Dynamics at Solid State Surfaces and Interfaces

Edited by Uwe Bovensiepen, Hrvoje Petek, Martin Wolf

Volume 2: Fundamentals

57268File AttachmentCover.jpg

Edited by

Uwe Bovensiepen, Hrvoje Petek,

and Martin Wolf

Dynamics at Solid State

Surfaces and Interfaces

Related Titles

Kolasinski, K. W.

Surface ScienceFoundations of Catalysis and

Nanoscience

2008

ISBN: 978-0-470-03304-3

Breme, J., Kirkpatrick, C. J., Thull, R. (eds.)

Metallic Biomaterial Interfaces2008

ISBN: 978-3-527-31860-5

Wetzig, K., Schneider, C. M. (eds.)

Metal Based Thin Films forElectronics2006

ISBN: 978-3-527-40650-0

Bordo, V. G., Rubahn, H.-G

Optics and Spectroscopy atSurfaces and Interfaces2005

ISBN: 978-3-527-40560-2

Butt, H.-J., Graf, K., Kappl, M.

Physics and Chemistry ofInterfacesSecond, Revised

and Enlarged Edition

2006

ISBN: 978-3-527-40629-6

Edited by

Uwe Bovensiepen, Hrvoje Petek,

and Martin Wolf

Dynamics at Solid State Surfacesand Interfaces

Volume 2: Fundamentals

The Editors

Prof. Dr. Uwe BovensiepenUniversity of Duisburg-EssenFaculty of PhysicsGermanyuwe.bovensiepen@uni-due.de

Prof. Hrvoje PetekUniversity of PittsburghDepartment of Physics and AstronomyUSApetek@pitt.edu

Prof. Dr. Martin WolfFritz-Haber Institute of the Max-Planck SocietyDepartment of Physical ChemistryBerlinGermanywolf@fhi-berlin.mpg.de

CoverThe cover figure depicts (i) a time-resolved experiment at aninterface using time-delayed pump and probe femtosecondlaser pulses (left) and the detected response (right) beingeither reflected light or a photoemitted electron. In addition(ii) charge transfer from an excited resonance of an alkaliatom to single crystal metal substrate is shown. The falsecolor scale represents the wave packet propagation whichwas calculated by A. G. Borisov including the many-bodyresponse of the metal. The figure was designed and createdby A. Winkelmann.

All books published by Wiley-VCH are carefully produced.Nevertheless, authors, editors, and publisher do not warrantthe information contained in these books, including this book,to be free of errors. Readers are advised to keep in mind thatstatements, data, illustrations, procedural details or otheritems may inadvertently be inaccurate.

Library of Congress Card No.: applied for

British Library Cataloguing-in-Publication DataA catalogue record for this book is available from the BritishLibrary.

Bibliographic information published bythe Deutsche NationalbibliothekThe Deutsche Nationalbibliothek lists this publication in theDeutsche Nationalbibliografie; detailed bibliographic data areavailable on the Internet at http://dnb.d-nb.de.

# 2012 Wiley-VCH Verlag & Co. KGaA,Boschstr. 12, 69469 Weinheim, Germany

All rights reserved (including those of translationinto other languages). No part of this book may be reproducedin any form – by photoprinting, microfilm, or any othermeans– nor transmitted or translated into a machine languagewithout written permission from the publishers. Registerednames, trademarks, etc. used in this book, even when notspecifically marked as such, are not to be consideredunprotected by law.

Composition Thomson Digital, Noida, IndiaPrinting and Binding Strauss GmbH, MörlenbachCover Design Adam Design, Weinheim

Printed in the Federal Republic of GermanyPrinted on acid-free paper

Print ISBN: 978-3-527-40924-2ePDF ISBN: 978-3-527-64649-4oBook ISBN: 978-3-527-64646-3ePub ISBN: 978-3-527-64648-7

Contents

Preface IXList of Contributors XIColour Plates XV (after page 154)

1 The Electronic Structure of Solids 1Uwe Bovensiepen, Silke Biermann, and Luca Perfetti

1.1 Single-Electron Approximation 21.1.1 The Drude Model of the Free Electron Gas 21.1.2 The Electronic Band Structure: Metals, Insulators,

and Semiconductors 41.2 From Bloch Theory to Band Structure Calculations 61.2.1 Bloch Theory 61.2.2 The Tight Binding Approach to the Solid 71.2.3 Band Structure Calculations 81.3 Beyond the Band Picture 81.3.1 Motts Hydrogen Solid 91.3.2 Mott Insulators in Nature 101.4 Electronic Structure of Correlated Materials 141.4.1 The Hubbard Model 141.4.2 Dynamical Mean Field Theory 161.4.3 Electronic Structure Calculations 171.4.4 Ordered States 181.4.5 Cooperation Between Lattice Instabilities and Electronic

