Biomedical Engineering Reference
In-Depth Information
shift Δ
f
:
f
0
caused by the tip-sample interaction and the damping is small
compared with the resonance frequency of the free cantilever (
=
f
−
⇒
f
/
f
0
≈
1and
2Δ
ff
0
). Second, we consider that the phase shift is typically set to 90
◦
in the FM-mode. In this case, the terms on the left side are given by cos
f
2
f
0
−
≈−
(
2π
ft
0
)=
0
and sin
1. Due to these simplifications, the frequency shift and the driving
amplitude can be calculated from
(
2π
ft
0
)=
d
+
A
f
0
2
f
0
π
k
cant
A
2
z
−
d
Δ
f
(
d
,
A
)=
−
I
+
=
−
F
ts
A
2
2
d
z
(2.32a)
−
(
z
−
d
)
d
−
A
A
Q
+
Δ
E
(
d
,
A
)
a
exc
=
(2.32b)
π
k
cant
A
These two equations are valid for every type of interaction as long as the result-
ing cantilever oscillation is nearly sinusoidal. For completeness we note that Equa-
tion 2.32a coincides with the result for the FM-mode with constant-oscillation ampli-
tude mostly used in vacuum (Giessibl, 1997; Durig, 1999).
2.4.6 F
ORCE
S
PECTROSCOPY
U
SING THE
FM-M
ODE
In the above subsection, we have calculated the influence of the tip-sample inter-
action on the cantilever oscillation with the assumption that the tip-sample force is
known. In practice, however, it is just the other way around: The tip-sample inter-
action is unknown. As in the AM-mode, however, it is of the great interest for the
experimentalist to measure the tip-sample interaction force as a function of the tip-
sample distance.
To solve this problem, we again simplify the integral Equation 2.22a to the form
given in Equation 2.25. Introducing now this equation into Equation 2.32a, we obtain
the following integral equation:
+
D
2
A
f
0
√
2π
k
cant
A
3
F
ts
(
z
)
(
,
)
≈
√
z
Δ
f
D
A
D
d
z
(2.33)
/
2
−
D
The inversion of this equation now leads to the following formula for the tip-sample
interaction potential (D urig, 1999; Holscher et al., 2000)
D
+
2
A
√
2
z
)
/
z
)
√
z
−
3
2
k
cant
A
(
Δ
f
(
D
d
z
V
ts
(
D
)=
(2.34)
f
0
D
Consequently, the tip-sample force is given by
D
+
2
A
√
2
∂
∂
D
z
)
√
z
−
Δγ
(
D
d
z
F
ts
(
D
)=
−
(2.35)
D
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