Chemistry Reference
In-Depth Information
the films are very different from each other. The alkali metal Na(110) and transition
metal Ag(111) films are the first type of stability, their surface energy become bulk-
like very quickly: beyond five monolayers (
L
5), their d
2
E
s
is already very small.
Another type is alkaline earth metal films, such as Mg(0001) and Be(0001) films;
the surface energy of Be(0001) films decays into bulk slower; despite the apparent
oscillations, Be(0001) films of different monolayers are stable since its second dif-
ference d
2
E
s
is almost always positive. Even its second difference d
2
E
s
becomes
negative at a few thicknesses, its absolute value is very small and does not indicate a
strong instability. This implies that, according to the computation, it is very hard to
grow atomically flat Be(0001) of any thickness in experiments. However, Al(111)
and Pb(111) films are very different from the above two types of films. First, their
second difference d
2
E
s
decays even slower. Second, their d
2
E
s
oscillates around
zero with d
2
E
s
possibly being negative. For example, Al(111) films have negative
d
2
E
s
at
L
>
5, 10, 13, and 16 monolayers, which means that Al(111) films are
unstable at these thickness. In other words, unlike Be(0001), the oscillations here in
d
2
E
s
imply oscillations in the film stability for Al(111) and Pb(111) films. Besides
the apparent similarity, Al(111) and Pb(111) films have different oscillation pat-
terns in the film stability. The stability of Pb(111) films oscillates in an even-odd
fashion interrupted by crossovers. This is exactly the oscillation pattern observed
in the stability of Pb(111) film in [
15
], although there exists difference in which
layers are stable due to the substrate effects in the experimental studies. Consider-
ing how crude the present model is, the agreement is quite amazing. Moreover, the
amplitude of the oscillations in d
2
E
s
matches well with the ab initio calculations
in. [
10
,
15
,
30
,
33
] and, more importantly, in thin film growth experiments [
8
,
31
].
The oscillation pattern in film stability is determined by the ratio between the Fermi
wavelength
=
λ
F
and the layer spacing
d
0
as we will discuss in the following sections.
For Pb(111) films, we have
λ
F
/
2
:
d
0
=
1
:
1
.
44
≈
2
:
3; for Al(111) films, we
have
4.
The model can be improved by changing the infinite potential barriers with finite
barriers, taking into account the lattice potential, allowing the electron wave func-
tion to spill into the vacuum and substrate regions [
27
,
30
]. Figure
4.6
presents the
Fermi energy of a Pb(111) film as a function of thickness with different boundaries.
In the infinite well model, when the thickness of the films is exactly equal to
Nd
0
,
E
F
(
λ
F
/
2
:
d
0
=
1
:
1
.
3
≈
3
:
rises drastically as the film thickness decreases to compensate for the areas
of electron depletion near the film boundaries. However, with a finite well, as the
energy barrier decreases, the less confinement felt by the free electrons and the
electrons can spill out of the film more easily, so such compensation is unnecessary.
Therefore, the Fermi energies
E
f
of the film decrease accordingly to its bulk value.
At the same time, the slopes of the energies decreasing with thickness become less
steep as finite well decreases. This makes the oscillations in Fermi energy
E
f
to
appear more pronounced. On the other hand, in the infinite well but allowing the
electron density to allow to spill out the classical film boundaries, the result is very
similar to the case of finite well. In addition, as indicated by the dashed line in
Fig.
4.6
, the barrier height of the well shifts the cusp positions, which is known as
phase shift [
32
].
d
)
