Chemistry Reference
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as shown in Fig.
5.8
c, d. Since two spatial derivatives are involved, the relative
variation in magnetic field curvature near edges and surface defects is even larger
than that in the actual field and is responsible for the high MFM resolution around
small structures. Note that since the MFM signal is proportional to the vertical mag-
netic field curvature, we expect the contrast to be maximum at edges parallel to the
current flow and, given a homogenous current density, this contrast should be sym-
metric (equal in magnitude but opposite in polarity). This illustrative analysis is a
good approximate method for determining the nature of current crowding. For more
quantitative analysis and for lines containing rather complicated defect structures,
a finite-element calculation may be used to determine the expected current density,
resulting field, and MFM signal.
A tapping AFM image and the corresponding MFM phase image for the 1
μ
m
×
1
m notch are shown in Fig.
5.9
. Given the vertical tip magnetization, there is
MFM contrast only at the line edges where the magnetic field must curve into or
out of the sample plane. In the presence of a uniform current density, the MFM
signal at the line edges must be symmetrical. This behavior is observed for lines of
constant width [
65
,
67
] and in the present lines for measurements away from the
notch. There is significantly higher contrast at the notch edge than at the line edge
on the side opposite the notch. The bold gray line profile in Fig.
5.9
b, averaged over
a0.5
μ
m segment (12 out of 512 line scans) along the symmetry point of the notch,
shows high asymmetry in the MFM peak heights. As a reference for comparison, the
thin line profile of Fig.
5.9
b, which shows the signal averaged along a 0.5
μ
mseg-
ment away from the notch, has the typical symmetry of a line with uniform current
density. This reference profile was used to determine the non-uniform background,
which was fixed by requiring that the reference line peaks have identical heights.
After background subtraction, the asymmetry in the peak heights at the notch is
found to be 1.5.
Although a rigorous inversion can be performed on the MFM, understanding of
the current crowding phenomena can be obtained from the simple model described
above. To introduce the effect of current crowding, these infinitesimally thin wires
were divided into groups as shown in the inset of Fig.
5.9
c. The current density in
each group was assigned with higher values closer to the notch. For our calculations,
five conductor segments were used: four of width 2
μ
m.
The curvature of the vertical component of the magnetic field was calculated for
each segment, the field curvatures from the segments were added together, and the
result was convolved with the estimated instrumental phase response [
68
] to account
for tip dimensional effects. Calculations using current densities with constant or
gently increasing gradients yielded near-unity asymmetries about 20% lower than
observed. A current density distribution that is much more localized to the notch
yields the calculated signal shown in the curve of Fig.
5.9
c, more consistent with
what was measured. The current distribution that yielded the calculated curve puts
about 70% of the current displaced by the 1
μ
m and one of width 1
μ
μ
μ
m adjacent seg-
ment. The observed asymmetry in the MFM signal is consistent with a highly local-
ized current crowding effect and is qualitatively similar to the numerical analysis of
Artz et al. [
10
,
69
].
m notch into the 1
