2D Materials: Properties and Devices 9781316681619

Table of contents :
Contents......Page 5
Contributors......Page 12
Introduction......Page 18
Part I......Page 22
1.1 Chemical Bonding and Ground-State Structure......Page 24
1.2 Thermal (In)Stability of 2D Crystals......Page 26
1.3 Electronic Structure of Single-Layer Graphene......Page 28
1.4 Electronic Structure of Bilayer Graphene......Page 34
1.5 Graphene as a Bridge between Condensed Matter and High-Energy Physics......Page 37
1.6 References......Page 38
2.1 Boltzmann Transport Theory......Page 42
2.2 Charged Impurities......Page 45
2.3 Resonant Scatterers......Page 48
2.4 Corrugations of the Graphene Sheet......Page 50
2.5 Phonons......Page 51
2.6 References......Page 53
3 Optical Properties of Graphene......Page 55
3.1 Tunable Interband and Intraband Transitions in Electrically Gated Graphene......Page 56
3.2 Landau Level Transitions in Graphene under a Magnetic Field......Page 60
3.3 Plasmon Excitations in Graphene......Page 61
3.4 Bilayer and Multilayer Graphene......Page 63
3.5 References......Page 65
4.1 Introduction......Page 69
4.2 Experiments......Page 70
4.3 Non-linear and Anisotropic Response of Graphene......Page 74
4.4 Experimental Validation......Page 78
4.5 Instabilities......Page 80
4.6 Defective Graphene......Page 81
4.8 References......Page 85
5.1 Structure and Vibrations of Monolayer Graphene......Page 88
5.2 Many-Layers Graphene and the Interlayer Vibrations in 2D Systems......Page 91
5.3 The Quantum Nature of Atomic Vibrations......Page 94
5.5 Probing Phonons Near Defects and Edges/Grain Boundaries......Page 96
5.6 References......Page 100
6.1 Thermal Conductivity of Graphene and Few-Layer Graphene......Page 107
6.2 Isotope and Rotational Engineering of Thermal Properties of Graphene......Page 110
6.3 Graphene Applications in Thermal Management Technologies......Page 113
6.4 Conclusions......Page 117
6.5 References......Page 118
7.1 Macroscopic Approach to Graphene Plasmonics......Page 121
7.2 Microscopic Approach......Page 128
7.3 Plasmon Damping......Page 132
7.4 Experimental Observation of Graphene Plasmons......Page 134
7.5 Applications......Page 151
7.6 References......Page 153
8.1 Introduction......Page 158
8.2 Basic Electrical Properties of p–n Junctions......Page 159
8.3 Photon Analogies for Carriers in Graphene......Page 165
8.4 Future Directions......Page 173
8.5 References......Page 174
9.1 Introduction......Page 176
9.2 Graphene RF Transistors and Circuits......Page 177
9.3 Graphene Nanostructures......Page 183
9.4 Bilayer Graphene Transistors......Page 186
9.5 Vertical Graphene Transistors......Page 188
9.6 Conclusion......Page 191
9.7 References......Page 192
10.1 Introduction......Page 197
10.2 Light to Current Conversion......Page 198
10.3 Photodetectors......Page 201
10.4 Light Modulators......Page 204
10.5 Ultra-Fast Lasers......Page 206
10.6 Thermal Radiation Sources......Page 208
10.7 Passive Optical Elements......Page 209
10.8 Transparent Conductive Electrodes......Page 210
10.9 References......Page 211
11.1 Introduction to Spintronics......Page 214
11.2 Advantages of Graphene for Spintronics......Page 215
11.3 How to Measure Spin Lifetimes in Graphene and 2D Materials......Page 217
11.4 New Spin Relaxation Mechanisms......Page 223
11.5 Proximity Effects and Spin Gating......Page 229
11.6 References......Page 232
12.1 Introduction......Page 236
12.2 Mechanical Assembly of Graphene–BN Heterostructures......Page 237
12.3 High-Performance Graphene......Page 242
12.4 Beyond Graphene......Page 249
12.5 References......Page 250
13.1 Introduction......Page 255
13.2 CVD Method for Graphene Growth......Page 256
13.3 Prospects......Page 267
13.4 References......Page 268
Part II......Page 274
14.1 Introduction......Page 276
14.2 Electronic Structure......Page 277
14.3 From Density Functional Theory to Tight-Binding Approximation......Page 281
14.4 Including Strain in the Tight-Binding Hamiltonian......Page 285
14.5 Low-Energy Model of Strained Transition Metal Dichalcogenides......Page 287
14.6 Strain Engineering in Transition Metal Dichalcogenides......Page 289
14.7 References......Page 293
15.1 Introduction......Page 296
15.2 Electronic Structure at the Band Edges......Page 297
15.3 Valley-Spin Physics in Monolayers......Page 300
15.4 Valley and Spin Physics in Bilayers......Page 306
15.5 References......Page 309
16.1 Introduction......Page 312
16.2 Ballistic Transport Simulations......Page 314
16.3 Scattering Mechanisms......Page 316
16.4 Point Defects......Page 320
16.5 References......Page 325
17.1 Fundamentals of 2D TMD Heterostructures......Page 327
17.2 Interlayer Exciton Properties......Page 332
17.3 Valley Optoelectronic Properties of 2D Heterostructure......Page 336
17.4 Outlook......Page 342
17.5 References......Page 343
18.1 Introduction......Page 346
18.2 Light-Emitting Diodes and Lasers......Page 347
18.3 Photovoltaic Devices......Page 350
18.4 Photodetectors......Page 353
18.5 Valley-Dependent Optoelectronic Devices......Page 357
18.6 References......Page 359
19.1 Introduction......Page 361
19.3 Sulfurization/Selenization of Transition Metal Oxides......Page 362
19.5 Physical Vapor Phase Transport......Page 368
19.7 References......Page 371
20.1 Introduction......Page 376
20.2 Point Defects......Page 377
20.3 Topological Defects: Dislocations and Grain Boundaries......Page 380
20.4 Dislocations in Bilayer Materials......Page 387
20.5 Other 1D Defects – Edges, Interfaces, and Nanowires......Page 389
20.6 Summary......Page 392
20.7 References......Page 393
Part III......Page 396
21.1 Crystal and Electronic Band Structures......Page 398
21.2 Electronic Properties......Page 406
21.3 Optical Properties......Page 409
21.4 Thermal Properties......Page 416
21.5 Mechanical Properties – Elasticity......Page 422
21.7 References......Page 425
22 Anisotropic Properties of Black Phosphorus......Page 430
22.1 Synthesis of Black Phosphorus......Page 431
22.2 Anisotropic Response of Black Phosphorus......Page 433
22.4 References......Page 446
23.2 Optical Properties......Page 452
23.3 Optoelectronic Devices......Page 459
23.4 Outlook and Remarks......Page 467
23.5 References......Page 469
24.2 The Advent of Silicene......Page 475
24.3 Epitaxial Silicene......Page 476
24.4 Electronic Structure of Silicene......Page 479
24.5 Functionalization of Silicene......Page 480
24.6 Multilayer Silicene......Page 482
24.7 Germanene and Stanene......Page 484
24.9 References......Page 486
25.1 Motivation and Methodology......Page 489
25.2 Group IV Elements: Silicene, Germanene......Page 491
25.3 Group III–V and II–VI Compounds......Page 495
25.4 Group V Elements: Nitrogene and Antimonene......Page 497
25.5 Transition Metal Oxides and Dichalcogenides......Page 498
25.7 References......Page 499
Index......Page 502

Citation preview

2D Materials Properties and Devices Learn about the most recent advances in 2D materials with this comprehensive and accessible text. Providing all the necessary materials science and physics background, leading experts discuss the fundamental properties of a wide range of 2D materials, and their potential applications in electronic, optoelectronic and photonic devices. Several important classes of materials are covered, from more established ones such as graphene, hexagonal boron nitride, and transition metal dichalcogenides, to new and emerging materials such as black phosphorus, silicene, and germanene. You will gain an in-depth understanding of the electronic structure and optical, thermal, mechanical, vibrational, spin, and plasmonic properties of each material, as well as the different techniques that can be used for their synthesis. Presenting a unified perspective on 2D materials, this is an excellent resource for graduate students, researchers, and practitioners working in nanotechnology, nanoelectronics, nanophotonics, condensed matter physics, and chemistry. Phaedon Avouris is an IBM Fellow Emeritus. He is a member of the National Academy of Sciences, and a Fellow of the American Academy of Arts and Sciences, the American Physical Society, the Institute of Physics, the IEEE, the Materials Research Society, and the American Association for the Advancement of Science. Tony F. Heinz is a Professor of Applied Physics and Photon Science at Stanford University and the SLAC National Accelerator Laboratory. He previously worked at Columbia University and IBM Research, USA. Tony Low is Assistant Professor of Electrical and Computer Engineering at the University of Minnesota. He previously worked at Yale University, Columbia University, and the IBM T. J. Watson Research Center.

“This book, edited by the top researchers who have been working on atomically thin materials in the past decade, contains the essential contents of our current scientific understanding of this novel form of materials. The authors have compiled comprehensive and contemporary reviews on various topics ranging from fundamental science to engineering applications, providing an excellent textbook for students as well as references for experts in the research field.” Philip Kim, Harvard University “This edited volume consists of 25 topical chapters contributed by scientists active in the growing field of 2D semiconductors, who summarize the most salient features of these intriguing materials. Contributions are grouped into three parts dedicated to graphene, transition metal dichalcogenides, and elemental group V layered semiconductors including phosphorene. Covered are the most actively researched topics : synthesis, stability, thermal and electronic properties including transport, optics, optoelectronics and spintronics, phonon structure, and mechanical properties of few-layer systems including heterostructures, as probed by state-of-the-art experimental and theoretical techniques. While emphasis is placed on the rigorous scientific representation of knowledge acquired to date, the contributors also offer a refreshing insight into potential applications of this new class of materials.” David Tomanek, Michigan State University “The field of 2D materials, which started with graphene, now includes dozens of one-atom thick crystals. Many of them demonstrate properties end effects which are equally as exciting as those found for the famous ancestor. And, judging from the recent progress, the field will be developing very fast for many years ahead. This book, written by scientists who are the leaders in their fields, is the most comprehensive and up-to-date attempt to review this fast-developing subject. Starting with an in-depth summary on graphene, it moves to other 2D crystals, such as transition metal dichalcogenides, black phosphorous, and others, providing probably the most complete reference on the topic at the moment.” Kostya Novoselov, University of Manchester

2D Materials Properties and Devices P H A E D O N AV O U R I S IBM T. J. Watson Research Center, New York

TONY F. HEINZ Stanford University and SLAC National Accelerator Laboratory

TONY LOW University of Minnesota

University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 4843/24, 2nd Floor, Ansari Road, Daryaganj, Delhi – 110002, India 79 Anson Road, #06–04/06, Singapore 079906 Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781107163713 10.1017/9781316681619 © Materials Research Society 2017 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2017 Printed in the United Kingdom by TJ International Ltd. Padstow Cornwall. A catalogue record for this publication is available from the British Library. Library of Congress Cataloging-in-Publication Data Names: Avouris, Phaedon, 1945- editor. | Heinz, Tony F., editor. | Low, Tony, editor. Title: 2D materials : properties and devices / edited by Phaedon Avouris (IBM T.J. Watson Research Center, New York), Tony F. Heinz (Stanford University and SLAC National Accelerator Laboratory), Tony Low (University of Minnesota). Other titles: Two dimensional materials Description: Cambridge, United Kingdom ; New York, NY : Cambridge University Press, 2017. | Includes bibliographical references and index. Identifiers: LCCN 2016043383| ISBN 9781107163713 (Hardback ; alk. paper) | ISBN 1107163714 (hardback ; alk. paper) Subjects: LCSH: Graphene. | Nanostructured materials. Classification: LCC TA455.G65 A15 2017 | DDC 620.1/15–dc23 LC record available at https://lccn.loc.gov/2016043383 ISBN 978-1-107-16371-3 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party Internet Web sites referred to in this publication and does not guarantee that any content on such Web sites is, or will remain, accurate or appropriate.

Contents

Contributors Introduction

Part I 1

page xi 1

5 Graphene: Basic Properties 1.1 Chemical Bonding and Ground-State Structure 1.2 Thermal (In)Stability of 2D Crystals 1.3 Electronic Structure of Single-Layer Graphene 1.4 Electronic Structure of Bilayer Graphene 1.5 Graphene as a Bridge between Condensed Matter and High-Energy Physics 1.6 References

7 7 9 11 17

2

Electrical Transport in Graphene: Carrier Scattering by Impurities and Phonons 2.1 Boltzmann Transport Theory 2.2 Charged Impurities 2.3 Resonant Scatterers 2.4 Corrugations of the Graphene Sheet 2.5 Phonons 2.6 References

25 25 28 31 33 34 36

3

Optical Properties of Graphene 3.1 Tunable Interband and Intraband Transitions in Electrically Gated Graphene 3.2 Landau Level Transitions in Graphene under a Magnetic Field 3.3 Plasmon Excitations in Graphene 3.4 Bilayer and Multilayer Graphene 3.5 References

38 39 43 44 46 48

Graphene Mechanical Properties 4.1 Introduction 4.2 Experiments

52 52 53

4

20 21

v

vi

Contents

4.3 4.4 4.5 4.6 4.7 4.8

Non-linear and Anisotropic Response of Graphene Experimental Validation Instabilities Defective Graphene Conclusion References

57 61 63 64 68 68

5

Vibrations in Graphene 5.1 Structure and Vibrations of Monolayer Graphene 5.2 Many-Layers Graphene and the Interlayer Vibrations in 2D Systems 5.3 The Quantum Nature of Atomic Vibrations 5.4 Phonon Coherence Length in Graphene 5.5 Probing Phonons Near Defects and Edges/Grain Boundaries 5.6 References

71 71 74 77 79 79 83

6

Thermal Properties of Graphene: From Physics to Applications 6.1 Thermal Conductivity of Graphene and Few-Layer Graphene 6.2 Isotope and Rotational Engineering of Thermal Properties of Graphene 6.3 Graphene Applications in Thermal Management Technologies 6.4 Conclusions 6.5 References

90 90 93 96 100 101

7

Graphene Plasmonics 7.1 Macroscopic Approach to Graphene Plasmonics 7.2 Microscopic Approach 7.3 Plasmon Damping 7.4 Experimental Observation of Graphene Plasmons 7.5 Applications 7.6 References

104 104 111 115 117 134 136

8

Electron Optics with Graphene p–n Junctions 8.1 Introduction 8.2 Basic Electrical Properties of p–n Junctions 8.3 Photon Analogies for Carriers in Graphene 8.4 Future Directions 8.5 References

141 141 142 148 156 157

9

Graphene Electronics 9.1 Introduction 9.2 Graphene RF Transistors and Circuits 9.3 Graphene Nanostructures 9.4 Bilayer Graphene Transistors

159 159 160 166 169

Contents

9.5 9.6 9.7

vii

Vertical Graphene Transistors Conclusion References

171 174 175

10

Graphene: Optoelectronic Devices 10.1 Introduction 10.2 Light to Current Conversion 10.3 Photodetectors 10.4 Light Modulators 10.5 Ultra-Fast Lasers 10.6 Thermal Radiation Sources 10.7 Passive Optical Elements 10.8 Transparent Conductive Electrodes 10.9 References

180 180 181 184 187 189 191 192 193 194

11

Graphene Spintronics 11.1 Introduction to Spintronics 11.2 Advantages of Graphene for Spintronics 11.3 How to Measure Spin Lifetimes in Graphene and 2D Materials 11.4 New Spin Relaxation Mechanisms 11.5 Proximity Effects and Spin Gating 11.6 References

197 197 198 200 206 212 215

12

Graphene–BN Heterostructures 12.1 Introduction 12.2 Mechanical Assembly of Graphene–BN Heterostructures 12.3 High-Performance Graphene 12.4 Beyond Graphene 12.5 References

219 219 220 225 232 233

13

Controlled Growth of Graphene Crystals by Chemical Vapor Deposition: From Solid Metals to Liquid Metals 13.1 Introduction 13.2 CVD Method for Graphene Growth 13.3 Prospects 13.4 References

238 238 239 250 251

Part II 14

257 Electronic Properties and Strain Engineering in Semiconducting Transition Metal Dichalcogenides 14.1 Introduction 14.2 Electronic Structure

259 259 260

viii

Contents

14.3 14.4 14.5 14.6 14.7

From Density Functional Theory to Tight-Binding Approximation Including Strain in the Tight-Binding Hamiltonian Low-Energy Model of Strained Transition Metal Dichalcogenides Strain Engineering in Transition Metal Dichalcogenides References

264 268 270 272 276

15

Valley-Spin Physics in 2D Semiconducting Transition Metal Dichalcogenides 15.1 Introduction 15.2 Electronic Structure at the Band Edges 15.3 Valley-Spin Physics in Monolayers 15.4 Valley and Spin Physics in Bilayers 15.5 References

279 279 280 283 289 292

16

Electrical Transport in MoS2: A Prototypical Semiconducting TMDC 16.1 Introduction 16.2 Ballistic Transport Simulations 16.3 Scattering Mechanisms 16.4 Point Defects 16.5 References

295 295 297 299 303 308

17

Optical Properties of TMD Heterostructures 17.1 Fundamentals of 2D TMD Heterostructures 17.2 Interlayer Exciton Properties 17.3 Valley Optoelectronic Properties of 2D Heterostructure 17.4 Outlook 17.5 References

310 310 315 319 325 326

18

TMDs – Optoelectronic Devices 18.1 Introduction 18.2 Light-Emitting Diodes and Lasers 18.3 Photovoltaic Devices 18.4 Photodetectors 18.5 Valley-Dependent Optoelectronic Devices 18.6 References

329 329 330 333 336 340 342

19

Synthesis of Transition Metal Dichalcogenides 19.1 Introduction 19.2 Mechanism of Growth 19.3 Sulfurization/Selenization of Transition Metal Oxides 19.4 Metal Organic Chemical Vapor Deposition 19.5 Physical Vapor Phase Transport 19.6 Summary and Outlook 19.7 References

344 344 345 345 351 351 354 354

Contents

20

Defects in Two-Dimensional Materials 20.1 Introduction 20.2 Point Defects 20.3 Topological Defects: Dislocations and Grain Boundaries 20.4 Dislocations in Bilayer Materials 20.5 Other 1D Defects – Edges, Interfaces, and Nanowires 20.6 Summary 20.7 References

Part III

ix

359 359 360 363 370 372 375 376

379

21

Theoretical Overview of Black Phosphorus 21.1 Crystal and Electronic Band Structures 21.2 Electronic Properties 21.3 Optical Properties 21.4 Thermal Properties 21.5 Mechanical Properties – Elasticity 21.6 Concluding Remarks 21.7 References

381 381 389 392 399 405 408 408

22

Anisotropic Properties of Black Phosphorus 22.1 Synthesis of Black Phosphorus 22.2 Anisotropic Response of Black Phosphorus 22.3 Conclusion 22.4 References

413 414 416 429 429

23

Optical Properties and Optoelectronic Applications of Black Phosphorus 23.1 Introduction 23.2 Optical Properties 23.3 Optoelectronic Devices 23.4 Outlook and Remarks 23.5 References

435 435 435 442 450 452

24

Silicene, Germanene, and Stanene 24.1 Introduction 24.2 The Advent of Silicene 24.3 Epitaxial Silicene 24.4 Electronic Structure of Silicene 24.5 Functionalization of Silicene 24.6 Multilayer Silicene 24.7 Germanene and Stanene 24.8 Summary 24.9 References

458 458 458 459 462 463 465 467 469 469

x

Contents

25

Predictions of Single-Layer Honeycomb Structures from First Principles 25.1 Motivation and Methodology 25.2 Group IV Elements: Silicene, Germanene 25.3 Group III–V and II–VI Compounds 25.4 Group V Elements: Nitrogene and Antimonene 25.5 Transition Metal Oxides and Dichalcogenides 25.6 Conclusions 25.7 References

472 472 474 478 480 481 482 482

Index

485

Contributors

Thierry Angot Aix-Marseille Université Phaedon Avouris IBM T. J. Watson Research Center Alexander A. Balandin University of California, Riverside S. Cahangirov Bilkent University Luiz Gustavo Cançado Federal University of Minas Gerais Andres Castellanos-Gomez Instituto Madrileño de Estudios Avanzados en Nanociencia Andrey Chaves Universidade Federal do Ceara Jian-Hao Chen Peking University S. Ciraci Bilkent University Aron W. Cummings ICN2 – Catalan Institute of Nanoscience and Nanotechnology (CSIC and the Barcelona Institute of Science and Technology) Cory.R. Dean Columbia University xi

xii

Contributors

C. DiMarco Columbia University Yuchen Du Purdue University Xiangfeng Duan University of California, Los Angeles Xidong Duan Hunan University Traian Dumitrică University of Minnesota Annalisa Fasolino Radboud University Dechao Geng National University of Singapore Francisco Guinea Instituto Madrileño de Estudios Avanzados en Nanociencia (IMDEA-nanociencia) Tony F. Heinz Stanford University and SLAC National Accelerator Laboratory J. Hone Columbia University Wei Ji Renmin University of China Ado Jorio Federal University of Minas Gerais Mikhail I. Katsnelson Radboud University Andras Kis École Polytechnique Fédérale de Lausanne (EPFL) Frank H.L. Koppens ICFO Institut de Ciències Fotòniques

Contributors

J.W. Kysar Columbia University Guy Le Lay Aix-Marseille Université Lain-Jong Li King Abdullah University of Science and Technology Mo Li University of Minnesota R. Li Columbia University Han Liu Purdue University Kian Ping Loh National University of Singapore Tony Low University of Minnesota Mark B. Lundeberg ICFO Institut de Ciències Fotòniques Zhe Luo Purdue University Jesse Maassen Dalhousie University Leandro M. Malard Federal University of Minas Gerais Thomas Mueller Vienna University of Technology Frank Ortmann Technische Universität Dresden Marco Polini NEST, Istituto Nanoscienze – CNR and Scuola Normale Superiore

xiii

xiv

Contributors

S. Rastogi Columbia University Pasqual Rivera University of Washington Stephan Roche Catalan Institute of Nanoscience and Nanotechnology Rafael Roldán Instituto de Ciencia de Materiales de Madrid, CSIC Eric Salomon Aix-Marseille Université Sufei Shi Rensselaer Polytechnic Institute Yumeng Shi Shenzhen University Sergio O. Valenzuela Catalan Institute of Nanoscience and Nanotechnology Chen Wang University of California, Los Angeles Feng Wang University of California, Berkeley Lei Wang Cornell University James R. Williams University of Maryland Xianfan Xu Purdue University Xiaodong Xu University of Washington Wang Yao The University of Hong Kong

Contributors

Boris I. Yakobson Rice University Peide D. Ye Purdue University Hongyi Yu The University of Hong Kong Xiaolong Zou Rice University and Tsinghua–Berkeley Shenzhen Institute

xv

Introduction Phaedon Avouris, Tony F. Heinz, and Tony Low

The study and use of layered materials whose structure involves the stacking of individual platelets has a very long history. Among these materials, graphite, an allotrope of carbon, is perhaps the best-known example. It was already used by Neolithic Danubians around 4,000 BC as paint for pottery [1]. Its chemical composition was determined by Carl Wilhelm Scheelein 1779 [2], with details of its atomic structure in 1924 [3], and electronic structure calculations in the 1950s [4, 5]. Other important families of layered materials include the transition metal dichalcogenides (e.g., MoS2, MoSe2), certain metal halides (e.g., PbI2 and MgBr2), and oxides (e.g., MnO2, MoO3), perovskites (general form ABO3), layered III-VIs (e.g., GaS, InSe), and V-VIs (e.g., InSe, GaS) materials and layered silicates (clays, micas). The insulating hexagonal boron nitride (h-BN) system is another important layered material, one isostructural with graphite, but exhibiting very different properties. Currently, around 500 different layered materials have been identified [6, 7]. Until relatively recently, research and applications of layered materials involved their bulk solids. It was the mechanical exfoliation of a single graphene layer from graphite in 2004 by Geim, Novoselov, and co-workers [8] that focused the attention of the scientific community on the study of single or few layers of these materials. A considerable variety of other layered 2D materials has now also been mechanically exfoliated using adhesive tape [9]. Chemical exfoliation in liquid dispersions is another widely used technique. Ancient Mayas applied such an approach with clays for use as pigments, while in the 1960s Boehm [10] isolated thin graphite films in this fashion, and Frindt [11] exfoliated metal dichalcogenides thin films. Typically, chemical exfoliation involves dispersion of the material in high-surface tension solvents, oxidation, or intercalation by a variety of agents that lead to exfoliation [12]. Chemical techniques can produce large quantities of 2D layers in a solvent, appropriate for depositing films that can be used in industry. Increasingly, for electronics and more high-end applications, 2D layers are directly synthesized using catalytic chemical vapor deposition (CVD) or van der Waals epitaxy techniques [13]. Heterostructures involving atomic layers of different materials can also be produced, by either sequential transfers or direct growth. In this book, the structure and properties of several classes of single or few layers of 2D materials are discussed. In the monolayer or few-layer regime, many layered materials can acquire new and distinctive properties compared with their bulk parent. For example, when graphite reduces to its monolayer, electrons acquire a linear 1

2

Introduction

massless Dirac energy spectrum, leading to interesting physical phenomena such as the anomalous half-integer quantum Hall and Klein tunneling. Indeed, the anomalous quantum Hall signatures provide a smoking gun experiment for the confirmation of single-layer graphene [14, 15]. Chapter 1 provides an introduction to the electronic structure and Dirac physics in graphene, including its AB stacked bilayer. Massless Dirac fermions have extremely high carrier mobilities. Chapter 2 surveys the various scattering mechanisms that can limit carrier mobilities in graphene, such as intrinsic and substrate phonons and impurities. Despite being atomically thin, graphene can absorb about 2% of incident light in the visible range and its absorption exhibits distinctive characteristics from the UV to THz frequencies. Chapter 3 discuss the electrically tunable optical properties of graphene. In its perfect crystalline form, graphene is also the strongest material. The mechanical properties of graphene and the experimental techniques used to establish these properties are described in Chapter 4. A formal introduction to the vibrational modes and their characteristics in graphene are provided in Chapter 5. This is followed by a discussion of the exceptional thermal conductivity of graphene in Chapter 6. While new physical phenomena are still being discovered and interesting technological advances are continually being made, research in this field has arguably reached a juncture where opportunities for real applications are being considered and weighed against today’s state-of-the-art solutions. Chapter 7 discusses graphene device physics and the development of graphene electronics especially for radio frequency applications. Chapter 8 provides a different perspective on graphene electronics, exploiting electrostatic p–n junctions to realize electronic analogues of optical effects such as waveguiding and negative index of refraction. Due to the negligible spin–orbit coupling in graphene, it is also touted as an excellent channel for spintronics as discussed in Chapter 9. Chapter 10 discusses the use of h-BN as a substrate for graphene and its encapsulation, yielding electronic properties in graphene close to the intrinsic limits. Chapter 11 discusses the application of graphene for optoelectronics, including as photodetectors and light modulators. By exploiting plasmons in graphene, light–matter interactions can be strongly enhanced at the mid- to far-infrared frequencies by effective light confinement. Chapter 12 discusses the basics of graphene plasmons and some emerging applications. We conclude the discussion of graphene with Chapter 13 on its growth using chemical vapor deposition. Another important class of 2D materials presented in this book is that of the transition metal dichalcogenides (TMDCs). The stoichiometry of these materials is MX2, where M = Ti, Zr, Hf, V, Nb, Ta, Mo, W, Re, and X = S, Se, Te. They form layered 2D crystal structures, each layer having a thickness of 0.6–0.7 nm, with the metal atoms forming a hexagonally packed layer sandwiched between two adjoining layers of chalcogen atoms. The stacked layers are held together by van der Waals forces. As bulk solids, these TMDC materials have many interesting properties. They can be metals (e.g., NbS2, VSe2), semiconductors (e.g., MoS2, WS2), and insulators (e.g., HfS2), as well as displaying superconductivity and charge-density waves at low temperatures (e.g., NbSe2, TaS2). The second part of this book is devoted to the electronic and optical properties of monolayer TMDCs and their potential applications. Chapter 14 discusses the evolution

Introduction

3

of the electronic structure of TMDCs, such as MoS2 or MoSe2, from indirect gap semiconductor in bulk to direct gap at monolayer thickness. The effective Hamiltonian for electrons, the band gap, and the materials’ dependence on strain are examined. Unlike graphene, electrons in TMDCs are described by a massive Dirac Hamiltonian and exhibit interesting coupled spin–valley physics. Chapter 15 introduces these basic theoretical concepts and the associated physical effects, such as spin–valley Hall currents, valley selective optical properties, and exciton physics. Experimental results on the electronic transport properties of MoS2 are presented in Chapter 16. This is followed by a discussion of the materials’ optical properties, focusing primarily on circular dichroism and strongly bound excitons, in its vertical heterostructures in Chapter 17. In Chapter 18, we provide a survey of emerging applications in optoelectronics, including light-emitting devices and photodetectors. Chapter 19 discusses the experimental realization of vertical heterostructures and their corresponding optical and excitonic properties. Chapter 19 provides an introduction into the synthesis of large-area TMDC crystals and their heterostructures by means of CVD. We conclude the second part of the book with a discussion about common sources of defects in 2D materials, particularly in TMDCs, in Chapter 20. In the final part of this book, emerging new 2D materials are discussed. A layered material that has recently generated great excitement is black phosphorus (BP). It is an allotropic form of phosphorus produced from the common red phosphorus at very high pressures [16]. Like graphene, 2D sheets can be formed by exfoliation. However, its monolayer structure is not planar, but buckled and featuring highly anisotropic properties. Sheets of BP are semiconducting, with a direct band gap that varies significantly with layer thickness. In Chapter 21, we provide a broad theoretical perspective on the anisotropic electronic, optical, excitonic, plasmonic, thermal, thermoelectric, transport, and mechanical properties in BP. This is followed in Chapter 22 with a review of recent experimental studies and of our current understanding of the electronic, mechanical, and thermal properties of BP. Experimental investigations of the optical properties of BP, particularly of its excitonic signatures and applications in optoelectronics are discussed in Chapter 23. In a different research direction, honeycomb lattice analogues of graphene involving group IV elements, such as Si (silicene), Ge (germanene), and Sn (stanene), have recently been synthesized. Unlike graphene, these layers have strong spin–orbit coupling and are predicted to be topological insulators, displaying an interesting quantum spin Hall effect. The structure and properties of these new materials are discussed in Chapter 24. The book then ends (Chapter 25) with a discussion about how theoretical predictions based on first-principle calculations can guide the search for 2D layered materials with desired properties.

References [1] Ehrich RW. Chronologies in Old World Archaeology. University of Chicago Press, 1992. [2] Partington JR. A History of Chemistry. New York, 1962.

4

Introduction

[3] Bernal JD. The structure of graphite. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character. 1924 December 1, 106(740): 749–73. [4] Slonczewski JC, Weiss PR. Band structure of graphite. Physical Review. 1958 January 15, 109(2): 272. [5] McClure JW. Band structure of graphite and de Haas–van Alphen effect. Physical Review. 1957 November 1, 108(3): 612. [6] Miró P, Audiffred M, Heine T. An atlas of two-dimensional materials. Chemical Society Reviews. 2014, 43(18): 6537–54. [7] Grasso V (ed.) Electronic Structure and Electronic Transitions in Layered Materials. Springer Science and Business Media, 2012 December 6. [8] Novoselov KS, Geim AK, Morozov SV, Jiang D, Zhang Y, Dubonos SA, Grigorieva IV, Firsov AA. Electric field effect in atomically thin carbon films. Science. 2004 October 22, 306(5696): 666–9. [9] Novoselov KS, Jiang D, Schedin F, Booth TJ, Khotkevich VV, Morozov SV, Geim AK. Two-dimensional atomic crystals. Proceedings of the National Academy of Sciences of the United States of America. 2005 July 26, 102(30): 10451–3. [10] Boehm HP, Clauss A, Fischer GO, Hofmann U. The adsorption behavior of very thin carbon films. Zeitschrift für Anorganische und Allgemeine Chemie. 1962, 316: 119–27. [11] Frindt RF. Single crystals of MoS2 several molecular layers thick. Journal of Applied Physics. 1966 March 15, 37(4): 1928–9. [12] Nicolosi V, Chhowalla M, Kanatzidis MG, Strano MS, Coleman JN. Liquid exfoliation of layered materials. Science. 2013 June 21, 340(6139): 1226419. [13] Novoselov KS, Fal VI, Colombo L, Gellert PR, Schwab MG, Kim K. A roadmap for graphene. Nature. 2012 October 11, 490(7419): 192–200. [14] Zhang Y, Tan YW, Stormer HL, Kim P. Experimental observation of the quantum Hall effect and Berry’s phase in graphene. Nature. 2005 November 10, 438(7065): 201–4. [15] Novoselov KS, Geim AK, Morozov S, Jiang D, Katsnelson M, Grigorieva I, Dubonos S. Two-dimensional gas of massless Dirac fermions in graphene. Nature. 2005 November 10, 438(7065): 197–200. [16] Bridgman PW. Two new modifications of phosphorus. Journal of the American Chemical Society. 1914 July, 36(7): 1344–63.

Part I

1

Graphene: Basic Properties Mikhail I. Katsnelson and Annalisa Fasolino

1.1

Chemical Bonding and Ground-State Structure Carbon is the sixth element in the periodic table. The carbon atom has six electrons, two of them in the 1s2 shell and four filling 2s and 2p states. The ground-state atomic configuration is 2s2 2p2 , with total spin S = 1, total orbital moment L = 1, and total angular moment J = 0 (the ground-state multiplet 3 P0 ). The first excited state with J = 1, the 3 P1 multiplet, has energy 16:4 cm
1  2 meV [1] which gives an estimate of the spin–orbit coupling strength. The lowest energy state with 2s1 2p3 configuration has excitation energy 33735:2 cm
1  4:2 eV [1], representing the promotion energy of a 2s electron to a 2p state. Therefore, we might expect carbon to be always divalent, due to the two 2p-electrons and the closed 2s shell. This conclusion is, however, wrong. Normally, carbon is tetravalent, due to the formation of hybridized sp states, according to the concept of “resonance” by Linus Pauling [2, 3]. When atoms form molecules or solids, the total energy decreases due to overlap of the electron wave functions at different sites and the formation of molecular orbitals (in molecules), or energy bands (in solids). This energy gain can be sufficient for the energy necessary to promote electrons from 2s to 2p states in the carbon atom. In non-ionic solids, there are two kinds of strong chemical bonding: covalent and metallic. In covalent bonds, the electrons are shared between neighboring atoms with a strong dependence of the cohesive energy on the angles between bonds, whereas in metallic bonding, the electrons are redistributed over the whole crystal in a more isotropic way. In diamond, four sp3 covalent bonds with tetrahedral orientation are formed, with bond length a  1:54 Å. Alternatively, three electrons can form sp2 covalent bonds with planar minimum energy configuration and 120o angles, whereas one p-electron per atom forms a metallic bond. The prototype is the conjugated bond in the benzene molecule where the interatomic distances are a  1:42 Å shorter than in diamond. Graphite, the most common form of elemental carbon at ambient conditions, consists of flat layers with this type of covalent and metallic character. When isolated, a single layer is called graphene. In graphite, the graphene layers are weakly coupled by van der Waals interactions with an equilibrium interlayer distance 3.35 Å [4]. The minimum energy corresponds to Bernal stacking AB where even layers (B) are rotated on odd layers by 60o, as shown in Fig. 1.1. A slightly less favorable configuration is the rhombohedral graphite with three periodically repeated layers ABC, each rotated by +60o with respect to the adjacent ones. Structures with a random distributions of +60o and –60o rotations are called 7

8

Graphene: Basic Properties

Fig. 1.1 Bernal stacking: carbon atoms belonging to sublattices A and B are shown as

dark and light gray

turbostratic graphite. The AA stacking is energetically costly, and is not found in nature. The energetics of different types of stacking are discussed, e.g., in [5, 6]. Similar sp2-type bonding is also found in nanotubes and fullerenes [7]. Nanotubes can be thought of as graphene rolled at given angles and leading to a structure similar to the planar hexagonal structure of graphene but with slightly bent hexagons. Fullerenes are large molecules (C60, C72, ...) which can be considered as graphene pieces curved into objects with the topology of a sphere. For topological reasons (different Euler characteristics for sphere and cylinder), they cannot be built from hexagons only and require the introduction of 12 pentagons per molecule [7]. All these allotropes of elemental carbon, with the exception of diamond, are called graphitic forms. As seen in the phase diagram shown in Fig. 1.2, at ambient conditions graphite is the stable structure, which is why it is much more common on the Earth’s surface whereas at large pressures, like in the Earth’s mantle, diamond is more favorable. The difference between graphite and diamond is very small but with a large barrier to be overcome for the transformation so that diamond exists as a metastable structure up to high temperature. In Fig. 1.3, we show how the energy change when pressure is applied perpendicularly to graphite layers leads to the transformation to diamond [8, 9]. The stability of the sp2 bonding with respect to sp3 for carbon is the reason why layers can be easily separated. In 2004, it was shown that single layers could be exfoliated and deposited on other substrates [10, 11] leading to the beginning of the “‘graphene boom” and opening the door to new two-dimensional (2D) materials. Other elements of the same fourth column of the periodic table strongly prefer sp3 hybridization. For Si, Ge, and Sn, the diamond structure corresponds to the minimal energy. For Sn, this is the ground state at low temperature (grey tin) competing with the metallic phase (white tin) at higher temperatures (T > 286 K), whereas Pb is a metal with an fcc structure. Nevertheless, we can obtain 2D analogues of graphene for Si [12] and Ge [13], e.g. by epitaxial growth on the (111) surfaces of fcc metals. Exfoliation is of course impossible because, as said, there is not a bulk graphitic form for these elements. While for graphene in its ground state, the chemical bond is of pure sp2 character and the structure is exactly planar, for silicene and germanene there is an essential sp2–sp3 mixture of the bonding resulting in a buckled configuration. As shown in Fig. 1.4, due to the buckling in germanene the electronic density is strongly modulated in the

9

1.2 Thermal (In)Stability of 2D Crystal

7000

6000

L Temperature [K]

5000

4000

D

G

3000

Calc. Graphite melting line Calc. Graphite–Diamond coex. Calc. Diamond melting line

2000

Exp. Graphite melting line [15] Exp. Graphite melting line [16] Exp. Graphite–Diamond coex. [17]

1000

0 K Graphite–Diamond coex. [15] 0 0

10

20

30

40

50

60

Pressure [GPa] Fig. 1.2 Temperature–pressure phase diagram of carbon including the graphite, diamond,

and liquid phase (adapted from [14]).

E (eV)

–7

–7.2

–7.4 1.5

2

2.5

3

3.5

rcc (Å) Fig. 1.3 Reaction path of the bulk diamond to graphite transformation as a function of the

carbon–carbon distance perpendicular to the layers (adapted from [9]). Dotted line (DFT results from [8], dashed line [9], solid line [29]).

z direction, contrary to graphene. This results in particular in a much stronger spin–orbit coupling in silicene and germanene with potential relevance for spintronics [13].

1.2

Thermal (In)Stability of 2D Crystals Many properties of graphene are studied for graphene deposited on a substrate, such as SiO2, hexagonal boron nitride (h-BN), and mica. Nevertheless, it is instructive to

10

Graphene: Basic Properties

(a)

(b)

Fig. 1.4 Side view of charge-density distribution in germanene (a) and graphene (b) (courtesy

A. N. Rudenko).

consider the properties of freely suspended graphene that can give general results for 2D structures embedded in 3D space. This is not only relevant for all other 2D crystals but also for the general long-standing problem of mechanics and statistical physics of thin membranes [18, 19, 20]. In crystals, the phonon spectrum always includes acoustic modes, with linear dispersion ω / q for q ! 0, where ω is the phonon frequency and q is the wavevector. As a result, in the harmonic approximation, at finite temperature the contribution of these modes to the mean square atomic displacement is divergent as
2 uq / 1=q2 . In
2D structures, this leads to divergence of the total mean square displacements as u2 / lnðL=aÞ, where L is the sample size. This was first noticed in the 1930s by Peierls [21] and Landau [22] and led them to the conclusion that 2D crystals cannot be stable. Strictly speaking, this divergence only implies that the harmonic approximation, based on the assumption that thermal displacements are much smaller than interatomic distances, is not valid in 2D crystals. However, this statement on destruction of the long-range crystalline order in 2D crystals turned out to be correct and rigorously proved as a part of the Mermin–Wagner theorem [23]. Therefore, the statement that graphene is a 2D crystal requires some explanation and clarification. If we embed 2D crystals in 3D space, the situation seems to be even worse. In fact, in this case, acoustic out-of-plane phonons turn out to have quadratic dispersion ω / q2 [24, 19, 20] leading to thermal
 instability of the flat state due to strong divergence of out-of-plane fluctuations h2 / L2 . Statistical mechanics of fluctuating membranes (for a review see [18, 19, 20]) show that an harmonic coupling of in-plane and outof-plane modes is crucially important. This coupling can be described by scaling considerations similar to those used in the theory of critical phenomena. As a result, out-of-plane acoustic modes become less soft q2 ! q2
η but in-plane modes become

1.3 Electronic Structure of Single-Layer Graphene

11

Fig. 1.5 Snapshot of graphene at room temperature as calculated by molecular dynamics

simulations with the reactive potential LCBOPII [29] for a sample of 12,096 carbon atoms. The gray scale gives the amplitude of out-of-plane fluctuations, which, for this sample, ranges between +3 Å and –3 Å.

even softer q ! q2
η . The best estimate of the parameter η from both atomistic Monte Carlo simulations and the functional renormalization group is 0.85 [20]. As a result, the
 membrane becomes flatter h2 / L2
η but the thermal instability
 of in-plane modes becomes even stronger than in the harmonic approximation u2 / L2
2η . For this reason, long-range crystal order does not exists also for 2D crystals embedded in 3D space. The long-range crystal order is observable through the existence of Bragg peaks in the structure factor, which become infinitely sharp in the limit L ! ∞. For 2D crystals in 3D, these peaks have a finite broadening [19]. This broadening is small if kB T  κ, where κ is the bending rigidity, and the centers of the broadened Bragg peaks still form the ideal reciprocal lattice. For graphene at room temperature, kB T=κ  1=40, thus justifying the use of the name crystal for graphene [20].
 In turn, the behavior predicted by scaling theory h2 / L2
η means that at any finite temperature graphene has a rippled structure due to out-of-plane fluctuations. This is clearly seen in atomistic Monte Carlo simulations [25, 26], where ripples reversibly appear with temperature with amplitude following the above scaling with L. A typical snapshot for graphene at room temperature is shown in Fig. 1.5. Rippling of suspended graphene was experimentally observed by electron diffraction [27], although it is still under debate if these ripples are intrinsic due to dynamical thermal fluctuations or static, e.g., due to thermal compression [28].

1.3

Electronic Structure of Single-Layer Graphene Graphene has the honeycomb crystal lattice shown in Fig. 1.6(a). The lattice vectors of the triangular Bravais lattice are pffiffiffi a  pffiffiffi a ! ! 3; 3 , 3;
3 , a1 ¼ a2 ¼ (1.1) 2 2 where a  1:42 Å is the nearest-neighbor distance. The unit cell contains two atoms belonging to two sublattices, A and B. Each atom of sublattice A is surrounded by three atoms of sublattice B, and vice versa. The vectors connecting each atom to the three nearest neighbors are

12

Graphene: Basic Properties

(a)

(b) b1

ky

A

B

K

Γ

a1

M

kx

K'

a2 b2 Fig. 1.6 (a) Honeycomb lattice with lattice vectors. Sublattices A and B are shown as black and

gray. (b) Reciprocal lattice vectors and some special points in the Brillouin zone. ! δ1

¼

a  pffiffiffi 1; 3 , 2

!

δ2 ¼

pffiffiffi a 1;
3 , 2

!

δ 3 ¼ að
1; 0Þ:

(1.2)

The reciprocal lattice of the triangular Bravais lattice is also triangular, with lattice vectors pffiffiffi ! ! 2π  pffiffiffi 2π  (1.3) b1 ¼ 1; 3 , 1;
3 : b2 ¼ 3a 3a In Fig. 1.6(b), we show the Brillouin zone with the special high-symmetry points K, K0 , and M       ! ! ! 2π 2π 2π 2π 2π 0 ; pffiffiffi , ;
pffiffiffi , ;0 : (1.4) K ¼ K¼ M¼ 3a 3 3a 3a 3a 3 3a The electronic structure of graphene and graphite are discussed in detail in [30]. In Fig. 1.7, we show the graphene band structure. The sp2 hybridized states (σ-states) form occupied and empty bands with a huge gap, whereas π-states form a single band with a conical self-crossing point at K (and by symmetry also at K0 ). The assignment to σ- and pz-bands is done as in [31]. This band structure was obtained for the first time by Wallace in 1947 [32], by a simple tight-binding model, later developed by McClure [33] and Slonczewski and Weiss [34]. The Fermi energy of undoped graphene falls exactly at the crossing of the conical points, separating a completely filled valence band and an empty conduction band with no band gap in between. Therefore, at first sight, graphene might be classified as a gapless semiconductor [35]. Three-dimensional crystals, such as HgTe or α-Sn (gray tin), are known to be gapless semiconductors. What makes graphene unique is not the gapless state itself but the very unusual chiral nature of the electron states, which will be described later, as well as the high degree of electron–hole symmetry. As a result of subtle quantum relativistic effects (conductivity via evanescent modes of the Dirac operator), undoped graphene has a finite conductivity at zero

1.3 Electronic Structure of Single-Layer Graphene

13

10

Energy (eV)

5 pz-bands

0

EF

–5 –10 s-bands –15 –20 G

M

G

K

Fig. 1.7 Band structure of graphene (courtesy A. N. Rudenko).

temperature, despite the vanishing Fermi surface, becoming a semimetal rather than a gapless semiconductor [36, 19]. Following [32], we consider only the π-states and nearest-neighbor hopping terms (parameter t), so that there are only hopping processes from one sublattice to the other (bipartite lattice). The basis of the electronic states contains two π-states per unit cell belonging to atoms of sublattices A and B. The tight-binding Hamiltonian is therefore described by a 2  2 matrix 0 1 ! ! tSð k Þ A ^ ðk Þ ¼ @ 0 H ; (1.5) !  tS ð k Þ 0 !

where k is the wavevector and !

Sð k Þ ¼

X !

δ

!!

e

ik δ

  pffiffiffi ik x a ky a 3 ¼ 2 exp cos þ exp ð
ik x aÞ: 2 2 

(1.6)

The resulting energy spectrum is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! Eð k Þ ¼ tjSð k Þj ¼  t 3 þ f ð k Þ, !

where

!

pffiffiffi    pffiffiffi  ! 3 3 k y a cos kx a : f k ¼ 2 cos 3ky a þ 4 cos 2 2 !

!

(1.7)

(1.8)

Since SðK Þ ¼ SðK 0Þ ¼ 0, the two bands cross at these points. By expanding the Hamiltonian near these points and eliminating a constant phase factor by a proper gauge choice, we find !   3at 0 qx
iqy ! ^ HK q  , (1.9) qx þ iqy 0 2

14

Graphene: Basic Properties



T ! ! ! ! ! ! ^ K 0 ð! ^ K 0 ð! where q ¼ k
K, H qÞ ¼ H q 0 Þ with q 0 ¼ k
K 0 and the superscript T indicates the transpose of the matrix. We can interpret the prefactor in the right-hand side of Eq. (1.9) as ħv, where v¼

3at 2ħ

(1.10)

is the electron velocity at the conical points. Experimentally, v  c=300  106 m=s where c is the velocity of light [37, 38]. Taking into account the next-nearest-neighbor hopping t0 we find, instead of Eq. (1.7) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! ! ! ! ! Eð k Þ ¼ tjSð k Þj þ t 0 f ð k Þ ¼ t 3 þ f ð k Þ þ t 0 f ð k Þ: (1.11) The term in t0 breaks the electron–hole symmetry shifting the conical point from E = 0 to E = –3t0 but it does not change the behavior of the Hamiltonian near the conical points. Experimentally, t 0 is negative (the sign of t is arbitrary since it can be eliminated by a gauge transformation of the wave function in one of the sublattices) and t 0 
0:1jt j [39]. The existence of the conical points in graphene is a consequence of symmetry [34], and therefore it does not depend on any specific assumptions on the number of bands, hopping parameters etc. The protection of this feature relies on two discrete symmetries: time reversal, and spatial inversion [40]. The latter assumes equivalence of the sublattices. The points K and –K0 are equivalent because they differ by the reciprocal lattice vector ! ! ! b ¼ b 1 þ b 2 . This equivalence becomes evident using a representation such as the one shown in Fig. 1.8, with a larger unit cell in reciprocal space and six conical points.

Fig. 1.8 Electron energy spectrum of graphene in the nearest-neighbor approximation (Eq. (1.7))

in a larger unit cell in reciprocal space with six conical points. The points K1 and K2 are equivalent to K and –K0 .

1.3 Electronic Structure of Single-Layer Graphene

15

For any realistic doping, the Fermi energy in graphene is close to the energy at the conical point, jEF j  jt j. To describe electron and hole states in this regime, we need to expand the effective Hamiltonian near the special points K and K0 and replace qx !
i ∂x∂ , qy !
i ∂y∂ which corresponds to the effective mass approximation, ! ! or k  p perturbation theory [41]. From Eq. (1.9), we obtain ^ K ¼
iħv ! H σ r,

(1.12)

where  σ0 ¼

1 0

  0 0 , σx ¼ 1 1

  1 0 , σy ¼ 0 i

 
i 1 , σz ¼ 0 0

0
1

 (1.13)

are the Pauli matrices (only the x- and y-components enter Eq. (1.12). A complete low-energy Hamiltonian consists of 4  4 matrices taking into account both sublattices and two conical points (in terms of semiconductor physics, two valleys). On the basis that 0 1 ψ KA Bψ C B KB C Ψ¼B (1.14) C, @ ψK0 A A ψK0 B where, e.g., ψ KA labels the component of the electron wave function corresponding to valley K and sublattice A, the Hamiltonian is a 2  2 block super matrix   ^ 0 ^ ¼ HK H : (1.15) ^ K0 0 H A more symmetric form of the Hamiltonian is obtained using another basis 0 1 ψ KA B ψ C B KB C Ψ¼B C @ ψK0 B A

(1.16)


ψ K 0 A so that the Hamiltonian (Eq. (1.15)) becomes ^ ¼
iħvτ 0 ⊗ ! H σ r,

(1.17)

where τ 0 is the unit matrix in valley indices. We will use different notations for the same ! ! Pauli matrices acting on different indices, namely σ in the sublattice space and τ in the valley space. For ideal graphene, the valleys are decoupled. In the presence of inhomogeneities, which are smooth at the atomic scale (external electric and magnetic fields, disorder, etc.), the valleys remain independent since the Fourier component of the external ! potential with the Umklapp wavevector b is very small, and intervalley scattering is improbable. However, we should keep in mind that any sharp (atomic-scale) inhomogeneity, like boundaries or vacancies, would mix the states from different valleys.

16

Graphene: Basic Properties

The Hamiltonian (Eq. (1.12)) is a 2D analogue of the Dirac Hamiltonian for massless fermions [42]. Instead of the velocity of light c, there is a parameter v  c=300. The internal degree of freedom, which is just spin for “true” Dirac fermions, is the sublattice index in the case of graphene. Dirac “spinors” consist here of the components giving the distribution of electrons in sublattices A and B. We will call this quantum number pseudospin, so that pseudospin “up” means sublattice A and pseudospin “down” means sublattice B. Beside the pseudospin, there are two more internal degrees of freedom, the valley label (sometimes called isospin) and the real spin, so that the most general low-energy Hamiltonian of electrons in graphene is an 8  8 matrix. Spin–orbit coupling leads to a mixture of pseudospin and real spin and to gap opening [43]. However, the value of the gap is supposed to be very small, of the order of 10
2 K for pristine graphene [44]. The reason is not only the lightness of carbon atoms but also the orientation of the orbital moments of π-states perpendicular to the graphene plane. For the case of “true” Dirac fermions in 3D space, the Hamiltonian is a 4  4 matrix, due to two projections of the spin and two values of the charge degree of freedom (particle versus antiparticle). For the 2D case, separation in particle and antiparticle is not independent of the pseudospin direction. Electrons and holes in graphene are just linear combinations of the states from the sublattices A and B. The 2  2 matrix !! ħv σ k (the result of the action of the Hamiltonian (Eq. (1.12)) on a plane wave with ! wavevector k ) is diagonalized by the unitary transformation ! ! ^ ! ¼ p1ffiffiffi ð1 þ im ! σ Þ, U k k 2

(1.18) !

!

where m!k ¼ ðcos ϕ!k ,
sin ϕ!k Þ and ϕ!k is the polar angle of the vector k . The eigenfunction ! exp ð
iϕ!k =2Þ 1 ðK Þ ! ψ e, h ð k Þ ¼ pffiffiffi (1.19) 2 exp ðiϕ!k =2Þ corresponds to electron (e) and hole (h) states, with energies Ee, h ¼ ħvk: For the valley K0 , the corresponding states (in the basis (Eq. (1.14)) are ! exp ðiϕ!k =2Þ 1 ðK 0 Þ ! ψ e, h ð k Þ ¼ pffiffiffi : 2 exp ð
iϕ!k =2Þ

(1.20)

(1.21)

For the electron (hole) states, by definition !!

