Chemistry Reference
In-Depth Information
The results shown in Fig.
5.12
provide clear evidence for current crowding in the
vicinity of the inner corner of the 90
◦
bend, where the maximum current density
is 1.74 as compared to regions far from the bend. There is a corresponding current
depletion in the vicinity of the outer corner of the bend. The total normalized current
density,
J
total
, across the line connecting the two corner edges (shown as a dotted red
line in Fig.
5.12
a) varies as 1
.
74
±
0
.
05 at
x
=
0 (inner corner), 1.41 at
x
=
L
/
4,
√
2
0.76 at
x
=
L
/
2, 0.45 at
x
=
3
L
/
4, and 0 at
x
=
L
, where
L
=
w
(the
45
◦
)
projection of the total current density. The horizontal current density component,
J
x
,isthesin
line width). The vertical current density component,
J
y
, is essentially a cos
(
±
45
◦
)
projection of the total current density, and we can therefore see
the horizontal current direction change at the bend (where
J
x
changes sign).
The results can be compared with the results of an a priori finite-element cal-
culation of the sheet current density for the conducting 90
◦
bend that has been
published previously [
11
]. The difficulty of this calculation lies in the mathematical
divergence at the bend, which formally results in a current density divergence near
the inner corner. As a result, the calculated maximum current density at the inner
corner varies, depending on the computational step size from 2 to 4.5 [
11
], mak-
ing comparison with experiment dubious. However, this difficulty does not affect
the calculated values away from the corner, which show an estimated normalized
current density,
J
total
, variation: 1
(
±
.
36
±
0
.
05 at
x
=
L
/
4, 0.77 at
x
=
L
/
2, 0.41 at
=
√
2
x
=
3
L
/
4, and 0 at
x
=
L
, where
L
w
(the line width). The measured current
density variation is in excellent (
10%) agreement with this calculation.
Current crowding can be qualitatively described as electrons following the path
of least resistance, that is the shortest path available, and collectively crowding along
this path, e.g., at the inner corners of bends. This kind of current crowding is particu-
larly important for the semiconductor industry, because all integrated circuits neces-
sarily incorporate changes in current path into their design. The degree of crowding
can be mitigated by creating gentler transitions and directional changes, but this
constrains designs that must also be optimized for device density and manufacturing
efficiency.
<
5.3.5.3 Effects of Defect Shape and Size
The effects of the shape of a constriction, i.e., how rapidly the change occurs, and
of the length (parallel to the current path) of the constriction are both of interest in
assessing the effect of defects on current crowding in an electromigration context.
Therefore a series of structures of systematically varying shapes were fabricated on
a single current line for direct comparison. Here we illustrate the effects of taper and
extent of the constriction.
Pang et al. have shown that the use of slowly tapering transition segments,
between the contact pads and the lines of interest in a sample, will tend to increase
the conducting lifetime of the sample [
74
], over the lifetime of samples that use
square transitions with strong bottleneck effects. Such results would suggest that the
current behavior is more severe in regions where the current density must change
abruptly. To observe the dependence of current crowding on how abruptly the cur-
