9789814282598_fmatter.pdf

An Introduction to Theory and Experiment

Molecular Electronics

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World Scientific Series in Nanoscience and Nanotechnology

Series Editor: Mark Reed (Yale University)

Vol. 1 Molecular Electronics: An Introduciton to Theory and ExperimentJuan Carlos Cuevas (Universidad Autónoma de Madrid, Spain) &Elke Scheer (Universität Konstanz, Germany)

Rhaimie - Molecular Electronics.pmd 5/7/2010, 1:54 PM2

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Wor

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Volume

1

N E W J E R S E Y • L O N D O N • S I N G A P O R E • B E I J I N G • S H A N G H A I • H O N G K O N G • TA I P E I • C H E N N A I

World Scientific

An Introduction to Theory and Experiment

Juan Carlos CuevasUniversidad Autónoma de Madrid, Spain

Elke ScheerUniversität Konstanz, Germany

Molecular Electronics

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

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World Scientific Series in Nanoscience and Nanotechnology — Vol. 1MOLECULAR ELECTRONICSAn Introduction to Theory and Experiment

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May 5, 2010 9:20 World Scientific Book - 9in x 6in book

To our families

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Preface

The trend in the miniaturization of electronic devices has naturally led

to the question of whether or not it is possible to use single molecules

as active elements in nanocircuits for a variety of applications. The re-

cent developments in nanofabrication techniques have made possible the

old dream of contacting individual molecules and exploring their electronic

transport properties. Moreover, it has been shown that molecules can in-

deed mimic the behavior of some of today’s microelectronic components,

and even strategies to interconnect molecular devices have already been

developed. These achievements have given rise to what is nowadays known

as Molecular Electronics. There are still many problems and challenges

to be faced to make this novel electronics a viable technology, but the

exploration of molecular-scale circuits has already led to the discovery of

many fundamental effects. In this sense, molecular electronics has become

a new interdisciplinary field of science, in which knowledge from traditional

disciplines like physics, chemistry, engineering and biology is combined to

understand the electrical and thermal conduction at the molecular scale.

This book provides a comprehensive overview of the rapidly developing

field of molecular electronics. It focuses on our present understanding of

the electrical conduction in single-molecule circuits and presents a thorough

introduction to the experimental techniques and the theoretical concepts.

To be precise, our goal in this monograph is two-fold. On the one hand, we

want to provide a true textbook for advanced undergraduate and graduate

students both in physics and chemistry who are interested in the field of

molecular electronics or nanoelectronics in general. Our idea is to take

a student with a good background in quantum mechanics all the way to

be able to follow the specialized literature in molecular electronics or to

start working in this field. On the other hand, we also want provide a

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viii Molecular Electronics: An Introduction to Theory and Experiment

thorough review of the recent activities in molecular electronics from which

newcomers and specialists in the field can benefit.

Bearing these goals in mind, this book has been written in a self-

contained and unified way. It contains four parts that can be read indepen-

dently. In the first two ones we review the basic experimental techniques

and the main theoretical concepts concerning the electronic transport in

atomic-scale junctions. These two parts are meant to be textbook material

for an advanced course in molecular electronics. In particular, we have in-

cluded a collection of exercises at the end of most chapters, which in many

cases are motivated by recent experiments in the field. On the other hand,

Part 3 contains two chapters in which we describe at an introductory level

the physics of metallic atomic-size contacts and we also point out some of

the remaining challenges and open problems in this context. Finally, Part

4 is devoted to the electrical and thermal transport in molecular circuits,

with special emphasis on single-molecule junctions. Here, we do not only

review the recent activities in the field of molecular electronics, but we also

introduce the addressed topics at a basic level. In this sense, we have often

included unpublished material and additional exercises to help the reader

to gain a deeper insight into the fundamental concepts involved in the field

of molecular electronics.1

We have tried to cover in this monograph as many aspects of molecular

electronics as possible, but obviously the selection is limited for space rea-

sons and it reflects unavoidably our own research interests. We also want

to apologize with those authors that feel that their contribution was not

properly highlighted in the review part of this monograph, but it is by now

impossible to include all the huge amount of work done in this field. Fi-

nally, we just hope to have achieved, at least partially, the goal that truly

motivated the writing of this book, namely the sincere will to provide a

useful book for the new generation of researchers that should consolidate

molecular electronics as a solid pillar of the emerging nanoscience.

1See section 1.3 for a more detailed description of the structure and scope of the book.

