INSTITUTE OF PHYSICS PUBLISHING
NANOTECHNOLOGY
Nanotechnology 13 (2002) 33–37
PII: S0957-4484(02)27598-4
Measuring the spring constant of atomic
force microscope cantilevers: thermal
fluctuations and other methods
R Lévy and M Maaloum
Institut Charles Sadron-Université Louis Pasteur, 6 rue Boussingault 67083 Strasbourg cedex,
France
E-mail: [email protected] and [email protected]
Received 3 August 2001, in final form 5 November 2001
Published 12 December 2001
Online at stacks.iop.org/Nano/13/33
Abstract
Knowledge of the interaction forces between surfaces gained using an
atomic force microscope (AFM) is crucial in a variety of industrial and
scientific applications and necessitates a precise knowledge of the cantilever
spring constant. Many methods have been devised to experimentally
determine the spring constants of AFM cantilevers. The thermal fluctuation
method is elegant but requires a theoretical model of the bending modes.
For a rectangular cantilever, this model is available (Butt and Jaschke).
Detailed thermal fluctuation measurements of a series of AFM cantilever
beams have been performed in order to test the validity and accuracy of the
recent theoretical models. The spring constant of rectangular cantilevers can
also be determined easily with the method of Sader and White. We found
very good agreement between the two methods. In the case of the V-shaped
cantilever, we have shown that the thermal fluctuation method is a valid and
accurate approach to the evaluation of the spring constant. A comparison
between this method and those of Sader–Neumeister and of Ducker has been
established. In some cases, we found disagreement between these two
methods; the effect of non-conservation of material properties over all
cantilevers from a single chip is qualitatively invoked.
(Some figures in this article are in colour only in the electronic version)
1. Introduction
The atomic force microscope (AFM) is not only an excellent
tool for recording the surface topography of a sample, but
thanks to recent technical advances, it has proven to be a good
means studying the strength of chemical bonds [1,2], the elastic and mechanical properties of macromolecules [3–6] and the
intermolecular interactions at the single molecular level [7–9].
The AFM can record the amount of force felt by the cantilever
as the probe tip is brought close to—and even indented into—a
sample surface and then pulled away. It is a great advantage,
for example, to be able to derive the rupture forces involved in
the experiments from an exact calibration and precise knowledge of the cantilever spring constant of each cantilever. The
force sensed by the AFM probe is calculated by multiplying the
deflection of the cantilever by the cantilever’s spring constant.
Many authors have noted that the spring values indicated by
0957-4484/02/010033+05$30.00
© 2002 IOP Publishing Ltd
the manufacturer are incorrect [10–14]. Manufacturers of cantilevers use wide tolerances in the specified values of the force
constant of cantilevers, due to the difficulties in controlling the
thickness of the cantilever at a particular stage of the manufacturing process and the structural defects and deviations in
geometry from cantilever to cantilever which are not taken into
account. Because of this, several methods have been developed
to measure or calculate the cantilever spring constant [10–26].
The most simple and elegant method used by different authors involves a study of the thermal fluctuation of the cantilever [7, 10, 16, 25, 26]. Many authors have modelled the
cantilever [7, 10] as a simple harmonic oscillator with one degree of freedom (neglecting higher modes of oscillation). The
potential energy of this system takes the form of 21 mω02 z2 =
1
Kz2 . The thermal method takes advantage of the equipar2
tition theorem, applied to the cantilever potential energy:
Printed in the UK
33
Power spectrum density (µV2/Hz)
R Lévy and M Maaloum
Frequency (kHz)
Figure 1. Power spectrum density plot of fluctuations in the output
of the cantilever deflection detector. The curve has a Lorentzian
form (solid curve) around the angular resonance frequency ωf . The
values of ωf , Q factor and the area below the peak are given.
Figure 3. The noise power spectral density of the detector output
plotted against frequency. Because of bandwidth limitation of our
spectrum analyser (0–100 kHz), we can see just two modes at
angular resonance frequencies ω1 = 14.1 kHz and ω2 = 90.1 kHz.
(ii) The optical lever deflection measures the inclination
dz/dx rather than the displacement (x is the coordinate
in the longitudinal direction of the cantilever). Then
the relation between z (nm) and zV (V) is not simply
zi = zV i /s. In order to evaluate the real displacement
in each free mode, the shape of the deformation in each
free mode and in the compliance region must be known.
(iii) Only the first modes are accessible in the experiment
(bandwidth limitation). For this reason, the repartition
of the displacement is needed in the different modes to
deduce z2 and thus, the spring constant.
Figure 2. Typical force curve, showing the deflection of a cantilever
as a function of sample displacement. To avoid the hysteresis, we
apply a very small voltage to the piezoelectric element in the region
where the tip and the sample were in contact (see the figure inset).
