Decay Mechanisms of Excited Electrons in Quantum-Well States of Ultrathin Pb
Islands Grown on Si(111): Scanning Tunneling Spectroscopy and Theory
I-Po Hong1, Christophe Brun1, Fran¸ cois Patthey1, I. Yu. Sklyadneva2,3, X. Zubizarreta2,4, R. Heid5,
V. M. Silkin2,4, P. M. Echenique2,4, K. P. Bohnen5, E. V. Chulkov2,4, and Wolf-Dieter Schneider1
1´Ecole Polytechnique F´ ed´ erale de Lausanne (EPFL),
Institut de Physique de la Mati` ere Condens´ ee, CH-1015 Lausanne, Switzerland
2Donostia International Physics Center (DIPC), Paseo de Manuel Lardizabal,
4, 20018 San Sebasti´ an/ Donostia, Basque Country, Spain
3Institute of Strength Physics and Materials Science,
Prospekt Academicheski 2/1, 634021, Tomsk, Russia
4Departamento de F´ ısica de Materiales and Centro Mixto CSIC-UPV/EHU,
Facultad de Ciencias Qu´ ımicas, UPV/EHU, Apartado 1072,
20080 San Sebasti´ an/ Donostia, Basque Country, Spain
5Forschungszentrum Karlsruhe, Institut f¨ ur Festk¨ orperphysik, P.O. Box 3640, D-76021, Karlsruhe, Germany
(Dated: August 17, 2009)
Using low-temperature scanning tunneling spectroscopy at 5 and 50 K, we studied the linewidth
of unoccupied quantum-well states in ultrathin Pb islands, grown on Si(111) on two different Pb/Si
interfaces. A quantitative analysis of the differential conductance spectra allowed us to determine
the electron-electron (e − e), electron-phonon (e − ph) and the interface and defect contributions
to the lifetime. Layer-dependent ab initio calculations of the e − ph linewidth contribution are in
excellent agreement with the data. Importantly, the sum of the calculated e−e and e−ph lifetime
broadening follows the experimentally observed quadratic energy dependence.
PACS numbers: 73.21.Fg, 68.37.Ef, 71.15.Mb, 79.60.Dp, 73.50.Gr
Understanding the basic processes governing the de-
cay of elementary electronic excitations in metals and
at metal surfaces is important because these excitations
play a major role in a large variety of chemical and phys-
ical phenomena, including chemical reactions or catalysis
at surfaces, molecule-surface interactions and transport
properties. A clear picture of the decay mechanisms oc-
curring in several types of bulk metals (simple, noble,
paramagnetic and some ferromagnetic transition metals)
has been obtained . The analysis of the dynamics of
surface and image potential states (SS, IPS) also clarified
the decay processes at the surface of various metals .
Thin metal films are interesting from a fundamen-
tal point of view and for technological applications. In
a thin metal film electrons occupy discrete eigenstates
with a quantized wave vector perpendicular to the sur-
face, known as quantum-well states (QWS) [2–4]. These
states, forming two-dimensional (2D) bands, are interme-
diate between bulk states and SS. Due to technical limita-
tions, few studies have reported so far detailed contribu-
tions to the QWS lifetime. For example, photoemission
(PES), two-photon PES (2PPE) and time-resolved 2PPE
(TR-2PPE) require homogeneous films over macroscopic
areas. Nevertheless, the electron-electron (e − e) contri-
bution Γe−ewas determined in Ag/Fe(100) by PES and
TR-2PPE [5, 6] and in Pb/Si(111) . The electron-
phonon (e − ph) contribution Γe−ph was extracted by
PES in Ag/Fe(100) , and in Ag/Cu(111) . Layer-
dependent or electronic structure dependent e − ph con-
tributions were also reported [9–12].
Scanning tunneling spectroscopy (STS) benefits from
being a local probe but suffers from the lack of k resolu-
tion. However, detailed quantitative lifetime studies were
achieved for SS [13–16] and IPS . Up to now only one
STS study reported a quantitative linewidth analysis of a
QWS metal system, Yb(111)/W(110) . A quadratic
energy dependence of the linewidth was found, in agree-
ment with three-dimensional (3D) Fermi-liquid (FL) the-
ory, and a large e − ph coupling constant. Both results
were subsequently questioned by a TR-2PPE study on
bulk Yb . These controversial results illustrate the
difficulties and limits encountered in STS experiments to
retrieve reliable quantitative QWS lifetime data.
