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Calibration of higher eigenmode spring constants of atomic force microscope cantilevers

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2010 Nanotechnology 21 465502(http://iopscience.iop.org/0957-4484/21/46/465502)
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IOP PUBLISHINGNANOTECHNOLOGY

Nanotechnology 21 (2010) 465502 (7pp)

doi:10.1088/0957-4484/21/46/465502

Calibration of higher eigenmode spring
constants of atomic force microscopecantilevers
Jose R Lozano1 , Daniel Kiracofe2 , John Melcher2, Ricardo Garcia1
and Arvind Raman2
1
IMM—Instituto de Microelectrónica de Madrid (CNM-CSIC), Isaac Newton 8, PTM,
E-28760 Tres Cantos, Madrid, Spain2
Birck Nanotechnology Center and School of Mechanical Engineering, Purdue University,
West Lafayette, IN 47904-2088, USA

E-mail: raman@purdue.edu

Received 21 April 2010, in final form 10 August 2010
Published 25 October 2010
Online at stacks.iop.org/Nano/21/465502
Abstract
Standard spring constant calibration methods are compared when applied to higher eigenmodesof cantilevers used in dynamic atomic force microscopy (dAFM). Analysis shows that Sader’soriginal method (Sader et al 1999 Rev. Sci. Instrum. 70 3967–9), which relies on a priori
knowledge of the eigenmode shape, is poorly suited for the calibration of higher eigenmodes.
On the other hand, the thermal noise method (Hutter and Bechhoefer 1993 Rev. Sci. Instrum. 64
1868–73) does not require knowledge of the eigenmode and remains valid for higher
eigenmodes of the dAFM probe. Experimental measurements of thermal vibrations in air for
three representative cantilevers are provided to support the theoretical results.

the equivalent spring constants of the participating eigenmodes
to be calibrated. However, the close similarity between thespring constant of the fundamental eigenmode and the static
spring constant [15] has resulted in the two rarely being
distinguished from one another in the literature. The evolutionof dynamic AFM to incorporate higher eigenmodes requiresthat distinction is made between the static spring constant,kstatic , and the equivalent spring constant of higher eigenmodes,
kn (n > 1). Moreover, we must reconsider calibration
methods, which were originally developed specifically tocalibrate kstatic , and how they can be modified to producekn [16].
Several methods have been suggested to calibrate
kstatic [17–31], although two of them have been particularly
popular [32].
The first was proposed by Sader [19].
Remarkably, this method uses the predicted hydrodynamic
damping affecting a rectangular AFM probe immersed ina fluid to calibrate kstatic in terms of the measured quality
factor and resonance frequency, as well as the geometricaldimensions of the beam. On the other hand the thermal noise
method, originally proposed by Hutter and Bechhoefer [17] butlater modified by Butt and Jaschke [16], takes advantage of theequipartition theorem which relates the thermal fluctuations

1. Introduction
Atomic force microscopy (AFM) stands out among othermicroscopy techniques for its capability to render imagesof heterogeneous surfaces with nanometric resolution [1–3].
In recent years several attempts have been proposed toalso convert AFM into an analytical tool, able to measurequantitatively and precisely the local properties.
The
problem of force inversion (i.e. to extract the actual valueof the interaction between tip and sample from experimental
observables) has been solved under different approximationsin amplitude- [4–7] and frequency-modulation [8, 9] modes.
To combine both quantitative material information andhigh resolution imaging a number of groups have opted forthe excitation and/or detection of higher harmonics [10] of thecantilever motion and/or higher eigenmodes [11] which are
related to other flexural [12] or torsional [13] resonances ofthe probe. However, the quest for quantitative information in
AFM greatly depends on accurate calibration of the relevant
spring constant(s) for the application. Quasi-static AFM modes
require the static spring constant, i.e. the constant relating apoint-load applied at the tip and the resulting deflection of
the tip, to be calibrated [14]. Dynamic AFM modes require0957-4484/10/465502+07$30.00

