front matter

9948_9789814749978_tp.pdf 1

An Introduction to Graphene Plasmonics Downloaded from www.worldscientific.com by 2.107.6.126 on 09/12/16. For personal use only.

Graphene Plasmonics An Introduction to

P.A. Gonça

30/3/16 8:58 AM


May 2, 2013 14:6 BC: 8831 - Probability and Statistical Theory

This page intentionally left blank

PST˙ws


An Introduction to

Graphene Plasmonics

P.A.D Gonçalves University of Minho, Portugal

N.M.R Peres Univeristy of Minho, Portugal

World Scientific NEW JERSEY

•

LONDON

9948_9789814749978_tp.pdf 2

•

SINGAPORE

•

BEIJING

•

SHANGHAI

•

HONG KONG

•

TAIPEI

•

CHENNAI

•

TOKYO

30/3/16 8:58 AM

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601


UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

Library of Congress Cataloging-in-Publication Data Names: Gonçalves, P. A. D., author. | Peres, N. M. R., 1967– author. Title: An introduction to graphene plasmonics / P.A.D. Gonçalves (University of Minho, Portugal), N.M.R. Peres (University of Minho, Portugal). Description: Singapore ; Hackensack, NJ : World Scientific Publishing Co. Pte. Ltd., [2016] | 2016 | Includes bibliographical references and index. Identifiers: LCCN 2016003760| ISBN 9789814749978 (hardcover ; alk. paper) | ISBN 9814749974 (hardcover ; alk. paper) | ISBN 9789814749985 (softcover ; alk. paper) | ISBN 9814749982 (softcover ; alk. paper) Subjects: LCSH: Plasmons (Physics) | Graphene--Optical properties. | Nanophotonics. Classification: LCC QC176.8.P55 G66 2016 | DDC 530.4/4--dc23 LC record available at http://lccn.loc.gov/2016003760 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

Copyright © 2016 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. In-house Editor: Christopher Teo Typeset by Stallion Press Email: [email protected] Printed in Singapore

Christopher - An Introduction to Graphene Plasmonics.indd 1

15/2/2016 1:33:59 PM


April 5, 2016

14:13

An Introduction to Graphene Plasmonics

9in x 6in

To our families: (Inês and João Nuno, and Augusta, Augusto, and Filipa)

And our parents: (Clementina and João Paulo, and Cecília and Luís)

v

b2445-mas

page v




PST˙ws

April 5, 2016

14:13


9in x 6in

b2445-mas


Foreword

The consequences of plasmons, collective excitations of the electron plasma in conductive materials, have been appreciated for centuries — if not millenia, just witness the beautiful ancient colored glasses which are found in museums and churches. There, the colors originate from localized surface plasmonic excitations in nanoscale metallic particles. This old example crystallizes the essence of modern plasmonics: on one hand, one needs surfaces, and on the other hand the active objects should be in the nanometer scale. Graphene, the wonder material of the 21st century, is twodimensional and therefore all surface; it is just one atomic layer thick and by using modern nanofabrication technologies it can be sculpted laterally to nanoribbons or nanoflakes. Graphene has very special optical properties, which stem from its fascinating Dirac-like electronic dispersion law its charge carriers obey. It is no wonder then that graphene plasmonics is an important and growing area of research. Most importantly, it holds the promise of many practical applications, for example in plasmonic circuitry, nanophotonics, and sensing, just to mention a few. This is a book on graphene plasmonics. It originates from a group that is world-known for its research in this area — who would be better qualified to write such a book? Students entering graphene plasmonics have had a difficult time in navigating between original research articles, which rarely are particularly pedagogical in their approach, and text books addressing plasmonics (many of which are excellent), but usually having conventional metals as the working material. The present book thus fills in an important gap. Now, all the relevant material is collected in a single volume. What is essential is that the material is thoroughly worked through: the approach is extremely systematic, all equations follow from previous equations in a manner that a dedicated student can easily follow. The technical