Correlations: The Example of Vanadium Dioxide 21References 23

2 Quasi-Particles and Collective Excitations 27Evgueni V. Chulkov, Irina Sklyadneva, Mackillo Kira, Stephan W. Koch,Jose M. Pitarke, Leonid M. Sandratskii, Paweł Buczek, Kunie Ishioka,Jörg Schäfer, and Martin Weinelt

2.1 Introduction 272.2 Quasi-Particles 302.2.1 Electrons and Holes 30

V

2.2.2 Phonons 312.2.2.1 Adiabatic Approximation 312.2.2.2 Harmonic Approximation 312.2.2.3 Lattice Dynamics 322.2.2.4 Phonons at Surfaces 332.2.3 Electron–Phonon Coupling in Metals 342.2.4 Excitons: Electron–Hole Pairs in Semiconductor Quantum Wells 382.2.4.1 Microscopic Theory 392.2.4.2 Excitonic Resonances and Populations 412.2.4.3 Terahertz Spectroscopy of Exciton Populations 432.2.4.4 Excitonic Signatures in the Photoluminescence 442.2.5 Polarons: Electron–Phonon Coupling in Polar and Ionic Solids 462.3 Collective Excitations 492.3.1 Plasmons: Electron Density Oscillations 492.3.1.1 Surface Plasmons 512.3.1.2 Acoustic Surface Plasmons 522.3.2 Magnons: Elementary Excitations in Ferromagnetic Materials 532.3.2.1 Spin Waves in the Heisenberg Model 542.3.2.2 Itinerant Electrons 572.3.2.3 Conclusions 642.4 Experimental Access to Quasi-Particle and Collective Excitations 652.4.1 Coherent Phonons 652.4.1.1 Coherent Optical Phonons 652.4.1.2 Coherent Acoustic Phonons 742.4.2 High-Resolution Angle-Resolved Photoemission 782.4.2.1 Photoemission Spectral Function of Quasi-Particles 782.4.2.2 Experimental Considerations for Photoelectron Spectroscopy 802.4.2.3 Quasi-Particles from Electron–Phonon Interaction 812.4.2.4 Quasi-Particles from Electron–Magnon Interaction 822.4.2.5 Conclusions and Implications 882.4.3 Time-Resolved Photoelectron Spectroscopy 892.4.3.1 Experiment 892.4.3.2 Electron Lifetimes 912.4.3.3 Electron–Phonon Coupling 942.4.3.4 Surface Exciton Formation 972.4.3.5 Magnon Emission 1012.4.3.6 Magnon–Phonon Interaction 1032.5 Summary 105

References 106

3 Surface States and Adsorbate-Induced Electronic Structure 115Thomas Fauster, Hrvoje Petek, and Martin Wolf

3.1 Intrinsic Surface States 1153.1.1 Basic Concepts of Surface States 1153.1.2 Scattering Model of Surface States 116

VI Contents

3.2 Crystal-Induced Surface States 1193.2.1 Tamm and Shockley Surface States 1193.2.2 Dangling Bond States 1203.3 Barrier-Induced Surface States 1213.3.1 Image Potential States 1213.3.2 Quantum Well States 1243.4 Experimental Methods 1253.4.1 Photoemission 1253.4.2 Two-Photon Photoemission 1283.4.3 Scanning Tunneling Methods 1333.5 Adsorbate-Induced Electronic Structure 1353.5.1 Bonding at Surfaces 1353.5.2 Energy-Level Alignment: Alkali–Metal Interfaces as a

Model System 1383.5.3 Electronic Band Structure: Chemisorbed and Physisorbed

Adsorbates 147References 151

4 Basic Theory of Heterogeneous Electron Transfer 155Daniel Sanchez-Portal, Julia Stähler, and Xiaoyang Zhu

4.1 Resonant Charge Transfer in Chemisorbed Systems 1554.1.1 Anderson–Grimley–Newns Hamiltonian 1564.1.2 Main Factors that Determine RCT Decay Rates 1584.1.3 Theoretical Approaches to Calculate RCT Rates in