In this Letter we present a detailed low-temperature
STS study of the linewidth of unoccupied QWS in Pb
islands of thicknesses 7-22 monolayers (ML) grown on
Si(111). Using a simple model with tunneling allowed
through a trapezoidal barrier for a set of discrete QWS,
a quantitative analysis of the differential conductance
dI/dV spectra allows us to determine the QWS lifetime
broadening as a function of energy, and the e − ph con-
tribution between 5 and 50 K. The interface and de-
fect scattering contribution to the QWS linewidth from
the disordered Pb/Si(111)-7×7 (hereafter 7 × 7) inter-
face is 90 meV larger than the one from the crystalline
Pb-√3 ×√3/Si(111) (in short Pb√3) interface. Layer-
dependent ab initio calculations of Γe−phwere performed
for 4-10 ML free-standing Pb(111) films, taking full ac-
count of the quantum-size effects on the electron and
phonon band structures and on the e − ph coupling
[20, 21]. The theoretical results are in very good agree-
ment with the experimental findings.Γe−e was esti-
FIG. 1: (Color online).
tures of Pb islands grown on Si(111)-7×7 (a-c) and on Pb-
√3 ×√3/Si(111) (d-f). (a,d) Large scale overview. The in-
dicated island thickness includes the wetting layer (WL). (b)
Disordered WL ? 1 ML high. (c) Atomic resolution of the
surface Pb lattice. Buried 7×7 interface seen through a 8ML
island. (e) High resolution of the crystalline Pb WL, a satu-
rated Pb ML. (f) Moir´ e pattern on a 8 ML island.
STM images showing typical fea-
mated from ab initio calculation of Γe−efor the parent
bulk band dispersing along Γ − L . The calculated
Γe−ph+ Γe−eis convincingly fitted by a quadratic equa-
tion in agreement with the experimental results. The ef-
fect of spin-orbit coupling (SOC) on the electronic band
energies and on Γe−eis small in the probed energy range.
The e − ph coupling constant calculated for the unoccu-
pied QWS, λ ? 1.45 − 1.60, is generally larger than λ
calculated at the Fermi energy (EF) for the correspond-
ing films .
The measurements were performed in a homebuilt
scanning tunneling microscope (STM) operated at 50 and
5 K in ultrahigh vacuum using cut PtIr tips . dI/dV
spectra were measured using currents of 200 ≤ I ≤ 500
pA, with open feedback loop via lock-in technique with a
modulation amplitude of 10 mVppat a frequency of 1.4
kHz. Pb was thermally evaporated on the Si(111)-7×7 or
on the Pb√3 substrate kept at room temperature favor-
ing the growth of Pb single crystals with their (111) axis
perpendicular to the surface [23, 24]. All dI/dV mea-
surements were performed on large Pb islands far from
steps or island boundaries to avoid additional broadening
of the QWS linewidths.
Figure 1 shows Pb islands grown on 7 × 7 (a-c) and
Pb√3 (d-f). Thicknesses given in ML include the wetting
layer (WL). Large islands of several hundreds of nm are
formed. Figure 1c) reveals the buried Si-7×7 interface
superimposed with the atomic resolution of the Pb lattice
indicating that the island surface is atomically flat [25,
26]. As shown in Fig. 1b) the Pb WL formed on Si-7×7
is disordered. In contrast, a crystalline topography is
FIG. 2: (Color online). Experimental (dots) and calculated
(full line) dI/dV spectra measured at 5 K by tunneling to
a large atomically flat Pb island of selected thickness grown
on (a) Si(111)-7×7 and (b) Pb-√3 ×√3/Si(111). The arrow
indicates negative differential conductance. (c) Schematic en-
ergy diagram of the tunnel junction used to model the exper-
iment. CBM: Conductance band minimum, VBM: Valence
band maximum, d: film thickness, z0: vacuum gap.
(EF,t): sample (tip) Fermi level. Evac,s (Evac,t): sample (tip)
vacuum level. Φ(z): vacuum potential drop between tip and
sample. V : tip-sample bias voltage.
observed both on the island and on the WL on Pb√3 (see
Fig. 1d-f). The WL displays a striped-incommensurate
superstructure (see Fig. 1e) corresponding to a saturated
Pb phase [27, 28]. Fig. 1f) shows a Moir´ e pattern on
a 8 ML island, caused by interfacial strain due to the
difference between the Si and Pb lattice constant .