ðk σ Þ ψ e, h ¼ ψ e, h : k

(1.22)

This means that the electrons (holes) in graphene have a definite pseudospin direction, namely, parallel (antiparallel) to the direction of motion. Thus, these states are chiral

1.4 Electronic Structure of Bilayer Graphene

17

(helical), as it should be for massless Dirac fermions [42]. This is of crucial importance for “relativistic” effects, such as Klein tunneling [45, 19], which will be considered briefly later. The Dirac model for electrons in graphene results from the lowest-order expansion of the tight-binding Hamiltonian (Eq. (1.12)) near the conical points. Taking into account the next, quadratic, term, we find (in the basis (Eq. (1.16)), instead of the Hamiltonian (Eq. (1.17)) h  i ! ^ ¼ ħvτ 0 ⊗ ! H σ k þ μτ z ⊗ 2σ y kx ky
σ x k 2x
k2y , (1.23) where μ ¼ 3a2 t=8. The additional term in Eq. (1.23) corresponds to a trigonal warping ! [46]. Diagonalization of the Hamiltonian (Eq. (1.23)) gives the spectrum Ee, h ð k Þ ¼ ! εð k Þ, where !

ε2 ð k Þ ¼ ħ2 v2 k2 ∓ 2ħvμk3 cos ð3ϕ!k Þ þ μ2 k4 :

(1.24)

The signs ∓ in Eq. (1.24) correspond to valleys K and K0 . The dispersion is no more isotropic but has a three-fold symmetry.

1.4

Electronic Structure of Bilayer Graphene The process of exfoliation leads often to samples made of more than one graphene layer. Bilayer graphene [47] is especially interesting. To describe a bilayer, we need to extend the tight-binding model to consider interlayer terms [33, 34, 48]. The crystal structure of bilayer graphene with Bernal stacking is shown in Fig. 1.9. The atoms in the sublattices A of the two layers lie exactly on top of each other, with a finite hopping parameter γ1 between them, whereas there are no significant hopping processes between the sublattices B of the two layers. Data on the electronic structure of graphite [49] indicate that the parameter γ1 t ⊥ is about 0.4 eV, an order of magnitude smaller than the nearest-neighbor in-plane hopping parameter γ0 ¼ t. The simplest model, which takes into account only these processes, is described by the Hamiltonian

g4 g

0

g3 g1

Fig. 1.9 Crystal structure of bilayer graphene, hopping parameters are shown.

18

Graphene: Basic Properties

(a)

(b)

Fig. 1.10 (a) Electronic structure of bilayer graphene within the model Eq. (1.25). (b) The same, for the model Eq. (1.32).

0

!

0

B B  ! tS ð k Þ ^ ðk Þ ¼ B H B B t @ ⊥ 0 !

1

tSð k Þ

t⊥

0

0

0

0

0

tSð k Þ

!

0

C C C ! C  tS ð k Þ C A 0 0

(1.25)

!

with Sð k Þ from Eq. (1.6). The basis states are ordered as first layer, sublattice A; first layer, sublattice B; second layer, sublattice A; second layer, sublattice B. The matrix (Eq. (1.25)) can be easily diagonalized, with four eigenvalues rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! ! 1 1 2 Ei ð k Þ ¼  t⊥  t þ t 2 jSðk Þj2 (1.26) 2 4 ⊥ with two independent  signs. The spectrum is shown in Fig. 1.10. It contains two degenerate bands at the points K and K0 with parabolic dispersion given by !

t 2 jSð k Þj2 ħ2 q2 E 1, 2 ð k Þ    , t⊥ 2m !

(1.27)

where the effective mass m ¼ jt ⊥ j=ð2v2 Þ. Experimentally, m  0:028me , where me is the mass of a free electron [50]. If we neglect intervalley scattering, we can construct the effective Hamiltonian describing only the two touching parabolic bands (Eq. (1.27)) by replacing ħqx and ħqy with operators p^x ¼
iħ ∂x∂ and p^y ¼
iħ ∂x∂ . The result reads [47, 48]  2 ! 0 p^x
i^ py 1 ^K ¼ H : (1.28)  2 2m p 0 p^ þ i^ x

y

The eigenstates of this Hamiltonian have very special chiral properties [47], resulting in a special Landau quantization, special scattering, etc. [19]. Electron and hole states corresponding to the energies

1.4 Electronic Structure of Bilayer Graphene

ħ2 k 2 2m

E e, h ¼ 

19

(1.29)

have a form similar to Eq. (1.19), with the replacement ϕ!k ! 2ϕ!k !
iϕ! 1 e k ðK Þ ! : ψ e, h ð k Þ ¼ pffiffiffi 2 eiϕ!k

(1.30)

.Instead of Eq. (1.22) the eigenfunctions Eq. (1.30) are characterized by the property ! 2 ! kσ ψ e, h ¼ ψ e, h : (1.31) k2 By applying a voltage V perpendicular to the carbon planes, we can open a gap in the energy spectrum [48, 51]. In this case, instead of the Hamiltonian (Eq. (1.25)), we have 0 V=2

B B tS ð! kÞ B Hð k Þ ¼ B B t @ ⊥ !

0

!

1

tSð k Þ

t⊥

V=2

0

0


V=2

C C C ! C  tS ð k Þ C A

0

tSð k Þ


V=2

!

0

0

(1.32)

and, instead of eigenvalues Eq. (1.26), one has ! E2i ð k Þ

t⊥2 V 2 þ  ¼ t jSðk Þj þ 2 4 2

! 2

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! t⊥4  2 þ t⊥ þ V 2 t 2 jSðk Þj2 : 4

(1.33)

For the two low-lying bands in the vicinity of the K (or K0 ) point, the spectrum has the “Mexican hat” dispersion   ! V Vħ2 v2 2 ħ4 v4 4 Eð k Þ  
k þ 2 k , (1.34) 2 t⊥2 t⊥ V where we assume, for simplicity, ħvk  V  jt ⊥ j. The expression in parenthesis has a maximum at k = 0 and a minimum at k ¼ pVffiffi2ħv (see Fig. 1.10(b)). The possibility to tune the gap in bilayer graphene by an electric field was experimentally confirmed [51, 52]. Consider now the effect of hopping processes at larger distances, namely, between B sublattices (γ3  0:3 eV [49]). Higher-order terms, such as γ4  0:04 eV, are supposed to be negligible. The term γ3 leads to a qualitative change in the spectrum near the K (K0 ) point. The effective Hamiltonian is modified by γ3 terms as [48] 0 1  2 py p^x
i^ 3γ3 a  p^x þ i^ þ py C 0 B C ħ 2m ^K ¼ B H B C (1.35) 2 @ p^
i^ A py 3γ3 a  x ^ 0 þ
i^ p p x y ħ 2m

20

Graphene: Basic Properties

Fig. 1.11 Effect of hopping parameter γ3 on the band structure of bilayer graphene.

with the energy spectrum !

E 2 ð kÞ  ð3γ3 aÞ2 k2 þ

  ħ2 k 2 2 3γ3 aħ2 k 3 ! cos 3ϕ : þ k m 2m

This means that at small enough wavevectors

γ3 γ1

ka 2

e 10
2 γ0

(1.36)

(1.37)

the parabolic dispersion law (Eq. (1.29)) is replaced by the linear one with four conical touching points instead of one parabolic touching point (see Fig. 1.11).

1.5

Graphene as a Bridge between Condensed Matter and High-Energy Physics The many exceptional properties of graphene discussed in this chapter make it interesting in many respects and it will therefore be the subject of following chapters. Here we will underline the relevance of graphene for fundamental physics, beyond condensed matter. The key point is that charge carriers in graphene can be described as massless Dirac fermions. In particle physics, to reach this ultra-relativistic regime we need energies much higher than the particle rest energy whereas in graphene one can reach it by default. Moreover, to open a gap in the spectrum, which corresponds to the rest energy in particle physics, some efforts are needed, like chemical functionalization [53], combination of strain with electric gating [54], size quantization in nanoribbons [55]. Massless Dirac fermions appear also in other situations in condensed matter physics, for example in superfluid 3He [56] and on the surface of topological insulators [57], but in

1.6 References

21

graphene the Dirac regime describes the energy spectrum in an particularly broad range up to a few eV. Since in graphene there are two Dirac cones instead of one as in topological insulators, the Dirac physics can be in principle destroyed by intervalley scattering processes due to atomically sharp defects. However, since graphene has one of the strongest chemical bonds in nature, such defects (vacancies, dislocations, Stone–Wales defects) have high-formation energy [29] and rarely occur spontaneously. Therefore, the observation of phenomena related to the Dirac nature of the charge carriers is not very challenging. An important difference between Dirac fermions and charge carriers in graphene is that the light velocity c is replaced by the Fermi velocity v which is 300 times smaller. As a result, instead of the fine structure constant α ¼ e2 =ħc  1=137 determining the strength of interaction in quantum electrodynamics, in graphene there is a much stronger interaction with adimensional coupling constant e2 =ħv  2:2. For this reason, some quantum relativistic effects, which are difficult to observe in particle physics, become accessible in the physics of graphene. Probably the best example is the relativistic atomic collapse which is supposed to happen for the elements with Z > 170, determining the natural limit of the periodic table [58] which is still far from being reached. For graphene, an analogous phenomenon was predicted [59, 60, 61] and experimentally observed [62]. Funnily, the true fine structure constant also plays a role in graphene, determining its optical transmission with absorption coefficient η ¼ πα  2:3% per graphene layer [63]. Probably the most relevant quantum relativistic phenomenon for the physics and applications of graphene is Klein tunneling [45] which was also experimentally observed [64, 65]. This is a property of relativistic quantum particles, to be transmitted across a potential barrier of any height and thickness. This property is a direct consequence of chirality (Eq. (1.22)). On the one hand, Klein tunneling does not allow to make a graphene transistor based just on p–n junctions as in semiconductors because we cannot lock the junction. On the other hand, it is the Klein tunneling that protects the mobility of charge carriers in graphene, making the unavoidable presence of electron– hole puddles [66] not so detrimental to charge transport. A detailed discussion of the relation of the physics of graphene to high-energy physics can be found in [19].

1.6

References [1] A. A. Radzig and B. M. Smirnov, Reference Data on Atoms, Molecules and Ions (Berlin: Springer, 1985). [2] L. Pauling, The Nature of the Chemical Bond (Ithaca, NY: Cornell University Press, 1960). [3] H. Eyring, J. Walter, and G. E. Kimball, Quantum Chemistry (Ithaca, NY: Cornell University Press, 1946). [4] L. A. Girifalco and R. A. Lad, Energy of cohesion, compressibility, and the potential energy of the graphite system. J. Chem. Phys. 25 (1956) 693–7. [5] G. Savini, Y. J. Dappe, S. Oberg et al., Bending modes, elastic constants and mechanical stability of graphitic systems. Carbon 49 (2011) 62–9.

22

Graphene: Basic Properties

[6] M. Reguzzoni, A. Fasolino, E. Molinari, and M. C. Righi, Potential energy surface for graphene on graphene: Ab initio derivation, analytical description, and microscopic interpretation. Phys. Rev. B 86 (2012) 245434. [7] M. Dresselhaus, G. Dresselhaus, and P. Eklund, Science of Fullerenes and Carbon Nanotubes (New York: Academic Press, 1996). [8] S. Fahy, S. T. Louie, and M. L. Cohen, Pseudopotential total-energy study of the transition from rhombohedral graphite to diamond. Phys. Rev. B 34 (1986) 1191–9. [9] J. H. Los and A. Fasolino, Intrinsic long-range bond-order potential for carbon: Performance in Monte Carlo simulations of graphitization. Phys. Rev. B 68 (2003) 024107. [10] K. S. Novoselov, A. K. Geim, S. V. Morozov et al., Electric field effect in atomically thin carbon films. Science 306 (2004) 666–9. [11] K. S. Novoselov, D. Jiang, Y. Zhang et al., Two-dimensional atomic crystals. Proc. Natl. Acad. Sci. USA 102 (2005) 10451–3. [12] A. Kara, H. Enriquez, A. P. Seitsonen et al., A review on silicene: New candidate for electronics. Surf. Sci. Rep. 67 (2012) 1–18. [13] A. Acun, L. Zhang, P. Bampoulis et al., Germanene: The germanium analogue of graphene. J. Phys.: Cond. Mat. 27 (2015) 443002. [14] L. M. Ghiringhelli, J. H. Los, E. J. Meijer, A. Fasolino, and D. Frenkel, Modeling the phase diagram of carbon. Phys. Rev. Lett. 94 (2005) 145701. [15] F. P. Bundy, W. A. Bassett, M. S. Weathers, R. J. Hemley, H. K. Mao, and A. F. Goncharov, The pressure–temperature phase and transformation diagram for carbon; updated through 1994. Carbon 34 (1996) 141–53. [16] M. Togaya, Pressure dependences of the melting temperature of graphite and the electrical resistivity of liquid carbon. Phys. Rev. Lett. 79 (1997) 2474–7. [17] F.P. Bundy, H.P. Bovenkerk, H.M. Strong, and J. R. H. Wentorf, Diamond–graphite equilibrium line from growth and graphitization of diamond. J. Chem. Phys. 35 (1961) 383. [18] D. R. Nelson, T. Piran, and S. Weinberg (eds.), Statistical Mechanics of Membranes and Surfaces (Singapore: World Scientific, 2004). [19] M. I. Katsnelson, Graphene: Carbon in Two Dimensions (Cambridge: Cambridge University Press, 2012). [20] M. I. Katsnelson and A. Fasolino, Graphene as a prototype crystalline membrane. Acc. Chem. Res. 46 (2013) 97–105. [21] R. E. Peierls, Bemerkungen über Umwandlungstemperaturen. Helv. Phys. Acta 7 (1934) 81–3. [22] L. D. Landau, Zur Theorie der Phasenumwandlungen II. Phys. Z. Sowjetunion 11 (1937) 26–35. [23] N. D. Mermin, Crystalline order in two dimensions. Phys. Rev. 176 (1968) 250–4. [24] I. M. Lifshitz, О тепловых свойствах цепных и слоистых структур при низких температурах [On thermal properties of chained and layered structures at low temperatures]. Zh. Eksp. Teor. Fiz. 22 (1952) 475–86. [25] A. Fasolino, J. H. Los, and M. I. Katsnelson, Intrinsic ripples in graphene. Nature Mater. 6 (2007) 858–61. [26] J. H. Los, M. I. Katsnelson, O. V. Yazyev, K. V. Zakharchenko, and A. Fasolino, Scaling properties of flexible membranes from atomistic simulations: Application to graphene. Phys. Rev. B 80 (2009) 121405 (R). [27] J. C. Meyer, A. K. Geim, M. I. Katsnelson et al., The structure of suspended graphene sheets. Nature 446 (2007) 60–3.

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[28] R. Zan, C. Muryn, U. Banger et al., Scanning tunneling microscopy of suspended graphene. Nanoscale 4 (2012) 3065–8. [29] J. H. Los, L. M. Ghiringhelli, E. J. Meijer, and A. Fasolino, Improved long-range reactive bond-order potential for carbon. I. Construction. Phys. Rev. B 72 (2005) 214102. [30] F. Bassani and G. Pastori Parravicini, Electronic States and Optical Transitions in Solids (Oxford: Pergamon, 1975). [31] A. N. Rudenko, F. J. Keil, M. I. Katsnelson, and A. I. Lichtenstein, Exchange interactions and frustrated magnetism in single-side hydrogenated and fluorinated graphene. Phys. Rev. B 88 (2013) 081405(R). [32]. P. R. Wallace, The band theory of graphite. Phys. Rev. 71 (1947) 622–34. [33] J. W. McClure, Band structure of graphite and de Haas–van Alphen effect. Phys. Rev. 108 (1957) 612–18. [34] J. S. Slonczewski and P. R. Weiss, Band structure of graphite. Phys. Rev. 109 (1958) 272–9. [35] I. M. Tsidilkovsii, Electron Spectrum of Gapless Semiconductors (Berlin: Springer, 1996). [36] M. I. Katsnelson, Zitterbewegung, chirality, and minimal conductivity of graphene. Eur. Phys. J. B 51 (2006) 157–60. [37] K. S. Novoselov, A. K. Geim, S. V. Morozov et al., Two-dimensional gas of massless Dirac fermions in graphene. Nature 438 (2005) 197–200. [38] Y. Zhang, Y.-W. Tan, H. L. Stormer, and P. Kim, Experimental observation of the quantum Hall effect and Berry’s phase in graphene. Nature 438 (2005) 201–4. [39] A. Kretinin, G. L. Yu, R. Jalil et al., Quantum capacitance measurements of electron–hole asymmetry and next-nearest-neighbor hopping in graphene. Phys. Rev. B 88 (2013) 165427. [40] J. L. Mañes, F. Guinea, and M. A. H. Vozmediano, Existence and topological stability of Fermi points in multi-layered graphene. Phys. Rev. B 75 (2007) 155424. [41] I. M. Tsidilkovsii, Band Structure of Semiconductors (Oxford: Pergamon, 1982). [42] J. D. Bjorken and S. D. Drell, Relativistic Quantum Mechanics (New York: McGraw-Hill, 1964). [43] C. L. Kane and E. J. Mele, Quantum spin Hall effect in graphene. Phys. Rev. Lett. 95 (2005) 226801. [44] D. Huertas-Hernando, F. Guinea, and A. Brataas, Spin–orbit coupling in curved graphene, fullerenes, nanotubes, and nanotube caps. Phys. Rev. B 74 (2006) 155426. [45] M. I. Katsnelson, K. S. Novoselov, and A. K. Geim, Chiral tunneling and the Klein paradox in graphene. Nature Phys. 2 (2006) 620–5. [46] T. Ando, T. Nakanishi, and R. Saito, Berry’s phase and absence of back scattering in carbon nanotubes. J. Phys. Soc. Japan 67 (1998) 2857–62. [47] K. S. Novoselov, A. McCann, S. V. Morozov et al., Unconventional quantum Hall effect and Berry’s phase of 2π in bilayer graphene. Nature Phys. 2 (2006) 177–80. [48] E. McCann and V. I. Fal’ko, Landau level degeneracy and quantum Hall effect in a graphite bilayer. Phys. Rev. Lett. 96 (2006) 086805. [49] M. S. Dresselhaus and G. Dresselhaus, Intercalation compounds of graphite. Adv. Phys. 51 (2002) 1–186. [50] A. S. Mayorov, D. C. Elias, M. Mucha-Kruczynski et al., Interaction-driven spectrum reconstruction in bilayer graphene. Science 333 (2011) 860–3. [51] E. V. Castro, K. S. Novoselov, S. V. Morozov et al., Biased bilayer graphene: Semiconductor with a gap tunable by the electric field effect. Phys. Rev. Lett. 99 (2007) 216802. [52] K. B. Oostinga, H. B. Heersche, X. Lie et al., Gate-induced insulating state in bilayer graphene devices. Nature Mater. 7 (2008) 151–7.

24

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[53] D. W. Boukvalow and M. I. Katsnelson, Tuning the gap in bilayer graphene using chemical functionalization: DFT calculations. Phys. Rev. B 78 (2008) 085413. [54] T. Low, F. Guinea, and M. I. Katsnelson, Gaps tunable by electrostatic gates in strained graphene. Phys. Rev. B 83 (2011) 195436. [55] Y.-W. Son, M. L. Cohen, and S. Louie, Energy gaps in graphene nanoribbons. Phys. Rev. Lett. 97 (2006) 216803. [56] G. E. Volovik, The Universe in a Helium Droplet (Oxford: Clarendon Press, 2003). [57] X.-L. Qi and S.-C. Zhang, Topological insulators and superconductors. Rev. Mod. Phys. 83 (2011) 1057–110. [58] Ya. B. Zel’dovich and V. S. Popov, Electronic structure of superheavy atoms. Sov. Phys. Uspekhi 14 (1972) 673–94. [59] V. M. Pereira, J. Nilsson, and A. H. Castro Neto, Coulomb impurity problem in graphene. Phys. Rev. Lett. 99 (2007)166802. [60]. A. V. Shytov, M. I. Katsnelson, and L. S. Levitov, Vacuum polarization and screening of supercritical impurities in graphene. Phys. Rev. Lett. 99(2007) 236801. [61] A. V. Shytov, M. I. Katsnelson, and L. S. Levitov, Atomic collapse and quasi-Rydberg states in graphene. Phys. Rev. Lett. 99 (2007) 246802. [62] Y. Wang, D. Wong, A. V. Shytov et al., Observing atomic collapse resonances in artificial nuclei on graphene. Science 340 (2013) 734–7. [63] R. R. Nair, P. Blake, A. N. Grigorenko et al., Fine structure constant defines visual transparency of graphene. Science 320 (2008) 1308. [64] N. Stander, B. Huard, and D. Goldhaber-Gordon, Evidence for Klein tunneling in graphene p–n junctions. Phys. Rev. Lett. 102 (2009) 026807 [65] A. F. Young and P. Kim, Quantum interference and Klein tunneling in graphene heterojunctions. Nature Phys. 5 (2009) 222–6 [66] J. Martin, N. Akerman, G. Ulbricht, T. Lohmann, J. H. Smet, K. von Klitzing, and A. Yacoby, Observation of electron–hole puddles in graphene using a scanning single-electron transistor. Nature Phys. 4 (2007) 144–8.

2

Electrical Transport in Graphene: Carrier Scattering by Impurities and Phonons Jian-Hao Chen

Charge carrier transport in graphene has been one of the major focuses of attention since the experimental realization of isolated graphene [1]. Much of the interest has arisen from the prospect of fabricating graphene into high-speed electronic devices, which rely on the exceptional carrier mobility of the electronic material as well as consistency in the performance of such devices. The large surface-to-bulk ratio of graphene/few-layer graphene provides exceptional tunability of the electron system to external perturbations, including electric field, magnetic field, surface modifications, etc. The meaning of such tunability is two-fold: it could be exploited in technological applications, while the effects of disorders (impurities, ripples, phonons of graphene, and the device substrate) become an important factor in determining the performance of graphene-based devices. Indeed, graphene field-effect devices fabricated on various substrates (e.g. silicon dioxide substrate, hexagonal boron nitride substrate) as well as suspended graphene devices (the active area of graphene suspended in vacuum) have shown varying field-effect mobilities ranging from 0.1 to 20 m2/V s [2, 3, 4], which are mainly caused by scattering of electrons by structural defects [5, 6, 7, 8], charged impurities [9, 10], and neutral physisorbed species [11]; phonons [12, 13, 14, 15, 16] have measurable effects on the performance of the device, with most of the effects coming from the scattering of electrons by substrate phonons [13, 14]; the corrugation of the two-dimensional graphene layer could cause some scattering, but the effects are small compared with those of other scattering sources in realistic device configurations [17, 18, 19]. This chapter proceeds as follows: Section 2.1 introduces the basics of Boltzmann transport theory and early experimental findings; Section 2.2 explains the experimental and theoretical understandings of charge carriers scattered by charged impurities; Section 2.3 covers the effects of resonance scatterers; Section 2.4 discusses the effects of corrugations of the graphene membrane (ripples); Section 2.5 examines the effects of phonons, including optical, acoustic, and substrate phonons on the performance of graphene devices.

2.1

Boltzmann Transport Theory The semiclassical picture is sufficient to understand most charge carrier scattering phenomena in graphene that are related to current state-of-the-art technological applications. In the semiclassical picture, the single-electron probability distribution is the 25

26

Electrical Transport in Graphene

key variable, and it is described by the Boltzmann transport theory. The Boltzmann equation has the general form [20] !

!

∂f ∂ k ∂r þ  r!k f þ  r!r f ¼ S½ f  ∂t ∂t ∂t ! !

and

!

∂r ∂t

!

!

!

∂k ∂t !

where f ð r , k , tÞ is the distribution function of charge carrier,




!

¼ 2πe Eþ h

(2.1) !

!

v ! B k

c

¼ v !k are the canonical equations of motion, E and B are the electric and !

magnetic field, v !k is the Fermi velocity of the charge carrier at a specific momentum

!

k , and h is the Planck constant. The collision integral S½ f  can be expressed as S½ f  ¼

X  !

k0

f !0 k



   !0 !     ! !0  1  f !k Θ k , k  f !k 1  f !0 Θ k , k , k

(2.2)

! !   ! ! ! where Θ k0 , k and Θ k , k0 are the probability of an electron to be scattered from k !

!

!

state to k 0 state and from k 0 state to k state, respectively. For simplicity, we consider only the spin-independent scattering process and omit the spin degree of freedom.  ! ! Assuming elastic scattering in the single-particle picture, we have Θ k0 , k ¼ ! !  ! ! ! !  ! ! ! !  Θ k , k0 ¼ 0 for E k 6¼ E k 0 and Θ k 0 , k ¼ Θ k , k 0 for E k ¼ E k0 . Thus, the collision term (Eq. (2.2)) becomes S½ f  ¼

X ! 0

k

  ! !  Θ k , k0 f !0  f !k : k

(2.3)

Assuming the electron scattering Hamiltonian has the form ^ ¼ H

X !! 0

kk

^ !0 : V !!0 a^þ !a kk

k

k

(2.4)

According to Fermi’s golden rule within the Born approximation 2   ! !  1   Θ k , k 0 ¼ V !!0 δ ε!k  ε !0 : kk k h

(2.5)

!

!

∂f Because ∂∂tk ¼  2πe h E , for stationary states ∂t ¼ 0 and in a spatially homogeneous conduction channel r!r f ¼ 0, the Boltzmann equation simplifies to



 X 1   2πe ! V ! ! 2 f !  f ! δ ε!  ε ! : ! E  r!k f ¼ k k k0 h k k0 k0 k0 h

(2.6)

One can put in the scattering matrix elements of various elastic scattering sources and obtain resistivity arising from respective types of isotropic scatterers. !   ! For inelastic scattering (e.g., scattering with phonons), we have E k 0 ¼ E k  hω 2π ! ! ! and k0 ¼ k  q (“+” for absorption of phonons and “–” for emission of phonons), thus Eq. (2.6) becomes

2.1 Boltzmann Transport Theory


X 1  2  2πe ! hω ! E  r!k f ¼  V !!0 : f !0  f !k δ ε!k  ε !0  k0 h kk k k h 2π

27

(2.7)

Within the Boltzmann transport theory, the conductivity of graphene can be expressed as [21]
 ð e ∂f 2 dε DðεÞvk τ ðεÞ  σ¼ , 2 ∂ε

(2.8)

where DðεÞ is the density of state in graphene at energy ε, and τ ðεÞ is the relaxation time for a specific scattering process. In the early dates of graphene research, there were three major experimental findings for the quasi-DC transport properties of graphene in zero magnetic fields [2, 22, 23, 24]: (1) conductivity of graphene depends roughly linearly on carrier density both in the electron conduction regime and in the hole conduction regime (see Fig. 2.1(a) and Fig. 2.2); (2) near the charge neutrality point, the conductivity of graphene devices does not go to zero, but instead has values near 4e2 =h (see Fig. 2.1(b) and the inset in Fig. 2.2); and (3) temperature-dependent resistivity of graphene is smaller than that of semiconductors such as silicon (see Fig. 2.3). For graphene devices with high fieldeffect mobilities, a sublinear dependence of the conductivity on carrier density is also observed [2, 10]. Theories on carrier scattering in graphene in zero magnetic fields at

(a)

(b)

Fig. 2.1 (a) The measured conductivity of graphene as a function of gate voltage Vg (or carrier density). The conductivity increases linearly with the density. (b) The relation between mobility and the minimum conductivity of various graphene samples (adapted with permission from [22]).

28

Electrical Transport in Graphene

Fig. 2.2 The measured conductivity of graphene as a function of Vg for five different samples. For clarity, curves are vertically displaced. The inset shows the detailed view of the densitydependent conductivity near the Dirac point for the data in the main panel (adapted with permission from [2]).

low temperature (Sections 2.2–2.4) have been proposed to try to explain these characteristics and were tested by experiments. The theoretical and experimental understanding of the effects of phonons is described in Section 2.5.

2.2

Charged Impurities Charged impurities are predicted to have dramatic effects on the transport properties of graphene. Several groups [9, 25, 26, 27, 28, 29] have shown theoretically that charged impurity scattering in graphene should produce a conductivity linear in charge density and inversely proportional to impurity density, i.e. n : σ ci ðnÞ ¼ C ci e nimp

(2.9)

This is equivalent to a constant mobility, inversely proportional to charged impurity density μ = C/nimp. The linear σci(n) arises from the 1/q dependence of the Coulomb potential on the wavevector q, leading to a 1/kF dependence of the scattering rate. A unique aspect of graphene, as opposed to other two-dimensional electron systems (2DES), is that the 1/kF dependence is preserved even for a screened Coulomb potential in graphene [28], creating a clear dichotomy in graphene between

2.2 Charged Impurities

29

Fig. 2.3 (a) The measured conductivity of graphene as a function of Vg at six different temperatures  (20 K,  100 K,180 K, 220 K, 260 K). The conductivity is split into two components 1 ; T ¼ ρ V g þ ρðT Þ (adapted with permission from [24]). (b) The measured ¼ ρ V g σ ðV g ;T Þ resistivity of graphene as a function of Vg at three different temperatures (30 mK, 77 K, 300 K) (adapted with permission from [23]). Both studies found that ρðT Þ is quite small compared to semiconductors such as silicon.

long-range and short-range scattering potentials. Hwang et al. [28] calculated the screened Coulomb potential within the random phase approximation (RPA), and used the results to determine Cci  5  1015 V–1 s–1. Novikov [30] noted that, beyond the Born approximation used by Hwang et al. [28], an asymmetry in Cci for attractive versus repulsive scattering (electron versus hole carriers) is expected for Dirac fermions. In the presence of charged impurities, at low carrier density, the conductivity does not vanish linearly, but rather saturates to a constant value, the minimum conductivity σmin, over a plateau of width ΔVg [9, 26, 28]. Numerical calculations [26, 28] showed a finite conductivity of order 4e2/h at zero charge density, which persisted over a plateau width roughly determined by the impurity density. Adam et al. [9] calculated the plateau width ΔVg analytically; they also found that the minimum conductivity ranges from 4e2/h to 20e2/h depending on the density of the charged impurities. Adapting the theory of semiconductor band tails [31] to this problem, they calculated the carrier density at which the minimum conductivity occurs (Vg,min), and predicted that σmin occurs not at the carrier density which neutralizes nimp, but rather at the carrier density at which the average impurity potential is zero [9]. This prediction suggests that the gate voltage of the minimum conductivity Vg,min would have an effective power law dependence on nimp, with an exponent not equal to one, but rather a function of the distance of the charged impurities to graphene [9]. The minimum conductivity problem was also treated by Cheianov et al. [27]. Their results are qualitatively consistent with Adam et al. [9], but they made no quantitative prediction on the magnitude or chargedimpurity-density dependence of the minimum conductivity.

30

Electrical Transport in Graphene

Fig. 2.4 The measured conductivity of graphene versus Vg for the pristine sample and three different doping concentrations (adapted with permission from [10]).

Chen et al. [10] performed an experiment in which graphene field-effect devices with clean surfaces were placed in an ultra-high vacuum environment and potassium atoms were absorbed on the surface of the graphene with the sample at low temperatures. The conductivity σ versus gate voltage Vg (proportional to carrier density n) were then measured in-situ for different potassium concentrations. The absorbed potassium atoms ionize and act as charged impurities. By measuring the dependence of σ versus Vg , Chen et al. extracted the scattering strength of the Dirac Fermions in graphene by charged impurities, and demostrated that (1) charged impurities give rise to linear dependence of σ versus n (see Fig. 2.4); (2) charged impurities are the major scatterers of high-quality graphene devices; and (3) the DC minimum conductivity of graphene is not a universal number, but, rather, determined by the details of local impurities (see Fig. 2.5). Further experiments have been done to verify the validity of the claim by Chen et al. [10]. The general idea of these follow up experiments is to alter the dielectric environment of the graphene sample so that the effects of charged impurities can be tuned. Ponomarenko et al. [32] covered a graphene sample with glycerol, ethanol, and water and put graphene onto various dielectric substrates such as SiO2, polymethylmethacrylate, spin-on glass, bismuth strontium calcium copper oxide, mica, and boron nitride. They found that the mobility of graphene increases modestly if it is put into an environment with stronger dielectric screening, but the magnitude of the increase could not match the dielectric constants of these substrate materials. This experiment points to the difficulties in increasing dielectric screening of low-dimensional materials without introducing other scattering sources: liquid dielectric such as water has an uncertain dielectric constant (it has been shown that water molecules absorbed on a graphite surface form a thin ice-like structure [33], with much less dielectric constant than liquid phase water), while high-k solid state dielectric materials generally have a surface plagued with dangling bonds and soft polar optical phonons that are activated at moderate temperatures; both of which are strong scattering sources that will lower the mobility of the graphene samples. Jang et al. [11] undertook a highly controlled in-situ experiment in an ultra-high vacuum environment, in which ice layers were absorbed on graphene

2.3 Resonant Scatterers

31

Fig. 2.5 The measured DC minimum conductivity of graphene with different potassium

densities (expressed in terms of 1/mobility for randomly distributed scatterers) in four in-situ experimental runs perform on a single graphene device. The dashed and solid lines are RPA theoretical predictions for charged impurities 1 nm and 0.3 nm away from the graphene surface. The clear variability of the minimum conductivity of graphene shows that the DC minimum conductivity of graphene does not have a universal value (adapted with permission from [10]).

field-effect devices with clean surfaces at low temperatures. The experiment found that, by changing the dielectric environment of the top surface of graphene from vacuum to ice: (1) the scattering strength of charged impurities was reduced; (2) the scattering strength of neutral “white noise” disorder was increased; and (3) the minimum conductivity of the device remained unchanged (see Fig. 2.6). These findings agree with theory qualitatively, and show unambiguously that charged impurity is indeed the dominant scattering source that produces the linear conductivity versus carrier density and limits the mobility of high-quality graphene samples.

2.3

Resonant Scatterers The concept of resonant scatterers was introduced to describe carrier transport in metals with impurities that have strongly attractive potentials which form bound states or quasi-localized states [34, 35]. It was first introduced in the context of carrier transport in graphene by Stauber et al. [5] and further developed by Titov et al. [36] and Weling et al. [37] as a type of inter-valley scatterers. Stauber et al. [5] proposed that vacancy defects in graphene are one kind of resonant scatterer and that they give rise to bound states at the Dirac point, which are also called mid-gap states. Mid-gap states are strong carrier scatterers with short-range interaction, which strongly perturb the system such that the Klein paradox [38] is not at work, and the scattering potential gives rise to a conductivity in graphene of the form that is similar to a non-relativistic electron gas [5, 39] n  pffiffiffiffiffi 2 σ mg ðnÞ ¼ C mg e ln πnR0 , (2.10) nd

32

Electrical Transport in Graphene

Fig. 2.6 The changes of mobility, short-range “white noise” disorder-induced conductivity, and the minimum conductivity of graphene as a function of the number of ice layers absorbed on the clean surface of a graphene field-effect device (adapted with permission from [11]).

where Cmg is a constant, nd is the vacancy defect density, and R0 is the effective radius of the vacancy (of the order of the bond length in graphene). The logarithmic term leads to slightly sublinear dependence of conductivity on charge density. Other resonance scatterers [36, 37, 40, 41] include the formation of various covalent bonds (such as C—C bonds, H—C bonds, and F—C bonds) on the surface of graphene. A number of experiments were reported in which resonance scatterers were unintentionally [42] and intentionally [6–8, 43, 44, 45]; [46, 47, 48] introduced into graphene samples. Chen et al. [6] performed an in-situ transport experiment in which the density of vacancy defects in graphene is controllably tuned during a single experimental run. In the experiment, transport properties of graphene devices were measured in an ultra-high vacuum environment while the devices were irradiated with low-energy inert gas ions to create vacancy defects. These defects are found to diminish the mobility of graphene and are four times as effective as the same amount of charged impurities (see Fig. 2.7), and produce a linear conductivity as a function of carrier density, consistent with the theory of resonant scatterers [5]. In another in-situ transport experiment performed in an ultra-high vacuum [44], atomic hydrogen was introduced to the graphene device at low temperature. It was found that the formation energy of the H—C bond on the basal plane for substrate-bounded single-layer graphene was small. The majority of absorbed hydrogen readily desorbs if the sample is warmed to room temperature. This experiment suggests that additional energy from electron beam irradiation or plasma bombardment

2.4 Corrugations of the Graphene Sheet

33

Fig. 2.7 The inverse mobility as a function of ion dosage (proportional to defect density) of

graphene samples under four different experimental runs. The predicted effects for the same density of charged impurities is plotted by the dashed line (adapted with permission from [6]).

might be crucial for ex-situ experiments to achieve hydrogenation of graphene at room temperature. (Here “ex-situ” means creating hydrogenation and measuring transport properties of the sample with two different apparatus, with likely exposure to air between the two experimental steps.) The ex-situ experimental findings of hydrogenated graphene and fluorinated graphene are similar. Temperature-dependent resistivity of hydrogenated graphene [7] and fluorinated graphene [8] are both consistent with variable range hopping, with the hopping temperature T0 tunable by carrier density. Magneto-transport of hydrogenated graphene [49] and fluorinated graphene [45] both show negative magneto resistance near charge neutrality. Chen et al. [50] studied the temperature-dependent resistivity of graphene with vacancy defects. After subtracting the effects of weak localization and an enhanced electron–electron interaction [51], a tunable Kondo effect is found in which resistivity at all carrier concentrations goes through a logarithmic divergence at around the Kondo temperature and saturates at the low temperature. The normalized resistivity to temperature curves at all carrier concentrations follows the universal Kondo behavior. Hong et al. [52] studied the low field magneto resistance of fluorinated graphene at high carrier density and extracted an anomalous electron dephasing time, which is attributed to the existence of magnetic impurities. McCreary et al. [53] studied the spin-polarized transport of the graphene spin valve. They found an anomalous V-shape in the non-local resistivity versus magnetic field curve around the zero magnetic field and attributed the observation to the existence of non-correlated magnetic impurities after hydrogenation of the graphene samples.

2.4

Corrugations of the Graphene Sheet Another proposal to explain the linear σ(n) has been the effect of geometric corrugation of graphene (i.e. “ripples”) present due to contact with a rough substrate [18, 54] or

34

Electrical Transport in Graphene

graphene’s “intrinsic” rippling [55], or as a result of proposed thermally activated outof-plane motion of the graphene sheet [56, 57], or the presence of local modification of the bonding in graphene [58]. Katsnelson and Geim [57] have suggested that ripples in graphene produce a conductivity of the form σ corr ðnÞ ¼ Ccorr en2H1 ,

(2.11)

where Ccorr is a constant which is proportional to ðR=zÞ2 , where R is the radius of the ripples, and z is the height of the ripples, and the exponent 2H is given by the dependence on the distance r of the height–heightDcorrelation function ofE a corrugated surface, e.g. gðr Þ / r 2H at small r, where gðr Þ ¼ ðhðr 0 þ r Þ  hðr 0 ÞÞ2 [18]. In this scenario, scattering by ripples would produce a linear σ(n) for 2H = 2, a situation that would, in principle, occur for equilibrium fluctuations of a flexible membrane in a planar confining potential [59]; or produce a constant σ(n) for 2H = 1, typical of the much more common case of a non-equilibrium structure with short-range correlations [60]; or produce a conductivity which has very weak density dependence, for 1< 2H < 2. The magnitude of the scattering from ripples can be estimated by Ccorr. Cullen et al. [19, 61] pointed out that the relationship, Ccorr / ðr=zÞ2 can be better formulated as C corr / ðqAðqÞÞ2 , where q is the wavevector of the Fourier spectrum of the corrugation, and A(q) is the Fourier amplitude which is a function of q. Cullen et al. argued that the maximum value of qA(q), readily obtained from the Fourier spectrum, should set an upper bound to the additional resistivity associated with the rippling of graphene [19, 61].

2.5

Phonons Longitudinal acoustic (LA) phonon scattering in graphene is expected to give rise to a resistivity independent of carrier density and linear in temperature, i.e.
 2 2 h π DA k B T ρLA ¼ 2 , (2.12) e 2h2 ρs v2s vF 2 where kB is the Boltzmann constant, ρs = 7.6  10–7 kg/m2 is the 2D mass density of graphene, vF = 106 m/s is the Fermi velocity, vs = 2.1  104 m/s is the sound velocity for LA phonons, and DA the acoustic deformation potential. For substrate-bound graphene devices, however, in addition to the LA phonon in graphene, the polar optical phonons of the SiO2 substrate are also expected to scatter electrons in graphene through surface polar phonon scattering [13, 62, 63, 64]. The two strongest surface optical phonon modes in SiO2 are calculated to have ħω  59 meV and 155 meV, with a ratio of coupling to the electrons of 1:6.5 [13, 63]. Surface polar phonon results in long-ranged potential, which gives rise to density-dependent resistivity in graphene, similar to charged impurity scattering. Specifically, in the simplest case, the electron–phonon matrix |Hkk0 |2 element is proportional to q–1, where q is the 1=2 scattering wavevector, and the resistivity is proportional to k 1 . However, F / Vg

2.5 Phonons

35

finite-q corrections to |Hkk0 |2 lead to a stronger dependence of ρB(Vg,T) on Vg [13], such that the resistivity arising from polar optical phonon scattering is
   1 6:5 α ρPO V g ; T ¼ C PO V g þ , (2.13) eð59meV Þ=kB T  1 eð155meV Þ=kB T  1 where CPO is a constant defining the strength of the scattering, α is the exponent on the density dependence, both of which can be experimentally determined; the terms in the brackets are the Bose–Einstein terms from the two strong polar optical phonons and the coupling ratio of 1:6.5 is determined by the oscillator strength and energy of these phonon modes [63]. Chen et al. [14] measured the temperature-dependent resistivity of pristine graphene devices in an ultra-high vacuum environment and between 16 K and 490 K. Two temperature-dependent components (ρA and ρB ) of the resistivity were extracted. ρA is density independent and linearly dependent on temperature, which dominates below 200 K; ρB has an activated behavior that is dependent on carrier density, which dominates above 200 K. By examining the details of Eqs. (2.12) and (2.13), Chen et al. confirmed that ρA ¼ ρLA and ρB ¼ ρPO . The findings point to an important conclusion that phonons in graphene are fairly ineffective in scattering charge carriers, which make graphene the most conductive material at room temperature at a technologically relevant density of 1012 cm–2; the choice of device substrates, on the other hand, is crucial in determining the room temperature performance of graphene devices (see Fig. 2.8). The drastic effect of substrate polar optical phonons as a scattering source for graphene and carbon nanotube devices has been independently confirmed by Raman spectroscopy on graphene [65] and carbon nanotube [64, 66] field-effect transistors

Fig. 2.8 Temperature dependence of the mobility in graphene and graphite. The

temperature-dependent mobilities of graphene Sample 1 (squares) and Sample 2 (triangles) at Vg = 14 V (n = 1012 cm–2) are compared with Kish graphite (solid black circles) and pyrolytic graphite (open black circles) [68]. The mobility limits in graphene determined in this work for scattering by LA phonons (dark solid line), remote interfacial phonon scattering (dark short-dashed line), and impurity scattering (light and dark long-dashed lines) are shown. The solid lines show the expected net mobility for each sample, according to Matthiessen’s rule (adapted with permission from [14]).

36

Electrical Transport in Graphene

under large bias, and by large bias transport measurements [16]. It is worth noting that the temperature-dependent resistivity of suspended graphene has shown an unexpected gate voltage dependent on a temperature below 200 K [15], pointing to other temperature-dependent mechanisms in suspended graphene, such as the interplay between flexural phonons in graphene and the strain of graphene from the application of a back-gate electric field [67]. To summarize, the resistivity of graphene can be decomposed into ρ ¼ ρci þ ρsr þ ρmg þ ρLA þ ρPO þ ρcorr ,

(2.14)

where the subscripts indicate the contributions due to charged impurities (ci), shortrange scatterers (sr), midgap states (mg), longitudinal acoustic phonons (LA), polar optical phonons (PO), and surface corrugations (corr). Understanding the relative strength of various scattering sources reveals the path for improving the mobility of substrate-bounded graphene and other low-dimensional electronic materials.

2.6

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27]

K. S. Novoselov et al., Science 306, 666 (2004). Y. W. Tan et al., Phys. Rev. Lett. 99, 246803 (2007). K. I. Bolotin et al., Solid State Commun. 146, 351 (2008). C. R. Dean et al., Nat. Nano. 5, 722 (2010). T. Stauber, N. M. R. Peres, and F. Guinea, Phys. Rev. B 76, 205423 (2007). J.-H. Chen et al., Phys. Rev. Lett. 102, 236805 (2009). D. C. Elias et al., Science 323, 610 (2009). F. Withers, M. Dubois, and A. K. Savchenko, Phys. Rev. B 82, 073403 (2010). S. Adam et al., Proc. Natl. Acad. Sci. USA 104, 18392 (2007). J.-H. Chen et al., Nat. Phys. 4, 377 (2008). C. Jang et al., Phys. Rev. Lett. 101, 146805 (2008). E. H. Hwang and S. D. Sarma, Phys. Rev. B 77, 115449 (2008). S. Fratini and F. Guinea, Phys. Rev. B 77, 195415 (2008). J.-H. Chen et al., Nat. Nanotechnol. 3, 206 (2008). K. I. Bolotin et al., Phys. Rev. Lett. 101, 096802 (2008). A. M. DaSilva et al., Phys. Rev. Lett. 104, 236601 (2010). M. I. Katsnelson and A. K. Geim, Philos. Trans. Royal Soc. London A 366, 195 (2008). M. Ishigami et al., Nano Lett. 7, 1643 (2007). W. G. Cullen et al., Phys. Rev. Lett. 105, 215504 (2010). N. H. Shon and T. Ando, J. Phys. Soc. Jpn 67, 2421 (1998). S. Das Sarma et al., Rev. Mod. Phys. 83, 407 (2011). K. S. Novoselov et al., Nature 438, 197 (2005). Y. W. Tan et al., Euro. Phys. J. Special Topics 148, 15 (2007). S. V. Morozov et al., Phys. Rev. Lett. 100, 016602 (2008). T. Ando, J. Phys. Soc. Jpn 75, 074716 (2006). K. Nomura and A. H. MacDonald, Phys. Rev. Lett. 98, 076602 (2007). V. V. Cheianov and V. I. Fal’ko, Phys. Rev. Lett. 97, 226801 (2006).

2.6 References

[28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65] [66] [67] [68]

37

E. H. Hwang, S. Adam, and S. Das Sarma, Phys. Rev. Lett. 98, 186806 (2007). S. Adam, E. H. Hwang, and S. Das Sarma, Physica E 40, 1022 (2008). D. S. Novikov, Appl. Phys. Lett. 91, 102102 (2007). P. Van Mieghem, Rev. Mod. Phys. 64, 755 (1992). L. A. Ponomarenko et al., Phys. Rev. Lett. 102, 206603 (2009). P. Cabrera Sanfelix et al., Surface Science 532–535, 166 (2003). J. C. Stoddart, N. H. March, and M. J. Stott, Phys. Rev. 186, 683 (1969). F. Mezei and G. Grüner, Phys. Rev. Lett. 29, 1465 (1972). M. Titov et al., Phys. Rev. Lett. 104, 076802 (2010). T. O. Wehling et al., Phys. Rev. Lett. 105, 056802 (2010). M. I. Katsnelson, K. S. Novoselov, and A. K. Geim, Nat. Phys. 2, 620 (2006). M. Hentschel and F. Guinea, Phys. Rev. B 76, 115407 (2007). D. W. Boukhvalov, M. I. Katsnelson, and A. I. Lichtenstein, Phys. Rev. B 77, 035427 (2008). A. Ferreira et al., Phys. Rev. B 83, 165402 (2011). Z. H. Ni et al., Nano Lett. 10, 3868 (2010). S. Ryu et al., Nano Lett. 8, 4597 (2008). J. Katoch et al., Phys. Rev. B 82, 081417 (2010). X. Hong et al., Phys. Rev. B 83, 085410 (2011). J. Guillemette et al., Phys. Rev. Lett. 110, 176801 (2013). J. Balakrishnan et al., Nat. Phys. 9, 284 (2013). A. A. Stabile et al., Phys. Rev. B 92, 121411 (2015). B. R. Matis et al., Phys. Rev. B 85, 195437 (2012). J.-H. Chen et al., Nat. Phys. 7, 535 (2011). J.-H. Chen et al., Nat. Phys. 8, 353 (2012). X. Hong et al., Phys. Rev. Lett. 108, 226602 (2012). K. M. McCreary et al., Phys. Rev. Lett. 109, 186604 (2012). E. Stolyarova et al., PNAS 104, 9209 (2007). V. Geringer et al., Phys. Rev. Lett. 102, 076102 (2009). A. Fasolino, J. H. Los, and M. I. Katsnelson, Nat. Mater. 6, 858 (2007). M. I. Katsnelson and A. K. Geim, Phil. Trans. R. Soc. A 366, 195 (2008). R. C. Thompson-Flagg, M. J. B. Moura, and M. Marder, Europhys. Lett. 85, 46002 (2009). H. Aranda-Espinoza and D. Lavallee, Europhys. Lett. 43, 355 (1998). J. L. Goldberg et al., Surf. Sci. Lett. 249, L285 (1991). J.-H. Chen et al., Solid State Commun. 149, 1080 (2009). K. Hess and P. Vogl, Solid State Commun. 30, 807 (1979). M. V. Fischetti, D. A. Neumayer, and E. A. Cartier, J. Appl. Phys. 90, 4587 (2001). V. Perebeinos et al., Nano Lett. 9, 312 (2009). M. Freitag et al., Nano Lett. 9, 1883 (2009). M. Steiner et al., Nat. Nano. 4, 320 (2009). H. Ochoa et al., Physica E 44, 963 (2012). K. Sugihara, K. Kawamura, and T. Tsuzuku, J. Phys. Soc. Jpn. 47, 1210 (1979).

3

Optical Properties of Graphene Feng Wang and Sufei Shi

The two-dimensional graphene crystal exhibits fascinating electrical, mechanical, and optical properties. In this brief review, we examine the distinctive behavior of light– matter interactions in graphene. These interactions are surprisingly strong, rendering optical spectroscopy a powerful tool for probing the unusual physics of graphene. Indeed, it was the strong optical absorption of single-layer graphene (with its absorbance of ~2.3%, as discussed below) that permitted the initial discovery of exfoliated monolayers by visual inspection under an optical microscope. In particular, the optical transitions in graphene can be controlled through electrical and magnetic fields, nanostructuring, and layer–layer interactions. Optical spectroscopy provides a powerful and versatile tool for probing the unique physics of various condensed matter systems and graphene in particular. The light– matter interaction in graphene is remarkably strong, which enables the observation of a broad range of unusual optical phenomena. It is interesting to note that the strong light–matter interaction allowed for seeing graphene, a monolayer of carbon atoms, with the naked eye under an optical microscope. This capability played a key role in the initial discovery of exfoliated graphene in 2004 [1]. The strong coupling between light and graphene originates from the delocalized π-orbital of graphene electrons [2, 3, 4]. Optical spectroscopy offers a sensitive probe of elementary excitations in graphene, ranging from electron–hole pairs and cyclotrons to phonons and plasmons. In addition, the light–matter interaction in graphene can be controlled through electrical gating [5, 6, 7], structural engineering [8, 9], and coupling between graphene layers [10]. This controllability provides exciting opportunities both for systematic understanding of graphene physics, and for technological applications in tunable photonics and optoelectronics. In the last few years, tremendous advances have been made in optical spectroscopy study of graphene. Broadly speaking, the studies can be divided into two categories based on the employed experimental techniques: (1) optical transmission/reflection spectroscopy that probes elastic optical process and (2) Raman spectroscopy that probes inelastic optical processes. Below we provide an introduction to optical transmission/ reflection spectroscopy of graphene, and Raman scattering will be covered in a separate chapter. Absorption/reflection spectroscopy is a basic but powerful optical technique. It yields direct information on the energy excitation spectrum of a system, and has long been an indispensable tool for atomic, molecular, and solid state physics. With the 38

3.1 Transitions in Electrically Gated Graphene

39

advent of modern lasers and synchrotron light sources, which provide tunable phonons ranging from the infrared and visible to ultra-violet and X-ray spectral ranges, absorption/reflection spectroscopy has becoming even more powerful in probing a wide range of elementary excitations [11]. In graphene research, absorption/reflection spectroscopy with infrared to visible and ultra-violet light has been most successful, because photons energy in this spectrum range match nicely with many important elementary excitations, including the interband [5, 6, 10, 12, 13, 14, 15, 16] and intraband [7, 17, 18, 19] electronic transitions, Landau level transitions [20, 21, 22], and plasmon excitations in graphene [8, 9]. Very often optical spectroscopy study of graphene is employed in a microscopy configuration so that one can readily probe exfoliated graphene samples of micrometer sizes. For large-area graphene, such as those produced through the chemical vapor deposition (CVD) method [23, 24], micro-spectroscopy is less of a necessity, but can still be valuable when examining spatial variation in optical properties in large-area samples. In this chapter, we will describe the unique tunable optical properties of graphene: (Section 3.1) Tunable Interband and Intraband Transitions in Electrically Gated Graphene; (Section 3.2) Landau Level Transitions in Graphene under a Magnetic Field; (Section 3.3) Plasmon Excitation in Graphene; (Section 3.4) Optical Transitions in Bilayer and Multilayer Graphene.