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Acknowledgments

It would not have been possible to write the book without the help of many

coworkers and colleagues. First of all, we want to thank Edith Goldberg

for encouraging one of us (JCC) to give a postgraduate course on molecular

electronics in the fall of 2008 in Santa Fe (Argentina). The excellent stu-

dents who attended that course demonstrated that, after a 50-hours course

and without any previous knowledge about this field, one can master the

basic concepts and techniques that now form the body of this monograph.

This fact provided the final boost that we needed to collect all our notes

and turn them into this book.

Similarly, for the experimental point of view of this book, the students

in the graduate course at Konstanz served as test candidates. Some of them

even got contaminated by this exciting field and went on asking questions

what finally resulted in contributions to this book. Very valuable input

came from my colleague Artur Erbe who was the real expert in molecular

electronics in our Department until he left to Dresden.

We also want to express our gratitude to Alvaro Martın Rodero, who

not only introduced one of us (JCC) to the exciting field of nanoelectronics,

but also contributed decisively to this manuscript with his personal notes,

which are the basis of several chapters of the theoretical background. The

same holds for Hilbert von Lohneysen and Cristian Urbina who sent the

other one of us (ES) to perform experiments with nanoelectronic circuits.

We would especially like to thank our coworkers Fabian Pauly, Janne K.

Viljas, Michael Hafner, SorenWohlthat, Stefan Bilan, Linda A. Zotti, Cecile

Bacca, Stefan Bachle, Tobias Bohler, Uta Eberlein, Stefan Egle, Daniel

Guhr, Ning Kang, Thomas Kirchner, Christian Kreuter, Shou-Peng Liu,

Youngsang Kim, Hans-Fridtjof Pernau, Olivier Schecker, Christian Schirm,

Dima Sysoiev, Simon Verleger, and Reimar Waitz. They have contributed

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x Molecular Electronics: An Introduction to Theory and Experiment

to this manuscript with many results, special figures and very important

suggestions and critical comments about the text.

Thanks go also sincerely to our colleagues who have read different parts

of the manuscript and have provided helpful comments: Douglas Natelson,

Abraham Nitzan, Wilson Ho, Latha Venkataraman, and Arunava Majum-

dar.

This monograph reflects our view of this field, which has emerged thanks

to the collaboration and exchange of ideas with many colleagues over the

years. So in this respect, we want to thank Alfredo Levy Yeyati, Gerd

Schon, Jan Heurich, Wolfgang Wenzel, Jan M. van Ruitenbeek, Nicolas

Agraıt, Gabino Rubio, Roel Smit, Oren Tal, Markus Dreher, Peter Nielaba,

Christoph Surgers, Maya Lukas, Christoph Strunk, Sophie Gueron, Richard

Berndt, Paul Leiderer, Wolfgang Belzig, Marcel Mayor, Thomas Huhn,

Andreas Marx, Ulrich Steiner, and Ulrich Groth.

We also want acknowledge the contribution of all the authors who have

kindly granted us the permission to reprint their work in this monograph.

Finally, I (JCC) want to thank my parents and brothers for being always

by my side. I also want to thank Ana for being so patient and share my

time with this book for too many nights and weekends. ES thanks her

family for continuous support and reminding me steadily of what is really

important in life.

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Contents

Preface vii

Acknowledgments ix

Brief history of the field and experimentaltechniques 1

1. The birth of molecular electronics 3

1.1 Why molecular electronics? . . . . . . . . . . . . . . . . . 5

1.2 A brief history of molecular electronics . . . . . . . . . . . 6

1.3 Scope and structure of the book . . . . . . . . . . . . . . 14

2. Fabrication of metallic atomic-size contacts 19

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2 Techniques involving the scanning electron microscope

(STM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3 Methods using atomic force microscopes (AFM) . . . . . 21

2.4 Contacts between macroscopic wires . . . . . . . . . . . . 22

2.5 Transmission electron microscope . . . . . . . . . . . . . . 23

2.6 Mechanically controllable break-junctions (MCBJ) . . . . 24

2.7 Electromigration technique . . . . . . . . . . . . . . . . . 31

2.8 Electrochemical methods . . . . . . . . . . . . . . . . . . . 35

2.9 Recent developments . . . . . . . . . . . . . . . . . . . . . 37

2.10 Electronic transport measurements . . . . . . . . . . . . . 38

2.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

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xii Molecular Electronics: An Introduction to Theory and Experiment