Figure 2 presents single dI/dV spectra obtained at 5
K on Pb islands of selected thickness. Remarkably, the
spectra consist of prominent maxima located at the QWS
energies. Previous STS studies of QWS in Pb/Si(111)
, in Yb/W(110)  and of lanthanide SS  sug-
gested that this line shape results from a high effective
mass near the 2D subbands onset, which was confirmed
by PES for Pb/Si(111) . The measured QWS energies
are similar for both interfaces, with larger dispersion on
the disordered one . If the WL thickness is assumed
to be 1 ML, agreement occurs between calculated QWS
energies for free-standing films  and our experimental
data for thicknesses ≥ 17 ML . At smaller thickness
a systematic deviation exists, increasing with decreasing
thickness. A comparison between the spectra shown in
Fig. 2a) and b) reveals a considerable narrowing of the
QWS linewidths on the Pb√3 interface with respect to
the ones on 7 × 7.
To extract the intrinsic QWS linewidth, dI/dV is mod-
eled based on a 1D WKB approach with a trapezoidal
potential barrier [25, 34]. Fig. 2c) depicts the schematic
energy diagram of the junction. The Pb island density
of states (DOS) ρsis simulated as a series of Lorentzian
peaks, whereas the tip DOS ρtis assumed to be constant.
As a function of bias voltage V , I is written 
The non-zero conductance observed between the QWS is
modeled by an additional exponential term. This analy-
sis describes convincingly the STS data (see Fig. 2a-b).
Possible causes of extrinsic broadening are the trans-
mission to the substrate, the QWS lateral dispersion and
the ac voltage modulation. The latter contributes a few
mV. Following Ref. , the reflectivity of both Pb/Si in-
terfaces was found to be very close to one in the studied
voltage range, contributing to negligible broadening. As
symmetric QWS peaks are observed on both interfaces,
tunneling of electrons with finite k?should contribute less
than 10 meV to the linewidth [18, 30].
Consequently, extrinsic linewidth contributions were
neglected in the following analysis. Γ(T,E) was further
decomposed as follows:
ρs(?)ρt(?−eV )[f(?)−f(?−eV )] ×
2m[Φ− ? + (1 −z
Γ(T,E) = Γ0+ Γe−e(E) + Γe−ph(E,T)
where T is the temperature and E is the QWS energy.
Γe−e(E) is the e − e interaction term and Γe−ph(E,T)
reflects the e − ph scattering. Γ0, independent of T and
E, describes interface and defects scattering.
Figure 3a shows that our calculated QWS energies are
almost lying on the parent bulk band. The calculated
effective masses are very close to the free electron mass.
Figure 3b shows Γe−e(E) computed for bulk band en-
ergy equal to the QWS energy with and without SOC.
A quadratic dependence Γe−e = α(E − EF)2is found,
leading to α = 0.023 eV−1with SOC (0.021 without),
which are very close to α = 0.02 eV−1obtained when
treating bulk Pb as a free electron gas (rs= 2.30 a.u).
Hence, in the probed energy range, the SOC effect on
band (QWS) energies and on Γe−e is small. Figure 3b
shows Γe−ph versus QWS energy calculated for 5 and
50 K. It varies with QWS energy, but the difference be-
tween the averaged Γe−ph’s (dashed lines), ∆Γe−ph =
FIG. 3: (Color online). (a) Calculated dispersion of the elec-
tronic band crossing EF along Γ − L for bulk Pb without
spin-orbit coupling (with SOC): dashed (solid) line. Dots:
computed QWS energies. (b) Calculated Γe−e and Γe−ph for
unoccupied QWS as a function of energy. Γe−e without SOC
(with SOC): open (full) triangles. Γe−ph at 5 K (50 K): dots
(squares) with their fit. In nm (n = 4, ..., 10; m = 1, 2) n
is the film number of monolayers, and m is the QWS number
counted from EF.
Γe−ph(50K) − Γe−ph(5K), remains nearly constant, in-
creasing from 23 meV close to EF to 26 meV at higher
energies. Since the energy dependence of Γe−eis much
stronger than that of Γe−ph, their sum Γe−e+ Γe−ph,
is fitted reasonably well by a quadratic equation with
α=0.025 eV−1(0.026) at 50 (5) K (see Fig. 4).
