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Solutions for a nanoscale world.™
Practical Advice on the Determination of
Cantilever Spring Constants
By: Ben Ohler, Ph.D.,
Veeco Instruments Inc.
INTRODUCTION
Atomic force microscopy (AFM) is
being used in a great variety of
force measurement applications 1,
including investigating the unfolding
pathways of native membrane
proteins 2, probing the structure of
single polysaccharide molecules 3, and
monitoring the response of living cells
to biochemical stimuli 4. All of these
techniques rely on the accurate
determination of the cantilever spring
constant in order to yield quantitative
results. For although the cantilever
deflection can be measured with great
accuracy and sub-Angstrom
sensitivity, converting these
measurements to units of force via
Hooke’s law, F = –k · x, requires that
the spring constant, k, be determined
for each cantilever.
It has often been noted in the
literature that spring constants can
vary greatly from the values quoted
by their manufacturers. In fact, these
values are only provided as nominal
indications of the cantilever properties
and the manufacturers often specify
the spring constant in a wide range
that may span values up to four
times smaller and four times larger
than the nominal value (cf. ref. 5).
This is because the techniques used
to fabricate the probes can result in
substantially different cantilever
dimensions, especially thickness,
from wafer to wafer and smaller
variations within a single wafer 6,7.
While it is sometimes possible to
achieve tighter tolerances8, this
generally is not practical for the
economical production of probes for
general imaging and force
measurement applications.
Fortunately, this issue was
recognized early while some of
the first AFM force measurement
applications were being developed
(e.g. ref. 9) and many techniques have
since been proposed to characterize
cantilever spring constants. These
can generally be grouped into three
categories: “Dimensional models”
where fully theoretical analysis or
semi-empirical formulas are used
to calculate the cantilever spring
constants based on their dimensions
and material properties, “Static
deflection measurements” where the
spring constant is determined by
loading the cantilever with a known
static force, and “Dynamic deflection
measurements” where the
resonance behavior of the cantilever
is related back to its spring constant.
Many articles have appeared that
have discussed and compared
some of the techniques 1,10-16.
Unfortunately, many of these
consider only a small subset of the
techniques, or the discussion of
existing techniques is secondary to
introduction of a new method, or
else they are simply outdated and
don’t include important recent work.
However, among these the recent
reviews by Sader 10, Cook et al. 11,
Hutter 12, and Butt et al.1 are very
good and are highly recommended
reading. In this Application Note we
hope to expand on these works by
including: 1) a review of the most
commonly used calibration methods,
including their recent refinements,
2) advice on selecting an appropriate
method, and, 3) practical advice on
implementing the methods
discussed.
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DIMENSIONAL MODELS FOR
PREDICTING CANTILEVER
SPRING CONSTANTS
There has been a great deal of work
done on developing dimensional
models to predict cantilever spring
constants. Clifford and Seah recently
published a very good review of many
of these methods 17. However, there
are a number of challenges that
have limited the popularity of these
techniques. In particular, the
thickness of cantilevers is difficult
to measure accurately. Even with
electron microscopy, uncertainty is
rarely better than 5% 18-20. These
difficulties led at least one group to
measure the thickness with AFM by
detaching the cantilever and laying it
on a flat substrate 21 and another to
predict the cantilever’s dimensions
based on the resonance frequencies
of several of its bending modes 22.
The elastic modulus is also a major
source of uncertainty, especially for
silicon nitride cantilevers where the
exact stoichiometry, SixNy, is often
unknown 10,23-25. The thin metal
films (i.e. Cr, Au, Al) often applied
to cantilevers to enhance their
reflectivity can also affect the spring
constant and must therefore be
considered 17, 26.
In spite of these issues, the basic
dimensional models are still useful
in situations where an estimate is
adequate and in special cases where
the thickness and modulus are
known. The result for a simple
cantilevered rectangular beam,
derived from Euler-Bernoulli beam
theory 27, is given by:
k=
Ewt 3
4 L3
(1)
where E is the elastic modulus of
the cantilever, w is its width, t is its
thickness, and L is its length.
For v-shaped cantilevers, the
simplifying assumption is often made
that the cantilever is equivalent to
two parallel beams with the same
dimensions. This “parallel beam
approximation” (PBA) was first
suggested by Albrecht et al.6 and
later considered by Butt 28. Sader
was the first to recognize the
ambiguity in defining w and L for
v-shaped levers and offered his own
refinements 29, 30, resulting in his
final form:
(2)
⎡
⎤−1
Ewt 3
4 w3
⎢
k=
cos θ 1 + 3 (3 cos θ − 2)⎥
⎢
⎥
b
2 L3
⎣
⎦
where b is the width at the base of
the “V”, ␪ is half the angle between
the two legs, w is the width of the
legs measured parallel to the front
edge of the substrate (not
perpendicular to the leg edge), and
L is the length of the cantilever
measured straight out to the apex
from the substrate (i.e. not parallel
to the leg). (cf. Figures 1 and 2 of
reference 30). Neumeister and
Ducker arrived at more complex
analytical results 31, though the
modest improvement in accuracy
may be offset by the additional
complexity. These various analytical
models are frequently compared
against finite element analysis
(FEA) models 23,29-31. The FEA
models are themselves dimensional
models and subject to many of the
same limitations as the simpler
analytical results.
