Biomedical Engineering Reference
In-Depth Information
recording - that is, the magnetization is parallel to the sample surface. In MFM
images, the magnetic transitions appear as dark or bright stripes corresponding
to attractive or repulsive phase gradients (Figure 15.2). However, a quantitative
characterization of the magnetic force microscope probe is especially desirable
when there is a need to ascertain the magnetic properties of the sample.
15.7.1
Quantitative Calibration of the Magnetic Force Microscope Probe
Several calibration functions have been defi ned using the magnetic force micro-
scope, with differences in variables such as probe properties, measurement mode,
stray- fi eld geometry, and so on. Among the theoretical models, we [2] and others
[46] have developed a model for the spatial distribution of a magnetic force micro-
scope tip. Transfer-function approaches have also been used to calculate the force
on a tip exerted by a sample with perpendicular magnetization [47, 48]. In numer-
ous experimental studies, the rather simple point probe model [49] has proven to
be quite successful in the understanding of MFM image formation. The point-
probe approximation idealizes a more or less complicated tip magnetization dis-
tribution by either an effective magnetic dipole moment
m
, or a magnetic monopole
moment
q
, located at an imaginary position
from the apex of the atomic force
microscope tip (Figure 15.4) [39]. For an oscillating cantilever, the phase shift
δ
is proportional to the force gradient acting on the tip (Equation 15.6), which results
from the fi rst and second derivatives of the magnetic fi eld felt by the tip (Equations
15.7 and 15.9). Based on experimental validation by SQUID [50], in almost all
models of tip calibration, it is considered that the easy axis of magnetization of
the magnetic coating lies within the plane of the coating. Thus, for a pyramidal-
shaped tip, the tip primarily has a magnetization in the
z
- direction with only a
small (10%)
y
- component, resulting in
m
x
=
m
y
= 0 and [39] :
Δ
Φ
Q
k
∂
∂
H
z
∂
∂
2
H
H
⎡
⎢
⎤
⎥
z
z
ΔΦ =
μ
0
−
q
+
m
(15.17)
z
2
which can be used to determine the tip dipole moment
q
or the monopole moment
m
z
from experimentally measured phase shifts, once the analytical expression for
the stray fi eld of the calibration sample are known. For a phase shift measured in
degrees:
180
Q
k
∂
∂
2
H
z
⎡
⎢
⎤
⎥
z
(15.18)
ΔΦ
dipole
=−
μ
m
0
z
π
2
and
180
Q
k
∂
∂
H
z
⎡
⎢
⎤
⎥
z
ΔΦ
monopole
=−
μ
−
q
(15.19)
0
π
