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downshift may be expected, depending on the metal, the size of the particles, etc. The
magnitude of the shift DE
F
of the Fermi level for a metal particle (or film) versus the
corresponding position for the bulk material may be very roughly estimated as
[Nagaev, 1992; Grigorieva et al., 1987]
DE
F
E
F
k
F
L
(15
:
6)
where k
F
is the Fermi momentum and L is the characteristic dimension, which for a
spherical particle is equal to its diameter and for a film to its thickness. Estimation
of DE
F
for a silver cluster with L ¼ 5 nm, k
F
0
:
5 nm and E
F
5 eV gives
DE
F
0.5 eV [Grigorieva et al., 1987]. Although this is a rough estimate, it shows
that Fermi level shifts may be expected even for relatively “large” particles of a few
nanometers in size. In the case of formation of surface electronic levels, yet stronger
shifts of E
F
may be expected. It should be stressed that the above-mentioned size
dependence of the Fermi energy is often neglected in the literature, and the size depen-
dence of the work function, which has been documented in numerous publications
(see, e.g., Knickelbein et al. [1990]; Ekardt [1984]), is entirely attributed to the
variation of the surface potential (due to an effective surface dipole) with particle
size. The reader is referred to the comprehensive review article by Nagaev for an
in-depth analysis of the influence of the size confinement on the Fermi energy
[Nagaev, 1992]. Experimentally, the electronic structures of supported clusters are
generally studied with photoemission spectroscopy: X-ray photoelectron spectroscopy
(XPS) and ultraviolet photoelectron spectroscopy (UPS). The broadening of the
valence band, and the shift of the core levels and the valence band toward the
Fermi level, have been observed with increasing cluster size [Henry, 1998, 2003].
The difficulties in interpreting the changes observed in photoemission spectra arise
from various contributions associated with initial and final state effects. The former
are related to the shift of the electronic levels, i.e., they are “true” size effects, while
the latter correspond to an increase in the binding energy that is due to the nonperfect
screening of the hole created during the photoemission process [Henry, 1998, 2003].
For example, for a free cluster, this additional energy w
c
may be estimated as the
Coulomb energy associated with the charge left on the particle upon photoemission:
e
2
4p11
0
d
w
c
¼
(15
:
7)
where w
c
scales with the inverse particle diameter and often outweighs the changes
occurring because of m
e
(initial state effect). Unfortunately, some researches neglect
final state effects in XPS and UPS, leading them to erroneous conclusions.
Differentiating between initial and final state effects is difficult, but not impossible.
For further details, the reader is referred to the review by Henry [1998] and references
therein. Overall, it should be mentioned that the understanding of the size dependence
of the electronic properties of metals is still in its infancy, and concerted efforts of the-
orists and experimentalists are necessary in order to further it.
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