For rectangular levers, Cleveland
et al. realized that the thickness
could be eliminated from
Equation (1) by combining it with
an expression for the resonant
frequency of a cantilevered beam 24:
/
t ⎛⎜ E ⎞⎟
⎟
f0 ≈
⎜
2π L2 ⎜⎝ ρ ⎟⎟⎠
1 2
(3)
k≈
(
)
These methods are based on the
simple premise that the spring
constant can be measured by
applying a known force to the
cantilever and measuring its
deflection. In practice, application
of these methods is complicated by
the difficulty in accurately applying
a known force of appropriate size.
Mass calibration standards, for
instance, are not generally available
under the milligram range 32, making
them several orders of magnitude too
large for use in calibrating most AFM
cantilevers. One early attempt
at static deflection calibration placed
a mass calibration standard in a
pendulum configuration, pressed the
cantilever to its side, and compared
the deflection to the calculated force
required to displace the mass from
its resting position 28. Another
attached small (i.e. 10-50µm
diameter) tungsten spheres to
cantilevers and compared the
resulting deflection to the calculated
mass of the spheres based on their
measured diameters 33.
However, by far the most popular
technique among the static deflection
measurements is use of a calibrated
reference cantilever. Here the
cantilever to be calibrated is used to
measure a force curve on the end of
a second cantilever that is calibrated.
The slope of the contact portion of
the force curve is compared to that
measured on a hard surface (i.e. the
deflection sensitivity in nm/V) and the
spring constant calculated from:
⎛S
⎞
k = k ref ⎜⎜⎜ ref −1⎟⎟⎟
⎟⎠
⎜⎝ Shard
where is the density of the
cantilever, to obtain:
2π 3 w f 0 L ρ
STATIC DEFLECTION MEASUREMENTS
OF CANTILEVER SPRING CONSTANTS
(5)
3
(4)
E
Note that this formula appears
incorrectly in Tortonese and Kirk8
and is derived using a more exact
expression for f0 in Clifford and
Seah 17, which results in the
3
prefactor 2π being replaced by a
constant α2= 59.3061.
where kref is the spring constant of
the reference cantilever, Sref is the
deflection sensitivity measured on
the reference cantilever, and Shard is
the deflection sensitivity measured
on a hard surface.
One complication, however, arises
in how the cantilever is positioned
over the reference cantilever. The
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cantilever must be positioned so that
it contacts the reference cantilever
as close to its end as possible
because the reference cantilever
appears progressively stiffer as
loads are applied closer to its base.
If the spring constant is measured
with the tip offset from the end of
the reference cantilever, it can be
corrected by:
⎛ L ⎞⎟3
k = koff ⎜⎜
⎜⎝ L − ∆L ⎟⎟⎠
(6)
where koff is the spring constant
measured offset from the end of the
cantilever, L is the length of the
reference cantilever, and ∆L is the
distance that the tip is offset from
the end of the reference cantilever 25.
Note that this correction is exact for
rectangular cantilevers and a good
approximation (i.e. less than about
4% error 30) for v-shaped cantilevers
provided that ∆L ≤ 0.10 L. Care
should also be taken to position
the tip near the midline of the
cantilever to avoid errors due to
torsional bending 34.
Of course the key requirement for
implementing the reference
cantilever technique is that one
has a calibrated reference cantilever.
The earliest reported uses of the
technique relied on reference
cantilevers fashioned from glass
fibers, which were themselves
calibrated by optically measuring
their deflection under the load of
small calibration masses 35, 36.
Later, handmade metal cantilevers
were calibrated similarly 9, 37, 38.
Others have used a regular AFM
cantilever calibrated via a
dimensional method 20.
Fortunately, calibrated reference
cantilevers are now commercially
available (i.e. model CLFC-NOBO
at Veeco Probes 5), which are
manufactured with very well
controlled dimensions and material
properties 8 such that their spring
constants are easily calculated from
Equation (4). These probes have three
cantilevers with spring constants
ranging from 0.157 to 10.4N/m.
This is important because the spring
constant of the cantilever to be
calibrated should generally be in
the range 0.3kref < k < 3kref such that
the deflection measured is not
dominated by just one of the
cantilevers 14. This allows the
technique to be applied to a wide
range of cantilever spring constants,
which is an advantage over some
other techniques.
Also noteworthy is work being
conducted by Cumpson and
colleagues at the National Physical
Laboratory in the U.K. 34,39-42 and
Pratt and coworkers at the National
Institute of Standards and Technology
in the U.S. 32,43,44. Both groups are
developing reference standards
utilizing the static deflection
approach with the ultimate goal of
producing a device for cantilever
calibration that is traceable to the
Système International d'Unites.
DYNAMIC DEFLECTION
MEASUREMENTS OF CANTILEVER
SPRING CONSTANTS
This category includes three of
the most widely used cantilever
calibration methods: the added
mass method 24, the thermal tune
method 45, and the Sader method 26.
The physical principles on which
these methods are based differ
greatly. However they all have in
common that each require high
speed measurement of the deflection
signal in order to characterize the
cantilever resonance behavior.