1© 2010 IOP Publishing Ltd Printed in the UK & the USA

Nanotechnology 21 (2010) 465502

J R Lozano et alof the cantilever to the equilibrium temperature of thesurrounding fluid through kstatic . While these methods aremostly used for calibration of the fundamental eigenmode, itremains unclear whether or not these methods can be extended
to higher eigenmodes.
In this paper, we investigate the calibration of higher
eigenmodes through adaptations of the Sader method and thethermal noise method. We find considerable limitations to theSader method for calibrating higher eigenmodes due to thenecessity of a priori knowledge of the eigenmode. Specifically,
the sensitivity of higher eigenmodes to the tip mass and thecantilever geometry introduces large errors in the Sader method
when applied to higher eigenmodes [33]. We find that thermal
calibration is much more amenable to the calibration of higher
eigenmodes, as long as the resonance frequencies, f n , areadequately spaced. Specifically, the spacing of the resonances,f 2 − f1 , should be greater than their bandwidths, f n /Q n ,with Q n the eigenmode quality factor. Experimental results
are presented for three representative cantilevers: a tipped
rectangular cantilever, a tipless rectangular cantilever, and atipless picket-shaped cantilever.

in the Euler–Bernoulli PDE, equation (1), and impositionof the boundary conditions, equations (2). Because thedamping in air is small, the undamped eigenmodes and thedamped eigenmodes are nearly identical [37]. Therefore, theimaginary part of the hydrodynamic function can be neglectedin equation (1), yieldingn (x; m ∗ ) = cos(αn x) − cosh(αn x)cos(αn ) + cosh(αn )(sin(αn x) − sinh(αn x))−sin(αn ) + sinh(αn )where αn is a solution of

1 + cos αn cosh αn + m ∗ αn (sinh αn cos αn − sin αn cosh αn )

=0

(4)
∗

where m ≡ m tip /(m beam + m hydro ) is the reduced or
dimensionless tip mass, and with m beam ≡ ρc bh L and m hydro ≡πρ f b2 L Re[(ω)]/4 the cantilever and hydrodynamic added
masses, respectively. Here, the real part of the hydrodynamic
function is denoted by Re[]. Equations (3) and (4) showthat the shapes of the eigenmodes, n (x; m ∗ ), are determinedby their eigenvalues, αn , which in turn are determined by
the value of m ∗ . We have omitted the dependence of αnon m ∗ , αn (m ∗ ), for simplicity of notation. At this point wewould like to remark that the value of m ∗ depends on the
eigenmode considered because the hydrodynamic added mass,
m hydro, is frequency dependent, according to Sader’s theory.
However, for cantilevers immersed in air, m hydro is negligible,
which leads to m ∗ = m tip /m beam , regardless of the eigenmodenumber. We will assume this situation during the present work.
Here we are concerned with the measurement of theequivalent spring constant related to the tip deflection of the n th
eigenmode, kn . Accordingly, we apply the energy equivalenceprinciple [15], relating kn to n (the prime denotes total
derivative with respect to x ),1
E I 0 dx [n (x; m ∗ )]2∗kn (m ) = 3.
(5)L[n (1; m ∗ )]2

2. Theory
We begin with the application of standard beam theory todemonstrate the influence of a particle mass at the free end ofa uniform rectangular cantilever in higher eigenmode spring
constants. The Euler–Bernoulli partial differential equation(PDE) describes the motion in time of a one-dimensional beamin the presence of external forces, among which we will only
consider the hydrodynamic loading of the surrounding fluid. In
the Fourier domain it is expressed as [34],EIπ
Wxxxx (x, ω) − ρc bhω2 W (x, ω) = ρ f b2 ω2 (ω)W (x, ω)
4
L
4
(1)
where W (x, ω) is the Fourier transform of the out-of-plane
displacement of a beam element, placed in its longitudinal
normalized coordinate, x . E is the Young’s modulus, I is
the moment of inertia and L the length of the cantilever.
The angular frequency is referred to as ω, while partial
derivatives with respect to x are denoted by (·)x . The cantileverdensity, width and thickness are denoted respectively by ρc ,b and h . The right-hand side of equation (1) represents thehydrodynamic loading, with ρ f and  (ω), respectively, thesurrounding fluid density and the (complex) hydrodynamic
function of the problem, as defined in [34].
We introduce the tip effect approximated as a particle
mass, m tip , which leads to a shear force balance at the free endof the cantilever. The boundary conditions of a beam clamped
at x = 0, and supporting a mass particle at x = 1 are given
by [33, 35, 36]