3.1

Tunable Interband and Intraband Transitions in Electrically Gated Graphene

3.1.1

Optical Absorption in Pristine Graphene The electronic structure of graphene can be well approximated by the linear Dirac cone close to the Fermi energy, where the conduction and valence bands touch at the Dirac point (Fig. 3.1(a)). Interband absorption [5] arises from direct optical transitions between the valence and conduction bands. At frequencies above the far-infrared region, these interband transitions typically define the optical response of graphene. Within the tight-binding model, the optical sheet conductivity from interband transitions can be readily calculated [25, 26, 27, 28, 29, 30, 31]. For pristine graphene at zero temperature, the optical conductivity in the linear dispersion regime of graphene is found to be independent of frequency. The corresponding “universal” conductance of graphene is 2 determined solely by fundamental constants and assumes the value of σ ðωÞ ¼ πe 2h [15, 25, 26, 27, 28, 29, 30, 31, 32, 33]. This conductivity corresponds to an absorbance of AðωÞ ¼ 4πc σ ðωÞ ¼ πα  2:29% [15, 32], where α denotes the fine structure constant. Figure 3.1(b) shows the frequency-dependent absorbance for three different graphene samples for photon energies between 0.5 and 1.2 eV. Over this spectral range, different samples show equivalent responses, not influenced by the detailed nature of the sample or its environment. Moreover, the absorbance is largely frequency independent, with an averaged value over the specified spectral range of A ¼ ð2:28  0:14Þ%, consistent with the universal value of πα = 2.29% [15, 34]. The slight departure from a completely

(b)

(a)

E

kx ky

2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0

pa (1 ± 0.1)

Sample 1 Sample 2 Sample 3

0.5

0.6

0.7

0.8

0.9

1.0

1.1

2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 1.2

Sheet conductivity (in units of pG0/4)

Optical Properties of Graphene

Absorbance (in units of pa)

40

Photon energy (eV) Fig. 3.1 Universal optical sheet conductivity of graphene. (a) Schematic of interband optical transitions in graphene. (b) The optical sheet conductivity (in units of πe2/2h, right scale) and the sheet absorbance (in units of πα, left scale) of three different samples of graphene over the photon energy range of 0.5 to 1.2 eV. The black horizontal line corresponds to the universal value of πα = 2.293% for the sheet absorbance (adapted with permission from [35]).

frequency-independent behavior, which becomes more pronounced at higher photon energies, is due to a deviation from the linear electronic band at high energy in graphene. The independence of the interband optical absorption on both frequency and the material properties (as encoded in the Fermi velocity vF ) can be understood in several ways. In general terms, it can be explained by dimensional analysis, since the graphene Hamiltonian describing the linear bands has no intrinsic energy scale with which to compare the photon energy. More directly, in terms of a calculation of the absorption in perturbation theory, we note the perfect cancellation of the ω and vF dependence
2 2in  the , the three important parameters – the square of the transition matrix element / v =ω F
 joint density of states / ω=v2F , and the photon energy ð/ ωÞ – the product of which defines the optical absorption.

3.1.2

Gate-Tunable Interband Optical Transitions Electrical doping can dramatically modify optical absorption in graphene. Because of the single-atom thickness of graphene and its linear dispersion with high Fermi velocity, the Fermi energy in graphene can be shifted by hundreds of meV through electrostatic gating. Such doping leads to a strong change in the interband absorption through Pauli blocking. As shown in Fig. 3.2(a), the interband transitions for photon energies below 2jϵ F j are suppressed, while those at energies above 2jϵ F j are unaffected. The optical response in graphene thus becomes highly tunable. Experimentally, the doping level can be tuned electrostatically in a field-effect transistor (FET) configuration by applying gating voltage across a SiO2 dielectric or an electrolyte layer. The doping concentration of the former structure is typically limited to e5  1012 cm2 by the breakdown of the oxide layer, while electrolyte gating can

3.1 Transitions in Electrically Gated Graphene

41

Fig. 3.2 Gate-induced interband transitions in graphene. (a) An illustration of interband transitions in hole-doped graphene. Optical transitions at photon energies greater than 2jϵ F j are allowed, while those at energies below 2jϵ F j are blocked. (b) The gate-induced change in transmission in hole-doped graphene as a function of gate voltage V g . The values of the gate voltage referenced to the voltage for charge neutrality,V g  V CNP , for the curves are –0.75 V, –1.75 V, –2.75 V, –3.5 V, from left to right (adapted with permission from [35]).

induce carrier concentrations as high as 1014 cm2 [36, 37]. Figure 3.2(b) displays the change in the optical transmission as a function of photon energy induced using gating with ionic liquid electrolyte. Increased optical transmission arising from Pauli blocking is observed up to a threshold characterized by a photon energy of 2jϵ F j. With increased carrier doping, this threshold energy shifts to a higher value, as expected. Using ionic liquid gating, one can reach a threshold energy above 1.7 eV, thus accessing the visible spectral range. Quantitatively, optical absorption in doped graphene can be described by h    i 2 ℏωþ2ϵ F F σ ðωÞ ¼ πe þ tanh ℏω2ϵ [29, 30, 31]. 4h tanh 4kB T 4k B T The tunability of the interband transitions in graphene offers new possibilities for probing fundamental physics and for various technological applications [5, 6, 38]. In addition, the underlying Pauli blocking process provides a direct approach to measuring the Fermi energy in graphene without the need of any electrical contacts. The threshold energy for increased optical absorption yields the value of 2jϵ F j from which the carrier 2 density n ¼ πℏE2Fv 2 can also be obtained. F

3.1.3

Gate-Tunable Intraband Optical Absorption Since the speed of light c is much higher than the Fermi velocity vF of graphene (c/vF ~ 300) [39, 40], direct absorption of a photon by an intraband optical transition does not satisfy momentum conservation. To conserve momentum, extra scattering with phonons or defects, as shown in Fig. 3.3(a), is required. The simplest description of the optical response of free carriers is then captured by a Drude model for the frequency-dependent sheet conductivity σ0 σ ðω Þ ¼ : (3.1) 1 þ iωτ Here σ0 and τ denote, respectively, the dc conductivity and the electron scattering time, and ω represents the (angular) frequency of the light [29, 33]. (Note that the

42

Optical Properties of Graphene

Fig. 3.3 Free-carrier absorption in graphene. (a) Schematic representation of the intraband absorption process. To conserve momentum, scattering with phonons or defects is needed. (b) Change in the optical sheet conductivity of graphene in the infrared range is controlled by electrostatic gating, which is at –1.0 V, –1.8 V, –2.2 V for the curves from left to right. In the far-infrared region, the conductivity is well described by the Drude model. In the mid- to near-infrared, Pauli blocking of interband transitions can be used to determine the Fermi energy. (c) The inferred Drude scattering rate as a function of the gating voltage. (d) Drude weight as a function of gating voltage. We see that the measured Drude weight from the free-carrier response 2 is suppressed with respect to the value predicted by Dinter ¼ eℏ ϵ F . (e) The integrated value of the change in optical sheet conductivity as a function of gating voltage. The change in the interband contribution is equal to that of the intraband parts, a consequence of the sum rule (adapted with permission from [35]).

optical absorbance at normal incidence is related to optical sheet conductivity by AðωÞ ¼ 4πc Re ½σ ðωÞ). To express the Drude conductivity in terms of microscopic material parameters, it is often convenient to introduce the Drude weight D ¼ πστ 0 , corresponding to the integrated oscillator strength of free-carrier absorption. In convenπ 2 tional semiconductors, or metals, the Drude weight is given by D ¼ πne m ¼ 2 ωp , with n and m denoting, respectively, the carrier density and carrier band mass, and ωp the plasma frequency [41]. For graphene with its massless electrons, the Drude weight pffiffiffiffiffi assumes a completely different form: D ¼ e2 vF πn [25, 28, 42, 43, 44]. Far-infrared spectroscopy can be used to probe the intraband absorption in large-area graphene grown by the chemical vapor deposition (CVD) method [23, 24]. In Fig. 3.3 (b), we show the free-carrier absorption spectra of graphene at different doping levels [37]. These spectra fit reasonably well using the Drude form, consistent with theoretical calculations based on the Kubo formalism that ignore many-body interactions [27, 29, 33]. The Drude scattering rates for different gate voltages are shown in Fig. 3.3(c). The different electron and hole scattering rates suggest that they experience different defect 2

3.2 Transitions in Graphene under a Magnetic Field

43

potentials. Near charge neutrality, we observe a Drude scattering rate of ~100 cm-1, which corresponds to a momentum relaxation lifetime of ~50 fs. From the Drude fit of the spectra in Fig. 3.3(b), we can obtain the free-carrier (or intraband) absorption Drude weight Dintra. This can be compared to the value pffiffiffiffiffi 2 Dinter ¼ e2 vF πn ¼ eℏ ϵ F , where the Fermi energy ϵ F is determined from inspection of Pauli blocking of the interband absorption (as discussed below). As can be seen in Fig. 3.3(d), the observed intraband transition Drude weight, Dintra, yields a somewhat lower value than Dinter. We note that the Drude weight of intraband absorption should equal the reduction of interband absorption oscillator strength, as required by the sum rule. This is confirmed in Fig. 3.3(e). The measured reduction in Dintra is, therefore, correlated to the imperfect Pauli blocking [6] of interband transitions at energies below 2ϵ F .

3.2

Landau Level Transitions in Graphene under a Magnetic Field Massless Dirac electrons in graphene have unique Landau level (LL) structures in a magnetic field. They give rise to an extraordinary “half-integer” quantum Hall effect [39, 40], the observation of which first ignited worldwide interest in graphene. In traditional 2D materials with parabolic band dispersions, the Landau level energies are equally spaced and are described by E n ¼ ðn þ 1=2Þℏωc ¼ ðn þ 1=2ÞℏeB=m , where ωc is the cyclotron frequency, n is the LL index, and m the electron effective mass. The linear dispersionp relation in graphene, however, leads to an unequally spaced ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 LL spectrum E n ¼ sgnðnÞ 2eℏvF Bjnj, where vF is the electron Fermi velocity, and n>0 or n20%Þ under the indenter tip of about 300 nm2, which is less than 0.05% of the total area of the specimen when accounting for nonlinear anisotropic elastic properties. We now consider the plausibility of obtaining a defect-free specimen suitable for mechanical testing. Graphene is a 2D material so it can contain only 0D and 1D defects. However, the exfoliated specimens are single crystals [15] so they can contain only 0D defects, of which atomic vacancies would have effect on strength. The  the largest  probability of an atomic vacancy is pv ¼ exp hf =kT , where hf is the enthalpy of formation of an atomic vacancy at constant pressure, k is the Boltzmann constant, and T is the temperature at which the graphene is formed [16]. Given hf ¼ 7:4 eV [17] for graphite, the probability of an atomic vacancy in the graphite prior to exfoliation of the graphene layers is exceedingly small. In agreement with this analysis, scanning tunneling microscopy (STM) studies of graphene (obtained from the same kish graphite source and prepared using the same methods as in our study) have found no point defects or dislocations over areas of hundreds of square nanometers [15]. We now return to the histogram in Fig. 4.3(b) of breaking force for both indenter tip radii. For pristine graphene, the force on the indenter at rupture is a function only of indenter tip radius, so the force at rupture would be constant for a given indenter. Thus, the distribution in the rupture force in Fig. 4.3(b) is due to instrument error alone, which is consistent with the observed normal distributions.

4.3 Non-linear and Anisotropic Response of Graphene

57

In order to determine the breaking strength from the measured fracture force, it is necessary to model the response of the membrane under tension to determine the maximum stress in the experiment. In our original work [6], we realized that a simple linear-elastic model was insufficient and added a (negative) quadratic term as an adjustable parameter in a radially isotropic description of graphene’s in-plane deformation. In this way, the intrinsic strength could be determined as the maximum point of the stress-strain curve. Finite element modeling was then used for comparative experiment; assuming fracture at the intrinsic strength allowed determination of the quadratic term, with intrinsic strength of 42 N=m (130 GPa). As will be discussed later and seen in Fig. 4.5, this constitutive model is still insufficient and the calculated strength is an overestimate. The accurate intrinsic strength turns out to be 34:5 N=m (103 GPa) with the appropriate model [5]. Subsequently, we expanded this analysis to go beyond the simple quadratic and isotropic description to create a constitutive model that describes the non-linear and anisotropic elastic response of graphene under arbitrary in-plane tensile deformations. This model, based on both a non-linear continuum description and ab initio calculations, allows for straightforward implementation in finite element modeling and provides excellent agreement with experiments.

4.3

Non-linear and Anisotropic Response of Graphene In principle, all crystalline materials have non-linear stress versus strain elastic responses for sufficiently large strains. Typically, however, defects activate inelastic deformation mechanisms (such as plastic deformation via dislocation motion or fracture via crack propagation) prior to non-linear elastic deformation. As we have seen above, however, for pristine graphene we must account for the non-linear elastic behavior. The work done on an homogeneously loaded specimen can also be written as dw ¼ σ dε, where w ¼ W=V 0 is the work per unit original volume V 0 , or the work density. Any such pair of stress and strain measures (here suitable for infinitesimally small deformations) is said to be work conjugate. The expression for work density can be readily generalized to dw ¼ σ ij dεij where the i and j indices refer to components of the second-rank stress and strain tensors. In this formulation, we consider only in-plane deformations within the sheet of graphene and neglect all out-of-plane bending contributions to the elastic strain energy density potential, so that the values of the i and j range over 1 to 2, where x1 is the zig-zag direction and x2 is the armchair direction. Figure 4.4(a) shows these coordinate directions with respect to graphene’s crystal lattice. The summation convention holds when the same index appears twice in an expression. In elastic materials under adiabatic conditions, this work is stored as elastic strain energy which can be recovered upon unloading. Thus an elastic strain energy density potential function, ψ, exists that is defined for infinitesimal deformations as dψ ¼ σ ij dεij where ψ has units of energy per unit original volume and represents the thermodynamic internal energy density. Rearranging this expression, the elastic energy density potential function determines the stress in an elastic material as a function of the deformation state by the expression σ ij ¼ ∂ψ=∂εij where all but one strain component

58

Mechanical Properties

Fig. 4.4 (a) Four-atom unit cell of graphene. (b) Lagrangian strain versus the second

Piola–Kirchhoff stress of the DFT results for uniaxial strain in the armchair and zig-zag directions, and equibiaxial strain with least-squares fittings of the fifth order Taylor series constitutive relation (adapted from [18]).

remains constant while taking the derivative. The fourth-rank elastic stiffness tensor is determined by the second derivative Cijkl ¼ ∂2 ψ=∂εij ∂εkl . For a linear elastic material, described by Hooke’s Law as σ ij ¼ Cijkl εkl , the elastic strain energy density potential is quadratic in strain as ψ ¼ 12 C ijkl εij εkl . The most common method to describe ψ is to choose a mathematical expression derived from a mechanistic model or phenomenological description, and then fit free parameters in the model to experiment [19, 20, 21, 22, 23, 24]. However, such a phenomenological approach is unnecessary for graphene. Instead, we expand the elastic strain energy density potential in a Taylor series in powers of strain as ψ¼

1 1 1 C ijkl ηij ηkl þ Cijklmn ηij ηkl ηmn þ C ijklmnop ηij ηkl ηmn ηop 2! 3! 4! 1 þ C ijklmnopqr ηij ηkl ηmn ηop ηqr þ , 5!

(4.3)

where we employ the Green–Lagrange strain measure, ηij , which together with its work P conjugate the second Piola–Kirchhoff stress measure, ij , has the frame invariance necessary to describe arbitrary finite deformations and rotations while normalizing all quantities relative to the undeformed configuration [18]. The Green–Lagrange  T reference  1 strain is calculated from η ¼ 2 F F  I , where F is the deformation gradient and I is the identity tensor. The second Piola–Kirchhoff stress is related to the more commonly used true or Cauchy stress measure, σ (e.g. force per unit current area or length) by σ ¼ 1J FΣFT , where J is the determinant of F. As before, the stress state is calculated P from ij ¼ ∂ψ=∂ηij and the tangent modulus (i.e. the slope of the stress–strain behavior) is calculated from the second derivative of ψ with respect to ηij . The zeroth- and firstorder terms of the Taylor series are neglected because they do not contribute to the second derivative and hence the elastic response. The quadratic term describes a linear elastic material and the higher-order terms introduce the non-linearity. As discussed below, a fifth-order expansion is necessary to accurately describe the behavior of graphene to breaking load, so that the stress–strain relationship becomes

4.3 Non-linear and Anisotropic Response of Graphene

X ij

¼ C ijkl ηkl þ þ

59

1 1 Cijklmn ηkl ηmn þ C ijklmnop ηkl ηmn ηop 2! 3!

1 C ijklmnopqr ηkl ηmn ηop ηqr : 4!

(4.4)

The number of non-zero and independent tensor components in the terms above is determined by the symmetry elements in the D6h point group that describes graphene’s crystal lattice, which consists of a six-fold rotational axis and six mirror planes. There are two independent components (that can be expressed as Young’s modulus and Poisson’s ratio) in the second-order elastic constants (SOEC) Cijkl , three in the third-order elastic constants (TOEC) C ijklmn , four in the fourth-order elastic constants (FOEC) C ijklmnop and five in the fifth-order elastic constants (FFOEC) Cijklmnopqr , for a total of 14 independent elastic constants [18]. The challenge of describing the elastic strain energy potential function thus reduces to finding suitable values for the 14 independent tensor components. The general expression for stress in Eq. (4.4) is lengthy and unwieldy after the summations are carried out. Therefore, we consider three simple deformation states: uniaxial strain in the zig-zag direction and armchair direction, and equibiaxial strain. In the case of uniaxial strain in the zig-zag direction, the graphene is strained with a tensile value of η11 in the x1 -direction while η22 ¼ η12 ¼ 0. The resulting stresses are Pzig 12 ¼ 0, Xzig 11

Xzig 22

1 1 1 ¼ C1111 η11 þ C 111111 η211 þ C11111111 η311 þ C1111111111 η411 , 2 6 24

(4.5)

1 1 1 ¼ C1122 η11 þ C 111122 η211 þ C11111122 η311 þ C1111111122 η411 , 2 6 24

(4.6)

P where zig 22 is referred to as the lateral constraint stress that is induced due to suppression of the Poisson contraction. In the case of uniaxial strain in the armchair direction, the graphene is strained with a tensile value of η22 in the x2 -direction while η11 ¼ η12 ¼ 0. The resulting stresses are Parm 12 ¼ 0, Xarm 11

Xarm 22

1 ¼ C1122 η22 þ ðC 111111  C 222222 þ C 111122 Þη222 2 1 1 þ ðC11111111 þ 2C11111122  C 22222222 Þη322 þ C1122222222 η422 , 12 24 1 1 1 ¼ C 1111 η22 þ C 222222 η222 þ C22222222 η322 þ C2222222222 η422 , 2 6 24

(4.7) (4.8)

P and arm 11 is the lateral constraint stress. We note that for infinitesimal strains for which all higher-order terms can be neglected, the elastic properties of graphene are linear and isotropic. However, the material exhibits an anisotropic elastic response when the symmetries within the crystal lattice are broken by finite deformations. For the case of equibiaxial strain for which η11 ¼ η22 ¼ η while η12 ¼ 0, the stresses P are bi 12 ¼ 0 and

60

Mechanical Properties

Xbi 11

¼

Xbi 22

1 ¼ ðC 1111 þC 1122 Þη þ ð2C111111 C222222 þ3C111122 Þη2 2
 1 3 1 C 11111111 þ 4C 11111122  C 22222222 þ 3C 11112222 η3 þ 6 2 2 1 ð3C 1111111111 þ10C1111111122 5C 1122222222 þ10C 1111112222 24 2C 2222222222 Þη4 , þ

(4.9)

for which the response is isotropic because the symmetry elements are not broken as a consequence of the applied deformation. All 14 elastic constants appear in Eqs. (4.5)–(4.9). In principle, they could be measured by devising suitable stress versus strain experiments with those deformation states, but such an approach is extremely challenging. Therefore, we calculate the elastic response using atomic scale calculations based on density functional theory (DFT). DFT is used to calculate the response of graphene at discrete values of the three deformation states discussed above, with the 14 elastic constants then determined by fitting the response with Eqs. (4.5)–(4.9) appropriately. While the primitive unit cell of graphene consists of two atoms, calculations are performed here with a four atom unit cell (which is outlined in Fig. 4.4(a)) for the convenience of the orthogonal coordinate system. Homogeneous deformation states are prescribed by specifying the positions of the boundaries of the unit cell, using periodic boundary conditions. The Vienna ab initio simulation package (VASP) is employed in the calculations. Since DFT performs 3D calculations, a large interplanar distance is chosen to eliminate interlayer interactions. The equilibrium lattice constant determined from the minimum energy position of atoms B within the unit cell is consistent with measured values within 0.5% [25, 26, 27]. The above deformation states are then calculated for a discrete number of strain increments by setting the displacement of the atoms along the edge of the unit cell and relaxing the inner atoms using a quasi-Newton method. The elastic strain energy density is given by the total energy within the supercell divided by the initial volume. The stresses in VASP are reported in the Cauchy stress measure which  is later T converted into P the second Piola–Kirchhoff stress measure by ¼ JF1 σ F1 . Likewise, the applied deformations are expressed in terms of the Green–Lagrange strain measure. The stress versus strain results of the DFT analyses for the five cases corresponding to Eqs. (4.5)–(4.9) are shown as discrete points in Fig. 4.4(b), with least-squares fits shown as solid lines. The 14 independent elastic constants given by this fitting are reported in Table 4.1. The fits are excellent, indicating that further terms beyond the fifth order are unnecessary, whereas lower-order versions of Eq. (4.4) (e.g. in our previous analysis and previous work demonstrating expansion to the cubic term [28]) are not sufficient to fully capture all five curves up to the maximum stress values. The 2D linear isotropic elastic Young’s modulus and the Poisson’s ratio under the infinitesimal strain assumption can be determined with the SOEC in Table 4.1 as  E ¼ C 21111  C21122 =C 1111 and ν ¼ C 1122 =C 1111 [18]. These values are 348 N/m and

4.4 Experimental Validation

61

Table 4.1 Fourteen independent elastic constants of the fifth-order Taylor series constitutive relation for graphene (adapted from [18]) SOEC (N/m)

TOEC (N/m)

ð2DÞ

C111111 ¼ 2817

ð2DÞ

C111122 ¼ 337:1

C1111 ¼ 358:1 C1122 ¼ 60:4

FOEC (N/m)

ð2DÞ

C11111111 ¼ 13416:2

ð2DÞ

C11111122 ¼ 759

ð2DÞ

C11112222 ¼ 2582:8

C222222 ¼ 2693:3

FFOEC (N/m)

ð2DÞ

C1111111111 ¼ 31383:8

ð2DÞ

C1111111122 ¼ 88:4

ð2DÞ

C1111112222 ¼ 12960:5

ð2DÞ

C1122222222 ¼ 13046:6

C22222222 ¼ 10358:9

ð2DÞ ð2DÞ ð2DÞ ð2DÞ ð2DÞ

C2222222222 ¼ 33446:7

0.169, respectively. These compare well with reported in-plane (basal plane) graphite values and graphene values [7, 29]. The TOEC and FFOEC components are all negative, such that graphene softens up to a point of maximum stress and after this point the stiffness is negative. The elastic properties in Fig. 4.4(b) are both non-linear and anisotropic. The nonlinearity becomes significant beyond about 5% strain and the anisotropy becomes evident after about 15% strain. Another notable feature is the presence of peak stress for each of the deformation states, which is interpreted as the intrinsic strength. The DFT results indicate that the intrinsic strength itself is a function of the stress measure and the deformation state. As can be seen in Fig. 4.4(b), when the stress is expressed in the second Piola–Kirchhoff stress measure, the peak stress is 31:4 N=m (93.7 GPa) and 29:5 N=m (88.1 GPa) for uniaxial strain in the zig-zag and the armchair directions, respectively, and is 33:1 N=m (98.8 GPa) for equibiaxial strain. When expressed in the more fundamental Cauchy stress, those peak values are 38:6 N=m (115 GPa), 34:7 N=m (104 GPa), and 33:1 N=m (98.8 GPa) for uniaxial strain in the zig-zag and the armchair directions and equibiaxial strain, respectively. Internal consistency of the higher-order continuum description is verified by computing deformation under uniaxial stress (as opposed to uniaxial strain). The results obtained by the continuum description correspond closely with those obtained directly by DFT calculation. The multiscale constitutive description is also compared to published ab initio results [29] and showed very good correspondence for strains even up to 30%.

4.4

Experimental Validation Validation of the non-linear anisotropic constitutive relation is achieved through implementation in ABAQUS finite element analysis (FEA) software to simulate the AFM and nanoindentation experiments. Since R
ɑ the contribution of bending stress is negligible compared to the in-plane stress, membrane elements are used to model the circular graphene sheet to utilize their zero bending rigidity. The indenter tip is modeled as a frictionless rigid sphere. The radius of the membrane and the indenter tip can easily be adjusted to match the various experiments performed. A zero displacement boundary condition is applied to the periphery of the membrane and the indenter incrementally

62

Mechanical Properties

Fig. 4.5 Force versus deflection response comparison between experiment, a linear elastic

model, and the non-linear elastic model presented here. (Inset: Cross-sectional view representation of the AFM indentation experiment simulation of a free-standing circular membrane in FEA.) (Adapted from [30].)

displaces the center of the membrane in a quasi-static manner at a constant velocity. Due to the large kinematic and material non-linearity, the equilibrium of the graphene membrane must be determined incrementally. The inset in Fig. 4.5 shows a representative cross section of the model during the simulation. As the simulation progresses and the stress approaches a maximum, the solution no longer numerically converges to equilibrium. This is a common issue in these types of problems and is combated by introducing a small amount of viscosity in the constitutive relation [31]. The negligible effect of the viscosity term on the solution in ABAQUS was verified by comparing the response under uniaxial and equibiaxial strain with ab initio calculations. The constitutive model is validated against experiments by comparing the forcedeflection response and the maximum force at fracture. Excellent consistency is seen in the force-deflection responses that is within the experimental error in Fig. 4.5 [30]. The maximum force beneath the indenter tip at failure is 1.818 and 2.988 μN (for the 16.5 and 27.5 nm radii tips, respectively) and is consistent within a 99% confidence interval of the experimental values, 1.8 and 2.9 μm. Figure 4.5 shows the material non-linearity at strains greater than 5%, and anisotropy at strains greater than 15% is well captured. These results provide evidence that the proposed non-linear anisotropic constitutive relation describes graphene’s deformation within the context of these nanoindentation experiments. Finally, we note that the linear elastic and non-linear models diverge for deflection beyond ~80 nm. Therefore, the linear constitutive relation initially used to fit the force–deflection response is accurate up to ~2/3 of the maximum deflection, but should not be used past this point. Figure 4.6 shows a representative contour map at varying indentation depths of the maximum in-plane stress in the area under the indenter tip where the largest stresses

4.5 Instabilities

63

Fig. 4.6 Magnified contour view of the max in-plane Cauchy stress directly beneath the indenter

tip from the FEA simulation immediately before the solver fails to converge to an equilibrium solution (adapted from [30]).

occur. Here, the anisotropy can readily be seen from the six-fold symmetry in the contour, corresponding to graphene’s hexagonal crystal lattice. It also shows that the stress state in the material point immediately under the indenter tip is less than the intrinsic value under equibiaxial strain. The material encircling that material point, however, is at the intrinsic stress, confirming that the material immediately under the indenter tip is deformed into a regime of negative tangent modulus.

4.5

Instabilities No defects exist in pristine graphene at which material failure can be initiated. Hence rupture will initiate due to one of at least three possible different classes of deformation instability. Elastic instability [32] occurs when the tangent modulus (the instantaneous slope of the stress versus strain response) approaches zero and becomes negative. For a uniformly strained sample, further load increments induce locally unbounded strain energy density leading to spontaneous fragmentation. This potential instability can be seen for deflections of 115 and 118 nm in Fig. 4.6 where the material has a negative tangent modulus [32]. Soft phonon mode instability is akin to a phase transformation and occurs when the frequency for a vibrational mode reaches zero, leading to a diverging vibrational amplitude at finite temperature. In 2010 Marianetti and Yevick predicted via DFT calculations that soft phonon mode instability can induce an accelerated softening that reduces the intrinsic strength at least in homogeneously deformed graphene [33]. The non-linear anisotropic elastic constitutive model discussed above does not account for the possibility of soft phonon mode instability, but could be generalized to do so.

64

Mechanical Properties

Elastic structural instability occurs when the potential energy of a structure (defined by considering the material, configuration of materials, and the applied loading) loses positive definiteness (e.g. [34]). Thus the structure can not store additional elastic energy as a consequence of additional load increment, and structural collapse ensues. In the simulations of Fig. 4.6, the local instability at the membrane center is constrained by a surrounding “ring” of graphene at stress below the peak. The good agreement between experiments and the maximum deflection and load predicted by structural instability indicate that this is likely to be the case in experiments, although more precise measurements are required for definitive confirmation. If so, this indicates that graphene can potentially be studied (e.g. through very local optical probes) in a locally unstable state.

4.6

Defective Graphene Utilizing graphene (or any 2D material) in practical applications requires the development of methods to scale the material to a usable size. Defects are an inevitable consequence of increased scale, so it is important to understand the effects of the defects that can exist (0D and 1D defects). The multiscale model described above provides a rigorous machinery by which experiments on defective graphene can be analyzed to extract elastic stiffness and breaking strength.

4.6.1

Point Defects The effects of point defects can be studied by oxidation of membranes using a weak oxygen plasma [35, 36, 37, 38]. The evolution of the defect type and density with plasma exposure time is determined through Raman spectroscopy: the ratio of the D peak to the D0 peak intensities discriminates between sp3-type defects from oxygen   ad-atoms IIððDD0ÞÞ > 7 and vacancy-type defects IIððDD0ÞÞ < 7 [39]. The defect density, quantified by average distance between defects ðLD Þ, is determined from the D peak – G peak intensity ratio,

I ðDÞ I ðGÞ

[38, 40].

Figure 4.7 shows the extracted value of E2D, as determined from analysis of loaddeflection curves using Eq. (4.1), for various plasma exposure times. E2D remains constant (within experimental uncertainty) over the entire sp3-type defect region, indicating that these defects do not appreciably change the stiffness; this is expected in the case of sparse defects, since the elastic stiffness averages over the bond stiffness of the whole sample. Once the exposure time crosses over into the vacancy-type defect region, E2D decreases with increasing defect density to a minimum of ~30% of the initial value, after which the membranes were not stable. As expected, the breaking load shows greater sensitivity to defects regardless of type. Þ An approximately 50% smaller breaking load is observed by the time the II ððD GÞ ratio reached ~1.0, with LD ~ 7 nm, or ~0.07% defect concentration. Because the elastic

4.6 Defective Graphene

65

Fig. 4.7 The upper half shows the change in (a) 2D Young’s modulus and (b) breaking load as a

function of increased oxygen plasma exposure time. The lower half shows the correlation of the Þ I ð2DÞ change in the Raman intensity ratios, II ððD GÞ and I ðGÞ , with the mechanical properties and oxygen plasma exposure time (adapted from [41]).

properties are dominated by the 99.93% of the graphene that is pristine, the non-linear model is still valid for modeling of the experiments, and can be used to generate a plot of maximum stress versus applied load (similar to the plot seen in Fig. 4.9 which is discussed in more depth later). We find that the two-fold decrease in breaking load corresponds to a decrease in strength of only ~14%. Once the graphene crosses into the vacancy-type region, the linear elastic modulus changes, so the non-linear model is no longer valid, and more detailed modeling will be required to analyze experimental data in this regime.

4.6.2

Line Defects Graphene produced by a scalable method such as chemical vapor deposition (CVD) possesses grain boundaries that can also reduce the breaking strength. Theoretical studies have argued that these grain boundaries can be nearly as strong as the pristine lattice, with the strength varying with tilt angle and arrangement of defects [42]. We have studied the mechanical behavior of two different types of CVD graphene: continuous films with small (1–5 μm) grains (SG), and isolated single crystals with large (>100 μm) grains (LG). These were transferred to Si substrates

66

Mechanical Properties

with 1 μm diameter holes to make suspended membranes, using transfer techniques that did not damage the graphene. Indentation tests were then performed on multiple membranes as described above. Figures 4.8(a), (b) and (c) show histograms of the derived values of E2D for exfoliated LG and SG graphene. No statistically significant

Fig. 4.8 Histograms showing and comparing the statistical distribution of 2D Young’s modulus

(N/m) between (a) mechanically exfoliated (ME), (b) large grain (LG), and (c) small grain (SG) graphene, and of the fracture load (nN) between (d) mechanically exfoliated (ME), (e) large grain (LG), (f) small grain (SG) graphene (adapted from [5]).

4.6 Defective Graphene

67

Fig. 4.9 Maximum 2D stress (N/m) as a function of indenter load (nN) for two different

indenter tip radii, 26 and 38 nm. The average point of failure is indicated for exfoliated, large grain (LG), small grain (SG), and indentation directly on the grain boundary for comparison

difference is seen in the elastic stiffness. Just as in the case of point defects, a dilute fraction of defective graphene in grain boundaries does not change elastic properties. Figures 4.8(d), (e) and (f) show histograms of the breaking force. LG graphene shows the same breaking force as exfoliated graphene. Therefore, we can conclude that the spacing between point defects (if any) in LG graphene must be much larger than the value of 7 nm that produced a 50% decrease in breaking strength in the previous study (Fig. 4.7). On the other hand, the SG graphene shows a ~25% decrease in the mean breaking force and a broader distribution of strengths. This distribution is presumably due to both random location of grain boundaries relative to the indentation point, and variation in the individual grain boundary strengths. In order to more precisely determine the properties of individual grain boundaries, we have performed AFM nanoindentation tests directly on grain boundaries identified by dark-field TEM imaging. Confirming the statistical results of the nanoindenter test, these show a ~30% decrease in average breaking force. However, it is important to note that a small fraction of grain boundaries is not “well-stitched” but instead consists of overlapping sheets. As can be expected, these show virtually no breaking strength. As in the study of point defects, the non-linear model can be used to extract the breaking strength in each of these cases. The ability to model experiments by FEA is particularly useful in this case, where the nanoindenter and AFM used tips with different radii of 26 nm and 38 nm, respectively. The computed maximum stress versus load for both tips is plotted in Fig. 4.9. The maximum strength given by the average breaking force of the SG graphene is only ~5% smaller than the intrinsic strength, and the grain boundary strength is only ~10% smaller. Thus we conclude that polycrystalline CVD graphene can be nearly as strong as single-crystal, provided that “overlap” grain boundaries can be eliminated.

68

Mechanical Properties

4.7

Conclusion We have developed an experimental regime to fabricate and experimentally test the mechanical properties of monatomically thin specimens of graphene. The experiments and concomitant analysis demonstrate that pristine graphene achieves its intrinsic strength, making graphene the strongest material yet characterized. From a materials perspective, the ability to isolate via mechanical exfoliation and also to grow graphene via scalable CVD methods creates potential avenues for applications. One such opportunity is to incorporate graphene as a strengthening element into composite materials based upon polymer, metal, or ceramic matrices. The high strength of graphene also is a requisite for incorporation of graphene into many electrical or optical applications. We have carried out an almost identical set of experiments and computations with suspended 2D MoS2 rather than graphene [26]. The close correspondence between the predicted behavior and experiments suggests that this framework for deriving a higherorder mulitscale anisotropic non-linear elastic continuum description would be valid for any 2D crystal. Multiple length scale modeling is required to obtain a predictive understanding of mechanical properties. The validated higher-order elastic constitutive model for graphene and MoS2 provides guidance for the development of rigorous models for 3D crystals as well. Graphene also provides the opportunity to study the initiation, suppression, and interactions of various instabilities in the absence of material defects. In the experiments with graphene and MoS2, the elastic and soft phonon mode instabilities are apparently suppressed, due potentially to a number of factors, including: out-of plane bending deformation; contact with indenter tip; presence of strain gradients; finite temperatures, chemical interactions with indenter tip, among others. Several avenues of future research would be of interest. The present constitutive model accounts only for in-plane deformation. While appropriate for the indentation experimental configuration, other applications in which large curvatures develop may require accounting for the strain energy density due to bending, and especially the coupling between the in-plane and bending deformations [43]. The finite deformations that can be achieved in graphene present the opportunity to study coupling between mechanical and other properties, such as catalysis.

Acknowledgement Support by the NSF–CMMI (Award Numbers: 1437450 and 1363093) is acknowledged.

4.8

References [1] C. Pedretti, The Codex Atlanticus of Leonardo Da Vinci: A Catalogue of Its Newly Restored Sheets, Johnson Reprint Corporation, 1978. [2] G. Galileo, Discorsi e dimostrazioni matematiche, intorno a due nuove scienze attenenti alla mecanica & i movimenti locali, 1638. Translated as Dialogues Concerning Two New Sciences, translated by H. Crew and A. de Salvio. Macmillan, 1914.

4.8 References

69

[3] E. Mariotte, Traité du mouvement des eaux et des autres corps fluides. C.-Jombert, 1718. [4] A. A. Griffith, “The phenomena of rupture and flow in solids,” Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol. 221, pp. 163–98, 1921. [5] G. H. Lee, R. C. Cooper, S. J. An, et al., “High-strength chemical-vapor-deposited graphene and grain boundaries,” Science, vol. 340, pp. 1073–6, 2013. [6] C. Lee, X. Wei, J. W. Kysar, and J. Hone, “Measurement of the elastic properties and intrinsic strength of monolayer graphene,” Science, vol. 321, pp. 385–8, 2008. [7] O. L. Blakslee, D. G. Proctor, E. J. Seldin, G. B. Spence, and T. Weng, “Elastic constants of compression-annealed pyrolytic graphite,” Journal of Applied Physics, vol. 41, pp. 3373–82, 1970. [8] M. Yu, B. Files, S. Arepalli, and R. Ruoff, “Tensile loading of ropes of single wall carbon nanotubes and their mechanical properties,” Physical Review Letters, vol. 84, pp. 5552–5, 2000. [9] W. Weibull, “A statistical theory of the strength of materials,” Proceedings of the Royal Swedish Institute of Engineering Research, vol. 151, 1939. [10] W. Weibull, “Wide applicability,” Journal of Applied Mechanics, vol. 18, 1951. [11] B. Lawn, Fracture of Brittle Solids, Cambridge University Press, pp. 1–178, 1993. [12] A. H. Barber, I. Kaplan-Ashiri, S. R. Cohen, R. Tenne, and H. D. Wagner, “Stochastic strength of nanotubes: An appraisal of available data,” Composites Science and Technology, vol. 65, pp. 2380–4, 2005. [13] N. M. Pugno, “On the strength of the carbon nanotube-based space elevator cable: From nanomechanics to megamechanics,” Journal of Physics: Condensed Matterials, vol. 18, pp. S1971–S1990, 2006. [14] N. M. Pugno and R. S. Ruoff, “Nanoscale Weibull statistics,” Journal of Applied Physics, vol. 99, pp. 0–4, 2006. [15] E. Stolyarova, K. T. Rim, S. Ryu, et al., “High-resolution scanning tunneling microscopy imaging of mesoscopic graphene sheets on an insulating surface,” Proceedings of the National Academy of Sciences of the United States of America, vol. 104, pp. 9209–12, 2007. [16] F. A. McClintock and A. S. Argon, Mechanical Behavior of Materials, Addison-Wesley, 1966. [17] A. A. El-Barbary, R. H. Telling, C. P. Ewels, M. I. Heggie, and P. R. Briddon, “Structure and energetics of the vacancy in graphite,” Physical Review B, vol. 68, 144107, 2003. [18] X. D. Wei, B. Fragneaud, C. A. Marianetti, and J. W. Kysar, “Nonlinear elastic behavior of graphene: Ab initio calculations to continuum description,” Physical Review B, vol. 80, pp. 1–8, 2009. [19] E. M. Arruda and M. C. Boyce, “A three-dimensional constitutive model for the large stretch behavior of rubber elastic materials,” Journal of the Mechanics and Physics of Solids, vol. 41, pp. 389–412, 1993. [20] R. W. Ogden, “Large deformation isotropic elasticity: On the correlation of theory and experiment for compressible rubberlike solids,” Proceedings of the Royal Society of London A: Mathematical and Physical Sciences, vol. 328, pp. 567–83, 1972. [21] M. Mooney, “A theory of large elastic deformation,” Journal of Applied Physics, vol. 11, pp. 582–92, 1940. [22] G. F. Smith and R. S. Rivlin, “The strain–energy function for anisotropic elastic materials,” Transactions of the American Mathematical Society, Vol. 88, pp. 175–93, 1958. [23] P. Tong and Y.C. Fung, “The stress–strain relationship for the skin,” Journal of Biomechanics, vol. 9, pp. 649–657, 1976.

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[24] S. Kumar and D. M. Parks, “A comprehensive lattice-stability limit surface for graphene,” cond-mat.mtrl-sci arXiv:1503.03944v2, March 16, 2014. [25] Y. Baskin and L. Meyer, “Lattice constants of graphite at low temperatures,” Physical Review, vol. 100, 544, 1955. [26] R. C. Cooper, C. Lee, C. A. Marianetti, X. Wei, J. Hone, and J. W. Kysar, “Nonlinear elastic behavior of two-dimensional molybdenum disulfide,” Physical Review B – Condensed Matter and Materials Physics, vol. 87, 035423, 2013. [27] R. C. Cooper, J. W. Kysar, and C. A. Marianetti, “Comment on ‘ideal strength and phonon instability in single-layer MoS2’,” Physical Review B – Condensed Matter and Materials Physics, vol. 90, pp. 1–3, 2014. [28] E. Cadelano, P. L. Palla, S. Giordano, and L. Colombo, “Nonlinear elasticity of monolayer graphene,” Physical Review Letters, vol. 102, pp. 1–4, 2009. [29] F. Liu, P. Ming, and J. Li, “Ab initio calculation of ideal strength and phonon instability of graphene under tension,” Physical Review B – Condensed Matter and Materials Physics, vol. 76, pp. 1–7, 2007. [30] X. Wei and J. W. Kysar, “Experimental validation of multiscale modeling of indentation of suspended circular graphene membranes,” International Journal of Solids and Structures, vol. 49, pp. 3201–9, 2012. [31] Y. F. Gao and A. F. Bower, “A simple technique for avoiding convergence problems in finite element simulations of crack nucleation and growth on cohesive interfaces,” Modelling and Simulation in Materials Science and Engineering, vol. 12, pp. 453–63, 2004. [32] S. Kumar and D. M. Parks, “On the hyperelastic softening and elastic instabilities in graphene,” Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 471, 2014. [33] C. A. Marianetti and H. G. Yevick, “Failure mechanisms of graphene under tension,” Physical Review Letters, vol. 105, pp. 1–4, 2010. [34] T. K. Caughey and R. T. Shield, “Instability and the energy criterion for continuous systems,” Journal of Applied Mathematics and Physics vol. 19, pp. 485–92, 1968. [35] S. P. Koenig, L. Wang, J. Pellegrino, and J. S. Bunch, “Selective molecular sieving through porous graphene,” Nature Nanotechnology, vol. 7, pp. 728–32, 2012. [36] L. G. Cançado, A. Jorio, E. H. M. Ferreira, et al., “Quantifying defects in graphene via Raman spectroscopy at different excitation energies,” Nano Letters, vol. 11, pp. 3190–6, 2011. [37] D. C. Kim, D.-Y. Jeon, H.-J. Chung, Y. Woo, J. K. Shin, and S. Seo, “The structural and electrical evolution of graphene by oxygen plasma-induced disorder,” Nanotechnology, vol. 20, 375703, 2009. [38] M. M. Lucchese, F. Stavale, E. H. M. Ferreira, et al., “Quantifying ion-induced defects and Raman relaxation length in graphene,” Carbon, vol. 48, pp. 1592–7, 2010. [39] A. Eckmann, A. Felten, A. Mishchenko, et al., “Probing the nature of defects in graphene by Raman spectroscopy,” Nano Letters, vol. 12, pp. 3925–30, 2012. [40] P. T. Araujo, M. Terrones, and M. S. Dresselhaus, “Defects and impurities in graphene-like materials,” Materials Today, vol. 15, pp. 98–109, 2012. [41] A. Zandiatashbar, G. H. Lee, S. J. An, et al., “Effect of defects on the intrinsic strength and stiffness of graphene,” Nature Communications, vol. 5, p. 3186, 2014. [42] R. Grantab, V. B. Shenoy, and R. S. Ruoff, “Anomalous strength characteristics of tilt grain boundaries in graphene,” Science, vol. 1456, pp. 10–13, 2010. [43] Q. Lu and R. Huang, “Nonlinear mechanics of single-atomic-layer graphene sheets,” International Journal of Applied Mechanics, vol. 01, pp. 443–67, 2009.

5

Vibrations in Graphene Ado Jorio, Luiz Gustavo Cançado, and Leandro M. Malard

5.1

Structure and Vibrations of Monolayer Graphene The organization of atoms in a crystalline structure imposes symmetry-related constrains on the net motion, which influences most properties of solid state systems. In the group theory framework [1], the hexagonal symmetry of monolayer graphene on isotropic medium belongs to the space group P6/mmm (D6h1), with two inequivalent C atoms in the unit cell [1, 2, 3, 4]. The unit cell is illustrated in the left sketch of ! ! Fig. 5.1(a), defined by the vectors a1 and a2 . Strain can break the hexagonal symmetry, as indicated by the other sketches in Figs. 5.1(a) and (b), with implications on the vibrational structure.

5.1.1

Graphene in an Isotropic Medium The graphene phonon dispersion is plotted in Fig. 5.2(a). The two unit-cell atoms moving in the three-dimensional space generate six phonon branches – three acoustic (A), with in-phase displacements of the two unit-cell atoms, and three optical (O), with out-of-phase displacements of the two unit-cell atoms. The phonon eigenvectors are described by the acoustic and optical unit-cell displacements multiplied by a wavevector q, which defines the phase modulation along the unit cells in the crystalline lattice. Figure 5.2(a) shows the phonon frequencies with q along the Γ–K direction in the hexagonal Brillouin zone. Figure 5.2(b) shows the 12 phonon eigenvectors at the Γ and K points. The atomic displacements in the graphene plane (i for in-plane) can be longitudinal (L) or transversal (T) with respect to the phonon wavevector direction. The atomic displacements perpendicular to the graphene plane (o for out-of-plane) are transversal (T) phonons, and they generally exhibit lower frequencies because the out-of-plane restoring forces are much weaker than the in-plane ones. At the Γ point (q = 0), the group of wavevector is isomorphic to the point group D6h, which has the special property of being homogeneous for in-plane lattice distortions [1, 2, 4]. Consequently, the in-plane longitudinal optical (iLO) and the in-plane transversal optical (iTO) phonons are degenerate, representing the only first-order Raman active mode of graphene, named G band, belonging to the double-degenerate irreducible representation E2g, appearing at 1584 cm–1 [4, 5, 6, 7]. The relatively high frequency of this optical phonon (~0.2 eV) allows the use of Raman spectroscopy to probe small environmental perturbations, including variations in strain [8], doping [9], and 71

72

Vibrations in Graphene

(a)

Black P

Graphene

a2 a1

a2

a1

D6h1

y

D2h7

D2h19

x z

Blue P

Graphene

(b)

a1

a2

D6h1

a1

a2

D3d3

Fig. 5.1 Atomic structure of graphene and the effect of lattice distortion [3]. (a) Starting from

graphene (left) and inducing a compressive strain along the ^x direction leads to a phase transition to the space group D2h19 (middle), and subsequently to the space group D2h7 (right), where light and dark gray atoms are at different levels along the ^z direction, which is the structure of black phosphorene. (b) Starting from graphene (left), a bi-axial compressive homogeneous strain leads to a phase transition to D3d3 space group (right), again with light and dark gray atoms at different levels along the ^z direction, which is the structure of blue phosphorene.

temperature [10]. The out-of-plane transversal optical mode (oTO) at the Γ point is active in infrared absorption spectroscopy [2]. However, the polar character of this vibrational mode is very weak because graphene is homoatomic, and infrared spectroscopy is mostly used to measure the vibrations from functionalization agents and other contaminants in graphene [11, 12], besides direct electronic effects near the K point [13, 14]. The lines in the graphene phonon dispersion plot in Fig. 5.2(a) are theoretical calculations [15], and the symbols are experimental data from inelastic light-scattering [16]. Force constant models are generally used for describing phonon dispersion in materials, but they rely on a limited number of atomic neighbor interactions and do not account for electron–vibration interactions [6]. Graphene, however, has long-range and non-adiabatic effects that had to be considered before an accurate description of the phonon dispersion could be achieved [15, 17, 18, 19, 20, 21]. Experimentally, due to momentum conservation requirements, most materials rely on either neutron or X-ray inelastic scattering to probe phonon dispersion [22, 23], necessary for achieving scattering by phonons in the interior of the Brillouin zone. Graphene, however, has a combination of factors that generate unique effects making it possible to probe the phonon dispersion with inelastic light-scattering in the visible

5.1 Structure and Vibrations of Monolayer Graphene

(a)

(b)

Γ iLA E1u D6h

Γ iTA E1u D6h

Γ iTO E2g D6h Γ oTO B2g D6h

K oTA E’’ D3h

K iLO E’ D3h

Γ oTA A2u D6h

Γ iLO E2g D6h

K iLA E’ D3h

K iTA A’2 D3h

K iTO A’1 D3h

K oTO E’’ D3h

73

(c)

Fig. 5.2 (a) Phonon dispersion of graphene [1, 16]. The solid lines stand for the phonon frequencies

(branch assignment near each curve) as a function of wavevector q in the Γ–K direction, taken from [15]. ◊ and □ symbols correspond to the frequencies of the peaks marked by * in the Raman spectra shown in panel (c) [29]. The ● symbols are experimental data obtained from [33], each data point being obtained using a different value of excitation laser energy. The ? and ■ symbols are experimental data obtained for intravalley and intervalley double-resonance Raman processes in monolayer graphene, respectively [31]. The ,, ⊕ and r symbols are experimental data obtained from [27, 34, 35, 36]. (b) Eigenvectors for the phonons at the high-symmetry Γ and K (K0 ) points of the Brillouin zone. Each of these 12 modes is labeled and their atom displacements are indicated. i/o stands for in-plane/out-of-plane; T/L stands for transversal/longitudinal; A/O stands for acoustic/optical. The other symbols are symmetry assignments according to group theory [2, 6]. (c) Raman spectra from twisted bilayer graphene (tBLG) [29]. The peaks marked with * are activated by the superstructure modulation. The peaks at 303 cm–1 and 512 cm–1 come from the Si substrate where the sample is sitting. The peak at 1584 cm–1 is the G band.

range [16, 24, 25, 26, 27, 28, 29]. The filled symbols in the phonon dispersion plotted in Fig. 5.2(a) come from the so-called double-resonance Raman scattering effect [24, 30], where q 6¼ 0 phonons that generate resonant electron scattering between different points in the K and K0 valleys can be probed [31]. Changing the excitation laser energy changes the double-resonance selected q, and different places in the phonon Brillouin zone can be probed [24]. Another interesting effect is the stacking of two graphene layers with a mismatch rotation angle θ between the lattice structures in each layer,

74

Vibrations in Graphene

building the so-called twisted bilayer graphene (tBLG) [16, 27, 28, 29]. The mismatch angle θ generates a superstructure where a potential modulation activates phonons with the modulation wavevector q(θ) [16, 27, 29, 32]. Figure 5.2(c) shows the Raman spectra from several tBLGs with different mismatch angles θ [29]. The peaks marked with * are Raman activated peaks from the interior of the Brillouin zone, and their frequencies are plotted in Fig. 5.2(a) with ◊ and □ symbols, allowing the use of Raman spectroscopy to measure the phonon dispersion in graphene. In the phonon dispersion of graphene shown in Fig. 5.2(a), notice the appearance of a new branch (dashed line) labeled iTO0 , fitting the □ and ● data. This branch is related to an interlayer vibration present only in many-layers graphene. N-layers graphene (N = number of layers) has 2N atoms in the unit cell, and they will be discussed in Section 5.2.

5.1.2

Graphene under Strain Small modifications in the graphene structure may lead to different symmetry groups, sometimes related to other 2D materials [3]. For example, the structure of monolayer black or blue phosphorus (phosphorene) [37], silicone, and germanene [38] can be achieved by a distortion of the graphene lattice. Figure 5.1(a) shows that starting from the D6h1 graphene a uniaxial compression induces a phase transition to subgroup D2h11 of strained graphene. The hexagonal symmetry is lost, and the resulting structure is orthorhombic with all atoms in the same plane. Such a strain breaks the hexagonal symmetry, and two G bands are observed in the Raman spectra [8]. In order to accommodate such strain in the orthorhombic lattice, a possible distortion is to displace lines of atoms perpendicular to the plane periodically up and down [3]. These zig-zag lines of atoms displaced periodically perpendicular to the plane generate the structure of black P, which belongs to the D2h7 subgroup. Otherwise, starting from the D6h1 graphene symmetry (Fig. 5.1(b)) and applying an isotropic strain, changes are expected in the C—C bond distances, and such changes can be measured through changes in the G band frequency [6]. Relatively small homogeneous strain does not affect the symmetry of the system. For higher pressures, in order to accommodate the anisotropic strain, one possible distortion is to displace the atoms periodically up and down, perpendicularly to the plane, generating a trigonal arrangement of atoms. Such a distortion generates the structure of blue P (also of silicene, germanene, and stanene), which belongs to the D3d3 subgroup [3]. Group theory has been used to gain insights into the symmetry aspects of graphene, black P, blue P, silicene, germanene, and stanene [3]. This analysis can be used to distinguish the different systems, and for a fast characterization of in-plane heterostructures that can be built to customize certain desired properties in these new materials.

