Biomedical Engineering Reference
In-Depth Information
the range of some nanonewton, the cantilever will be deflected by 0.01-100 nm. Con-
sequently, the precise detection of the cantilever bending is the key feature of AFM.
If a sufficiently sharp tip is directly attached to the cantilever, we could then measure
the interacting forces between the last atoms of the tip and the sample through the
bending of the cantilever.
2.1.1 E
XPERIMENTAL
S
ETUPS
During the last years, many experimental setups have been developed and today com-
mercial AFMs are offered from various manufactures. Although most of these instru-
ments are designed for specific applications and environments, they are typically
based on the following types of sensors, detection methods, and scanning principles.
Sensors.
Cantilevers are produced by standard microfabrication techniques (Binnig
et al., 1987; Wolter et al., 1991), mostly from silicon and silicon nitride as rectangular
or V-shaped cantilevers. Spring constants and resonance frequencies of cantilevers
depend on the actual mode of operation. For contact AFM measurements, they are
about 0.01-1 N/m and 5-100 kHz, respectively. In a typical force microscope, can-
tilever deflections in the range from 0.1 A to a few micrometers are measured. This
corresponds to a force sensitivity ranging from 10
−
13
to 10
−
5
N.
Figure 2.1 shows two scanning electron microscope (SEM) images of a typical
rectangular silicon cantilever. Using this imaging technique, the length (
l
), width
(
w
), and thickness (
t
) can be precisely measured. The spring constant
k
cant
of the
cantilever can then be determined from these values (Meyer et al., 2004)
4
t
l
E
Si
w
3
k
cant
=
(2.2)
10
1
1N/m
2
is the Youngs's modulus. Typical dimensions of sil-
icon cantilevers are as follows: lengths of 100-500
where
E
Si
=
1
.
69
×
μ
m, widths of 10-50
μ
m, and
thicknesses of 0.3-5
m.
The torsion of the cantilever due to lateral forces between tip and surface depends
also on the height of the tip
h
. The torsional spring constant can be calculated from
(Meyer et al., 2004)
μ
wt
3
lh
2
G
3
k
tor
=
(2.3)
10
11
N/m is the shear modulus of silicon.
Since the dimensions of cantilevers given by the manufacturer are only average
values, high accuracy calibration of the spring constant requires the measurement
of length, width, and thickness for each individual cantilever. The length and the
width can be measured with sufficient accuracy using an optical microscope, but
the thickness requires high-resolution techniques like SEM. To avoid this time- and
cost-consuming measurement, one can determine the cantilever thickness from its
eigenfrequency in normal direction (L uthi et al., 1995; Meyer et al., 2004; Bhushan
where
G
Si
=
0
.
68
×
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