Figure 4 shows Γ versus energy measured at 50 and
5 K on Pb√3 with the theoretical Γe−e+ Γe−ph. Both
experimental data sets are well fitted by 3D FL theory:
Γ(E) = α(E − EF)2, yielding the same value α = 0.033
eV−1. The difference ∆Γe−ph? 25 meV between the 50
and 5 K fit to the experimental data yields an estimate
of the average e − ph contribution to the QWS lifetime
in excellent agreement with the theoretical ∆Γe−ph ?
23−26 meV. A similar analysis was conducted on the 7×7
interface, which showed a larger linewidth dispersion due
to disorder at this interface. In contrast to the crystalline
Pb√3, the linewidths increase with decreasing thickness
(? 20 meV from 22 to 7 ML). This linewidth variation
was taken into account before Γe−e, Γe−phand Γ0were
extracted. Γ0is found to be about 90 meV larger on 7×7
(see Fig. 2). α = 0.028 eV−1at 50 K (0.037 at 5 K),
∆Γe−ph? 26 meV, which is consistent with the values
obtained on Pb√3 and with the theoretical results.
The large ∆Γe−phmeasured on both interfaces reflect
a strong e−ph coupling of the QWS in Pb thin films. A
Debye model  with λbulk= 1.55 yields ∆Γe−ph= 23
meV, which is close to the measured averaged ∆Γe−ph.
The present ab initio calculations yield for most QWS
1.45 ≤ λ ≤ 1.6. These values are larger than those com-
puted for Pb thin films at EF , but close to λbulk
at EF . The excellent agreement between theoretical
FIG. 4: (Color online).
cupied QWS in Pb islands grown on Pb-√3 ×√3/Si(111)
measured at 5 (50) K: full dots (full squares). The data are
fitted according to 3D Fermi-liquid theory (continuous lines).
Theoretical linewidth Γe−e + Γe−ph at 5 (50) K: open dots
(open squares) with corresponding fits (dashed lines). For
easier comparison the theoretical data have been shifted up so
that the theoretical fits coincide with the experimental ones
at low energy. The linewidth difference ∆Γe−ph ? 25 meV
between the 50 and 5 K fit to the experimental data agrees
very well with the corresponding calculated difference, yield-
ing the QWS e−ph coupling constant λ ? 1.45−1.60. Silicon
conduction band minimum is indicated.
Linewidth versus energy of unoc-
and experimental e−ph coupling terms allows us to dis-
criminate among the three contributions of Eq. 2. For
7-22 ML films the resulting electronic mean free path
at EF, vFτ0 (Γ0 = ¯ h/τ0) can be estimated for both
interfaces, yielding 3 − 4 nm for 7 × 7 and 11 nm for
Pb√3 (Fermi velocities vF are determined from the re-
constructed band dispersion along Γ − L ).
In a previous Yb/W(110) QWS linewidth study by
STS, the neglect of the interface and defect scattering
term in the low-energy residual linewidth and a lack of
temperature-dependent measurements, led to a strong
e − ph coupling constant λ ? 1.6 − 2.8 . In con-
trast, TR-2PPE measurements of the parent d-band in
bulk Yb found λ ? 0.4 . Moreover TR-2PPE results
together with ab initio calculations reported a linewidth
energy dependence far from being quadratic .
In conclusion,the combination of high-accuracy
temperature-dependent STS experiments with ab initio
calculations allowed us to identify individual QWS in
single ultrathin metal islands, to separate consistently
the different decay mechanisms of these electronic exci-
tations and to determine the QWS electron-phonon cou-
pling strength. These achievements open up an avenue
toward detailed investigations of the decay processes of
electronic excitations on a local scale, e.g. of individual
supported molecules, clusters or other nanostructures.
We thank J. H. Dil, P. S. Kirchman, U. Bovensiepen
and T.-C. Chiang for stimulating discussions. Financial
support from the Swiss National Science Foundation, the
University of the Basque Country, the Departamento de
Educaci´ on del Gobierno Vasco, and the Spanish Minis-
terio de Ciencia y Tecnolog´ ıa (MCyT) (Grant No.FIS
2004-06490-C03-01) is acknowledged.
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