5.2

Many-Layers Graphene and the Interlayer Vibrations in 2D Systems

5.2.1

In-Layer Vibrations The symmetries for N-layer graphene, with N even or odd (from now on, N 6¼ 1), are the same as for bilayer and trilayer graphene, respectively [1, 2, 3]. The main symmetry

5.2 Many-Layers Graphene

75

operation distinguishing the point groups between even and odd layers are the horizontal mirror plane, which is absent for N even, and the inversion, which is absent for N odd. At the Γ point, the point group is D3d for N even and D3h for N odd [2]. The consequences on the vibrational modes are mostly related to the fact that N-layers graphene has 2N atoms in the unit cell, thus multiplying the number of phonon branches. This aspect is behind the broad use of Raman spectroscopy as the main method when assigning the number of layers is a graphene sample [1, 5, 7, 39, 40]. Although the interlayer interactions are weak and no major change happens to the firstorder phonon spectra, the changes in the second-order spectra subjected to the double-resonance condition is considerable, and generates the well-known method for Nassignment [6, 39, 40]. Graphite belongs to the P63/mmc (D6h4) non-symorphic space group with only four atoms in the unit cell [2, 41]. The wavevector point groups are isomorphic to the wavevector point groups of monolayer graphene, but differ fundamentally for some ! ! ! classes where a translation of c/2 is present c is the unit cell vector normal to a1 and a2 .

5.2.2

Interlayer Vibrations Up to now, mostly vibrations within the two-dimensional planes have been discussed. These vibrations are usually characterized by relatively large phonon frequencies, and the stacking of multiple layers is of minor importance. There is another set of vibrations that are related to the coupling of two or more layers, which are the interlayer vibrations. The frequencies of the interlayer vibrations are much lower because the interlayer coupling in two-dimensional materials are governed by van der Walls bonds, which are much weaker than the in-plane covalent bonds [42, 43]. The physics of interlayer vibrations is independent on the specific in-plane symmetry, being general for twodimensional materials with two layers or more, and can be used to monitor the number of layers and to study the evolution from the two-dimensional system to the bulk form. There are two main types of interlayer vibrations: the shear (in-plane) and the breathing (out-of-plane) between two adjacent layers [44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60]. Figure 5.3(a) shows both types of vibrations for N = 2, 3, and 4. The number of different modes increases with (N – 1). In bilayer graphene, the shear mode vibrates at 28 cm–1, while the interlayer breathing-mode frequency is close to 80 cm–1 [44, 47, 48, 49, 50, 51, 52]. Different works have reported the observation of shear modes from bilayer to bulk graphite using optical techniques such as Raman spectroscopy and coherent phonon spectroscopy [47, 48]. Figure 5.3(b) shows the measured differential reflectivity as a function of time for graphene samples with different numbers of layers. This experiment was accomplished using coherent phonon spectroscopy, where two pulsed laser beams (one pump, the other probe) were used, and the time delay between them could be tuned [48, 53]. The excitation generated by the pump pulses perturbs the interlayer potential in the fewlayer graphene samples, launching the shear-mode oscillations. Since the duration of the pump pulse is shorter than the shear-mode period, the shear mode can be excited coherently. These coherent oscillations modulate the optical dielectric function of the

Vibrations in Graphene

(a)

(c)

(b)

Frequency (cm–1)

Differential reflectivity (arb. units)

76

Time delay (ps)

Number of layers

Fig. 5.3 (a) The shear vibrations, shown by the horizontal arrows, and the out-of-plane

interlayer vibrations, shown by the vertical arrows, for graphene with number of layers N = 2, 3, and 4. (b) Differential reflectivity as a function of time for graphene samples with different number of layers, from two (2L) up to “infinite” (graphite) layers. The oscillations are due to the shear vibrations. (c) Dependence of the shear-mode frequency on the number of layers in multi-layer graphene samples. The symbols are experimental data and the line is a fit to the data using Eq. (5.1), with ω∞ = 44 cm–1 [48].

graphene layers, the amplitude of the modulation being proportional to the amplitude of the vibration [48, 53]. The Fourier transform of the experimental differential reflectivity data shown in Fig. 5.3(b) provides a measure of the shear-mode frequency as a function of the number of layers, as shown in Fig. 5.3(c). As shown in the graphics, the shearmode frequency increases monotonically with the sample thickness [47, 48]. The shear and out-of-plane modes in few-layer graphene (as well as for other two-dimensional systems) are theoretically described by considering a coupled oscillator model, in which only interactions between identical adjacent layers are considered. The branch with the higher frequency corresponds to the shear motion of adjacent layers vibrating in opposite directions. The frequency of this branch for N-layers is given by ωN ¼ ω∞ cosðπ=2N Þ,

(5.1)

where ω∞ is the shear-mode frequency in the bulk limit [47, 48]. This model describes the trend of the experimental data (solid line in Fig. 5.3(c)), and the asymptotic behavior

5.3 The Quantum Nature of Atomic Vibrations

77

predicts the shear-mode pffiffiffi frequency of bilayer graphene to be lower than for bulk graphite by a factor 2. Although we have focused our attention on the graphene case, the interlayer vibrations occur for all different classes of two-dimensional materials [47, 51, 54, 55, 56, 57, 58, 59, 60]. These vibrations provide a signature for the thicknesses of these two-dimensional systems, and open up new possibilities for studying interactions between adjacent layers or between the layers and the external environment (e.g. supporting substrate).

5.3

The Quantum Nature of Atomic Vibrations Because of energy quantization, the low-energy acoustic phonon states are generally more populated than the high-energy optical phonon states. At a given temperature, the phonon population is described by the Bose–Einstein distribution function n0 ¼

1 eEq =kB T

1

,

(5.2)

where Eq and kBT are the phonon and thermal energies, respectively [6, 61]. In the harmonic oscillator theory, the increase and decrease in the population number n is achieved by applying the creation (a†) and annihilation (a) operators, respectively, on an eigenstate |ni. These operations introduce normalization factors given by a†|ni = (n + 1)½|n + 1i and a|ni = n½|n – 1i. Therefore, the phonon population can be measured in the inelastic scattering of light by phonons, where a photon creates or annihilates a phonon, in the so-called Stokes (S) or anti-Stokes (aS) Raman scattering processes, respectively (see Fig. 5.4(a)). The intensity ratio between the Stokes and anti-Stokes signals (IaS/IS) is determined by the quantum mechanics normalization factors, i.e. IaS/IS / n/(n + 1) [6, 61]. Replacing n by the Bose–Einstein population function n0 gives I aS ¼ CeEq =kB T : IS

(5.3)

This equation can be inverted to obtain the effective phonon temperature in a material by measuring their Stokes and anti-Stokes–Raman spectra, given by T¼

1 h  i , k B ln C  ln IIaSS

(5.4)

where the constant C accounts for the optical properties of the system and can be found by setting T at room temperature for very low laser power (and zero bias in a graphene electronic device). This procedure has been largely used to study the phonon population dependence of different nanomaterial properties, such as electrical power [62], thermal conductivity [63], effective temperatures on biased nanostructures [64], phonon anharmonicities and lifetimes [65, 66], and optical transitions [67, 68]. However, Eq. (5.3) it is not universal. Special care should be taken when using Eq. (5.3) in materials with high-energy phonons like the G band in graphene. In this

78

Vibrations in Graphene

(b)

(a) S

ωS

ν aS

ω0

ωaS

ω0

ωS ν

SaS ω0

ωaS

Fig. 5.4 (a) Feynman diagrams for the Stokes (S), anti-Stokes (aS) and Stokes–anti-Stokes (SaS) Raman scattering processes. Wavy and straight arrows stand for photons and phonons, respectively. Dashed circles represent electron–hole pairs, while black dots and black squares represent electron–photon and electron–phonon interactions, respectively. ω0 and ν are the incident and the phonon fields frequencies, respectively, and ωS,aS are the S, aS scattered field frequencies [70]. (b) Excitation laser power dependence for the IaS/IS intensity ratio from ABstacked bi-layer graphene (open circles) and twisted-bilayer graphene (filled bullets) [70, 71]. The experimental data (symbols) were fit using Eq. (5.5) (see the text).

case, the thermally activated phonon population at room temperature is very low, and the anti-Stokes scattering may be dominated by a correlated phenomenon, called Stokes–anti-Stokes (SaS) scattering, where the same phonon is exchanged in the Stokes and the anti-Stokes scattering [see SaS in Fig. 5.4(a)] [69, 70]. To account for this phenomenon, Eq. (5.3) has to be generalized to include this effect, and the IaS/IS ratio is given by [71] I aS n0 ¼C ½1 þ CSaS PL , IS n0 þ 1

(5.5)

where the term explicitly dependent on the excitation laser power (PL) accounts for the SaS correlated scattering. The constant CSaS measures the importance of the SaS phenomenon, and if CSaS is negligibly small, Eq. (5.5) goes back to Eq. (5.3). Evidence for the SaS process has been accumulating in materials science [70, 72, 73, 74, 75], generating interest in quantum optics [74, 75]. Since phonons have a significant lifetime (in the order of picoseconds [65, 66]), it has been proposed that these systems can work as a solid-state quantum memory, storing information between the write (Stokes) and read (anti-Stokes) processes [74]. For the optical phonons in graphene, the quantum memory would work at room temperature, because Eq  kBT at T ~ 300K. The correlated character of the Stokes and anti-Stokes photons can be continuously varied from purely quantum to purely classical, as demonstrated in diamond [75]. While many experiments have been performed in diamond with ultra-fast pulsed lasers to enhance the response of the non-linear SaS event, there is evidence for the observation of the dominant SaS event in graphene using a few miliwatts continuum wave (CW) laser, i.e. achievable even with a simple laser pointer. This result was obtained in a tBLG, specially engineered to exhibit resonance with the anti-Stokes photon emission [70]. In Fig. 5.4(b), the excitation power dependence of an AB-stacked

5.5 Probing Phonons Near Defects and Edges/Grain Boundaries

79

bilayer graphene exhibits an IaS/IS power dependence that can be fit using Eq. (5.5) with CSaS = 0 and an effective phonon temperature increasing linearly with the increasing excitation laser power. The excitation power dependence of the tBLG, however, clearly exhibits a linear dependence on PL for larger excitation power values, and a fit using Eq. (5.5) shows that anti-Stokes photon production is dominated by the correlated SaS scattering [70, 71].

5.4

Phonon Coherence Length in Graphene !

The phonon coherence length c is another important aspect of vibrations, playing an important role in the optical and transport properties of materials. For example, the high-energy optical phonons are known to limit electrical transport [62, 63, 64, 65]. Since phonons are studied in graphene mostly using light-scattering, the phonon coherence length will rule the length scale where spatial coherence takes place. Spatial coherence defines the ability of two distinct points of a wave to interfere. The classic example of this phenomenon is the fringe visibility in a Young’s double-slit interferometer. While broadly explored in optics, spatial coherence has been neglected in the light-scattering by phonons, because the field emitted by an incoherent source at a given wavelength λ is spatially uncorrelated on length-scales larger than λ/2 [76]. However, this approach is not valid if the non-radiating near-field components in the light–matter interaction are taken into account [77, 78, 79, 80]. It has been shown that, because of coherence, Raman intensities at the nanoscale depend significantly on the mode symmetry and spatial confinement of the vibration [81, 82]. In the case of graphene, it was theoretically predicted [81] and experimentally demonstrated [82] that the G band signal (E2g symmetry) is subjected to destructive interference in the near-field regime. The results obtained from near-field Raman experiments revealed that these optical phonons in graphene are subject to coherence ! lengths c  30 nm [82]. For nanostructured materials whose average crystallite size La is smaller than the ! coherence length c of the phonons, the quantum confinement can generate frequency shift, broadening, and asymmetry of the peaks in the Raman spectra [6]. These changes are caused by the uncertainty in the phonon momentum associated with the finite size of ! crystallites. In other words, when La < c , the Raman-allowed phonon wavevector q is relaxed by the amount Δq / La–1. For this reason, the crystallite size La can be extracted ! from the width of the Raman lines in polycrystalline graphene systems with La < c , as discussed in Section 5.5.

5.5

Probing Phonons Near Defects and Edges/Grain Boundaries

5.5.1

Disorder-Induced-Raman Processes in Graphene Raman spectroscopy is considered as one of the most useful techniques for graphene characterization, with strong emphasis on defects [5, 6, 85, 86]. The key factor is the

80

Vibrations in Graphene

Fig. 5.5 Raman spectrum obtained from an ion-bombarded graphene sample with average distance !

between point defects c  5 nm [87].

occurrence of disorder-induced bands in the presence of defects, and their evolution with the amount and type of disorder. Figure 5.5 shows a Raman spectrum obtained from an ion-bombarded graphene sample [87]. This spectrum shows the first-order allowed G band (~1584 cm–1), the second-order (or two-phonon) G0 (~2700 cm–1, also called 2D band in the literature), and 2D0 (~3120 cm–1) bands. The spectrum also shows three other additional disorder-induced bands, namely: the D band (~1350 cm–1), the D0 band (~1620 cm–1), and the combination D+D0 (~2970 cm–1). Some important aspects and definitions about these features are: 



The D band originates from a totally symmetric vibration mode occurring near the corners (K or K0 points) of the first Brillouin zone (see eigenvector iTO at K in Fig. 5.1(b)) [88]. This mode belongs to the iTO phonon branch [89], and its Raman activity is mediated by a double-resonance mechanism in which the photoexcited electron or hole is scattered by a phonon (with finite wavevector qD 6¼ 0), and back-scattered by a defect that provides momentum conservation in the process [30]. Because the phonon scatters an electron/hole from one Dirac cone to another, this process is called intervalley [24]. The mechanism giving rise to the D0 band is similar to the one originating the D band (double resonance). The differences are: (i) the associated vibration mode belongs to the iLO phonon branch, occurring near the center of the firstBrillouin zone, although the phonon wavevector is still finite, i.e. qD0 6¼ 0; (ii) since the phonon wavevector is much shorter than the one giving rise to the D band, the electron is scattered by the phonon from one eigenstate to another belonging to the same Dirac cone. For this reason, this process is called intravalley [5, 6, 7, 24, 30].

5.5 Probing Phonons Near Defects and Edges/Grain Boundaries





5.5.2

81

The G0 (2D) and 2D0 bands originate from the overtone of the phonon modes that give rise to the D and D0 bands, respectively [5, 6, 7, 24, 30, 90]. Since the electron (or hole) is scattered twice by the same phonon (with opposite wavevectors), the G0 and 2D0 do not require the defects to be observed. Unlike the G0 and 2D0 bands, the combination D+D0 requires the presence of defects for momentum conservation, since the D and D0 phonon wavevectors have different magnitudes (|qD| 6¼ |qD0 |) in the scattering process [91].

Types of Defects and Their Influence on the Disorder-Induced Bands Long-Range versus Short-Range Defects Because the magnitude of the wavevector of the phonon giving rise to the D band is relatively large (if compared to the total extension of the first Brillouin zone), the D band is only activated by short-range defects [15]. In other words, the defect has to be confined to a small region in the real-space in order to provide such a large momentum in the reciprocal space. Structural defects such as vacancies, edges, and crystallite borders belong to this class of short-range defects, and therefore the D band is activated. On the other hand, the wavevector of the phonon associated with the D0 band is considerably shorter, and therefore the D0 band can be activated by long-range defects such as charged impurities absorbed on the graphene sheet [15]. For these reasons, the ratio between the D and D0 intensities can be used to probe the nature of the defects, being larger for long-range defects than for short-range ones [92].

Edges/Grain Boundaries Regarding graphene edges, and considering that the grain boundary is a type of graphene edge, two important points are highlighted. 

Both D and D0 intensities are strongly dependent on the direction of the polarization vector P of the exciting field relative to the edge direction, following the relation I D, D0 / cos 4 α,



(5.6)

where ID,D0 is the intensity of the D, D0 scattered field, and α is the angle between P and the edge direction [93, 94]. Therefore, these two features present maximum intensities for incident light with the polarization field parallel to the edge, and minimum (null for perfect edges) for incident light with the polarization field perpendicular to the edge [94, 95]. The D band exhibits maximum intensity for armchair edges, and cannot be activated by zig-zag edges. This property can be used to probe the crystallographic orientation of graphene edges [94, 95]. The atomic structure of the edges plays no role in the D0 scattering.

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5.5.3

Quantifying the Amount of Disorder by Raman Spectroscopy The ratio between the D and G band integrated intensities (ID/IG) has been broadly used to measure the amount of disorder in nanostructured graphitic samples [4, 5, 6, 7]. Care should be taken when reading the literature because some authors use the intensity (peak height) ratio and others use the integrated intensity (peak area) ratio. The first approach was originally introduced by Tuinstra and Koenig [88]. By noticing that the D band is activated by the borders of crystallites in nanographitic samples, the authors proposed that its intensity should be scaled with the perimeter of the crystallites, that is, ID / La, with La being the lateral dimension of the crystallite. On the other hand, the G band intensity should be proportional to the crystallite area (IG / La2). Based on these two geometrical assumptions, Tuinstra and Koenig proposed the proportionality relation ID/IG / 1/La. Later on, Ferrari and Robertson noticed that the relation proposed by Tuinstra and Koenig was no longer valid for samples with a high-degree of disorder, for which they proposed the relation ID/IG / La2 [89]. Both relations (Tuinstra and Koenig (ID/IG / 1/La), and Ferrari and Robertson (ID/IG / La2)) are based in a single parameter, which is the crystallite size La. In order to introduce a unified approach, Lucchese and collaborators [87] proposed a phenomenological model to explain the evolution of the ID/IG ratio in a graphene sheet with points defects, as a function of the average distance LD among defects. In this model, two new parameters are introduced: (i) the radius (rS) of the structurally disordered area, which is a measure of the defect size; and (ii) the radius (rA) of the area where the D band scattering takes place, rA > rS. While rS is defined as a purely structural parameter, rA is a dynamical variable related to the coherence length of the photoexcited electrons or holes involved in the D band scattering. A similar model was applied for polycrystalline graphene samples with average crystallite size La [84]. In this case, the length of the D band activation area (LA) near the borders of the crystallites replaces rA, and the thickness LS of the structurally disordered area surrounding the crystallite (LS  LB/2, where LB is the thickness of the grain boundary between two merged crystallites) replaces rS. The two models have the same geometrical reasoning, but in one case the defects are point-like (zero-dimensional, 0D) and in the other the defects are line-like (one-dimensional, 1D). The experimental values obtained for these parameters were rA  LA  4 nm [84, 87], rS  1 nm [87], and LS  2 nm [84]. The ID/IG ratio depends on the wavelength of the excitation laser source, λL [96, 97, 98]. While the G band intensity is proportional to λL-4 (as usual for first-order Raman scattering processes [6]), the D band intensity does not depend considerably on λL [96], and therefore the general trend ID/IG / λL4 was experimentally observed for λL, within the visible range [84, 91, 96, 98]. Figures 5.6(a) and (b) show the plots of the relations obtained for the ID/IG ratio as a function of LD (Fig. 5.6(a)) and La (Fig. 5.6(b)), considering different excitation laser wavelengths. These plots summarize the results presented in [87, 91] and [84], and can be used as a guide to estimate LD and La, respectively. For samples with point defects separated by LD  10 nm, the experimental results presented in [87, 91] lead to   L2D ðnmÞ ¼ 1:8  109 λ4L ðI D =I G Þ1 : (5.7)

5.6 References

(a)

83

(b)

Fig. 5.6 Plots of the relations obtained for the ID/IG ratio as a function of LD (0D defects, panel (a)) [87, 91] and La (1D defects, panel (b)) [84], considering four different excitation laser wavelengths (as indicated in the graphics).

For LD < 10 nm, ID/IG deviates from this simple equation [87] and the broadening effects take place [99]. In the case of the nanostructured graphene with the crystallite sizes larger than the phonon coherence length, i.e. La  30 nm (see Section 5.4), the Tuinstra and Koenig proportionality relation holds, and the experimental results yield [84, 98]   La ðnmÞ ¼ 2:4  1010 λ4L ðI D =I G Þ1 : (5.8) For La < 30 nm, the ID/IG deviates from this simple equation [84], and the G band broadening provides a more accurate figure of merit, obeying the relation [84] ΓG ðLa Þ ¼ 15 þ 95e2La =30 :

(5.9)

Equations 5.7–5.8 and Fig. 5.6 can be used for quantifying defects/disorder in graphene systems. For the dependence on the number of layers, see [99, 100].

5.6

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[75] M. Kasperczyk, A. Jorio, E. Neu, P. Maletinksy, and L. Novotny, Stokes–anti-Stokes correlations in Raman scattering from diamond membranes. Optics Letters, 40 (2015), 2393. [76] W. H. Carter and E. Wolf, Coherence properties of Lambertian and non-Lambertian sources. Journal of the Optical Society of America, 65 (1975), 1067. [77] R. Carminati and J.-J. Greffet, Near-field effects in spatial coherence of thermal sources. Physical Review Letters, 82 (1999), 1660. [78] A. V. Shchegrov, K. Joulain, R. Carminati, and J.-J. Greffet, Near-field spectral effects due to electromagnetic surface excitations. Physical Review Letters, 85 (2000), 1548. [79] H. Roychowdhury and E. Wolf, Effects of spatial coherence on near-field spectra. Optics Letters, 28 (2003), 170. [80] A. Apostol and A. Dogariu, Spatial correlations in the near field of random media. Physical Review Letters, 91 (2003), 093901. [81] L. G. Cançado, R. Beams, A. Jorio, and L. Novotny, Theory of spatial coherence in nearfield Raman scattering. Physical Review X, 4 (2014), 031054. [82] R. Beams, L. G. Cançado, S.-H. Oh, A. Jorio, and L. Novotny, Spatial coherence in nearfield Raman scattering. Physical Review Letters, 113 (2014), 186101. [83] H. Richter, Z. P. Wang, and L. Ley, The one phonon Raman spectrum in microcrystalline silicon. Solid State Communications, 39 (1981), 625. [84] J. Ribeiro-Soares, M. E. Oliveros, C. Garin, M. V. David, L. G. P. Martins, C. A. Almeida, E. H. Martins-Ferreira, K. Takai, T. Enoki, R. Magalhães-Paniagoa, A. Malachias, A. Jorio, B. S. Archanjo, C. A. Achete, and L. G. Cançado, Structural analysis of polycrystalline graphene systems by Raman spectroscopy. Carbon, 95 (2015), 646–52. [85] R. Beams, L. G. Cançado, and L. Novotny, Raman characterization of defects and dopants in graphene. Journal of Physics: Condensed Matter, 27 (2015), 083002. [86] M. A. Pimenta, G. Dresselhaus, M. S. Dresselhaus, L. G. Cançado, A. Jorio, and R. Saito, Studying disorder in graphite-based systems by Raman spectroscopy. Physical Chemistry Chemical Physics, 9 (2007), 1276. [87] M. M. Lucchese, F. Stavale, E. H. Ferriera, C. Vilane, M. V. O. Moutinho, R. B. Capaz, C. A. Achete, and A. Jorio, Quantifying ion-induced defects and Raman relaxation length in graphene. Carbon, 48 (2010), 1592. [88] F. Tuinstra and J. L. Koenig, Raman spectrum of graphite. Journal of Chemical Physics, 53 (1970), 1126. [89] A. C. Ferrari and J. Robertson, Resonant Raman spectroscopy of disordered, amorphous, and diamondlike carbon. Physical Review B, 64 (2001), 075414. [90] L. G. Cançado, M. A. Pimenta, R. Saito, A. Jorio, L. O. Ladeira, A. Grueneis, A. G. SouzaFilho, G. Dresselhaus, and M. S. Dresselhaus, Stokes and anti-Stokes double resonance Raman scattering in two-dimensional graphite, Physical Review B, 66 (2002), 035415. [91] L. G. Cançado, A. Jorio, E. H. Martins Ferreira, F. Stavale, C. A. Achete, R. B. Capaz, M. V. O. Moutinho, A. Lombardo, T. S. Kulmala, and A. C. Ferrari, Quantifying defects in graphene via Raman spectroscopy at different excitation energies. Nano Letters, 11 (2011), 3190. [92] A. Eckmann, A. Felten, A. Mishchenko, L. Britnell, R. Krupke, K. S. Novoselov, and C. Casiraghi, Probing the nature of defects in graphene by Raman spectroscopy. Nano Letters, 12 (2012), 3925. [93] A. Grüneis, R. Saito, Ge. G. Samsonidze, T. Kimura, M. A. Pimenta, A. Jorio, A. G. Souza Filho, G. Dresselhaus, and M. S. Dresselhaus. Inhomogeneous optical absorption around the K point in graphite and carbon nanotubes. Physical Review B, 67 (2003), 165402.

5.6 References

89

[94] L. G. Cançado, M. A. Pimenta, B. R. A. Neves, M. S. Dantas, and A. Jorio, Influence of the atomic structure on the Raman spectra of graphite edges. Physical Review Letters, 93 (2004), 247401. [95] C. Casiraghi, A. Hartschuh, H. Qian, S. Piscanec, C. Georgi, A. Fasoli, K. S. Novoselov, D. M. Basko, and A. C. Ferrari, Raman spectroscopy of graphene edges. Nano Letters, 9 (2009), 1433. [96] L. G. Cançado, A. Jorio, and M. A. Pimenta, Measuring the absolute Raman cross section of nanographites as a function of laser energy and crystallite size. Physical Review B, 76 (2007), 064304. [97] P. Klar, E. Lidorikis, A. Eckmann, I. A. Verzhbitskiy, A. C. Ferrari, and C. Casiraghi. Raman scattering efficiency of graphene, Physical Review B, 87 (2013), 205435. [98] L. G. Cançado, K. Takai, T. Enoki, M. Endo, Y. A. Kim, H. Mizusaki, A. Jorio, L. N. Coelho, R. Magalhães-Paniago, and M. A. Pimenta, General equation for the determination of the crystallite size L[a] of nanographite by Raman spectroscopy. Applied Physics Letters, 88 (2006), 163106. [99] E. H. Martins Ferreira, M. V. O. Moutinho, F. Stavale, M. M. Lucchese, R. B. Capaz, C. A. Achete, and A. Jorio, Evolution of the Raman spectra from single-, few-, and many-layer graphene with increasing disorder. Physical Review B, 82 (2010), 125429. [100] R. Giro, B. S. Archanjo, E. H. Martins Ferreira, R. B. Capaz, A. Jorio, and C. A. Achete. Quantifying defects in N-layer graphene via a phenomenological model of Raman spectroscopy. Nuclear Instruments and Methods in Physics Research Section B, 319 (2014), 71–4.

6

Thermal Properties of Graphene: From Physics to Applications Alexander A. Balandin

6.1

Thermal Conductivity of Graphene and Few-Layer Graphene Graphene [1], in addition to its unique optical [2] and electronic [3, 4] properties, demonstrated extremely high thermal conductivity [5, 6, 7, 8, 9, 10]. The first experimental studies of the thermal conductivity of graphene were made possible with development of the optothermal Raman technique (see Fig. 6.1). In this technique, a Raman spectrometer acts as a thermometer measuring the local temperature rise in graphene in response to the laser heating from the same instrument [5, 6, 7, 8, 9, 10]. Graphene has distinctive Raman signatures, e.g. G peak and 2D band, which exhibit strong temperature dependence [11, 12, 13, 14, 15, 16, 17]. This means that the shift in the position of G peak in response to the laser heating can be used to measure the local temperature rise. The correlation between temperature rise and the amount of power dissipated in graphene, for the sample with given geometry and proper heat sinks, allows one to extract the value of the thermal conductivity K. A small amount of power dissipated in graphene is sufficient to induce a measurable shift in the G peak position due to the atomic thickness of the material. The suspended portion of graphene serves several essential functions in this method (Fig. 6.2). It is needed to accurately determine the amount of power absorbed by graphene, forming an in-plane two-dimensional (2D) heat front propagating toward the heat sinks, and reducing thermal coupling to a substrate through increased micro- and nano-scale corrugations (Fig. 6.1). In recent years, the optothermal Raman technique developed at UC Riverside was extended to other 2D materials, such as layered transition metal dichalcogenides (TMDs) [18, 19, 20] and macroscopic suspended films [21]. In the case of TMD, the thermal conductivity values are much smaller, which allows to induce sufficient heating with a Raman spectrometer laser even in relatively thick films. The first optothermal Raman measurements of suspended graphene [5, 6, 7, 8, 9, 10] demonstrated that the thermal conductivity can exceed that of the bulk graphite, which is ~2000 W/mK at room temperature (RT). It was determined that the electronic contribution to heat conduction in graphene, which is not gated, is much smaller than that of phonons near RT. The phonon mean-free path (MFP) in graphene was estimated to be ~775 nm near RT [6, 10]. Several independent studies, which utilized modified versions of the Raman optothermal technique [22, 23], found that the thermal conductivity of suspended high-quality chemical vapor deposited (CVD) graphene is in the range of K ~ 1500 to ~5000 W/mK [22, 23]. At temperature T ~ 600 K the thermal conductivity

90

6.1 Thermal Conductivity of Graphene

91

Fig. 6.1 Illustration of the optothermal Raman experimental technique developed

for measuring thermal conductivity of suspended graphene. The suspended graphene is heated with the laser light while the corresponding temperature rise is determined from the shift in the G or 2D peaks in the Raman spectrum of graphene. The knowledge of temperature rise in response to the dissipated power and geometry of the graphene layer are sufficient for determining the in-plane thermal conductivity. The corrugations in the partially suspended graphene, which are depicted in the schematic, reduce graphene thermal coupling to the substrate. The technique was subsequently modified with metal heat sinks and a detector placed under the suspended graphene. Recently, the optothermal Raman technique was extended to other two-dimensional van der Waals materials that have clear Raman signatures (for details, see [10]).

Fig. 6.2 Scanning electron microscopy images of bilayer (a) and tri-layer (b) graphene suspended across trenches in Si/SiO2 wafers. Note the nearly ideal rectangular shape of the suspended graphene flakes, which simplifies the thermal data analysis in the optothermal Raman measurements. In the experiments, 2D material is heated in the middle of the suspended part (data from [10]).

92

Thermal Properties of Graphene

extracted under the assumption of 2.3% light absorption was determined to be K ~ 630 W/mK [24]. One should note though that, due to the many-body effects, the absorption in graphene is the function of wavelength λ, when λ > 1 eV [25, 26, 27]. The absorption of 2.3% is observed only in the near-infrared at ~1 eV, and it steadily increases with increasing energy. Taking into account actual absorption at 488 nm under the conditions of the experiments would increase the extracted values of thermal conductivity. A recent study using a different experimental technique – electrical thermal bridge – of the residue-free suspended graphene also obtained thermal conductivity above the bulk graphite limit: K ~ 2430 W/mK at T = 335 K [28]. Naturally, defects and grain boundaries characteristic for CVD graphene can substantially reduce the thermal conductivity [10]. The thermal conductivity of the supported graphene is lower than that in suspended graphene due to coupling of graphene phonon modes with substrate modes and enhanced interface phonon scattering. The thermal conductivity of graphene on the Si/SiO2 substrate was measured using the thermal bridge technique and found to be K  600 W/mK near RT [29]. Encasing graphene within two layers of SiO2 leads to further reduction of the thermal conductivity due to the phonon–boundary and disorder scattering [30]. Theoretically, the thermal conductivity in graphene has been investigated using a range of approaches from the Boltzman transport equations (BTE) to molecular dynamics (MD) simulations [31–54]. The BTE approaches gave K values from 1000 W/mK to 8000 W/mK, depending on the defect nature and density, edge quality and size of graphene flakes. The MD simulations typically result in lower values from ~500 W/mK to ~2900 W/mK owing to the small simulation domain sizes and underestimation of the contribution of the long wavelength acoustic phonons. Theoretical studies confirmed that phonon scatterings on edge roughness and lattice defects, as well as the strain characteristics for suspended samples can strongly affect the thermal conductivity of graphene. The fact that graphene can have higher thermal conductivity than that of the basal planes of graphite, despite similarities in phonon dispersion and the crystal lattice inharmonicity, was explained by the specifics of the long-wavelength phonon transport in 2D crystal lattices [10]. The long-wavelength phonons in 2D graphene have exceptionally long MFP, limited by the size of the sample even if the thermal transport is diffusive. In different terms, this means that the phonon Umklapp scattering is ineffective in restoration of the thermal equilibrium in 2D crystal lattice such as in 3D crystal lattices. The latter results in an anomalous dependence of the thermal conductivity of few-layer graphene (FLG) on the number of atomic planes in the samples [17]. The most recent theoretical studies suggest that the size of graphene samples have to be in the 100-μm or 1-mm range in order to converge to the intrinsic value of the thermal conductivity, limited by the crystal lattice inharmonicity alone [55, 56]. In both reports, the K values were substantially larger than the bulk graphite limit (K = 5800 W/mK at T = 300 K [55]). This conclusion is in line with the theoretical predictions of logarithmic divergence of the thermal conductivity of strictly 2D crystal lattices without defects [57, 58]. Another interesting question in the theory of heat conduction in graphene, which continues to attract major attention, is the relative contributions of the longitudinal acoustic (LA), transverse acoustic (TA), and out-of-plane (ZA) phonons to thermal conductivity.

6.2 Thermal Properties of Graphene

6.2

93

Isotope and Rotational Engineering of Thermal Properties of Graphene We now turn to a discussion of the effects of isotopes on the thermal conductivity of graphene and the possibility of tuning thermal properties of FLG by the atomic plane rotations. Naturally occurring carbon materials are made up of two stable isotopes of 12C (~99%) and 13C (~1%). It is known from theoretical studies that a relative concentration of isotopes and crystal lattice defects can have a dramatic affect on the thermal conductivity. The first experimental study of the isotope effect on thermal properties of graphene [59] used material with various percentages of 13C, which were synthesized using the CVD technique. The thermal conductivity of the isotopically pure 12 C (0.01% 13C) graphene, measured by the optothermal Raman technique [10], was higher than 4000 W/mK at temperature ~320 K, and more than a factor of two higher than the thermal conductivity of graphene sheets composed of a 50%–50% mixture of 12 C and 13C (see Fig. 6.3). The evolution of thermal conductivity with the isotope content was attributed to the changes in the phonon – point defect scattering rate via the mass-difference term [60]. The ability to control thermal transport via isotope engineering can help in understanding the specifics of heat conduction in 2D systems. It may also have practical implications for situations when the thermal conductivity has to be reduced without degrading the electrical conductivity. One of the most interesting recent developments in the graphene thermal field is the investigation of the phonon and thermal properties in twisted bilayer graphene (T-BLG). Single-layer graphene reveals four in-plane phonon branches – transverse acoustic (TA), longitudinal acoustic (LA), transverse optical (TO), and longitudinal optical (LO) with the atomic displacements in the graphene plane – and two out-ofplane branches – acoustic (ZA) and optical (ZO) with the displacements perpendicular to the graphene plane. The in-plane acoustic branches are characterized by the linear

Fig. 6.3 Thermal conductivity of the suspended graphene film with 13C isotope concentrations of 0.01%, 1.1% (natural abundance), 50%, and 99.2%, respectively, as a function of temperature (data from [59]).

94

Thermal Properties of Graphene

energy dispersions over the most part of the Brillion zone (BZ) except near the zone edge, while the out-of-plane ZA branch demonstrates a quadratic dispersion near the zone center q = 0 (q is the phonon wavenumber). However, it deviates from the quadratic dispersion as q increases. The number of the phonon branches in bilayer graphene (BLG) is doubled: six additional branches possess non-zero frequency at q = 0 and at the low frequencies are affected by the interlayer interactions [61]. The emergence of many folded hybrid phonon branches in T-BLG was explained by the change in the unit cell size and a corresponding modification of the reciprocal space geometry. The number of the polarization branches and their dispersion in T-BLG depend strongly on the rotation angle [61]. The first measurements of the thermal conductivity of suspended twisted bilayer graphene were also performed using the optothermal Raman technique [62]. It was found that the thermal conductivity of the twisted bilayer graphene is lower than that of monolayer graphene and the reference Bernal stacked bilayer graphene in the entire examined temperature ranges from ~300 K to 700 K (see Fig. 6.4). This finding indicates that the heat carriers – phonons – in the twisted bilayer graphene do not behave as in individual graphene layers. The decrease in the thermal conductivity in twisted bilayer graphene was explained by the modification of the Brillouin zone due to plane rotation and the emergence of numerous folded phonon branches that enhance the phonon Umklapp and normal scattering [61, 62]. A possibility of engineering the specific heat of T-BLG has been studied theoretically [63, 64, 65]. In Fig. 6.5(a), the temperature dependences of the phonon specific heat cv in SLG are shown for different phonon branches: ZA, TA+LA and ZO+TO+LO. The contribution of ZA phonons to the specific heat is dominant up to T ~ 200 K. Nevertheless, both the ZA contribution to the specific heat and the total value of cv

Fig. 6.4 Thermal conductivity of the suspended single-layer graphene, Bernal stacked

bilayer graphene and twisted bilayer graphene as a function of the measured temperature. The thermal conductivity of the wrinkled single-layer graphene is shown for comparison (data from [62]).

6.2 Thermal Properties of Graphene

95

demonstrate deviation from the linear T-dependence beginning from T ~ 15 K. This is a manifestation of the anisotropy and a deviation from the parabolic of the ZA dispersion [64]. The power index n of Tn-dependence of the specific heat increases faster for the total specific heat than for the ZA component of cv due to the contributions from LA and TA phonons revealing ~T2 dependence for T < 100 K. The contribution of the in-plane phonons to the total cv increases with the temperature and becomes comparable with that of the ZA phonon contribution for T ~ 250–300 K. This illustrates that the simplified isotropic models for the phonon density of states in graphene do not capture all the characteristics of the specific heat and thermal conductivity [64, 65]. For temperatures

Fig. 6.5 Phonon branch-dependent heat capacity as a function of temperature in graphene

(a) and T-BLG with rotation (b). In panel (a), the contributions from different branches, ZA, TO+LO+ZO, and TA+LA are shown. Panel (b) shows the contributions from the out-of-plane and in-plane phonons for graphene (solid curves) and 13.20 T-BLG (dashed curves).

96

Thermal Properties of Graphene

T ~ 300 K, the contribution of the in-plane acoustic phonons to the specific heat is larger than that of ZA phonons. The contribution from optic phonons is very small ( jE F j=ħ, pushing down the plasmon frequency. Competing with this are intraband conductivity modifications at low phase velocity (for ω=kx not much larger than Fermi velocity), which decrease kinetic inductance and push up the plasmon frequency. Figure 7.2 compares the Drude model predicted plasmon to the fully calculated dispersion for a particular case. Retardation effects, violating the quasi-electrostatic limit that was assumed for (7.11).

Comparison with Plasmons in Other Systems Comparison with Other 2D Plasmons Graphene plasmons can be compared to a variety of other plasmonic systems. The plasmons in semiconducting two-dimensional electron gases (2DEGs) are the closest analogue. All 2DEG plasmons rely on inductive sheet conductivity σ, and hence follow the electrodynamics described in Section 7.1.2. As a result, all 2DEG plasmons have the pffiffiffiffi ω / k x dispersion when the 2DEG is between two simple dielectrics, as described in Section 7.1.3. The main distinction between graphene and other 2DEGs is quantitative: the plasmon wavelengths differ greatly, as do the frequency ranges for plasmonic behavior. There are two reasons for this: first, a high Drude weight is possible in graphene where E F e 0:4 eV is easily achievable, whereas in a semiconductor 2DEG one is hard pressed to exceed 0:1 eV;3 second, the bulk semiconductor permittivity around a 2DEG is quite large, typically ϵ a þ ϵ b > 20ϵ 0 , whereas graphene on a typical substrate has ϵ a þ ϵ b 5ϵ 0 . The different Drude weights and permittivities combine (see Eq. (7.11)) 3

Note that for a two-valley semiconductor 2DEG with degenerate electron gas the formula for Drude weight in terms of EF is identical to that of graphene, Eq. (7.3).

7.1 Macroscopic Approach to Graphene Plasmonics

109

to give graphene a much longer graphene plasmon wavelength (smaller k x ) for a given frequency, or in other words, a higher graphene plasmon frequency ω for a given k x . This allows graphene plasmonic structures to oscillate at much higher frequencies than semiconducting 2DEGs, for the same spatial dimensions. Graphene plasmons, however, operate at lower frequency than thin film metal plasmons; this is true even in the hypothetical case of a single-atom-thick metal film, which would follow many of the same equations just discussed but with a much higher Fermi energy of order 3–10 eV. Each material also has a different relaxation time, τ, which limits plasmonic operation at low frequencies: below f ¼ 1=ð2πτ Þ, the plasmons are overdamped. At room temperature, the large τ in graphene allows plasmonic operation down to approximately 0.3 THz, whereas in metals the lower limit is around 8 THz. Note that dielectric properties also influence plasmons losses, as further examined in Section 7.3.

Comparison with Surface Plasmons Two-dimensional plasmons are quite different in nature from surface plasmons, which instead occur at the surface of a bulk conductor (without a 2DEG). In this case, one of the bulk materials provides the necessary inductance, appearing as a negative permittivity. For example, the permittivity of a bulk conductor (metal or otherwise) is typically represented by Drude law permittivity 
 ϵ b ðωÞ ¼ ϵ ∞ 1  ω2p = ω2 þ iωτ 1 , which is a combination of background permittivity (ϵ ∞ ) and free-carrier Drude response; ωp is the bulk plasma frequency. The analysis of Section 7.1.2 is easily adapted to derive surface plasmons: we take σ ¼ 0 and look for resonances given by Y a þ Y b ¼ 0. The surface modes can occur when they have dispersion given by (via (7.10), taking ϵ b negative) pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kx ¼ ω μ0 ϵ a = 1 þ ϵ a =ϵ b : Note the essential appearance of μ0 in the surface plasmon dispersion, indicating that full electromagnetic forces are at play, in contrast to the electrostatic forces required for the 2DEG plasmon dispersion in Eq. (7.11). A surface plasmon is therefore typically referred to as surface plasmon polariton (SPP) reflecting that it exists as a hybridization of electromagnetic light plane waves with surface plasma resonance. For most frequencies, the surface mode phase velocity ω=kx only differs slightly from the speed of light pffiffiffiffiffiffiffiffiffi 1= μ0 ϵ a . In the quasi-electrostatic limit, the surface plasmons are non-dispersive at a fixed frequency of ffiωLSP , defined by ϵ a =ϵ b ¼ 1 (in the metal example, ωLSP ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ωp = 1 þ ϵ a =ϵ ∞ , a slightly lower frequency than the bulk plasma ωp ). Since this mode is non-dispersive, it does not propagate and is known as the localized surface plasmon pffiffiffiffi (LSP). This is in striking contrast to the dispersive ω / kx for 2DEG plasmons in the quasi-electrostatic limit.

110

Graphene Plasmonics

7.1.5

Influence of Dielectric Environment: Permittivity Dispersion and Thin Films We have seen that the dielectric environment has direct impact on the plasmon dispersion: in Eq. (7.11), the plasmon kx for a given frequency has a direct proportional scaling with permittivity ϵ. However, to arrive at Eq. (7.11) we have assumed a very simple dielectric environment. To accurately characterize the dielectric environment around the graphene, one may need to include (1) phonons, (2) birefringence, or (3) thin film effects. As an example of practical interest, the highest quality plasmons in graphene to date have been observed when graphene was encapsulated between thin films of hexagonal boron nitride [97]. In this case, one needs to take into account all three of the effects just mentioned: hexagonal boron nitride is a birefringent dielectric with highly polarizable phonons, and in these devices it is usually encountered in nanometer-thick films. Dielectrics can be modeled as a macroscopic and local permittivity tensor that is frequency-dependent, ϵ ij ðωÞ, implying a constitutive relation for polarization given by P Pi ðωÞ ¼ j ϵ ij ðωÞEj ðωÞ. This captures the effects of both phonons and birefringence. One can then solve Maxwell’s equations as usual, and taking into account thin films with the appropriate boundary conditions.4 The modes are, as before, found as zeros of ðY a þ Y b þ σ Þ, see Section 7.1.2. Assuming the quasi-electrostatic limit and also kx > 0 and ω > 0, below are a few notable examples: 1.

For a half-space of isotropic medium with dispersion (e.g., phonons, free electrons, or other dynamic polarizable degrees of freedom), the admittance expression in Section 7.1.3 remains valid, except ϵ ðωÞ is frequency-dependent. We repeat this formula Y ðk x ; ωÞ ¼

2.

4

Re ϵ 0 :

(7.12)

Note that this expression may be invalid nearby phonon frequencies, where Re ϵ < 0 can occur. For a half-space of uniaxial birefringent material, such as h-BN, having a diagonal permittivity tensor of ϵ x component in-plane and ϵ z component out-ofplane Y ð k x ; ωÞ ¼

3.

iω ϵ ðωÞ, kx

iω pffiffiffiffiffiffiffiffi ϵxϵz, kx

Re ϵ x , Re ϵ z > 0 :

(7.13)

In other words, this case behaves the same as an isotropic dielectric of permittivpffiffiffiffiffiffiffiffi ity ϵ x ϵ z [91]. For a half-space consisting of a nearby finite film of birefringent dielectric (ϵ x , ϵ z ) of thickness d, followed by a perfect conductor (approximating a metal, for example)

The transfer matrix approach provides a convenient framework for analytically or numerically evaluating the effects from an arbitrary number of layers.

7.2 Microscopic Approach


 Y ðkx ; ωÞ ¼ iωϵ x coth βz d =βz ,

111

(7.14)

pffiffiffiffiffiffiffiffiffiffi where βz ¼ ϵ x =ϵ z k x .

An interesting limit of practical interest is small βz d  1 (long wavelength and thin dielectric); this gives Y iωϵ z =k2x d and hence one obtains a linear dispersion of the plasmon instead of the usual square root. This linear dispersion persists to arbitrarily low frequencies but with a phase velocity much lower than the speed of light [4]. 4.

For a half-space consisting of a nearby finite film of birefringent dielectric ðϵ x ; ϵ z Þ of thickness d, followed by an isotropic medium ðϵ Þ Y ðk x ; ωÞ ¼ i

ωϵ x ð1 þ αÞeβz d  ð1  αÞeβz d , βz ð1 þ αÞeβz d þ ð1  αÞeβz d

Re ϵ > 0 ,

(7.15)

pffiffiffiffiffiffiffiffiffiffi where βz ¼ ϵ x =ϵ z k x and α ¼ kx ϵ x =βz ϵ. For a metal, one typically has a very large jϵj ⋙ jϵ x j and hence α 0, and one recovers Eq. (7.14). Any of the above formulas may be used with dispersive dielectrics, i.e., with a frequency-dependent permittivity. Regarding the last two examples, an unsusual phenomenon occurs when ϵ x =ϵ z is negative, which characterizes so-called hyperbolic materials or indefinite materials. This gives an imaginary βz and hence the waves in the dielectric are propagating in z, instead of evanescent. This phenomenon occurs naturally in hexagonal boron nitride, in two frequency bands [11, 22]. One then finds that numerous confined electromagnetic modes of differing k x can exist at a single value of ω, not all of which are related to the graphene plasmonic response.

7.2

Microscopic Approach In this section, a more in-depth theory of the plasmonic response is provided. In particular, we summarize a simple but highly successful approximation: the random phase approximation (RPA) [8, 40], which captures plasmon collective excitations. In connection with the previous section (analyzing plasmons via macroscopic electrodynamics), here we provide the microscopic approach to calculate the conductivity function σ ðk; ωÞ of graphene, which goes beyond Drude’s law. Electrons in graphene, as in any other material, do not move as independent particles. Their motions are instead highly correlated
 due
to pairwise  interactions [42]. These are described by an interaction potential v rij ¼ v jri  rj j (with physical dimensions of energy), which depends only on the absolute value of the relative distance rij ¼ ri  rj between two electrons. The non-relativistic (instantaneous) Coulomb interaction between two electrons in graphene is sensitive to the dielectric media surrounding the graphene sheet. For graphene with one side exposed to a uniform and isotropic medium with permittivity ϵ 1 and the other to one with permittivity ϵ 2 , a simple electrostatic calculation yields

112

Graphene Plasmonics


 v r ij ¼

1 e2 : 2π ðϵ 1 þ ϵ 2 Þ r ij

with 2D Fourier transform vq given by ð vq  d 2 r eiq r vðr Þ ¼

e2 : ðϵ 1 þ ϵ 2 Þq

(7.16)

(7.17)

In more complicated dielectric environments, the potential may be non-instantaneous (frequency-dependent) due to dielectric internal degrees of freedom, such as phonons. Then, we must replace vq ! V q, ω in the expressions that follow. For arbitrary dielectric structures, we have, in general, in the quasielectrostatic limit that the reciprocal of the potential is given by the added responses of the half-spaces above and below. In terms of the notation of the previous section vq ! V q , ω ¼

iω=q e2 : Y a ðq; ωÞ þ Y b ðq; ωÞ q

(7.18)

This expression conveniently allows the dielectric responses above and below to be calculated independently. We first consider the electron density δnðr; t Þ induced by an external potential V ext ðr; t Þ. The relation between the Fourier components of δnðr; t Þ and V ext ðr; t Þ when electron–electron interactions are taken into account reads δnðq; ωÞ ¼ χ nn ðq; ωÞV ext ðq; ωÞ,

(7.19)

and defines the causal (or retarded) density–density response function χ nn ðq; ωÞ of the interacting system. A finite-induced density δnðr; t Þ generates a mean field, which, for a system of charged particles, corresponds to a large instantaneous electric potential given by ð V Hartree ðr; t Þ ¼ d2 r vðjr  r0 jÞδnðr0 ; t Þ, (7.20) or, in Fourier transform V Hartree ðq; ωÞ ¼ vq δnðq; ωÞ:

(7.21)

The sum of the external potential and the Hartree potential is usually called the “screened” potential W sc ðq; ωÞ

W sc ðq; ωÞ ¼ V ext ðq; ωÞ þ V Hartree ðq; ωÞ ¼ V ext ðq; ωÞ 1 þ vq χ nn ðq; ωÞ : (7.22) On the general grounds of linear response theory [42, 79], we can also define the density–density response function χ~nn ðq; ωÞ related to the non-local frequency-dependent conductivity σ ðq; ωÞ as follows: χ~nn ðq; ωÞ ¼

q2 σ ðq; ωÞ, ie2 ω

(7.23)

7.2 Microscopic Approach

113

giving the ratio of the mean density to the mean total field (compare Eq. (7.19)) δnðq; ωÞ ¼ χ~nn ðq; ωÞW sc ðq; ωÞ:

(7.24)

The dependences of the three functions χ nn , χ~nn , and σ on ðq; ωÞ are only known in an approximate fashion, however, they are exactly related as shown above. The ratio between the Fourier components of screened and external potentials defines the inverse of the dynamical dielectric function εðq; ωÞ 1 W sc ðq; ωÞ χ ðq; ωÞ 1  ¼ 1 þ vq χ nn ðq; ωÞ ¼ nn ¼ : εðq; ωÞ V ext ðq; ωÞ χ~nn ðq; ωÞ 1  vq χ~nn ðq; ωÞ

(7.25)

Self-sustaining oscillations, such as plasmons, can be found as zeros of εðq; ωÞ, indicating a non-zero oscillation W sc with zero driving potential V ext . Note that, due to Eqs. (7.18) and (7.23), the condition ε ¼ 0 is equivalent to Eq. (7.9), Y a þ Y b þ σ ¼ 0, as identified in the macroscopic analysis.

7.2.1

Random Phase Approximation (RPA) In the random phase approximation (RPA), the system of interacting electrons is assumed to respond as a non-interacting gas to the mean-field screened potential, i.e. to W sc ðq; ωÞ. Mathematically, the RPA approximation for the density–density response function is specified by ~χ nn ¼ χ ðnn0Þ , RPA

(7.26)

where χ ðnn0Þ ðq; ωÞ ¼ N f

limþ

ϵ!0

X λ, λ0 ¼1

ð

nk, λ  nkþq, λ0 2 ħω þ ε ð2π Þ k, λ  εkþq, λ0 d2 k

þ iϵ

  hχ ðkÞjχ 0 ðk þ qÞi2 (7.27) λ λ

is the non-interaction density–density response function of a doped graphene sheet [6, 45, 81, 98], which is a complex function of momentum q and frequency ω. In Eq. (7.27), N f ¼ 4 is the number of fermion species (or “flavors”) in graphene, εk, λ ¼ λħvF k are single-particle Dirac band energies, vF is the Fermi velocity, nk, λ are Fermi–Dirac band-occupation factors, and   1 1 χ λ ðkÞ ¼ pffiffiffi (7.28) iφ 2 λe k are two-component (pseudo)spinors. Here λ ¼ þ1 (λ ¼ 1) labels the conduction (valence) band and φk is the angle between k and the x^ axis, which physically denotes the momentum-dependent phase difference between wave function amplitudes on the A and B sublattices of graphene’s honeycomb lattice. Detailed analytical expressions of χ ðnn0Þ ðq; ωÞ are available [98, 45, 6, 81] at arbitrary values of q and ω, for an arbitrary value of the Fermi energy εF and at zero temperature. The Maldague identity [42] can be conveniently utilized to calculate χ ðnn0Þ ðq; ωÞ at finite temperature T

114

Graphene Plasmonics

χ ðnn0Þ ðq; ωÞ ¼



ð∞ dE ∞

χ ðnn0Þ ðq; ωÞ

T¼0, εF !E

4kB T cosh ½ðE  μÞ=ð2kB T Þ 2

:

(7.29)

At this point it is worth reconnecting with the previous analysis in terms of macroscopic electrodynamics. The random phase approximation is equivalent to representing the graphene by a macroscopic conductivity function given by5 σ ðk; ωÞ ¼ i

e2 ω ð0Þ χ ðk; ωÞ: k2 nn

(7.30)

This RPA conductivity function can be used as an element in standard electrodynamic treatments, for example in Section 7.1.2. Equivalently, the plasmon is found as a pole in the dynamical density–density response function (see Eq. (7.25)), which occurs when 1  vq χ ðnn0Þ ðk; ωÞ ¼ 0:

7.2.2

Beyond RPA We conclude this section with a few comments on the physics of Dirac plasmons beyond the RPA. (i)

(ii)

5

As explained in [3], because the 2D massless Dirac fermion (MDF) model Hamiltonian is not invariant under an ordinary Galilean boost, the RPA is not exact for interacting systems of MDFs even in the limit q ! 0. This is in striking contrast to what happens in the conventional 2D parabolic-band electron gas [42], where for q ! 0 the plasmon dispersion is protected from many-body renormalizations by Galilean invariance. When interactions between MDFs are treated beyond RPA, even the leading long-wavelength term in the plasmon dispersion acquires a non-trivial density and coupling-constant dependence [3]. In the RPA, a Dirac plasmon has an infinite lifetime at low momentum and energy and zero temperature ðT ¼ 0Þ, i.e. when its dispersion lies in the triangular region near the origin of the ðq; ωÞ plane where single-electron–hole pairs cannot exist

due to Pauli blocking, Im χ ðnn0Þ ðq; ωÞ ¼ 0. When the plasmon dispersion hits the continuum of interband electron–hole excitations, Landau damping kicks in. This simply means that a plasmon can decay by exciting single interband electron–hole pairs. At finite temperature, Landau damping “leaks” into the T ¼ 0 Pauli-blocked region of the ðq; ωÞ plane. Going beyond RPA, the plasmon acquires a finite intrinsic lifetime τ p1 ðqÞ, also at small energies and momenta [86]. A longwavelength plasmon can decay by emitting, for example, two electron–hole pairs

Note that in a full electrodynamic model of graphene, σ ðq; ωÞ at non-zero q becomes a tensor, containing longitudinal and transverse parts. The quantity examined here is the longitudinal conductivity involved in plasmonic behavior, i.e., in transverse magnetic modes. The transverse conductivity, related to magnetization and relevant for transverse electric modes, has also been calculated for non-interacting electrons in graphene in [87].

