Table Of ContentSolid State Physics of Finite Systems
Advanced Texts in Physics
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Springer-Verlag Berlin Heidelberg GmbH
R.A. Broglia G. Colo
G. Onida H.E. Roman
Solid State Physics
of Finite Systems
Metal Clusters, Fullerenes, Atomic Wires
With 54 Figures and 31 Tables
' Springer
Prof. Ricardo A. Broglia Prof. Giovanni Onida
Dipartimento di Fisica Dipartimento di Fisica
Universita di Milano Universita di Milano
and INFN Sez. Milano andINFM
Via Celoria 16 Via Celoria 16
20133 Milano, Italy 20133 Milano, Italy
and e-mail: [email protected]
Niels Bohr Institute
University of Copenhagen Dr. H. Eduardo Roman
Blegdamsvej 17 Dipartimento di Fisica
2100 Copenhagen Universita di Milano
Denmark and INFN Sez. Milano
e-mail: [email protected] Via Celoria 16
20133 Milano, Italy
Dr. Gianluca Colo
e-mail: [email protected]
Dipartimento di Fisica
Universita di Milano
and INFN Sez. Milano
Via Celoria 16
20133 Milano, Italy
e-mail: [email protected]
ISSN 1439-2674
ISBN 978-3-642-05911-7 ISBN 978-3-662-09938-4 (eBook)
DOI 10.1007/978-3-662-09938-4
Library of Congress Control Number: 2004105249
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Preface
Quantum mechanics is the set of laws of physics which, to the best of our
knowledge, provides a complete account of the microworld. One of its chap
ters, quantum electrodynamics (QED), is able to account for the quantal
phenomena of relevance to daily life (electricity, light, liquids and solids, etc.)
with great accuracy. The language of QED, field theory, has proved to be uni
versal providing the theoretical basis to describe the behaviour of many-body
systems. In particular finite many-body systems (FMBS) like atomic nuclei,
metal clusters, fullerenes, atomic wires, etc. That is, systems made out of a
small number of components.
The properties of FMBS are expected to be quite different from those of
bulk matter, being strongly conditioned by quantal size effects and by the
dynamical properties of the surface of these systems. The study of the elec
tronic and of the collective behaviour (plasmons and phonons) of FMBS and
of their interweaving, making use of well established first principle quantum
(field theoretical) techniques, is the main subject of the present monograph.
The interest for the study of FMBS was clearly stated by Feynman in his
address to the American Physical Society with the title "There is plenty of
room at the bottom". On this occasion he said among other things: "When we
get to the very, very small world -say circuits of seven atoms -we have a lot of
new things that would happen that represent completely new opportunities
for design" [1].
Furthermore, FMBS can be used as building blocks to construct new
(nanostructured) materials, with properties which are controlled by those of
the individual atomic aggregates, in particular by their phonon spectrum,
and by the electronic density of levels and plasmons of the solid as a whole.
The material presented here is an outgrowth of lectures given to fourth
years students at the University of Milan, based on research carried out by
our group. We have, therefore, placed special emphasis on the role played by
the general concepts which are at the basis of the physics of FMBS in general,
and of atomic and molecular aggregates in particular. We have supplemented
this basic subject matter in an organic fashion, with material taken from work
going on at the forefront of research on the structure of atomic and molecular
clusters. This monograph should thus be useful to advanced students as well
as to young researchers alike, both theorists and experimentalists.
VIII Preface
We want to thank F. Alasia, P. Arcagni, F. Barranco, G. Benedek, M.
Bernasconi, M. Bianchetti, P. F. Buonsante, N. Breda, F. Cinelli, A. Loren
zoni, F. Marini, C. Milani, J. Pacheco, D. Provasi, Ll. Serra, E. Vigezzi and
C. Yannouleas, for many discussions and fruitful collaboration. Discussions
with G. F. Bertsch, S. Bj¢rnholm, P. F. Bortignon, 0. Gunnarson, P. Milani,
B. R. Mottelson and R. Smalley are gratefully acknowledged. We also express
our gratitude for the inspiring atmosphere of the Department of Physics of
the University of Milan and of the Niels Bohr Institute of the University
of Copenhagen, institutions where the research which is at the basis of the
present monograph was carried out. The financial support of the Istituto
Nazionale di Fisica Nucleare (INFN) Sez. di Milano, and Istituto Nazionale
di Fisica della Materia (INFM) Sez. G, is gratefully acknowledged.
