Angle-resolved photoemission of ultrathin Pb films on Si(111)-(6×6)Au: quantum size effect
ABSTRACT The electronic band structure of extremely thin (from 1 to 8 monoatomic layer (ML) thick) epitaxial Pb(111) films grown at low temperatures in ultrahigh vacuum (UHV) condition on Si(111)-(6×6)Au substrate is studied with angle-resolved photoelectron spectroscopy (ARPES). The morphology of the Pb film is determined with scanning tunneling microscopy (STM). Normal-emission photoelectron spectra recorded at the sample temperature of 130 K reveal quantum well states (QWS) characteristic of quantization perpendicular to the film surface. The energies of these states as a function of the number of the Pb(111) monoatomic layers are determined and compared with calculated in terms of the Bohr–Sommerfeld phase accumulation model.
Optica Applicata, Vol. XXXV, No. 3, 2005
Angle-resolved photoemission of ultrathin Pb films
on Si(111)-(6×6)Au: quantum size effect
MARCIN KISIEL*, KAZIMIERZ SKROBAS, MIECZYSŁAW JAŁOCHOWSKI
Institute of Physics and Nanotechnology Center, Maria Curie-Skłodowska University
pl. M. Curie-Skłodowskiej 1, PL-20031 Lublin, Poland
*Corresponding author: Marcin Kisiel, firstname.lastname@example.org
The electronic band structure of extremely thin (from 1 to 8 monoatomic layer (ML) thick)
epitaxial Pb(111) films grown at low temperatures in ultrahigh vacuum (UHV) condition on
Si(111)-(6×6)Au substrate is studied with angle-resolved photoelectron spectroscopy (ARPES).
The morphology of the Pb film is determined with scanning tunneling microscopy (STM).
Normal-emission photoelectron spectra recorded at the sample temperature of 130 K reveal
quantum well states (QWS) characteristic of quantization perpendicular to the film surface. The
energies of these states as a function of the number of the Pb(111) monoatomic layers are determined
and compared with calculated in terms of the Bohr–Sommerfeld phase accumulation model.
Keywords: quantum wells, Pb, angle-resolved photoelectron spectroscopy.
When the thickness of the film is comparable with the de Broglie wavelength of electron
confined in, the motion of the electrons is quantized in the direction perpendicular to
the surface. Theoretical works show (SCHULTE  and TRIVEDI et al. ) that in such
two-dimensional systems the energy spectrum splits into the subbands and the work
function, the electron density, and the inner potential depend on the film thickness.
These effects are well known as so-called quantum size effect (QSE). Several
experiments confirmed occurrence of such phenomenon. Already JAKLEVIC et al. [3, 4]
were studying tunneling in metal–insulator–metal junctions. They observed electron
standing wave states in 100 to 1000 Å Pb, Mg, Au, Ag thick texturized films. Other
experimental techniques show electrical resistivity oscillations (JAŁOCHOWSKI et al.
[5–7]) as a function of the thickness of ultrathin Pb films. A suitable experimental
method for the determination of the QSE electron energy levels in ultrathin films is
angle-resolved photoelectron spectroscopy (ARPES). This method was recently
successfully applied for studying the QSE in thin Ag [8–10], Cu , Mg ,
444M. KISIEL, K. SKROBAS, M. JAŁOCHOWSKI
Pb [13, 14] metallic films. In the first photoemission experiments on Pb quantized
films, JAŁOCHOWSKI and BAUER  studied QSE during the growth of ultrathin Pb
films onto Si(111)-(6×6)Au substrate at low temperatures. Their study clearly showed
the thickness dependent band structure quantization.
In the present work, we extend studies of the Pb ultrathin films grown on
Si(111)-(6×6)Au substrates by performing energy and angle highly resolved
photoemission measurements. The experimentally determined energies of the quantum
well states (QWS) are compared with calculated in terms of the Bohr–Sommerfeld
phase accumulation model.
The experiments were performed in the ultrahigh vacuum (UHV) chamber with the
base pressure below 8×10–11mbar. The ARPES apparatus consisted of a high-intensity
helium lamp with a polarizer as a HeI photon source (E = 21.22 ev) and electron energy
analyzer VGX900IC. The Si(111) sample was mounted on a precise manipulator which
enabled its rotation around two axes – in the plane of the sample, and perpendicular
to it. The samples were cooled with liquid nitrogen to about 130 K.
Before experiments the Si(111) substrate was cleaned by heating shortly up to
about 1500 K, applying the direct current heating. This procedure was repeated several
times, until the Si(111)-(7×7) surface reconstruction was well developed and the
sample was free of residual SiC. The cleaning of the substrate was fully controlled by
reflection high electron energy diffraction (RHEED).
Si(111)-(6×6)Au reconstruction was obtained by deposition of about 1.3monoatomic
layers (ML) of Au onto Si(111) at room temperature, and subsequent annealing at
about 1000 K. The amount of deposited Au and Pb was determined with the aid of a
precise quartz crystal monitor.
During photoemission measurements the electron energy analyzer was set to full
width half maximum (FWHM) energy resolution equal to 60 meV, and FWHM of the
analyzer acceptance angle was equal to 2 deg. The morphology of the ultrathin Pb
films deposited onto Si(111)-(6×6)Au at low temperatures was studied using scanning
tunneling microscopy (STM) type Omicron VT.
