Biomedical Engineering Reference
In-Depth Information
where
A
is the mean-square amplitude of the driven cantilever vibration and
B
is
the detection bandwidth [30]. In FM mode, the force sensitivity is limited not only
by the cantilever thermal motion, but may also be affected by electronic noise
arising from the cantilever defl ection sensor, photodiode shot noise, and Johnson
noise.
Thus, we can see that the spring constant of a cantilever is a critical
factor in determining how sensitive a cantilever is - that is, how small a force
can be measured. Interatomic force constants in solids are in the range from
10 to about 100 N m
− 1
, and in biological samples they can be as small as
0.1 N m
− 1
. Thus, typical values for
k
in the static mode [33] are 0.01 to 5 N m
− 1
.
In the dynamic mode, if the cantilever is soft (small
k
), the point at which
the gradient of the interaction force becomes equal to
k
is reached earlier than
for a stiff cantilever. At that point, the tip jumps into contact to the sample
surface (called “ snap - down ” ), and the defl ection of the cantilever is coupled to
its deformation. Thus, the useful range of
k
for MFM is restricted to
0.1 N m
− 1
(Table 15.1). On the other hand, it is also necessary to maintain a high
resonant frequency to decrease sensitivity to mechanical disturbances and
track the sample surface more rapidly in dynamic mode. These confl icting require-
ments are generally solved by reducing the mass of the cantilever (
m
), so that
the ratio
k
/
m
is as large as possible. Further, as discussed above in dynamic
mode, a high quality factor of the cantilever
Q
is preferred to reduce thermal noise.
While increasing
Q
reduces the cantilever's sensitivity and reduces the noise [34],
it also increases the response time
t
AM
and thus restricts the bandwidth of the
system.
>
15.5.2
Magnetic Versus Topographic Signals
Besides thermal noise, various probe-sample interactions such as electrostatic
forces (proportional to 1/
z
2
), van der Waals forces (proportional to 1/
z
7
), damping
and capillary forces can contribute to the force derivative. These nonmagnetic
interactions are also present in topographic AFM imaging, and can give rise
to changes in cantilever defl ection (in static mode) or amplitude, phase and
frequency changes in the vibrating cantilever (in dynamic mode). In MFM, it
is important to separate the long-range magnetostatic coupling between probe
and sample from the contribution of other (topographic) interactions. This
is especially important when the tip is brought very close to the sample (in order
to improve MFM resolution), as the nonmagnetic forces become increasingly
stronger. A number of factors in cantilever design and/or selection and in
experimental conditions are employed to distinguish between MFM and topo-
graphic imaging.
The primary solution to this problem is to keep the topography infl uence con-
stant by letting the tip follow the sample height profi le at a user-specifi ed distance
above the sample, commonly called the “ lift height ” [27] . This mode is typically
called the “constant distance mode”, and involves measuring the topography on
