Chemistry Reference
In-Depth Information
12.3.1 Step Dynamics with Slow Attachment and Detachment
of Atoms
In the first limit (
d
s
l
where
l
is the average terrace width) the simplified step
dynamics equations are
−
n
s
x
j
+
n
s
x
j
−
1
+
l
j
+
l
j
−
1
+
(
)
n
s
x
j
2
d
x
j
d
t
=
K
ν
j
=
2
kT
l
j
n
s
x
j
−
n
s
x
j
−
1
K
τ
s
F
l
j
−
1
(12.17)
where the equilibrium concentrations of adatoms are
1
+
A
1
n
s
x
j
=
1
l
j
n
s
l
j
−
1
−
(12.18)
g
kT
.Here
g
is the strength of the entropic and stress-mediated
repulsion between the steps (see (
12.11
)), whereas
A
where
=
2
n
s
x
j
=
n
s
x
j
+
1
−
n
s
x
j
,
n
s
x
j
.
It is instructive to look at (
12.17
) in the special case of uniform distribution of
steps at the vicinal surface (i.e. when all terraces are equal
l
j
n
s
x
j
=
n
s
x
j
+
1
+
l
j
=
x
j
+
1
−
x
j
, and
=
l
). Then one obtains
n
s
l
τ
s
=
ν
j
(12.19)
which is the rate of a train of equidistant steps during sublimation. The other two
terms in the curly brackets describe the impact of the step-step repulsion (the first
square brackets) and electromigration of adatoms (the product of
F
and the last
square brackets). These terms (and the physical phenomena they reflect) are essen-
tial for the stability of the step train as well as for the shape of the step bunches in
the case of instability of the vicinal crystal surface.
Expression (
12.17
) provides a ground to write down equations for the time
evolution of the terrace widths
l
j
x
j
. These equations are non-linear
because of the non-linear dependence of the equilibrium adatom concentration on
the terrace widths
l
j
(see (
12.18
)). Leaving the non-linear dynamics for the next
section, here we focus our attention on the linear stability analysis. It is convenient
to introduce dimensionless variables
=
x
j
+
1
−
l
j
l
and consider small fluctuation of
η
j
=
the uniform terrace distribution
η
j
=
1
+
η
j
where
η
j
1. The time evolution
of
η
j
is approximately governed by linear equations having a solution in the form
η
j
=
where
i
is the imaginary unit and
q
is the wave number.
As seen the vicinal surface will be unstable (the fluctuations will grow with the
evaporation time) when the real part of the parameter
s
is positive. For small values
of the wave number
q
exp
(
i
jq
)
exp
(
st
)
<
1 the real part of
s
is approximately given by