Each also has distinct advantages
and disadvantages, which will be
discussed below.
The added mass method, also known
as the Cleveland method after one of
its creators, is based on the following
formula relating a cantilever’s
fundamental resonance frequency,
spring constant, and mass:
f =
1
2π
k
M + m*
Where m* is the “effective mass” of
the cantilever, a quantity proportional
to the actual mass of the cantilever,
and M is an additional mass applied
to the end of the cantilever 24. This
additional mass consists of small
(i.e. 3-10 micron diameter) tungsten
microspheres 46 that are placed as
close to the tip as possible. From
Equation (7) we see that adding
additional mass to a cantilever will
reduce its resonance frequency. Of
course the resonance frequency of
the cantilever is easily measured by
either performing a low amplitude
TappingMode™ frequency sweep or
else by looking at a power spectral
density analysis of the cantilever’s
thermally driven oscillations.
Rearranging Equation (7) gives:
M=
k
( 2π f )
2
− m*
(8)
Clearly if we were to add several
masses and measure the new
resonance frequency after each
addition we would be able to plot
2
M vs. 1/(2πf) and find k from the
slope of that line. However, one can
also calculate the spring constant
based on just one mass addition by
comparing the resonance frequency
of the original cantilever, f0, with
that of the cantilever after addition
of one mass, f1, by substituting
the appropriate quantities into
Equation (7) for the two cases and
combining them to eliminate m*:
For f=f0 and M=0:
m* =
k
2
(2π ) f 02
(9a)
For f=f1 and M=M1:
k = (2π ) f12 ( M 1 + m* )
2
(9b)
then combining to eliminate m*:
k=
(1
2
( 2π ) M 1
f12 −1 f 02 )
(9c)
(7)
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There are two major factors that
limit the accuracy of this method. The
first is that the position of the applied
masses is critical. Masses applied
closer to the base of the cantilever
have less effect on the resonance
frequency, effectively having a
smaller mass. A correction for this
effect is available, based on Equation
(6), where:
⎛ L − ∆L ⎞⎟
= M meas ⎜⎜
⎜⎝ L ⎟⎟⎠
3
M eff
(10)
where Meff is the effective mass of
the particle, Mmeas is the measured
mass of the particle (discussed
below), L is the length of the
cantilever, and ∆L is the distance
that the particle is offset from the
tip of the cantilever. Note that this
correction can be quite significant, for
instance a 20µm offset on a 200µm
long cantilever reduces the effective
mass by almost 30%.
The second significant source of
error is in measurement of the
tungsten masses. This is usually
done by measuring the diameter
of the particles with an optical
microscope, calculating their volume
based on the formula for a sphere,
3
V = (1/6)π D , and using the bulk
density of tungsten (19300 kg/m3)
to convert to mass. However, the
particles are not perfectly spherical
so it is preferable to measure their
diameter in two axes and take the
1/2
geometric average, Davg=(D1D2) .
Better yet would be to examine the
particles by SEM and calculate
the volume as an ellipsoid, though
that obviously increases the
effort required.
Despite these potential limitations
on accuracy, this method has gained
a reputation as a sort of “gold
standard” for cantilever calibration.
Considering that measurements of
the position of the sphere and its
diameter could easily each have 5%
error and each enters cubically in
the calculation, it is unclear if this
reputation is deserved. Certainly it is
a very time intensive procedure
with a high risk of damage to the
cantilevers. Overall we believe that
other methods are better choices
for routine use.
The second technique to be
discussed here is the so-called
Sader method. Note that Sader has
been quite prolific in his study of
cantilever calibration 25,26,29,30 and
dynamics47,48, but here we specifically
consider his technique for
rectangular cantilevers that requires
only their plan view (i.e. top down)
dimensions, resonance frequency,
quality factor (Q), and the density
and viscosity of the fluid in which
these are measured (typically air) 26.
This method is quite convenient, but
as formulated here it is only
applicable to rectangular cantilevers
and the frequency spectrum must
generally be measured in air such
that the response is not highly
damped (i.e. Q>>1). More
specifically, the theory requires that
L>>w>>t, though in practice it is
found that ratios of at least L/w>3
are acceptable 49.
Sader’s result then is:
k = 7.5246ρ f w2 LQ f 02Γi ( Re)
(11)
where:
Re =
2πρ f f 0 w2
4η f
where f is the density of the fluid in
which the measurement is taken
(typically air), f is the viscosity of
that fluid, Q is the quality factor of
the cantilever oscillation, and Γi is
the imaginary component of the
hydrodynamic function, which is a
function of the Reynolds number, Re.
While the other variables are familiar
from the previous discussion, these
new ones require explanation. Note
that the prefactors here differ by
2
factors of (2π) and 2π, respectively,
because Sader used the radial
frequency in his original formulas.
