University of Groningen
Contact mode Casimir and capillary force measurements
Zwol, Pieter Jan van
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Zwol, P. J. V. (2011). Contact mode Casimir and capillary force measurements Groningen: s.n.
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2
Electrostatic calibration procedure for
force measurements with AFM
2.1 AFM force measurements
The atomic force microscope or AFM (fig. 2.1 and 2.2) is an instrument
capable of imaging specimen surfaces with a horizontal and vertical resolution
down to a fraction of a nanometer. The instrument works by measuring the
deflection produced by a sharp tip on a micron-sized cantilever as it scans
across the surface of the specimen. However, the AFM is not only a tool to
image the topography of solid surfaces at high resolution, but it can also be
used to measure force-versus-distance curves [1]. It enables measurements
in a short time span with a high spatial resolution, and also allows for nontransparent materials to be investigated. Force curves can provide valuable
information on local material properties such as elasticity, hardness,
adhesion, van-der-Waals and Casimir forces, and surface charge densities.
For this reason the measurement of force curves has become essential in
different fields of research such as surface science and engineering, and even
biology [2-6].
During force measurements with an AFM, the cantilever, having spring
constant k, bends under the load of a force and as a consequence of Hooke’s
law it obeys F=kx, if the bending x is much smaller than the length of the
lever. Nowadays pico-Newton sensitivity is well in reach of commercial AFMs.
In this chapter we describe the calibration procedure for our AFM system
used for the force measurements.
It is worth noting that besides the AFM, there exists an historically
important machine called the surface force apparatus (SFA) [2]. The SFA
employs only surfaces of known geometry, thus leading to precise surface
force measurements. The main difference between the two is the scale of the
interacting surfaces or probes, which for the SFA are much larger, giving it
higher force sensitivity and lower spatial resolution. To increase the force
sensitivity of an AFM one can attach a microsphere to the end of the lever
(see also chapter 3). Such a probe is used for the measurements presented
here.
2.2 The PicoForce system
The Picoforce system from VEECO is a multimode AFM specifically designed
for force measurements. It includes a low-noise AFM head that achieves
thermally limited pico-Newton scale performance. In addition it can perform
any form of force or contact or tapping mode surface morphology analysis.
9
The closed loop Z scanner is separated from the X and Y scanners and has a
noise level of <0.5 nm. Furthermore Piezo creep and hysteresis are reduced
to 0.1%. The Piezo is positioned under the sample instead of on the
cantilever. So the cantilever does not move with respect to the deflection
sensor and the laser. Below (fig 2.1) the Picoforce AFM is shown.
AFM
Angler
Figure 2.1: A typical Veeco Picoforce AFM. The angler tool allows ‘feeling’ of contact forces
and enables precise manual nano-positioning of the cantilever.
2.3 Stiffer or softer cantilevers?
A soft cantilever is more sensitive to weak forces since the force to noise ratio
increases with lowering k. Besides that applied coatings (for electrical contact)
result in unwanted bending (see §2.5.5), soft levers also jump to contact more
easily. Therefore, the maximum force that can be measured with a soft
cantilever will naturally be lower. If Casimir force measurements at small
separations are to be performed (i.e. strong forces), a stiffer cantilever is
needed. Stiff cantilevers will bend less, resulting in a lower force to noise ratio.
A proper way to increase the force to noise ratio without changing the
measurement range is to use bigger spheres with stiffer cantilevers. For
example a 5 times stiffer cantilever with a 5 times larger sphere will result in a
√5 increase in the force to noise ratio. Or one can just increase the amount of
statistics if time allows this.
2.4 Thermal noise and drift
Two important sources of error in force measurements are thermal noise and
drift. However the obvious solutions to these two are mutually exclusive. A
longer integration time reduces Brownian noise but increases the drift problem
and vice versa. In our setup for one force curve, drift is not a serious problem,
since it is measured within 1 second. However, drift is observed when one is
10
measuring multiple times to average noise. The point of contact, i.e. when the
cantilever hits the surface, (lowest point in the curves shown in figure 2.3)
does not remain at the same place but drifts somewhat horizontally, typically a
few nm over time. This drift can be easily corrected by shifting the point of
contact of all curves to the same place. Vertical drift seen in a graph is
probably related to cantilever, laser, mirror, and photodiode, since all these
parts may drift somewhat. This drift can be corrected by adding a fitting
constant to the equation that is fitted to the data.
