Chemistry Reference
In-Depth Information
surface which transforms to a different 2-d phase during deposition without anneal-
ing. In Fig.
3.8
1-d diffraction patterns along the [11 2] direction are shown at 190 K.
Initially the surface is prepared in the
β(
√
3
×
√
3
)
phase with
θ
=
1
/
3ML (bottom
of Fig.
3.8
). After the addition of
∼
2
/
3ML the surface transforms to a phase with
(1
×
1) diffraction pattern (the second curve). With the addition of a total of
∼
1ML
the pattern changes to a pattern characteristic of the
-phase (with the 2/3, 2/3 spot
stronger than the 1/3, 1/3 spot, the third curve). This phase is comparable to the
phase that is obtained after thermal annealing of an initial amount 1.7ML at 500 K
(top pattern in Fig.
3.8
). It has lower intensity because the domains are smaller and
there is a large network of domain walls. The sequence of phases is the same as
the phases observed during hole refilling, by the diffusion of the outside layer on
both interfaces. They showed that the same ordered patterns are observed once the
correct coverage of Pb is present in the hole irrespective of how the atoms arrive,
by deposition or diffusion. As discussed below in detail this shows that the unusual
kinetics are not reaction front limited.
α
3.4 Modeling of the LEEM Experiments
3.4.1 Coverage-Dependent Coefficients
In the 1-d diffusion geometry of Fig.
3.1
, different profile shapes can be used to
deduce the coverage-dependent coefficient
D
c
(θ)
by the BM method if the scaling
t
1
/
2
condition
c
is met. In the LEEM experiments, the geometry of the
vacant circular hole is 2-d, so the BM can still be used if the change of the profile
shape is small compared to the hole radius
r
0
. However, since the scaling condition
is clearly not met (because of the constant speed
(
r
,
t
)
=
c
(
r
/
)
t
) the more fundamental
question was whether a still coverage-dependent coefficient
D
c
(θ)
x
/
can reproduce
the novel features described earlier.
The non-linear diffusion equation (
3.1
) in 2-d can be written as
1
∂θ
∂
2
2
∂θ
∂
r
∂θ
r
+
∂
θ
+
∂
D
c
(θ)
∂θ
t
=
D
c
(θ)
(3.7)
∂
∂
r
2
r
By applying the physical argument described before in connection with (
3.4
)
(that a flat profile implies fast diffusion and a steep profile implies slow diffusion)
the Boltzmann function is a possible functional form
D
1
−
D
2
D
c
(θ)
=
exp
θ
−
θ
c
w
+
D
2
(3.8)
1
+
This coverage-dependent
D
c
(θ)
is shown in Fig.
3.9
a.
D
c
increases rapidly from
a small value at low
to reflect the observations: the wetting
layer moves rapidly outside the hole where the coverage is high, but once it enters
θ
to a high value at high
θ