vii

page vii

April 5, 2016

14:13


viii


9in x 6in

b2445-mas


level of the book is accessible to an advanced undergraduate or a beginning graduate student, but experienced researchers will also benefit from the many insights it offers. It starts with an introduction to Maxwell’s equations, and thereafter works its way towards graphene, and the nanostructures made of graphene. The very important issue of interfaces and two-dimensionality is given an exhaustive treatment. Underway the quantum mechanical description of graphene’s electronic structure is developed, including effects due to applied magnetic fields, and central topics in manybody physics are discussed (and further explained and elaborated in useful appendices). With the basic concepts at hand, the reader is taken to the frontier of today’s research: plasmons in nanostructured two-dimensional materials, such as arrays of microribbons, nanorings, and gratings. The excitation mechanisms needed to launch plasmons and plasmon-polaritons are discussed, as well as methods that are required to observe the excited plasmons. The authors should be congratulated for having accomplished an important task: to offer a self-contained and complete, yet accessible, treatment of a modern area under rapid progress. Armed with the knowledge extracted from this book tomorrow’s students are ready to tackle the many exciting opportunities and challenges that plasmonics in two-dimensional materials offer.

Antti-Pekka Jauho, Technical University of Denmark, 2016

page viii

April 5, 2016

14:13


9in x 6in

b2445-mas


Preface

Over the last few years, graphene plasmonics has emerged as a new research topic, positioned at the interface between condensed matter physics and photonics. This still young but rapidly growing field, resulting from the overlap between graphene physics and plasmonics, deals with the investigation and exploration of graphene surface plasmon-polaritons (GSPs) for controlling light-matter interactions and manipulating light at the nanoscale. In 2011, experimental research in plasmonics in graphene was in its infancy. Some theoretical papers had already been written, but the field was awaiting for some sort of experimental boost. In 2011 a seminal experimental paper (to be discussed in Section 7.1) emerged, demonstrating that plasmonic effects in graphene could be controlled optically, by shining electromagnetic radiation onto a periodic grid of graphene micro-ribbons. This opened a new avenue in graphene physics, launching the new flourishing field of graphene plasmonics. Since then, the field has been witnessing enormous developments at a rapid pace (to be described in Section 1.2). This is a book on plasmonics in graphene. It grew out of the authors’ own interest in the field. Many of the topics covered here refer to problems that were amongst the first to appear in the scientific literature, to which one of the authors has contributed to some extent. Throughout this monograph we have tried to make this book as self-contained as possible, and to provide substantial detail in all derivations. Indeed, with a pencil, some sheets of paper, a dose of enthusiasm, and, at times, a bit of effort, the interested reader should find it easy to reproduce all the results in the book (the Appendices will be most useful for such endeavor). Therefore, we hope that this book will serve as a springboard for any newcomer in graphene

ix

page ix

April 5, 2016

14:13


x


9in x 6in

b2445-mas


plasmonics. Hopefully, the advanced researcher may also find reasons to keep this book in her/his shelf and to recommend it to her/his students. This is also a book about theoretical techniques to deal with plasmonics in graphene. Many of these techniques have been applied to metal plasmonics before and have been adapted here to tackle plasmonic effects in graphene. The main difference is that metals are described by a complex dielectric function, whereas graphene is described by its complex (non-local whenever necessary) optical conductivity and its sheet (two-dimensional) current. The book is organized as follows: in Chapter 1 we introduce the theme of the book and present a short review of the literature. We have surely missed some important references and can only hope for the indulgence and comprehension of their authors. This might be specially true in what concerns theoretical and computational work (the literature is already too vast to be covered in a comprehensive way). In Chapter 2 we review some elementary concepts about electronic transport in solids, give a short introduction to graphene’s electronic properties, and present some tools to be used later in the book, with particular emphasis on a method to compute the non-local optical conductivity of graphene from the electronic susceptibility in the relaxation-time approximation. Chapter 3 discusses plasmonics at metaldielectric interfaces and in metal thin films. The results obtained there will be later compared with those found in graphene. Both the semi-infinite metal and the thin metal-film are discussed. In Chapter 4, we finally dive into the field of graphene plasmonics and discuss plasmons both in singleand double-layer graphene. The interesting problem of coupled surface plasmon-phonon-polaritons is also discussed in that Chapter. A short introduction of magneto surface plasmon-polaritons is included, considering both the semi-classical and quantum regimes. We close the chapter with the study of guided (bounded) surface phonon-polaritons in a crystal slab of hexagonal boron-nitride (hBN), a problem very recently considered in the literature and one for which many future developments are expected. As we shall see, surface plasmons cannot be directly excited in graphene by free propagating electromagnetic waves (as is also the case in planar metal-dielectric interfaces). Thus, in Chapter 5 we review some of the most popular methods of inducing surface plasmon-polaritons in this twodimensional material (which are also used in traditional metal-based plasmonics). We then move to a detailed study of two methods for exciting plasmons in graphene. These are discussed in Chapters 6 and 7, encompassing the excitation of GSPs by a metallic antenna and by coupling light