Note that for old AFMs piezo hysteresis and creep can be a big
problem, which is reduced with the advent of the closed loop scanner. It is
also advisable to wait a few hours before starting the measurement with a
typical AFM in order thermally stabilize the system, because the laser
generates heat.
2.5 Electrostatic calibration of the AFM for force measurements
In a typical AFM, sensing of cantilever bending due to a force can be
performed with an interferometer or the deflection of a laser beam as shown
in fig. 2.2. The latter method is used in the Multimode Picoforce AFM. The
reflected laser goes into a photodiode which generates a voltage. Thus we
still do not know the actual force. Therefore a calibration is needed. For this
the electrostatic force is used.
We will discuss the calibration procedure for our force measurement
setup with a sphere (Duke Scientific 100±1.5 micron Au coated polystyrene)
attached on the cantilever (450 µm long, k=0.2 N/m, and Au coated). The
setup and calibration parameters are shown in fig. 2.2. For the plates we use
Si wafers with Au coatings of thicknesses 100 -1600 nm. The thicker the
evaporated gold film (deposited at room temperature and 10-6 mbar), the
rougher it becomes. The sphere is first plasma sputtered with gold for good
electrical contact, and then an 100 nm Au layer was evaporated resulting in a
roughness of 3.5 nm RMS (see also chapter 3). The detailed description of
the various calibration parameters depicted in fig. 2.2 is presented in the
following.
2.5.1 Deflection sensitivity and deflection correction
The deflection sensitivity ‘m’ relates the voltage of the photodiode to cantilever
deflection. It can be measured when the cantilever is in contact with the
surface as shown in figure 2.3, or by varying the force strength and fitting
through the contact points of different curves (the latter method is more
susceptible to drift). The piezo will continue to move while the cantilever
bends as it is in contact with the surface resulting in a voltage change. The
distance correction due to cantilever deflection should be added to the piezo
movement in a typical force measurement. Veeco states that the typical
11
uncertainty in m is 3 percent [8]. Note that a variation in ‘m’ from
measurement to measurement will result in a variation in the calibrated
stiffness of the cantilever, and the force measurement, due to an uncertainty
in distance from lever-bending.
Figure 2.2: Sphere, cantilever and laser movement are indicated with thick arrows. Calibration
parameters are also indicated in italic with question marks.
0
Deflection (nm)
−100
−200
−300
−400
−500
−600
−700
1200
1400
1600
1800
2000
Piezo (nm)
Figure 2.3: Here from left to right means that the sphere is moved closer to the surface, and at
some point contact is made. Once in contact with the surface the piezo still moves and the
cantilever will therefore further bend. In this linear regime, between the vertical dashed lines,
one can relate the voltage of the diode to the linear piezo movement and obtain the deflection
sensitivity. The fit through the contact point of different curves (thick black lines) with different
forces (for example due to different applied voltages between sphere and plate) should give the
same slope as in the contact (linear) region. (Negative deflection means that the lever bends
towards the surface)
12
2.5.2 Cantilever spring constant
The cantilever spring constant k can be measured using thermal tuning [7,8].
However, this method is only accurate within 10%, and one requires that the
sphere is much smaller than the length of the lever. Using the electrostatic
force one can determine k within roughly a few percent, but obviously that
does not work for insulating samples. The equation for the electrostatic force
Fe between a sphere and a plate is given by [9]
F e = 2πε (V1 − V0 ) 2 [∑ csc −1 ( nα )(coth α − n. coth nα )]
(2.1)
n
where α=cosh[1+(d+d0)/R], d the distance between sphere-plate, R the
sphere radius, Vo the contact potential, and V1 the applied voltage between
sphere and plate. This equation is not particularly easy to fit. Interpolating this
equation with polynomials increases fitting speed.