3. Contacting single molecules: Experimental techniques 45

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.2 Molecules for molecular electronics . . . . . . . . . . . . . 46

3.2.1 Hydrocarbons . . . . . . . . . . . . . . . . . . . . 47

3.2.2 All carbon materials . . . . . . . . . . . . . . . . . 50

3.2.3 DNA and DNA derivatives . . . . . . . . . . . . . 51

3.2.4 Metal-molecule contacts: anchoring groups . . . . 52

3.2.5 Conclusions: molecular functionalities . . . . . . . 52

3.3 Deposition of molecules . . . . . . . . . . . . . . . . . . . 53

3.4 Contacting single molecules . . . . . . . . . . . . . . . . . 55

3.4.1 Electromigration technique . . . . . . . . . . . . . 56

3.4.2 Molecular contacts using the transmission electron

microscope . . . . . . . . . . . . . . . . . . . . . . 58

3.4.3 Gold nanoparticle dumbbells . . . . . . . . . . . . 59

3.4.4 Scanning probe techniques . . . . . . . . . . . . . 60

3.4.5 Mechanically controllable break-junctions (MCBJs) 64

3.5 Contacting molecular ensembles . . . . . . . . . . . . . . . 66

3.5.1 Nanopores . . . . . . . . . . . . . . . . . . . . . . 66

3.5.2 Shadow masks . . . . . . . . . . . . . . . . . . . . 68

3.5.3 Conductive polymer electrodes . . . . . . . . . . . 69

3.5.4 Microtransfer printing . . . . . . . . . . . . . . . . 70

3.5.5 Gold nanoparticle arrays . . . . . . . . . . . . . . 71

3.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

Theoretical background 75

4. The scattering approach to phase-coherent transport in

nanocontacts 77

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.2 From mesoscopic conductors to atomic-scale junctions . . 79

4.3 Conductance is transmission: Heuristic derivation of the

Landauer formula . . . . . . . . . . . . . . . . . . . . . . . 81

4.4 Penetration of a potential barrier: Tunnel effect . . . . . . 83

4.5 The scattering matrix . . . . . . . . . . . . . . . . . . . . 88

4.5.1 Definition and properties of the scattering matrix 88

4.5.2 Combining scattering matrices . . . . . . . . . . . 91

4.6 Multichannel Landauer formula . . . . . . . . . . . . . . . 92

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Contents xiii

4.6.1 Conductance quantization in 2DEG: Landauer

formula at work . . . . . . . . . . . . . . . . . . . 97

4.7 Shot noise . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.8 Thermal transport and thermoelectric phenomena . . . . 104

4.9 Limitations of the scattering approach . . . . . . . . . . . 106

4.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5. Introduction to Green’s function techniques for systems

in equilibrium 111

5.1 The Schrodinger and Heisenberg pictures . . . . . . . . . 112

5.2 Green’s functions of a noninteracting electron system . . . 113

5.3 Application to tight-binding Hamiltonians . . . . . . . . . 118

5.3.1 Example 1: A hydrogen molecule . . . . . . . . . 118

5.3.2 Example 2: Semi-infinite linear chain . . . . . . . 122

5.3.3 Example 3: A single level coupled to electrodes . 124

5.4 Green’s functions in time domain . . . . . . . . . . . . . . 128

5.4.1 The Lehmann representation . . . . . . . . . . . . 131

5.4.2 Relation to observables . . . . . . . . . . . . . . . 134

5.4.3 Equation of motion method . . . . . . . . . . . . 136

5.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

6. Green’s functions and Feynman diagrams 143

6.1 The interaction picture . . . . . . . . . . . . . . . . . . . . 144

6.2 The time-evolution operator . . . . . . . . . . . . . . . . . 146

6.3 Perturbative expansion of causal Green’s functions . . . . 148

6.4 Wick’s theorem . . . . . . . . . . . . . . . . . . . . . . . . 149

6.5 Feynman diagrams . . . . . . . . . . . . . . . . . . . . . . 151

6.5.1 Feynman diagrams for the electron-electron inter-

action . . . . . . . . . . . . . . . . . . . . . . . . . 152

6.5.2 Feynman diagrams for an external potential . . . 157

6.6 Feynman diagrams in energy space . . . . . . . . . . . . . 158

6.7 Electronic self-energy and Dyson’s equation . . . . . . . . 162

6.8 Self-consistent diagrammatic theory: The Hartree-Fock

approximation . . . . . . . . . . . . . . . . . . . . . . . . 167

6.9 The Anderson model and the Kondo effect . . . . . . . . . 170

6.9.1 Friedel sum rule . . . . . . . . . . . . . . . . . . . 171

6.9.2 Perturbative analysis . . . . . . . . . . . . . . . . 173

6.10 Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . 175

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xiv Molecular Electronics: An Introduction to Theory and Experiment