page x

April 5, 2016

14:13



Preface

9in x 6in

b2445-mas

xi

to GSPs in a patterned array of graphene ribbons. The discussion of the periodic graphene ribbons system sets the stage for the discussion of the spectrum of surface plasmons in graphene micro-ribbons, micro-rings, and disks. This is done in Chapter 8, where the case of a micro-deformation of the graphene sheet is also discussed, taking as an example the problem of a Gaussian groove. This latter case can be considered an example of nano-structured graphene. These three problems are considered in the electrostatic limit, which turns out to be a rather good approximation when dealing with plasmons in graphene. As we shall show they exhibit scale invariance, a property of the electrostatic limit. It then follows, in Chapter 9, a discussion of the excitation of plasmons by a corrugated dielectric grating, which is a generalization of the single micro-deformation studied in the preceding chapter to the case of a periodic deformation. In this case, the calculation takes retardation into account, but could very well be treated within the electrostatic approximation. Despite some differences, the four methods discussed in Chapters 6, 7, 8, and 9 have in common the decomposition of the fields in a superposition of Fourier modes, either using a Fourier series, for periodic structures, or a Fourier integral for non-periodic systems. In Chapter 10 we discuss how an emitting dipole (for example, a dye, a quantum dot, or a Rydberg atom) can excite graphene plasmons by non-radiative energy transfer from the emitter to graphene. We will see that this is in principle possible, and we compute the non-radiative decaying rate of an emitter. The role of particle-hole processes and excitation of plasmons is discussed separately. We end this Chapter with the calculation of the Purcell factor which encodes information about the total decaying rate of the emitter in the presence of graphene. Finally, in Chapter 11, we conclude with a brief outlook and with the discussion of some topics that have been left out in this book, and which we believe will become rather important in the near future. Also, we have written a number of appendices with either details of the calculations or extensions of the materials discussed in the bulk of the book. To get a flavor of the main ideas, reading the appendices is unnecessary. However, they are a requisite for a working knowledge on the topics discussed in the main text. Our ultimate hope is that readers may find this book a valuable tool for their study and research in the exciting field of graphene plasmonics and nanophotonics, whether they are currently working on the topic or are about to enter on it. The book’s intended purpose is to bring together some of the main topics that are scattered throughout the already vast literature, and to do justice to this alluring field of graphene plasmonics.

page xi

April 5, 2016

14:13

xii


9in x 6in

b2445-mas


To conclude, we further hope that the readers may read this book with the same enthusiasm that we had writing it. Comments, suggestions, and corrections aiming at improving the book for future usage will be most welcome (these can be sent to [email protected]).


Paulo André Dias Gonçalves Nuno Miguel Machado Reis Peres Department of Physics University of Minho Braga, 2016

page xii

April 5, 2016

14:13


9in x 6in

b2445-mas


Acknowledgments

The authors acknowledge B. Amorim, Y. V. Bludov, E. V. Castro, A. H. Castro Neto, A. Chaves, A. Ferreira, F. Guinea, J. Lopes dos Santos, V. Pereira, R. M. Ribeiro, J. E. Santos, T. Stauber, and M. I. Vasilevskiy for many years of collaboration and discussions about the physics of graphene in general and on plasmonics in graphene and other materials in particular. We thank B. Amorim, Y. V. Bludov, A. Chaves, A. Ferreira, J. Lopes dos Santos, R. M. Ribeiro, J. E. Santos, and M. I. Vasilevskiy for reading parts of the manuscript at different moments of writing. B. Amorim is acknowledged for providing the derivations in Appendices D, O, and P. A. Chaves is acknowledged for providing the derivation in Appendix A J. E. Santos is acknowledged for providing the derivation in Appendix K. Any error and/or inaccuracy that may remain in the text of these five Appendices (and, for that matter, in the rest of the text) is the authors’ own responsibility. Paulo André Gonçalves acknowledges the Calouste Gulbenkian Foundation for financial support through the grant “Prémio Estímulo à Investigação 2013 "1 (No. 132394) and the hospitality of the Centre of Physics of the University of Minho, where most of this book was written. Nuno Peres acknowledges financial support from the Graphene Flagship Project (Contract No. CNECT-ICT-604391).

