Chemistry Reference
In-Depth Information
and above room temperature with diffusion along the step edge being the dominant
mechanism [
40
,
44
-
46
].
The temporal correlation function of a step in thermal equilibrium, for the case
where fluctuations are dominated by step edge diffusion, is
2
kT
β
2
3
4
(
1
−
1
/
4
)
4
t
1
/
4
G
(
t
)
=
(
x
(
t
)
−
x
(
0
))
=
(
n
t
)
=
g
eq
(
t
)
(5.2)
π
(
)
n
is the step mobil-
where
x
t
is the experimentally measured step edge position,
β
ity, and
is the step stiffness. Given a measurement or calculation [
47
]ofthestep
stiffness, the step mobility and thus the time constant for step edge diffusion can
be extracted from fits to (
5.2
). For close-packed steps on Ag(111), the resulting
hopping time constant is
s at room temperature. For misoriented steps, the
hopping time constant is much larger
∼
31
μ
s[
41
].
The addition of an electromigration force creates a biasing effect on the atoms
diffusing along the step edge, which can be understood by analogy to the Bales-
Zangwill fingering instability that occurs during growth [
48
]. In this case, a diffu-
sional bias perpendicular to the step edge results in spontaneous increased deviations
from the equilibrium wandering when the bias favors diffusion in the step-downhill
direction. For an uphill bias, an anomalous straightening of the step edge occurs.
This effect is quantified in a modified step continuum model that includes the effect
of an electromigration force acting perpendicular to a step that is fluctuating via step
edge diffusion [
49
]. The correlation function deviates from the equilibrium result as
∼
704
μ
t
1
/
4
1
1
/
2
3487
t
τ
EM
G
(
t
)
=
g
eq
(
t
)
±
0
.
(5.3)
where the
signs correspond to a downhill and uphill direction of force,
respectively. The electromigration force gives rise to an additional time constant,
τ
EM
, which is the time when the time correlation function begins to deviate signifi-
cantly from its equilibrium behavior:
+
and
−
a
x
a
x
a
y
β
(
k
B
T
)
τ
EM
=
τ
h
(5.4)
2
(
Fa
x
)
τ
EM
is expected to
be much longer than the fundamental time constants of step motion.
The experimental test of the prediction of (
5.3
) is shown in Fig.
5.4
[
25
]. Devi-
ations from the pure
t
1
/
4
behavior of the equilibrium fluctuations were observed
as the current direction was switched from the uphill to downhill direction. The
deviations occur at shorter times than expected based on the expected value of the
effective valence,
Z
∗
∼
Because the electromigration force is weak, the time constant
38 for steps on Ag(111). A quantitative analysis using (
5.3
)
and the approximate expression for the electromigration force,
F
Z
∗
ρ
=
j
,using