7.3 Plasmon Damping

115

with nearly opposite momenta: ðp1 ; p1 þ kÞ and ðp2 ; p2  kÞ. In a real system, other physical effects, such as scattering by disorder and phonons, can substantially affect the plasmon lifetime, as we discuss in Section 7.3.

7.3

Plasmon Damping From any defining dispersion relation of graphene plasmons, such as (7.11), it is clear that if there are losses ðRe σ > 0 or Im ϵ > 0Þ, then either k x or ω must be a complex quantity. This means that the mode must either decay in time, or in space. In this section, we provide the common metrics used to describe this decay, and discuss the common physical sources.

7.3.1

Quantifying Lifetime and Propagation Length In the case of plasmonic resonators, one considers standing waves (kx real) with a complex ω. Depending on the situation, one may be concerned with the amplitude lifetime, 1=Im ω, or the energy lifetime 1=ð2 Im ωÞ. A useful dimensionless figure of merit is the lifetime quality factor Q, which is defined as

Re ω Q : (7.31) 2 Im ω real kx This is 2π times the number of oscillation periods required for the standing wave to decay to 1=e of its starting energy. In the case of propagating plasmons, one may consider ω real (continuous-wave excitation) and examine the amplitude attenuation length, 1=Im kx , which determines the propagation losses away from a localized excitation source; in this case, the standard dimensionless figure of merit, known as the inverse damping ratio, is given by

Re kx γ1  : (7.32) k Im kx real ω This is a sort of propagation quality factor (not to be confused with Q): γ1 k is 4π times the number of wavelengths that have travelled away from the source before the wave decays in energy by 1=e. Provided the losses are small, the temporal and spatial losses will be related via the group velocity vg ¼ dω=dk x of the mode6 6

This generic relation for wave phenomena occurs since plane wave modes can be defined as a complex zero of some analytic, two-parameter response function F ðk; ωÞ. The mode condition F ðk; ωÞ ¼ 0: defines a three-dimensional surface in a four-dimensional k, ω space. Movements along this surface follow the exact ∂F ∂F ∂F differential 0 ¼ dF ¼ ∂F ∂k dk þ ∂ω dω, implying dω=dk ¼  ∂k = ∂ω from which we can define vg ¼ dω=dk. If ð Þ ¼ 0 with real ω we have a mode F k ; ω and a complex k 0 0 0 0 , it should be possible to find a nearby solution
 F k 0 þ Δk; ω0 þ vg Δk 0 at real k by shifting by a small imaginary amount Δk ¼ i Im k 0 . This new mode has real k and Im ω ¼ vg Im k 0 .

116

Graphene Plasmonics

½Im ωreal kx vg ½Im k x real ω ,

(7.33)

giving the following relationship between lifetime quality factor Q and relative propagation length γ1 k Q

vp 1 γ , 2vg k

(7.34)

where vp 
ω=kx is the  phase velocity. Remarkably, for the simple square-root plasmon dispersion vg ¼ 12 vp we have that Q ¼ γ1 k . It is stressed however that square-root dispersion is not universal for graphene plasmons (see Section 1.3 and the following subsection), and in general one should not conflate the lifetime/propagation quality factors.

7.3.2

Sources of Damping, Influence on Plasmon The graphene conductivity losses are often the dominant cause of plasmon damping. A non-zero value of Re σ can come from extrinsic disorder such as electron scattering from charged impurities in the substrate, or ripples in the graphene. Intrinsically, thermal vibrations in the graphene lattice also cause scattering. Even without scattering processes, one finds Re σ 6¼ 0 in some regions of ω, kx due to Landau damping (from allowed intraband or interband single-particle electronic transitions excited by the plasmon), and this is captured in the RPA. The Drude formula (7.1) can be crudely used to capture scattering losses via the parameter τ, where it is considered as a constant and thus independent of ω and k. This is a simplification, however, and the effective dependence of both τ and D on ω, k is relevant for most of the plasmon damping process. Most significantly, the τ relevant for plasmonics is not the same τ as obtained from low-frequency transport measurements; this is further discussed in the next subsection. In general, the complex-valued k x or ω should be found as a complex pole of the impedance function discussed in Section 7.1.2, or equivalently as a complex zero of the RPA dynamical dielectric function discussed in Section 7.2.1. It is, however, informative to consider the case of Drude conductivity with infinite dielectrics of arbitrary ϵ ðωÞ in the quasielectrostatic limit, Eq. (7.11). For small losses

Re kx 1 Im ðϵ a þ ϵ b Þ 1 γ1 þ ¼ (7.35) k ωτ Re ðϵ a þ ϵ b Þ Im kx and the lifetime quality factor can be found using relationship (7.34), leading to

Re ω 1 ω ∂ðϵ a þ ϵ b Þ 1 1þ Q¼ þ (7.36) γk : 2 Im ω 2 ϵa þ ϵb ∂ω In the special case of non-dispersive and lossless dielectrics, one obtains simply Q ¼ γ1 k ¼ ωτ. In other words, the energy lifetime of the plasmon is simply the conductivity relaxation time τ. If there is dielectric dispersion, however, the lifetime may easily exceed τ, since the mode would no longer be purely plasmonic (it would

7.4 Experimental Observation of Graphene Plasmons

117

hybridize with, e.g., a phonon). The large dielectric losses near phonons will, on the other hand, tend to decrease the mode lifetime.

7.3.3

Extrinsic Plasmon Damping within RPA Framework Microscopic calculations of the plasmon lifetime require microscopic calculations of the proper non-local conductivity σ ðq; ωÞ (or proper density–density response χ~nn ðq; ωÞ) in the presence of agents external to the 2D electron liquid [85, 88]. The RPA response approximates the proper response, χ~nn χ ðnn0Þ – see Eq. (7.26). For small wavevectors, we can approximate the proper non-local conductivity in the Drude-like form [85, 87] σ ðq; ωÞ ’

D τ ðq; ωÞ1  iω

,

(7.37)

where now τ ðq; ωÞ is no longer a constant (compare with Eq. (7.1)). We stress that the plasmon lifetime does not coincide with the Drude transport time τ ¼ lim q!0 τ ðq; 0Þ, which is probed in a dc transport experiment. In the presence of long-range scatterers, which play an important role in graphene on SiO2 or at low densitities in high-quality encapsulated devices, the plasmon lifetime increases with increasing carrier density, remaining substantially shorter than the dc Drude transport time, even at high carrier densities [85]. The authors of [87] carried out a detailed study of the inverse damping ratio of plasmon–phonon polaritons in high-quality van der Waals stacks comprising graphene and h-BN. They considered two possible sources of scattering that limit the lifetime of these hybrid modes: scattering against graphene’s acoustic phonons and h-BN optical phonons. It was found that scattering against intrinsic acoustic phonons is the dominant limiting factor (yielding theoretical quality factor Q and propagation factor Re kx =Im k x in the range 50–70 at room temperature). Including thin film effects, dielectric dispersion, and dielectric losses – see Section 7.1.5 – it was found [97] that the plasmon lifetime displays a very weak dependence on carrier density and a strong dependence on frequency.

7.4

Experimental Observation of Graphene Plasmons Plasma waves and surface plasmon polaritons have been studied experimentally in metallic systems, doped semiconductors, graphene or semiconductor electron gases, employing a wide palette of experimental techniques. The most common technique for probing and exciting plasmons is based on radiation fields of visible, infrared or THz frequencies, depending on the plasma frequency to be addressed. Various alternative techniques include electron beams or photon emission. As discussed before in the theory section, the momentum of plasma waves in graphene is much larger than the momentum of free-space photons. Therefore, several methods are employed for

118

Graphene Plasmonics

launching and detecting plasmons, either by overcoming this momentum mismatch or by employing direct electrical coupling to the plasma waves.

7.4.1

Probing Plasmons by EELS and ARPES Before discussing the optical experiments on graphene plasmons, we will briefly address techniques which have been used for studying collective excitations in graphene, in particular angle-resolved photoemission spectroscopy (ARPES), and electron energy loss spectroscopy (EELS). The first experimental reports on plasmons in graphene, and in particular the interaction between plasmons and single-particle electron excitations, were based on ARPES [9, 95]. ARPES is a spectroscopic technique for measuring the electron distribution in reciprocal space. This is achieved by detecting the energetic and angular distribution of ejected electrons which are excited by high-energy photons (usually soft X-ray). Instead of the usual interactionless crossing of the two bands at a single point (Dirac point), these pioneering works reported the observation of the four bands which cross in a diamond-like shape. This modification of the electron/hole momentum distribution is well understood by considering interaction effects, which can be very strong in graphene [56, 81]. For example, electron–electron interactions can lead to the formation of quasiparticles, such as plasmarons which consists of electrons (or holes) and plasmons, coupled by Coulomb interactions between single electrons/holes and the plasma excitations in the electron sea. The existence of the plasmaron changes the topology of the bands near the Dirac crossing as theoretically predicated [47, 81]. The key signature of the involvement of graphene plasmons in the renormalized band structure was the observation of linear scaling of the dielectric function (and therefore the plasmon dispersion) with Fermi energy E F . The first experimental observation of the dispersion relation (energy versus momentum) for graphene plasmons for a wide range of energies was obtained by energy loss spectroscopy (EELS). EELS is a powerful tool for studying the interactions of electrons with plasmons, phonons, or single-particle excitations (inter- and intraband transitions). The working principle of EELS can be summarized as follows: when an electron beam impinges on a solid-state material, it loses part of its energy by inelastic scattering (e.g., by phonons, plasmons, or electronic single-particle excitations). By analyzing this energy loss in addition to the associated momentum transfer – employing an electron spectrometer – information about the inelastic interactions can be obtained. In contrast to optical absorption spectroscopy, EELS is not limited to zero-momentum transfer (q 0 for photons but electrons), thus providing information about the energy
not for 1  loss spectrum Im ϵ ðω; qÞ . For graphene, the observed energy loss spectrum showed a peak at the energy of the two-dimensional π plasmon (or sheet-plasmon). From these peaks, the energy and momentum of the plasmon resonances can be extracted. Based on this method, several reports on EELS or high-resolution EELS pffiffiffi (HREELS) applied to graphene confirmed the characteristic ω / q dispersion [54, 60, 63] for plasmons of 2D electron gases. Two plasmon modes were reported in [78], possibly due to multipole plasmons. Several studies on epitaxially grown graphene

7.4 Experimental Observation of Graphene Plasmons

119

on top of the polar substrate SiC [54, 60, 63] showed a strong deviation from the 2Dplasmon dispersion, explained by plasmon–phonon hybridization, as was predicted theoretically [47]. The coupling of collective electronic modes with optical phonons has been studied extensively on semiconducting structures using EELS spectroscopy [26, 53, 83]. More details on plasmon–phonon hybridization will be discussed below.

7.4.2

Plasmons in Patterned Graphene Structures and Substrates The size of the plasmonic material can strongly influence how it interacts with light, especially when it is much smaller than the free-space wavelength [57]. Discontinuities in the electric permittivity can set up standing localized plasmon waves confined to the plasmonic material, when the light frequency is tuned to its plasmon resonance. In a similar fashion, 2D localized plasmons can also be excited in lithographically designed graphene ribbons, disks, their periodic arrays, and anti-dots [10, 50, 55, 89, 102, 104, 107]. Grating structures can also be etched onto the substrate to couple light into graphene plasmons [38, 43, 80, 106]. Figure 7.3 illustrates these various plasmonic nanostructures and grating structures in graphene.

(a)

(c) 200 nm

GPC1 Yeung14 GPC4

D

Dielectric Graphene

a

D´ Eper

a

SiO2 Epar

GPC3 Grahene

Yan13

GPC2

To

T

(b) y

z

Si x

Indium wire for Hall measurements

(d)

Graphene

Piper14 Fang13 Fig. 7.3 Techniques for exciting graphene plasmons employing nanopatterned graphene and

substrates, i.e. (a) graphene nanoribbons array [102], (b) graphene dots array [30], (c) graphene antidot lattice [104], and (d) graphene on patterned silicon photonic crystal [80].

120

Graphene Plasmonics

As an example, let us consider plasmons in graphene nanoribbons. When illuminated at normal incidence with electric field polarized perpendicularly to the ribbon axis, plasmon modes can be excited. They propagate back and forth between the ribbon edges, setting up Fabry–Pérot modes. In a single homogeneous ribbon with width W, these modes should satisfy k p W þ ΦR ¼ πn, where W is the ribbon’s width, and ΦR is the reflection phase picked up by the plasmon as it is reflected off the ribbon edge, known to be reflected with virtually 100% amplitude attributed to the large density of states in graphene versus the photonic radiating modes [39]. If the fields vanishes exactly at the ribbon edge, then one obtains ΦR ¼ π . However, recent work has found a non-trivial reflection phase ΦR 3π=4 in the case of graphene due to the excitation of highly evanescent modes [74]. Hence, the fundamental mode n ¼ 0, will have the wavevector kp ¼ ΦR =W. Subsequent excited modes follow with n ¼ 2, 4 . . . as modes characterized by odd integers are antisymmetric with respect to the ribbon axis and do not couple to normal incident light. These modes have zero dipole moments, and are also known as dark modes. Experiments on these graphene nanoribbon plasmonic resonators reveal resonant absorption in their measured extinction spectra [10, 30, 50, 102], defined as
 1  T ⊥ =T k , where T ⊥ T k are the transmission spectra for perpendicular (parallel) polarized light with respect to the ribbon axis. Since these resonances reside in the terahertz to mid-infrared frequencies, the experiments typically involve an infrared microscope coupled to a Fourier transform infrared spectrometer (FTIR), used in conjunction with a broadband polarizer. Figure 4.4 illustrates the measured extinction spectra for graphene nanoribbon arrays with different widths. The observed resonances correspond to the plasmonic “cavity” modes we discussed earlier. With the ribbon width, which translates into the plasmon wavevector kp , a reasonable estimate of the ribbon plasmon resonances can be obtained from its 2D graphene plasmon dispersion sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e2 E F k p wpl ¼ : 2πħ2 ϵ eff E F is the Fermi energy and ϵ eff is the effective dielectric constant of the environment. This approximation gives reasonable agreement with experiments [50, 74, 102], as depicted in Fig. 7.4(b). The absorption resonance can be modeled by considering the h i iωD absorption cross-section of one disk [88, 106] 4π ðω=cÞ Im D3 A= ϵ 12L  σ ðωÞ . Here, ϵ 2

.Here,

D is the disk diameter, σ the optical conductivity of graphene, and the constants A = 0.65 and L = 12.5 are for disks, while other constants are used for other geometries [88]. Typically in these experiments on graphene plasmonic resonators, the plasmons reside outside the Landau damping regime. Nevertheless, these plasmons have a finite lifetime [10, 50, 102, 104, 107]. This is because several mechanisms can provide the momentum or energy to couple the plasmons with the single-particle phase space (i.e. Landau damping), where the plasmons will decay into electron–hole pairs. For example,

7.4 Experimental Observation of Graphene Plasmons

121

Fig. 7.4 Plasmons in graphene nanoribbons on a diamond-like substrate. (a) Extinction spectra

of graphene ribbons with varying ribbon widths. Inset shows the measurement scheme, performed using FTIR spectrometer. (b) Plasmons resonance plotted as a function of wavevector q ¼ π=W e , where W e is the effective electrical width of the ribbon. Plasmon data for a lower doping sample are also displayed (adapted from [102]).

high-energy plasmons can emit optical phonons in graphene (~0.2 eV) or scatter off short-range defects along the edges which provide large momentum transfer [48, 102]. Other phonons such as the acoustic phonons in graphene or the polar substrate phonons can also play an important role in the plasmon lifetimes [87, 97]. In sum, the damping rate Γpl in graphene nanoribbons can be phenomenologically described by Γpl ¼ Γ0 ðωÞ þ

a , W

where Γ0 ðωÞ refers to the damping due to the various phonons and the latter is due to scattering off the edges, where a is of the order of the Fermi velocity, and was experimentally found to be a 2  106 m s1 [102]. We note that the latter effect is also important in small plasmonic metal particles [51]. This simple model gives a good account of the dominant plasmon damping observed in experiments [102]. The damping rate can be obtained from the experimental spectra, through the full-width half maximum of the Lorentzian-like resonant profile, δω, and is related via δω ¼ 2Γpl . It is interesting to note graphene also accomodates peculiar one-dimensional plasmons modes propagating along the edges [70, 96]; however, these do not ordinarily couple to light. In fact, the fundamental mode in a graphene nanoribbon consists of such localized edge modes [75], while the higher-order modes are usual cavity modes as we discussed earlier. These edge modes were observed recently in an infrared near-field scattering optical microscope [35, 108]. By exploiting resonant excitation of plasmons, as discussed above, light absorption in patterned graphene can be strongly enhanced. This enhanced absorption is related to the oscillator strength of the plasmon resonance, which can be relatively strong, and even lead to absorption cross sections that are higher than the size of the resonant structure

122

Graphene Plasmonics

[15, 55]. Mid-infrared absorption spectra [10, 29, 100] on highly doped periodic patterns of graphene nanodiscs revealed absorption fractions of around 10%, taking into account the coverage fraction of the graphene, and up to 30% was observed by the implementation of a Salisbury screen [49]. This is remarkably strong absorption considering that this plasmonic system is only one atom thick, and usually absorbs only 2.3% [71]. Higher cross-sections are relevant for applications such as sensing and light-harvesting or photodetection. A more intuitive and quantitative insight is provided by the consideration that the absorption crosssection σ ext is governed by the oscillator strength of the resonant plasmon mode, which is in general proportional to the plasmon lifetime and the ratio of the carrier density and effective mass n=meff . Due to the small effective mass of graphene charge carriers, relatively large absorption cross sections are attainable. For a single-layer graphene pffiffiffi meff / n and thus σ ext / ω2p τ. In order to increase the oscillator strength, one can increase the plasmon resonance frequency (by reducing the size of the nanostructures or by increasing n), the plasmon lifetime, or one can employ multi-layer graphene/insulator stacks. It was shown in [100] that stacks of graphene disks exhibit enhanced resonant absorption due to the coulomb interaction between the disks. We remark that compared to conventional 2DEGs with typical densities up to 1012 cm2 , the density of graphene can be tuned to much larger values, up to 1014 cm2 [23, 52]. In combination with the smaller effective mass for graphene, compared with the effective mass of charge carriers in semiconductors, the accessible oscillator strength is much stronger for graphene resonant plasmon systems than for conventional 2DEGS. Additional advantages for confining light to very small volumes are the prospects for enhancing light–matter interactions, e.g., for sensing and enhanced non-linear response [16, 17, 18, 67]. One of the key parameters for quantifying the strength of light–matter interactions is the Purcell enhancement which is traditionally defined for a dipolar emitter coupled to an optical cavity. The Purcell factor P is proportional to Q=V, where Q is the quality factor and V is the mode volume normalized by the free-space mode volume λ30 . The ratio Q=V was quantified in [10] for localized plasmon modes in graphene ribbons 40 nm width. While the quality factor was fairly low, between 2.5 and 15, the mode volume was found to be extremely small, down to 106 λ3 . The ratio Q=V was reported to be within the range 106107 , in agreement with predictions by [55]. This Q=V ratio is 2 orders of magnitude larger than ratios reported so far for metallic nano wires [58, 61].

Plasmon–Phonon Coupling in Far-Field Experiments As we have discussed in Section 7.1.5, many dielectrics have a dispersive permittivity due to phonons, which strongly disturbs the electrodynamics of the plasmon. This can be fully captured by the frequency-dependent permittivity, which in the case of phonons is typically approximated by adding damped oscillator responses εðωÞ ¼ ε∞ þ

X α

δεα

ωTOα 2 , ωTOα 2  ω2  iγα ω

(7.38)

7.4 Experimental Observation of Graphene Plasmons

123

Table 7.1 Parameters for the permittivity function in Eq. (7.38) of the common polar substrate used for graphene, i.e. SiO2 [76], h-BN [11], 6h-SiC [76]. εð0Þðε∞ Þ is the static (high-frequency) dielectric constant, ωTO is the bulk transverse optical phonon frequency, δεα is the oscillator strength, and γα is the damping rate of the phonon mode α. Angular frequencies and damping rates have been divided by 2πc and expressed in units of cm1 . Dielectric SiO2 (amorphous)
!  h-BN ϵ x , ϵ y E ⊥ c
!  h-BN ϵ z E k c
!  6H-SiC E ⊥c

εð0Þ

ðε∞ Þ

3.9

2.3

6.90

4.90

3.48

2.95

10.0

6.7

ωTO1

ωTO2

ωTO3

δε1

δε2

δε3

γ1

γ2

γ3

451

752

1098

1.1

0.08

0.45

17.6

28

24

1360

-

-

2.00

-

-

7

-

-

-

760

-

-

0.53

-

-

2

-

793

-

-

3.3

-

-

4.8

-

-

where ε∞ is the high-frequency dielectric constant, ωTOα is the natural vibration frequency of the bulk optical phonon in the absence of long-range repulsion, δεα is the oscillator strength, and γα the damping rate of the phonon mode α. Table 7.1 summarizes some of these paramaters for the common dielectric substrates for graphene. By plugging this permittivity into (7.11), or its extension discussed in Section 7.1.5, one finds the full mode dispersion function kx ðωÞ that exactly includes the plasmon– phonon hybridization. In far field experiments on resonant nanostructures, however, one does not measure k x but rather one measures the mode frequency for a fixed k x determined by the nanostructure size. One is more interested in the function ωðkx Þ. In this section, we consider the special case of graphene on top of an isotropic dielectric with a single phonon mode, and show how for fixed kx the “bare plasmon,” which would have frequency ωpl , hybridizes with the “bare surface phonon mode” at frequency ωSO , leading to a hybridization and splitting which is analogous to harmonic oscillator coupling. The plasmon–phonon hybridization can also be understood as a the long-range Frohlich coupling through the surface optical (SO) phonons [37]. In the simple case of a dielectric-vacuum surface, thep surface phonon mode ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi (SO) is related to its bulk phonon frequency via ωSO ¼ ðεð0Þ þ 1Þðε∞ þ 1ÞωTO . The dynamical dielectric function (see Section 7.2.1) including these SO phonons can be written as εðωÞ ¼ εenv  εenv

X ω2pl ω 2SO  ε , env 2 2 2 ω2 SO ðω þ iγSO Þ  ωSO þ ω SO

where εenv is the average dielectric constant of the two half spaces, γSO is the damping rate, and ω 2SO is the Frohlich oscillator strength of the SO phonon mode given by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   1 1  ω SO ¼ ωSO 2 : ε∞ þ 1 ε0 þ 1 The dispersion of these coupled plasmon–phonon modes can then be obtained by solving for εðωÞ ¼ 0. In the simple case of only one SO mode, and γSO ¼ 0, the plasmon–phonon modes reduce to a simple biquadratic equation given by

124

Graphene Plasmonics

Fig. 7.5 Plasmons in graphene nanoribbons in SiO2 . (a) Extinction spectra of graphene ribbons with varying ribbon widths, vertically displaced for clarity. Vertical dashed line indicates graphene optical phonon frequency. (b) Plasmon frequency as a function of wavevector q ¼ π=W e , where W e is the effective electrical width of the ribbon. Two substrate surface polar phonons across the frequency range of interest, and the graphene optical phonon, are indicated as dashed lines (adapted from [102]).

ω2 ¼

ω2pl þ ω2SO  2

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2
 ω2pl þ ω2SO  4ω2pl ω2SO  ω 2SO 2

:

One can easily recover the results ωþ ! ωSO and ω ! ωpl in the limit of zero coupling, i.e. ω SO ! 0. Figure 7.5 shows the measured extinction spectra [102] for graphene nanoribbons array on a polar substrate SiO2 . Across the measured frequency range, there are two SO phonon modes at ωSO1 ¼ 806 cm1 and ωSO2 ¼ 1168 cm1 . The coupling of these phonons with the graphene plasmon results in three coupled plasmon–phonon modes, where the experimentally obtained dispersions (resonance position of these spectra) are plotted in Fig. 7.4. Another SO phonon mode at ωSO3 ¼ 406 cm1 resides outside the experimentaly frequency range.

Classical Harmonic Oscillators In the literature, plasmon coupling with other infrared active mode (i.e. phonons, plasmons) are often understood in classical terms with coupled harmonic oscillators. Consider a driven damped harmonic oscillator as illustrated in Fig. 7.6(a). Its equation of motion can be written as m1 x€1 þ

m1 x_ 1 þ k 1 x1 ¼ F cosðωt Þ, τ 1 =2

7.4 Experimental Observation of Graphene Plasmons

125

Fig. 7.6 (a) and (b) Simple damped harmonic and coupled two harmonic oscillators systems.

(c) and (d) Extinction spectra obtained from bilayer graphene nanoribbon array (adapted from [44]). (c) Measured extinction spectra of a ribbon array with ribbon width of 100 nm, and fitted to a coupled oscillator model as described in main text. (d) Evolution of extinction spectra with increasing doping.

where m1 is the mass of the oscillator, τ 1 is its lifetime, and k 1 is the spring constant. The dynamics of the oscillator has the solution xðt Þ ¼ a1 cos ðωt þ φ1 Þ, with amplitude a1 ¼ 

F 1 ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 m1
2 2 ω  ω1 þ ðω δω1 Þ2

and the phase shift with respect to the driving force is   δω1 1 φ1 ¼ tan : ω2  ω21 pffiffiffiffiffiffiffiffiffiffiffiffi Here, ω1 ¼ k1 =m1 is the resonance frequency of the undamped oscillator, and δω1 ¼ 2=τ 1 is the damping factor. The presence of damping affects the resonance pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi frequency, i.e. the frequency where a1 is maximum, ωmax ¼ ω21  δω21 ω1 , when ω1  δω1 . We note that the full-width half maximum of the ampliude spectrum is also given by δω1 . Figure 7.6(b) illustrates the coupling between two harmonic oscillators. Here, one considers a driven oscillator 1, coupled to oscillator 2, via the coupling parameter given by κ. In this case, one obtains the following expression for the amplitude of the driven oscillator [12, 44]

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Graphene Plasmonics


 F α22 þ β22 a1 ¼ q
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2
2ffi , κ2 α2  α1 α22  α1 β22 þ κ2 β2  β1 α22  β1 β22
 where αj ¼ mj ω2j  ω2 þ κ=mj and βj ¼ mj ω δωj . We consider a typical physical situation where such a model can be applied. Figure 7.6(c) shows the measured extinction spectra of a AB-stacked bilayer graphene ribbon array for two incident light polarizations [44]. The optical response of bilayer graphene has serveral interesting infra-red responses, such as mid-infrared plasmons, the Γ point optical phonon at 0.2 eV, and interband response at ~ 0.4 eV [64]. In the perpendicular polarization, the plasmons resonance is tuned to coincide with that of the optical phonon. Their largely mismatched lifetimes and weak coupling results in an induced transparency phenomenon, and can be largely reproduced by the simple coupled oscillator model we describe here. As the plasmons and phonon detuning increases, the induced transparency evolved into a Fano-like structure in the spectra as shown in Fig. 7.6(d).

7.4.3

Exciting Plasmons by a Dipolar Emitter A (point) dipolar emitter, placed close to the graphene, can couple efficiently to plasmons because its momentum covers a wide range [55, 73]. In particular, for graphene, this coupling can be very strong, leading to a significant modification of the emitter lifetime. The emitter decay is enhanced relative to the free-space emission by
rate 3 3π f =2½ðϵ þ 1Þ λ0 =λsp , where
λ0 is the 3 light wavelength and f = 1 ( f = 2) for parallel (perpendicular) polarization. As λ0 =λsp is about 102 to 107 for graphene, the plasmon generation rate (and thus also emitter decay rate) can be extremely high. The characteristic plasmon–emitter coupling strength decays with ek⊥ d, where d is the emitter– graphene distance and k ⊥ the inverse decay in the plasmon away from the surface. In [91], the coupling between graphene plasmons and erbium emitters was observed. The emission energy of erbium is 0.8 eV, and thus plasmons of 0.8 eV were excited, which is promising for the prospects of data communication applications. The coupling between emitter and graphene plasmons was experimentially verified by measuring the energy transfer rate from emitter to graphene Γeg , which is given by ð

 Γeg (7.39)  1 / dk k B kk Im rp kk , Γ0
 2 with r p kk ¼ ϵþ1þ2σik the Fresnel reflection coefficient, Γ0 the emission rate without k λ0 the graphene, and k the parallel wavevector. The bell-shaped weight function k
 2k k d represents the distribution of the wavevectors that contribute to the B kk ¼ e emitter–graphene coupling when they are separated by a distance d. The modified emitter decay rate, due to the presence of doped graphene, is shown in Fig. 7.7(a) for different emitter–graphene spacer layers. The emission contrast decreases for large Fermi energies, unless a spacer layer of 12 nm or more is present. This reveals the confinement of the plasmon to the graphene sheet below a lengthscale of about 12 nm.

7.4 Experimental Observation of Graphene Plasmons

(a)

(b)

0.9

(c) 15 nm distance

Wave vector, k|| (nm−1)

Emission contrast, Fg/F0

{rp} 12 nm spacer

5 nm spacer

No spacer 0.4 0.0 0.2

0.4

0.6

Fermi energy, EF (eV)

0.8

127

2

Graphene

Intraband

10 nm distance 1

Plasm

on

e–h pair excitation 0 0.0

d = 2 nm d = 5 nm

Photon emission 0.2

0.4

0.6

0.8

5 nm distance

d = 12 nm B(k||)

Fermi energy, EF (eV)

Fig. 7.7 (a) From [90]: plasmon launching by dipolar emitters with emission wavelength 1.5 μm (= 0.8 eV). The emission contrast as a function of Fermi energy for different spacer layers. The plasmon coupling regime is characterized by a downward-sloping emission with Fermi energy. (b) Calculated Fresnel coefficient r p as a function of E F and k k . The plasmon resonance is clearly visible for high Fermi energies. On the right side, we show the normalized bell shape that represents the contributing wavevectors, for three emitter–graphene distances d. (c) Snapshots of the instantaneous electric-field amplitude, demonstrating the near-field interaction between an emitter and graphene, for emitter–graphene distances of 15 nm, 10 nm, and 5 nm.

It is also insightful to inspect the Fresnel reflection coefficient in a colorplot as a function of wavevector and Fermi energy (as shown in Fig. 7.7). One can see the intraband and interband excitations, as well as the plasmon resonance emerging at Fermi energies above the emission energy of the emitter.

7.4.4

Real-Space Imaging of Plasmons by Scanning Probe Techniques An appealing technique for studying and imaging high-momentum plasmons in realspace is scattering near-field microscopy (s-SNOM). This imaging technique is based on light-scattering from a metal-coated and sharp AFM tip. This tip has an elongated shape along the z-direction perpendicular to the surface, with an apex rounding radius r 20 nm, which is sharp compared to the light wavelength r 110 μm. By illuminating the tip, the light-scattering at the tip apex produces a nanoscale-induced electric dipole oriented perpendicular to the surface. This dipole can excite optical fields with the wavevector up to kmax 1=r. Additionally, due to the strong optical field confinement under the apex of the tip, this technique enables imaging with a resolution of order r, which is far below the diffraction limit. The first experiment exploiting this technique on graphene was performed by [34] for the mid-infrared region 880–1270 cm1 (8–11 μm). Spectroscopic measurements of the near-field scattered light revealed a blue-shift of the optical phonon resonance of the SiO2 substrate. This was explained by considering the hybridization of the SiO2 phonon with the graphene plasmon. As discussed above, the plasmon–phonon interaction splits the dispersion into two branches. The observed blue-shift was in agreement with the energy shift associated to the higher branch and this energy shift was tunable by gate voltages, confirming the tunability of the plasmon–phonon hybridization, as expected by theory [46, 54].

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Graphene Plasmonics

The first observation of the spatial profile of propagating graphene plasmons [13, 32] enabled direct probing of the plasmonic properties such as its dispersion, losses, and scattering at edges and defects. These near-field real-space imaging experiments are based on the following principle, which is illustrated in Figs. 7.8(a) and (b). First, light scattered at the tip launches plasmons that propagate along the surface of the graphene sheet (or nanostructure). Next, these plasmons reflect at the edges of the graphene structure (or grain boundaries). Part of these plasmons propagate back to the tip, and are subsequently elastically scattered into photons, which are then detected with a pseudoheterodyne interferometer. The scattered intensity is mainly sensitive to the z-component of the field, as the tip is comparatively much less efficient in producing scattering of fields oriented along the other two remaining directions. Therefore, the s-SNOM setup is collecting a complex amplitude (including information on the phase) that is proportional to the zcomponent of the field produced by a dipole along z at its own position. The imaginary part of this field is the local density of optical states (LDOS). Thus, the detected signal is qualitatively correlated with the vertical component of the LDOS [13], also called partial LDOS. The experimental data and calculations show interference fringes which run parallel to the edge and decay away from the edge. This is consistent with the aforementioned physical picture of the near-field measurement technique. Interestingly, the spatial profile away from the edge (as shown in Fig. 7.8(c)) deviates from what is expected from basic interference effects and decay due to plasmon damping as well as additional decay in the signal due to the circular character of the waves. These deviations can be attributed due to the inhomogeneous doping profile at the edge, the finite size of the scattering tip leading to multiple reflections close to the edge, edge roughnesses or some additional intrinsic damping mechanisms. A more elaborate model was developed [32] to take some of these effects into account. Further from the edge, about one plasmon wavelength away, the near-field signal fits reasonably well with a simple interference model, including an additional (non-lossy) decay factor due to the cylindrical character of the plasmon wave pffiffiffi sðd Þ ¼ e2ik1 d2k2 d = d, where k ¼ k 1 þ ik 2 is the complex plasmon wavevector. Propagation losses are characterized by the ratio k 1 =k2 , discussed in Section 7.3. Near-field data and a fit to this basic equation are shown in Fig. 7.8(d). In [97], this was used to show that the lifetime of plasmons in graphene can be as high as 500 fs, in the case of pristine graphene encapsulated inside h-BN layers. Because the near-field imaging experiments probe the interference of forward- and backward-propagating plasmons, the maxima of the interference fringes are separated by half the plasmon wavelength λp =2. For excitation wavelengths of the range 811 μm, a plasmon wavelength of about 70–500 nm was found [13, 97], which is about a factor of 40 smaller than the free-space excitation wavelength. This is consistent with the 4 EF compression factor λp =λ0 ¼ α ϵþ1 E 0 (see theory section), with α ¼ 1=137 the fine structure constant. From these precise measurements of kp ¼ 2π=λp , the dispersion relation can be extracted. In [13], the dispersion relation for epitaxial graphene on carbon-terminated

7.4 Experimental Observation of Graphene Plasmons

129

(c) 1

0 100

λp/2

Re ξopt (a.u.)

Distance from edge (nm)

(a)

200 300 400 500

200 nm

ħω = 116 meV 12 −2 ns = 7.4 × 10 cm

0

(d)

(b) λ0 =

Graphene

h-BN

SiO2

10.6

μm

90 nm

7 nm

20 nm

46 nm 0

Ex

γ p−11 = 70

0

200

400

600

800

Distance from edge (nm) Fig. 7.8 (a) Diagram of a typical experimental configuration for launching and detecting plasmons in graphene. The metal-coated AFM tip is illuminated by a laser beam (adapted with permission from [32]). (b) Simplified side-view schematic of the s-SNOM measurement on encapsulated graphene (by h-BN) including probe tip, excitation (with laser source of wavelength λ0 ) and detection. Plasmons are launched radially from the tip. The dark and lighter gray features show the simulated in-plane component of the electric field of a dipole source oscillating at a photon energy of 116 meV coupling to graphene plasmons. The simulated field confinement in the x-direction (E x ) of the plasmon in the out-of-plane direction of 20 nm full-width at half-maximum can be seen on the right. (c) s-SNOM optical signal from a two-dimensional scan of the tip position, near the graphene edge (dashed line) at room temperature. Edge-reflected plasmons appear as interference fringes. (d) Black crosses show the s-SNOM optical signal from (c), averaged along the edge, with a smooth background subtracted. The shaded regions show the exponential decay envelopes for the measured damping ðγ1 ¼ 25Þ and for the case limited only by electron scattering from thermal phononsðγ1 ¼ 25Þ. (Panels (b),(c), and (d) have been adapted from [97].)

pffiffiffi 6H-SiC was extracted and strongly deviated from the basic relation ω / k owing to the coupling of plasmons to the SiC surface phonons [13] (further discussed in Section 7.4.2. Due to the strong dispersion of the SiC dielectric constant around the phonon resonance, the plasmon wavelength was tuned over a very wide range while varying the excitation wavelength only over a small range. This relatively “flat” dispersion was extracted from the near-field experiments. Intriguingly, the coupling between plasmons and phonons was shown to be tunable. The authors of [21] showed that hyperbolic polaritons can be effectively modulated in a van der Waals heterostructure. The near-field imaging technique is also suitable for probing localized plasmon modes, such as standing waves between the edges of a ribbon. Due to the resonant character of these modes, an enhancement of the near-field signal was observed at the resonance conditions where the ribbon width is comparable to λp or λp =2. This is

130

Graphene Plasmonics

Fig. 7.9 Near-field amplitude image acquired for a tapered graphene ribbon on top of 6H-SiC.

The imaging wavelength is λ0 ¼ 9:7 μm. The tapered ribbon is 12 μm long and up to 1 μm wide. Bottom: Gray-scale image of the calculated local density of optical states (LDOS) at a distance of 60 nm from the graphene surface (adapted from [13]).

indicated by the white and red arrows in Fig. 7.9. These observations were consistent with the LDOS calculations as shown in Fig. 7.9. Interestingly, the field intensity is maximum at the edges for the lowest order mode (indicated by the red arrows). This suggests that the mode-profile is equivalent to a Fabry–Pérot model with a zero-phase reflection coefficient. This situation is also encountered in plasmonic slot waveguides [7] and plasmonic nanoantennas [25], and it originates in the complexity of the reflection of 3D electric fields at the edges. An important aspect of graphene plasmons is the strong optical field confinement. This feature was directly reflected by the observed resonant modes. Experiments revealed a mode volume of about 106 below the free-space mode volume λ30 [13]. This mode volume is extracted by measuring directly the mode-profile in-plane. The decay in the mode-profile out-of-plane, with typical distance 1=k⊥ can be extracted from the relation k 2k k 2⊥ , and thus the plasmon decays exponentially away from the graphene sheet with a typical length scale k k =2π. Here, kk can be extracted experimentally from the in-plane mode-profile. Due to the strong plasmonic field confinement, it is possible to use the near-field imaging technique as a probe for nanoscale interactions. For example, grain boundaries in CVD-grown graphene have been visualized using infrared s-SNOM. This has been realized by infrared nanoimaging of reflected surface plasmons from the graphene grain boundaries, and thus causing plasmon interference [14, 33]. In addition, the dynamical response of graphene on a crystal of hexagonal boron nitride is significanly altered due to superlattic structures. In [72, 82], the formation of these collective modes is discussed.

7.4.5

Plasmon Tuning An important advantage of graphene plasmons compared to metallic plasmonic systems is the tunability of the carrier density which governs the plasmonic properties such as

7.4 Experimental Observation of Graphene Plasmons

131

wavelength and damping. Various techniques are available for introducing charge carriers in graphene: 





The most common technique is electrostatic doping by a gate: the application of a gate voltage leads to a potential difference between gate and graphene, inducing additional charge carriers in the graphene. This shifts the Fermi energy up or down, relative to the intrinsic Fermi energy without voltage applied. For pristine graphene, the intrinsic doping is zero and therefore the Fermi energy is at the Dirac (or charge-neutrality) point. However, contaminations on the surface, or graphene–substrate interactions [41, 59], can lead to residual charge carriers in graphene, shifting the Fermi energy away from the Dirac point. A major disadvantage of gating techniques is dielectric breakdown, which limits the Fermi energy to about 0.3 eV An alternative gating technique to reach higher doping levels is electrolyte gating. Due to the nanometer-thick Debye layer, the electrical potential drops over a distance of only a few nanometers, and thus a much larger gate capacitance compared to oxides can be achieved. In this way, doping levels of up to 1014 cm2 have been realized [103] using a liquid electrolyte or a solid polymer electrolyte gate [23, 28]. Even higher doping levels can be achieved by chemical doping techniques [63, 31], such as surface transfer doping or substitutional doping. Unintentional chemical doping is often encountered by species absorbed from the surroundings or residual polymers used during device fabrication. Chemical treatments such as intercalation can also be used to create intentional doping, e.g., for the purpose of plasmonics or highly conductive transparent electrodes. Intercalation is the reversible inclusion of a molecule (or group) between two other molecules (or groups). An example is intercalating few-layer graphene with ferric chloride (FeCl3), for which Fermi energies up to 1.5 eV and densities up to 91014 cm2 have been realized, while maintaining charge carrier mobility of 2000–3500 cm2/V s [52].

Experimentally, in-situ plasmon tuning is exhibited by near-field imaging experiments as well as optical spectroscopy (see Fig. 7.10). Specifically, plasmon absorption peaks shift to higher energies for higher n, due to “stiffening” of the plasmon wave. Additionally, the amplitude of the absorption resonance increases for higher n due to the increase in oscillator strength which scales with ω2p . Quantitatively, the relation ωp / n1=4 was reported for many experiments, based on spectroscopic or near-field techniques [10, 13, 29, 32, p50]. ffiffiffiffiffi This is as expected because for resonant plasmons the wavevector kp / 1= W is defined (and thus independent of n) by the geometry and size W of the pffiffiffi structure while the relation between kp and ωp depends on n: k p / ω2p =EF / ω2p = n. Thus, the plasmon resonance shifts to higher frequencies for higher E F , as kp is constant. These observations have also been exploited to confirm the Dirac fermion nature of graphene plasmons (Dirac plasmons), as the resonance frequency depends on carrier density according to ωp / EF / n1=4 . In contrast, classical plasmons follow the relation ωp / n1=2 because the effective mass is not dependent on n and thus the Drude weight is proportional to the ratio of the carrier density and mass of the charge

Graphene Plasmonics

(c)

(b)

Frequency (cm–1)

ħ = 116 meV 120

200

400 −50

0

(nm)

0

Re xopt (a.u.)

Distance from edge (nm)

(a)

80

p

132

40 0

0

2 4 6 8 10 ns ( 1012 cm−2)

50

Vg (V)

Fig. 7.10 Gate tuning of plasmon properties. (a) Far-field plasmon resonance tuned by varying the carrier density with a gate (adapted with permission from [49]). (b) s-SNOM optical signal, from a scan of tip position perpendicular to the graphene edge (dashed line) and gate voltage, showing the gate dependence of plasmon fringes. (c) Plasmon wavelength dependence on carrier density. (Panels (b) and (c) adapted from [97].)

carriers: n=m. For graphene, the Drude weight n=meff is governed by effective mass meff ðk Þ (dynamical mass) of graphene charge carriers, which is proportional to momenpffiffiffi tum and thus n: meff ðkÞ ¼ ħk=vF / n. Therefore, the Drude weight (and thus ω2pl ) is pffiffiffi pffiffiffi proportional to n=meff / n= n ¼ 1= n. We remark that deviations from the relation ωpl / n1=4 due to carrier–carrier scattering have been predicted [62, 81], but have not been confirmed experimentally. The tunability of the wavelength and losses of plasmons by in-situ tuning of the carrier density by gate voltages was also revealed by near-field measurements in various ways. First, the location of the resonant “cavity-like” plasmon mode in the tapered ribbon was tuned by gate voltages as discussed in [13]. For larger carrier concentration, the resonant mode shifts to the wider part of the tapered ribbon, consistent with the pffiffiffiffiffi ffi expected scaling λp / jEF j / jnj. Moreover, the interference fringes close to an edge exhibit an increase of fringe amplitude and periodicity with increasing n, as shown in Fig. 7.10(b). By tuning the carrier density into the interband excitation regime E < 0:7EF , strong damping was observed and the near-field signal disappears (see red band of Fig. 7.10(b)). In this way, it is possible to switch plasmons “on” and “off.”

7.4.6

Magnetoplasmons Magnetoplasmons have been studied extensively for low frequencies (0–100 MHz) in two-dimensional electron systems (e.g., Si and GaAs), both experimentally [68, 3] and theoretically [36, 69, 92, 93]. Due to the low cyclotron mass of graphene carriers (about 2 orders of magnitude lower than in noble metals), plasmon resonances respond strongly to the magnetic field [19, 101], leading to a distinct splitting of the plasmon mode at higher frequencies and for moderate fields of 1 T. This splitting was observed for graphene nanodiscs at high excitation energies of up to 5 THz [19, 101].

7.4 Experimental Observation of Graphene Plasmons

133

Fig. 7.11 Splitting of the plasmon modes due to a magnetic field. For lower frequencies, the edge magnetoplasmon emerges and for higher frequencies the bulk magnetoplasmon. Adapted from [101].

Interestingly, the magnetoplasmons were observed not only for well-defined circular discs, but also for large-area CVD graphene containing natural nanoscale inhomogeneities, such as substrate terraces and wrinkles. These form natural scatterers for plasmons, and can even lead to a strong THz plasmon resonance [19]. The two branches of the field-induced split plasmon mode exhibit dispersion relation: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω4 ω ¼ 4c þ ω20  jω2c j [36], where ω0 is the plasmon frequency at zero field, and ωc ¼ eB=mc is the cyclotron frequency. The higher frequency branch is the bulk mode which increases with field due to the field-induced gap (cyclotron energy ħωc ), eventually approaching the usual cyclotron resonance which depends linear on field. The lower branch is a gapless collective mode confined to the edge (edge magnetoplasmon, also abbreviated as EMP) which decreases with the field [36]. A remarkable change in resonance line width was observed by [101]: the upper branch broadens for increasing field while the lower branch strongly sharpens for increasing field. The latter is due to the suppression of backscattering by the magnetic field. At even higher magnetic fields (and lower densities), the quantum hall regime can be reached where edge electron transport is chiral and backscattering is fully suppressed. Graphene edge magnetoplasmons in this quantum hall regime have been reported in the time domain by Petkovic [77]. Ultra-fast voltage pulses, applied to a voltage probe attached to a circular sheet of graphene, were used to launch EMP and ultra-fast voltage

134

Graphene Plasmonics

measurements to probe the propagation in the time domain. A constant propagation velocity was found, in contrast to the strong dispersion for bulk plasmons. Chiral propagation of EMP along the edge was observed with extremely low attenuation, corresponding to a plasmon relaxation time of 50 ps. This long plasmon lifetime reveals additional advantages of EMP above bulk graphene plasmons.

7.5

Applications Vibrational spectroscopy is a powerful, non-destructive analytical technique that provides a fingerprint of molecular and bio-molecular structures. However, the interaction of infrared light with wavelengths in the μm range with nm scale molecules is weak and thus requires rather a large amount of the analyte. The large confinement that graphene plasmons offer at infrared wavelengths makes them promising candidates for the enhancement of vibrational absorption spectra. The first studies involved thin organic layers such as MMMA and PVP on graphene nanoribbon arrays [99]. These polymers contain C=O chemical groups, so the studies concentrated on the C=O vibrational absorption at about 1700 cm1 . Tuning of the plasmon frequency was achieved by changing the width of the nanoribbons. When the plasmon resonance frequency was far away from the vibrational frequency, normal vibrational absorption was observed. As the overlap of the two excitations commenced, characteristic lineshape changes were observed for light polarized perpendicular to the ribbons, and at resonance an EMT was clearly observed (Fig. 7.5). For an 8-nm-thick PMMA film, excitation resulted in a transparency dip of 0.63% compared with a 0.25% absorption without the plasmon. Taking into account the filling factor of the nanoribbon structure, this corresponds to a five-fold enhancement of the absorption of the C=O in the film. Deposition of PMMA on the nanoribbon led to a red-shift of the plasmon absorption that increased with increasing film thickness due to the change in the dielectric environment. This type of shift is utilized in refractive index sensors for organic and bio-organic systems. At saturation coverage, the observed 260 cm1 shift corresponds to a high refractive index sensitivity of ~2650 nm/RIU, a figure of merit used in plasmonic sensors [65]. From the change of the red shift as a function of PMMA thickness, the 1/e decay in the plasmon E-field into the dielectric was obtained to be d0 ¼ 10  2 nm. Considering that the incident light wavelength was λ0 ¼ 6 μm, this decay length corresponds to a confinement factor in the normal direction of λ0 =ð2πd0 Þ 100, and similar results were obtained with even thinner films of PVP [100]. A similar approach was utilized to study the vibrational spectra of proteins [20]. The proteins, recombinant protein A/G and goat anti-mouse IgG antibody, were deposited on graphene nanoribbons and the vibrations associated with the C=O stretch and N—H wag/C—N were monitored. In this case, the tuning of the plasmon was performed electrostatically. By depositing the protein bilayer on the nanoribbons, a 160 cm1 redshift of the plasmon was observed. More recently, experiments in the submonolayer molecular coverage utilizing PTCDA molecules on graphene nanoribbons (GNRs) were performed. The plasmon resonance was controlled using chemical doping. At a

7.5 Applications

135

thickness below 0.6 nm, when no vibrations were observable in the conventional FTIR spectrum, clear EMT transparencies were observed when the plasmon was in resonance with the PTCDA C=O, ring C—C and C—H modes. Studies of the dependence on the detuning of the plasmon and vibrational excitations showed that the C—C ring vibrations couple stronger to the graphene plasmons through ππ interactions than the dipole coupling of C=O. Given that the plasmon field extends to a distance of about the plasmon wavelength above the graphene surface, the spectra of gases whose molecules are within that distance from the surface would be enhanced. This indeed was demonstrated recently for acetone and hexane vapors [24]. All the studies of vibrational absorption enhancement by graphene plasmons that have been performed so far involved a blanket coverage of the entire device surface. However, the field enhancement is mainly concentrated in certain regions, the so-called “hot spots.” For example, in a GNR the field is highest at the edges of the ribbon. Simulations show that if molecules could be directed to such hot spot sites, for example by factionalization, enhancements of many orders of magnitude would result. In a recent computational study of the plasmonic enhancement of the vibrational absorption of monolayers of the molecule pyridine over graphene nanodots, enhancements of 3–4 orders of magnitude were predicted [2]. It was concluded that using broadband light for excitation and scanning the graphene Fermi energy (through a gate), the transmitted intensity would reproduce the infrared spectra without the need of a spectrometer. Since the graphene plasmons can enhance the infrared absorption, it can also be employed to enhance infrared photodetectors [66]. Photodetection is driven by the carrier heating effect, i.e. bolometric, and can be further optimized by eliminating heat loss pathways as is commonly done in bolometers. Kirchhoff’s law tells us that the absorptivity should equal the emissivity of a material which would imply an enhancement of the infrared emission. Thermal emission from a graphene nanoribbon array on SiNx has been studied [94]. Along with the broad thermal emission, two narrow band emissions at 1360 cm1 and 730 cm1 were observed. The 1360 cm1 emission was found to be polarized perpendicular to the ribbon axis and to be tunable by changing the gate voltage. The 730 cm1 band was essentially unchanged by gate bias. It was concluded that the high-energy band is due to emission from the confined plasmon of the ribbons, while the low-energy emission is due to emission from the phonon of the underlying SiNx. Another interesting application of graphene nanoresonators is to design electrically tunable metasurfaces that can control the angle of reflection of an incident mid-infrared beam [1, 27]. By utilizing nanoribbons with different widths (or doping), one can design plasmon resonators that impart a spatially varying scattering phase to the impinging light. In our classical harmonic oscillator discussion, the oscillator can have a different phase shift with respect to the driving force, which also depends on the detuning frequency. Such a phase gradient, dθ dx , can result in an angle of reflection that differs from the angle of incident via the generalized Snell’s law described by sinðθr Þ  sinðθi Þ ¼ 2πλpffiε dθ dx [105]. However, since the resonator can only impart a phase from zero to π, a cavity resonator was introduced to provide the required zero to 2π phase shifts [27].