From the above results we deduce that when the
eigenmode shapes are modified by m ∗ , the equivalent spring
constants will be affected too. On the other hand, if m ∗ = 0 werecover the eigenmode shapes and equivalent spring constants
corresponding to rectangular tipless cantilevers [15].
Figure 1(a) shows the dependence of the ratio kn /kstaticfor eigenmodes n = 1, 2 with respect to the reduced mass, m ∗ ,with kstatic ≡ 3 E I /L 3 (constant) the cantilever static spring
constant [38]. While k1 /kstatic is found to slowly decrease from1.03 to 1.00 when varying m ∗ between 0 and 0.2, k2 /kstatic
dramatically increases from 40.2 to 125. Thus, the presenceof a tip mass at the free end of the cantilever does not modify
the first eigenmode spring constant significantly, but causes
major changes in k2 . For a tip mass that is 10% of thecantilever mass, m ∗ = 0.1, the equivalent spring constantwill be k2 = 74.9kstatic . The increase in k2 with respect
to m ∗ occurs primarily because 2 (1; m ∗ ) becomes small as
2 (x; m ∗ ) begins to resemble the fundamental eigenmode ofa clamped–pinned beam for large values of m ∗ . On the other
hand, 1 (x; m ∗ ) will approach the profile of a cantilever witha static point-load applied at the free end, and thus k1 → kstatic .

W (0, ω) = Wx (0, ω) = Wxx (1, ω) = 0;
EI
Wxxx (1, ω) = −m tip ω2 W (1, ω).
L3

(3)

(2)

The eigenmodes, n (x), and the characteristic equation
for the eigenvalues, αn , follow from separation of variables
2

Nanotechnology 21 (2010) 465502

J R Lozano et al

Figure 1. (a) Normalized equivalent spring constants, kn /kstatic, and (b) effective mass factors, μn , for first (circles) and second (solid line)
eigenmodes are plotted with respect to the reduced mass, m ∗ .

(slope) of the cantilever rather than its deflection or velocity.
Thus, additional optical lever sensitivity calibration must be
performed to relate the bending angle at the laser spot location
to the deflection of the tip in each eigenmode. This can be
performed either theoretically, with knowledge of the shape ofthe eigenmode [16, 36, 40], or experimentally [41]. However,in our experimental setup (see section 3), laser Doppler
vibrometry was used to measure the cantilever’s velocity
directly, which did not require the optical lever sensitivity
calibration [42]. Consequently, in order to apply the thermal
noise method to higher eigenmodes, the equipartition theorem
had to be applied to the velocity, yielding the eigenmode
effective mass instead of the spring constant itself [43]. Finallythe latter could be calculated by mere multiplication by ωn2 ,Once we have calculated the eigenmodes of the problem,n (x; m ∗ ), we can implement the general form of the Sader
formula to calculate the fundamental and higher eigenmodes
equivalent spring constant, n  1,

kn Sader = μn

π 2
Lb ρ f Im[(ωn )]ωn2 Q n4(6)

where ωn and Q n are, respectively, the measured resonancefrequency and quality factor of the eigenmode considered. The
imaginary part of the hydrodynamic function is denoted byIm[]. The effective mass factor of the beam, μn , is defined as

1∗

μn (m ) ≡0dx [n (x; m ∗ )]2,[n (1; m ∗ )]2(7)

and will depend on the tip mass through m ∗ . Note that if inequation (6) μn is set to μstatic = 0.2427 and n = 1, we recover
the original Sader formula for kstatic [19]. On the other hand, wewould like to remark that calibration of the tipless rectangularcantilever’s first mode spring constant technically requires theapplication of equation (6) with n = 1 and μ1 = 0.25;
however, the difference is clearly small.
In figure 1(b) the dependence of μn for n = 1, 2 withrespect to m ∗ is shown. As with the equivalent spring constant,the first eigenmode effective mass factor, μ1 , tends to μstaticin the limiting case of large m ∗ . However, μ2 increases from0.25 to 1 when m ∗ changes from 0 to 0.2. Consequently,
equations (6) and (7) imply that the application of the original
Sader formula to determine k2 would entail an error of about
100% for a tip mass m tip = 0.1m beam .
The correct application of equation (6) would involve an
accurate knowledge of the eigenmode shape (equation (3)) orof the reduced mass, m ∗ . Because this is rarely the case formost commercially available probes, we recommend use of
the thermal noise method proposed by Hutter and Bechhoefer
to calibrate spring constants of higher eigenmodes of nonrectangular and/or tipped cantilevers. This method provides
the spring constant of a given eigenmode by measuring
the temperature and thermal fluctuations under the desiredresonance in the cantilever response power spectral density(PSD).
Most AFM experimental setups use a quadrature
photodiode system [39], which measures the bending angle

kn thermal =ωn2 kB T
kB T=
qn2 
q̇n2 (8)

where qn is the modal projection of the tip deflection, andwhere the symbols q and q̇ denote, respectively, statistical
average and temporal derivative of q . The Boltzmann
constant is denoted by kB , while T represents the equilibrium
temperature.