The resonance frequency and Q of
the cantilever are measured by
performing a power spectral density
analysis of the cantilever’s thermally
driven oscillations. The resonance
peak is then fit with the simple
harmonic oscillator (SHO) model:
A = Awhite +
A0 f 04
(f
2
−f
2
0
)
2
⎛f f ⎞
+ ⎜⎜ 0 ⎟⎟⎟
⎜⎝ Q ⎟⎠
2
(12)
where Awhite is a white noise fit
baseline, A0 is the zero frequency
amplitude, and f0 and Q are again
the resonance frequency and quality
factor. These four parameters are fit
to the data using a least-squares
method. We have found that this
method gives more consistent results
than attempting to use values taken
from a TappingMode tuning curve.
The hydrodynamic function is a more
complicated calculation. Sader gives
an analytical expression that is an
approximate correction to the exact
solution for a cylindrical beam 47.
That solution is a complex function
using modified Bessel functions of
the third kind. The function is shown
graphically in Sader et al. 26.
However, Sader also makes available
on his website 50 a Java applet to
perform the calculation and
downloadable Mathematica code. A
posting also appeared on the Veeco
SPM Digest that provided tabulated
results versus Reynolds number and
simple fit equations for these data 51.
Note that the Reynolds number
depends on the air viscosity and
density. While the viscosity will be
-5
-1 -1
1.84·10 kg m s at 20°C for all
ambient pressures, the density is
directly proportional to atmospheric
pressure and can vary significantly
from its sea level value of
-3
1.18 kg m at higher elevations 11.
Although mathematically complex,
the Sader method is very convenient
experimentally. The resonant
frequency and Q can be measured
very accurately and do not depend
on any calibration of the AFM.
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The cantilever dimensions can be
measured optically and the air
density and viscosity are essentially
constants for a given laboratory
location. Clearly the most severe
limitation is that the method is only
applicable to rectangular cantilevers.
Even nominally rectangular
cantilevers where the cantilevers
taper to a long point (e.g. OTESPA
probes 5) have been reported to give
inaccurate results 15. However, minor
deviations from rectangular shape
like clipped corners on the cantilever
ends (e.g. Veeco TESP, FESP, ESP
probes) do not seem to introduce
significant errors 15. Though this
method is not applicable to v-shaped
cantilevers, Sader does mention a
way to calibrate v-shaped cantilevers
as long as there is a rectangular
cantilever on the same probe. By
calibrating the rectangular cantilever
using the method here, the elastic
modulus and thickness of the
cantilever can be deduced from
Equation (1) and then the spring
constant of v-shaped cantilevers on
the same probe calculated with
Equation (2) using those same
values 26. However, this method is
not generally applicable because
it is rare for a probe to have both
rectangular and v-shaped
cantilevers. The known exceptions
are Veeco Microlevers, which are
silicon nitride probes that have
five v-shaped cantilevers and one
rectangular cantilever (e.g. model
MLCT-AUNM 5).
The last calibration method to be
discussed here is the thermal tune
method. This is probably the most
popular and widely available method.
It comes as a standard feature on all
Veeco systems that use the new
NanoScope® V controller. It is also
available on the MultiMode®
PicoForce™ on any NanoScope
controller. The method is based on
modeling the cantilever as a simple
harmonic oscillator. Making use of
the equipartition theorem, Hutter and
Bechhoefer related the thermal
(i.e. Brownian) motion of the
cantilever’s fundamental oscillation
mode to its thermal energy, kBT, via
the formula:
k T
k = B2
(13)
zc
where kB is the Boltzmann constant
-23
(1.38·10 J/K), T is the temperature,
2
and 〈z c 〉 is the mean square
displacement of the cantilever 45.
This quantity is found by performing
a power spectral density analysis of
the cantilever oscillations (i.e. of
the vertical deflection signal) and
integrating the area under the peak
of the fundamental mode. Analysis in
the frequency domain also has the
effect of eliminating external noise
sources from the measurement since
they will likely be either broadband
noise (e.g. white noise) that can be
baseline subtracted or else they will
occur at discrete frequencies
separated from the cantilever’s
resonance and can simply be ignored.
Later, it was realized that two
important corrections are necessary.
These corrections are sometimes
lumped together in the literature,
which makes it somewhat confusing
to compare different methods. The
first correction takes into account
that cantilevers do not behave as
ideal springs and therefore the
energy of their oscillatory modes
differs from that of a simple
harmonic oscillator. To correct for
this, Butt and Jaschke derived a
formula similar to Equation (13) by
using beam theory to explicitly
consider the actual bending modes
of the cantilever:
k=
12k BT
αi4 zi2
k BT
z12
More significantly, Butt and Jaschke
also recognized that the cantilever
displacement as measured by the
optical lever detection scheme
(i.e. using a reflected laser spot on a
photodetector) is different from the
actual displacement of the cantilever
because it is proportional to angular
changes in the cantilever position,
not its absolute deflection, and
these angular changes depend on the
bending mode of the cantilever.