Milan, R. A. Broglia
April 2004 G. Colo
G. Onida
H. E. Roman
Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1 k-Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 Electrons in Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.1 Jellium Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3 Electron Fluctuations (Plasmons) . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4 Electron-Plasmon Coupling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.5 w-Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.6 Total Effective Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2. 7 Induced Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.8 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.9 Minimal Mean Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.9.1 Non-locality in Space of the Exchange Contribution. . . 24
2.9.2 Non-locality in Time of the Correlation Contribution . 26
2.10 Phonons ............................................... 26
2.11 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.11.1 Second Quantization.............................. 28
2.11.2 Single-Particle Self-Energy......................... 32
3. Electronic Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.1 Born-Oppenheimer Approximation . . . . . . . . . . . . . . . . . . . . . . . 35
3.2 Density Functional Theory: Generalities . . . . . . . . . . . . . . . . . . . 38
3.3 The Local Density Approximation . . . . . . . . . . . . . . . . . . . . . . . . 41
3.3.1 Discussion of Kohn-Sham Method.................. 43
3.3.2 Pseudopotentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.4 Kohn-Sham Equations in a Spherical Basis . . . . . . . . . . . . . . . . 45
3.4.1 The Expansion of Vext in Spherical Harmonics . . . . . . . 47
3.4.2 The Expansion of g(r), VH and Vxc . . . . . . . . . . . . . . . . . 50
3.5 Kohn-Sham Equations in a Cylindrical Basis . . . . . . . . . . . . . . 51
3.6 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.6.1 Basics on Pseudopotentials . . . . . . . . . . . . . . . . . . . . . . . . 54
3.6.2 Matrix Elements in a Spherical Basis . . . . . . . . . . . . . . . 57
X Contents
4. Electronic Response to Time-Dependent Perturbations . . . 61
4.1 Linear Response: RPA and TDLDA . . . . . . . . . . . . . . . . . . . . . . 61
4.2 Strength Function and Sum Rules . . . . . . . . . . . . . . . . . . . . . . . . 65
4.3 Transition Density and Plasmons . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.4 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.4.1 Linear Response. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.4.2 TDLDA or RPA in the Configuration Space . . . . . . . . . 72
4.4.3 Plasmon Wavefunction Components
for a Separable Interaction . . . . . . . . . . . . . . . . . . . . . . . . 73
4.4.4 Spherical TDLDA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.4.5 Cylindrical TDLDA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5. Applications to Carbon Structures and Metal Clusters . . . 79
5.1 C6o Fullerene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.1.1 Kahn-Sham States for C6o . . . . . . . . . . . . . . . . . . . . . . . . 79
5.1.2 The Optical Absorption of C60 . . . . . . . . . . . . . . . . . . . . . 81
5.2 Carbon Nanotubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.2.1 Electronic Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.2.2 Electromagnetic Response and Static Polarizabilities. . 87
5.3 Linear Carbon Chains................................... 89
5.3.1 Electronic Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.3.2 Optical Response and Plasmon Resonances . . . . . . . . . . 93
5.4 Metal Clusters: the Case of Na8 . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.5 Appendices ............................................ 100
5.5.1 Role of Symmetries ............................... 100
6. Phonons: Harmonic Approximation ....................... 105
6.1 Harmonic Approximation ................................ 106
6.1.1 A Practical Example: a Linear Triatomic Molecule ... 111
6.2 Phonon Calculations .................................... 112
6.2.1 Introduction ..................................... 112
6.2.2 Bond Charge Models ............................. 114
6.2.3 Phonons from ab initio Calculations ................ 128
7. The Car-Parrinello Method .............................. 131
7.1 An Efficient Tool for Solving the Electronic Problem ........ 131
7.2 Car-Parrinello Molecular Dynamics ....................... 133
7.3 Ionic Motion: Parameter-Free Molecular Dynamics Simulation 136
7.3.1 Frozen-Phonon Calculations ....................... 141
8. Coupling of Electrons to Phonons
and to Plasmons .......................................... 145
8.1 Definition of the Electron-Phonon Interaction .............. 146
8.2 Diagonal Matrix Elements ............................... 148
8.3 Electron-Phonon Coupling in Cij0 ........................ 149
Contents XI
8.4 Signatures of the Electron-Phonon Interaction:
Photoemission in C(i0 ................................... 154
8.4.1 Electron-Phonon Coupling in Other Fullerenes ....... 160
8.4.2 Electron-Phonon Coupling in Sodium Clusters ....... 163
8.5 The Effective Electron-Electron Interaction ................ 165
8.6 Coupling of Electrons to Plasmons ........................ 166
8.7 TDLDA with Electron-Plasmon
and Electron-Phonon Couplings .......................... 169
8.8 An Example: Electron-Phonon and
Electron-Plasmon Couplings in Nat ...................... 171
8.9 Appendices ............................................ 173
8.9.1 Terms Neglected in the Born-Oppenheimer
Approximation ................................... 173
8.9.2 The Electron-Phonon Coupling Constant .\ ......... 175
8.9.3 Outline of the Derivation of (8.50) .................. 177
8.9.4 Equation (8.50) Using Second-Order
Perturbation Theory .............................. 178
9. Pairing in Atomic Aggregates ............................. 183
9.1 The BCS Solution ...................................... 184
9.2 Superconductivity in Metals ............................. 190
9.2.1 Finite Temperature ............................... 190
9.3 Superconductivity in Fullerides ........................... 194
9.3.1 C60 Fullerene Based Superconductors ............... 195
9.3.2 Nanometer Superconductors Based on Small Fullerenes 197
9.3.3 Thin Carbon Nanotubes .......................... 200
9.4 Selected Open Questions ................................ 201
9.4.1 Hole Doped Fullerites ............................. 202
9.4.2 Screening Effects ................................. 204
9.5 Appendices ............................................ 204
9.5.1 Simple Estimate of fL ............................. 204
9.5.2 Solution of the Pairing Hamiltonian ................ 206
9.5.3 Temperature Dependent BCS Solution .............. 213
10. Discussion ................................................ 221
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
Index ......................................................... 233