In this way the two slopes are completely superposed. We use this
value of the slope in order to calibrate the power spectrum density.
Kz2 = kT .
Here, z2 is the mean square deflection of the cantilever
caused by the thermal motion in the vertical direction, k is
the Boltzmann constant, T the temperature of the surrounding
heat bath, K the cantilever spring constant and m the effective
mass. K is deduced by measuring z2 . By transforming the
time series data to the frequency domain, the Lorentzian form is
obtained around the resonance frequency as shown in figure 1.
The area below this peak is taken to be equal to the mean
square z2 of the fluctuations in the time series data; the area
is indicated by zV and measured in volts. The displacement
z is deduced by the relation z = C · zV . In the static case
(cantilever not moving), the constant C is given by the inverse
of the slope, s, of the compliance region (figure 2): C = s −1 .
This argument poses three major problems:
(i) We cannot take just the first peak (see figure 3), as stated
by different authors, and neglect the higher modes, as this
may lead to errors of about 30% in the estimation of the
spring constant of the cantilever.
34
Recently Butt and Jaschke [17] have calculated the thermal
fluctuation of the rectangular cantilever using the equipartition
theorem and considering all possible vibrations. They
attempted to correlate dz/dx with the displacement using the
optical lever technique.
In this work, we have described the cantilever dynamics
and thermal fluctuation behaviour of a cantilever. Thermal
fluctuation measurements of a series of AFM cantilever beams
have been performed. We have determined the ratio of the
amplitudes of the first two modes as a function of the distance
from the anchorage point, in order to test the validity and
accuracy of the recent theoretical models [17]. Then we
compared the experimental spring constant of the rectangular
cantilevers derived from the thermal fluctuations and Sader
methods [14, 15].
Next, calibration of the various shapes of cantilevers
including the triangular (V-shaped) cantilevers has been
determined using different methods such as thermal
fluctuation, Sader–Neumester and Ducker models [16] and the
slope method. A comparison between these methods has been
established in order to investigate the validity and accuracy
of the different approaches to the evaluation of V-shaped
cantilever spring constants.
2. Experimental methods
All measurements were performed using a home-made
AFM. The cantilevers used (figure 4) were procured from
displacement
Measuring the spring constant of atomic force microscope cantilevers: thermal fluctuations and other methods
Figure 5. The shape of the first and second vibration modes of a
free cantilever. With our optical lever system we measured the
inclination (angle) dz/dx rather than the displacement z. The x-axis
represents the longitudinal direction of the cantilever.
Figure 4. Cantilevers procured from Thermomicroscopes
(Sunnyvale CA, USA).
Thermomicroscopes (Sunnyvale CA, USA). The dimensions
of the various cantilevers were determined by optical
microscopy. The free vibration spectra and the vibration
amplitude of the cantilevers have been examined, using a
spectrum analyser (Stanford Research System), in the spectral
range 0 Hz–100 KHz. The power spectrum density was
recorded, in units of V2 Hz−1 , by taking the signal from the
photodiode and feeding it directly into a spectrum analyser.
For the data presented here, we calibrated each cantilever
individually. With each cantilever we recorded force curves on
a mica surface, using a previously calibrated piezoelectric element (performed interferometrically), immediately after collecting the thermal motion data. The slope was collected from
the contact portions of the extending and retracting regions.
The measurement of the slope is critical in the accurate calibration of the cantilever spring constant. As the spring constant
varies with the square of the slope, an error of 10% on the slope
results in an error of 20% on the spring constant. Often the
force curves used from the contact portions of the extending
and retraction regions present an hysteresis. To compensate
for this, some authors [11] have used the average value of the
slopes together. Consequently, this manner to estimate the
slope may lead to an error of 10% on the slope.
To avoid this hysteresis, we applied a very small voltage
on the piezoelectric element (very small displacement) in the
region where the tip and sample were in contact (figure 2). In
this way the two slopes are completely superposed. The value
of the slope (V nm−1 ) allows us to convert the power spectrum
density from V2 Hz−1 to nm2 Hz−1 .
Since the oscillating shape is different from the static
shape, the relationship between the end position and the end
angle is also different, as stated by Butt and Jaschke [17]. With
our optical lever system we measure the inclination (angle)
dz/dx rather than the displacement z. x is the coordinate
in the longitudinal direction of the cantilever as shown in
figure 5. Consequently, what we actually measure with the
optical lever technique is a virtual height z∗ , caused by the
inclination dz/dx.
3. Results and discussion
Rectangular cantilever. In this section, we assess the
accuracy and validity of the theoretical model by presenting a
comprehensive and detailed comparison with the experimental
measurements.
