Chemistry Reference
In-Depth Information
12.3.2 Surface Diffusion Limited Regime
In this regime of sublimation the simplified equations of step motion are
⎨
λ
s
⎬
−
n
s
x
j
l
j
+
n
s
x
j
−
1
l
j
−
1
2
n
s
x
j
2
l
j
+
l
j
−
1
+
2
kT
n
s
x
j
F
λ
s
+
α
d
x
j
d
t
D
s
λ
s
λ
s
kT
d
s
=
−
n
s
x
j
l
j
+
n
s
x
j
−
1
l
j
−
1
+
n
s
x
j
−
1
+
n
s
x
j
−
1
+
⎩
⎭
(12.25)
F
λ
s
Here the term in the first square brackets stabilizes the regular step distribution,
the second term accounts for the motion of the step train due to the sublimation, and
the last term destabilizes the vicinal surface when
F
0.
The linear stability analysis of (
12.25
) gives expression (
12.20
)for
s
real
with
<
v
drift
+
2
l
2
λ
d
s
l
2
n
s
B
2
=−
V
cr
α
s
V
cr
1
2
6
l
2
λ
d
s
−
α
2
l
2
n
s
B
4
=
−
2
v
drif
t
+
(12.26)
s
2
l
2
λ
s
Here again we can define an interval
V
cr
α
v
drif
t
V
cr
where the
wave number of the most unstable mode is
Fd
s
l
3
12
v
drif
t
V
cr
=
q
max
=
(12.27)
g
12.4 Non-linear Dynamics and Self-Similarity of the Bunch
Shape
The linear stability analysis provides information about the very early stages of
the step-bunching instability evolution. At long sublimation time the role of the
non-linear effects becomes essential and the theoretical treatment of the instability
should be based on (
12.17
)or(
12.25
).
12.4.1 Sublimation Controlled by Slow Surface Diffusion
Following the history of the research activity [
12
,
30
] we first analyze the subli-
mation in the surface diffusion limited regime, i.e. the case described by (
12.25
).
A relatively simple way to study the non-linear dynamics at the crystal surface is
provided by the continuum model equation, where the function
z
(
,
)
describes the
shape of the vicinal surface in the moment
t
. Going back to the very first paper on
the step bunching at crystal surfaces [
31
] we take the equation
x
t