1 "Gulbenkian

Prize for Stimulating Young Researchers 2013". xiii

page xiii




PST˙ws

April 5, 2016

14:13


9in x 6in

b2445-mas


Contents

Foreword

vii

Preface

ix

Acknowledgments

xiii

List of Figures

xxiii

List of Tables

xxix

1.

2.

Introduction

1

1.1 1.2

1 4

Plasmonics: Generalities . . . . . . . . . . . . . . . . . . . Plasmonics: Recent Developments . . . . . . . . . . . . .

Electromagnetic Properties of Solids in a Nutshell

11

2.1

11 11 14 14 17 17 24 28

2.2 2.3

2.4

Classical Electrodynamics Basics . . . . . . . . . . . . . . 2.1.1 Maxwell’s equations . . . . . . . . . . . . . . . . . 2.1.2 Boundary conditions . . . . . . . . . . . . . . . . . Drude Model . . . . . . . . . . . . . . . . . . . . . . . . . Preliminaries to Graphene Plasmonics . . . . . . . . . . . 2.3.1 Elementary electronic properties . . . . . . . . . 2.3.2 The optical conductivity of graphene . . . . . . . 2.3.3 Lindhard function: Beyond the local approximation 2.3.4 Lindhard polarization function in relaxation-time approximation: The case of graphene . . . . . . . The Transfer-Matrix Method and the First Appearance of Plasmons in Graphene . . . . . . . . . . . . . . . . . . . . 2.4.1 Transfer-matrix for a graphene monolayer . . . . . xv

33 35 36

page xv

April 5, 2016

14:13


xvi

2.4.3 2.4.4 2.4.5


Transmittance, reflectance and absorbance of electromagnetic radiation by a graphene monolayer . Transfer-matrix for a graphene double-layer . . . . Transfer-matrix for multi-layer graphene structures Plasmons: A transfer-matrix approach . . . . . . .

Surface Plasmon-Polaritons at Dielectric-Metal Interfaces 3.1

3.2

3.3 4.

b2445-mas


2.4.2

3.

9in x 6in

Single Dielectric-Metal Interface . . . . . . . . . . 3.1.1 Dispersion relation . . . . . . . . . . . . . 3.1.2 Propagation length and field confinement Multilayer Structures . . . . . . . . . . . . . . . . 3.2.1 Double Interface . . . . . . . . . . . . . . 3.2.2 Dispersion relation . . . . . . . . . . . . . A Short Note on Perfect Conductors . . . . . . .

. . . . . . .

39 40 42 45 47

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

47 48 54 57 57 59 64

Graphene Surface Plasmons

67

4.1

67 68 75 78 81 85 86 87 91

4.2

4.3

4.4

4.5

Monolayer Graphene . . . . . . . . . . . . . . . . . . . . . 4.1.1 Spectrum of GSPs . . . . . . . . . . . . . . . . . 4.1.2 Field profile of GSPs in single-layer graphene . . . 4.1.3 Plasmon dispersion revisited . . . . . . . . . . . . 4.1.4 Loss function . . . . . . . . . . . . . . . . . . . . 4.1.5 A word on TE surface waves . . . . . . . . . . . . Double-Layer Graphene . . . . . . . . . . . . . . . . . . . 4.2.1 Dispersion relation . . . . . . . . . . . . . . . . . . 4.2.2 Field profile of GSPs in double-layer graphene . . 4.2.3 Plasmon dispersion beyond the Drude model approximation . . . . . . . . . . . . . . . . . . . . . Surface Plasmon-Phonon-Polaritons in Graphene . . . . . 4.3.1 Graphene on SiC . . . . . . . . . . . . . . . . . . . 4.3.2 Graphene on SiO2 . . . . . . . . . . . . . . . . . . Magneto-Plasmons in Monolayer Graphene . . . . . . . . 4.4.1 Derivation of the spectrum’s condition . . . . . . . 4.4.2 Solution of the semi-classical spectrum’s condition in a special limit . . . . . . . . . . . . . . . . . . . 4.4.3 Spectrum of magneto-plasmons in the quantum regime: Landau quantization . . . . . . . . . . . . A Detour: Surface Phonon-Polaritons in hBN . . . . . . . 4.5.1 Solution of the electromagnetic problem . . . . . .

