Chemistry Reference
In-Depth Information
electron density can further impose an oscillatory modulation onto the two previous
interactions.
Generally, for most metals, the Fermi wavelength is comparable to the lattice
spacing in a given direction. This means that if a metal film is only several atomic
layers thick, the motion of the nearly free electrons perpendicular to the film will
resemble that of the famous “particle-in-a-box” in introductory quantum mechanics.
When the “free” electrons are squeezed into a box with hard wall barriers (referring
to one dimension for simplicity), the momentum wave vector should become quan-
tized such that an integer multiple of half-wavelengths can fit inside the box. In a
very thin film the wave function of an itinerant electron is expected to obey the same
quantization condition along the film normal, the total phase difference accumulated
by the reflections must be a number of 2
π
according to the Sommerfeld-Bohr quan-
tization rule:
2
k
z
d
+
1
+
2
=
2
π
n
(4.1)
here
k
z
is the allowed wave vector component normal to the surface,
n
takes on
integer values,
d
is the thickness of the films, and
2
are the phase terms that
account for wavefunction leakage beyond the potential steps at the film boundaries,
for the vacuum films on both sides, the
1
and
1
and
2
are taken to be
π
. Furthermore we
have
k
z
=
2, where
n
is a non-zero, positive integer.
The parallel momentum states, however, are still represented by two-dimensional
n
π/
d
or, equivalently,
d
=
n
λ/
(2D) Bloch waves with a free electron-like dispersion,
E
k
0
2
m
∗
, where
m
∗
is the effective mass. In turn, the discretization of the electronic energy band can
lead to an oscillatory dependence of the film's total energy on its thickness, instead
of the linear dependence on thickness for very thick films. This oscillatory behavior
implies that a thin film of certain layers may be energetically favored over other
layers, opening a window of opportunity that an atomically flat film will form upon
annealing [
4
,
16
,
27
] (see Figs.
4.1
and
4.2
).
A more accurate analysis of quantum size effect physics requires first-principles
electronic structure calculations as a function of the film thickness. Such first-
h
2
k
0
/
=
¯
k
z
n=4
n=3
k
F
n=2
k
y
n=1
k
x
Fig. 4.1
Energy subbands of a metallic thin film
