Decay mechanisms of excited electrons in quantum-well states of ultrathin Pb islands grown on Si(111): Scanning tunneling spectroscopy and theory
ABSTRACT 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.
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ABSTRACT: A theoretical study of the surface energy-loss function of freestanding Pb(111) thin films is presented, starting from the single monolayer case. The calculations are carried applying the linear response theory, with inclusion of the electron band structure by means of a first-principles pseudopotential approach using a supercell scheme. Quantum-size effects on the plasmon modes of the thinnest films are found in qualitative agreement with previous work based on the jellium model. For thicker films, results show a dispersionless mode at all thicknesses, in agreeement with electron energy-loss measurements. For sizeable values of the momentum, the raising of the surface plasmon with increasing thickness is retrieved.04/2013;
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ABSTRACT: The electron–phonon coupling parameters in the vicinity of the point, , for electronic quantum well states in epitaxial lead films on a Si(1 1 1) substrate are measured using 5, 7 and 12 ML films and femtosecond laser photoemission spectroscopy. The values in the range of 0.6–0.9 were obtained by temperature-dependent line width analysis of occupied quantum well states and found to be considerably smaller than the momentum averaged electron–phonon coupling at the Fermi level of bulk lead, (λ = 1.1–1.7). The results are compared to density functional theory calculations of the lead films with and without interfacial stress. It is shown that the discrepancy can not be explained by means of confinement effects or simple structural modifications of the Pb films and, thus, is attributed to the influence of the substrate on the Pb electronic and vibrational structures.Journal of Physics Condensed Matter 08/2014; 26(35):352001. · 2.22 Impact Factor
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ABSTRACT: The unoccupied electronic structure of epitaxial Pb films on Si(111) is analyzed by angle-resolved two-photon photoemission in the Γ¯ → M¯ direction close to the Brillouin zone center. The experimental results are compared to density functional theory calculations and we focus on the nature of the interaction of the 6p z states with the Si substrate. The experimentally obtained dispersion E (k ||) of the unoccupied quantum well states is weaker than expected for freestanding films, in good agreement with their occupied counterparts. Following E (k ||) of quantum well states as a function of momentum at different energies, which are degenerate and non-degenerate with the Si conduction band, we observe no influence of the Si bulk band and conclude a vanishing direct interaction of the Pb 6p z states with the Si band. However, the momentum range at which mixing of 6p z and 6p x,y derived subbands is found to occur in the presence of the Si substrate is closer to Γ¯ than in the corresponding freestanding film, which indicates a substrate-mediated enhancement of the mixing of these states. Additional femtosecond time-resolved measurements show a constant relaxation time of hot electrons in unoccupied quantum well states as a function of parallel electron momentum which supports our conclusion of a px,y mediated interaction of the pz states with the Si conduction band.Journal of Electron Spectroscopy and Related Phenomena 08/2014; · 1.55 Impact Factor
Decay mechanisms of excited electrons in quantum-well states of ultrathin Pb islands grown on
Si(111): Scanning tunneling spectroscopy and theory
I-Po Hong,1Christophe Brun,1François Patthey,1I. Yu. Sklyadneva,2,3X. Zubizarreta,2,4R. Heid,5V. M. Silkin,2,4
P. M. Echenique,2,4K. P. Bohnen,5E. V. Chulkov,2,4and Wolf-Dieter Schneider1
1École Polytechnique Fédérale de Lausanne (EPFL), Institut de Physique de la Matière Condensée, CH-1015 Lausanne, Switzerland
2Donostia International Physics Center (DIPC), Paseo de Manuel Lardizabal, 4, San Sebastián/Donostia,
20018 Basque Country, Spain
3Institute of Strength Physics and Materials Science, Prospekt Academicheski 2/1, 634021 Tomsk, Russia
4Departamento de Física de Materiales and Centro Mixto CSIC-UPV/EHU, Facultad de Ciencias Químicas, UPV/EHU, Apartado 1072,
San Sebastián/Donostia, 20080 Basque Country, Spain
5Forschungszentrum Karlsruhe, Institut für Festkörperphysik, P.O. Box 3640, D-76021 Karlsruhe, Germany
?Received 5 August 2009; published 20 August 2009?