136

Graphene Plasmonics

7.6

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8

Electron Optics with Graphene p–n Junctions James R. Williams

8.1

Introduction Quantum materials offer the opportunity to investigate electronic behavior beyond electron-like excitations, and graphene is (from an electrical perspective) our highestquality quantum material. Its honeycomb arrangement of carbon atoms produces a gapless, linear energy dispersion, and electron mobility can now reach values near 1,000,000 cm2/V s [1–3]. Aside from reaching mobility near the best two-dimensional semiconductor materials, its energy dispersion allows for novel device phenomena, enhancing device capabilities beyond electronics based on conventional material electrical devices. Within a few years of isolating single-layer graphene, several groups developed methods to create adjacent p (hole-like) and n (electron-like) regions within a single sheet of graphene [4–6] (Fig. 8.1). Around the same time, the role of disorder on charge transport at low carrier densities was understood to be dominated by puddles of electrons and holes [7, 8]. Hence, this junction is essential to many electrical properties of graphene, and a thorough understanding of the role the p–n interface plays in electron transport is essential. Small devices, where the mean-free path is of the order of or greater than the device size, enable the investigation of optics-like phenomena using electrons rather than photons. For example, focusing on ballistic electrons using an electrostatic lens has been demonstrated in GaAs two-dimensional electron gases [9, 10]. However, in typical semiconductors, only carriers of the same sign of electrical charge are allowed in adjacent regions – carriers of a different sign are separated by an insulating space where the Fermi energy transitions through the band gap. For gapless materials such as graphene, carriers are allowed to switch sides at the interface; electrons can be transmitted or scattered as holes (and vice versa). From this come new phenomena, augmenting the optics-like phenomena available to electrons. Further, the relative immunity of the mean-free path to temperature in graphene [11] allows some of the novel optical phenomena in graphene to persist to high temperatures (>100 K). In this chapter, I will survey the current status of research in graphene p–n junctions. In Section 8.2, the basic theory of electrical transport through a single p–n interface at B = 0 and B > 0 (B is the magnetic field) will be described. Section 8.3 will review three aspects of the optical-like phenomena capable in graphene: Veselago lensing, confinement and guiding, and snake states. Recent experimental progress in this area will be detailed. Finally, some key future research directions will be discussed in Section 8.4. 141

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n-type

transition p-type region

Fig. 8.1 (Above) Schematic of dual-gated graphene device used to create a p–n junction within

a single sheet. (Below) The electric field provided by the gates in the above device produces a potential difference in the graphene sheet, creating a transition region between electrons and holes: a p–n junction.

8.2

Basic Electrical Properties of p–n Junctions

8.2.1

Sublattices, Valleys, and Pseudospin Graphene is known to posses a linear-in-energy band structure and the Hamiltonian of a single sheet of graphene resembles the relativistic Dirac equation. At six high-symmetry points in the Brillouin zone, the conduction and valence band meet at a single point called a Dirac point; two of these reciprocal lattice points or “valleys” are unique, termed the K and K0 points or Dirac points. The Dirac equation can be obtained from tight-binding approximation [12] using nearest-neighbor hopping between the inequivalent lattice sites A and B (A to 3B atoms and B to 3A atoms). Using a small momentum k expansion for values around the K and K0 points gives E ¼ ℏνF k,

(8.1)

where ℏ is Planck’s reduced constant and vF is the Fermi velocity. Embedded in the hexagonal arrangement of carbon atoms is a degree of freedom associated with the amplitude of the wave function on each of the A and B sublattices. This degree of freedom can be viewed in a similar light to the spin associated with an electron, and assume an “up” value for A sublattice and a “down” value for the B sublattice. For this reason, the sublattice degree of freedom is called the pseudospin. The hopping motion of electrons between the two lattices produces an artificial magnetic field that is proportional to the electron’s momentum. This gives a “handedness” or chirality to the direction of the pseudospin that is opposite for electrons and holes. The pseudospin texture for the K and K0 valleys are shown in Fig. 8.2(a). This has a fantastic effect on transport of electrons in graphene: electrons are protected from backscatter by their pseudospin and, for any pseudospin-preserving barrier, carriers are transmitted with unit

8.2 Basic Electrical Properties of p–n Junctions

143

(a) σ+

σ–

K electrons

K´ electrons

σ–

σ+

K holes

K´ holes E+V

(b) σ+

σ–

σ+

σ+ E Fig. 8.2 (a) Pseudospin up ðσ þ Þ and down ðσ  Þ for electrons and holes in the K and K0 valleys.

Note that the direction of motion and pseudospin are linked. (b) An illustration of the Klein phenomena in graphene; conservation of pseudospin protects against changes in direction of propagation from potential variations.

probability (Fig. 8.2(b)). This is one description (albeit oversimplified) of the Klein phenomena in graphene. However, the situation is complicated if carriers are allowed to scatter between valleys. A right mover in the K valley has the same pseudospin as a left mover in the K0 valley; hence for any scatterer that allows for the transfer of electrons between valleys – intervalley scattering – the arguments above no longer hold.

8.2.2

Angular Dependence of Transmission, Sharp, and Smooth Barriers The angular transmission of carriers across a p–n interface will be considered for two situations: a sharp and a smooth barrier. A barrier is defined as sharp or smooth by comparing the Fermi wavevector kF to the width of the transition region between the n and p regions, d. For kF d < 1, the barrier is defined as sharp, while for kF d > 1 the barrier is smooth. The transmission characteristics of a sharp barrier will be investigated first for each case of a symmetric junction: |n| = |p|. A simple tight-binding approximation allows for an accurate construction of the graphene band diagram and electron eigenstates. For a nearest-neighbor, tight-binding approximation, the Hamiltonian is of the form H ¼ vF p  σ, which is identical to the Dirac equation, with the role of spin being replaced by pseudospin. Eigenstates for this Hamiltonian will therefore have the same spinor form   1 eiθ=2 eiθ=2 ikr ψðkÞ ¼ pffiffiffi ¼ pffiffiffi χ s eikr , (8.2) þiθ=2 e 2 se 2

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Electron Optics with Graphene p–n Junctions

where tanðθÞ ¼ k y =k x is the angle of the pseudospin vector (defined below), s ¼ 1 for electron/hole states, and χ  are the spinors for electron and hole states. The sharp limit consists of a step function barrier at the origin and is independent of y. The transmission of an incident electron can be calculated by conserving the transverse component of momentum k y , as demanded by the symmetry of the barrier, and by matching the incident, reflected and transmitted spinor amplitudes at the interface x ¼ 0. First, the former demands both specularity in the reflection angle (θr ¼ π  θi , where θi and θr are the angle of incidence and reflection, respectively), giving rise to an equation similar to Snell’s Law for the law of refraction in optics sin ðθi Þ kt Ep ¼ ¼ , sin ðθt Þ ki En

(8.3)

where θt is the angle of transmission and En , E p are the Fermi energies in the n and p regions. Here is evidence of the first demonstration of the novelty of graphene p–n junctions. The ability to tune the Fermi energy to negative values (by tuning the region to p-type carriers) allows for the index of refraction η to be negative, and the consequences of this property will be considered later in this chapter. Matching the spinor amplitude at the interface gives χ i þ rχ r ¼ tχ t ! r ðθi Þ ¼

si eiθi  st eiθt , si eiθi þ st eiθt

(8.4)

where the relationship between θi and θr from above was used, and the incident and reflected wave were taken to be in the n region. Solving for t for a symmetrically bias junction gives jt j2 ¼ cos2 ðθi Þ, and perfect transmission is obtained at normal (θi ¼ 0) incidence, regardless of the height or length of the barrier. The perfect transmission holds for conditions when the p–n junction is not symmetrically biased [13]. The angular dependence of transmission can also be viewed through the lens of pseudospin. An electron approaching the barrier at normal incidence from the right has group velocity dE=dk > 0, with momentum ℏkx and right-pointing pseudospin. It is transmitted as a hole with momentum ℏkx and right-pointing pseudospin (Fig. 8.2), matching the incoming wave perfectly. For finite values of k y , which must be matched on either side of the junction, there is a mismatch of incoming and outgoing pseudospin of angle. The resulting reflection amplitude is related to the scalar product of the two spinors: χ †i  χ t e sin ðθi Þ. For a potential that varies smoothly, the transmission properties of the barrier can be treated within a WKB approximation. As can be seen in Fig. 8.3, the dispersion along kx acquires a “k y -dependent gap”, resulting in a mimicked classical forbidden region between n and p. The size of the forbidden region increases as ky increases, thus a decreasing transmission probability with increasing ky is expected. The case of a linear transition region between n and p was treated in [14] and it was found that the transmission probability as a function of angle is jt ðθi Þj2 ¼ eπ ðkF dÞsin θi : 2

(8.5)

8.2 Basic Electrical Properties of p–n Junctions

(kx, ky = 0)

(kx, ky

145

0)

E

kx ky E

E kx

kx n

p

n

p

Fig. 8.3 An example of the “momentum-dependent gap” acquired by electrons approaching the

p–n junction with a smooth transition region. For k y ¼ 0 (left), the process of k x (electron) ! k x (hole) cuts through the Dirac point, i.e. available states occur for all values of k x during the transition. (Right) For finite values of k y , a classically forbidden region occurs and the carrier must tunnel across the k y momentum gap.

Thus, in either case – smooth or sharp – the barrier acts to decrease the transmission probability. For example, in the case of a sharp junction symmetrically biased (|n| = |p|), the effect of reduced transmission probability at angles of incidence away from zero is to lessen the conductance by 2/3 [15] ! k2y 2e2 X 2 Gnp ¼ 1  2 ! Gnp ¼ Gnn : (8.6) ky 3 h kF Further, jt ðθi Þj2 in both the sharp and smooth junctions acts as a collimator: only carriers within a window around θi ¼ 0 have an appreciable chance to propagate through the interface. This effect allows for some of the optical analogies that will be discussed below. However, this collimation and other novel phenomena are lost when an appreciable amount of disorder is added at the interface, as was studied in [16], hence the peculiar properties of these junctions must be investigated under low-disorder conditions.

8.2.3

Experimental Properties: Device Design and Basic Transport Properties Fabrication of graphene p–n junctions have now gone through a number of generations. Common to most devices is a two-gate configuration consisting of a global back-gate (either V BG or V B in this chapter) and a local top-gate (either V TG or V T in this chapter). The global back-gate typically consists of the degenerately doped silicon wafer with a

146

Electron Optics with Graphene p–n Junctions

silicon oxide on which sits the graphene device and controls the carrier density everywhere on the sheet of graphene. A local top-gate covers only part of the graphene device and the carrier density underneath is controlled by a linear combination of the top-gate and back-gate. The coefficients of this linear combination are controlled by the capacitive coupling of the device at each gate. Early devices were created using gate oxide which covered the sheet of graphene and electrically isolated the local gate. Careful attention was paid to the oxide to ensure uniform coverage and the preservation of the graphene sheet [5]. The next generation moved away from oxides to suspended gates [17], avoiding the degradation of the graphene mobility in the oxide from the topgate geometries. The most recent version of graphene fabrication have eliminated the oxide from both the back-gate and top-gate and replaced it with either a suspended gate [18, 19] or boron nitride (substrate) [20] and either a suspended gate [21–24] or boron nitride top-gate dielectric [25, 26]. This experimental progress has elevated the mobility from around 10,000 cm2/V s to above 100,000 cm2/V s and has allowed for the investigation of optics-like phenomena in graphene. The transmission properties of a p–n junction fabricated using the clean methods above were first reported by [23, 24]. An air bridge device micrograph and resistance data are reproduced in Figs. 8.4(a) and (b). This air-bridge produces two back-to-back p–n junctions, each of which can be characterized by the above jt ðθi Þj2 for smooth junctions (Eq. (8.5). Figure 8.4(b) shows the increase in resistance produced by driving the top-gated region to an opposite polarity (p–n–p), resulting in an increase in resistance above the unipolar resistance associated with the angle-average transmission. In addition to this increase in resistance, regular conductance oscillations are observed and these oscillations can be useful in decoding the nature of transmission at the interface. These are the electronic analog of the Fabry–Pérot effect in optical cavities, whereby multiple reflections at the two parallel p–n interfaces interfere and give rise to the coherent addition of electron waves, producing regular peaks in conductance. Measuring the conductance provides only an indirect method of probing the Klein paradox, since conductance measures the angle-average of jt ðθi Þj2 , not jt ðθi ¼ 0Þj2 . However, Fabry–Pérot resonances measured in small magnetic fields do provide a direct measure of Klein tunneling [27], where a phase shift occurs directly as a result of the Klein paradox in graphene [28]. With the increase in quality of graphene devices, Fabry–Pérot resonances can now be observed in micron-sized gated regions, indicating top-gate devices can be fabricated with mean-free paths greater than 1 micron (Fig. 8.4(c)) [23, 24].

8.2.4

Transport Properties in a Magnetic Field At magnetic fields large enough to produce the quantum Hall effect in graphene, edge states form and the p–n junction plays an important role in determining the conductance of the device. The n and p regions have opposite directions of propagation in the quantum Hall regime, and at the interface each edge state copropagates along the junction with its partner (Fig. 8.5(a)). Should there be no crosstalk between the two adjacent edge states, the two-terminal conductance will be zero. Any amount of

8.2 Basic Electrical Properties of p–n Junctions

(a)

147

(b) VBG VTG

R (kW)

30 20 10

5 10 –10 –5 0 VBG (V), VTG (V) (c)

6

n

p

4

n

3

dg/dnTG (arb. units) –1

0

1

2

nBG (1011 cm–2)

5

1 0 –10

–9

–8

–7

–6

–5

–4

–3

–2

–1

0

nTG (1011 cm–2) Fig. 8.4 (a) SEM micrograph of a hybrid device with a boron nitride substrate and a suspended

“air bridge” top-gate. (b) A comparison of a back-gate (V BG ) sweep (black) of resistance R that controlled the density in the entire graphene sheet and a top-gate (V TG ) sweep (gray) that only controls the density in a region of the device. Since the device is intrinsically p-type at V TG ¼ V BG ¼ 0, application of a positive V TG produces a p–n junction, whereas no junction is formed in sweeps of VBG. The effect on resistance from the p–n junction can be seen in the increase in resistance of the gray curve above the black curve for positive gate voltage values. (c) Plot of dg=dV TG (where g ¼ 1=R) showing regular oscillations from Fabry–Pérot resonances, demonstrating phase coherent mean-free paths of several microns. (Adapted with permission from [23].)

crosstalk will cause equilibration between the left edge state at potential V and right edge state at potential 0, and the amount of equilibration will determine the conductance. In the case of full mixing – defined by the condition that a carrier approaching the junction in one edge state has equal probability of leaving the junction in either edge state – the conductance of a p–n junction is given by the filling factors of the Landau levels on the right and left side ðv1 ; v2 ¼ 2; 6; 10:::Þ Gnp ¼

v1 v2 , v1 þ v2

(8.7)

i.e. the series addition of the conductance of each region [5, 6, 29]. In early measurements, full mixing was achieved in this scenario for Landau levels 2 and 6 (Fig. 8.5(b)), and this mixing also has consequences for the shot noise measured across the junction [29, 30]. For high-mobility graphene devices, moderate magnetic fields allow for spin and valley resolved edge states. In the bipolar regime, the lowest Landau

148

Electron Optics with Graphene p–n Junctions

n1

Equilibration

(a)

n2

n2 = −2

n2 = 2

(b)

g [e2/ h]

3.0

n1 = 2

2.0 1.0 0.0

n2 = −6 –1.0

–0.5

0.0

n2 = 6 0.5

VTG [V] Fig. 8.5 (a) Schematic of edge state propagation in the quantum Hall regime. At the p–n interface, transport is dominated by the mixing of copropagating edge states at the p–n interface. (b) For full mixing of edges, additional plateaus in the quantum Hall conductance g can be observed, as plotted here in the p–n regime for υ2 ¼ 2 and –6. (Adapted with permission from – [5].)

levels are always separated by a small gap, produced by the spin-split υ ¼ 0 state, and the mixing between copropogating edge states falls to zero, as does the conductance [31].

8.3

Photon Analogies for Carriers in Graphene

8.3.1

Introduction Electron-light analogies are provided by the similarities between the Helmholtz equation for light waves and the time-independent Schrödinger equation for electrons. Diffusive electron motion in disordered, and interacting conductors can spoil this analogy. Some conductors can now be synthesized with extremely low disorder levels and devices can be fabricated with dimensions below the typical length scale for scattering and phase relaxation. In such ballistic coherent devices at low temperatures, the electrons can be described by a wave function that satisfies a time-independent Schrödinger whose quantum wave function propagates in a similar fashion to geometric ray optics (à la Huygens construction). For example, the ability to locally tune densities in nanostructures in conventional two-dimensional electron gases produces a Snell-like Law [32]. This analogous equation allows for micron and nanoscale devices to mimic ray optical devices such as lens, prisms, beam splitters, and waveguides. In addition, the charge degree of freedom of electrons allows for manipulation of electron trajectories with magnetic fields [33]. In ballistic graphene devices, similar optics-like devices are possible except with the added degree of freedom that both positive and negative values of η in adjacent regions.

8.3 Photon Analogies for Carriers in Graphene

149

This addition allows for a variety of optical phenomena with electrons to either enable or enhance compared to conventional semiconductors. For example, optics-like devices have been proposed in graphene to study collimation [34], waveguides [35–38], snake states [39–42], Vesalago lenses [43], whispering gallery modes [44, 45], and the quantum Goos–Hänchen effect [46], to name a few. Soon after the implementation of local gates to produce p–n junctions in graphene, experimental evidence for novel optical phenomena were observed including Fabry– Pérot oscillations [27], optical guiding of carriers [47], and snake states [48]. The clarity of the observation of optics-like phenomena has been enhanced by rising mobility values from the creation of suspended graphene and graphene–boron nitride heterostructures. After boron nitride was used as a substrate to enhance the mobility of graphene [20], boron nitride was also implemented as a nanometer-thin tunneling barrier [49], and gate dielectric [26] and p–n junctions with transition regions less than the Fermi wavelength were possible. In conjunction with the micron-scale mean-free path, devices were now capable of exploring the effects on transport of a negative index material. Below three optical phenomena – Vesalago lensing, confinement and guiding, and snake state – will be reviewed, illustrating both theoretical and experimental works, elucidating the physics explored therein.

8.3.2

Veselago Lensing Materials with negative index of refraction can occur when the group velocity (dE=dk) is opposite in sign to the phase velocity (momentum) as is the case for electrons and holes in graphene. The reformulation of optical laws for negative index materials was first considered by Vesalago [50]. That Snell’s Law can accommodate negative index of refraction materials by simply allowing for negative values of η is by no means obvious. A consideration of Fermat’s principle of least time for geometric rays which accounts for negative index materials led to the postulate that the path of light is a local extremum of the length of the optical path [50, 51]. With this addition, the laws of geometric optics can describe negative index materials taking the refractive indices to include values less than zero. For example, an extremum for the optical path for two materials with opposite values of η, η1 ¼ η2 , paths which satisfy the modified Fermat’s principle are shown in Fig. 8.6(a). Applications of graphene for use in negative refractive index materials was first proposed in [43]. Using the modified Snell’s law for negative index materials, the authors were able to show the focusing of electrons for convergent parallel plate (Vesalago) lenses and investigate the wave singularities (caustics) in asymmetric junctions (η1 > 0, η2 < 0, η1 6¼ η2 ). Device schematics for observing Vesalago lensing were also proposed (for similar devices, see Fig. 8.6(b)). Typical device operation is as follows: current is provided to the device through the injector and the current extracted at the “focused” contact (β) is compared to that at one or two other contacts (α and γ). At the gate voltage (typically the top-gate voltage V T ) such that η1 ¼ η2 , the extracted current in β is enhanced compared with α and γ. Using boron nitride-encapsulated devices, Lee et al. [52] have observed the first evidence in transport of negative refraction in graphene. A schematic of the device and

150

Electron Optics with Graphene p–n Junctions

(a)

(b) n

p

n

Top Gate

Collectors a

Injector b θi = θt

θi = θt

g

Fig. 8.6 (a) Focusing of electrons in a parallel plate geometry by Veselago lensing for |p| = |n|.

(b) Schematic of a typical device proposed to investigate Veselago lensing. Current density (shown in a black/white scale with black being high and white being low current density) is injected and directed to an adjacent contact β resulting in an increase in current collected from β, compared with other nearby contacts α and γ.

the lensing effect is shown in Fig. 8.7(a) where graphene is first deposited on top of a boron nitride substrate, following by deposition of an additional boron nitride on top of the graphene serving as the gate dielectric (Fig. 8.7(b)). The thickness of the boron nitride top-gate dielectric was estimated to be 14 nm, resulting in a junction sharpness of ~12 nm. Current is injected into the upper left corner of the device and signatures of lensing were obtained by analyzing the current extracted from the lower right-hand corner, I 2 . Vesalago focusing peaks are observed on the bipolar side of the normalized extract current ΔI ¼ I 2  I 2 ðHDÞ ðI 2 ðHDÞÞ is taken at high density where no focusing is observed), indicated by gray triangles in Figs. 8.7(c) and (d). The data are taken at a temperature of 100 K, suggesting that the observed peaks are a result of classical electron trajectories, consistent with expectations from lensing. The focusing peak moves linearly with back-gate voltage V B and diminishes as the magnitude of V B increases; the latter is a result of the decreasing ratio of the Fermi wavelength to width of the p–n junction and the former an indication that the peak arises from changes in density (rather than a random disorder-induced increase in I 2 ). Further evidence of negative index refraction is obtained in measurements of a non-local resistance (Rnl ). Electrons are injected from a contact on the lower left and the voltage difference between two contacts, one roughly aligned with the injected electrons and one aligned with the path expected from Snell’s Law with a negative η (Fig. 8.8(a)). At the value of V T , where electrons are expected to refract towards the lower contact, an increase in non-local resistance (Fig. 8.8(b), white triangle) is seen, consistent with expectations for η < 0.

8.3.3

Confinement and Guiding In addition to lensing, p–n junctions permit confinement of carriers, a phenomenon made difficult by the gapless nature of carriers in graphene. Carriers approaching a p–n junction at large angles of incidence have a high probability of being reflected and this effect can be used to confine carriers in graphene, and interesting trapped state behaviors in 1D channels [53] and 0D whispering gallery modes [45] have been reported. For example, placing two adjacent, parallel interfaces (Fig. 8.9(a)) along a device creates a channel capable of confining carriers that approach the barrier with large angles of

151

8.3 Photon Analogies for Carriers in Graphene

(a) IN

(b)

Ibias

n(p) IN

p(n)

I1

a

VT¢

Top gate

b

VB b

OUT

I3

I2

1 mm

a (c)

D I (nA)

20 15 10

(d) VB (V)

20 VB (V)

4.5 15

3.5 D I (nA)

25

2.5 1.5 0.5 –20

–4.5 10 –5.5 –6.5 5 –7.5 –8.5

5 0

–3.5

–10

0 VT (V)

10

20

0

–20

–10

0

10

20

VT (V)

Fig. 8.7 (a) Schematic of the operation of a Veselago device. Current is injected through a small

constriction and focusing on the diagonal contact. (b) SEM micrograph of a device used to investigate Veselago lensing in graphene. (c) and (d) Extracted current ΔI as a function of V T for various values of V B shows an increase in the p–n junction region (gray triangles), indicative of Veselago lensing (Adapted with permission from [52].)

incidence, or equivalently, large values of k y =k x , where ky and k yx are the components of momentum parallel and perpendicular to the channel. The resulting guiding has been studied theoretically [35–38], numerically [47], and experimentally [47, 53]. For channels in graphene with index of refraction η1 surrounded on either side by regions with η2 6¼ η1 , three effects combine to promote confinement of carriers to the channel. A fiber-optic analogue is where carriers experience total internal reflection called optical guiding OPG. This effect occurs for angles of incidence above a critical angle defined as     1 k 2 1 E 2 θc ¼ sin ¼ sin , (8.8) k1 E1 where E 1 , E2 are the in-channel and out-of-channel Fermi energies, respectively. In addition, reflection can occur due to the small transmission probability of a p–n junction at large angles, called p–n junction guiding PNG. Shown in Fig. 8.9(b) is a quantum transport simulation of the guiding values for PNG at a Fermi energy of –0.2 eV inside

152

Electron Optics with Graphene p–n Junctions

(b) 50

(a)

V+

40

T (K) 9 30 60 100 150

Rnl (W)

I+

VB = –18.4 V

30 I– 1 µm

V–

20

–20

–10

0 VT (V)

10

20

Fig. 8.8 (a) SEM micrograph of the device used to probe the non-local resistance in the Veselago regime. (b) The enhancement of the non-local resistance Rnl (white triangle with black outline) is consistent with focusing of electrons via Veselago lensing (Adapted with permission from [52].)

(a)

(b) Injector

(c) 1.0 J/Jmax

PNG Ω=0.54 γ=0.39 a

a

0.5

E1*E2 < 0 PNG

nout OPG PNG

|E1|>|E2| OPG

n

p Collector

n

E1 = –0.2eV p-type

OPG |E1|>|E2| OPG PNG PNG E1*E2 < 0

E2 = 0.2eV

n-type

0.0

nin

Fig. 8.9 (a) Schematic of a device used to confine carriers using two parallel p–n junctions. Current

from the injector is guided down the p channel to a collector. The guiding is measured by comparing the current in the injector to that of two contacts outside the p channel. (b) Simulation of the guiding of current density in the p–n regime for a balanced (the Fermi energy E1, 2 in each region has equal and opposite values of 0.2eV) shows a guiding efficiency (Ω) and a corrected guiding efficiency (γ) of 0.54 and 0.39, respectively. (c) Representation of the possible guiding mechanism as a function of the density inside (nin ) and outside (nout ) the channel region. PNG = p–n guiding, OPG = optical guiding. Parts (b) and (c) have been adapted with permission from [47].

the channel and –0.2 eV outside the channel. The guiding efficiencies Ω (a measure of the transmission probability inside the channel) and γ (normalized transmission probability, see [47] for details) can be extracted from simulations and compared to experiments. Finally, the two effects can combine in a regime labeled OPG+PNG. A summary of each effect, and their combination, for a range of densities in the channel (nin ) and out of the channel (nout ) is shown in Fig. 8.9(c). The guiding of electrons in optical fiber waveguides have been explored in the clean limit using suspended graphene [53]. Experimentally, the guiding efficiency is

8.3 Photon Analogies for Carriers in Graphene

153

(a) S

D2 nin

(b)

nout

D1 C

Fig. 8.10 Schematic (a) and SEM micrograph (b) of a high-mobility suspended graphene device used to investigate guiding of electrons using p–n junctions (Adapted with permission from [53].)

measured in a similar fashion to Veselago (Fig. 8.9(a)); current is injected into the channel through an injector contact and collected at a contact at the opposite end of the channel. This current is compared to the current lost to two contacts placed outside the channel. As the channel becomes smaller (or, equivalently, the Fermi wavevector becomes smaller as the channel is depleted), the number of propagating modes can become quantized due to confinement of carriers in the channel. Figures 8.10(a) and (b) shows an example of a high-quality graphene electron waveguide. In [53], a graphene sheet is suspended above a dual-gate geometry and the channel density can be controlled by the local bottom-gate of width 350 nm. In this work, guiding efficiencies were experimentally extracted as the ratio of the current collected at the collector contact C to the total current injected into the device: γ ¼ I SC =I TOT . Figures 8.11 (a) shows γ as a function of nin and nout . The grey dashed lines separate regions where guiding occurs. Light gray regions are where neither PNG nor OPG guiding occur, and arise from the finite transmission probability of electrons traversing the graphene device. In the right and left quadrants to the two crossed lines, guiding occurs: for nin * nout > 0, it is OPG and for nin * nout < 0 it is PNG (or a combination of p–n and OPG, although this distinction is not made in the paper). It is clear that the two effects enhance the guiding, and guiding is observed to be largest when the two mechanisms both contribute (black regions of Fig. 8.11(a)). As nin is increased from zero in the PNG regime, steps in the guiding efficiency are seen (indicated by arrows in Fig. 8.11(a)). One-dimensional cuts of the plot of γðnin ; nout Þ are shown in Fig. 8.11(b) where steps in γ are observed and attributed to a momentum-dependent filling and filtering effect of modes in the channel. This mode filtering effect allows for the transmission properties of single channels to be probed, and gives rise to quantum point contact in γ.

8.3.4

Snake States The charge degree of freedom allows for direct coupling of the electron to magnetic fields, augmenting the class of optics-like phenomena available in graphene like

Electron Optics with Graphene p–n Junctions

(a) 0.4 10 G

OP

0.3 5 ISC/Itot

nout (1010cm–2)

0

np

−5

−10

−15

−20 −15

−10

−5 nin

0

5

10

(1010cm–2)

(b) Experiment 3 GSC (e2/h)

154

2.5 2 −10

−8

−6

−4

−2

0

nin (1010cm–2) Fig. 8.11 (a) Extracted guiding current γ ¼ I SC =I TOT as a function of density in the channel (nin ) and outside the channel (nout ). Guiding is maximzed in the p–n junction regime (dark regions). (b) Steps in the guiding conductance GSC as a function of nin mimic plateaus seen in transport in one-dimensional channels (Adapted with permission from [53].)

magnetic focusing [54]. Application of moderate magnetic fields enables another mode of confinement at the interface of p and n regions in graphene resulting from the alternating sign of the Lorentz force on either side of the junction. Above a critical magnetic field – defined when the magnetic length becomes less than the width of the junction – no conductance is theoretically expected across the junction (in the absence of disorder) and the only propagating states are those which move parallel to the junction in skipping and snake-like trajectories [55] that have measurable effects on transport [48]. A detailed theoretical analysis of this phenomenon is given in [41]. Trajectories induced by the perpendicular magnetic field are matched at the p–n

8.3 Photon Analogies for Carriers in Graphene

155

Fig. 8.12 (a) Simulated skipping and snaking orbits are shown for a variety of radii of electron orbits r n, p injected a distance x0 from the p–n interface. For a balanced |n| = |p| junction (starred plot, second from the right), the transmission along the interface is dominated by snake states (Reproduced with permission from [41].)

interface using the above Snell’s Law, and, depending on the angle of incidence, are either reflected into skipping orbits or are transmitted across the interface. Transmitted carriers experience an opposite Lorentz force as a result of the change in sign of the charge and are bent back towards the interface. Successive transmission across the interface form an oscillating trajectory about the p–n junction called a “snake state,” Skipping and snake state trajectories (caustics and cusps) are shown in Fig. 8.12 for various initial injection conditions (x0 is the injection distance from the interface) and ratios of densities (quantifies through the cyclotron radius, r n, p ¼ ℏk n, p =eB). For example, for x0 ¼ r n ¼ r p (see starred plot in Fig. 8.12) only features arising from snake state trajectories occur. A semiclassical description of the case r n ¼ r p which accounts for the coherent quantum interference effects, predicts that the probability amplitude of finding the electron on either side of the junction oscillates, depending on the number of times the electron (classically) traverses the interface. Even and odd number of crossings puts the electron on either the same or opposite side of the junction (Fig. 8.13(a)), and in [41] this effect was predicted to have a impact on conductance, contributing to an oscillation component of the conductance with a period of 1/B, the inverse of the magnetic field. Oscillatory behavior unique to the bipolar regime has been explored in encapsulated [56] and suspended [57] graphene devices. Conduction along a p–n junction can be predicted by considering the electron motion in a semiclassical billiard model. For carriers approaching the interface from the injector side, four possible interface trajectories are possible depending on the density in each region. In the snake state regime, conductance is enhanced if the end point of the semiclassical trajectory is on the collector side (Fig. 8.13(a), left schematic). The shape of the conductance oscillations

156

Electron Optics with Graphene p–n Junctions

(a)

(b) BGR TGR

(c) 4

f

(d)

2T

2 VTG (V)

c

VTG (V)

2 i

0.3 dG/dVTG (a.u.)

0 4

0

B –2

0

–2

–4 –40

–20

0 VBG (V)

20

40

–4 –40

–20

0 VBG (V)

20

40

0

1

dG/dVTG

Fig. 8.13 (a) Snake state trajectories for different magnitudes of density on the p (left) and n (right) sides of a p–n junction. Depending on the magnitude of density, a carrier can end up either on the injector (i) or collector (c) side of the junction. (b) Simulation of the conductance from the snake state shows an oscillatory behavior, reflecting the expected termination of the snake state: for terminations on the collector side, the conductance is high (light gray) and for termination on the injector side, conductance is low (dark gray, black). Measurements of the oscillatory behavior of conductance are provided by snake states at the p–n junction. (Adapted with permission from [56].)

as a function of V BG and V TG simulated in Fig. 8.13(b) is unique to snake state behavior, distinct from both Shubnikov–de Haas and Fabry–Pérot oscillations. Experimental results of the differential conductance dG=dVTG plotted over the same voltage range are shown in Fig. 8.13(c) and show good agreement with the simulations. Remarkably, the snake state trajectory crosses the interface ~50 times over the 12.2 micron long p–n junction, showing an impressive immunity to the disorder, despite the susceptibility to disorder near the low-density transition region of the junction [58], a testament to the sharpness of the interface created in this encapsulated device. Further, as with other optical phenomena seen in this chapter, the oscillations persist above a temperature of 100 K.

8.4

Future Directions The introduction of new fabrication techniques have allowed for investigation of electron optics phenomena in graphene. However, clarity of experimental results is lagging behind that of high-mobility two-dimensional electron gases despite similar levels of disorder in the two systems. One source of this discrepancy is the lack of injection techniques in graphene with well-defined momentum, unlike GaAs where electron collimation is possible in quantum point contacts [59, 60]. Collimation techniques in graphene are difficult using point contacts, due to the difficulty of creating one-dimensional channels in the gapless material. However, collimation using an array of p–n junctions [34] or using one-dimensional channels of graphene [53] might now be possible, given the high quality of current graphene devices and the expected immunity to disorder in p–n junction arrays [61].

8.5 References

157

In addition, the Schrödinger equation is a function of a scalar quantity and hence can only be related to a single component of the electric or magnetic field in the Helmholtz equation. The wave function for graphene has a spinor form and more resembles a Jones vector representing polarized light, as was noted in [62]. In structures created from a p–n junction superlattice, collimation occurs through a collapse of the pseudospin vector in the direction perpendicular to the superlattice [34]. Hence linear and circularly polarized light analogues through pseudospin control in graphene should be possible, offering an interesting future direction of research.

8.5

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33]

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[34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62]

C.-H. Park, Y.-W. Son, L. Yang et al., Nano Lett. 8, 2920 (2008). J. Milton Pereira, Jr., V. Mlinar, F. M. Peeters et al., Phys. Rev. B 74, 045424 (2006). F.-M. Zhang, Y. He, and X. Chen, Appl. Phys. Lett. 94, 212105 (2009). R. R. Hartmann, N. J. Robinson, and M. E. Portnoi, Phys. Rev. B 81, 245431 (2010). D. Stone, C. Downing, and M. E. Portnoi, Phys. Rev. B 86, 075464 (2012). P. Carmier, C. Lewenkopf, and D. Ullmo, Phys. Rev. B 81, 241406(R) (2010). J.-C. Chen, X. C. Xie, and Q.-F. Sun, Phys. Rev. B 86, 035429 (2012). A. A. Patel, N. Davies, V. Cheianov et al., Phys. Rev. B 86, 081413(R) (2012). S. P. Milovanović, M. Ramezani Masir, and F. M. Peeters, Appl. Phys. Lett. 103, 233502 (2013). V. V. Cheianov1, V. Fal’ko, B. L. Altshuler, Science 315, 1252 (2007). J. Cserti, A. Pályi, and C. Péterfalvi, Phys. Rev. Lett. 99, 246801 (2007). Y. Zhao, J. Wyrick, F. D. Natterer et al., Science 348, 672 (2015). C. W. J. Beenakker, R. A. Sepkhanov, A. R. Akhmerov, and J. Tworzdlo, Phys. Rev. Lett. 102, 146804 (2009). J. R. Williams, T. Low, M. S. Lundstrom et al., Nature Nanotechnol. 6, 222 (2011). J. R. Williams and C. M. Marcus, Phys. Rev. Lett. 107, 046602 (2008). F. Amet, J. R. Williams, A. G. F. Garcia et al., Phys. Rev. B 85, 073405 (2012). V. G. Veselago, Usp. Fiz. Nauk 92 517 (1967) [Sov. Phys. Usp. 10 509 (1968)]. V. G. Veselago, Physics – Uspekhi 45, 1097 (2002). G.-H. Lee, G.-H. Park, H.-J. Lee, Nature Phys. 11, 925 (2015). P. Rickhaus, M.-H. Liu, P. Makk et al., Nano Lett. 15, 5819 (2015). T. Taychatanapat, K. Watanabe, T. Taniguchi et al., Nature Phys. 9, 225 (2013). C. W. J. Beenakker, Rev. Mod. Phys. 80, 1337 (2008). T. Taychatanapat, J. Y. Tan, Y. Yeo et al., Nature Commun. 6, 6093 (2015). P. Rickhaus, P. Makk, M.-H. Liu et al., Nature Commun. 6, 6470 (2015). L. M. Zhang and M. M. Fogler, Phys. Rev. Lett. 100, 116804 (2008). L. W. Molenkamp, A. A. M. Staring, C. W. J. Beenakker et al., Phys. Rev. B 41, 1274(R) (1990). K. L. Shepard, M. L. Roukes, and B. P. Van der Gaag, Phys. Rev. Lett. 68, 2660 (1992). S. K. Choi, C.-H. Park, and S. G. Louie, Phys. Rev. Lett. 113, 026802 (2014). D. Dragoman, J. Opt. Soc. Am. B 27, 1325 (2010).

9

Graphene Electronics Chen Wang, Xidong Duan, and Xiangfeng Duan

9.1

Introduction Graphene, a single atomic sheet of carbon atoms arranged in a honeycomb lattice, has taken centre stage in materials science both for fundamental studies and many potential applications, ever since its first discovery about a decade ago. In particular, graphene exhibits many unique characteristics that make it a highly attractive material for the new generation of electronics, with an unprecedented combination of speed and flexibility [1]. For example, graphene exhibits the highest carrier mobility up to 1,000,000 cm2/V
s (i.e., ~2–3 orders of magnitude higher than that of silicon) (Fig. 9.1(a)) [2–5] and the highest carrier saturation velocity of all known materials (Fig. 9.1(b)) [6], making it a potential channel material for ultra-high-speed transistors [7]. Additionally, the singleatom thickness represents the ultimate limit for down-scaling [8–13], while the large-area two-dimensional structure makes it suitable for large-scale integration [14–16], and the exceptional mechanical strength and elasticity make it desirable for highly robust flexible electronics [16]. However, the lack of an intrinsic band gap in graphene has limited the achievable on/off ratio in transistors using graphene as the active channel. Considerable efforts have been devoted to addressing this challenge, including the induction of a transport gap in graphene nanostructures [22–33] or bilayer graphene [34–37]. The creation of a transport gap in graphene nanostructures or bilayer graphene can improve the on/off ratio [22, 24, 38], but often at a cost of the electronic performance and the deliverable current density. For example, graphene nanoribbons obtained from a lithographic cutting approach typically exhibit carrier mobility of the order of 100–1,000 cm2/V
s or less, which is 2–3 orders of magnitude smaller than that of the pristine graphene [39–45]. With single-atom thickness, finite density of states (DOS), and weak electrostatic screening effect, graphene exhibits a field-tunable work-function and partial electrostatic transparency. It can thus function as an “active” contact in graphene-semiconductor junctions to enable entirely new possibilities for device design and engineering. Exploiting graphene as a tunable contact in the vertical heterostructures [46–51] with diverse semiconductors or insulators, new designs of devices, including vertical tunneling transistors [46, 47], vertical field-effect transistors [48], barristers [51], and vertical thin film transistor [49, 50] have been demonstrated with a greatly improved on/off ratio and current density. 159

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Fig. 9.1 Fundamental electronic properties of graphene versus other semiconductor

materials. (a) Electron mobility versus band gap for different materials as indicated. The mobility data relate to undoped material except for the Si MOS data [1, 17–19]. (b) Electron drift velocity versus electric field for common semiconductors (Si, GaAs, InAs, and InP), carbon nanotube and large-area graphene [1, 20, 21]. (Adapted from [1, 17–21].)

In this chapter, we review the recent progress in exploring graphene as the semiconducting channel for ultra-high-speed transistors or as an active contact for a whole new generation of tunable electronic and optoelectronic devices. Specifically, we will first discuss exploring pristine graphene as the active channel for radio frequency (RF) transistors and its potential application in analog circuits. For band-gap engineering, we will discuss graphene nanostructures, including graphene nanoribbons and graphene nanomeshes, and bilayer graphene as the active channel for transistors with an improved on/off ratio. Finally, we will discuss the exploration of graphene as a tunable contact for the design and creation of a variety of vertical graphene transistors, and finish with a brief summary and perspective.

9.2

Graphene RF Transistors and Circuits

9.2.1

Graphene RF Transistors With superior carrier mobility and current saturation velocity, graphene is well suited for RF analog applications. The performance of an RF transistor (see the Fig. 9.2(a) for the equivalent circuit diagram) is typically characterized by the cut-off frequency, fT, at which the magnitude of the small-signal current gain rolls off to unity (|h21| = 1). The cut-off frequency, fT, is generally determined by fT ¼ 

1 gm
 2π Cgs þ C gd
f1 þ ðRs þ RD Þ=Rds g þ C gd
gm ðRS þ RD Þ 1 g  m 2π C G

,

9.2 Graphene RF Transistors and Circuits

161

Fig. 9.2 Graphene RF transistors. (a) The equivalent circuit topology of a graphene transistor. The Cpg and Cpd are the gate and drain parasitic capacitances; Rs, Rd, and Rg are the resistances of the source, drain and gate electrodes. Ls, Ld, and Lg are the inductance of the source, drain and gate electrodes. Cgs, Cgd are the top-gate to the source and the top-gate to the drain capacitance, respectively; Cds and Rds are the capacitance and resistance between the drain and the source, respectively. Ri is the intrinsic resistance of dielectrics [6]. (b) RF performance of a 350 nm-gate GFET, showing a cutoff frequency fT of 50 GHz. Inset shows the SEM image of a double-channel graphene transistor. The channel width is 27 μm, and the gate length is 350 nm for each channel [55]. (c) RF performance of a 240 nm-gate (◇) and a 550 nm-gate (△) graphene transistor at VD = 2.5 V. fT of 53 and 100 GHz for the 550 nm and 240 nm devices, respectively. Inset shows the device schematics [8]. (d) RF performance of a self-aligned 144 nm graphene transistor at Vds = –1V; Inset shows schematic of the cross-sectional view of the device [13]. (e) RF performance for two 40 nm devices on diamond-like carbon (DLC) substrates with peak fT of 300 and 350 GHz. Inset shows the schematic view of a top-gated graphene RF transistor [56]. (f) RF performance of a 67 nm graphene transistor with transferred gate on peeled graphene device under two different dc bias voltages, highlighting a record-high cutoff frequency of 427 GHz. The inset shows cross-sectional TEM image of the overall device layout. Scale bar, 30 nm [54]. (g) Cut-off frequencies versus gate length for devices from different materials [1]. (h) Maximum oscillation frequency versus gate length for graphene and other transistors [57]. (i) The projected intrinsic cut-off frequency fT at Vds = 1 V calculated using fT = gm/2πCg, versus channel length of graphene transistors with a self-aligned GaN nanowire gate. The black square represents the theoretically simulated data [7, 58]. (Adapted from [1, 7, 8, 13, 54–58].)

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where gm is the transconductance; Cgs, Cgd, CG is the parasitic capacitance of gate-source, gate-drain, and gate, respectively, and RS, RD is the resistance of source and drain. While fT is of considerable importance in determining the intrinsic speed of RF devices, for analog applications, another important figure-of-merit is the maximum oscillation frequency fMAX, which is the highest possible operating frequency before a transistor loses its ability to amplify power. This value is typically defined as the frequency at which the unilateral gain (U) becomes unity and is generally governed by the following formula f MAX ¼

1 gm

1 , 2π C gs þ Cgd 4go ðRi þ RS þ RG Þ þ 4gm RG Cgd = C gs þ C gd 2

where go is output conductance and Ri is the intrinsic resistance. In general, the speed of an RF transistor is proportional to the intrinsic transconductance, gm, and inversely proportional to all the parasitic capacitances and resistances. Therefore, to achieve a high-frequency device, it is necessary to maximize the transconductance using high-mobility materials (e.g., graphene), reducing the channel length, optimizing the gate coupling (e.g., using high-k dielectrics), and reducing parasitic capacitance and access resistance (e.g., using optimized device geometry with self-aligned source and drain electrodes to minimize the source-gate and drain-gate gap or overlap). In particular, to create a functional RF transistor, it is necessary to integrate a local gate (typically top-gate) to minimize the large parasitic capacitance in global back-gated devices. The creation of top-gated graphene transistors requires effective integration of high-quality gate dielectrics with graphene, which is non-trivial since the most conventional dielectric integration approach is not compatible with the pristine clean graphene. Typical dielectric deposition approaches either result in poor dielectrics with considerable trapping states or severely damage the pristine graphene lattice and degrade its carrier mobility [52]. The first graphene RF transistor was reported in 2009 using epitaxial graphene and atomic layer deposition (ALD) deposited Al2O3 as the gate dielectrics, with a relatively low cut-off frequency 4.4 GHz [53]. The performance of the device is largely limited by the relatively low-quality gate dielectrics and severely degraded carrier mobility (~200 cm2 V–1 s–1) in graphene during the dielectric deposition process. By depositing a functionalization layer consisting of 50 cycles of NO2-trimethylaluminum (TMA) prior to the ALD growth of gate oxide, a uniform coating of oxide with less pinholes can be deposited on mechanically peeled graphene, which soon enabled the demonstration of faster graphene transistors [10]. The new device architecture consists of two parallel channels controlled by a single gate in order to increase the drive current and device transconductance. For a device with a gate length of 150 nm, a peak fT as high as 26 GHz is obtained. Furthermore, an fT = 50 GHz was demonstrated in a 350 nm gate length device by the same group (Fig. 9.2(b)) using a thin layer of naturally oxidized Al2O3 as the nucleation layer for the deposition of high-k gate dielectrics to minimize graphene mobility degradation and simultaneously using a back-gate modulation to electrostatically dope the graphene and reduce the access resistance. A further key

9.2 Graphene RF Transistors and Circuits

163

development in dielectric integration is to use an interfacial polymer layer as the buffer layer for the ALD of high-k dielectrics, which minimizes the mobility degradation to achieve a mobility value of 900 to 1520 cm2 V–1 s–1 in a top-gate graphene transistor. With this approach, a 240 nm channel length graphene transistor has been demonstrated with the fT as high as 100 GHz (Fig. 9.2(c)), marking an important milestone in the development of graphene RF transistors [8]. Despite the tremendous progress, the reported RF performance was still far from the potential that the graphene transistors can promise, which is primarily limited by two adverse factors in the device fabrication process. The first limitation is associated with the severe mobility degradation resulting from the graphene–substrate interaction and graphene–dielectric integration processes, which introduce considerable scattering sources which reduce carrier mobility. In particular, the dielectric integration processes have typically severely damaged the graphene lattice and reduced carrier mobility by 1–2 orders of magnitude. To mitigate the mobility degradation effect during the dielectric deposition process, a dry transfer approach has been developed for dielectric integration without any obvious impact to graphene mobility. For example, using dry transferred Al2O3 nanoribbons as the top-gate dielectrics, a highest carrier mobility of 23,000 cm2 V–1 s–1 has been achieved [5, 13, 54]. A second limitation of the top-gated graphene transistor is the large access resistance due to non-optimum alignment of the source, drain and gate electrodes, which can have particularly adverse effects on short channel devices that are necessary for highfrequency applications. To this end, a self-alignment process has been designed to ensure that the edges of the source, drain and gate electrodes are automatically and precisely positioned so that no overlapping or significant gaps exist between these electrodes, thus simultaneously minimizing parasitic capacitance and access resistance [13, 54]. A notable development in this regard is the utilization of a dry-transferred nanowire as the self-aligned gate for an ultra-high-speed graphene transistor [13]. In this device, a synthetic Co2Si/Al2O3 core/shell nanowire is used as the gate electrode to define the channel length on a mechanically exfoliated graphene flake, with the 5 nm Al2O3 shell functioning as the gate dielectrics, and the metallic Co2Si core functioning as the local top-gate. At the same time, the nanowire gate is used as a shadow mask for the deposition of an ultra-thin layer of platinum thin film pads, which naturally separates into two isolated parts at the left and right sides of the gate to function as the self-aligned source and drain electrodes. Using this approach, a record high cutoff frequency of 300 GHz is achieved in a 144 nm transistor (Fig. 9.2(d)). Its performance is about twice as fast as the best silicon MOSFETs of comparable sizes and is similar in speed to the best InP high electron mobility transistor (HEMT) and GaAs metamorphic HEMT (mHEMT) with similar channel lengths. Despite the considerable potential demonstrated with mechanically exfoliated graphene, it is intrinsically unsuitable for large-scale integration. To this end, large-area graphene by CVD or the epitaxial method has also been explored for RF transistors [59]. An fT up to 210 GHz has been achieved in 210 nm epitaxial graphene transistors [8, 60]. With the channel length shrinking to about 40 nm, the fT of the CVD graphene

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device on a diamond like carbon substrate can reach 300 GHz, and the epitaxial graphene device can offer an fT of 350 GHz (Fig. 9.2(e)) [56]. Additionally, a transfer-gate approach has also been developed to create 67 nm long self-aligned graphene transistors with the highest cut-off frequency of 427 GHz in mechanical exfoliated graphene [54]. Compared with the transferred nanowire gate described above, the transfer gate stack is first defined on a sacrificial substrate using a scalable lithography process; it is thus compatible for mass production of graphene RF devices when applied to large-area CVD graphene [13, 54]. In summary, graphene has been shown to exhibit considerable potential for RF transistors applications. In general, the intrinsic cutoff frequency of graphene RF transistors exceeds the best silicon MOSFET devices with comparable dimensions and is on par with the best group III–V HEMT devices (Fig. 9.2(g)), while the maximum oscillation frequencies are generally worse than those of silicon or group III–V transistors (Fig. 9.2(h)), largely due to the lack of band gap and current saturation. On the other hand, with the availability of large-area CVD graphene, graphene could offer an attractive material for low-cost large-area flexible RF electronics. Additionally, it should be noted that the performance of current graphene transistors is still limited by the practical issues in material integration and device fabrication processes, and the ultimate performance of graphene transistors could far exceed what is achieved today [52]. It has been predicted that an optimized graphene transistor can deliver an intrinsic cut-off frequency over 1 THz when the channel length is reduced to 70 nm or 5 THz at 10 nm channel length [58, 61]. Importantly, an experimental projection based DC study of ultra-short channel graphene transistors with a self-aligned GaN nanowire gate suggests such high-speed operation is indeed possible in the optimized graphene transistors (Fig. 9.2(i)) [12, 62, 63].

9.2.2

Graphene RF Circuits The demonstration of graphene RF transistors with both the intrinsic cutoff frequency and maximum oscillation frequency well exceeding 1 GHz can open up exciting opportunities for applications in RF electronics. Inspired by the rapid advancements in single graphene RF devices, graphene RF functional circuits have been fabricated, such as the graphene amplifier [56], frequency multiplier [64], frequency mixer [65], and oscillator [66]. For example, exploiting the unique ambiploar characteristics of graphene transistors, a frequency doubler can be readily created. Figures 9.3(a) and (b) show a typical graphene frequency doubler, which doubles the input frequency from 1.05 GHz to 2.10 GHz [67]. In practice, frequency multiplication is a key signal generation technique, where a signal of frequency f0 is introduced to a non-linear element to generate harmonics at higher frequencies, which is central for all major areas of analog communications, radio astronomy, and terahertz sensing. A graphene frequency mixer has also been demonstrated (Figs. 9.3(c) and (d)), where the input frequency is 4.0 GHz and 3.8 GHz, and the detected output frequency includes the frequency superposition (7.8 GHz) and frequency difference (0.2 GHz) [65]. In addition, graphene RF receiver integrated circuits (IC) have also been developed to perform signal amplification, filtering, and down conversion mixing [65]. It has been shown that

165

9.2 Graphene RF Transistors and Circuits

(a)

(b)

Oscilloscope

VG

2

0.01

0

0.00

RF

L2

RL

VDD

Voltage (V)

C2 L1 C1

–2

2

0

(c) Inductor (M3)

Gate (M2)

Spacer

–0.01

fRF + fLO (7.8 GHz)

fLO (4 GHz)

1 2

Metal Level 2 Metal Level 1

3

fRF – fLO (200 MHz)

Inductor (M3)

6

Time (ns)

(d) Metal Level 3

4

fRF (3.8 GHz) 4

Spacer Dielectrics Source (M1) Graphene Drain (M1) SiC

(f)

(e) RF Carrier (4.3 GHz) 0

1 On 20.0mV/

1

0

1

0

2 On

3 On

4

On

Graphene RF Receiver output signal

IF (100 MHz)

Rectified/LPF 0

01001001 01000010 01001101 I H

500.ns

B

M –281.3200ns

T –2.0mV

Fig. 9.3 Graphene RF circuits. (a) The circuit diagram of a graphene transistor-based RF

frequency doubler [7]. (b) Measured input (black) and output (gray) signals of the frequencydoubling circuit when the graphene device is gated near the Dirac point, the input frequency is 1.05 GHz and the output frequency is 2.10 GHz [67]. (c) Schematic illustration of a graphene mixer circuit [65]. (d) A snapshot of output spectrum, between 0 and 10 GHz, of the mixer taken from the spectrum analyzer with fRF = 3.8 GHz and fLO = 4 GHz. Each x and y division corresponds to 1 GHz and 10 dBm, respectively. The frequency mixing was visible with two peaks observed at frequencies of 200 MHz and 7.8 GHz [65]. (e) Measured waveforms of RF input signal amplitude modulated at a rate of 20 Mb s–1, IF output signal and restored binary code after rectifying and low-pass filtering IF signal [68]. (f) A screenshot of receiver output waveforms taken from the oscilloscope, with LO power of –2 dBm at 4.2 GHz. The original bit stream comprising three letters (24 bits) was recovered by graphene receiver with very low distortion [68]. (Adapted from [7, 65, 67, 68].)