94 101 104 106 112 113 115 120 124 125

page xvi

April 5, 2016

14:13


9in x 6in

b2445-mas

Contents

4.5.2 5.


5.2 5.3 5.4 5.5

. . . . of SO . . . . . . . . . . . . . . . . . . . . . . . .

7.3

7.4 7.5

8.2

132 134 134 140 142 143

Theoretical Model and Integral Equation . . . . . . . . . . 149 Approximate Solution via Fourier Expansion . . . . . . . 153 Results and Discussion . . . . . . . . . . . . . . . . . . . . 156

A Seminal Paper . . . . . . . . . . . . . . . . . . . . . Theoretical Model . . . . . . . . . . . . . . . . . . . . 7.2.1 Setting up the model . . . . . . . . . . . . . . 7.2.2 The scattering problem . . . . . . . . . . . . . Applications and Results . . . . . . . . . . . . . . . . . 7.3.1 Periodic array of graphene ribbons . . . . . . . 7.3.2 Theory versus experiment . . . . . . . . . . . . A THz Polarizer . . . . . . . . . . . . . . . . . . . . . Scattering From a Periodic Grid in the Regime kw < 1

163 . . . . . . . . .

. . . . . . . . .

Plasmons in Graphene Nanostructures and in Onedimensional Channels 8.1

131

149

Plasmonics in Periodic Arrays of Graphene Ribbons 7.1 7.2

8.

Grating Coupling . . . . . . . . . . . . . . . . . . . 5.1.1 Graphene on Gallium-Arsenide: The role phonons . . . . . . . . . . . . . . . . . . . . Prism Coupling . . . . . . . . . . . . . . . . . . . . 5.2.1 Otto configuration: Single-layer graphene . Near-field Excitation and Imaging of GSPs . . . . . Others . . . . . . . . . . . . . . . . . . . . . . . . . Excitation of SPP’s by a Moving Line of Charge .

131

Launching Plasmons Using a Metallic Antenna 6.1 6.2 6.3

7.

Guided phonon-polariton modes . . . . . . . . . . 128

Excitation of Graphene Surface Plasmons 5.1

6.

xvii

Edge Plasmons in a Graphene Nanoribbon . . . . . . . . 8.1.1 Prelude: Green’s functions for a 2-layered medium 8.1.2 Plasmonic spectrum of a graphene nanoribbon . . 8.1.3 An extension: magneto-plasmons in a graphene ribbon . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.4 Scattering of THz radiation by a graphene microribbon . . . . . . . . . . . . . . . . . . . . . . . . . Localized Surface Plasmons in a Graphene Ring . . . . .

163 167 167 170 172 172 178 183 185

193 195 196 199 210 211 218

page xvii

April 5, 2016

14:13


xviii


b2445-mas


8.3 8.4

8.5

9.

9in x 6in

8.2.1 Spectrum of graphene plasmons in a graphene ring Plasmonic Excitations in a Graphene Nanodisk . . . . . . Scattering of Graphene Plasmons by a Conductivity Step 8.4.1 The mathematical problem . . . . . . . . . . . . . 8.4.2 Solution of the Riemann-Hilbert problem . . . . . 8.4.3 Further details . . . . . . . . . . . . . . . . . . . . 8.4.4 Calculation of the reflection amplitude . . . . . . 8.4.5 An alternative derivation of Eq. (8.137) . . . . . . Localized Surface Plasmons in a Graphene Sheet with a Gaussian Groove . . . . . . . . . . . . . . . . . . . . . . . 8.5.1 Few useful definitions . . . . . . . . . . . . . . . . 8.5.2 Formulation of the problem . . . . . . . . . . . . . 8.5.3 Green’s theorem, Green’s functions, and the eigenvalue problem . . . . . . . . . . . . . . . . . . . . 8.5.4 Spectrum of surface plasmon-polaritons in the presence of a Gaussian groove . . . . . . . . . . .