6.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

7. Nonequilibrium Green’s functions formalism 179

7.1 The Keldysh formalism . . . . . . . . . . . . . . . . . . . 180

7.2 Diagrammatic expansion in the Keldysh formalism . . . . 184

7.3 Basic relations and equations in the Keldysh formalism . 186

7.3.1 Relations between the Green’s functions . . . . . 186

7.3.2 The triangular representation . . . . . . . . . . . 187

7.3.3 Unperturbed Keldysh-Green’s functions . . . . . . 189

7.3.4 Some comments on the notation . . . . . . . . . . 191

7.4 Application of Keldysh formalism to simple transport

problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

7.4.1 Electrical current through a metallic atomic contact193

7.4.2 Shot noise in an atomic contact . . . . . . . . . . 199

7.4.3 Current through a resonant level . . . . . . . . . . 200

7.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

8. Formulas of the electrical current: Exploiting the Keldysh

formalism 205

8.1 Elastic current: Microscopic derivation of the Landauer

formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

8.1.1 An example: back to the resonant tunneling model 211

8.1.2 Nonorthogonal basis sets . . . . . . . . . . . . . . 212

8.1.3 Spin-dependent elastic transport . . . . . . . . . . 213

8.2 Current through an interacting atomic-scale junction . . . 215

8.2.1 Electron-phonon interaction in the resonant tun-

neling model . . . . . . . . . . . . . . . . . . . . . 217

8.2.2 The Meir-Wingreen formula . . . . . . . . . . . . 222

8.3 Time-dependent transport in nanoscale junctions . . . . . 224

8.3.1 Photon-assisted resonant tunneling . . . . . . . . 231

8.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

9. Electronic structure I: Tight-binding approach 237

9.1 Basics of the tight-binding approach . . . . . . . . . . . . 237

9.2 The extended Huckel method . . . . . . . . . . . . . . . . 241

9.3 Matrix elements in solid state approaches . . . . . . . . . 242

9.3.1 Two-center matrix elements . . . . . . . . . . . . 244

9.4 Slater-Koster two-center approximation . . . . . . . . . . 246

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Contents xv

9.5 Some illustrative examples . . . . . . . . . . . . . . . . . . 247

9.5.1 Example 1: A benzene molecule . . . . . . . . . . 248

9.5.2 Example 2: Energy bands in line, square and cubic

Bravais lattices . . . . . . . . . . . . . . . . . . . . 250

9.5.3 Example 3: Energy bands of graphene . . . . . . 252

9.6 The NRL tight-binding method . . . . . . . . . . . . . . . 253

9.7 The tight-binding approach in molecular electronics . . . 257

9.7.1 Some comments on the practical implementation

of the tight-binding approach . . . . . . . . . . . . 258

9.7.2 Tight-binding simulations of atomic-scale trans-

port junctions . . . . . . . . . . . . . . . . . . . . 259

9.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 260

10. Electronic structure II: Density functional theory 263

10.1 Elementary quantum mechanics . . . . . . . . . . . . . . . 264

10.1.1 The Schrodinger equation . . . . . . . . . . . . . . 264

10.1.2 The variational principle for the ground state . . 265

10.1.3 The Hartree-Fock approximation . . . . . . . . . . 266

10.2 Early density functional theories . . . . . . . . . . . . . . 268

10.3 The Hohenberg-Kohn theorems . . . . . . . . . . . . . . . 269

10.4 The Kohn-Sham approach . . . . . . . . . . . . . . . . . . 271

10.5 The exchange-correlation functionals . . . . . . . . . . . . 273

10.5.1 LDA approximation . . . . . . . . . . . . . . . . . 273

10.5.2 The generalized gradient approximation . . . . . . 275

10.5.3 Hybrid functionals . . . . . . . . . . . . . . . . . . 277

10.6 The basic machinery of DFT . . . . . . . . . . . . . . . . 277

10.6.1 The LCAO Ansatz in the Kohn-Sham equations . 278

10.6.2 Basis sets . . . . . . . . . . . . . . . . . . . . . . . 280

10.7 DFT performance . . . . . . . . . . . . . . . . . . . . . . 282

10.8 DFT in molecular electronics . . . . . . . . . . . . . . . . 284

10.8.1 Combining DFT with NEGF techniques . . . . . 285

10.8.2 Pluses and minuses of DFT-NEGF-based methods 291

10.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 292

Metallic atomic-size contacts 293

11. The conductance of a single atom 295

11.1 Landauer approach to conductance: brief reminder . . . . 296

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xvi Molecular Electronics: An Introduction to Theory and Experiment

