Chemistry Reference
In-Depth Information
m
2
Fig. 5.15
Notches with different radii of curvature. (
a
) The sample topography shows (2
×
6)
μ
notches with corners of radius of curvature
R
m. (
b
) Corresponding MFM
image. (
c
) Inverted current density, calculated from the MFM measurement in (
a
). From
left
to
right
, the panels correspond to the parallel current density component, the perpendicular current
density component, and the total current density. In the image of
J
total
, the color scale was chosen
so that the brightest color corresponds to 2.0, the darkest to 0 (in units of current density that have
been normalized to the reference current density far from the slit); the scale for the horizontal
component
J
perp
=
J
x
is such that the brightest (darkest) color corresponds to 1.6 (
−
1
.
6)
=
0
.
050 and
R
=
1
.
0
μ
parameters used will only penetrate to
50-100 nm, damaged-induced effects do
not sufficiently account for the lack of a significant difference between the
R
∼
=
0
.
05
and
R
m corners. It is thus likely that the similarity between measured
current densities is a real geometric effect, where crowding is dominated by the
steep angle at which both structures deflect current flow, somewhat analogous to the
90
◦
bend and tapered structures discussed earlier.
=
1
.
0
μ
5.3.6 Current Crowding Effects
The MFM phase measurement, deconvolution, and inversion technique presented
here allows quantification of spatially variable current density with sub-micron res-
olution. Although the analytical methodology involved may introduce systematic
error into the final results, this technique remains highly suited for current density
measurements on scales relevant to electromigration studies. The measurements are
generally in good agreement with theoretical calculations which exclude disconti-
nuities due to the (unphysical) infinitely sharp edges modeled.
For all the structures examined, the magnitude of current crowding is primar-
ily determined by the abruptness of current deflection. However, the pattern of
current crowding, and in particular the dipolar behavior of the perpendicular dis-
placements, is strongly dependent on the size and sharpness of the structures. Since
electromigration-driven diffusion is highly dependent upon local current density,
current-deflecting defects will be particularly susceptible to nonlinearities, which
accelerate mass flux and line failure. Similar behavior should be observed in nano-
electronic devices, with increasing impact as even atomic-scale structural defects
will become a significant part of the cross section of a nanostructure.
