Biomedical Engineering Reference
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(a)
11
(b)
10
10
Bistable regime
8
9
6
8
4
Tap
ping (Q = 300)
Without force
Force for d = 8.5 nm
Low to high sweep
High to low sweep
Tapp
ing mode (Q = 300)
A(d)-curve
Approach
Retraction
7
2
6
0
299.6
300.0
300.4
0
2
4
6
8
10
12
Driving frequency (kHz)
Distance d (nm)
FIGURE 2.11
(See color insert.)
(a) Resonance curve for AM-mode operation if the can-
tilever oscillates near the sample surface with
d
10 nm, thereby experi-
encing the model force field given by Equation 2.9. The solid lines represent the analytical
result of Equation 2.24a, while the symbols are obtained from the numerical solution of the
equation of motion Equation 2.12. The dashed lines reflect the resonance curves without tip-
sample force and are shown purely for comparison. The resonance curve exhibits instabilities
(“jumps”) during a frequency sweep. These jumps take place at different positions (marked by
arrows) depending on whether the driving frequency is increased or decreased. (b) A hystere-
sis is also observed for amplitude versus distance curves. The dashed line shows the analytical
result, and the symbols show the numerical solutions for approach and retraction using a driv-
ing frequency of 300 kHz and the same parameters as in (a).
=
8
.
5nmand
A
0
=
2.4.4 F
ORCE
S
PECTROSCOPY
U
SING THE
AM-M
ODE
In the above subsections, we have outlined the influence of the tip-sample interac-
tion on the cantilever bending and oscillation based on the assumption of a specific
model force. However, in practical imaging, the tip-sample interaction is not
apriori
known. In contrast, the ability to measure the continuous tip-sample interaction force
as a function of the tip-sample distance would add a tool of great value to the force
microscopist's toolbox. Since the cantilever reacts to the interaction between tip and
sample in a highly nonlinear way, one might wonder how that could be done.
Surprisingly, despite the long time that the AM-mode is already used, it was only
recently that solutions to this inversion problem have been found (Holscher, 2006;
Lee & Jhe, 2006; Hu & Raman, 2008). We start our analysis by applying the trans-
formation
D
Equation 2.22a, where
D
corresponds to
the nearest tip-sample distance as defined in Figure 2.8. Next, we note that due to
the cantilever oscillation, the current method intrinsically recovers the values of the
force that the tip experiences at its lower turning point, where
F
=
d
−
A
to the integral
I
+
necessarily equals
↓
F
. We thus define
F
ts
=(
F
↓
+
F
↑
)
/
2
,
and Equation 2.22a subsequently reads as
↑
D
+
2
A
2
π
k
cant
A
2
z
−
D
−
A
A
2
I
+
=
F
ts
2
d
z
(2.25)
−
(
z
−
D
−
A
)
D
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