Realistic Systems 1614.1.4 Effect of the Adsorbate Motion 1634.2 Electron Transfer in the Presence of Polar/Polarizable Media 1664.2.1 Nonadiabatic (Outer Sphere) Electron Transfer 1674.2.1.1 Continuum of Accepting States 1694.2.2 Adiabicity and the Effect of Strong Electronic Coupling 1704.2.3 Intermediate Coupling and the Impact of Solvent Relaxation 1714.2.3.1 Classical Description and a Wide Band Acceptor 1744.3 Transient Electronic Coupling: Crossover between Limiting Cases 1744.4 Conclusions 177

References 178

5 Electromagnetic Interactions with Solids 181Ricardo Díez Muiño, Eugene E. Krasovskii, Wolfgang Schattke,Christoph Lienau, and Hrvoje Petek

5.1 Dielectric Function of Metals 1825.1.1 Calculations of Dielectric Functions 1835.2 Band Mapping of Solids by Photoemission Spectroscopy 1865.2.1 Nonlinear Photoemission as a Band Mapping Tool for

Unoccupied States 1895.3 Optical Excitations in Metals 191

Contents VII

5.3.1 Optical Response of Noble Metals 1935.3.2 Interband Absorption 1955.3.3 Intraband Absorption 1975.3.4 Extended Drude Model 1995.3.5 Frequency-Dependent Scattering Rate 2015.3.6 Surface Absorption 2045.3.7 Summary 2065.4 Plasmonic Excitations at Surfaces and Nanostructures 2065.4.1 Drude Model for Optical Conductivity 2075.4.2 Interaction of Light with a Planar Metallic Surface 2085.4.3 Surface Plasmon Polariton Fields 2105.4.3.1 Planar Interfaces 2105.4.4 Surface Plasmons in Nanostructured Metal Films 2155.4.4.1 Spherical Nanoparticles 2155.4.4.2 Elliptical Nanoparticles 2195.4.4.3 Diffraction Gratings 2205.4.4.4 Adiabatic Metallic Tapers 2245.4.5 Exciton–Plasmon Coupling 2275.4.6 Summary 230

References 231

Index 239

VIII Contents

Preface

The dynamics of elementary processes in solids are decisive for various physicalproperties of solid state materials and their application in devices. This book intendsto provide an introductory and comprehensive overview of the fundamental con-cepts, techniques and underlying elementary processes in the field of ultrafastdynamics of solid state surfaces and interfaces. While the first volume addressesrecent research on quasiparticle dynamics, collective excitations, electron transferand photoinduced dynamics, the focus of this second volume lies on fundamentalsand provides introductory information on elementary processes and light-matterinteraction. Our goal is to make these concepts accessible also to non-experts and, inparticular, to newcomers and younger researchers in the field of ultrafast dynamicsof solids, their interfaces and nanostructured materials. We hope that both volumeswill help to further new research directions and developments in this field.

We acknowledge support from our funding agencies, important contributionsfrom our co-workers, stimulating discussions with colleagues and the understand-ing from our families that were essential to realize this book.

Uwe Bovensiepen,Duisburg, Pittsburgh and Berlin, January 2012Hrvoje Petek,

and Martin Wolf

IX

List of Contributors

XI

Silke BiermannÉcole PolytechniqueCentre de Physique Theorique91128 Palaiseau CedexFrance

Uwe BovensiepenUniversität Duisburg-EssenFakultät für PhysikLotharstr. 147048 DuisburgGermany

Paweł BuczekMax-Planck-Institut fürMikrostrukturphysikWeinberg 206120 HalleGermany

Evgueni V. ChulkovDonostia International Physics Center(DIPC)Paseo de Manuel Lardizabal 420018 San Sebastián/DonostiaBasque CountrySpain

and

Centro de Física de Materiales (CFM)Departamento de Física de MaterialesPaseo de Manuel Lardizabal 420018 San Sebastian/DonostiaBasque CountrySpain

Thomas FausterUniversität Erlangen-NürnbergLehrstuhl für FestkörperphysikStaudtstr. 791058 ErlangenGermany

Kunie IshiokaUniversity of TsukubaNational Institute for Materials ScienceGraduate School of Pure and AppliedSciencesAdvanced Nano Characterization Center1-2-1 SengenTsukuba 305-0047Japan