Excitation of Surface Plasmon-Polaritons Using Dielectric Gratings 9.1 9.2 9.3 9.4

9.5

Some Basic Definitions and Results . . . . . . . . . . . . . Tangent and Normal Vectors, and Boundary Conditions . A Trivial Example: Recovering Previous Results . . . . . The Fields in D+ and D− . . . . . . . . . . . . . . . . . . 9.4.1 Reflectance and transmittance efficiencies . . . . . 9.4.2 Particular limits for the transmittance and the reflectance . . . . . . . . . . . . . . . . . . . . . . . A Non-Trivial Example: A Grating with a Sine-Profile . .

10. Excitation of Plasmons by an Emitting Dipole 10.1 Statement of the Problem and a Bit of Electrostatics . . . 10.2 Calculation of the Non-Radiative Transition Rate: Particle-Hole Excitations Pathway . . . . . . . . . . . . . 10.3 Calculation of the Total Transition Rate: Full Electromagnetic Calculation . . . . . . . . . . . . . . . . . . . . . . . 10.4 Purcell Effect in Hyperbolic Materials . . . . . . . . . . .

218 232 239 241 242 244 245 247 250 251 253 254 259

267 268 270 271 272 274 274 275 281 282 287 289 292

11. Concluding Remarks

295

Appendix A

301

Derivation of the Susceptibility of Graphene

page xviii

April 5, 2016

14:13


9in x 6in

b2445-mas

Contents

A.1


A.2

A.3

A.4

xix

Undoped Susceptibility . . . . . . . . . . . . . . . . . . . A.1.1 The imaginary part of the undoped susceptibility A.1.2 The real part of the undoped susceptibility . . . Doped Part: Fermi Sea Contributions . . . . . . . . . . A.2.1 Intraband term . . . . . . . . . . . . . . . . . . . A.2.2 Interband term . . . . . . . . . . . . . . . . . . . A.2.3 Imaginary part of χ1D . . . . . . . . . . . . . . . A.2.4 Imaginary part of χ2D . . . . . . . . . . . . . . . A.2.5 Real Part of χ1D and χ2D . . . . . . . . . . . . . Summary for the Real and Imaginary Parts of Graphene Susceptibility . . . . . . . . . . . . . . . . . . A.3.1 Real part of the susceptibility . . . . . . . . . . A.3.2 Imaginary part of the susceptibility . . . . . . . Some Useful Integrals . . . . . . . . . . . . . . . . . . .

. . . . . . . . .

301 302 303 303 303 304 306 306 307

. . . .

309 310 311 312

Appendix B Derivation of the Intra- and Inter-band Conductivity of Graphene

313

Appendix C

319

C.1 C.2

Inhomogeneous Drude Conductivity

Longitudinal Conductivity . . . . . . . . . . . . . . . . . . 320 Transverse Conductivity . . . . . . . . . . . . . . . . . . . 321

Appendix D Derivation of the Expression Relating the Longitudinal Conductivity with the Polarizability

323

Appendix E Derivation of the Polarization of Graphene in the Relaxation-Time Approximation

327

E.1 E.2 E.3 E.4 E.5

Hamiltonian and Particle and Current Densities . . . Local Equilibrium Density Matrix . . . . . . . . . . . A Useful Identity . . . . . . . . . . . . . . . . . . . . Equation of Motion for the Density Matrix . . . . . Polarization in the Relaxation-Time Approximation

Appendix F F.1 F.2

RPA for Double-Layer Graphene

. . . . .

. . . . .

. . . . .

327 328 330 330 332 333

Equation of Motion Method . . . . . . . . . . . . . . . . . 333 Diagrammatic Approach . . . . . . . . . . . . . . . . . . . 337

Appendix G

Effective Dielectric Constants for Coulomb-

page xix

April 5, 2016

14:13


xx

9in x 6in

b2445-mas


Coupled Double-Layer Graphene Appendix H Appendix I


I.1 I.2

Magneto-Optical Conductivity of Graphene Supplementary Material for Chapter 7

339 343 347

Derivation of the Expressions for the Transmittance and Reflectance . . . . . . . . . . . . . . . . . . . . . . . . . . 347 Modes Contributing to Each Resonance . . . . . . . . . . 349