Measured electrostatic force curves (applied voltage of a few volts) are
fitted at separations above a few microns, where the Casimir force is
negligible (fig 2.4). In this case also the uncertainty in distance due to
unknown contact point (§2.5.4) will be small. However from the roughness
scans we obtain a rough estimate for the separation upon contact. Also there
will be no roughness effect on the electrostatic force since the roughness (in
the order of ≤10nm RMS) is negligible compared to the distance d. The
deflection correction m must be applied to the curves before fitting. Since the
electrostatic force scales with separation distance d as Fe~1/d, an error for
example of 30 nm in d will result in only 1% error in the cantilever spring
constant k. From roughness scans of sphere and plate one can estimate the
separation upon contact (d0). If ‘m’ is known then measuring k is rather
straightforward. This fitting process was found to be reproducible within a few
percent for different voltages and separations. The value can be double
checked with that obtained from the tuning method. (having the cantilever
much larger than the sphere radius in this case).
Electrostatic calibration of the cantilever spring constant is depicted
with details in figure 2.4. The data is fitted at large separations by a vector
function for +V and –V (in the order of a few Volt). In this manner the k
constant and contact potential can be obtained simultaneously. Standard
deviations can be obtained by using multiple curves and repeating the
procedure. Typical calibration curves for different films show a small variation
of the contact potential as will be shown in the next section. In fig. 2.4 inset, at
separations d>3000 nm, for the weakest electrostatic forces, drift effects
become more visible (larger difference between +V and –V curves). For
smaller separations, and larger potentials, the difference becomes smaller
(large force compared to drift). When correcting for this ‘drift offset’ by fitting,
13
the obtained contact potentials were constant with separation distance and
applied voltage, within 5 mV (standard deviation) in the range 500-7000 nm.
Fitting the data in the range 700-1500 or 2000-7000nm did not yield
differences in k beyond 1% or V0 beyond 5mV. At the smaller plate-sphere
separations the relative uncertainty in distance due to roughness becomes
larger. Below 400nm, the cantilever jumps to contact (sharp change in
scaling) due to the strong attractive force.
4 mV
1
Deflection (nm)
−10
10 mV
2
−10
−−−− +V
−−−− −V
2
−10
3
10
1000
2000
3000
4000
5000
6000
7000
Separation (nm)
Figure 2.4: Electrostatic calibration of the cantilever spring constant on a semilog scale. Fit in
the range 500-8000nm. Horizontal error is shown by grey lines. The data (dotted line) is fitted
(solid line) at large separations by a vector function for +V and –V (in the order of a few Volt)
since in this way the stiffness and contact potential can be obtained simultaneously, contact
potentials are indicated. The inset shows another sample (10mV contact potential) on log-log
scale.
2.5.3 Contact potential
An electrostatic potential Vo exists between samples of two dissimilar
electrically conductive materials (with different electron work functions) that
have been brought into thermal equilibrium with each other, usually through a
physical contact. Although Vo is normally measured between two surfaces that
are not in contact, this potential is called the contact potential difference. A
contact potential Vo is probably always present since the Au coatings on
sphere and plate can have slightly different work functions (even when both
surfaces are grounded). The contact potential Vo may not be small if badly
conducting materials are used where contributions due to trapped charges
can exist. Even in the case of Au surfaces it may be a few mV (Large
potentials between Au surfaces may indicate bad grounding however).
Actually, the parameters m, k and V0 can be measured with the same
14
electrostatic force curves. For this purpose we used eight curves with positive
and negative voltages. First m is measured and applied to the curves, and
then a vector function is fitted to two measurement curves with +V and –V (at
a separation d it gives as output the vector [V-V0, -V-V0]). In this way k and Vo
can be obtained simultaneously. For different voltages, k and Vo must of
course be the same. The latter was confirmed for k within 2 % error and for Vo
within 5 mV upon repeating the procedure (see table 2.1). If V0 became large
>100mV the procedure was somewhat less accurate upon repeating.