Mackillo KiraPhilipps-UniversitätFachbereich PhysikRenthof 535032 MarburgGermany

Stephan W. KochPhilipps-UniversitätFachbereich PhysikRenthof 535032 MarburgGermany

Eugen E. KrasovskiiUniversität KielInstitut für Theoretische Physik undAstrophysikLeibnizstraße 1524098 KielGermany

Christoph LienauCarl von Ossietzky UniversitätInstitut für Physik26129 OldenburgGermany

Ricardo Díez MuiñoDonostia International Physics Center(DIPC)Paseo de Manuel Lardizabal 420018 San Sebastián/DonostiaSpain

Luca PerfettiÉcole Polytechnique91128 Palaiseau CedexFrance

Hrvoje PetekUniversity of PittsburghPhysics and Astronomy Department100 Allen Hall3941 O’Hara StreetPittsburgh, PA 15260USA

Jose M. PitarkeEuskal Herriko UnibertsitateaMateria Kondentsatuaren Fisika SailaZientzi Fakultatea644 Posta kutxatila48080 BilboBasque CountrySpain

Daniel Sánchez-PortalDonostia International Physics Centre(DIPC)Paseo Manuel Lardizábal 420018 San SebastiánSpain

and

Dep. Física de Materiales (UPV/EHU)Facultad de QuímicaApartado 107220080 San SebastiánSpain

Leonid M. SandratskiiMax-Planck-Institut fürMikrostrukturphysikWeinberg 206120 HalleGermany

Jörg SchäferUniversität WürzburgPhysikalisches InstitutAm Hubland97074 WürzburgGermany

Wolfgang SchattkeUniversität KielInstitut für Theoretische Physik undAstrophysikLeibnizstraße 1524098 KielGermany

XII List of Contributors

Irina SklyadnevaDonostia International Physics Center(DIPC)Paseo de Manuel Lardizabal 420018 San Sebastián/DonostiaBasque CountrySpain

and

Tomsk State Universitypr. Lenina 36634050 TomskRussian Federation

Julia StählerFritz-Haber-Institut der Max-Planck-GesellschaftAbteilung für Physikalische ChemieFaradayweg 4-614195 BerlinGermany

Martin WeineltFreie Universität BerlinFachbereich PhysikArnimallee 1414195 BerlinGermany

Martin WolfFritz-Haber-Institut der Max-Planck-GesellschaftAbteilung für Physikalische ChemieFaradayweg 4-614195 BerlinGermany

Xiaoyang ZhuUniversity of Texas at AustinDepartment of Chemistry &Biochemistry1 University Station A5300Austin, TX 78712USA

List of Contributors XIII

Colour Plates

Figure 1.9 Fermi surface of CeTe3 obtained byARPES at the temperature of 25 K. In thehorizontal direction, the Fermi surface isgapped due to charge density wave formation

along this directiondue to Fermi surfacenestingwith a momentum qCDW. Reprinted withpermission from Ref. [30]. Copyright (2004) bythe American Physical Society.

Figure 1.10 Spectral function of the insulatingphase of vanadium dioxide, as calculated fromLDA þ DMFT. The origin of energy has beenchosen as the Fermi level. (a) Themaxima of thespectral function follow closely the poles ofthe one-particle spectral function (blue dots),stressing that sharply defined one-particleexcitations exist in this phase. (b) Starting from

the full LDA þ DMFT solution, a static orbital-dependent potential has been constructed.The band structure corresponding to thispotential is plotted in red dots and comparedwith the DMFT result of (a). Reprinted withpermission from Ref. [37]. Copyright (2008) bythe American Physical Society.

XV

Dynamics at Solid State Surfaces and Interfaces: Volume 2: Fundamentals, First Edition.Edited by Uwe Bovensiepen, Hrvoje Petek, and Martin Wolf.� 2012 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2012 by Wiley-VCH Verlag GmbH & Co. KGaA.

q

Figure 2.9 Schematic picture of the magnetic configuration corresponding to a spin wave withwave vector q.

Figure 2.12 Enhanced susceptibility ofCo2MnSi, an example of the spectrum of theloss matrix for q ¼ 0:28ð1; 1; 0Þ2p=a. a standsfor the lattice constant. The three largesteigenvalues are shown, and other eigenvaluesare of vanishing magnitude. Three clear peaks

(EV 1, 2, and 3 corresponding to the labels inFigure 2.13) can be discerned. The panels showcorresponding eigenvectors; arrows indicatethe deviations ofmagneticmoments. Reprintedwith permission from Ref. [65]. Copyright(2009) by the American Physical Society.