3. Methodology
To compare the Sader and thermal noise methods, thermal
vibration time series were measured by laser Doppler
vibrometry (Polytec MSA-400 Micro System Analyzer
from Polytec Gmbh, Waldbronn, Germany, laser spot size
approximately 1 μm) at the free end of three different
cantilevers: tipped rectangular, tipless rectangular and tiplesspicket, respectively. Their manufacturer, model, type, length
and width are listed in table 1 (see figure 2 for scanningelectron microscopy (SEM) images of the three cantilevers).
For each cantilever, n r velocity time series containing N points
and sampled at a frequency f s (see table 1) were measured. The
PSD of the velocity time series was estimated using Welch’s
periodogram method [44]. Each time series was divided into
n seg segments and the overall estimate of the PSD of eachtime series was taken as the average of the estimates fromeach segment. The sampling frequency f s was selected basedon each cantilever’s second natural frequency and the numberof points was chosen such that the number of points inside3

Nanotechnology 21 (2010) 465502

J R Lozano et al

Figure 2. SEM images of the cantilevers discussed in section 4. (a) Planar view of cantilever 1 (tipped rectangular). (b) Detail of the tip from
cantilever 1. (c) Planar view of cantilever 2 (tipless rectangular). (d) Planar view of cantilever 3 (tipless picket).
Table 1. Manufacturer specifications and PSD parameters for the AFM probes analyzed in this paper. L and b are, respectively, the cantilever
length and width. For each cantilever, n r time series containing N data points were sampled at a frequency f s . Each time series was thendivided into n seg segments and the PSD of each time series was estimated as the average of the PSD of each segment.
Cantilever
1
2
3

Manufacturer
Mikromasch
SandiaAppNanoModel
NSC12 F
CADP
Forta TL

L (μm)

Type
TippedTipless
Picket

250
300
213

b (μm)Nn seg

fs (MHz)

nr

35
29
36

21

128
128
128

1.024
0.512
2.560

5
4
5

2
219
223

respectively, yielding the first and second eigenmode
equivalent spring constants corresponding to eigenmodeshapes where m ∗ = 0. In that way, k1 and k2 wereobtained from the original Sader method, rather than kstatic .
Finally, to estimate kn by means of the thermal noise method,equations (8) and (9) were used.
Sader’s method is frequently applied to a quality factor
obtained from a driven/forced response (tuning curves),
whereas in this experiment a quality factor obtained froma thermal response is used. These quality factors should
theoretically be the same [19]. We have confirmed that theseare similar by comparing the thermal and forced response for
cantilever 3. An agreement in quality factor of 7% (not shown)
was found between the two values.
A third method is provided as a reference to verify theexperimental procedure and that the spring constants predictedfrom the thermal noise protocol are correct. This was achievedby using the scanning laser Doppler vibrometer to measure
the PSD at multiple points along the x -axis of the cantilever(see insets in figure 3(a) for the normalized PSD profile along
cantilever 1—tipped rectangular—at its first two eigenmodes
respectively). Then, the experimental data were fitted to thesquare of the eigenmode given in equation (3) with n =2, to directly determine the eigenvalue α2 . From this andthe resonance bandwidth is approximately the same for eachcantilever (see table 1 for the parameters used to calculate the
PSD).
Figure 3(a) depicts the experimental PSD velocity thermal
fluctuations measured at the free end of a tipped cantilever. The
first two flexural resonances are marked by arrows and also
enlarged in figures 3(b) and (c) respectively. Other peaks inthe spectrum correspond to spurious resonances and electronic
or mechanical noise. To estimate the thermal noise containedin the desired eigenmode resonance, the latter was normalized
to its maximum, PSDmax , and fitted to the PSD of a single
harmonic oscillator (see figures 3(b) and (c)). The quality
factor, Q n , and resonance frequency, f n = ωn /2π , wereobtained as fitting parameters. In order to avoid the influence
from the noise floor, the eigenmode’s thermal fluctuations invelocity were determined as the integral of the fitted PSD, ∞( f / f n )2
q̇n2  = PSDmax
df 2
. (9)
Q n [ 1 − ( f / f n )2 ] 2 + ( f / f n )20
Once the quality factor, resonance frequency and thermal
fluctuations were obtained for each eigenmode, Sader andthermal noise methods were applied for both eigenmodes of
all the three cantilevers. In particular, for the Sader method
equation (6) was used with μn = 0.25 and n = 1, 2
4