Graphene Electronics

Vout (V)

166

Fig. 9.4 Graphene nanoribbons. (a) Schematic illustration of two GNRs with armchair (left) and zig-zag (right) edges [28]. (b) Theoretical energy band gap versus width for GNRs with different edge configuration [72]. (c) Schematic illustration of the top-gated GNR transistors using Si/HfO2 core–shell nanowires as the etching mask and top-gate [24], and (d) the cross-section TEM image of the top-gate stack [24]. (e) The transfer characteristics Ids–VTG at Vds = 0.10 (down curve) and 1.0 V (up curve) for a top-gated GNR transistor with a Si/HfO2 core–shell nanowire top-gate [24]. (f) Schematic illustration of the gradual unzipping of one wall of a carbon nanotube to form a GNR [73]. (g) STM image with partly overlaid structural model of bottom-up synthetic GNRs, scale bar: 2 nm [63]. (h) Band gap versus GNR width relationship. [1, 23, 28, 72]. (i) Carrier mobility versus GNR width [1, 75, 76]. (j) The logic OR gate output characteristics made from a GNR with double top-gates. The inset shows the schematic circuit diagram [29]. (Adapted from [1, 50, 52, 58, 59, 61–65].)

a graphene RF IC can be implemented for practical wireless communication functions, receiving and restoring digital text transmitted on a 4.3 GHz carrier signal (Figs. 9.3(e) and (f)) [68].

9.3

Graphene Nanostructures Although attractive for high-frequency analog applications, graphene transistors typically exhibit a rather lower on/off ratio (100 cm2/V s has been achieved, far higher than that of typical thin film semiconductors, making GNM an attractive thin film semiconductor material for large-area flexible electronics.

9.4

Bilayer Graphene Transistors Although the creation of graphene nanostructures can effectively induce a transport gap in graphene to enable room temperature transistors with a significantly increased on/off

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ratio, a sufficient gap can only be achieved at extremely narrow channel width (typically 103 (Fig. 9.7(c)) [48]. It should be noted that in the graphene–MoS2–metal vertical transistor design, the unique part is graphene, which is an active tunable contact, and the MoS2 merely functions as a thin film semiconductor. This approach thus offers a general strategy for the vertical integration of graphene with various semiconductors to obtain both pand n-channel transistors, or even with insulators to create vertical tunneling transistors. For example, using the layered p-type Bi2Sr2Co2O8 (BSCO) nanosheet, a p-type VFET has been demonstrated [48]. Integrating with silicon, graphene barristors have been created on 6-inch silicon wafers with essentially the same working mechanism [51], where graphene can be used as both the p-type and n-type Schottky contact to silicon, resulting in complementary inverters and half-adder logic circuits (Fig. 9.7(d)). Integrating graphene with thin film oxide semiconductor has enabled the scalable fabrication of vertical thin film transistors (VTFTs) and logic circuits (Fig. 9.7(e)) [49]. Integrating graphene with insulating thin films (e.g., BN) has enabled the creation of vertical tunneling transistors [46, 47]. More importantly, this strategy of vertical integration can be readily extended to multiple-transistor stacking in the vertical direction, opening a new dimension for 3D integration at the device level. For example, by vertically stacking two graphene-based vertical transistors, with a Bi2Sr2Co2O8 layer as the p-channel and a MoS2 layer as the n-channel material (Fig. 9.7(f)), a complementary inverter has been demonstrated with a larger-than-unity voltage gain (Fig. 9.7(g)). In this way, the concept of vertical transistors can open up a totally new strategy for logic integration in the third dimension, distinct from the two-dimensional integration in traditional planar silicon electronics. The design of vertical transistors is fundamentally different from the traditional “planar transistor” structure that has been used in conventional silicon electronics for more than half a century, and exhibits two important features that make it highly attractive for large-area flexible electronics. In a planar transistor, the charge carriers travel in the planar (lateral) direction from source to drain electrode, with the speed largely defined by the time required for electrons to travel from the source to drain electrode, which in turn is determined by the carrier speed of the semiconductor materials (determined by the intrinsic carrier mobility and applied field, or carrier saturation velocity) and the distance between the source and drain electrodes

9.5 Vertical Graphene Transistors

173

Fig. 9.7 Vertical transistors with graphene an active contact. (a) Schematic illustration of the vertically stacked graphene–MoS2–metal FETs, with the graphene and top metal thin film functioning as the source and drain electrodes, and the MoS2 layer as the vertically stacked semiconducting channel, with its thickness defining the channel length [48]. (b) The band structure at negative source bias at graphene (up, Vsd 0) with the top metal electrode connected to ground under positive (solid) or negative (dashed) Vg [48]. (c) The on/off current ratio of the vertical transistors with various channel lengths (thicknesses of MoS2). The on- and off-state band diagrams for thin (solid) and thick (dashed) MoS2 layers are represented in the insets [48]. (d) Inverter characteristics obtained from integrated n- and p-type graphene barristors and schematic circuit diagram for the inverter. Positive supply voltage (VDD) is connected to p-type GB, and the gain of the inverter is ~1.2 [51]. (e) Transfer characteristics of a vertical thin film transistor made of a graphene–IGZO–Ti stack at 0.1 V (down curve), 0.5 V (medium curve) and –1.0 V bias (down curve). The band diagram of the corresponding ON- and OFF- state at negative bias is shown in the down inset. The up inset is a schematic of the crosssectional view of the device [49]. (f) Cross-sectional view of the vertically stacked multiheterostructures of layered materials for complementary inverters that integrate a p- and n-channel VFET to the vertical stack [48]. (g) The inverter characteristics from vertically stacked p- and n-type VFETs [48]. (h) Schematic illustration of the cross-sectional view of a graphene–P3HT–Au organic VTFT on glass, where the glass serves as the substrate, Ti/Au as the gate, ALD Al2O3 as the dielectric, and the graphene and top gold thin film functioning as the source and drain electrodes, respectively [50]. (i) Operation frequency of a graphene–P3HT–Au OVTFT on glass. Gate and drain AC current as a function of input frequency signal. The dot is the measured drain current, and the square is the measured gate current. The triangle is the calibrated gate current considering the effective gate area and gate capacitance [50]. (j) Schematic illustrating the operation of planar TFT (up) and VTFT (down) on a flexible plastic substrate. Arrows indicate current flow across the channel, which can be severely limited by cracks in the planar TFT but is not significantly affected in the VTFT [49]. (k) Normalized conductance at various bending cycles for the planar structure (down curve) and vertical structure (up curve), highlighting significant greater robustness of the VTFT architecture [49]. (Adapted from [48–51].)

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(determined by the lithographically defined gate length). With the same device architecture, the speed of the planar transistors is inevitably limited by the lateral size or the ultimate resolution of the lithography used to fabricate the devices. The current largearea electronics and thin film transistors are typically dominated by low performance materials with relatively large device dimensions, and thus typically exhibit a rather low speed. Importantly, with the vertical transistor concept, the channel length is no longer determined by the lithography, but by the thickness of the semiconductor thin film. It can therefore enable ultra-short channel transistors beyond the limitation of traditional lithographic resolution, opening a new pathway to high-speed devices (even using lowresolution lithography). This attribute of vertical transistors can be highly significant for low-cost, large-area flexible electronics, where a moderate device operating speed is required but the high-resolution lithography is too costly or too difficult to implement. For example, using a simple shadow mask approach without sophisticated lithography, a graphene–poly(3-hexylthiophene-2,5-diyl) heterostructure vertical device with a size over 10,000 μm2 has been produced with megahertz frequency response, comparable to the highest speed P3HT transistors fabricated using the high-resolution lithography (Figs. 9.7(h) and (i)) [50]. Additionally, for flexible electronic applications, the repeated bending cycles can induce planar cracks that may block the current transport in traditional planar TFTs (Fig. 9.7(j)). Importantly, the unique vertical (out-of-plane) charge transport across the large-area vertical junction makes the source–drain current much less affected by the in-plane cracks in the semiconductor thin films to afford unprecedented tolerance to in-plane cracks (Fig. 9.7(j)) [49]. It can thus enable exciting opportunities for highly flexible electronics with exceptional electrical performance and superior mechanical robustness. Indeed, the repeated bending cycle test of VTFTs made on a plastic substrate confirms the highly robust nature of the VTFTs (Fig. 9.7(k)). A bending test shows that the current drops by about five orders of magnitude from their initial state within 50 bending cycles for the planar TFTs due to the formation of small cracks or dislocations inside the channel under repeated bending cycles, while the VTFTs show highly stable operation current with little change up to 1000 bending cycles. These studies clearly demonstrated the overall robustness of VTFTs against repeated bending cycles, which can open up exciting opportunities in large-area, flexible, highperformance macroelectronics.

9.6

Conclusion Graphene, as an extraordinary member of a large family of layered materials, displays rich physical properties and represents an exciting material system for exploring the fundamental physics and for developing exciting technologies. With the unique two-dimensional geometry, ultra-high mobility, high carrier saturation speed, linear dispersion relationship, and exceptional mechanical properties, graphene has attracted considerable interest as an alternative material for a new generation atomically thin electronics with unique functions or unprecedented performance. With the potential for

9.7 References

175

terahertz RF circuits, current graphene RF transistors already show comparable performance with the best devices made from traditional materials. Though the zero band gap of graphene limits its potential for logic applications, the band-gap opening in nanostructures or bilayer graphene may greatly expand the application horizon towards digital circuits. By integrating with other two-dimensional materials or semiconductor thin films and exploiting graphene as a unqiue tunable contact, vertical graphene transistors show much improved properties (on/off ratio) for logic electronics. Despite the significant potential demonstrated to date, there are also substantial challenges to bringing these potentials into practical technologies. To overcome the limitations of the semimetal nature of graphene, recent interest is expanding to other 2D materials, particularly transition metal dichalcogenides (TMDs) with tunable intrinsic band gaps. In particular, exploiting atomically thin MoS2 as the active semiconductor channel, RF transistors with an intrinsic cutoff frequency >40 GHz, and maximum power gain frequency up to 50 GHz have been demonstrated [79]. However, further improvement of the RF performance in such devices is limited by the intrinsically lower carrier mobility (~100 cm2/V s) and the challenge in creating low resistance contacts. To this end, black phosphorous has recently attracted considerable interest because of its moderate intrinsic band gap and considerably higher carrier mobility (~1000 cm2/V s) than TMDs [80]. Additionally, the creation of van der Waals heterojunctions by combining several different 2D materials could allow atomic scale integration of high disparate materials and high distinct properties. It could enable totally new device concepts, and open up exciting technological opportunities beyond the reach of existing materials.

Acknowledgment The authors acknowledge the financial support by NSF DMR1508144 (materials synthesis) and ONR Grant N00014-15-1-2368 (device studies).

9.7

References [1] Schwierz, F. Graphene Transistors. Nat Nanotechnol 5 487–496, (2010). [2] Bolotin, K. I. et al. Ultrahigh Electron Mobility in Suspended Graphene. Solid State Commun. 146, 351–355 (2008). [3] Elias, D. C. et al. Dirac Cones Reshaped by Interaction Effects in Suspended Graphene. Nat Phys 7, 701–704 (2011). [4] Castro, E. V. et al. Limits on Charge Carrier Mobility in Suspended Graphene due to Flexural Phonons. Phys Rev Lett 105, 266601 (2010). [5] Liao, L. et al. High-κ Oxide Nanoribbons as Gate Dielectrics for High Mobility Top-Gated Graphene Transistors. Proc Natl Acad Sci USA 107, 6711–6715 (2010). [6] Meric, I. et al. Current Saturation in Zero-Bandgap, Topgated Graphene Field-Effect Transistors. Nat Nanotechnol 3, 654–659 (2008).

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10

Graphene: Optoelectronic Devices Thomas Mueller and Phaedon Avouris

10.1

Introduction The unique optical properties of graphene coupled with its outstanding electrical properties make it an ideal system for optoelectronic and photonic applications. Graphene’s absorption spectrum extends from the ultra-violet to the far-infrared and terahertz regions. In the near infrared and visible part of the spectrum, the absorption is the result of interband excitations at the K and K 0 -points of the π-band structure of graphene, where the dispersion is essentially linear. In pristine graphene and T ¼ 0, the optical conductivity σ ðωÞ in this region is quantized, independent of the exact frequency, and given by σ ðωÞ ¼ πe2 =2h, where e and h denote the electron charge and Planck’s constant, respectively [1–3]. This conductivity corresponds to an absorbance for normal incidence of AðωÞ ¼ ð4=πcÞσ ðωÞ ¼ πα  2:3%, where α is the fine structure constant and c is the speed of light. At higher frequencies, trigonal warping and many-body effects affect the interband transitions. Near the M-point of the band structure, a saddle-point singularity gives an absorption maximum at about 4.3 eV [4]. Absorption at the Γ-point occurs at very high energies (e20 eV). Doped graphene absorbs in the far-infrared region as a result of intraband transitions involving free carriers. These transitions are momentum forbidden but become possible due to scattering by phonons and defects. The absorption in this region is not quantized but is determined by the free-carrier density. Its oscillator strength is given by the Drude pffiffiffiffiffi weight D, which for graphene is D ¼ e2 vF πη, where η is the carrier density and vF the Fermi velocity. The sum rule dictates that an increase in the Drude weight of the intraband transitions should lead to a corresponding decrease in the strength of the interband transitions; providing an important tool for modulating the optical absorption of graphene [5]. Along with the above single-particle excitations, collective excitations (plasmons) are possible in graphene. Due to the mismatch in the light and plasmon dispersions, direct excitation of plasmons in extended graphene is not possible. However, there are a number of approaches by which momentum conservation can be achieved and light can couple to plasmons [6]. The intrinsic plasmon of the π-electrons of monolayer graphene is centered at e5 eV while the σ þ π plasmons at e10 eV [7]. More relevant for device applications are the intraband plasmons arising from collective free-carrier excitations in doped graphene. In monolayer graphene, the frequencies of these

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10.2 Light to Current Conversion

181

excitations scale with free-carrier density η as ωpl / η1=4 / jEF j1=2 , with E F being the Fermi energy [8, 9]. In addition to the extended plasmons, localized plasmons in structures smaller than the relevant wavelength, as in metallic particles, can be directly excited. Such structures may involve graphene nanoribbons, nanodots, or have other shapes. The localized plasmon resonance frequency depends not only on η, but also on particle size and shape. For example, light polarized perpendicular to the axis of nanoribbons can excite localized plasmons with frequencies that scale with the ribbon width W as W 1=2 [10]. Typically micron size graphene confined structures have plasmon resonances in the far-infrared region, while nanostructures have them in the nearinfrared region. Graphene in general does not exhibit photoluminescence. This is due to the absence of a band gap and the ultra-fast relaxation and cooling of the photoexcited carriers. Hot band luminescence at a given frequency is observed in doped graphene when 2jE F j approaches the emission energy and disappears when 2jE F j surpasses the excitation energy [11]. The value of graphene as a photonic material is due to its wide range of absorption and the ways this absorption can be controlled. Furthermore, the large carrier mobilities in graphene further enhance its potential in optoelectronics. Key to the control of the optical properties is the ability to change the carrier density by electrostatic or chemical doping. In this way, the charge density can be varied from about 1011 to 1014 cm–2. The tunable doping leads to Pauli blocking of interband excitations in the near infrared, allowing for light switching [12, 13]. In the case of localized plasmons, it determines the plasmonic resonance frequency. The plasma frequency can also be controlled by confinement, while the absorption strength can be enhanced by the excitation of intrinsic plasmons or the plasmons of other attached materials. Furthermore, the properties of other optical materials such as photonic crystals can be modified by combing with graphene layers [14–16].

10.2

Light to Current Conversion Most photonic applications of graphene involve light to current conversion. This can be accomplished through different physical mechanisms, illustrated in Fig. 10.1(a) [17]. The simplest perhaps is through the photoelectric (PE) effect, where the electron–hole (e–h) pairs produced by photoexcitation are separated and accelerated by the internal electric field present at doping (e.g. p–n) or metal–graphene junctions. The junctions are created either electrostatically by using split gates, or chemically by doping, or by using contact metals with work functions that are higher/lower than that of graphene. Another current generation mechanism involves the photothermoelectric (PTE) effect. A photovoltage V PTE is generated at a doping junction 1–2 that is given by V PTE ¼ ðS1  S2 ÞΔT, where S1 and S2 are the Seebeck coefficients of the two sides of

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Graphene: Optoelectronic Devices

Fig. 10.1 (a) Schematic representation of photocurrent generation mechanisms in graphene. (b) Drawings of the temperature distribution T ðxÞ (top), Seebeck coefficient SðxÞ (middle), and the term SðxÞrT ðxÞ (bottom) in a graphene p–n junction. The produced photovoltage corresponds to the area underneath the bottom curve. (c) Top: Gate voltage dependence of the Seebeck coefficient in graphene. V D is the Dirac point voltage. Bottom: Gate voltage dependence of photocurrent in a graphene junction. In this example, it is assumed that the doping in region 2 is fixed (position 3) and the doping in region 1 is controlled by the gate voltage (positions 1–4). The photocurrent changes sign twice (positions 2 and 3).

the junction and ΔT is the temperature difference. In general, a photovoltage is generated when irradiation generates a temperature gradient rT ðxÞ in a material with Ð a spatially varying Seebeck coefficient, then V PTE ¼ SðxÞ  rT ðxÞdx. The Seebeck coefficient of graphene can be obtained from the Mott formula by differentiating the DC conductance with respect to the Fermi level. The coefficient can be tuned by

10.2 Light to Current Conversion

183

varying the chemical potential and can reach values of e100 μV=K [18]. A typical gate voltage dependence is shown in Fig. 10.1(c). Due to the slow scattering of excited carriers in graphene by acoustic phonons, a substantial number of hot carriers survives and contributes to the photoresponse. Figure 10.1(b) shows an example of a graphene p–n junction that is locally illuminated with a laser beam. The dominance of the PE or PTE effect in the photocurrent generation in a graphene junction can be ascertained by examining the sign change in the current as the doping in the two regions changes [19– 21]. In case of the PE effect, the sign changes once – namely, when the two regions have the same chemical potential. Instead, if the PTE effect dominates, the sign changes twice – one sign change occurs when the chemical potentials in both regions are the same, the other arises from the non-monotonic dependence of the Seebeck coefficient on the chemical potential (see Fig. 10.1(c)). Irradiation-induced heating can also change the photocurrent by changing the transport characteristics of graphene, i.e. the mobility of the carriers or by thermally exciting additional carriers. This effect is referred to as bolometric (BE). A temperature increase in graphene reduces carrier mobility leading to a current reduction, while additional carriers increase the current. The relative importance of the two factors is largely determined by the Fermi energy. Near the Dirac point, the number of carriers is low and additional carriers have a significant effect, while at high doping the reduction in mobility dominates. Detection of terahertz (THz) radiation by graphene can also be achieved based on a mechanism proposed by Dyakonov and Shur [22, 23]. In this mechanism, a twodimensional electron system in a field-effect transistor (FET) structure acts as a resonant cavity for plasma waves. When an AC voltage is applied between source and drain and the channel is irradiated with THz radiation, a DC component appears in the potential difference between source and drain. This rectification is due to the non-linear response of the cavity. The detection characteristics depend on whether the device operates in the resonant mode (plasmon transit time shorter than the momentum relaxation time) with very high-peak responsivity at the resonance frequency (up to 106 V/W predicted), or in the non-resonant mode. Resonant operation requires mobilities of at least several thousand cm2/V s at frequencies >1 THz. In the non-resonant (overdamped) regime, the plasma waves decay at a distance smaller than the channel length, and broadband THz detection is expected. Graphene can also be utilized in a FET configuration as a sensitive photogate. In this case, light may be absorbed by the graphene, e–h pairs are created, and then one of the carriers is transferred to another material in contact with it, e.g. a quantum dot, a thin insulator with traps, or adsorbed atoms or molecules. The reverse process where the primary e–h pair generation takes place on the second material is also possible. The charge separation is the critical step. The resulting strong field leads to photogating that drastically increases the number of carriers drawn in the FET channel and therefore the current in the FET. The photoconductive gain G is given by G ¼ τ trap =τ transit , where τ trap is the trapping lifetime and τ transit is the transit time from source to drain. While longer τ trap gives higher gain, the photoresponse time also becomes slower.

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Graphene: Optoelectronic Devices

10.3

Photodetectors Photodetectors convert light into electrical signals and are at the heart of a multitude of technologies, such as optical communications, video and biomedical imaging, and many more. For photodetection applications, graphene offers two main advantages. (i) Graphene exhibits ultra-fast carrier dynamics, which enables picosecond conversion of photons into electrical signals, and could be applied in high-data rate optical communication networks. (ii) Being gapless, graphene absorbs light over a wide range of electromagnetic frequencies, which makes it an appealing material for photodetection applications in spectral regimes that are currently not well covered by other materials (e.g. terahertz and mid-infrared).

10.3.1

Metal–Graphene–Metal Photodetectors Photocurrents of opposite polarities are generated by local illumination of the drain and source contacts in a graphene FET, operated under short-circuit conditions. In early studies, this current was attributed to the e–h separation at the built-in electric fields at the drain and source contacts [24–26]. The fields arise from the charge transfer between the contact metal and the graphene due to their work-function differences. It can be adjusted by proper choice of the metal and can be further enhanced by electrostatic doping. The photoresponse near graphene p–n junctions and single/bilayer graphene interfaces were shown to be dominated by the PTE effect [19–21]. Recent wavelengthdependent measurements of the photoresponse near graphene–metal junctions were used to quantify the relative contributions of PTE and PE effects, both adding to the overall photoresponse, with the PTE effect being more important [27, 28]. As the photocurrent in a graphene transistor is generated only in the vicinity of the contacts, an interdigitated metal finger configuration was used to increase the effective photodetection area (Fig. 10.2(a)). Breaking the mirror symmetry of the device by applying a drain-source voltage, as is done in traditional metal-semiconductor-metal detectors, is not practical, because graphene’s low resistance would result in a large dark current. An asymmetric metallization scheme was thus employed to break the symmetry under short-circuit conditions [29]. This enabled near zero-dark current operation and responsivities (the responsivity relates the electric current produced by a photodetector to the incident optical power) of up to 6 mA/W. Several approaches have been pursued to enhance the interaction between light and monolayer graphene and to increase the responsivity. One approach is based on the integration of graphene into an optical cavity. Incident light is trapped inside the cavity and passes multiple times through the graphene sheet, giving rise to enhanced absorption. Cavity-integrated graphene photodetectors were realized using distributed Bragg reflectors [30], metallic mirrors [31], and photonic crystals [32]. Another approach involves the use of the field enhancement resulting from the excitation of plasmons, either by placing metallic nanostructures near the contacts [33] or by exploiting collective charge oscillations of the electron gas inside the graphene itself. Graphene nanoribbon arrays with different widths were demonstrated as tunable mid-infrared photodetectors [34].

10.3 Photodetectors

10.3.2

185

Waveguide-Integrated Photodetectors Figure 10.2(b) shows the coplanar integration of graphene with an optical waveguide for photonic integrated circuits [35]. The optical mode in the silicon waveguide couples through its evanescent tail to the graphene on top. The photocurrent is generated at the interface between the signal-electrode and graphene and is driven towards the ground leads. In order to avoid losses due to light absorption in the signal-electrode, other studies employed a pair of electrodes located on each side of the waveguide [36, 37]. One of the electrodes was positioned sufficiently close to the waveguide edge to efficiently separate the photoexcited electron–hole pairs at the junction, and the ground electrode was placed at larger distance to break the device symmetry. An almost flat photoresponse across all optical telecommunication windows, from the O-band to the U-band, was demonstrated, which is well beyond the wavelength range of germanium detectors. A bandwidth of 42 GHz and responsivities of 0.36 A/W and 0.08 A/W under bias and zero-bias operation, respectively, were reported [37].

10.3.3

Sensitized Graphene Photodetectors Sensitive photon detection requires a gain mechanism that can provide multiple electrical carriers per incident photon. One way to accomplish gain in a graphene transistor is by sensitizing its surface with light absorbing particles, such as quantum dots, molecules, or metallic nanoparticles (Fig. 10.2(c)). Photons are absorbed in these particles, followed by charge (either electron or hole) transfer into the conducting graphene channel. As a result, the carrier density in graphene changes by Δn and therefore its conductance by ΔG ¼ eμ Δn. As the carrier mobility μ is large in graphene, it is a natural candidate for highly sensitive photodetection. Moreover, since graphene is a two-dimensional material, the charge transfer can be very efficient. The photoconductive gain results from recirculation of free carriers in graphene during the lifetime of the trapped carriers in the particles. A gain of up to 108 was observed for graphene decorated with PbS quantum dots, corresponding to an ultra-high responsivity of 107 A/W [38]. However, the trade-off between responsivity and speed, which is associated with the trapping time (typically milliseconds), makes these devices only suitable for applications that do not require a fast response.

10.3.4

Bolometers A bolometer measures the power of electromagnetic radiation by absorbing the incident radiation, ΔP, and reading out the resulting temperature increase, ΔT. The thermal resistance Rh ¼ ΔT=ΔP defines the sensitivity of a bolometer, and the heat capacity determines its response time. Graphene has a high thermal resistance and a low heat capacity, which makes it an ideal material for a low temperature bolometer. The weak temperature dependence of the electrical resistance in monolayer graphene, however, imposes a challenge for electrical read-out of the temperature increase ΔT. Yan et al. thus used a dual-gated bilayer graphene sheet with an optically transparent top-gate

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Graphene: Optoelectronic Devices

Fig. 10.2 (a) Schematic drawing of a metal–graphene–metal photodetector with asymmetric contacts (reproduced with permission from [29]). (b) Scanning electron microscope image of a waveguide-integrated graphene photodetector (reproduced with permission from [35]). (c) Schematic drawing of a sensitized graphene photoconductive photodetector (reproduced with permission from [38]). (d) Dual-gated bilayer graphene bolometer (reproduced with permission from [39]). (e) Antenna-coupled graphene terahertz detector (schematic drawing, left; scanning electron microscope image of FET channel, right) (reproduced with permission from [40]).

10.4 Light Modulators

187

electrode to open a band gap and obtain temperature-dependent resistance (Fig. 10.2(d)) [39]. This device was characterized under mid-infrared illumination (10.6 μm wavelength). The thermal resistance showed approximately the T 3 temperature dependence expected for (disorder-free) phonon cooling and the photoresponse mechanism was identified to be bolometric. The detector exhibits a noise-equivalent-power of 33 fW/Hz0.5 at 5 K (lower than that of commercial silicon bolometers) and an intrinsic bandwidth of >1 GHz (orders of magnitude higher than in silicon). A challenge that remains is the matching of the high impedance of a gapped graphene bilayer to that of free space for more efficient photon coupling.

10.3.5

Terahertz Detectors Commercial THz detectors are mostly based on thermal sensing elements or fast Schottky diodes. However, both technologies have limitations in terms of performance. Thermal detectors exhibit a slow response and Schottky diodes are limited to low THz frequencies. Newer generations of THz detectors can overcome these limitations by relying on the Dyakonov–Shur mechanism, i.e. the excitation of plasma waves in a high-electron-mobility transistor (HEMT) channel by the incoming THz radiation [22, 23]. Graphene supports plasma waves that are weakly damped, making it a promising candidate for such devices. Room temperature detection of 0.3 THz radiation was demonstrated in a top-gated graphene FET, in which a log-periodic circular-toothed antenna at the source and gate electrodes was used to couple the radiation (Fig. 10.2(e)) [40]. The dependence of the photovoltage on doping was found to be in agreement with the predictions of a diffusive theoretical model, thereby proving that the detector operates in the overdamped plasma wave regime. In an improved buried-gate device, responsivity of e1:2 V=W and noise equivalent power of e2  109 W=Hz0:5 were achieved, competitive with the performance of commercially available systems [41]. Resonant detection is still to be demonstrated. An ultra-fast THz detector based on the PTE effect with an array of asymmetric metal contacts has also been demonstrated [42].

10.4

Light Modulators Optical modulators allow for the electrical control of the amplitude, frequency, phase, or polarization of light. Electro-absorption modulators based on bulk semiconductors or semiconductor quantum wells exploit the Franz–Keldysh or quantum-confined Stark effects to shift the material’s optical absorption edge by an electric field. The optical absorption in graphene can be controlled by shifting the chemical potential with a gate voltage V G . At V G ¼ 0, the Fermi level E F resides close to the Dirac point and interband transitions can occur, giving rise to light absorption (Fig. 10.3(a), middle image). At large negative V G , the Fermi level is lowered and the graphene becomes transparent as no electrons are available for interband transitions (Fig. 10.3(a), left). At large positive V G , the Fermi level is raised which prevents interband transitions due to Pauli blocking (Fig. 10.3(a), right). Optical interband transitions can thus be turned on

188

Graphene: Optoelectronic Devices

Fig. 10.3 Gate voltage control of interband optical transitions (a) in graphene and (b) in a graphene double-layer structure. (c) Schematic drawing of a waveguide-integrated graphene electro-absorption modulator, and (d) electro-optical response of the device for different drive voltages (reproduced with permission from [13]). (e) Graphene-based terahertz modulator (left) that utilizes intraband optical transitions (right) for absorption modulation (reproduced with permission from [45]).

and off, provided that the photon energy hv is smaller than 2jE F j. Modulation by monolayer graphene is only about 2.3% at normal incidence, but integration with an optical waveguide (Fig. 10.3(c)) greatly increases the interaction length through coupling between the evanescent light wave and graphene, resulting in strong electroabsorption modulation of e0:1 dB=μm (Fig. 10.3(d)). In the first graphene-based modulator, demonstrated by Liu et al., the waveguide itself was utilized as a back-gate electrode to dope the graphene sheet [13]. A 4 dB modulation depth was achieved in a 40 μm long device. The modulator showed a broad spectral bandwidth (1.35–1.6 μm),

10.5 Ultra-Fast Lasers

189

small device footprint (25 μm2), and high operation speed (1.2 GHz). An improved design, demonstrated later by the same group, used a graphene–oxide–graphene structure [43]. Under bias, both electrons and holes are injected into the graphene layers to form a p–oxide–n junction (Fig. 10.3(b)) and both layers become transparent simultaneously. Without bias, both graphene sheets become absorbing. This design avoids the participation of electrons/holes in the (usually highly resistive) silicon waveguide, and therefore increases its operation speed. In addition, using two graphene sheets increases the optical absorption and modulation depth, resulting in smaller device footprint and lower power consumption. Graphene modulators face fundamental tradeoffs between operation speed and efficiency. The speed can be increased using a thicker gate oxide, but the lower capacitance then results in reduced efficiency because of a lower carrier concentration change with voltage. To overcome this tradeoff, Phare et al. integrated a graphene double-layer over a ring resonator that is coupled to a bus waveguide [44]. The resonator was designed to be critically coupled for low losses. When losses in graphene are increased, the resonator becomes undercoupled, increasing transmission through the bus waveguide. This device operated with a 30 GHz bandwidth and with state-of-the-art modulation efficiency of 15 dB per 10 V. In addition to interband transitions, which dominate in the infrared and visible range, intraband optical transitions can occur in graphene, resulting in an optical conductivity described by the Drude model. The latter dominate the optical properties of graphene at terahertz frequencies. By electrically tuning the density of carriers available for intraband transitions, the transmission of terahertz radiation through graphene can be controlled (Fig. 10.3(e), right). On the basis of this principle, Sensale-Rodriguez et al. demonstrated a graphene-based terahertz modulator [45]. The device, depicted in Fig. 10.3(e), consists of large-area monolayer graphene that is electrostatically gated by a SiO2/Si back-gate. An intensity modulation depth of 15%, an intrinsic insertion loss of 5%, and a modulation frequency of 20 kHz were demonstrated at 0.57 THz.

10.5

Ultra-Fast Lasers Ultra-fast lasers produce nano-, pico-, or femtosecond optical pulses and are deployed in a variety of applications, ranging from optical spectroscopy to materials processing and medical surgery. To produce a train of ultra-short optical pulses, most laser designs exploit a technique called passive mode-locking, in which a non-linear optical element causes self-modulation of the intracavity light intensity. A commonly used non-linear optical device is a saturable absorber, in which the absorption of light decreases with increasing intensity. Saturable absorbers for ultra-fast lasers are either based on semiconductor saturable absorber mirrors, called SESAMs, or various kinds of artificial saturable absorbers that exploit non-linear optical effects (e.g. Kerr-lensing). In general, saturable absorption results from depletion of the ground-state population and most materials show some degree of saturable absorption if the incident light intensity is high enough. However, the key requirements for using an optical absorber

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Graphene: Optoelectronic Devices

in mode-locked laser – ultra-fast response time and low saturation fluence – can only be met by a few materials. In addition, low optical insertion loss, broad spectral response, and high power handling capabilities are desirable. The ultra-fast carrier relaxation in graphene, combined with strong optical absorption, makes it an ideal material for fast saturable absorption. Further, due to graphene’s broadband optical response, there is no need for band-gap engineering as in SESAMs. Figure 10.4(b) shows the optical transmission of graphene (more precisely, a graphene–polymer composite) as a function of pump power at different wavelengths. The transmission is almost independent of pump intensity for low input powers and increases by 1.3% due to absorption saturation when the incident power is raised. The wavelength independent behavior demonstrates that graphene can be used for saturable absorption over a broad spectral range. The origin of the saturation can be understood by considering the photoexcited carrier dynamics in graphene (Fig. 10.3(a)). Optical interband excitation by an ultra-short optical pulse produces non-equilibrium carrier distributions in the conduction and valence bands. The excitation is followed by an equilibration process [46]. After photogeneration, the electrons and holes thermalize among themselves on a time scale

Fig. 10.4 (a) Hot carrier dynamics in graphene. (b) Saturable absorption in a graphene–polymer

composite for different wavelengths between 1548 and 1568 nm. (c) Ultra-fast fiber laser. The mode-locker consists of a graphene–polymer composite, sandwiched between two fiber connectors (reproduced with permission from [47]).

10.6 Thermal Radiation Sources

191

of tens of femtoseconds via carrier–carrier scattering, leading to a hot electron–hole plasma with elevated electron and hole temperatures. As the carrier–carrier collisions preserve the total electronic energy, the carrier temperature scales with pump power and can reach thousands of kelvins. Pauli blocking of interband optical transitions gives rise to the saturation behavior. Subsequently, a phonon-mediated cooling process sets in and the system returns to equilibrium on a picosecond time scale. An example of an ultra-fast laser that comprises a graphene saturable absorber is shown in Fig. 10.3(c). This specific device is entirely fiber-based and uses a ring cavity design [47]. An erbium-doped fiber, pumped by a diode laser, is used as the gain medium. The optical isolator and polarization controller maintain unidirectional pulse propagation for optimized mode-locking. The mode-locker itself consists of a graphene–polymer composite, sandwiched between two fiber connectors. Other works have utilized mechanically exfoliated flakes, CVD-grown films, functionalized graphene, or reduced graphene oxide flakes instead. The laser produces 464-fs-long pulses at a 1.55 μm wavelength and with a 20 MHz repetition rate. A more recent, optimized, design has allowed the generation of pulses as short as 29 fs [48]. Graphene saturable absorbers have also enabled mode-locking of free-space solid-state lasers [49] and semiconductor lasers [50].

10.6

Thermal Radiation Sources Light sources in the far-infrared and THz regions of the spectrum are scarce. This is largely due to free electron or phonon absorption in the materials used in optics. This, for example, limits the operation of quantum cascade lasers in the region of phonon absorption of III–IV materials. As a result, this important spectral regime is often referred to as an “optical desert” [51]. Thermal radiation could be used in this frequency regime but, unlike laser radiation which is monochromatic, coherent, and directional, thermal radiation is generally considered to be broad, incoherent, and determined solely by the material properties. However, not all of this is necessarily true [52–54]. One way of achieving coherence and directionality in thermal radiation involves the thermal excitation of coherent propagating modes, plasmons in free electron systems, or phonon polaritons in polar dielectrics. If the material is appropriately patterned into a grating, momentum conservation can be achieved and these evanescent modes can become radiative [55]. Similar results can be obtained by coupling to other types of antennae systems [56]. Graphene with its continuum of quantum states, controllable carrier concentration, and high thermal stability seems to be an ideal thermal radiation source. Spatially resolved thermal radiation from electrical heated graphene on SiO2 in a FET configuration was studied by Freitag et al. [57] and Berciaud et al. [58]. The temperature was measured using the Stokes/anti-Stokes ratio and the shift of the G0 band in Raman measurements. They found that the location of the maximum temperature along the length of the graphene channel and the position of the Dirac point could be controlled by the gate voltage. The infrared radiation in general follows Planck’s law, but local hot

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Graphene: Optoelectronic Devices

spots were also observed. By suspending the graphene so as to reduce heat dissipation, graphene could be electrically heated to even higher temperatures (e2800 K) leading to bright visible light emission [59]. Emission from electrically heated graphene incorporated inside a planar metal microcavity was first reported by Engel et al. [31]. The cavity was fabricated with a resonance in the near infrared. Unlike free-space thermal radiation, thermal radiation from the cavity had a 140 times narrower spectral distribution, and was concentrated in a 240 angular lobe (FWHM). An effort to enhance thermal emission and make it directional and tunable was reported by Brar et al. [60]. They studied thermal emission from a graphene nanoribbon array on SiNx. Along with the broad thermal emission, two narrow band emissions at 1360 cm–1 and 730 cm–1 were observed. The 1360 cm–1 emission was found to be polarized perpendicular to the ribbon axis and be tunable by changing the gate voltage. The 730 cm–1 band was largely unchanged by gate bias. It was concluded that the high-energy band is due to emission from the confined plasmon of the ribbons, while the low-energy emission is due to emission from the phonon of the underlying SiNx. The authors estimated that the device can modulate a 2 μW of power over a 100 cm–1 bandwidth, which is comparable or better than commercial LEDs over similar bandwidths. The spectral observations in the above study can be understood on the basis of Kirchhoff’s law connecting thermal absorption and emission [61] and the fact that the nanoribbons allow plasmon excitation with perpendicularly polarized light [62], and that these plasmons can couple effectively with the underlying substrate phonons [60] Brar et al. also point out that spontaneous thermal emission is subject to the Purcell effect and therefore can be modulated in plasmonic cavities. Thermal emission from graphene at even longer wavelengths (few THz range) from a FET was reported by J. Tong et al. [63]. The emission was found to be optimized at a particular gate voltage suggesting a charge-density dependence of the THz generation mechanism. From the above, we can conclude that graphene in the form of gratings, or coupled to various antennae, or in metamaterial structures could potentially produce directional, thermal emitters that operate throughout the infrared frequency range.

10.7

Passive Optical Elements Graphene is becoming the basis of a number of passive optical elements such as filters and polarizers, which are scarce particularly in the THz region. It is relatively simple to form notch filters by patterning graphene into nanostructures which exhibit localized plasmon resonances. Using nanoribbon structures and utilizing the fact that only light polarized perpendicular to the ribbons can excite the plasmons, polarizers can be produced (Fig. 10.5(a)) [62]. The frequency and attenuation of such devices can be tuned by doping, changing the confinement, or the stacking of multiple graphene layers [64]. The limited absorption of graphene, however, limits practical applications. For this reason, graphene is usually coupled with waveguides, optical fibers or as metamaterials structures. For example, Bao et al. [65] demonstrated a TE optical fiber graphene polarizer with an up to 27 dB extinction ratio at 1.55 μm (Fig. 10.5(b)), while

10.8 Transparent Conductive Electrodes

193

Fig. 10.5 (a) Graphene nanoribbon polarizer (ribbon widths: 50–130 nm) (reproduced with permission from [62]). (b) Optical fiber graphene polarizer (reproduced with permission from [65]).

Kim et al. [66] produced a polymer waveguide which changes its polarization from TE (e10 dB) to TM (e19 dB) by adding a dielectric cladding on the graphene. A graphene metamaterial, tunable THz filter, which is based on the shift of the resonance frequency, was proposed by Yang et al. [67].

10.8

Transparent Conductive Electrodes A very active area of graphene research involves its use in transparent and conductive electrodes. Such electrodes are employed in touch screens [68], organic light-emitting diodes (OLED) [69, 70], and organic photovoltaic (OPV) cells [71, 72]. Currently the dominant technology is based on indium tin oxide (ITO), which has a low sheet resistance of 10–25 Ω/sq, 90% transmittance, and can be deposited conformally by sputtering. However, ITO is expensive and has poor mechanical properties when bent or stretched. Other conductive polymer systems used for the same purpose, such as PEDOT:PSS films, have good optical properties, but poor environmental stability. Graphene, on the other hand, has excellent optical transparency (e97%), high carrier mobility, is inert and stable, and has a high Young’s modulus [73]. Furthermore, in principle, it can be produced cheaply by catalytic CVD processes, and large-area deposition can be achieved using roll-to-roll technology [68]. In practical devices, several graphene layers need to be stacked to reduce the sheet resistance, which, for

194

Graphene: Optoelectronic Devices

Cu CVD graphene, is typically about 300 Ω/sq and also for structural stability. At four layers, this resistance is already reduced to about 50 Ω/sq. However, as the number of graphene layers increases the optical transmittance decreases by roughly 3% (at 550 nm). Further reduction in sheet resistance can be induced by doping using chemical dopants such as HNO3 or AuCl3 or electrostatic doping using, for example, poled ferroelectric fluoropolymers [74]. A number of different devices utilizing graphene electrodes have been successfully demonstrated. These include both resistive and capacitive touch screens. The former require 90% transmittance at 550 nm and a sheet resistance up to 550 Ω/sq [68], while the latter must have a lower resistance up to 100 Ω/sq. Graphene OLEDs have been demonstrated for both display applications and lighting. In a recent study, green graphene OLEDs with an enhanced out-coupling and the addition of a hole transporting layer were demonstrated and had external quantum efficiency >60%. Phosphorescent white organic light-emitting diodes with an external quantum efficiency >45% at 10,000 cd/m2 with color rendering index of 85 were also demonstrated [75]. In OPVs, graphene is used as a counter electrode and a low resistance and high work function are required [76–78]. Current research in the graphene conductive electrode area is focusing on increasing the sheet conductivity of graphene, developing doping schemes that provide better environmental stability [79, 80], and efforts to decrease contact resistances [81]. In terms of processing the graphene films, improvements in the roll-to-roll process are required and also ways of transferring CVD graphene on non-flat structured surfaces.

Acknowledgements T.M. acknowledges financial support by the Austrian Science Fund FWF (START Y-539).

10.9

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

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11

Graphene Spintronics Aron W. Cummings, Sergio O. Valenzuela, Frank Ortmann, and Stephan Roche

11.1

Introduction to Spintronics Charge and spin are two fundamental properties of the electron which are currently exploited in advanced technologies, but to date they have been used separately in information processing and data storage, respectively. Charge currents drive the operation of elementary electronic devices and logic circuits that encode and process binary or analogue information. Meanwhile, the spin degree of freedom is used in its collective form of magnetic domains for switching magneto resistance signals and realizing long-term data storage, from ferrite core memories to modern hard disk drives [1]. The field of spintronics aims to combine the charge and spin of electrons to create novel functionalities [2]. In the simplest spintronic device, called a spin valve, an electronic current flows between two ferromagnetic electrodes through a non-magnetic channel. A spin signal is carried along with the charge current and is normally detected through its magneto resistance [3], which is influenced by the magnetic ordering of the electrodes. The requirement for non-magnetic channels is to transport spin currents with minimum spin information loss due to spin-scattering events, which in most cases are caused by spin–orbit coupling. The field of spintronics emerged from scientific discoveries in the 1980s, which concerned spin-dependent electron transport phenomena in solid-state devices. Following the observation in 1985 by Johnson and Silsbee [4] of spin-polarized electron injection from a ferromagnetic metal to a normal metal, the foundational step of the field of spintronics was the discovery, by Albert Fert et al. [5] and Peter Grünberg et al. [6], of giant magneto resistance in thin film structures composed of alternating ferromagnetic and non-magnetic conductive layers. Control of magneto resistance has required the use of various magnetic and non-magnetic metallic and semiconducting materials, and has resulted in a massive technological impact on magnetic field sensors, which today are used in hard disk drives, biosensors, microelectromechanical systems (MEMS), and magneto resistive random-access memory (MRAM) [7]. Following the discovery of the spin transfer torque effect, which permits the control of the magnetization with an electrical current, a second revolution in spintronics is currently underway [8]. As will be discussed in this chapter, graphene spintronics has been envisioned as highly promising, owing to its unique electronic band structure of so-called massless 197

198

Graphene Spintronics

Dirac fermions, which are robust to backscattering and allow charge to travel at room temperature at an unprecedented speed (1/300 of the speed of light c) over long distances (tens of microns). Additionally, the weak spin–orbit coupling (SOC) in sp2 carbon also suggests that the electron spin should be carried nearly unaffected over considerable distances, making practical applications of lateral spintronics feasible [9, 10]. Pioneering works have echoed such high expectations, heralding graphene as a unique material [11]. Current state-of-the-art research confirms the potential of graphene for transporting spin signals over tens of micrometers at room temperature, which is sufficient for the realization of magnetic sensors, nanooscillators, or spin-based logic circuits [12–15]. Graphene also provides solutions for the integration of several circuit elements on the same platform. For instance, non-volatile graphene field-effect transistors with ferroelectric gates have been demonstrated to operate as three terminal resistive memories, while graphene-based memristors are interesting since they may act both as memory and logic elements [16]. From this perspective, graphene, being compatible with more-than-Moore CMOS and non-volatile low-energy MRAM, brings revolutionary opportunities for achieving efficient spin manipulation and for the creation of a full spectrum of spintronic nanodevices, including ultra-low-energy devices and circuits comprising (re-)writable microchips, transistors, logic gates, and more.

11.2

Advantages of Graphene for Spintronics

11.2.1

SOC in Graphene – Expectation of Long Spin Lifetimes The unique electronic band structure of graphene leads to a strong suppression of backscattering and allows electrons to travel over long distances at a constant speed of c/300, even at room temperature. This makes graphene highly interesting for a variety of electronic applications, where high carrier mobility plays an important role. Also different than other materials (metal and semiconductors), the electronic structure in graphene is electron–hole symmetric with a linear energy dispersion relation E
ðpÞ ¼
vF jpj. The total momentum close to the Dirac point, i.e., near the two inequivalent Kþ and K points of the zone, is K
þ pħ. By linearization of
!Brillouin


the dispersion relation for small p , one can write an effective Hamiltonian   H K þ ¼ H †K  ¼ vF σ  p, which describes massless Dirac fermions (σ ¼ σ x ; σ y ; σ z , defined by the Pauli matrices). The corresponding eigenvectors for the K+ valley are !   ψA   p eiθ=2
1 1 p1ffiffi py =px and the labels A/B given by ψp ¼ pffiffi2 ¼ iθ=2 , with θ ¼ tan B 2
e ψp refer to different sublattices of graphene’s honeycomb crystal structure, also referred to as the pseudospin degree of freedom. Additionally, the absence of hyperfine coupling and the small intraatomic spin–orbit coupling in carbon atoms should yield long spin relaxation times [17]. The spin–orbit coupling interaction can be derived from the weakly relativistic limit of the Dirac

11.2 Advantages of Graphene for Spintronics

199

Hamiltonian. In its simplest explanation, it can be seen as an effective magnetic field in the rest frame of an electron moving through an electric field E. This effective magnetic field couples to the spin angular momentum of the electron as a Zeeman term, so that the SOC Hamiltonian can be written as H so ¼ gμ2B BR  σ, where the Rashba SOC magnetic Ep field is generically expressed as BR ¼ 2mc 2 . Such a field will induce a spin precession

2μB jBR j ħσ given by dS dt ¼ Ω  S (S ¼ 2 ) with spin precession frequency jΩj ¼ ħ . We finally note that the SOC Hamiltonian is invariant under the time reversal operation [17]. The size of the intrinsic spin–orbit coupling in graphene was a matter of some debate, with the earliest estimates of 200 μeV [18] conflicting with tight-binding studies (s and p orbitals only) [19] that predicted a spin–orbit gap as small as 1 μeV. All-electron firstprinciple calculations gave a much higher value of 50 μeV [20], but it was later shown that the spin–orbit coupling is influenced by the (nominally unoccupied) d orbital and higher orbitals [21], which give a gap of about 25 μeV. It is currently accepted that the intrinsic SOC is a value ranging between 10 and 25 μeV. Additionally, in a device geometry, graphene is generally placed onto a substrate, which usually includes screened charged impurities and is gated during the transport measurement. While free-standing graphene ideally has a center of inversion symmetry, making its states doubly (spin) degenerate at a given momentum, even in the presence of intrinsic spin–orbit coupling, graphene on a substrate or under a gate bias voltage loses this property and the bands are split. This splitting is akin to the one encountered in semiconductor physics under the name of Bychkov–Rashba splitting or structure inversion asymmetry-induced splitting [22, 23]. In the presence of Rashba splitting, only Kramers degeneracy is left, meaning that the energies of the states of opposite spins and momenta are equal. The Stark effect is the mechanism describing the extrinsic splitting of the graphene bands, driven by the SOC-induced hybridization of pz and s orbitals combined with the intraatomic splitting of the p-orbitals. Tight-binding and ab-initio calculations show that d orbitals give a contribution of only about 1% [19, 22]. The corresponding extrinsic gap is typically found to reach about 10 μeV for an electric field of 1 V/nm, and it scales linearly with the field. For the continuum model approximation of linear massless Dirac  fermions, the expression of the Rashba SOC Hamiltonian reads H R ¼ λR τσ x sy  σ y sx whereas the intrinsic SOC Hamiltonian is H I ¼ λI τσ z sz with τ ¼
1 a valley index, and σ, s are Pauli matrices for pseudospin and spin, respectively. It is also  useful to note that if one computes theeigenstates of the full Hamiltonian H ¼ vF τσ x px þ σ y py þ λI τσ z sz þ λR τσ x sy  σ y sx , at low values of momentum   0 1 (vF p  λI , R ), one gets two solutions ψIp
¼ ⊗j"i
i ⊗j#i and ψIIp
¼ 1 0   1 0 ⊗j"i
i ⊗j#i, where here we define two “up/down” or “1/0” states, respect0 1 ively, for the spin and pseudospin degrees of freedom. This particular correlation between spin and pseudospin is referred to as spin–pseudospin coupling, and it has further consequences for the time-dependent dynamics of spin and pseudospin polarization and spin relaxation mechanisms.

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Graphene Spintronics

Finally, it is worth mentioning that a significant enhancement of the extrinsic spin splitting has been reported for supported graphene [24–26]. The estimations of the splitting are obtained by assuming the Elliot–Yafet spin relaxation mechanism, which consistently results in a splitting of the order of 1 meV in samples of different origins and processing [27]. In the case of chemisorbed hydrogen, such an enhancement was argued to result from charge transfer and local sp3 hybridization and distortion caused by the defect/graphene interaction [28]. This may have consequences for the formation of the spin Hall effect, as will be discussed later.

11.2.2

Lateral Spintronics – Room Temperature Record High Spin Diffusion Lengths The first spin injection measurements based on non-local spin valve geometry revealed surprisingly short spin relaxation times of 100–200 ps, which were only weakly dependent on the charge density and temperature [11]. However, after significant effort to improve device integration, and interface engineering such as encapsulating graphene with hexagonal boron nitride substrates, room temperature spin lifetimes in single, bi, and trilayer graphene devices have been found to reach up to 12 nanoseconds with spin diffusion lengths up to 30 μm, combined with carrier mobilities exceeding 20,000 cm2/(V s) [12–14, 77]. The spin relaxation lengths of graphene compare favorably with those obtained in metallic thin films such as aluminum [29], silver [30], or copper [31]. For metallic thin films, the relaxation lengths are about 1 μm. The transmission of spin information has been demonstrated over much longer distance in crystalline aluminum (several hundreds of micrometers [4]) and semiconductors, albeit at cryogenic temperatures. In GaAs, spin lifetimes of a few tens of nanoseconds and spin relaxation lengths of the order of 10 μm have been observed [32]. The results are more favorable in intrinsic silicon: spin lifetimes of several hundred of nanoseconds have been reported [33] whereas the measurements of Hanle precession was identified over millimeter distances [34].