XVI Colour Plates

(a) (b)

Figure 2.13 Energies (a) and inverse lifetimes C (b) of three SW modes in Co2MnSi. Reprintedwith permission from Ref. [65]. Copyright (2008) by the American Physical Society.

Figure 2.15 Spectral power Imxðqjj; �hvÞ of spin-flip excitations in three-monolayer iron film on Cu(100) obtained fromLRDFT calculations. Reprintedwithpermission fromRef. [17]. Copyright (2010)by the American Physical Society.

Colour Plates XVII

XVIII Colour Plates

(a) (b)

(c)

EF

hω0B

indi

ng E

nerg

y

Crystal Momentum k

ε(k)without

interaction

quas

i-par

ticle

spec

tral

wei

ght

ReΣ(ω)

m*(1+λ) m*

Figure 2.22 Schematic display of quasi-particle formation. (a) The electron isdressed with the excitation (phonon orspin excitation) up to an energy v0 belowthe Fermi energy, leading to a mass

enhancement. (b) Electron–phonon couplingimplies a distortion of the crystallattice surrounding the electron. (c) Electron–magnon coupling implies spin scatteringmediated by magnetic interactions.

Figure 2.23 Spectral function generatednumerically (T ¼ 20 K, Tdebye ¼ 340 K, impurityscattering 10 meV). The shift of spectral weightnear the Fermi level and a reduced slope

become obvious. The self-energy S implies abroadening (given by ImS) and an energy shift(given by ReS) of the spectrum compared to thenoncoupling case.

Colour Plates XIX

Figure 2.24 The H:W(110) system allowsthe study of adsorbate vibrations in thepresence of interactions with a metallicelectron bath. (b) Spectra of the surface

state band (T ¼ 150 K, hn ¼ 100 eV).They reflect the electron–phonon couplingin an energy window of � 160 meV thatcorresponds to the H stretching vibration.

Figure 2.25 (a) ARPES data of the metallicFe(110) surface state near �S at T ¼ 85 K.It is a surface resonance overlapping withthe projection of bulk bands of opposite spin.(b) Peak position extracted from the band mapusing a fitting procedure (thick dots). The

smooth line is a parabolic interpolation for thenoninteracting band. A deviation from this bandextends over a large energy scale below theFermi level. Reprinted with permission fromRef. [131]. Copyright (2004) by the AmericanPhysical Society.

Figure 2.27 ARPESdata ofNi(110) recorded athn ¼ 100 eV (corresponding to a high-symmetry plane). The bandmap shows the bulkband near the K-point. The momentumdistribution curves are fitted to yield thepeak positions. At low and high binding

energy (as indicated), indicationsfor kinks in the dispersion are discernible,corresponding to the phonon andmagnon energy scale, respectively. Reprintedwith permission from Ref. [124]. Copyright(2009) by the American Physical Society.

XX Colour Plates

Colour Plates XXI

Figure 2.28 Self-energy analysis of the K-pointband in Ni(110) derived from ARPES. (a) Realpart ReS and (b) imaginary part ImS. In bothpanels, peaks in the self-energy are seen atenergy scales that correspond to the phononsand magnons: in terms of ImS, this occurs at� 30 meV and � 340 meV, respectively. Notethat in (a) a Gutzwiller reference band is used,which includes the electron correlation effects,

so that they are effectively removed from ReS.In (b) for ImS, electron correlations have to beexplicitly considered. The model curves for ReSand ImS are Kramers–Kronig transformable,consistent with the quasi-particle picture.Reprinted with permission from Ref. [124].Copyright (2009) by the American PhysicalSociety.