Using low-temperature scanning tunneling spectroscopy at 5 and 50 K, we studied the linewidth of unoc-
cupied 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
DOI: 10.1103/PhysRevB.80.081409PACS number?s?: 73.21.Fg, 68.37.Ef, 73.50.Gr, 79.60.Dp
Understanding the basic processes governing the decay of
elementary electronic excitations in metals and at metal sur-
faces is important because these excitations play a major role
in a large variety of chemical and physical phenomena, in-
cluding chemical reactions or catalysis at surfaces, molecule-
surface interactions and transport properties. A clear picture
of the decay mechanisms occurring in several types of bulk
metals ?simple, noble, paramagnetic and some ferromagnetic
transition metals? has been obtained.1The analysis of the
dynamics of surface and image potential states ?SS and IPS?
also clarified the decay processes at the surface of various
Thin metal films are interesting from a fundamental point
of view and for technological applications. In a thin metal
film electrons occupy discrete eigenstates with a quantized
wave vector perpendicular to the surface, known as
quantum-well states ?QWS?.2–4These states, forming two-
dimensional ?2D? bands, are intermediate between bulk
states and SS. Due to technical limitations, few studies have
reported so far detailed contributions 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? contribution ?e-e was determined in Ag/
Fe?100? by PES and TR-2PPE ?Refs. 5 and 6? and in
Pb/Si?111?.7The electron-phonon ?e-ph? contribution ?e-ph
was extracted by PES in Ag/Fe?100? ?Ref. 5? and in
Ag/Cu?111?.8Layer-dependent or electronic structure depen-
dent e-ph contributions were also reported.9–12
Scanning tunneling spectroscopy ?STS? benefits from be-
ing a local probe but suffers from the lack of k resolution.
achieved for SS ?Refs. 13–16? and IPS.17Up to now only
one STS study reported a quantitative linewidth analysis of a
QWS metal system, Yb?111?/W?110?.18A quadratic energy
dependence of the linewidth was found, in agreement with
three-dimensional ?3D? Fermi-liquid ?FL? theory, and a large
e-ph coupling constant. Both results were subsequently ques-
tioned by a TR-2PPE study on bulk Yb.19These controver-
sial results illustrate the difficulties and limits encountered in
STS experiments to retrieve reliable quantitative QWS life-
In this Rapid Communication we present a detailed low-
temperature STS study of the linewidth of unoccupied QWS
in Pb islands of thicknesses 7–22 monolayers ?MLs? 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 contribution between 5
and 50 K. The interface and defect scattering contribution to
the QWS linewidth from the disordered Pb/Si?111?-7?7
?hereafter 7?7? interface 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 ?e-phwere per-
formed for 4–10 ML free-standing Pb?111? films, taking full
account of the quantum-size effects on the electron and pho-
non band structures and on the e-ph coupling.20,21The theo-
retical results are in very good agreement with the experi-
mental findings. ?e-ewas estimated from ab initio calculation
of ?e-efor the parent bulk band dispersing along ?-L.20The
calculated ?e-ph+?e-eis convincingly fitted by a quadratic
equation in agreement with the experimental results. The ef-
fect of spin-orbit coupling ?SOC? on the electronic band en-
ergies and on ?e-eis small in the probed energy range. The
e-ph coupling constant calculated for the unoccupied QWS,
??1.45–1.60, is generally larger than ? calculated at the
Fermi energy ?EF? for the corresponding films.21
The measurements were performed in a homebuilt scan-
ning tunneling microscope ?STM? operated at 50 and 5 K in
ultrahigh vacuum using cut PtIr tips.22dI/dV spectra were
measured using currents of 200?I?500 pA, with open
PHYSICAL REVIEW B 80, 081409?R? ?2009?
©2009 The American Physical Society 081409-1
feedback loop via lock-in technique with a modulation am-
plitude of 10 mVppat a frequency of 1.4 kHz. Pb was ther-
mally evaporated on the Si?111?-7?7 or on the Pb?3 sub-
strate kept at room temperature favoring the growth of Pb
single crystals with their ?111? axis perpendicular to the
surface.23,24All dI/dV measurements 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 WL.
Large islands of several hundreds of nm are formed. Figure
1?c? reveals the buried Si-7?7 interface superimposed with
the atomic resolution of the Pb lattice indicating that the
island surface is atomically flat.25,26As shown in Fig. 1?b?
the Pb WL formed on Si-7?7 is disordered. In contrast, a
crystalline topography is observed both on the island and on
the WL on Pb?3 ?see Figs. 1?d?–1?f??. The WL displays a
striped-incommensurate superstructure ?see Fig. 1?e?? corre-
sponding to a saturated Pb phase.27,28Figure 1?f? shows a
Moiré pattern on a 8 ML island, caused by interfacial strain
due to the difference between the Si and Pb lattice constant.29
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.
