Chemistry Reference
In-Depth Information
analytical expressions will be given later. From these quantities, and from a fact that
the closed-loop phase delay is approximately twice the open-loop phase delay
feedback bandwidth, f
B
, is approximately expressed by
f
B
¼
a
f
c
8
2Q
c
p
2f
c
Q
s
p
=
1
þ
þ
f
s
þ
2f
c
ðt
þ t
Þ
ð
12
:
3
Þ
I
p
where
is a factor by which the derivative operation contained in feedback control
compensates for a phase delay in the feedback loop. According to our experience,
a¼
a
/(8f ) (here, the compen-
sation effect is taken into consideration), we can estimate the feedback bandwidth
and the minimum time delay for a given imaging condition. For video-rate imaging
(30 frames/s) with L
2.8. From Equation 12.2 and the relationship
Dt ¼a
¼
200 nm, N
¼
100, and
l¼
10 nm, we require f
B
¼
121 kHz and
Dt ¼
2.89
m
s.
12.4
Feedback Operation and Parachuting
To maintain the amplitude of an oscillating cantilever at a constant level while the
sample stage is being raster-scanned in the xy-directions, the detected amplitude is
compared with the set point amplitude. Their difference (error signal) is fed to a
proportional-integral-derivative (PID) feedback controller. The PID output is fed to a
voltage ampli
er to drive the z-piezoactuator. This is repeated until the error signal is
minimized. To reduce the tapping force exerted by the oscillating tip onto the sample,
the set point amplitude should be set close to the cantilevers free oscillation
amplitude. However, under this condition, the tip tends to detach completely from
the sample surface, especially at a steep down-hill region of the sample. Once
detached, the error signal is constant (i.e. saturated at a small level) irrespective of
how far the tip is separated from the sample surface. The gain parameters of the PID
controller can be increased to reduce the parachuting time. However, such large gains
in turn produce an overshoot in an up-hill region of the sample, which promotes
parachuting around the top region of the sample and leads to instability in the
feedback operation.
As seen above, parachuting is problematic, especially for high-speed bioAFM in
which the tapping force has to beminimized. During parachuting, the information of
sample topography is completely lost. The parachuting time is a function of various
parameters such as themaximumsample-height, h
0
, the peak-to-peak free oscillation
amplitude, 2A
0
, of the cantilever, the amplitude set point, r, the cantilevers resonant
frequency and the phase delay,
, in the feedback operation.
Let us
find a condition under which parachuting occurs. When the phase of the
feedback operation is delayed by
j
j
, a cantilever tip senses the residual sample
topography
D
S(t) as a function of time.
D
S(t) is expressed as
h
0
2
h
0
sin
2
cos 2
2
D
St
ðÞ¼
½
sin 2
ð
p
ft
Þ
sin 2
ð
p
ft
j
Þ
¼
p
ft
ð
12
:
4
Þ
