INFLUENCE OF THE MEASUREMENTBANDWIDTH

The thermal noise of the cantileveroccurs in a bandwidth near theresonance frequency of the cantilever.
Therefore the measurement bandwidth,
determined by the sampling rate andany data averaging or other ﬁltering,
can have a large effect on the observed
noise in force data. In aqueoussolutions, where many of thesepiconewton regime measurements arebeing made, the resonance frequencies
of most cantilevers are quite low,
typically less than 10 kHz. This is wellwithin the possible measurement
bandwidth of the AFM. For instance,
Veeco AFMs running on the NanoScope
V controller can capture standard force
curves at data rates of up to 40 kHz.
Using the High Speed Data Capturefeature this can be increased all theway up to50 MHz.
However, the measurement bandwidthis also determined by any data ﬁlteringthat occurs on the deﬂection signal.
This can include analog ﬁlters, digitalﬁlters, and basic data averaging (i.e.
“moving” or “boxcar” averaging).
Analog ﬁltering on the NanoScope
V controller is primarily intended toreduce aliasing effects caused byfrequency components in the signal thatexceed the Nyquist frequency, whichis half the sampling rate. Therefore,
the normal “low-speed” deﬂection
signal, sampled at 500 kHz, is ﬁltered
at 200 kHz. The “high-speed” deﬂection
signal, which can be sampled at up to50 MHz, is AC-coupled and low-passﬁltered at about 6.5 MHz.
In addition to this analog ﬁltering, adigital ﬁlter on the deﬂection signal canoptionally be used. This can be selectedusing the “LP Deﬂection” parameterfound under the “Other” parameter list.
The cutoff frequency for this ﬁlter isselectable within the range of 2–20 kHz
using the “LP Deﬂection” parameterfound under the “Feedback”
parameter list.
Finally, the sampled data can befurther ﬁltered by applying a moving

average. This can be implemented in
two different ways. First, the ramp rate
and the number of points per curve(“Number of Samples” parameter)
determine an overall data capture
rate. For instance, a 1 Hz ramp rateand 19968 points per curve combine togive the maximum data capture rate ofabout 40 kHz (1 Hz = 0.5 s per direction,
which at 19968 points per curve is about
25 µs per point, or about 40 kHz). Using
fewer points per curve simply averages
more points to downconvert thesignal to a lower bandwidth. Second,a moving average can be applied tothe data using the “Average Points”
parameter found under each “Channel”
group in the ramp mode parameter
lists. This results in similar ﬁltering of
the data but retains more points percurve, which is important when ﬁtting
functions to the data and in order tomaintain good resolution in the distanceaxis of the data.
The combined effect of the “ScanRate”, “Number of Samples”, and“Average Points” parameters results
in a parameter called “Effective BW,”
which is found under the ramp channel
parameter group. This is calculated by:Eff. BW =2  Scan Rate #Samples
Average Points

This is an estimate of the measurement
bandwidth for that channel of force
curve data. Note that the rolloff of theﬁltering that results from a moving
average differs from that of a usualﬁrst order low pass ﬁlter. That is,
the attenuation of the signal begins
at frequencies well below the cutofffrequency calculated in Eqn. ( 3 )whereas the attenuation of a normal
ﬁrst order low pass ﬁlter would only be–3 dB at the cutoff frequency. The rateof rolloff is increased as more points
are used in the average, similar to using
a higher order ﬁlter. The practical effectof these differences is that the effective
bandwidth calculated will be somewhat
higher than the actual bandwidth, whichmeans that frequency components near
the high end of the bandwidth will be
substantially attenuated.