3. Results and discussion
The Pb ultrathin films under investigation were composed of polycrystalline domains
with well defined thickness. The samples deposited at 130 K grow in a layer-by-layer
mode. Figure 1 shows STM scans of two samples with the fractional top layer coverage
– in the intermediate state, when two surfaces with different thicknesses are exposed.
In Fig. 1a the 1 ML thick islands are formed on the Si(111)-(6×6)Au substrate. The
average diameter of these islands is about 5 nm. Single Pb atoms between islands, not
attached yet to the islands, can also be seen. Further deposition results as first in the
formation of smooth 1 ML thick film and then new 1 ML thick islands grow, Fig. 1b.
Angle-resolved photoemission of ultrathin Pb films...
At low temperatures the scenario repeats several times, thus allowing production of
the Pb film with desired integer number of the Pb monolayers. Such samples are
suitable for study of the quantum size effects with ARPES.
Figure 2 shows examples of the ARPES spectra for the Pb films with thickness
equal to 1, 2, and 8 ML. The polar angle (the angle between the sample normal and
the analyzer entrance slit) varies from –12 to 19.5 deg and lies in the  plane. The
spectra show clearly thickness-dependent features which are well described as the QSE
Fig. 1. STM topographic images of the ultrathin Pb films deposited on Si(111)-(6×6)Au surface. The
samples were deposited and measured at 170 K. The average Pb thickness is equal to 0.45 and 1.35 ML
in (a) and (b), respectively. The sample bias and the tunneling current were equal to: –1.6 V and
0.1 nA (a), and –1.2 V and 0.1 nA (b).
Fig. 2. Angle-resolved photoemission spectra of the Pb films deposited on Si(111)-(6×6)Au with
thickness 1, 2 and 8 ML. The off-normal angle varies from –12 to 19.5 deg. The bars indicate positions
of the QWS peak. For 1 ML thick sample the structures originating from the substrate are marked with S.
446M. KISIEL, K. SKROBAS, M. JAŁOCHOWSKI
states. The energies of these states are marked in Fig. 2 by short bars. For the 1 ML
thick sample only one QSE state at about 0.55 eV of binding energy is visible. In this
sample, the other strongly dispersive feature visible at about 1.5 eV (marked by S)
originates from the Si(111)-(6×6)Au substrate. The experimentally determined
energies of the QSE states for all measured samples with the thicknesses ranging from
1 to 8 ML of Pb are presented in Fig. 3. These energies are marked by triangles.
In order to describe the quantization condition in the quantum wells (QW) we use
the Bohr–Sommerfeld phase accumulation model . According to this model the
electron plane wave propagates inside the quantum well and it is reflected backward
with additional phase shifts ΦB and ΦC, for surface–vacuum and surface–substrate
interface reflection, respectively. Thus, the quantization condition is given by:
where k(E) is the wave vector perpendicular to the surface, d is the thickness of a single
monolayer taken to be 2.685 Å, N is the number of monolayers, ΦB and ΦC are the
phase shifts at the boundaries, and n is the principal quantum number of the QWS.
The phase shift at the vacuum side is well known and approximated by Wentzl–
Kramers–Brillouin formula :
where EV is the work function, and E is the energy of the QWS. Both energies are
expressed in eV unit. The phase shift at the substrate can be determined experimentally
if one finds QWS at the same binding energy for various film thicknesses. These states
Fig. 3. Measured (triangles) and calculated (dots) energies of the QWS states in the ultrathin Pb samples.
The lines connect the states with i = 3N – 2n = –1, 0, 1 and 2.
Angle-resolved photoemission of ultrathin Pb films...
must have the same phase shifts since the total phase shift depends only on energy.
Applying Eq. (1) for 2 films with different thicknesses we obtain wavevector k for the
QSE level with energy E:
After inserting this value, and the phase shift from Eq. (2), back into Eq. (1), we
obtain the phase shift for a particular energy at the substrate–sample interface. Figure 4
shows phase shifts determined for three pairs of the QWS. The numerical fit of
the linear dependence ΦC= Ea + b allows coefficients a and b to be determined.
They are equal a = 0.2262 rad/eV and b = 1.8971 rad. For comparison, MANS et al.
 obtained linear dependence with a = 0.34 rad/eV and b = 3.1 rad for the
Si(111)-(7×7)–Pb interface. For the Pb–Si(111)-(6×6)Au interface these values are
slightly different. Knowing ΦC and ΦB we calculated the QSE energies for all the
thicknesses under study. These energies are marked in Fig. 3 by circles. The number
near each circle indicates the corresponding principal quantum number. The QWS are
connected by the thin lines labeled i = –1, 0, 1, 2 satisfying the condition 3N – 2n = i.
Each of these lines has the property that it brings a new quantum state as the thickness
increases by two monolayers. The number of antinodes for each bilayer increment
increases by three.
The electronic structure of the QWS in the ultrathin Pb grown onto Si(111)-(6×6)Au
films has been extensively studied. The energies of the QWS are determined and
Fig. 4. Phase shift of electrons reflected at QW–substrate interface as a function of the QWS binding