Yb/W?110?,18and of lanthanide SS ?Ref. 30? suggested that
this line shape results from a high effective mass near the 2D
Pb/Si?111?.31The measured QWS energies are similar for
both interfaces, with larger dispersion on the disordered
one.32If the WL thickness is assumed to be 1 ML, agreement
occurs between calculated QWS energies for free-standing
?17 ML.32At smaller thickness a systematic deviation ex-
ists, increasing with decreasing thickness. A comparison be-
tween the spectra shown in Figs. 2?a? and 2?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 modeled
based on a 1D WKB approach with a trapezoidal potential
barrier.25,34Figure 2?c? depicts the schematic energy diagram
of the junction. The Pb island density of states ?DOS? ?sis
simulated as a series of Lorentzian peaks, whereas the tip
DOS ?tis assumed to be constant. As a function of bias
voltage V, I is written35
???s????t?? − eV??f??? − f?? − eV??exp?−2
Re??2m?? − ? +?1 −z
FIG. 1. ?Color? STM images showing typical features of Pb
??3/Si?111? ?d?–?f?. ?a? and ?d? Large scale overview. The indi-
cated island thickness includes the wetting layer ?WL?. ?b? Disor-
dered 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 saturated Pb ML. ?f? Moiré
pattern on a 8 ML island.
FIG. 2. ?Color online? Experimental ?dots? and calculated ?full
line? dI/dV spectra measured at 5 K by tunneling to a large atomi-
cally flat Pb island of selected thickness grown on ?a? Si?111?-7
?7 and ?b? Pb-?3??3/Si?111?. The arrow indicates negative dif-
ferential conductance. ?c? Schematic energy diagram of the tunnel
junction used to model the experiment. CBM: conductance band
minimum, VBM: valence-band maximum, d: film thickness, z0:
vacuum gap. EF,s?EF,t?: sample ?tip? Fermi level. Evac,s?Evac,t?:
sample ?tip? vacuum level. ??z?: vacuum potential drop between tip
and sample. V: tip-sample bias voltage.
HONG et al.
PHYSICAL REVIEW B 80, 081409?R? ?2009?
The nonzero conductance observed between the QWS is
modeled by an additional exponential term. This analysis
describes convincingly the STS data ?see Figs. 2?a? and
Possible causes of extrinsic broadening are the transmis-
sion to the substrate, the QWS lateral dispersion and the ac
voltage modulation. The latter contributes a few mV. Follow-
ing Ref. 5, the reflectivity of both Pb/Si interfaces was found
to be very close to one in the studied voltage range, contrib-
uting 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 ne-
glected in the following analysis. ??T,E? was further decom-
posed as follows:
??T,E? = ?0+ ?e-e?E? + ?e-ph?E,T?,
where T is the temperature and E is the QWS energy. ?e-e?E?
is the e-e interaction term and ?e-ph?E,T? reflects the e-ph
scattering. ?0, independent of T and E, describes interface
and defects scattering.
Figure 3?a? shows that our calculated QWS energies are
almost lying on the parent bulk band. The calculated effec-
tive masses are very close to the free electron mass. Figure
3?b? shows ?e-e?E? computed for bulk band energy equal to
the QWS energy with and without SOC. A quadratic depen-
dence ?e-e=??E−EF?2is found, leading to ?=0.023 eV−1
with SOC ?0.021 without?, which are very close to ?
=0.02 eV−1obtained when treating bulk Pb as a free elec-
tron gas ?rs=2.30 a.u?. Hence, in the probed energy range,
the SOC effect on band ?QWS? energies and on ?e-eis small.
Figure 3?b? shows ?e-phversus QWS energy calculated for 5
and 50 K. It varies with QWS energy, but the difference
=?e-ph?50K?−?e-ph?5K?, remains nearly constant, increasing
from 23 meV close to EFto 26 meV at higher energies. Since
the energy dependence of ?e-eis much stronger than that of
?e-ph, their sum ?e-e+?e-ph, is fitted reasonably well by a
quadratic equation with ?=0.025 eV−1?0.026? at 50 ?5? K
?see Fig. 4?.
