Chemistry Reference
In-Depth Information
Fitted Envelope Function
First-Principles
0.22
0.20
0.18
0.16
4.0
3.9
3.8
0
5
10
15
20
25
N
(ML)
Fig. 4.18
Surface energy
E
s
per surface atom and work function
W
as a function of thickness
N
of
freestanding Pb(111) films from a first-principles calculation. Also shown are envelope functions
derived from a model fit to highlight the beating patterns. The two
vertical lines
are lined up with
two adjacent nodes in
E
s
and highlight the out of phase relationship between the envelope functions
of
E
s
and
W
. Reproduced from [
53
]
where
N
,
are the film thickness, the electron density, and the density
of states per unit volume at the chemical potential, respectively.
ρ
, and
D
(μ)
μ
bulk
is the bulk
chemical potential. Equations (
4.8
) and (
4.9
) indicate that the rate of change of the
chemical potential for increasing
N
is related to the inverse of the density of states
at the chemical potential, and the rate of change of the surface energy is related to
the chemical potential.
The numerical results for Pb(111) are shown in Fig.
4.19
. The top panel shows
the evolution of the quantized electron structure
E
n
. The subbands cross
μ
with a
period of
7ML; the crossings are marked by the vertical dashed lines.
The second panel shows
D
N
=
0
.
. Each subband has a constant density of states, and
the total density of states is a series of steps. Upon normalizing by the volume, it
becomes a series of diminishing sawteeth. The third panel is
D
(μ)
(μ)
−
1
. The next panel
displays the work function relative to the bulk limit:
.
The bottom panel displays
E
s
. Clearly, the oscillations in surface energy lead the
oscillations in the work function by 1/4 of a thickness period.
W
=−
μ
≡
μ
bulk
−
μ