In the case of a rectangular cantilever with a free end, Butt
and Jaschke [17] have shown that the virtual deflection z∗2 is correlated with the real one using the following relation:
4kT
4 2
z∗2 =
=
z .
3K
3
In this equation the contributions of all vibration modes have
been considered. The virtual deflection is caused by the
inclination dz/dx. The virtual and real deflections for each
vibration mode are given by the following relations:
sin αi sinh αi 2
16kT
(1)
zi∗2 =
3Kαi2 sin αi + sinh αi
zi2 =
12kT
.
Kαi4
(2)
Here, L/αi represents the wavelength of the ith vibration mode
and L is the length of the rectangular cantilever (figure 5).
Consequently, the amplitude of thermal noise in the deflection
falls by a factor of 100 and the virtual deflection decreases
by a factor of about six when going from the first to the sixth
vibration modes. The αi are dimensionless quantities fixed
by the boundary conditions [17]. The resonance frequency ωi
associated with each mode depends on αi , αi4 /αj4 = ωi2 /ωj2 .
The authors have calculated the αi and the corresponding
expressions of the spring constant K for each of the two modes:
first mode
kT
α1 = 1.88 and K = 0.82 ∗2
z1 second mode
kT
(3)
α2 = 4.9 and K = 0.25 ∗2 .
z2 Our spectrum analyser allows us to study the two first
modes (bandwidth limitation). From figure 3 we measure the
35
R Lévy and M Maaloum
Table 1. The table gives the spring constants of rectangular
cantilevers measured with the thermal fluctuation method (first and
second modes, Kth1 and Kth2 ) and with the method of Sader (KSader ).
Kman corresponds to the spring constant indicated by the
manufacturer. R1 to R6 are rectangular cantilevers
(Thermomicroscopes, cantilever B) from the same wafer.
R1
R2
R3
R4
R5
R6
Figure 6. The curve presents the ratio z2∗2 /z1∗2 as a function of
the laser spot position on a rectangular cantilever. There is very
good agreement between the theoretical model [17] (dotted curve)
and the experimental data. The dotted curve is the actual theoretical
curve, which has also been reproduced and shifted by 9 µm to show
its overlap with the experimental points (solid curve). When
z2∗2 = 0, the curve z2∗2 /z1∗2 presents a minimum. This illustrates
perfectly the fact that we measure the inclination dz/dx rather than
the displacement, since the latter is a maximum (see the figure
inset). The black squares and circles correspond to two independent
experiments showing the reproducibility of the measurements.
resonance frequencies associated with each mode. The ratio
dz/dx is in good agreement with the values given by the theory,
α12 /α22 = 0.161. Note that the first vibration mode contributes
only 70% to z∗2 .
According to relation (3), we can make independent
measurements of the spring constant with different peaks.
In figure 6 we plotted the ratio z2∗2 /z1∗2 as a function
of the laser spot position on a rectangular cantilever. We have
compared it to the theoretical model (dotted curve). Again
we found very good agreement between the theoretical model
and the experimental data. The shift between the theoretical
and experimental curves of 9 µm could be explained by air
damping, by the imperfect geometry or imperfect anchorage
(small variation of the cantilever inclination at point x = 0).
The profile curve z2∗2 /z1∗2 presents a minimum at the point
(85 µm, z2∗2 /z1∗2 = 0). This point corresponds to the
zero difference of angles when the amplitude is maximum in
the second mode (figure 6 inset). This illustrates perfectly
the fact that what we measure is not the displacement but
the inclination dz/dx. All these results demonstrate clearly
that the model of Butt and Jaschke can be used to calibrate
rectangular cantilevers with the thermal fluctuations method.
We now turn our attention to the comparison between the
thermal fluctuations and Sader methods [14, 15]. The method
of Sader relies on the measurement of the resonance frequency,
on the quality factor in air, and on the knowledge of the plan
view dimensions of a rectangular cantilever. Note that the
plan view dimensions have to be precisely measured for each
cantilever because from cantilever to cantilever the dimensions
are not conserved.
36
Kth1
(mN m−1 )
Kth2
(mN m−1 )
KSader
(mN m−1 )
Kman
(mN m−1 )
22.9
22.1
21.5
16.8
17.8
16.9
22.2
22.9
22.0
16.0
17.6
17.0
22.7
22.1
21.5
18.5
18.4
18.0
20.0
20.0
20.0
20.0
20.0
20.0
Table 1 gives the spring constant of a rectangular cantilever measured with the thermal noise method (in the case of
the first and second modes) and with the method of Sader. The
six cantilevers have been taken from the same wafer. As can be
seen from table 1, excellent agreement is obtained between the
two methods. In contrast, it should be noted that the cantilever
spring constant varies from cantilever to cantilever. This reflects the expectation that the material properties and geometry
of all cantilevers are not uniform across the entire wafer.