However, it was found to be quite robust and flexible for different cantilever
spring constants k and contact potentials V0. Note that, if V0 is large enough to
produce significant electrostatic force with respect to the Casimir force, then
the effect of Vo cannot be subtracted from the Casimir force curves, due to an
interfering signal of laser-surface scattered light which also has to be filtered
out at larger separations (§2.5.5). Thus the effect of Vo should be removed
while performing the force measurement by applying a compensating voltage.
V0 calibration for different fit ranges
(mV)
Electrostatic Scaling Parameter
Au Film
Thickness
(nm)
100
700-1500nm
2000-7000nm
700-2500
700-5000nm
1000-7500
29(2)
24(4)
1.01(0.01)
1.015(0.003)
1.017(0.003)
200
400
11(2)
27(2)
11(4)
26(6)
1.00(0.01)
1.00(0.01)
1.007(0.004)
0.999(0.004)
1.014(0.004)
1.009(0.003)
800
1600
3(2)
0(1)
5(1)
-4(2)
0.99(0.01)
0.96(0.01)
0.985(0.004)
0.980(0.005)
0.996(0.004)
0.997(0.006)
Table 2.1. The contact potential and the scaling parameter are shown for different fit ranges;
see description in the main text.
In table 2.1 the contact potential Vo is shown for different fit ranges, and
it appears to be stable within 5 mV standard deviation. The electrostatic
scaling with distance d has the form Fel~V2/dn with an expected n=1. If we
define n as a free parameter in the fitting procedure we obtain n=0.99±0.02.
This indicates that there are no large patch potentials or impurities present on
either sphere or plates [10]. A small change of the contact potential with
distance of a few mV cannot be ruled out. However this effect does not
significantly affect the Casimir force measurements presented in this thesis at
distances below 200nm.
2.5.4 Distance upon contact due to surface roughness
Although we can measure pico-Newton forces, if the distance between sphere
and plane is not known we will not reach high measurement accuracy [10].
The plate-sphere separation, which is measured with respect to the point of
contact with the surface, is given by
15
d =d piezo +d o - d defl .
(2.2)
dpiezo is the piezo movement, do is the distance on contact due to substrate
and sphere roughness, and ddefl=mFpd is the cantilever deflection correction.
Fpd is the photodiode difference signal and m the deflection coefficient. The
problem related to the measurement of d is surface roughness as it is
manifested via do. Nowadays the smoothest surfaces have in many cases
only a few nm top to bottom roughness. But even such a small roughness
may lead to large errors in the force measurement. We will discuss this topic
in relation to our Casimir force measurements. If one realizes that the Casimir
force, taking into account the finite conductivity, scales with one over the
distance squared to cubed, F∼dc (c=2-3), then to reach a 1% error in the force
measurement one requires a 0.33-0.5% error in determining the distance.
This can be understood if we consider the relative error ∆F/F=c(∆d/d).
Therefore, by simple calculation one obtains that the separation distance d
must be known to within 1 Å for example at d=30 nm. This is a difficult
requirement and a technical challenge.
The value of d0 can also be found electrostatically (fig. 2.5). Since k m,
and V0 are known, we can determine d0 by fitting electrostatic curves, using
potentials of a few hundreds of milliVolts and plate-sphere separations of up
to a few hundreds of nanometers. Since the roughness of the deposited Au
films is random (typical during deposition under conditions far from
equilibrium) there is variety in local peaks. Unfortunately the spheres typically
used in AFM measurements are also not smooth. Any applied gold coating
will result in at least ~10 nm top to bottom. The typical error in d0 is therefore
of the order of a nanometer depending on the surface roughness (tables 2.2
and 2.3).
When fitting electrostatic force curves to obtain d0 (figs. 2.5-2.7, tables
2.2, 2.3), one must first shift all curves with contact points to zero and then
substract the zero voltage or Casimir force curve from the curve with applied
potential. Now eq.(2.1) can be fitted to the data with two free parameters; one
is an offset to the force in eq.(2.1) and the other is d0. The offset parameter is
necessary to correct for any vertical drift in the data. Notably more time is
needed for the measurement of 60 electrostatic curves for the determination
of d0 (measurement time~2-3 minutes) than for the measurement of the actual
Casimir force (where 30 curves were used, measurement time < 1 minute).