Nanotechnology 21 (2010) 465502

J R Lozano et al

Figure 3. (a) Velocity power spectral density (PSD) under thermal motion of cantilever 1 (tipped rectangular). The first two flexural
resonances are marked by arrows and enlarged in (b) first eigenmode and (c) second eigenmode. The solid lines in (b) and (c) correspond to
the fitting performed on each resonance to a single harmonic oscillator, with the eigenmode resonance frequency and quality factor as the only
free parameters. The other two cantilevers yielded similar frequency responses (see tables 1 and 2 for specific details). The insets in figure 3(a)show the PSD profiles measured along cantilever 1’s x -axis, normalized to the value at the free end, for the first and second eigenmodes.
(This figure is in colour only in the electronic version)

Table 2. First and second eigenmode resonance frequencies, f n , quality factors, Q n , and equivalent spring constants, calibrated by thermal
noise, kn,thermal , and Sader methods, kn,Sader , for the cantilevers described in table 1. Error bars are calculated as the standard deviation of theestimates, obtained from the different time series taken for each cantilever. For the resonance frequency, error bars were calculated as the
maximum between half of the PSD frequency resolution,  f = f s n seg /N , and the standard deviation.
Cantileverf 1 (kHz)Q1

k1,thermal(N m−1 )k1,Sader(N m−1 )f 2 (kHz)Q2

k2,thermal(N m−1 )k2,Sader(N m−1 )1
2
3

44.88 ± 0.03
33.00 ± 0.06
72.62 ± 0.02

150 ± 2082 ± 8139 ± 71.21 ± 0.04
0.46 ± 0.02
1.64 ± 0.04

1. 2 ± 0. 1
0.43 ± 0.04
1.63 ± 0.08

308.20 ± 0.08
204.50 ± 0.06
454.6 ± 0.1

390 ± 40270 ± 30420 ± 2076 ± 222.6 ± 0.5
55 ± 250 ± 519 ± 271 ± 4equation (5), the value of k2 /k1 was calculated, and denoted by
k2 /k1,fit (see [36] for details of the method). Note that, while
this method is using thermal noise measurements, it is the noise
profile along the beam, rather than the overall magnitude at thecantilever free end, that ultimately determines k2 . Also, thismethod is not practical for implementation on an AFM, butdoes provide a useful check of our experiments.
Finally, the SEM images in figure 2 were used to estimatem ∗ = m tip /m beam . This estimate is denoted m ∗SEM and is
calculated from the ratio of the volumes of tip and cantilever,assuming the same mass density in both elements. For
the picket cantilever, m ∗SEM (negative) is estimated from the
top view as the effective area removed from a rectangularbeam with the same length. For the tipped cantilever
the tip is approximated as half a cone and is estimatedfrom top, side, and front views. From this estimate, it ispossible to numerically compute the ratio k2 /k1 (m ∗SEM ) fromequations (3)–(5).

4. Results and conclusions
The main results are summarized in table 2. Equivalent modal
spring constants, k1 and k2 , calculated by the thermal noise andSader methods, as well as resonance frequencies and quality
factors for the first and second eigenmodes, are shown for thethree cantilevers. The corresponding value of each magnitude
was calculated as the mean of the results obtained in the
n r replicates, while the random error was characterized by
the standard deviation. The differences between stiffnesses
from Sader’s method and the thermal method reported in5

Nanotechnology 21 (2010) 465502

J R Lozano et al

The data in tables 2–4 demonstrate that there are important
limitations to the Sader method when applied to higher
eigenmodes due to their sensitivity to mass inhomogeneities at
the free end, such as tip mass or picket geometry. On the other
hand, the thermal noise method does not require knowledgeof the eigenmode shape, and, as a result, is better suited forthe calibration of higher eigenmodes. Furthermore, table 4shows that mass inhomogeneities have a strong effect on thespring constant ratio k2 /k1 , a fact which may be importantfor probe selection in AFM methods which utilize the multiple
eigenmodes [47, 48].
In conclusion, we have compared the two most popularcalibration methods, Sader and thermal noise methods,
and their application to higher eigenmode spring constantdetermination in air, by means of laser Doppler vibrometry.
Results were presented for AFM probes with two subtle, butcommon, mass inhomogeneities (tip mass and picket shape).
We have found both protocols to agree within 7% in thefirst eigenmode, while the effect of mass inhomogeneities
could lead to errors of up to 60% in the second eigenmodewhen using Sader’s method, due to deviations in the
eigenmode shape with respect to tipless rectangular cantilever
eigenmodes. Because the Sader method relies on geometricaspects and strictly considers tipless rectangular cantilevers,
we propose the thermal noise method to be the method of
choice when calibrating the spring constant of AFM probehigher eigenmodes.

