Chemistry Reference
In-Depth Information
of kinks when
θ
=
0, the density of which is fixed by geometry (and so are pro-
portional to tan
θ
). In a bond-counting model, the energetic portion of the
β(θ)
is
canceled by its second derivative with respect to
, so that the stiffness is due to the
entropy contribution alone. Away from close-packed directions, this entropy can be
determined by simple combinatoric factors at low temperature
T
[
51
-
53
].
Interest in this whole issue has been piqued by Dieluweit et al.'s finding [
54
]
that the stiffness as predicted in the above fashion, assuming that only NN inter-
actions
E
1
are important, underestimates the values for Cu(001) derived from two
independent types of experiments: direct measurement of the diffusivity on vicinal
Cu surfaces with various tilts and examination of the shape of (single-layer) islands.
The agreement of the two types of measurements assures that the underestimate
is not an anomaly due to step-step interactions. In that work, the effect of NNN
interactions
E
2
was crudely estimated by examining a general formula obtained
by Akutsu and Akutsu [
55
], showing a correction of order exp
θ
, which
was glibly deemed to be insignificant. In subsequent work the Twente group [
56
]
considered steps in just the two principal directions and showed that by including
an attractive NNN interaction, one could evaluate the step-free energies and obtain
a ratio consistent with the experimental results in [
54
]. They later extended their
calculations [
57
] to examine the stiffness.
To make contact with experiment, one typically first gauges the diffusivity along
a close-packed direction and from it extracts the ratio of the elementary kink energy
ε
k
to
T
. Arguably the least ambiguous way to relate
(
−
E
2
/
k
B
T
)
ε
k
to bonds in a lattice-gas
model is to extract an atom from the edge and place it alongside the step well away
from the new unit indentation, thereby creating four kinks [
58
]. Removing the step
atom costs energy 3
E
1
+
2
E
2
while its replacement next to the step recovers
E
1
+
2
E
2
. Thus, whether or not there are NNN interactions, we identify
ε
k
=−
2
E
1
=
0.
In the low-temperature limit, appropriate to the experiments [
54
], we have shown
that
|
E
1
|
(since the formation of Cu islands implies
E
1
<
0); thus, as necessary,
ε
k
>
2
m
m
2
k
B
T
β
(
1
−
m
)
2
+
4
me
E
2
/
k
B
T
m
+
−→
m
a
=
(2.10)
1
m
2
3
/
2
(
+
m
2
)
3
/
2
1
0
+
→
+
where
m
is the step-edge in-plane slope.
Figure
2.3
compares (
2.10
) to corresponding exact solutions at several temper-
atures when
E
2
=
E
1
/
10. We see that (
2.10
) overlaps the exact solution at tem-
peratures as high as
T
c
/
6. As the temperature increases, the stiffness becomes more
isotropic, and (
2.10
) begins to overestimate the stiffness near
0
◦
.InFig.
2.4
θ
=
ε
k
(using the experimental value [
59
]
=
128meV
⇒
E
1
=−
256meV), we com-
pare (
2.10
) to the NN Ising model at
T
320 K, as well as to the experimental
results of [
54
]. For strongly attractive (negative)
E
2
,
k
B
T
=
/ β
a
decreases signifi-
cantly. In fact, when
E
2
/
E
1
is 1/6, so that
−
E
2
/
2
k
B
T
=
(
E
2
/
E
1
)(ε
k
/
k
B
T
)
≈
/ β
(
1
/
6
)
4
.
64, the model-predicted value of
k
B
T
a
has decreased to less than half its
E
2
=
0 value, so about 3/2 the experimental ratio. For the NNN interaction alone to
