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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

dependence.

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

metals.1

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.

However,detailed quantitative

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

lifetimestudies were

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-

time data.

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

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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.

PreviousSTS studiesof

Yb/W?110?,18and of lanthanide SS ?Ref. 30? suggested that

this line shape results from a high effective mass near the 2D

subbandsonset, which was

Pb/Si?111?.31The measured QWS energies are similar for

QWS in Pb/Si?111?,25

in

confirmed by PESfor

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

films33

andour experimental

?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

datafor thicknesses

IWKB?V? =?

−?

???s????t?? − eV??f??? − f?? − eV??exp?−2

??

0

z0

Re??2m?? − ? +?1 −z

z0?eV??dz??d?.

?1?

FIG. 1. ?Color? STM images showing typical features of Pb

islands grownonSi?111?-7?7

??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.

?a?–?c?

andon Pb-?3

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.

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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

2?b??.

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?,

?2?

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

between the averaged

?e-ph’s

=?e-ph?50K?−?e-ph?5K?, remains nearly constant, increasing

?dashed lines?,

??e-ph

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…

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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

In conclusion,thecombination

temperature-dependent STS experiments with ab initio cal-

of high-accuracy

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

acknowledged.

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