They derived another formula,
accounting for both this effect and
the effect of considering only a single
bending mode:
(14a)
where for i =1 (fundamental mode):
k = 0.971
of these theoretical calculations
were experimentally verified by
independently calibrating the same
cantilever with both the fundamental
resonance and the resonance of the
second bending mode, which
resulted in good agreement between
the two results 53. For v-shaped
cantilevers this correction is more
difficult to calculate because there
is no analytical expression for the
shape of the cantilever bending
modes. Using finite element analysis,
Stark et al. examined one particular
v-shaped cantilever, the Veeco
Microlever cantilever “E”, which is
140µm long and has a nominal
spring constant of 0.1N/m. They
found that the appropriate correction
was very nearly the same, 0.965
instead of 0.971 for the rectangular
cantilever 54. While this is the result
of a numerical simulation on a
specific v-shaped cantilever, it is
probably safe to assume that there
will also not be large variations in
this correction for other common
v-shaped cantilevers.
(14b)
where αi is a constant equal to
1.8751 for i = 1, the fundamental
2
mode of a rectangular lever, and 〈 zi 〉
is the mean squared displacement of
a single bending mode 52. The results
k=
16k BT
3αi2 zi*2
⎛ sin α sinh α
i
i
⎜⎜
⎜⎜⎝ sin α + sinh α
i
⎞⎟2
⎟⎟ (15a)
⎟
i⎠
where for i =1:
k = 0.817
k BT
z1*2
(15b)
*2
where the asterisk in 〈 z1 〉 indicates
that it is the “virtual” cantilever
displacement (i.e. the displacement
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as measured by the optical lever
scheme). Note that this second
correction frequently appears as
a correction to the deflection
sensitivity, such as in Walters et al. 55
where the deflection sensitivity was
adjusted upwards by a factor of
1.09. This factor comes from
comparison of Equations (14b) and
(15b), which we can then rewrite as:
k=
0.971 k BT
χ 2 z1*2
(16)
which, when ␹ = 1.09, is equivalent
to Equation (15b). Again note that
this value is calculated based on
the assumption of a rectangular
cantilever. As with the first correction,
it is difficult to calculate the proper
correction for a v-shaped cantilever.
Though again Stark et al. have
addressed the question numerically
for the same Veeco Microlever
cantilever “E” and found that ␹ = 1.12,
which when combined with the first
correction (0.965), results in an
overall prefactor for Equation (15b) of
0.764 instead of 0.817 for rectangular
cantilevers 54. It is unknown how
much ␹ may vary for other v-shaped
cantilevers, though we don’t expect it
to vary greatly.
There have been reports noting that
the original value of ␹ = 1.09 is based
on the assumption of an infinitely
small laser spot and that more
accurate values can be calculated by
taking into account the finite spot
size, the cantilever size, and the spot
position on the cantilever. The value
of ␹ = 1.09 is the asymptotic limit
of this calculation as the spot size
goes to zero 56, 57. However we find
that this calculation only results in
significantly different values of ␹
when the cantilever length is
significantly shorter than normal, the
spot size is significantly larger than
normal, or the laser is not properly
aligned on the cantilever. For Veeco
MultiMode and Dimension™ systems
the laser spot size is typically <20µm,
so when aligned at the end of a
cantilever at least 100µm in length
the value of ␹ only varies between
1.08 and 1.09, which is insignificant
compared to other uncertainties in
the measurements. We do
acknowledge that this correction will
become more important if so-called
“small” cantilevers are used
(cf. ref. 55) where the cantilever
length is similar to the laser spot
size] or if using an instrument with a
large spot size. In these cases an
online calculator for ␹ can be found
at Schaffer’s website 58. It should also
be noted that this calculation is only
applicable to rectangular cantilevers.
Since v-shaped cantilevers tend to
have relatively small areas near the
apex on which the laser spot can be
positioned, this tends to dictate that
the spot must be small and very near
the end of the cantilever. Therefore
we predict that these considerations
would not be significant when using
these cantilevers.
Overall then the thermal tune
method is an attractive method for
spring constant calibration because
of its simplicity and its general
applicability to both rectangular and
v-shaped cantilevers. The only real
limitation to the technique is that it
is best applied to relatively soft
cantilevers where the thermal noise
is well above the noise floor of the
deflection measurement 61.
EXPECTED UNCERTAINTY IN SPRING
CONSTANT CALIBRATION
It is very difficult to define the
absolute error for any of these
calibration methods. Often,
estimates of error in the literature
are based on comparisons of one
method to another. However, we
know that the overall error is due
A NOTE ON THE EFFECT OF CANTILEVER TILT
AFM cantilevers are typically mounted at a slight angle from
horizontal in order to increase the distance between the sample
and probe substrate. It has been noted that this can lead to errors
in force measurements if proper corrections are not made 59, 60.
Specifically, if the intrinsic spring constant is used then force
2
measurements are overestimated by a factor of cos α, where
α is the tilt angle off horizontal 60, typically about 12°.
As it turns out, however, the calibrated spring constant is often
2
underestimated by a factor of cos α due to the tilt and is
therefore not the intrinsic spring constant. This occurs for both
the reference cantilever and thermal tune methods. So for these
two methods the errors cancel each other out and no further
correction is required.
The dimensional, added mass, and Sader methods, however,
result in the correct intrinsic spring constants. So spring constants
calibrated with those methods should be modified to obtain the
“effective” spring constant:
k
keff =
cos2 α
Also, note that if the tip is long compared to the cantilever length
then this can also result in a non-trivial error 60. However, most
tips only protrude <20µm from the cantilever for Si probes and
<4µm for SiN probes, so this correction is not often significant.
