Chemistry Reference
In-Depth Information
equivalently, 2
k
F
, perpendicular to the film [
20
]. That means, the charge density
close to the surface does not share the same periodicty with the crystal along the
direction normal to the film. Hence, one would expect the ion cores at the top few
layers to be displaced slightly with respect to bulk equilibrium positions [
22
,
42
,
43
].
This behavior is observed as bilayer oscillations in the atomic layer spacing
d
12
and
d
23
in Pb(111) films via LEED-IV (intensity versus electron energy) studies [
42
].
There it was shown that for a few selected Pb thicknesses the first layer spacing
d
12
(as counted from the surface) is contracted compared to its bulk value (
d
12
is
negative) while the second layer spacing
d
23
is slightly expanded. Actually, these
results are very similar to the relaxation of the top few layers of bulk Pb as expected
from Friedel oscillations [
43
,
44
]. On the other hand, QSE might add a thickness-
dependent variation in the layer spacing to this picture due to confinement. This
effect, however, turns out to be quite small and cannot be detected by LEED method
due to limited probing depth.
Much larger relaxations have been reported based on HAS [
34
,
45
,
46
] and STM
[
24
] measurements, with values up to
−
+
15% for
d
23
around the
bulk equilibrium spacing [
45
]. Due to the nature of these techniques, however, the
observed relaxations should be attributed to the strong electronic charge density
variations around the vacuum boundary of the quantum well instead of the actual
ion core displacemets.
Floreano et al. [
46
] have measured the atomic layer relaxations inside thin films
with X-ray diffraction and X-ray reflectivity methods and concluded that the out-
ermost layer
d
12
exhibits much smaller oscillations (about 5%) compared to those
given by HAS measurements. Large number of parameters used in the fitting of
their rod scans introduce limited confidence on the results. Czoschke et al. [
42
]
have reduced the number of the parameters employing a theroretical model. They
have calculated the derivative of the charge density
30% for
d
12
and
z
inside the film using free
electron model. Assuming the lattice displacements should be proportional to this
gradient, they have obtained a displacement pattern to which the X-ray reflectivity
da
ta
is t
o
be fitted. The result obtained in this fashion for a 10ML Pb film on Si(111)
∂ρ/∂
√
3
×
√
3
R
30
◦
−
Pb gave a
d
12
contraction of 9% (a 5% contraction was found for a
9ML film (Chiang, Private Communication)). So, this model ends up with smaller
relaxations compared to HAS and STM measurements, although 9% relaxation is
still quite large. Although this model is physically intuitive and seemingly relevant,
it is probably too simplistic. Not only the charge density in the film is model depen-
dent but also the linear response of the ion cores to the charge density gradient is
presumably not valid for the large relaxations obtained.
DFT calculations of ultrathin Pb(111) slabs on a variety of substrates confirmed
the oscillatory nature of the interlayer spacing due to the 2
k
F
modulation of the
charge density propagating into the bulk, as well as the superimposed oscillations as
a function of thickness [
9
]. Figure
4.13
shows the interlayer relaxations
d
12
and
d
23
as a function of the Pb film thickness on a Ge(111) substrate [
9
]. The interlayer
relaxation in a film of
N
monolayer is defined as
d
m
,
m
+
1
−
d
0
d
m
,
m
+
1
=
×
100
d
0
(
d
m
,
m
+
1
is the interlayer spacing between the
m
and the
m
1 layers). The by now
familiar oscillation pattern including the even-odd crossovers is clearly reproduced.
DFT results show that the first interlayer spacing
d
12
is contracted by about 5.5%
+
