFM system (Multimode series, equipped with
Nanoscope 4 controller, signal break-out box, and torsional vibration detection
electronics). Also used were an additional lock-in amplifier (Stanford Research
Systems, SRS-844), a digital oscilloscope and a data acquisition card (NI DAQ,
S-series). The sample was placed on a commercially available controlled heatingstage. The cantilever was maintained at the same temperature as the sample.

Blended polymer film We used polystyrene (Mw ¼ 81 K) andpolymethylmethacrylate (Mw ¼ 73 K and 81 K). Both are products of Polymer
Source. The polymers were dissolved in hot toluene and cast as a thin film on Si
substrate. The thickness of the film was 40 – 50 nm as estimated by AFM heightmeasurements of a scratch made on this film by a sharp copper needle.

Received 20 October 2006; accepted 28 June 2007; published 29 July 2007.

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Acknowledgements
We thank N. Yerina of Veeco Instruments for preparing the polymer samples. The authors acknowledge
support from the Center for Probing the Nanoscale (CPN), and NSF NSEC, NSF Grant No. PHY0425897. O. Sahin acknowledges support from the Rowland Junior Fellows Program.
Correspondence and requests for materials should be addressed to O. Sahin.
Supplementary information accompanies this paper on www.nature.com/naturenanotechnology.

Competing financial interests
The authors declare competing financial interests: details accompany the full-text HTML version of thepaper at www.nature.com/naturenanotechnology.
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