11.2 Conductance of atomic-scale contacts . . . . . . . . . . . 297

11.3 Conductance histograms . . . . . . . . . . . . . . . . . . . 300

11.4 Determining the conduction channels . . . . . . . . . . . . 304

11.5 The chemical nature of the conduction channels of one-

atom contacts . . . . . . . . . . . . . . . . . . . . . . . . . 308

11.6 Some further issues . . . . . . . . . . . . . . . . . . . . . . 316

11.7 Conductance fluctuations . . . . . . . . . . . . . . . . . . 319

11.8 Atomic chains: Parity oscillations in the conductance . . . 322

11.9 Concluding remarks . . . . . . . . . . . . . . . . . . . . . 331

11.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 332

12. Spin-dependent transport in ferromagnetic atomic

contacts 335

12.1 Conductance of ferromagnetic atomic contacts . . . . . . 336

12.2 Magnetoresistance of ferromagnetic atomic contacts . . . 343

12.3 Anisotropic magnetoresistance in atomic contacts . . . . . 347

12.4 Concluding remarks and open problems . . . . . . . . . . 353

Transport through molecular junctions 355

13. Coherent transport through molecular junctions I: Basic

concepts 357

13.1 Identifying the transport mechanism in single-molecule

junctions . . . . . . . . . . . . . . . . . . . . . . . . . . . 359

13.2 Some lessons from the resonant tunneling model . . . . . 364

13.2.1 Shape of the I-V curves . . . . . . . . . . . . . . . 366

13.2.2 Molecular contacts as tunnel junctions . . . . . . 368

13.2.3 Temperature dependence of the current . . . . . . 369

13.2.4 Symmetry of the I-V curves . . . . . . . . . . . . 371

13.2.5 The resonant tunneling model at work . . . . . . 373

13.3 A two-level model . . . . . . . . . . . . . . . . . . . . . . 374

13.4 Length dependence of the conductance . . . . . . . . . . . 377

13.5 Role of conjugation in π-electron systems . . . . . . . . . 381

13.6 Fano resonances . . . . . . . . . . . . . . . . . . . . . . . 382

13.7 Negative differential resistance . . . . . . . . . . . . . . . 385

13.8 Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . 388

13.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 389

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Contents xvii

14. Coherent transport through molecular junctions II:

Test-bed molecules 391

14.1 Coherent transport through some test-bed molecules . . . 392

14.1.1 Benzenedithiol: how everything started . . . . . . 392

14.1.2 Conductance of alkanedithiol molecular junctions:

A reference system . . . . . . . . . . . . . . . . . 395

14.1.3 The smallest molecular junction: Hydrogen

bridges . . . . . . . . . . . . . . . . . . . . . . . . 401

14.1.4 Highly conductive benzene junctions . . . . . . . . 405

14.2 Metal-molecule contact: The role of anchoring groups . . 408

14.3 Tuning chemically the conductance: The role of

side-groups . . . . . . . . . . . . . . . . . . . . . . . . . . 412

14.4 Controlled STM-based single-molecule experiments . . . . 416

14.5 Conclusions and open problems . . . . . . . . . . . . . . . 420

15. Single-molecule transistors: Coulomb blockade and

Kondo physics 423

15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 423

15.2 Charging effects in transport through nanoscale devices . 425

15.3 Single-molecule three-terminal devices . . . . . . . . . . . 429

15.4 Coulomb blockade theory: Constant interaction model . . 432

15.4.1 Formulation of the problem . . . . . . . . . . . . . 432

15.4.2 Periodicity of the Coulomb blockade oscillations . 435

15.4.3 Qualitative discussion of the transport

characteristics . . . . . . . . . . . . . . . . . . . . 436

15.4.4 Amplitudes and line shapes: Rate equations . . . 439

15.5 Towards a theory of Coulomb blockade in molecular tran-

sistors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445

15.5.1 Many-body master equations . . . . . . . . . . . . 447

15.5.2 A simple example: The Anderson model . . . . . 449

15.6 Intermediate coupling: Cotunneling and Kondo effect . . 451

15.6.1 Elastic and inelastic cotunneling . . . . . . . . . . 451

15.6.2 Kondo effect . . . . . . . . . . . . . . . . . . . . . 453

15.7 Single-molecule transistors: Experimental results . . . . . 456

15.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 468

16. Vibrationally-induced inelastic current I: Experiment 473

16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 473

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xviii Molecular Electronics: An Introduction to Theory and Experiment