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to both measurement errors and
intrinsic errors and approximations
in the methods themselves. Lacking
a true “gold standard,” perhaps the
best we can do is to try to examine
the relative uncertainty of each
method separately. This will only tell
us about the absolute error of the
calibration to the extent that we
trust the model on which the method
is based. However, it is useful and
instructive to answer the question
of how much uncertainty there is in
a single calibration, i.e. how much
scatter would we expect in the
spring constant if we calibrated the
same lever many times with the
same method?
To examine this issue, Monte Carlo
calculations were performed by
preparing 10,000 data sets for each
method with estimated average
errors that were normally distributed
for each measurement type. The
spring constants for each set were
calculated and the standard deviation
of those taken as representative of
uncertainty resulting for each
method due to measurement errors.
The estimated average errors for
each measurement type are listed
in Table 1.
The results of the analysis are shown
in Table 2, which lists the estimated
uncertainty for each method due to
measurement errors along with the
particular measurements that
dominate the overall uncertainty.
The estimates for the PBA and
thermal tune methods are based
on typical values for the long, thin
(k~0.06N/m) cantilever on a silicon
nitride DNP probe. The estimates
for the other methods are based
on typical values for a silicon FESP
(k~2N/m) probe. This choice is
especially significant to the error
estimate for the PBA formula, which
suffers from the large uncertainty in
the silicon nitride elastic modulus.
The other surprisingly large error
is that for the added mass method.
Here even modest 5% uncertainty in
the particle diameter contributes
almost 15% uncertainty to the
Table 1: Estimated measurement errors
Average
error
Measurement
Note
Cantilever length
1%
Optical (e.g. 2µm out of 200µm)
Cantilever width
4%
Optical (e.g. 1µm out of 25µm)
Cantilever thickness
5%
SEM (e.g. 150nm out of 3µm)
Length offsets (∆L)
10%
Optical (e.g. 1µm out of 10µm)
Particle diameter
5-10%
Elastic modulus
5% (Si)
20% (SixNy)
Estimate
5%
Estimate
Density (Si or W)
Resonance frequency
Optical (e.g. 1µm out of 10-20µm)
0.1%
Cantilever tune, reference [26]
Quality factor
1%
SHO fit, reference [26]
Deflection sensitivity
3%
Estimate
Reference lever k
3.3%
Density of air
5%
Viscosity of air
2.5%
Temp. at cantilever
2
<z >, voltage PSD
Reference [8]
Typical atmospheric pressure changes
Changes with temperature, 20-40°C
3%
Uncertain due to laser heating
3%
Estimate
Table 2: Overall uncertainty in spring constants
Method
Uncertainty
Main source of error
Simple beam, Eqn. (1)
~16%
Cantilever thickness
PBA, Eqn. (2)
~26%
Elastic modulus of SiN
Freq. scaling, Eqn. (4)
~9%
Si density
Reference cantilever,
Eqn. (5) and (6)
~9%
Deflection sensitivity
Added mass,
Eqn. (9c) and (10)
15-30%
Particle diameter
Sader, Eqn. (11)
~4%
Cantilever width
Thermal tune, Eqn. (15a)
~8%
Deflection sensitivity
spring constant. In order to reduce
the error it would be important to
use larger particles where the
relative uncertainty in diameter is
lower. For any of the methods, it is
important to realize that the main
source of error listed in Table 1 is
the largest contributor to uncertainty
by a large margin. Any efforts to
improve accuracy should be directed
toward reducing uncertainty in
these measurements.
Although we believe that these are
reasonable uncertainty estimates,
we do emphasize that they do not
consider any weaknesses of the
methods themselves, but rather only
uncertainty in measurements. For
instance, the thermal tune method
has a fair amount on uncertainty in
its various corrections. The Sader
method assumes a perfect
rectangular cantilever, which is often
only an approximation of reality.
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The added mass method is actually
based on simple beam theory and
is only extended to v-shaped levers
by the basic parallel beam
approximation. While difficult to
quantify the uncertainty that they
contribute, it is best to understand
these various limitations.
RECOMMENDATIONS ON
SELECTING A SPRING CONSTANT
CALIBRATION METHOD
Any of the experimental methods are
well accepted and appear frequently
in the literature. Increasingly,
equipment availability is becoming
less of a limiting factor. In the past
the dynamic deflection measurement
methods frequently required external
hardware capable of higher sampling
rates than the standard AFM
controller. However, especially with
the introduction of Veeco’s new
NanoScope V controller, researchers
can now perform all of these
measurements with no additional
hardware. Even analysis has become
much simpler, now that the thermal
tune method is directly supported by
the NanoScope V and PicoForce and
the Sader method calculations can
be performed using Sader’s online
calculator 50. Selecting a spring
constant calibration method should
then be based on the limitations
of the methods, their accuracy,
and convenience.
For relatively stiff, (i.e. k > 1N/m),
rectangular cantilevers the Sader
method has a lot of advantages. The
uncertainty of the method is very
good and the measurements are
convenient to make. The reference
cantilever method can also be a
good choice, though it is somewhat
less convenient and generally has
more uncertainty.