By limiting the bandwidth in any ofthese ways it is possible to exclude
a portion of the thermal noise fromforce measurements. This is perhaps
best illustrated by considering a power
spectral density plot of the deﬂection
signal, as shown in Figure 1. Thisshows the thermal noise (blue points)
ﬁt to a simple harmonic oscillator
function (red line). Obviously the noiseoccurs in a peak centered on theresonance frequency of the cantilever.
By integrating the area under this peak
we can calculate the RMS force noise.
If we set the limits of integration to thebandwidth of our measurement we canobtain the theoretical RMS force noisefor a given bandwidth. Experimentally,
however, it is impossible to achieve ameasurement bandwidth that precisely
limits the bandwidth to the desired
range because the ﬁltering cutoff
frequencies are not inﬁnitely sharp.
Experimental RMS force noise
measurements were made with thesame cantilever used to measure
the data in Figure 1, which was therectangular cantilever on a Veeco MLCT
probe2 with a spring constant 24.2 pN/
nm. A series of measurements weremade in order to demonstrate theeffect of each method of limiting themeasurement bandwidth. According toEqn. ( 2 ), a cantilever with this spring
constant should have RMS force noiseof about 10 pN.
Figure 2A shows the effect of changing
the digital low pass ﬁlter cutoff
frequency while keeping the data
capture rate ﬁxed at 40 kHz and without
any data averaging. Because the rolloffof the digital ﬁlter is not as sharp
as that obtained by data averaging,
we see that the force noise is only
modestly decreased even at the lowest
cut off frequency. While effective for
reducing high frequency noise, thedigital low pass ﬁlter is not very well
suited for reducing the low frequency
thermal noise. Note that the total noise
measured in a bandwidth up to 20 kHz,10.9 pN, agrees well with the predictedvalue of 10 pN.

PAGE 2 FORCE RESOLUTION IN FORCE SPECTROSCOPY EXPERIMENTS: THERMAL NOISE AND THE EFFECT OF MEASURMENT BANDWIDTH

Figure 2B shows the effect of changing
the data capture rate by adjusting thenumber of points per curve. Recall that
this results in data averaging to reduce
the number of data points. We seethat for data capture rates beyond theresonance frequency that there is little
variation in the force noise. However,at rates at or below the resonancefrequency the noise levels begin to drop
substantially, ultimately to less thanhalf the original value.

Figure 1: The power spectral density plot of the deﬂection signal shows the thermal noise occuring at
the resonance frequency of the cantilever, here about 4 kHz. The columns show the RMS force noise thatshould be measured in a bandwidth from DC to the frequency shown.

Finally, Figure 2C shows the effect ofusing the “Average Points” parameterto reduce the measurement bandwidth
while holding the data capture rate anddigital low pass ﬁlter ﬁxed at 20 kHz.
The resulting RMS noise values are very
similar to those obtained for equivalent
bandwidths in Figure 2B, as theyshould be since they are essentiallythe same type of ﬁltering. However,this method offers the advantage ofkeeping the total number of points per
curve constant even as the bandwidth
is changed. As previously noted, this isoften advantageous in order to maintain
sufﬁcient distance resolution in theforce curves and to provide more points
for curve ﬁtting operations (e.g. wormlike chain ﬁts to extension data). We ﬁndthat this is the most generally useful
method of reducing the measured
thermal noise in force
spectroscopy data
INFLUENCE OF CANTILEVER SELECTION

The measured thermal noise canbe further reduced by selecting
a cantilever with a resonancefrequency beyond the measurement
bandwidth. This concept has been
exploited by groups working on “smallcantilevers”3,4. These cantilevers havemuch higher resonance frequencies
and lower viscous damping, which
reduces the measured force noise in
the measurement bandwidth compared
to conventional cantilevers.
Figure 2: Experimental measurements of RMS force noise under different measurement conditions. (A)40 kHz data capture rate, no averaging, digital low pass ﬁlter enabled (B) No averaging, 20 kHz digital
low pass ﬁlter, different data capture rates determined by number of points per curve (C) 20 kHz datacapture rate, 20 kHz digital low pass ﬁlter, different bandwidths determined by averaging points.

Though true “small cantilevers” andcompatible hardware are not currently
commercially available, some current
cantilevers do have considerably
higher resonance frequencies, even inﬂuid. In particular, Veeco’s OBL series

FORCE RESOLUTION IN FORCE SPECTROSCOPY EXPERIMENTS: THERMAL NOISE AND THE EFFECT OF MEASURMENT BANDWIDTH PAGE 3