16.2 Inelastic electron tunneling spectroscopy (IETS) . . . . . 475

16.3 Highly conductive junctions: Point-contact spectroscopy

(PCS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483

16.4 Crossover between PCS and IETS . . . . . . . . . . . . . 490

16.5 Resonant inelastic electron tunneling spectroscopy

(RIETS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493

16.6 Summary of vibrational signatures . . . . . . . . . . . . . 499

17. Vibrationally-induced inelastic current II: Theory 501

17.1 Weak electron-phonon coupling regime . . . . . . . . . . . 501

17.1.1 Single-phonon model . . . . . . . . . . . . . . . . 502

17.1.2 Ab initio description of inelastic currents . . . . . 512

17.2 Intermediate electron-phonon coupling regime . . . . . . . 520

17.3 Strong electron-phonon coupling regime . . . . . . . . . . 524

17.3.1 Coulomb blockade regime . . . . . . . . . . . . . . 524

17.3.2 Interplay of Kondo physics and vibronic effects . . 532

17.4 Concluding remarks and open problems . . . . . . . . . . 534

17.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 535

18. The hopping regime and transport through DNA

molecules 537

18.1 Signatures of the hopping regime . . . . . . . . . . . . . . 538

18.2 Hopping transport in molecular junctions: Experimental

examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 541

18.3 DNA-based molecular junctions . . . . . . . . . . . . . . . 546

18.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 552

19. Beyond electrical conductance: Shot noise and thermal

transport 553

19.1 Shot noise in atomic and molecular junctions . . . . . . . 554

19.2 Heating and heat conduction . . . . . . . . . . . . . . . . 560

19.2.1 General considerations . . . . . . . . . . . . . . . 561

19.2.2 Thermal conductance . . . . . . . . . . . . . . . . 562

19.2.3 Heating and junction temperature . . . . . . . . . 565

19.3 Thermoelectricity in molecular junctions . . . . . . . . . . 569

20. Optical properties of current-carrying molecular

junctions 579

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Contents xix

20.1 Surface-enhanced Raman spectroscopy of molecular

junctions . . . . . . . . . . . . . . . . . . . . . . . . . . . 580

20.2 Transport mechanisms in irradiated molecular junctions . 583

20.3 Theory of photon-assisted tunneling . . . . . . . . . . . . 585

20.3.1 Basic theory . . . . . . . . . . . . . . . . . . . . . 586

20.3.2 Theory of PAT in atomic contacts . . . . . . . . . 590

20.3.3 Theory of PAT in molecular junctions . . . . . . . 592

20.4 Experiments on radiation-induced transport in atomic and

molecular junctions . . . . . . . . . . . . . . . . . . . . . . 594

20.5 Resonant current amplification and other transport phe-

nomena in ac driven molecular junctions . . . . . . . . . . 601

20.6 Fluorescence from current-carrying molecular junctions . 604

20.7 Molecular optoelectronic devices . . . . . . . . . . . . . . 608

20.8 Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . 613

20.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 614

21. What is missing in this book? 617

Appendixes 621

Appendix A Second Quantization 623

A.1 Harmonic oscillator and phonons . . . . . . . . . . . . . . 624

A.1.1 Review of simple harmonic oscillator quantization 624

A.1.2 1D harmonic chain . . . . . . . . . . . . . . . . . 626

A.2 Second quantization for fermions . . . . . . . . . . . . . . 628

A.2.1 Many-body wave function in second quantization 628

A.2.2 Creation and annihilation operators . . . . . . . . 630

A.2.3 Operators in second quantization . . . . . . . . . 632

A.2.4 Some special Hamiltonians . . . . . . . . . . . . . 634

A.3 Second quantization for bosons . . . . . . . . . . . . . . . 637

A.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 638

Bibliography 639

Index 699

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