For relatively soft cantilevers,
(i.e. k < 1N/m), the thermal tune
method is convenient and its
uncertainty is relatively low.
Moreover, many of these soft
cantilevers are v-shaped, so the
Sader method cannot be used. The
reference cantilever method can be a
reasonable alternative.
such that the nominal spring
constant of the cantilever to be
calibrated is in the range of
0.3k ref < k < 3k ref
In general, we see few reasons to
recommend the dimensional
methods and the added mass
method. There is simply too much
uncertainty in the required
measurements and the added mass
method is experimentally very
time consuming.
2. Mount the reference cantilever on
a sample disc using double-sided
tape. Align the reference
cantilever on the AFM sample
stage such that its long axis is
aligned with the cantilever to be
calibrated, but facing the
opposite direction, as shown
below in Figure 1.
Although we haven’t discussed it
here, calibration of the lateral spring
constant can also be done. This is
important for quantifying frictional
forces, for instance. While an
in-depth discussion of the methods is
beyond the scope of this Application
Note, we will note that the method
by Ogletree et al.62 and refined by
Varenberg et al.63 is well accepted.
The more recent contribution by
Green et al.64, which is very similar
to the Sader method discussed here,
also appears convenient.
IMPLEMENTING THE CALIBRATION
METHODS ON VEECO AFM SYSTEMS
REFERENCE CANTILEVER METHOD
Implementing the reference
cantilever method is straightforward
on all Veeco AFM systems. We
recommend the use of Veeco
reference cantilevers (i.e. model
CLFC-NOBO at Veeco Probes 5).
1. Calibrate the reference cantilever
as directed according to its
documentation. Briefly, the
resonance frequency of the
cantilever is measured using the
TappingMode cantilever tune and
Equation (4) is then used to
calculate the spring constant
based on the cantilever size and
material properties, which are
specified in the documentation.
Recall that Equation (4) appears
incorrectly in Tortonese and
Kirk 8. Be sure to select a
reference cantilever
Figure 1: Initial alignment of reference cantilever
3. Align the cantilever to be
calibrated over the substrate of
the reference cantilever, as
shown below in Figure 2, and
engage normally in contact
mode. Measure the deflection
sensitivity in ramp mode. It is
good practice to keep the vertical
deflection signal near 0V and
measure the deflection sensitivity
over about 100nm of deflection.
Repeat the deflection sensitivity
measurement a few times and
average the values.
Figure 2: Deflection sensitivity on hard surface
4. Withdraw and realign the
cantilever close to the end of the
reference cantilever as shown
below in Figure 3. Engage
normally in contact mode and
PAGE 8 PRACTICAL ADVICE ON THE DETERMINATION OF CANTILEVER SPRING CONSTANTS
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repeat the deflection sensitivity
measurement. Take care that the
vertical deflection signal is near
the same value as for the last
measurement and measure the
deflection sensitivity over the
same range. Again, take several
measurements and average
the results.
Figure 3: Measurement on reference cantilever
5. Measure the offset of the tip
from the end of the reference
cantilever. This is most
conveniently done by using the
top down optics of the AFM.
You can use the length of the
reference cantilever as a
standard to calibrate the
optical view.
6. Use equations (5) and (6) to
calculate the spring constant.
SADER METHOD
The Sader method requires a
power spectral density analysis
of the cantilever’s thermal noise.
This is easily measured on a Veeco
NanoScope V system by using the
included thermal tune interface,
as follows:
3. From the workspace bar, enter
the thermal tune view. If it
doesn’t show up there, you
can access it through the
Acquire>Thermal Tune menu.
If you get an error that the
deflection sensitivity is not
calibrated, you should simply go
to Tools>Calibrate>Detector…
and enter a reasonable value
(e.g. 60 nm/V). The Sader method
does not require that the
deflection signal be calibrated, so
there is no reason to do a real
deflection sensitivity calibration.
4. In the thermal tune view, select
the appropriate range for your
cantilever, either 1-100 kHz or
5-2000 kHz. Click the “Get Data”
button and wait for the
acquisition to complete.
5. You should see a peak on the
graph near the nominal
resonance frequency for the
cantilever. Sometimes you will
see more than one peak. These
might include resonances of
higher modes of the cantilever.
6. Zoom in on the peak by holding
down the Ctrl key and dragging
a box over the peak. If you need to
zoom back out, click on the
magnifying glass icon in the
bottom left corner of the graph.
The peak should look something
like the one in Figure 4 below.
7. The resonance frequency and
quality factor must be determined
by fitting the peak with the SHO
equation (Equation 12). Drag two
markers in onto the graph by
clicking between the edge of the
graph box and the vertical axis of
the graph and dragging. Position
them on each side of the peak, as
shown in Figure 4. Select the
"Simple Harmonic Oscillator
(Fluid)" fit option and click "Fit
Data." A red line showing the fit
function will appear over the
peak. The resonance frequency
and Q will appear among the
parameters to the right.
8. If you are far from sea level,
calculate the density of air using
the ideal gas equation:
ρair =
PM
RT
where P is the pressure, M is
the average molar mass of air
-1
(0.02897kg mol ), R is the molar
-1 -1
gas constant (8.314 J mol K ),
and T is the temperature. At sea
-3
level, the density is 1.18kg m .
