Biology Reference
In-Depth Information
8.2.1 Feedback Bandwidth and Imaging Rate
Supposing that an image is taken in time
t
for a scan range
W
s
W
with scan
lines
N
, the scan velocity
V
s
in the
x
-direction is simply given by
V
s
= 2
WN
/
t
.
For
V
s
becomes 1.6 mm/s. Here, assuming
that the sample surface has a sinusoidal shape with a periodicity
W
= 240 nm,
N
= 100 and
t
= 30 ms,
λ
in the
x
-
direction, the sample stage should move in the
z
-direction with a frequency
of
f
=
V
s
/
λ
to keep the tip-sample distance constant. When
λ
= 10 nm and
V
s
=
1.6 mm/s,
f
becomes 160 kHz. The feedback bandwidth
f
B
should be equal to
f
or higher and thus can be expressed as
f
B
r
2
WN
/
λt
(8.1)
Equation (8.1)
gives the relationship between the image acquisition time
t
f
B
. Because of the chasing-after nature of
feedback control, sample topography is always traced with a phase delay,
θ
and the feedback bandwidth
is the open-loop time delay (the
sum of time delays of devices contained in the feedback loop). The main
delays in tapping-mode AFM are the reading time of the cantilever oscillation
amplitude, the cantilever response time, the
, which
is given by ~2
s
2π
fΔτ
, where
Δτ
-scanner response time, the
integral time of error signals in the feedback controller and the parachuting
time. Here, “parachuting” means that the cantilever tip completely detaches
from the sample surface at a steep down-hill region of the sample and
thereafter takes time until it lands on the surface again. It takes at least a time
of 1/(2
z
to measure the amplitude of a cantilever that is oscillating at its
resonant frequency
f
c
)
f
c
. The response time of second-order resonant systems
such as cantilevers and piezoactuators is expressed as
f
0
are the quality factor and the resonant frequency, respectively. The feedback
bandwidth is usually deined by the feedback frequency that results in a
phase delay of π/4. With this deinition, we obtain
Q
/π
f
0
, where
Q
and
f
B
= 1/(16
Δτ
).
8.2.2 Key Devices For High-speed AFM
8.2.2.1 Canlever
Cantilevers for fast and low-invasive imaging should have a high resonant
frequency and a small spring constant. Regarding the feedback bandwidth,
it is most important that the amplitude detection time and the cantilever's
response time decrease in inverse proportion to the resonant frequency.
To realize both, i.e., a small spring constant and a high resonant frequency,
the size of cantilevers must be reduced. The small cantilevers most recently
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