Chemistry Reference
In-Depth Information
10
−
5
eV/unit cell at a current density of
10
7
A/cm
2
[
35
] is so small that it may result in a directional bias in fewer than
one in a hundred thousand diffusion steps. However, at large current densities,
∼
spacing. A typical estimate of the bias of
∼
10
8
A/cm
2
, the cumulative effects of this small bias can generate micron-scale
voids in conductors [
3
,
13
,
36
], and observable atomic-scale rearrangements in
nanoscale, electromigration structures [
27
,
30
,
31
,
34
] occur at current densities
2-3 orders of magnitude smaller.
The effective valence,
Z
∗
, is a convenient notation that encompasses the underly-
ing physics of electron scattering. This can be quantified using a simple description
of the electromigration force in terms of ballistic scattering [
2
]. A local electron
flux
j
/
e
, with electron mass
m
and speed
v
F
, the Fermi velocity, impinging on an
atom with scattering cross section
, will impart an average force
F
equal to the
product of the momentum transfer per collision and the number of collisions per
time,
F
σ
=
m
v
F
σ
j
/
e
. Relating the current density
j
to the field through the resis-
E
, yields an expression for the effective valence
Z
∗
tivity
e
.
Theoretical difficulty arises in this simple picture because the macroscopic current
density depends on the charge density
n
, drift velocity
ρ
,
j
=
ρ
=
m
v
F
σρ/
v
d
, and electron scattering
ne
2
time
τ(
j
bulk
=
nev
d
=
τ
E
/
m
=
E
/ρ
, with
ρ
the bulk resistivity), where the
τ
quantities
n
and
are known for the bulk [
37
] but poorly understood for electrons
near surfaces and interfaces [
20
,
21
]. Despite these issues, the effective valence is
a useful metric for quantifying and comparing electromigration effects in different
materials and structures. As an example, the predicted effective valence of an iso-
lated Ag adatom on Ag(111) is
Z
∗
19 [
20
]. For atoms in a close-packed site
along a step edge, with a current direction perpendicular to the step edge, the direct
force per step atom may be two times higher than the force on an adatom [
38
,
39
],
which would yield a predicted valence of
Z
∗
∼
=−
38.
One consequence of electron scattering from surface defects is a change in the
resistivity of the material. The change in the surface resistivity
ρ
s
resulting from a
wind force
F
w
acting on a scattering site density
n
k
is [
20
]
⎛
⎞
j
δ
t
f
∂ρ
s
∂
1
en
c
J
F
w
⎝
F
w
+
⎠
n
k
=−
(5.1)
where
t
f
is the material thickness,
n
c
is the local carrier density,
F
w
is the wind force
acting per defect site, and
F
w
are additional changes in force on the carriers due to
the perturbation of atomic structure in the immediate vicinity of the surface defect
sites.
5.2.2 Experimental Observation of Electromigration Effects
One approach to assess the electromigration force is to observe the biased mass
transport that it causes. To do this in a way that can be interpreted in terms of
