Chemistry Reference
In-Depth Information
interaction force), which is performed by feedback operation. The resulting three-
dimensional movement of the sample stage traces the sample surface. A computer
constructs the topography image, usually from electric signals that are used to drive
the sample stage scanner in the z-direction. In the operation mode described above,
the cantilever tip is always in contact with the sample, exerting a relatively large,
undesirable lateral force on the sample due to a large cantilever spring constant in the
lateral direction. To avoid this problem, tapping-mode AFMwas invented [10] inwhich
the cantilever is oscillated in the z-direction at (or near) its resonant frequency. The
oscillation amplitude is reduced by a repulsive interaction between the tip and sample.
The amplitude signal is generated usually by an rms-to-dc converter and ismaintained
at a constant level (set point) by feedback operation.
12.3
Imaging Rate and Feedback Bandwidth
The frame acquisition time (T) is expressed by
T
¼
2NL
=
V
s
ð
12
:
1
Þ
where L is the scan size, N the number of scan lines, and V
s
the scan speed of the
sample stage in the x-direction. The maximum scan speed is limited by feedback
bandwidth. Suppose that the sample has a sinusoidal shape with a periodicity
l
.
Then, the sample stage is moved in the z-direction at a feedback frequency f (
)
tomaintain the cantilevers oscillation amplitude. Due to the chasing after nature of
the feedback operation, there is always a time delay in the feedback loop. Feedback
bandwidth ( f
B
) is usually de
ned by the feedback frequency at which the sample
topography is traced with a 45
phase delay. For a feedback bandwidth f
B
, the shortest
possible frame acquisition time, T
m
, is approximated by
T
m
¼
V
s
/
l
f
B
¼
2NL
= lf
ð
12
:
2
Þ
Of course, this is just a rough estimate as the very fragile samples are disrupted by
extra tip-sample interactions resulting from the 45
phase delay.
The 45
phase delay corresponds to a closed-loop time delay
1/(8f ). In the
feedback loop, there are various devices with
finite time delays (see Figure 12.1).
In tapping mode AFM, the delay mainly consists of the time (
Dt ¼
t
a
) required tomeasure
the cantilevers oscillation amplitude, the response time (
t
c
) of the cantilever, the
response time (
t
s
) of the z-scanner, the integral time (
t
I
) of error signals in the
feedback controller, and the parachuting time (
p
). Here, parachuting means that
the cantilever tip detaches completely from the sample surface at a steep down-hill
region of the sample and cannot quickly land on the surface again. The minimum
t
t
a
is given by 1/(2f
c
), where f
c
is the cantilevers fundamental resonant frequency.
Cantilevers and the z-scanner are second-order resonant systems. Therefore
t
c
and
t
s
are expressed by Q
c
/(
f
s
), respectively, where f
s
is the resonant frequency
of the z-scanner, andQ
c
andQ
s
are respectively the quality factors of the cantilever and
the z-scanner.
p
f
c
) and Q
s
/(
p
t
I
and
t
p
are functions of various parameters, of which approximate
