Chemistry Reference
In-Depth Information
maximum slope of the bunch during the cooling time is assumed to be negligible. To
produce bunches of different sizes (containing from 13 to 160 steps) the authors used
different sublimation pulse durations (from 10 to 180 s at 1250
◦
C and from 15 to 200
min at 1145
◦
C). STM images and cross sectional profiles of step bunches provide
a ground to check the predicted by (
12.38
) and (
12.43
) scaling relation
L
H
1
/
3
.
∼
The experimental data obtained at 1250
◦
C fit the scaling relation
L
H
α
with a
∼
scaling exponent
03 in good agreement with the theory. Compar-
ing the expression for the pre-factor in (
12.38
) with the experimentally obtained
value the authors arrived to the estimation
F
α
=
0
.
68
±
0
.
10
−
6
nm
−
2
. They even went
further to estimate the electromigration force
F
and the effective electric charge
of the adatoms. In fact, they used the value
g
/
g
=
2
×
=
0
.
04 eV nm to obtain effective
electric charge
z
ef
. Having in mind the theoretical results of Akutsu
[
33
] for a strong temperature dependence of the step repulsion coefficient
g
=
0
.
14
|
e
|
(
T
)
the
1 eV nm at 1250
◦
C and obtained
z
ef
|
|
authors assumed the value
g
.
Similar experiments (but in wider temperature range) were carried out by Homma
and Aizawa [
2
]. The obtained results were in good agreement with the theoretical
expression (
12.38
).
A different kind of experiments were carried out by Stoyanov et al. [
19
]. These
authors measured the width of a bunch, containing a fixed number of steps, at dif-
ferent temperatures and interpreted the results on the basis of (
12.38
) rewritten in a
normalized form
=
0
.
=
0
.
35
e
L
3
g
(
T
)
(
T
)
I
(
T
)
T
0
)
=
(12.44)
g
(
L
(
T
0
)
I
(
T
0
)
To obtain the last relation the authors made use of the findings of Ichikawa and
Doi [
34
] that the electromigration force
F
(
T
)
is proportional to the electric current
through the Si crystal. The most reliable data were obtained in the case of
crystal-vapour equilibrium since the REM image of the Si surface in this case is
rather static - there is no steps crossing the large terraces and the requirement for
a constant number of steps in the bunch is easy to meet. An Arrhenius plot of the
temperature dependence of the right-hand side of (
12.44
) results in an activation
energy of 4.2 eV [
19
].
I
(
T
)
12.6 Concluding Remarks
The evolution of non-equilibrium morphologies via kinetic step bunching at vici-
nal crystal surfaces is an interesting phenomenon which received recently a lot of
attention. Contrary to the classical studies of surface shape evolution, driven by the
reduction of the surface free energy, here one deals with the evolution, governed
by an external driving force. The electromigration of the adatoms at the surface
of the growing (or evaporating) crystal is a key feature of the mechanism of the
step-bunching instability, induced by the electric current heating of Si crystals. The
experiments on step bunching at step-down direction of the electric current [
1
,
2
,
15
]