No drift in the calibrated value of d0 is visible in the time frame in which we
perform our measurements (it appears more or less random).
16
Deflection (nm)
0
−5
−10
Casimir force
Electrostatic force
Fit
−15
−20
0
100
200
300
400
500
Separation (nm)
Figure 2.5: Determination of d0 by electrostatic calibration. The Casimir force and an eventual
(close to) linear signal should be subtracted from the electrostatic curves prior to fitting.
By comparison, the measured height profiles of the surfaces match
with the point of contact do found from the electrostatic calibration (figs. 2.52.7 and tables 2.2 and 2.3). In practice if the topography distribution is
symmetric one can estimate do by adding the top to bottom roughness and
divide it by two (Note that this is only a rough estimate, since the roughness is
random and the RMS roughness adds up squared). The topography
distribution for the thicker films becomes more and more skewed (more high
peaks). Then, one should estimate do from the mean and the distribution on
the right side (fig. 2.7) of the mean (corresponding to the peaks, while the left
side corresponds to the valleys).
Vapplied (mV)
V0 (mV)
3000
27.45
3500
21.90
4000
28.107
4500
20.32
K (N/m)
0.245301
0.24397
0.24195
0.242837
Applied voltage, and contact points do found from electrostatic fit of 60 curves
250
300
350
400
450
500
16.6004
17.9023
16.6795
15.8400
16.8534
16.8947
18.2507
16.8545
15.9478
17.2073
15.9018
17.0145
17.7187
18.0503
18.8961
17.6095
17.5917
18.5418
18.5290
17.2613
17.6723
17.5362
17.3446
18.3995
16.4955
17.8540
17.8284
17.4679
17.0206
16.8928
16.5543
18.4878
17.4599
17.4061
17.4984
17.3774
16.2645
17.5568
16.7768
17.5256
17.2500
17.0839
17.9401
17.2010
16.1211
17.9758
18.0913
17.1202
17.8019
17.7299
18.5228
18.8480
18.6787
18.3178
18.4538
18.2278
19.0830
17.9972
18.6736
17.7129
Vo=24±4mV, k=0.243±0.0014N/m, do=17.6±0.8nm
Table 2.2: Typical values for the spring constant, contact potential and distance calibration. In
this case the sphere had ~25-30 nm top to bottom roughness, and the plate ∼8 nm. The
calibrated value for the contact distance do is in agreement with the roughness scans.
17
d0 values for different fit ranges d0 + (nm)
Au film
100
20-400nm
17.9(0.8)
17.8(0.8)
19.4(1.3)
20.1(0.8)
23.7(0.7)
23.8(0.7)
33.6(1.6)
33.2(1.6)
50.5(1.0)
50.7(0.8)
200
400
800
1600
40-400
17.7(1.1)
18.1(0.9)
20.2(1.2)
20.8(0.9)
23.0(0.9)
23.6(0.7)
34.5(1.7)
33.9(1.5)
50.8(1.3)
50.9(0.9)
60-600
17.3(1.9)
18.3(1.0)
21.0(1.8)
22.0(1.1)
22.9(1.6)
24.4(0.7)
36.1(2.4)
34.9(1.7)
51.7(1.6)
51.5(1.0)
100-1000
16.9(3.7)
18.5(1.2)
24.0(3.2)
25.4(1.8)
25.6(3.4)
28.0(1.5)
38.3(3.0)
38.7(2.5)
53.5(3.2)
52.5(1.8)
Table 2.3: Obtained values (with standard deviation) for the contact distance do when fitting
electrostatic force curves in different ranges. Two values are shown in each cell. The first is
performed with all 60 curves, and the second with 40 curves for higher applied potentials to
improve force sensitivity.
16
12
100nm film
Number of events
1600nm film
800nm film
14
12
10
12
10
8
10
8
6
8
6
6
4
4
4
2
2
0
14
16
18
20
22
0
30
2
32
34
36
0
50
52
54
Electrostatically callibrated distance upon contact d0 (nm)
Figure 2.6: Values found for d0 for different rough films (shown in figure 2.8). The normal
distribution calculated from the standard deviation and the mean found is also shown. Errors in
the determination of the contact point do range between 0.8 and 1.6nm.