Triangular cantilever. Often V-shaped cantilevers are used.
Recently, Stark et al [26] have investigated theoretically the
thermomechanical noise of a triangular cantilever by means
of a finite element analysis. The modal shapes of the first ten
eigenmodes are given together with the numerical constants
needed for a calibration using the thermal noise method. The
cantilever of type E was used in their calculation (figure 7).
They found that the spring constant of this cantilever can be
expressed by the following equation:
K = 0.778
kT
.
z1∗2 Recently Neumeister and Ducker [16] have established
equations involving geometry and material constants to
describe the stiffness of a V-shaped AFM cantilever in three
dimensions. An experimental determination of the stiffness
of the cantilever using this model necessitates a precise
knowledge of the geometry and above all the material constants
such as the Young’s modulus E. In addition, the spring
constant depends strongly on the cantilever thickness t, which
is difficult to measure. As in the equation, established by
the authors, the stiffness of the cantilever is proportional to
Et 3 . We can use the Sader method to deduce the value of
Et 3 from the measurements with the rectangular cantilever
assuming that this value remains constant over all cantilevers
(rectangular and triangular) of a single chip. The procedure
involves calibration of the rectangular cantilever, from which
the product Et 3 , is evaluated using
Et 3 = K
4L3
.
b
We have determined the spring constant K using the Sader–
Neumeister and Ducker method (SND). In table 2, we present
results for the calibrated cantilevers, determined by the manufacturer and by the cited methods. In the case of cantilevers E,
Measuring the spring constant of atomic force microscope cantilevers: thermal fluctuations and other methods
(a)
(b)
depends strongly on the cantilever thickness t, a thickness
variation of 5% results in a change in the spring constant
of 15%. This may explain the discrepancy, regarding the
measured value of the V-shaped cantilever (type E), between
the SND method and the thermal fluctuation method.
Conclusion
Figure 7. The AFM cantilever (type E) shape with its typical
dimensions. (a) The geometry used by Neumeister and
Ducker [15]. (b) The real geometry of the cantilever as viewed by
the optical microscope with high magnification. It is clear that if we
take the real geometry of the cantilever the spring constant will be
stiffer than if we take the dimension of figure (a).
Table 2. The table presents the measured values of the V-shaped
cantilever spring constant of type E and D from two chips of the
same wafer (Thermomicroscope) using the thermal fluctuation and
SND methods.
RE1
RE2
RD1
RD2
Kth1
(mN m−1 )
KNeu
(mN m−1 )
Kman
(mN m−1 )
82
87
29
30
96
110
28
30
100
100
40
40
there is clearly a discrepancy between the thermal fluctuation
and SND methods. The spring constant measured using SND
methods is stiffer than those measured using the thermal fluctuation model. This disagreement would have been more pronounced if we had taken into account the real geometry of
the cantilever, because the measured spring constant using the
SND method would be stiffer than the indicated values (see
figure 7). In contrast, we found very good agreement between
the two methods in the case of V-shaped cantilevers D. To
investigate the disagreement regarding the lever E, we have
adopted the slope method to calibrate the V-shaped E cantilever
[19–21]. This method consists of mounting on the piezoelectric element a calibrated rectangular cantilever so that its free
end will come into contact with the V-shaped cantilever. The
deflection of the triangular cantilever is measured as a function
of the piezoelectric displacement. The spring constant is determined from the slope of the force curve, using linear region.
Using the calibrated rectangular cantilever, we measured
the deflection of the AFM cantilever (type E). The measured
spring constant value obtained is K = 90±10 mN m−1 , which
is in agreement with the measured value using the thermal
fluctuation method.
It should be noted that, in the case of the SND method, the
assumption that Et 3 is constant over all cantilevers on a single
chip could have a dramatic consequence on the evaluation of
the spring constant. In fact, as the stiffness of the cantilever
The thermal fluctuation measurements of a series of AFM cantilever beams have been performed and compared to a recent
theoretical model. We found a very good agreement between
the experimental data and the theory of Butt and Jaschke. We
have confirmed that the values of the calibrated AFM cantilevers varies from cantilever to cantilever and differs from
those given by the manufacturer. In the case of the rectangular
cantilever, a good agreement was found between the thermal
fluctuation and Sader methods. We have shown that the thermal fluctuation method is a valid and accurate approach to the
evaluation of the spring constant of V-shaped AFM cantilevers.
In some cases, the calibration of V-shaped cantilevers using the
SND method gives values different to those determined using
thermal fluctuation. This disagreement is probably due to the
non-conservation of the material properties of all cantilevers
from a single chip.
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