Δt(p

s)

0

0.5

1.01.00.5 2.0

Energy ( eV )

(a)

(b)

Inte

nsity

(ar

b. u

nits

)

Si(111)-(7x7)

Energy (eV)

1.5 2.5

Tstart = 90 K

Inte

nsi

ty (

arb

. un

its)

1.0 1.5

0.4

0.5

0.6

0.7

0.80.91.0

Delay (ps)

CBM

1 2 30

Ele

ctro

nte

mpe

ratu

re(K

)

2000

1000

Delay (ps)

Si(111) 7 x 7Tstart = 300 K

Figure 2.34 (a) Series of 2PPE spectrameasured for Si(111) 7� 7 for s-polarized2.21 eV pump pulses (fluence � 0:5mJ cm�2)and p-polarized 4.95 eV probe pulses. The insetshows the temporal evolution of the CB peak inan expanded scale from0.4 to 1 ps delay. (b) Theelectron temperature at the CBM as a function

of pump–probe delay. The solid curve is the fit ofa single exponential decay component and aconstant value of 296 K. The time constant is240 fs. Reprinted with permission from Ref.[197]. Copyright (2009) by the AmericanPhysical Society.

Figure 2.35 (a) The cð4� 2Þ reconstruction ofthe Si(100) surface.Dimers are arranged in rowsalong the ½0�11�-direction and are alternatelytilted along and perpendicular to the rows.The unit cell (solid rhombus) of the cð4� 2Þstacking comprises two dimers. The surfaceexciton is visualized by the contour plot ofthe probability density of the electron bound tothe hole, located in the center of the depictedarea Reprinted with permission from Ref. [214].Copyright (2005) by Springer. (b) Measured(symbols) and calculated (solid lines, shadedareas) surface-projected band structure of the

Si(100) cð4� 2Þ surface. CY 0 and CJcorrespond to the directions along andperpendicular to the dimer rows. Thepresence of two dimers per unit cellleads to two occupied (Dup and D0up)and two unoccupied (Ddown and D0down)dangling bond bands. X is the signatureof a surface exciton in the single-particle bandstructure. The silicon valence and conductionbands are indicated by the gray-filled areas.Reprinted with permission from Ref. [211].Copyright (2004) by the American PhysicalSociety.

XXII Colour Plates

Figure 2.39 (a) Linear magnetic dichroism ofthe 4f photoemission line (surface and bulkcomponents) of Gd(0001) (adapted fromRef. [234]). (b) Sketch of the measurementgeometry. Spectra are recorded close to normalemission (0�) for opposite directions ofthe in-plane magnetization (M" and M#).(c) Transient, relative change in the

magnetization: measured 4f dichroism (opencircles), calculated from lattice temperature andmagnetization (dashed line), and convolvedwith a Gaussian of 50 ps FWHM duration (solidline). Reprintedwith permission fromRef. [170].Copyright (2008) by the American PhysicalSociety.

Colour Plates XXIII

Figure 3.19 The (a) 2PP spectra and (b) 2PPexcitation scheme for Cu(111) surface during acontinuous deposition of alkali atoms. Thebinding energy of the unoccupied ns resonanceof alkali atoms is determined by the image–charge interaction. The formation of a surface

dipole layer reduces the vacuum level byDW andthe binding energy of ns resonance by�(DW)3/2,as explained in Ref. [87]. Reprinted withpermission from Ref. [87]. Copyright (2008) bythe American Physical Society.

XXIV Colour Plates

Figure 3.22 The effective electron potentialsfor single alkali atom (Li throughCs) onCu(111)surface. The total potential combines theChulkov potential for Cu(111), which includesthe image potential Vip, the pseudopotential forfree alakli [92] atoms, and the main mode ofinteraction through the image–charge repulsionDE [87]. Both the Vip and DE potential have

the universal form expected for the Coulombinteraction. The adsorption height ismeasured from the image plane z0, while zsrepresents the geometrical boundary of thesurface. Reprinted with permission from Ref.[87]. Copyright (2008) by the American PhysicalSociety.

Colour Plates XXV

Figure 4.2 (a and b) Electronic structure of theCu(111) and Cu(100) surfaces. Surface states,resonances, and the first image potential statesare indicated by dashed lines. The thickhorizontal lines indicate the energy of theresonances localized in Cs adsorbates at lowcoverage, while the thin horizontal line indicatesthe Fermi energy. (c) Logarithmof the electronicdensity associated with a Cs-localizedresonance for Cs/Cu(111). The Cs atom is

located at the origin of coordinates andthe horizontal line indicates the position ofthe image plane of the metal. The electronicdensity decreases when going from red toviolet (white corresponds to very small valuesof the density). Notice that the projectedbandgap inCu(111)preventspropagation alongthe normal to the surface. Reprinted withpermission from Ref. [18]. Copyright (2001) bythe American Physical Society.

XXVI Colour Plates