-5
-1 -1
The viscosity is 1.86·10 kg m s .
9. Calculate k using the online
calculator at Sader’s website 50,
or using the other methods for
calculating the hydrodynamic
function discussed earlier.
Note: Be sure to use a recent version of the
NanoScope software. You can tell if you have a
new enough version by comparing your thermal
tune view to the one in Figure 4. The fluid fit
(circled) should be labeled "Simple Harmonic
Oscillator (Fluid)." If it is labeled simply "Fluid,"
please contact Veeco customer support for
an upgrade.
1. Measure the length and width
of the cantilever you wish to
calibrate. You could use the top
down optics of the AFM after
calibrating them with a standard
calibration grating.
2. Setup the AFM with the same
cantilever, zero the vertical
deflection, and configure the
software for contact mode.
You do not need to engage.
Figure 4: Using the thermal tune view to collect data for the Sader method.
PRACTICAL ADVICE ON THE DETERMINATION OF CANTILEVER SPRING CONSTANTS PAGE 9
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THERMAL TUNE METHOD
As previously mentioned, the thermal
tune method is fully supported on
Veeco NanoScope V systems. The
procedure is as follows:
1. Engage in contact mode on a
hard surface. Enter ramp mode
and calibrate the deflection
sensitivity. Again, it is good
practice to keep the vertical
deflection signal near 0V and
measure the deflection
sensitivity over about 100nm of
deflection. It is very important
for the deflection sensitivity to
be accurate.
2. Withdraw the tip by clicking the
withdraw button 2-3 times.
3. From the workspace bar, enter
the thermal tune view. If it
doesn’t show up there, you
can access it through the
Acquire>Thermal Tune menu.
4. In the thermal tune view, select
the appropriate range for your
cantilever, either 1-100kHz or
5-2000kHz. You can also enter
the temperature if it is
significantly different from the
default values. You will not
generally need to change the
“PSD bin width” and “Median
filter width” parameters.
5. The “Deflection sensitivity
correction” parameter is applied
the same way as the ␹ correction
discussed earlier. However, it
must also include the thermal
tune correction of 0.971 or 0.965
from Equation (14b). Therefore we
recommend the following values:
rectangular cantilevers: 1.106
v-shaped cantilevers: 1.144
These values result in the
appropriate prefactors as
discussed for Equation (15b).
wait for the data collection to
finish. The PSD will appear in
the graph.
7. Find the fundamental resonance
peak. There may be additional
peaks (e.g. resonances of higher
bending modes) at higher
frequencies. The nominal
resonance frequency provided
by the manufacturer should help
you find the correct peak.
8. Drag markers in onto the graph
by clicking between the edge of
the graph box and the vertical
axis of the graph and dragging.
Bracket the resonance peak such
that the markers are position on
each side and far enough away
from the peak to include some of
the baseline.
9. Select the “Air” or “Fluid” button.
This selects either a Lorentzian
or SHO model fit, respectively.
The Lorentzian fit generally fits
the data well in either air or fluid.
10. Click the “Fit data button.” A fit
curve will appear on top of the
PSD data. Verify that the fit
appears valid. Readjust the
marker positions as needed if
the fit is not good.
11. Click the “Calc spring constant”
button. A box will pop up with
the calculated spring constant.
The thermal tune view varies slightly
for PicoForce and Dimension Hybrid
systems on NanoScope IIIa and IV
controllers. The frequency range is
limited to 30kHz. However, this will
still allow you to calibrate most soft
cantilevers. Note that on Dimension
Hybrid systems you will also be
asked to manually toggle a switch
on the thermal tune adaptor.
ADDED MASS METHOD
As we noted before, this method is
not generally recommended because
of its considerable uncertainties and
the time consuming nature of
attaching the particles. However, the
general procedure is as follows:
1. Measure the resonance
frequency of the cantilever by
using the TappingMode cantilever
tune window. Use very small
drive amplitudes. You only want
to be able to accurately find
and measure the frequency of
the resonance.
2. Attach a tungsten sphere near
the end of the cantilever. It is
usually best to attach it to the tip
side of the cantilever so that you
can still align the laser on the
backside. Refer to Veeco Support
Note 226, “Attaching particles to
AFM cantilevers”, for helpful
hints on this procedure. Note that
capillary forces may be sufficient
to hold the particle in place
and therefore allow you to avoid
using an adhesive. Measure the
diameter of the particle as
accurately as possible. Also
measure the distance between
the particle’s center and the end
of the cantilever.
3. Re-measure the resonance
frequency of the cantilever.
4. Using Equations (9c) and (10),
calculate the spring constant.
5. Alternatively, remove the mass,
attach another, and re-measure
the resonance frequency. Repeat,
collecting a small data set of M
vs. f values. Perform a linear
regression analysis on this data
set, extracting k as the slope
according to Equation (8). The
averaging obtained may improve
the overall uncertainty.
6. Click the “Get data” button and
PAGE 10
PRACTICAL ADVICE ON THE DETERMINATION OF CANTILEVER SPRING CONSTANTS
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