Appendix J

The Method of Toigo

351

Appendix K

Boundary Condition in the Dipole Problem

355

Appendix L

Fourier Transform of the Dipole Potential

359

Appendix M

Non-Radiative Decaying Rate: Electrostatic Calculation361

M.1 Doped Graphene . . . . . . . . . . . . . . . . . . . . . . . 361 M.2 A Detour: Neutral Graphene . . . . . . . . . . . . . . . . 363 Appendix N Reflection Amplitude of an Electromagnetic Wave due to a Graphene Interface N.1 N.2

The Simple Case of Free Standing Graphene . . . . . . . 367 Reflection and Transmission Amplitudes: Fresnel Coefficients369 N.2.1 TE (transverse electric) / s−components . . . . . 372 N.2.2 TM (transverse magnetic) / p−components . . . . 373

Appendix O O.1 O.2

Green’s Functions in the Rope Problem

The Homogeneous Rope . . . . . . . . . . . . . . . . . . . O.1.1 Response to a localized force . . . . . . . . . . . . The Inhomogeneous Rope . . . . . . . . . . . . . . . . . . O.2.1 Reflection and transmition coefficients . . . . . . . O.2.2 Response to an external force in the inhomogeneous regime . . . . . . . . . . . . . . . . . . . . .

Appendix P Derivation of the Transition Rate of a Quantum Emitter Near an Interface P.1

367

375 375 375 377 377 378

383

Polarization Vectors and Green’s Functions . . . . . . . . 384

page xx

April 5, 2016

14:13


9in x 6in

b2445-mas

Contents

Definition of s− and p−polarization vectors and angular spectrum representation . . . . . . . . . . P.1.2 The free space Green’s function . . . . . . . . . . P.1.3 Green’s functions for problems with flat interfaces I Dipole Decaying Rate . . . . . . . . . . . . . . . . . . . . Explicit Form of the Tensor Product of the Polarization Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Free Green’s Function in Real Space . . . . . . . . . . Green’s Function for Problems With Flat Interfaces II . . P.5.1 Free-space Green’s function revisited . . . . . . . . P.5.2 Electric field from a dipole in terms of Green’s functions . . . . . . . . . . . . . . . . . . . . . . . . . P.5.3 Green’s functions: s−component . . . . . . . . . . P.5.4 Green’s functions: p−component . . . . . . . . . . P.5.5 Derivation of the matrix structure of the Green’s functions . . . . . . . . . . . . . . . . . . . . . . .

xxi

P.1.1


P.2 P.3 P.4 P.5

384 386 388 391 392 393 394 394 395 396 398 399

Bibliography

403

Index

427

page xxi




PST˙ws

April 5, 2016

14:13


9in x 6in

b2445-mas


List of Figures

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8

Real space and Brillouin zone of graphene. . . . . . . . . . . . Graphene’s band structure. . . . . . . . . . . . . . . . . . . . . Image of graphene viewed through an optical microscope. . . . Optical conductivity of graphene . . . . . . . . . . . . . . . . . Regions of the (q, ~ω)-space and contributions to the polarizability. Graphene’s 2D polarizability in the RT approximation. . . . . . Electromagnetic scattering at a single graphene 2D layer . . . . Transmittance, reflectance and absorbance of light through suspended single-layer graphene at normal incidence . . . . . . . . 2.9 Electromagnetic scattering at a graphene double-layer . . . . . 2.10 Transfer-matrix for multi-layer graphene structures . . . . . . . 2.11 Transmitance and reflectance of THz radiation across a N -layer graphene structure . . . . . . . . . . . . . . . . . . . . . . . . . 2.12 Intensity plots of the imaginary part of the Fresnel reflection coefficient indicating plasmons in single- and double-layer graphene

18 20 23 27 32 35 37

3.1 3.2

47

3.3 3.4 3.5 3.6 3.7 3.8

Dielectric-metal interface. . . . . . . . . . . . . . . . . . . . . . Dispersion relation of TM SPPs at a dielectric-metal interface (zero damping). . . . . . . . . . . . . . . . . . . . . . . . . . . . Dispersion relation of TM SPPs in air/Ag and SiO2 /Ag interfaces (finite damping). . . . . . . . . . . . . . . . . . . . . . . .