Figure 2.7: AFM topography scans and height distribution of the Au coated sphere, and the Au
coated Si plates. The measured height profiles match with the point of contact found from the
electrostatic calibration (figs. 2.6, 2.7 and tables 2.1 and 2.2).
2.5.5 The non-linear signal
A non-linear signal is present when performing force measurements with an
AFM due to additional backscattering of laser light from the approaching
surface into the photodiode. The non-linearity is not large but it cannot be
18
neglected at large separation range. If the measurement range is only a few
hundreds of nm, then this signal is approximately linear. At first glance one
might think that it completely spoils the force measurement. However, besides
that it is relatively small, in case of fitting any electrostatic curve for calibration
purposes it can be simply filtered out (by fitting with zero voltage and
subtracting the fitted result from the electrostatic curve, see fig. 2.8) together
with the Casimir force. When the Casimir force is measured, one can
approximate the signal by fitting a linear function at larger separations, where
the Casimir force is small or negligible. This is valid since the non-linearity is
only visible in the range of a few microns and above. It is thus approximately
linear at small ranges up to several hundreds of nanometers (see fig 2.5).
2
Data
Fit
Deflection (nm)
0
−2
−4
−6
−8
−6000
−4000
−2000
0
2000
Z−Piezo (nm)
Figure 2.8: Non linear signal during force measurements with an AFM due to additional
backscattering of light from the approaching surface into the photo diode.
There are several factors causing the non-linear signal. For uncoated
levers this signal was not observable. However, when a coating was applied
we did see this effect of non-linearity (depending on the cantilever stiffness
and coating thickness). A cantilever usually makes an angle of a few degrees
with the substrate. But due to induced bending (from stress in the coating, i.e.
due to elevated deposition temperatures and different material expansion
coefficients) this may not be the case anymore. As a result any reflective
surface, which may now be more parallel to the cantilever, may scatter light
into the diode as well. A solution to this problem is to use cantilevers with high
width (100 um) or large k constant (depending on the aimed force resolution)
or one can use thinner coatings. Therefore, stiffer cantilevers, thinner
coatings, and lower temperatures when evaporating or sputtering, may reduce
this signal. Coatings on both sides of the cantilever can also help.
19
2.6 Conclusion
We have outlined in detail the electrostatic calibration procedure for a
cantilever with a metal coated sphere and a metal plate. Several cross checks
for the calibration method were made, yielding consistent results. This
electrostatic calibration procedure is used throughout chapters 5-7.
References
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Cambridge (1994)
[2] J. Israelachvili, Intermolecular and Surface Forces (Academic, New York, 1992), Vol. 2, p. 330.
[3] A. Ata, Y.I. Rabinovich, R. K. Singh, J. Adhesion Sci. Technol. 16, 337 (2002)
[4] H.J. Butt, B. Capella, M. Kappl, Surf. Sci. Rep. 59 (2005)
[5] B. W. Harris, F. Chen, U. Mohideen, Phys. Rev. A. 62, 052109 (2000)
[6] M. Bordag, U. Mohideen, V. M. Mostepanenko, Phys. Rep. 353 (2001)
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[8] Practical Advice on the Determinaton of Cantilever Spring Constants (AN94)
http://www.veeco.com/library/appnotes.php?page=application&id=15; For rough surfaces this
contact method may not work. In this case due to friction the sphere cannot move laterally on the
surface. Then the deflection sensitivity may not be correct. We calibrated the deflection sensitivity
only once for the smoothest surface. From the contact points, and the linear contact regime the
values we found appear similar (Fig. 3.3 left). Any systematic error in the deflection sensitivity
would be the same for all our measurements on different rough films. The difference in slope from
the 2 methods in figure 3 is 4%.
[9] W. R. Smythe, “Electrostatics and Electrodynamics” (McGraw-Hill, New York 1950)
[10] W. Kim, M. Brown-Hayes, D. Dalvit, J. Brownell, R. Onofrio, Phys. Rev. A 78, 020101(R) (2008)
20