Chemistry Reference
In-Depth Information
Fig. 4.17
Work function of Pb(111) films measured as a function of film thickness. Reproduced
from [
51
]
So, the connection between work function and surface relaxation is established
(Fig.
4.18
).
Why the surface energy exhibits a different beating pattern compared to work
function and lattice relaxations is not that transparent. According to Jellium calcula-
tions by Schulte, the charge spilling is minimum when a 2D subband is just touching
the Fermi sphere and it exhibits a maximum when the Fermi level is located in the
middle of the highest occupied and the lowest unoccupied subbands at
[
18
]. This
largest charge spilling case coincides with a local minimum of
E
s
(
L
)
. Therefore
one tentatively would expect
E
s
(
and work function to oscillate in phase. How-
ever, the Fermi energy also oscillates as well as the dipole strength. So, these two
phenomena are entangled to each other and the physics of the beating would be
revealed after a self-consistent treatment of the problem. Apparently, the result of
such an analyses turns out to be a phase shift of about 4-5ML, i.e., half of a beating
wavelength.
Very recently, Miller, Chou, and Chiang presented an analytic derviation and
numerical examples for the phase relations and the beating functions in terms of
subband crossing of the Fermi level. Based on the standard quantum well model [
33
,
54
,
55
], one can derive the following central equations for the chemical potential
L
)
μ
and surface energy per unit area
E
s
:
1
N
2
d
d
N
(
ρ
ND
3
N
2
μ)
=
(4.9)
(μ)
1
N
4
d
d
N
(
3
2
ρ(μ
−
μ
bulk
)
N
4
E
s
)
=
(4.10)
