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the even layer thicknesses N = 6 and 8 in the same energy range. These peaks represent quantum
well states formed by confinement of the Pb p-band electrons by the Si band gap. No such
quantum well states exist for N = 6 and 8, as will be explained below.
The appearance of intense peaks for odd N only has to do with the specific band structure
of Pb and the Si band gap. From data taken over a wide thickness range, it is concluded that the
Fermi level of the Pb is at 0.5 eV above the Si valence band edge. Electrons in the Pb film with
binding energies within 0.5 eV of the Fermi level are thus confined by the Si band gap, giving
rise to sharp and intense quantum well peaks. Electrons at higher binding energies are not
confined. Nevertheless, partial reflection at the Pb-Si boundary can give rise to resonances which
appear in photoemission as weak and broad peaks [3]. One such weak resonance peak at a
binding energy of 0.63 eV can be seen in the three-dimensional plot of Fig. 1 at N = 8. Further
evidence for the confinement edge is seen in the line scan of the peak at N = 5 in Fig. 1. This
peak, with a binding energy very close to the confinement edge, is asymmetric. Its higherbinding-energy
side is substantially broader because this portion of the spectral weight lies
outside the confinement range. From the confinement range of 0.5 eV and the known Si band
gap of 1.2 eV, a Schottky barrier height of 0.7 eV is deduced for our n-type Si substrate. This is
consistent with a recent reported value of 0.72 0.02 eV based on electrical measurements [4].
The binding energies of quantum well states are determined by the Bohr-Sommerfeld
quantization rule [2, 3, 5]:
2kNt
s
i
2n
, (1)
6 where k is the wave vector, t is the monolayer
0
thickness,
s
is the phase shift at the surface, i
is
5
the phase shift at the interface, and n is a quantum
1
number. For a given N and integer quantum
4 numbers, this equation determines the allowed k
values, which in turn determine the binding energies
2
3 of the quantum well states through the band
dispersion relation E(k). For Pb(111), the relevant
3
2 band is the p valence band with a known dispersion
that extends from 4.2 eV below the Fermi level to
Binding Energy (eV)
4 n = 0
0 2 4 6 8 10 12
Thickness N (ML)
Figure 2: Binding energies of quantum
well states (circles connected by lines)
deduced from a fit to the experimental
results (crosses). The band structure of
Pb and the phase shifts enter the model
calculation. The only fitting parameter
is an unknown constant offset in the
interface phase shift. The quantum
number n for each branch is indicated.
1
8.0 eV above the Fermi level [5]. The surface phase
shift
s
as a function of energy has been computed
by a first-principles method [5], and this is used in
the present analysis. The interface phase shift i
is
given by
1
E E
L
Re
cos
2
1
0
, (2)
EU
EL
where Re refers to the real part, E is the energy, EL
is the lower edge of the Si band gap, E
U
is the upper
edge, and 0
is a constant [6, 7]. It is easy to verify
that
i
changes by across the Si band gap. The
only unknown quantity in Eq. (1) is the constant
0
.
This is treated as a fitting parameter in a fit of the