Table 3. Thermal noise method versus Sader method spring constantcalibration comparison for both eigenmodes, n = 1, 2. The first andthird columns show the kn,Sader /kn,thermal ratio from table 1 values.
The second and fourth columns list the normalized deviation between
the thermal noise and Sader methods, Rn = (kn,thermal − kn,Sader )/kn,Sader , with kn,Sader representing the error bars for kn,Sader fromtable 1.
Cantilever

k1,thermal /k1,Sader

R1

k2,thermal /k2,Sader12
3

1.03 ± 0.09
1.07 ± 0.07
1.00 ± 0.04

0.1
0.7
0.1

1. 5 ± 0. 1
1. 2 ± 0. 2
0.78 ± 0.05

R2
5.2
1.8
−4.0

table 2 are summarized in table 3. The first column showsthat, regardless of the cantilever type, both methods agree
within 7% for k1 , and the second column indicates that thedifference is not statistically significant (i.e. the difference inthe means is small compared to the standard deviations)—a
result previously shown by other groups [42, 45, 46]. However,differences of up to 50% are observed when comparing thecalibration protocols for k2 [33] and these are statisticallysignificant (i.e. the difference in the means is large comparedto the standard deviation). Moreover, the greatest differences
happen for cantilever 1 (tipped rectangular, 50% error), whilecantilever 2 (tipless rectangular) exhibits a deviation of about
20%. Finally, cantilever 3 (tipless picket) presents a 22%
error between the two methods in the second eigenmode spring
constant. Our results show a clear deviation between thethermal noise and Sader methods for the second eigenmode
depending on the type of cantilever considered: the tipped
rectangular one presents greater deviations than the tiplesspicket-shaped one, and the latter shows greater deviations than
the tipless rectangular one.
Table 4 shows k2 /k1 ratios calculated from table 2 for
Sader and thermal methods, compared to the values calculatedby the third method, k2 /k1,fit . It can be seen that k2 /k1,fit andk2 /k1,thermal agree within less than 12% error, while differences
up to 35% are observed between k2 /k1 and k2 /k1,Sader ,confirming our experimental results.
Finally, the values for m ∗ estimated from the SEM images
in figure 2, m ∗SEM , and the theoretical ratio, k2 /k1 (m ∗SEM ), arepresented in the last two columns of table 4. Theoretical andthermally measured k2 /k1 ratios agree within 20% in all thecases, the main sources of error being attributed to the nonuniformity of the beam thickness along its length [33] as wellas the approximation of the mass inhomogeneity as a particle
at the free end. However, differences up to 60% are observed
between k2 /k1 (m ∗SEM ) and k2 /k1,Sader .

Acknowledgments
We acknowledge fruitful discussions with Jin W Lee andProfessor Reifenberger and assistance with data collection
from R Wagner. AR gratefully acknowledges financial
support from the National Science Foundation via grantCMMI-0927648. JRL acknowledges the hospitality andfunding offered to him by A Raman’s group at the
Birck Nanotechnology Center, Purdue University, during thecompletion of this work, as well as financial support from
Ministerio de Ciencia e Innovación (Contract No. MAT200908650) and Comunidad de Madrid (Contract No. S2009/MAT1467) in Spain.

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Table 4. First and second eigenmode equivalent spring constant ratios, k2 /k1 , from thermal noise and Sader calibration methods, arecompared with ratios predicted by equation (5), in two different ways. k2 /k1 (m ∗SEM ) is calculated by estimating the reduced tip mass, m ∗ , as
the ratio of tip and cantilever volumes, from the SEM images in figure 2. k2 /k1,fit is calculated by fitting measured second eigenmode shapes
to equation (3), to determine the eigenvalue α2 . Then, equation (5) is applied to calculate k2 /k1 . The brackets indicate upper and lower boundsof the corresponding estimate.
Cantilever

k2 /k1,thermal

k2 /k1,Saderk2 /k1,fit

k2 /k1 (m ∗SEM )

1
2
3

62.8
49.1
33.5

41.7
44.2
43.6

57.8
43.8
32.4

(45.0, 65.8)
39.0
27.7

6

m ∗SEM(0.02, 0.08)
0.00
−0.06

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