Figure 4 shows ? versus energy measured at 50 and 5 K
on Pb?3 with the theoretical ?e-e+?e-ph. Both experimental
data sets are well fitted by 3D FL theory: ??E?=??E−EF?2,
yielding the same value ?=0.033 eV−1. The difference
??e-ph?25 meV between the 50 and 5 K fit to the experi-
mental data yields an estimate of the average e-ph contribu-
tion to the QWS lifetime in excellent agreement with the
theoretical ??e-ph?23–26 meV. A similar analysis was
conducted on the 7?7 interface, which showed a larger line-
width dispersion due to disorder at this interface. In contrast
to the crystalline Pb?3, the linewidths increase with decreas-
ing thickness ??20 meV from 22 to 7 ML?. This linewidth
variation was taken into account before ?e-e, ?e-ph, and ?0
were extracted. ?0is found to be about 90 meV larger on
7?7 ?see Fig. 2?. ?=0.028 eV−1at 50 K ?0.037 at 5 K?,
??e-ph?26 meV, which is consistent with the values ob-
tained on Pb?3 and with the theoretical results.
The large ??e-phmeasured on both interfaces reflect a
strong e-ph coupling of the QWS in Pb thin films. A Debye
model36with ?bulk=1.55 yields ??e-ph=23 meV, which is
close to the measured averaged ??e-ph. The present ab initio
calculations yield for most QWS 1.45???1.6. These val-
ues are larger than those computed for Pb thin films at EF
?Ref. 21? but close to ?bulkat EF.36The excellent agreement
between theoretical and experimental e-ph coupling terms
FIG. 3. ?Color online? ?a? Calculated dispersion of the electronic
band crossing EFalong ?-L for bulk Pb without spin-orbit coupling
?with SOC?: dashed ?solid? line. Dots: computed QWS energies. ?b?
Calculated ?e-eand ?e-phfor unoccupied QWS as a function of
energy. ?e−ewithout SOC ?with SOC?: open ?full? triangles. ?e-phat
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 num-
ber counted from EF.
FIG. 4. ?Color online? Linewidth versus energy of unoccupied
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 ?e-e
+?e-phat 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 ??e-ph?25 meV be-
tween the 50 and 5 K fit to the experimental data agrees very well
with the corresponding calculated difference, yielding the QWS
e-ph coupling constant ??1.45–1.60. Silicon conduction band
minimum is indicated.
DECAY MECHANISMS OF EXCITED ELECTRONS IN…
PHYSICAL REVIEW B 80, 081409?R? ?2009?
allows us to discriminate among the three contributions of
Eq. ?2?. For 7–22 ML films the resulting electronic mean free
path at EF, vF?0??0=?/?0? can be estimated for both inter-
faces, yielding 3–4 nm for 7?7 and 11 nm for Pb?3 ?Fermi
velocities vFare determined from the reconstructed band dis-
persion along ?-L ?Ref. 32??.
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 con-
stant ??1.6–2.8.18In contrast, TR-2PPE measurements of
the parent d band in bulk Yb found ??0.4.19Moreover TR-
2PPE results together with ab initio calculations reported a
linewidth energy dependence far from being quadratic.19
Inconclusion, the combination
temperature-dependent STS experiments with ab initio cal-
culations allowed us to identify individual QWS in single
ultrathin metal islands, to separate consistently the different
decay mechanisms of these electronic excitations and to de-
termine the QWS electron-phonon coupling strength. These
achievements open up an avenue toward detailed investiga-
tions 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 Educación del
Gobierno Vasco, and the Spanish Ministerio de Ciencia y
Tecnología ?MCyT? ?Grant No. FIS 2004-06490-C03-01? is
1E. V. Chulkov, A. G. Borisov, J. P. Gauyacq, D. Sánchez-Portal,
V. M. Silkin, V. P. Zhukov, and P. M. Echenique, Chem. Rev.
106, 4160 ?2006?.
2R. C. Jaklevic, John Lambe, M. Mikkor, and W. C. Vassell, Phys.
Rev. Lett. 26, 88 ?1971?.
3T.-C. Chiang, Surf. Sci. Rep. 39, 181 ?2000?.
4M. Milun, P. Pervan, and D. P. Woodruff, Rep. Prog. Phys. 65,
5J. J. Paggel, T. Miller, and T.-C. Chiang, Science 283, 1709
6S. Ogawa, H. Nagano, and H. Petek, Phys. Rev. Lett. 88, 116801
7P. S. Kirchmann and U. Bovensiepen, Phys. Rev. B 78, 035437
8K. Takahashi, A. Tanaka, M. Hatano, H. Sasaki, S. Suzuki, and
S. Sato, Surf. Sci. 433-435, 873 ?1999?.
9M. Kralj, A. Siber, P. Pervan, M. Milun, T. Valla, P. D. Johnson,
and D. P. Woodruff, Phys. Rev. B 64, 085411 ?2001?.
10J. J. Paggel, D.-A. Luh, T. Miller, and T.-C. Chiang, Phys. Rev.
Lett. 92, 186803 ?2004?.
11Y. F. Zhang, J. F. Jia, T. Z. Han, Z. Tang, Q. T. Shen, Y. Guo, Z.
Q. Qiu, and Q. K. Xue, Phys. Rev. Lett. 95, 096802 ?2005?.
12S. Mathias, M. Wiesenmayer, M. Aeschlimann, and M. Bauer,
Phys. Rev. Lett. 97, 236809 ?2006?.
13J. Li, W. D. Schneider, R. Berndt, O. R. Bryant, and S. Crampin,
Phys. Rev. Lett. 81, 4464 ?1998?.
14L. Bürgi, O. Jeandupeux, H. Brune, and K. Kern, Phys. Rev.
Lett. 82, 4516 ?1999?.
15J. Kliewer, R. Berndt, E. V. Chulkov, V. M. Silkin, P. M. Ech-
enique, and S. Crampin, Science 288, 1399 ?2000?.
16L. Vitali, P. Wahl, M. A. Schneider, K. Kern, V. M. Silkin, E. V.
Chulkov, and P. M. Echenique, Surf. Sci. Lett. 523, L47 ?2003?.
17P. Wahl, M. A. Schneider, L. Diekhöner, R. Vogelgesang, and K.
Kern, Phys. Rev. Lett. 91, 106802 ?2003?.
18D. Wegner, A. Bauer, and G. Kaindl, Phys. Rev. Lett. 94,
19V. P. Zhukov, E. V. Chulkov, P. M. Echenique, A. Marienfeld, M.
Bauer, and M. Aeschlimann, Phys. Rev. B 76, 193107 ?2007?.
20See EPAPS Document No. E-PRBMDO-80-R26932 for details
of this kind of calculation, which can be found in Ref. 1. For
more information on EPAPS, see http://www.aip.org/pubservs/
21C. Brun, I. Po Hong, F. Patthey, I. Y. Sklyadneva, R. Heid, P. M.
Echenique, K. P. Bohnen, E. V. Chulkov, and W. D. Schneider,
Phys. Rev. Lett. 102, 207002 ?2009?.
22R. Gaisch, J. K. Gimzewski, B. Reihl, R. R. Schlittler, M. Ts-
chudy, and W. D. Schneider, Ultramicroscopy 42-44, 1621
23M. Jalochowski and E. Bauer, Phys. Rev. B 38, 5272 ?1988?.
24H. H. Weitering, D. R. Heslinga, and T. Hibma, Phys. Rev. B 45,
25I. B. Altfeder, K. A. Matveev, and D. M. Chen, Phys. Rev. Lett.
78, 2815 ?1997?.
26I. B. Altfeder, D. M. Chen, and K. A. Matveev, Phys. Rev. Lett.
80, 4895 ?1998?.
27L. Seehofer, G. Falkenberg, D. Daboul, and R. L. Johnson, Phys.
Rev. B 51, 13503 ?1995?.
28M. Hupalo, J. Schmalian, and M. C. Tringides, Phys. Rev. Lett.
90, 216106 ?2003?.
29I. B. Altfeder, V. Narayanamurti, and D. M. Chen, Phys. Rev.
Lett. 88, 206801 ?2002?.
30A. Bauer, A. Mühlig, D. Wegner, and G. Kaindl, Phys. Rev. B
65, 075421 ?2002?.
31J. H. Dil, J. W. Kim, T. Kampen, K. Horn, and A. R. H. F.
Ettema, Phys. Rev. B 73, 161308?R? ?2006?.
32I.-P. Hong et al. ?to be published?.
33C. M. Wei and M. Y. Chou, Phys. Rev. B 66, 233408 ?2002?.
34E. L. Wolf, Principles of Electron Tunneling Spectroscopy ?Ox-
ford University Press, New York, 1985?.
35We assume that z0=10 Å with sample and tip work functions
equal to 4 eV ?Ref. 37 and 38?.
36G. Grimvall, The Electron-Phonon Interaction in Metals ?North-
Holland, New-York, 1981?.
37P. S. Kirchmann, M. Wolf, J. H. Dil, K. Horn, and U. Boven-
siepen, Phys. Rev. B 76, 075406 ?2007?.
38H.-C. Ploigt, C. Brun, M. Pivetta, F. Patthey, and W. D.
Schneider, Phys. Rev. B 76, 195404 ?2007?.
HONG et al.
PHYSICAL REVIEW B 80, 081409?R? ?2009?