11.3

How to Measure Spin Lifetimes in Graphene and 2D Materials

11.3.1

Two-Terminal Resistance Measurements To investigate the potential of graphene for spintronics, two-terminal resistance measurements are of fundamental importance. This approach allows for the investigation of interface effects, and for optimizing spin injection efficiency. In addition, using the phenomenological theory developed by Fert and co-workers, an estimate of the spin diffusion length can be obtained [35]. Finally, two-terminal devices will be key for the further construction of spin-logic-based circuits, so that their complete understanding will be necessary for future technologies. As discussed above, graphene exhibits a variety of electronic properties suitable for spintronics, including the capability for efficient spin injection from ferromagnets owing to the tunability of the Fermi level [35]. The large electron velocity already

201

11.3 Spin Lifetimes in Graphene and 2D Materials

implies the propagation of spin polarized currents over long distances, meaning that the optimization of graphene spin valve devices necessarily includes a focus on the specific role of magnetic contacts, including their spin injection and detection efficiencies and the possible role of interface effects in spin relaxation and spin dephasing. Interface engineering to optimize spin injection and detection between a ferromagnet and graphene still remains a very challenging issue and represents one of the main bottlenecks for graphene spintronics. Systematic investigations of the role of the tunnel barrier have been slow because of difficulties in growing uniform, ultra-thin insulating layers on graphene. Any irregularity during growth favors the formation of pinholes which may alter the spin injection efficiency. Contact effects and general environmental factors can have an important impact on the estimated spin relaxation lengths and thus bring an important contribution to the device efficiency. A fundamental mechanism known as “conductivity mismatch” was identified in 2000 as a roadblock for efficient spin injection from ferromagnets into semiconductors [4]. To circumvent this effect and to prevent the “backflow” of the spins back into the contact, efforts have been made to increase the contact resistance above the “spin resistance” of the spin transport channel. This backflow is generally accompanied by spin relaxation in the metallic ferromagnet, which has a very short spin relaxation time. It has been demonstrated that the effect can be controlled technologically by tuning the dwell time of the spins in the semiconducting channel, giving only a narrow window for the contact resistance [28]. In graphene, this phenomenon is well described in [35]. A two-terminal spin valve device made of graphene on a silicon carbide substrate is shown in Fig. 11.1(a), and the local magneto resistance spin signals measured at 4 K are shown in Fig. 11.1(b). Variations of about 10% of the magneto resistance in the range of 6 MΩ are observed.

(a) (b)

SiC

V

12 6.4

10 8

6.2

6 4

6.0

2

DR=MR (%)

200 mm

42

Resistance (MW)

Epitaxial graphene

0

Co

5.8 Epitaxial graphene SiC

1 mm

–1000

–500

0

500

1000

Magnetic field (Oe)

Fig. 11.1 (a) Typical two-terminal local spin valve device. The width of the graphene channel on SiC is 10 μm, and the distance between the two Al2O3/Co electrodes is L = 2 μm. The optical image shows the entire structure, including contact pads. (b) Large local magneto resistance spin signals measured at 4 K (adapted from [35] with permission).

202

Graphene Spintronics

11.3.2

Non-Local Hanle Spin Precession Measurements The spin diffusion coefficient and the spin relaxation time are generally estimated from Hanle measurements (Fig. 11.2), which are non-local transport measurements in which the spin diffusion far from the source–drain contact is tuned by an external and perpendicular magnetic field, inducing spin precession [4, 36–37]. The basic physical principles of the non-local devices are electrical spin injection, the generation of non-equilibrium spin accumulation, and electrical spin detection using ferromagnetic electrodes as spin polarizers. Sketches of the device are shown in Figs. 11.2(a) and (b). Figure 11.2(c) shows a scanning electron microscope image of an actual device based on a suspended graphene flake [37]. The injected current I on the source contact (FM1) produces spin accumulation in the graphene layer, which is quantified by the detector voltage VNL. The current I is injected from FM1 and away from FM2, and the electron spins diffuse isotropically from the injection point. The sign of VNL is determined by the relative magnetization orientations of FM1 and FM2 (Fig. 11.2(d)). The spin accumulation and VNL can then be quantified from the spin splitting in the electrochemical potential induced by spin injection, which decays over a characteristic length λs. The spin orientation can be manipulated by applying an external magnetic field Bperp [35–37], which is perpendicular to the substrate and induces coherent spin precession in the plane of the graphene layer (Fig. 11.2(b)). In this situation, spins that are polarized along the FM1 magnetization rotate around an axis that is parallel to the field with a period determined by the Larmor frequency Ω = γBperp, where γ is the gyromagnetic ratio of the electron. During the time t that it takes the electron to travel to FM2, the spin will rotate an angle ϕ = γt. Because VNL is sensitive to the projection of the spins along the FM2 magnetization, it oscillates as a function of Bperp (Fig. 11.2(e)). The measured non-local magneto resistance RNL = VNL/I is usually modeled with a one-dimensional spin-Bloch diffusion equation [35–37], which assumes a diffusive propagation of spin and relates the resistance to microscopic parameters through þ∞ ð

RNL  0

1 L2 t pffiffiffiffiffiffiffiffiffiffi e4Ds t cosðΩ t Þeτs , 4πDt

(11.1)

where Ds = vF2τs is the spin diffusion coefficient, τs is the spin relaxation time, and L is the distance between electrodes. An important observation is that this approach cannot tackle the situation of ballistic or quasi-ballistic spin propagation and needs further generalization for describing clean graphene, for which mean-free paths can be several hundreds of nanometers long [38], and are thus comparable to the typical electrode spacing. Additionally, for more disordered graphene the contribution of quantum interferences and localization phenomena, which in certain materials have been shown to be robust up to 100 K, are neglected and could affect the estimates of Ds and τs. Finally, τs has been also estimated independently from two-terminal spin valve

11.3 Spin Lifetimes in Graphene and 2D Materials

203

Fig. 11.2 (a) Schematic of a non-local spin injection/detection device. An in-plane magnetic field is applied to change the relative orientation of the magnetization of the ferromagnetic contacts. (b) Electrical detection of spin precession in a perpendicular magnetic field. (c) Scanning electron microscope image of a suspended graphene device. Only the two inner contacts in (a) are visualized. The scale bar is 200 nm. (d) Non-local measurements in a graphene device using the configuration shown in (a) as a function of the in-plane magnetic field. (d) Spin precession measurements for parallel and antiparallel configuration of the electrode magnetizations.

measurements, and was found to be at least one order of magnitude larger than those obtained from Hanle measurements [35–37]. This difference was attributed to the high contact resistances in the two-terminal measurements, compared with those typically found in four-terminal Hanle measurements [12, 35]. Spin transport has been studied in non-local devices (see illustration in Fig. 11.2(a)) where the graphene was attached to, or supported by, hexagonal boron nitride (h-BN) [13, 14, 77]. Significant increases of the spin lifetime (up to 12 ns) and length (exceeding 30 micrometers at room temperature) have been reported as compared with graphene on SiO2 establishing a new record for spin relaxation length at room temperature [77]. Others have performed experiments where the graphene was fully encapsulated by h-BN, showing indications that the ratio of spin lifetimes for spin directions perpendicular and parallel to the graphene sheet can be tuned by the use of top and bottom gate electrodes [13]. These results suggest an electrically induced Rashba spin– orbit coupling, and open up new possibilities for electric control of spin transport in

204

Graphene Spintronics

graphene. However, subsequent studies of this ratio using a much smaller perpendicular magnetic field showed no evidence of Rashba spin–orbit coupling, albeit for a SiO2 substrate and not for graphene fully encapsulated in h-BN [78]. These results identify the Rashba spin–orbit coupling as one of the factors which could be responsible for the observed short spin relaxation times. Another bottleneck for spin transport is inhomogeneous oxide barriers which may result in conducting pinholes within the otherwise insulating spin injection and detection barriers [39]. In graphene/ MgO/Co spin valve devices, it was found that the pinholes cause inhomogeneous current flow through the MgO barrier. Next to the actual spin signal, this results in an additional charge accumulation signal which is measured as a magnetic field-dependent background signal in the non-local voltage [40]. This background signal, which is often observed in non-local spin transport studies, is thus a hallmark for the quality of the oxide barriers. Additional background signals that depend on the carrier density can originate from thermoelectric effects that are known to be strong in graphene and that can be further enhanced by the presence of hot electrons [41]. Finally, it is worth observing that spin injection is enhanced by the use of suitable interface materials. Because of the so-called conductance mismatch and the spin absorption at both injector and detector FMs, the spin injection efficiency is strongly suppressed for Ohmic contacts. Typical reported non-local spin magneto resistances in this case, i.e., the overall change ΔRNL in the non-local spin transresistance VNL/I between the parallel and antiparallel configuration of the electrode magnetizations, are in the range of a few mΩ to a few tens of mΩ [42]. Larger values of ΔRNL have been obtained by placing an insulator between graphene and the FMs, which helps circumvent the conductance mismatch and reduces the spin absorption in the contacts. The insulators are typically MgO or AlOx because of their success for tunnel magneto resistance [11]. In this way, ΔRNL was observed to increase to up to a few ohms (pinhole barrier) or a hundred ohms (tunnel barrier) [43]. However, high-resistance tunnel barriers are detrimental to highspeed and spin-torque applications and alternative approaches to increase ΔRNL and the spin accumulation have been proposed by, for example, adding a Cu layer at the metal– graphene interface [44]. Recently developed unconventional interfaces comprising amorphous carbon [45] or fluorinated graphene [46] are also very promising.

11.3.3

Spin Relaxation Mechanism – DP versus EY The analysis of scattering mechanisms and their connection to spin relaxation are necessary in order to achieve optimal production methods for graphene-based spin valves. In materials such as metals and small band-gap semiconductors, two mechanisms are usually dominant, namely the Elliot–Yafet (EY) [47, 48] and Dyakonov–Perel (DP) [49] mechanisms. In EY, the electron spin has a finite probability to flip off impurities or phonons during each scattering event [47, 48]. The DP mechanism is driven by the precession of electron spins about an effective magnetic field whose orientation depends on the momentum, with the direction and frequency of precession changing at each scattering event [47, 48]. A theoretical derivation of the EY mechanism

11.3 Spin Lifetimes in Graphene and 2D Materials

205

Fig. 11.3 Dynamics of spin polarization depending on the transport time τ p . Ultra-short τ p (a) and (d), medium τ p (b) and (e), and long τ p (c) and (f). (a)–(c) Illustration of spin vectors for electrons for increasing time as indicated in (d)–(f), respectively. (d)–(f) Time dependence of the polarization averaged over the Fermi circle. The time-dependent profiles suggest a scaling as t Sz  cos ð2πt=T Ω Þeτs , where T Ω is the spin precession time and τ s denotes the spin relaxation time.

in SLG, accounting for the Dirac cone physics, predicts a spin relaxation time that varies proportionally with both the transport time and the electron density [50]. Figure 11.3 provides an overview of the dynamics of spin polarization of electrons moving through graphene away from the Dirac point. The dependence of the polarization dynamics on disorder strength is dictated by the transport time τ p . An effective spin–orbit coupling λR (as induced by ad-atoms for instance) leads to spin precession when the injected electrons are polarized out of the plane or along the direction of momentum. The spin precession time is given by T Ω ¼ πħ=λR . Panel (a) shows the case when τ p  T Ω , a regime which is typically seen in experiments. Different snapshots at different times (T 1 , T 2 , T 3 ) show that the ensemble-averaged spin polarization (Sz in panel (d)) decays weakly, since elastic scattering interferes with the spin precession. A scattering event experienced by one propagating state randomizes its momentum and hence the orientation of the effective field about which the spin precesses. Being no longer coherent with the other states, such randomization drives spin relaxation in the ensemble. The spin relaxation time depends on how strongly the precession of the scattered state deviates from the rest of the ensemble. If the scattering time is short (τ p  T Ω ), the difference in the precession remains small.

206

Graphene Spintronics

By increasing the scattering time (τ p  T Ω ), faster spin decoherence is obtained (panel (b)), as confirmed by comparing the corresponding averaged polarization (panels (d) and (e)). This illustrates the Dyakonov–Perel mechanism, for which the spin relaxation time scales inversely with the transport time (τ s / 1=τ p ). Panel (c) shows the clean limit (with small residual disorder such that τ p T Ω ), where only a few scattering events occur during the time sequence. The average polarization (panel (f)) then clearly exhibits the spin precession phenomenon characterized by oscillations with period T Ω . On top of this, the spin signal decays continuously owing to scattering and dephasing effects. The decay is weaker for lower scattering, suggesting that τ s should be maximized for long τ p , usually observed at energies closest to the Dirac point. However, based on further quantum simulations, we find the opposite behavior, i.e. τ s is minimized at E  0 and that a simplified description of spin motion fails to capture the more subtle phenomena in place at the Dirac point, as detailed below.

11.4

New Spin Relaxation Mechanisms Original predictions of spin lifetimes in the micro- or even millisecond range [51], based on conventionally conceived mechanisms of spin relaxation, have heralded graphene as an exceptional and unique material for the development of lateral spintronics [15]. However, despite one decade of experimental efforts to improve the quality of the material and device fabrication, spin lifetimes have only reached the nanosecond scale. A variety of modeling techniques from ab-initio to efficient real-space spin propagation methods have to be used for an in-depth exploration and understanding of what makes spin dynamics in graphene unique and why spin relaxation times are much shorter than initially predicted. New spin relaxation mechanisms, beyond Dyakonov– Perel and Elliot–Yafet, have been recently discovered in graphene and are either related to the presence of magnetic moments [53, 54] or to the contribution of spin–pseudospin locking, the latter of which has been shown to yield minimum spin lifetimes at the Dirac point and at high energies [55–57]. Experimentally these mechanisms can be identified by studying the spin relaxation anisotropy ζ = τ s? s∥ , where τ s∥ τ s? is the lifetime of spins oriented in (perpendicular to) the graphene plane [9, 13]. The ratio ζ provides information on the dominant spin–orbit fields that are involved in the relaxation. A classical example is the two-dimensional electron gas with Rashba spin–orbit coupling, for which ζ = 0.5; this is due to the fact that spins along the in-plane Rashba field do not precess and do not dephase. Recently it has been demonstrated that ζ can be readily determined by measuring the response of non-local spin devices under oblique magnetic fields [78]. Fast spin relaxation can be related to the presence of isolated magnetic impurities that also act as resonant scatterers. A magnetic impurity locally produces a fixed (although arbitrarily polarized) magnetic moment [52, 54], and an enhanced spin flip probability is introduced due to the exchange interaction between propagating spins and the local

11.4 New Spin Relaxation Mechanisms

207

Fig. 11.4 (a) “Planetary” model of resonant scattering electrons, where the electrons tend to stay around (“orbit”) the impurity for a certain dwell time. If the dwell time is comparable to or greater than the spin precession time induced by the exchange interaction with the local magnetic moment, then spin-flip scattering is as likely as spin-conserved scattering. (b) Schematic of the entangled spin–pseudospin dynamics induced by local Rashba spin–orbit interaction related to adsorbed gold atoms.

moment, where resonant scattering occurs. While in conventional scattering (Born approximation) the probability for a spin flip is small (of the order of 0.01%), for a resonant scatterer the spin flip probability becomes comparable to that of spin conserving scattering (around 50%). Similarly to a Friedel virtual bound state, propagating spins will remain trapped by the resonant scatterer for a certain dwell time, during which the electron spin will precess due to the exchange interaction with the magnetic impurity (Fig. 11.4(a)). Such mechanism works as long as the spin precession time is shorter or comparable to the resonance dwell time, which is predicted to be the case for ad-atoms such as hydrogen [52]. The local magnetism induced by a hydrogen impurity has been calculated using the self-consistent Hubbard model in the mean field approximation for the dilute limit, while spin relaxation lengths and transport times were computed using an efficient realspace order-N wavepacket propagation method [17]. Typical spin lifetimes of the order of 1 ns are obtained for 1 ppm of hydrogen impurities, while charge transport times of about 50 ps lead to spin diffusion lengths far beyond tens of micrometers. The spin lifetime modification with impurity density suggests a scaling τ s / τ p as for the case of the Elliott–Yafet mechanism, even though the origin of the spin relaxation is unrelated to the spin–orbit interaction [54]. Fast spin relaxation can also be related to the complex interplay between the spin and pseudospin degrees of freedom when disorder does not mix valleys [55]. The contribution of spin/pseudospin entanglement is particularly important when defects or impurities at the origin of local Rashba spin–orbit coupling preserve the pseudospin symmetry and lead to very long mean-free path (schematics shown in Fig. 11.4(b)). Pseudospin-related effects (and the associated Berry’s phase) are known to drive most of the unique transport signatures of graphene (Klein tunneling, weak antilocalization, anomalous quantum Hall effect), but until recently the role of the pseudospin on spin relaxation has not been established. Pseudospin and spin dynamics are usually perceived as decoupled from one another, with pseudospin dynamics much faster than those for spins. However, this picture collapses in the vicinity of the Dirac

208

Graphene Spintronics

Fig. 11.5 (a) Convolution of an energy-dependent spin precession frequency (right axis) and a Lorentzian energy broadening (left axis). The HWHM of the Lorentzian is η, and the variation in the spin precession frequency is α. (b) Exponentially decaying cosine, with frequency ω0 and decay time 1/αη.

point, a region that is out of reach for semiclassical and perturbative approaches, but is particularly relevant for experiments. In the presence of SOC, spin couples to orbital motion, and therefore to pseudospin [24], so that spin and pseudospin dynamics cannot be treated independently. Below we present spin dynamics in graphene using more general quantum simulations while taking into account the specificities of the graphene band structure in the presence of spin–orbit coupling and disorder (energy broadening and electron–hole puddles). It has recently been found that the strength and size of electron–hole puddles can vary significantly for different substrates (such as SiO2 and h-BN), and in the situation of clean supported graphene it is generally assumed that a weak Rashba-SOC field is always present because of mirror symmetry breaking and environmental electric fields. Values of the Rashba spin–orbit coupling typically range from a few to a few tens of μeV. An important limit for spin relaxation in graphene is the limit of very weak momentum scattering, when charges propagate ballistically and the momentum scattering rate is no longer a relevant timescale for spin relaxation. In this limit, spin relaxation is dictated by the presence of energy broadening and non-uniform precession frequency. The presence of spin–orbit coupling generates an effective magnetic field Beff at the origin of the electron spin precession. When the magnitude of Beff (or its direction) depends on the energy or momentum of the charge carriers, then because of a certain distribution of energies or momenta (due to temperature or effective disorder-induced broadening), the total spin signal will be dictated by the interference and dephasing between different precession frequencies. To see this more explicitly, consider the example shown in Fig. 11.5, where we assume that the spin precession frequency varies linearly with energy, ωðEÞ ¼ ω0 þ αE. We also assume that the charge carriers occupy a Lorentzian distribution in energy, LðE Þ ¼ π E2ηþη2 , where ½ð Þ

η is the half-width at half-maximum (HWHM). It is straightforward to show that the total spin signal is

11.4 New Spin Relaxation Mechanisms

ð∞ sðt Þ ¼ LðEÞ∘ cos ðωðE Þt Þ ¼

209

η=π cos ððαE þ ω0 Þt ÞdE ¼ eαηt cos ðω0 t Þ: (11.2) E þ η2 2

∞

In general, Eq. (11.2) indicates that the combination of energy broadening and non-uniform spin precession leads to a decay in the spin signal due to dephasing, with a relaxation rate proportional to both the broadening η and the variation in the precession α. This decay will occur even in the ballistic limit as long as there is a mixing in energy or momentum. We note that the decay is not necessarily exponential, but rather depends on the broadening function and the variation in the precession frequency. For example, replacing the Lorentzian in Eq. (11.2) with a Gaussian distribution gives a 2 time decay of eðασtÞ =2 , where σ is the standard deviation, while a Fermi distribution yields a decay of ζ t= sinhðζ t Þ, where ζ ¼ απkT and kT is the thermal energy [57]. Figure 11.6 below shows how this phenomenon manifests itself in pristine graphene. We use a tight-binding representation of graphene with Rashba SOC (see Eq. (11.3) below) from which we calculate the spin dynamics of a large sample of k-points in the first Brillouin zone. The total spin signal is the sum of spin dynamics at each k-point weighted by a Lorentzian broadening function. Panels (a) and (b) show the complex spin dynamics that arise in graphene with Rashba spin–orbit coupling in the presence of energy broadening. A non-uniform and anisotropic spin precession frequency, arising from the tight-binding band structure, results in a collective spin signal

Fig. 11.6 Spin dynamics in clean graphene in the presence of Rashba SOC and energy broadening, at (a) the Dirac point and (b) at the energy of 200 meV. (c) Spin lifetimes calculated for this ballistic limit. Here we assumed a Rashba SOC of 25 μeV and an energy broadening of 13.5 meV.

210

Graphene Spintronics

exhibiting multiple exponential decays and beating patterns. Panel (c) shows the energy-dependent spin relaxation time extracted from the spin dynamics. Owing to the complexity of the spin dynamics, we calculate the spin relaxation time as the point where the envelope function of Sz(t) falls below e–1, as illustrated in panels (a) and (b). Here we have assumed a Rashba SOC strength of VR = 25 μeV and a broadening of η = 13.5 meV. There are two interesting features of the spin relaxation time shown in Fig. 11.6(c). The first is the M-shaped energy dependence of τs, which is a minimum at the Dirac point together with an independent decrease towards higher energies. Near the Dirac point, spin–pseudospin coupling leads to highly complex spin dynamics [55], which leads to fast dephasing of the spin signal in the presence of energy broadening. This effect is weaker away from the Dirac point, leading to the minimum of τs at the Dirac point. At higher energies, the spin splitting becomes anisotropic, resulting in increased dephasing due to the mixing over momentum and energy. Overall, these two effects yield the M-shape dependence of the spin lifetime on the energy or electron density. This shape is solely a consequence of the graphene band structure (including Rashba SOC), and is thus a signature of the dephasing-induced spin relaxation mechanism in graphene. The second interesting feature is the magnitude of the spin relaxation time. Even in the clean limit, relatively modest values of Rashba SOC (μeV) and energy broadening (meV) yield spin relaxation times of the order of nanoseconds. While we used a constant value of η, in realistic systems this energy broadening could arise from scattering (electron–electron, electron–phonon, etc.), electron–hole puddles [58], or even a finite temperature. The strength of the spin dephasing in the clean limit suggests that it could also play an important role in more disordered systems. To compare this limit with a situation of more realistic disorder, we simulate graphene supported on SiO2 and h-BN substrates using a tight-binding model given by  X X X ! ! † † ! V c c þ iV c z  s d (11.3) H ¼ γ0 hiji ci † cj þ ij cj : i i i R i i ij hi h i Electron/hole puddles are described by a screened Coulomb potential (long range Gaussian potential) with the onsite energy distribution of the π–orbitals 2 P α j!r α !r i j V i ¼ Nα¼1 εα e 2ζ 2 , and the puddle depths (εα ) are chosen at random so that the standard deviation for the local potential of h-BN (SiO2) is 5 meV (56 meV), similar to experiments [58]. We also assume a Rashba SOC strength of V R ¼ 25 μeV. Our calculations reveal that for graphene on SiO2, τ p =T Ω  1, whereas for graphene on h-BN τ p =T Ω  1. This will have profound consequences for the dominant spin relaxation mechanism (see below). To describe an electron spin moving in a randomly fluctuating in-plane magnetic field, we solve the time-dependent Schrödinger equation jψðt Þi ¼ eiHt=ħ jψð0Þi for an initial wavepacket which is prepared in a well-defined spin polarization, typically outof-plane or in-plane. We then compute the evolution of the spin polarization using the expectation value of the energy-projected spin operator [55]

11.4 New Spin Relaxation Mechanisms

211

Fig. 11.7 Spin lifetime versus 1/τp for graphene in the presence of SOC and e–h puddles. Squares (circles) are for graphene on the h-BN (SiO2) substrate. Closed (open) symbols are for spin relaxation at the Dirac point (at E = –200 meV). The dashed line shows the spin lifetime assuming only energy broadening. Inset: experimental results for spin lifetime versus electron mobility on h-BN and SiO2 substrates (from [14] with permission).


! 

!
ψðt Þ
s δðE  H Þ þ δðE  H Þ s
ψðt Þ S ðE; t Þ ¼ : 2hψðt ÞjδðE  H Þjψðt Þi

!

(11.4)

Calculations can be performed for wavepackets propagating in graphene supported on SiO2 or h-BN and for different initial spin polarizations [56]. From the numerical data, a t general form for fitting the time-dependent spin polarization is Sα ðt Þ  cos ð2πt=T Ω Þeτs , which allows us to estimate the spin precession time and the spin lifetime. Figure 11.7 shows a global summary of our results, where in the main frame we plot the spin lifetime τs as a function of the inverse momentum relaxation time, 1/τp. For the SiO2 substrate, we find that τs is inversely proportional to the density of e–h puddles, and, equivalently, to τp. In fact, we find that for E = –200 meV the scaling matches well with the traditional DP relation, τ s ¼ ðT Ω =2π Þ2 =τ p , where T Ω ¼ 2πħ=3V R . For the h-BN substrate, we find the opposite behavior, with τs proportional to τp, reminiscent of the EY mechanism. This transition in behavior can be connected to the relationship between τp and T Ω . For SiO2, one obtains τ p  T Ω ; in this regime, the momentum scattering strongly interrupts the spin dynamics, inducing motional narrowing of the precession and yielding the DP relaxation mechanism (cf. Figs. 11.3(a) and (d)). For the h-BN substrate, one finds that τ p  T Ω , allowing for spin precession between scattering events. In this case, the momentum scattering acts as an effective broadening of states, leading to increased dephasing and relaxation. This behavior can be qualitatively reproduced by tuning the broadening parameter in the case of clean graphene (dashed line in main frame). Interestingly, similar trends have been obtained in experiments that compare spin relaxation on SiO2 and h-BN substrates. As shown in the inset of Fig. 11.7, these

212

Graphene Spintronics

measurements indicate an inverse relationship between τs and the sample mobility for SiO2 substrates, suggesting the DP mechanism of spin relaxation. Meanwhile, spin lifetimes of single-layer graphene on h-BN substrates appear to show a positive correlation between τs and mobility. In conclusion, the numerical study of spin dynamics in ultra-clean graphene devices has revealed that a very weak uniform Rashba spin–orbit coupling (with a value of μeV) – as generally assumed for supported graphene – in conjunction with substrate-dependent electron–hole puddles, yields spin lifetimes in the range of few hundreds of picoseconds to the nanosecond scale. The spin relaxation is jointly governed by the Dyakonov–Perel mechanism together with a pure dephasing mechanism whose relative strength depends on the electron–hole puddle characteristics [56]. Spin dynamics in clean graphene on SiO2 are found to be dominated by the Dyakonov– Perel mechanism, in contrast with the case of puddles for the boron nitride substrate. One should note that the effect of a low density of ad-atoms, such as gold or nickel (down to 1012cm–2 or lower), yield similar conclusions [55]. By introducing a random Rashba field in real-space but vanishingly small intervalley scattering, long mean-free paths of the order of several hundreds of nanometers are obtained, but the timedependent spin polarization also reveals a short spin lifetime that is at a minimum at the Dirac point. The spin relaxation mechanism revealed in non-magnetic samples is interpreted as resulting from a quantum entanglement between spin and pseudospin driven by SOC, unique to graphene. The mixing between spin and pseudospin results in fast spin dephasing even when approaching the ballistic limit, with increasing relaxation times away from the Dirac point, as repeatedly observed experimentally. In the clean graphene limit, SOC can be caused by ad-atoms, ripples or even the substrate. Such a finding actually suggests inspiring ways to control the spin by modifying the pseudospin or vice versa. For example, spins could be manipulated by inducing pseudomagnetic fields by straining graphene [15].

11.5

Proximity Effects and Spin Gating In addition to ballistic transport and micron-long mean-free paths [38], spin relaxation lengths in graphene also reach unrivalled values. Thus, graphene’s intrinsic features are ideal not only for downscaling of conventional devices, but also to demonstrate radically new ideas, for instance harnessing spin manipulation. New concepts to be explored include tailoring spin degrees of freedom in graphene through magnetic proximity effects, torque effects, the spin Hall effect, mechanical strain, chemical functionalization, or nanopatterning, all of which require an interdisciplinary research community. By tackling these issues, one can anticipate the eventual demonstration of external ways to control the propagation of spin currents, achieving operational reliability at room temperature and architectural compatibility with Si technologies.

11.5 Proximity Effects and Spin Gating

11.5.1

213

Magnetic Proximity Effects Electronic and spin properties in graphene can be modulated by harnessing interfacerelated proximity effects with insulating magnetic materials, magnetic molecules, materials that have a large spin–orbit coupling, or ferroelectric materials. For example, spin polarization has been predicted to be induced by an interface with europium oxide [59] or yttrium iron garnet (YIG), or a material with strong spin–orbit interaction [60]. Such a mechanism would not require ferromagnetic metallic contacts to inject spinpolarized electrons, and thus could be a way to circumvent the conductivity mismatch problem. Similarly, it is appealing to study the interaction between spin-polarized currents in graphene and extrinsic sources of spin or magnetization.

11.5.2

Spin Hall Effect The spin Hall effect (SHE) is a relativistic spin–orbit coupling phenomenon that can be used to electrically generate or detect spin currents in non-magnetic systems. Spin current generation by means of the SHE in heavy metals for spin-torque applications has gained considerable attention in recent years, and magnetic memory prototypes have been demonstrated using this technology [65]. The SHE in graphene has been studied by means of the so-called Hall bar geometry (Fig. 11.8(a)) that relies on the SHE for spin injection and the inverse SHE for detection. In a first experiment, a large Hall response was observed near the graphene neutrality point in the presence of an external magnetic field [61]. The results were ascribed to spin currents that resulted from the imbalance of the Hall resistivity for the spin-up and spin-down carriers induced by the Zeeman interaction (thus entitled Zeeman Hall Effect). This process does not involve a spin–orbit interaction, i.e., is not of the SHE origin, and is largest in the cleanest graphene samples. More recently, spin Hall angles larger than 10% have, surprisingly, been reported in graphene after the addition of ad-atoms such as hydrogen or metallic particles, or by contacting graphene with WS2, suggesting that the SHE (in modified graphene) can also be an efficient way of generating spin currents [62]. The controlled addition of small amounts of covalently bonded hydrogen atoms has been reported to induce an enhancement of the spin–orbit interaction, estimated to be up to three orders of magnitudes

Fig. 11.8 (a) Hall bar device. A charge current in the top arm generates a spin current via the SHE that flows to the bottom bar, which results in a measurable voltage via the inverse SHE. (b) Non-local device where the spin current is achieved via spin injection from a ferromagnet (adapted from [65]).

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when compared with clean graphene [63]. Such large enhancement was estimated from non-local signals of up to 100 Ω, which are observed at zero external magnetic fields and at room temperature. From the magnetic field and the length dependence of the nonlocal signal, a spin–orbit strength of 2.5 meV was extracted for samples with 0.05% hydrogenation using an Elliot–Yafet scaling. However, the interpretation of the results as originating from the SHE has been recently criticized and is thus still an open question [64]. Additional experiments on hydrogenated graphene and graphene decorated by Au and Ir ad-atoms indicate that the spin Hall model fails to explain the observed phenomenon and an alternative interpretation might be required [64]. Moreover, recent quantum simulations found the presence of multiple background contributions to the non-local resistance, as measured experimentally, that are unrelated to SHE and question the spin origin of the measurements [79]. Therefore, the mechanisms involved in the generation of the detected signals are yet to be fully understood, and further optimization of the SHE has not yet been achieved; these will be key objectives for the years to come. Device geometries such as the one shown in Fig. 11.8(b), or the one proposed in [79], present an alternative for the exploration and control of the SHE in a modified graphene platform [65].

11.5.3

Quantum Spin Hall Effect In 2005, Kane and Mele predicted the existence of the quantum spin Hall effect (QSHE) in graphene due to intrinsic spin–orbit coupling (SOC) [66]. In the QSHE regime, the action of the spin–orbit interaction can be understood as a momentum-dependent magnetic field coupling to the spin of the electron, which results in the formation of chiral edge channels for spin up and spin down electron population. The observation of the QSHE is inhibited in clean graphene owing to increasingly small intrinsic spin–orbit coupling of order of μeV [67], but has been demonstrated in strong SOC materials (such as CdTe/HgTe/CdTe quantum wells or bismuth selenide and telluride alloys), giving rise to the new exciting field of topological insulators [68, 69]. Recent proposals to induce a topological phase in graphene include functionalization with heavy ad-atoms [70], proximity effect with topological insulators [71], or intercalation and functionalization with 5d transition metals [72]. In particular, the seminal theoretical study by Weeks and co-workers [70] has revealed that graphene endowed with a modest coverage of heavy ad-atoms (such as indium and thallium) could exhibit a substantial band gap and QSHE fingerprints (detectable in transport or spectroscopic measurements). For instance, one signature of such a topological state would be a robust quantized two-terminal conductance (2e2/h), with an ad-atom density-dependent conductance plateau extending inside the bulk gap induced by SOC [70, 73]. To date, such a prediction lacks experimental confirmation, despite some recent results on indium-functionalized graphene that have shown a surprising reduction of the Dirac point resistance with increasing indium density [74].

11.6 References

215

Ad-atoms deposited on graphene inevitably segregate, forming islands rather than a homogeneous distribution, and the impact of ad-atom clustering on the formation of the QSHE has been clarified recently by Cresti and co-workers focusing on thallium ad-atoms [75]. In this work, in addition to the disappearance of the QSHE upon atom segregation, two other physical regimes are identified, namely a SHE regime in which spins with opposite polarization still accumulate at opposite edges, but in the absence of a bulk band gap. The second regime of topological nature is manifested in a bulk minimum conductivity and suppression of localization effects. Additionally, such a bulk topological state is visualized by the real-space formation of locally and partially chiral bulk currents circulating around thallium islands, which are at the origin of the suppression of quantum interferences. Finally, the scale-invariant quantum conductivity is reminiscent of the usual topological insulator surface states but exists in a bulk extended state forming inside the material in the presence of locally strong and inhomogeneous spin–orbit coupling. The finite value of the quantum conductivity also point towards the existence of a quantum critical point, as suggested for instance theoretically by Mirlin and co-workers in the situation of similar symmetry breaking effects [76].

11.6

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12

Graphene–BN Heterostructures Lei Wang, James Hone, and Cory. R. Dean

12.1

Introduction Minimizing extrinsic sources of disorder is crucial to fully characterize and utilize the exceptional electronic properties of graphene. In the first studies of graphene exfoliated onto SiO2 substrates, it was shown that scattering from charged surface states and impurities [1], substrate surface roughness [2, 3], and SiO2 surface optical phonons [4–6] dominate the electron response, limiting device mobilities to less than ~20,000 cm2/V s. Near charge neutrality, substrate-induced disorder creates inhomogeneous patterns of electrons and hole puddles [7, 8] that obscure the low-density transport signatures. Suspending graphene above the substrate provided the first demonstration that carrier mobility in excess of 200,000 cm2 =V s could be achieved [9, 10]. However, this geometry imposes severe limitations on device architecture and functionality. Achieving both device requirements – namely high electron mobility and substrate-supported geometry – was made possible by the introduction of hexagonal boron nitride (h-BN) as an alternative substrate dielectric [11]. The hexagonal phase of BN is an insulating isomorph of graphite with boron and nitrogen atoms occupying the inequivalent A and B sublattices in the Bernal structure. The different on-site energies of the boron and nitrogen atoms result in a large (5.97 eV) band gap [12] and a small (1.7%) lattice mismatch with graphite [13]. Owing to the strong, in-plane, ionic bonding of the planar hexagonal lattice structure, h-BN is relatively inert and is expected to be free of dangling bonds or surface charge traps. Furthermore, the atomically planar surface may play a role in suppressing rippling in graphene, which has been shown to mechanically conform to both corrugated and flat substrates [2, 14]. The dielectric properties of h-BN are similar to those of SiO2, allowing the use of h-BN as an alternative gate dielectric with no loss of functionality [15]. Moreover, the surface optical phonon modes of h-BN have energies two times larger than similar modes in SiO2, suggesting the possibility of an improved hightemperature and high-electric-field performance of h-BN based graphene devices over those using typical oxide/graphene stacks [16, 17]. The first electrical transport measurements of graphene with h-BN dielectrics showed a dramatic decrease in disorder, exhibiting an order of magnitude enhancement in carrier mobility [11]. This technical advancement paved the way for numerous experiments exploring the intrinsic electron properties and device applications in the disorderfree limit, and, indeed, most modern studies of graphene now include h-BN as the 219

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standard substrate choice [18]. Many excellent reviews can be found discussing the details of these experiments and so we attempt to avoid repetition here. Instead, we focus on the mechanical assembly techniques that we have developed in recent years to maximize device performance. Our discussion centers on device characteristics that are equally important for both fundamental scientific pursuits as well as potential electronic device applications. These include carrier mobility, electrical contact resistance, environmental insensitivy, and flexibility in terms of the range of device geometries that may be realized. We conclude by briefly reviewing recent applications of these same techniques to the emerging study of other 2D van der Waals materials, beyond graphene.

12.2

Mechanical Assembly of Graphene–BN Heterostructures

12.2.1

van der Waals Assembly The first h-BN-supported graphene devices were constructed using a layer-by-layer assembly technique, in which a polymer layer was used to peel the graphene off of a Si/SiO2 substrate and place it onto an h-BN flake [11]. This technique was extended to enable more complex structures such as fully encapsulated graphene with h-BN as both the top and bottom dielectric [19]. However, accumulation of polymer residues at the interfaces has proved to be a significant drawback. A particularly striking consequence of this residue is the appearance of bubbles between layers (Fig. 12.1). These bubbles limit the size of h-BN/graphene/h-BN devices to e1 μ m. Moreover, the problem is made worse as more layers are assembled, making it impractical to realize a multi-layered device with sufficient area. The more recently developed van der Waals (vdW) transfer method eliminates interlayer polymer contamination using van der Waals adhesion to directly assemble heterostructures. This results in further enhancement of device performance while

Fig. 12.1 (a) AFM image of graphene transferred to h-BN using polymer tranfer technique. (b) AFM image of same sample after transfer of second h-BN layer on top. The dense bubbles in the area of the h-BN/graphene/h-BN stack are due to polymer residue on the top surface of the graphene.

12.2 Graphene-BN Heterostructures

221

Fig. 12.2 (a) Cartoon schematic of the van der Waals assembly technique. (b) Optical image showing the sequential assembly of a multi-layer structure, resulting in two monolayer graphenes separated by a few nm thick BN spacer, with the entire structure fully encapsulated in BN [20].

simultaneously allowing for increased complexity in the layer stacking sequence. Here we describe this process in detail. First, a bare Si chip is coated with e1 μ m of poly-propylene carbonate (PPC) (SigmaAldrich, CAS 25511–85-7). The PPC is manually peeled from the Si substrate and placed onto a transparent elastomer stamp (poly-dimethyl siloxane, PDMS) (Fig. 12.3). In parallel, flakes of graphene and h-BN are exfoliated onto Si/SiO2 (285 nm) wafers and examined by optical microscopy and atomic-force microscopy (AFM). For the highest quality devices, particle contamination and atomic step edges in the h-BN are avoided. To make an h-BN/graphene/BN stack, the slide with the PDMS stamp is inverted and attached to a micromanipulator. Typically, the micromanipulator is used on a wafer probe station that provides independent motion and temperature control of the wafer chuck, and an optical microscope for alignment (Fig. 12.3). Using this setup, the PDMS stamp is manipulated over a BN flake that will ultimately form the top of the layer stack. It is then slowly pressed onto the h-BN, and the PPC is heated slightly so that it flows around the h-BN. The sample is then cooled and the stamp is retracted, such that the PPC then peels the h-BN from the SiO2 surface. Next, the manipulator is used to position the h-BN flake over a chosen graphene flake, to bring the two flakes into contact, and then to lift the stack. The graphene adheres more strongly to the h-BN than the SiO2 and is lifted from the substrate. We have found that setting the stage temperature to 30
C produces the best results (nearly 100% yield). This process can be repeated by alternating the h-BN and graphene layers as desired. Once complete, the full layer stack is placed onto the final target substrate. This substrate is then heated to 90
C slowly to soften the PPC, which allows the glass slide and PDMS to be removed, leaving the PPC on the substrate. This PPC is removed with chloroform, followed by annealing in a vacuum furnace to 350
C for 15 min. A critical feature of this technique compared with previous methods is that the active interfaces do not contact any polymer throughout the process, reducing impurities trapped between the layers. Figure 12.4(a) shows an AFM image of a h-BN/graphene/h-BN heterostructure made by vdW assembly. The graphene appears clean and free of macroscopic contamination over the entire device area,  200 μm2 . In Fig. 12.4(b), a high-

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Fig. 12.3 (a)–(e) Technique for making PPC-coated PDMS on a glass slide. (q) A piece of PDMS placed on a glass slide. (b) A piece of Scotch tape with a hole in the center. (c) The tape is placed on a silicon chip coated with PPC film. (d) The scotch tape peels off the PPC film. (e) The PPC film is fixed on the PDMS surface by the tape. (f) A wafer probe station is used for alignment during stacking. The sample is affixed to the chuck, which allows x–y motion and controlled heating/cooling. The glass slide with the material to be transferred is held in a precision micromanipulator allowing independent xyz motion.

Fig. 12.4 (a) AFM image of a large-area, multi-layer heterostructure showing that it is pristine and completely free of wrinkles or bubbles except at its boundary [21]. (b) Cartoon schematic crosssection (left) of the device in (a). Right shows a high-resolution cross-section TEM image of this device. The BN–G–BN interface is found to be pristine and free of any impurities down to the atomic scale [20].

resolution cross-section STEM image shows that the resulting interface is pristine, with the graphene layer nearly indistinguishable from the adjacent BN lattice planes. Remarkably, this suggests that despite being assembled in the open laboratory environment, the interface between the graphene and BN is free of impurities at the atomic scale.

12.2.2

One-Dimensional Edge Contact Assembly of high-quality h-BN/graphene/h-BN stacks poses a new challenge to making electrical contacts since the encapsulated graphene layer is no longer directly accessible.

12.2 Graphene-BN Heterostructures

223

To address this, we developed a new device topology in which 3D metal electrodes are connected to a 2D graphene layer along the one-dimensional (1D) graphene edge [22–26]. This is achieved by etching the heterostructure to expose only the edge of the graphene layer, which is in turn metalized. The result is a similar contact geometry as in conventional semiconductor FETs where doped 3D bulk regions make lateral contact to a 2D electron gas (Fig. 12.5). Despite the fact that in in this geometry carrier injection is limited only to the 1D atomic edge of the graphene sheet, the contact resistance is remarkably low, reaching 100 Ω  μm. The edge-contact geometry additionally enables a complete separation of the layer assembly and contact metalization processes. Figure 12.5 illustrates the edge–contact fabrication process in detail. Beginning with a BN-G-BN heterostructure, a hard mask is defined on the top BN surface by electron– beam (e-beam) lithography of hydrogen-silsesquioxane (HSQ) resist. The regions of the heterostructure outside of the mask are then plasma etched to expose the graphene edge. Finally, metal leads (1 nm Cr/15 nm Pd/60 nm Au) are deposited by e-beam evaporation making electrical contact along this edge. Contact can also be made by both thermal evaporation and sputter deposition, making a variety of other metal combinations possible. In Fig. 12.5(b), a cross-section scanning transmission electron microscope (STEM) image of a representative device shows the resulting geometry of the edge contact. In the magnified region, electron-energy-loss-spectroscopy (EELS) mapping confirms that the graphene and metal overlap at a well-defined interface. From the angle of the

Fig. 12.5 (a) Schematic of the edge-contact fabrication process (see text). (b) High-resolution bright field STEM image showing details of the edge-contact geometry. The expanded region shows a magnified false-color EELS map of the interface between the graphene edge and metal lead [20]. (c) Two terminal resistance versus density for several channels in a TLM geometry [20]. (d) Resistance versus channel length at fixed density, shown for two densities. The solid line is a linear fit to the data [20]. (e) Contact resistance calculated from the linear fit at multiple carrier densities for two separate devices. Error bars represent uncertainty in the fitting [20]. (f) Contact resistance versus temperature showing no variation down to cryogenic temperatures [20].

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etch (~30
), we can expect that the graphene terrace is exposed only 1–2 atoms deep, as confirmed within the resolution of the STEM image. There is no evidence of metal diffusion into the graphene/BN interface, confirming the truly edge nature of the contact. The EELS map additionally indicates that contact is made predominantly to the Cr adhesion layer. To characterize the quality of the edge contact, we used the transfer-length method (TLM), as shown in Fig. 12.5(c). Multiple two-terminal graphene devices consisting of a uniform 2 μ m channel width but with varying channel lengths were fabricated, and their resistances were measured as a function of carrier density n induced by a voltage applied to a silicon back-gate. Figure 12.5(d) shows the resistance versus channel length, measured at two different carrier densities. In the diffusive regime, where the channel length remains several times longer than the mean-free path, the two-terminal resistance can be written as R ¼ 2RC ðW Þ þ ρL=W, where RC is the contact resistance, L is the device length, W is the device width, and ρ is the 2D channel resistivity; RC and ρ are extracted as the intercept and slope of a linear fit to the data. The contact resistance versus carrier density measured for two separate devices is shown in Fig. 12.5(e). RC is remarkably low, reaching e150 Ω  μm for n-type carriers at high density. As a comparison, this value is approximately 25% lower than the best reported surface contacts without additional engineering such as chemical [27] or electrostatic [28] doping. We further note that, since this value is obtained in a twoterminal geometry, it includes the intrinsic limit set by the quantum resistance of the channel, which can be subtracted to yield an extrinsic contact resistance close to 100 Ω  μm. The edge-contact resistance is asymmetric, being lower by a factor of 2–3 when the device is gated to be n-type versus p-type. This asymmetry is consistent with electrical contact being made primarily to the Cr adhesion layer, as suggested by the cross-section EELS map (Fig. 12.5), since the Cr work function is approximately 0.16 eV lower than that of graphene [29]. The contact resistance scales inversely with the contact width (inset in Fig. 12.5(e)), as expected for the edge–contact geometry. Finally, we find that the contact resistance is largely independent of temperature (Fig. 12.5(f)), in contrast with the linear temperature-scaling that has been reported for surface contacts [30]. We note that the contact resistance diverges near the charge neutrality point. This may be expected owing to a decrease in the density of states in the 2D graphene layer. We also observe an additional local peak in the contact resistance at finite negative density. We have observed that exposure of the graphene edge to a weak O2 plasma immediately prior to metalization can cause this satellite peak position to vary, indicating that the origin of the second peak may relate to the specific chemistry of the edge termination. A full understanding of the nature of the edge contact remains incomplete. The low junction resistance results at least in part from the favorable momentum matching across the geometry [22–26], compared with top contacts where the electron must be injected perpendicular to the channel. Ab initio simulation assuming a Cr metal electrode indicates that edge contacts lead to shorter bonding distance with larger orbital overlap than surface contacts, consistent with previous calculations for other common metals in

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225

Fig. 12.6 Metal–graphene edge-contact resistance as a function of carrier density of three devices: (a) metal layers immediately deposited after graphene edge has been exposed; (b) graphene edge exposed to 15 s O2 plasma before metal deposition; and (c) graphene edge exposed to 25 s plasma before metal deposition [20].

the same contact geometry [22]. Incorporation of some additional interfacial species, such as oxygen, which might result from the etch process, may additionally help to improve bonding and increase the transmission [20].

12.3

High-Performance Graphene

12.3.1

Electron Mobility Combining the vdW assembly and edge-contact techniques, we can fabricate a larger area ð> 100 μm2 Þ, with fully encapsulated devices with good contacts. These devices allow for assessment of device performance from the point of view of electron transport through the channel. Figure 12.7(a) shows resistance versus density from a 15 μm  15 μm, BN–G–BN, device. The transport characteristics indicate the graphene device to be remarkably pristine, with a narrow resistance peak (indicating low disorder) centered near zero back-gate bias (indicating low intrinsic doping). At carrier density jnj ¼ 4:5  1012 cm2 , the room temperature sheet resistivity is less than e40 Ω=W, corresponding to an equivalent 3D resistivity below 1:5 μΩ  cm, smaller than the resistivity of any metal at room temperature. The room temperature electronic mobility μ, calculated using the Drude model as σ ¼ neμ, is shown in Fig. 12.7(b). A remarkable feature of this device is the simultaneous realization of high mobility at both low and high carrier density: μ is 40,000 cm2 =V s at n e 4:5  1012 cm2 and increases beyond 140,000 cm2/V s at low density. We note that in the high-density regime the measured mobility is comparable to the acoustic phonon-limited mobility theoretically predicted for graphene [6, 31], indicating that we have reached the intrinsic limit. The room temperature response of the graphene device reported here outperforms all other 2D materials, including the highest mobility 2D heterostructures fabricated from III–V semiconductors [32, 33]

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Fig. 12.7 (a) Four-terminal resistivity measured from a 15 μm  15 μm device fabricated by the van der Waals assembly technique with edge contacts. Inset left shows an optical image of the device. Low temperature response is shown inset right. A negative resistance is observed, indicating ballistic transport [20]. (b) Room temperature mobility versus density (solid black curve). Dashed black curve indicates the theoretical mobility limit due to acoustic-phonon scattering [6, 31]. Remaining data points label the range of mobilities reported in literature for high-performance 2D semiconductor FETs [32, 33].

(Fig. 3.1(b)) by at least a factor of two over the entire range of technologically relevant carrier densities. At low temperatures, four-terminal measurement yields a negative resistance (inset in Fig. 12.7(a)), indicating quasi-ballistic transport [34]. In the diffusive regime, the meanfree path, Lmfp , can be calculated from the conductivity, σ, according to pffiffiffiffiffi Lmfp ¼ σh=2e2 kF where k F ¼ πn is the Fermi wavevector. In Fig. 12.8(a), Lmfp versus applied gate voltage is shown for selected temperatures from 300 K down to 20 K. The mean-free path increases with gate voltage until it saturates to a temperature-dependent value at high density. This maximum Lmfp increases monotonically with decreasing temperature until the mean-free path approaches the device size at T ~ 40 K (Fig. 12.8 (b)). In the low-temperature ballistic regime, four-terminal measurement is dominated by mesoscopic effects [35] and the calculated mean-free path exhibits large variation, depending on the measurement geometry. The temperature dependence therefore provides only a lower bound of the mean-free path. The negative resistance observed at base temperature indicates that electrons travel ballistically across the diagonal of the square, corresponding to a mean-free path as large as 21 μ m in this device. This corresponds to an electron mobility of approximately 1,000,000 cm2/V s at a carrier density of e3  1012 cm2 . Figure 12.8(c) shows the mean-free path versus carrier density for devices of varying sizes. In each case, the mean-free path initially increases with increasing carrier density, before saturating to a value similar to the device dimension (labeled in the figure). In Fig. 12.8(d), the maximum mean-free path is plotted against device size for devices ranging from 1 to 15 μm. Over this range, the mean-free path scales linearly

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227

Fig. 12.8 (a),(b) Calculated mean-free path (see text) versus density and temperature for the device shown in Fig. 12.7. Shaded region in (b) indicates the temperature below which the mean-free path exceeds the device size. Circle and squares correspond to the “a” configuration and triangles correspond to the “b” configuration of the van der Pauw measurement. (c), (d) Mean-free path at T = 1.7 K for devices with size varying from 1 μm to 15 μm. [20].

with device size. This result indicates that, in the low-temperature limit, mobility is due to edge scattering, suggesting we have not reached the intrinsic impurity-limited scattering length. Even higher mobility is therefore expected for larger area devices, which may be enabled by further progress in scalable growth techniques [36, 37].

12.3.2

Ballistic Transport The negative resistance measured in the van der Pauw geometry (inset Fig. 12.7(a)) is one characteristic feature indicative of ballistic transport [19]. A more direct probe is provided by magnetic focusing. Under application of a transverse magnetic field, electrons in the graphene undergo cyclotron motion with radius determined by the Lorentz force. A resonant conduction path (measured as a voltage peak) is realized when an integer number of half circles fit between electrodes, at a field pffiffiffiffiffi B ¼ j  2ħ πn=eL, where the mode number j is the number of half circles between the electrodes, e is the electron charge, B is the magnetic field, L is the distance between the electron emitter and voltage detector, and n is the charge carrier density [38]. Figure 12.9 shows an example of this measurement for the same device as in Fig. 12.7. The resonant peaks have perfect fit. Since this measurement requires that the electrons travel ballistically from source to probe electrodes, the path length of the corresponding half circle gives a further lower bound on the mean-free path. For the device geometry depicted here, this corresponds to approximately 23 μm.

12.3.3

Environmental Insensitivity Figure 12.10 shows four examples of the exceptional protection against environmental factors afforded by h-BN encapsulation. Figure 12.10(a) shows Raman spectra of exfoliated graphene on SiO2, transferred to h-BN, and encapsulated between h-BN flakes. Each sample is examined as fabricated: in air after annealing in Ar/H2 at 400
C for two hours, and in a vacuum. The encapsulated sample shows no change in the

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Graphene–BN Heterostructures

Fig. 12.9 (a) Transverse magnetic focusing measurement configuration. Solid lines indicate the

trajectory for the first and second resonant path paths. (b) Resistance versus gate bias. Resonant paths are apparent as resistance peaks. Dashed lines correspond to the expected resonant modes calculated from the sample geometry.

Raman spectrum from either the annealing process in a vacuum environment, whereas the SiO2-supported sample shows a large shift in the G peak position and decrease in the 2D peak intensity upon annealing, and the non-encapsulated graphene on h-BN shows a decrease in 2D peak intensity upon annealing. The position (1582 cm–1) and width (12 cm–1) of the G peak for the encapsulated sample indicates doping below a level of 1012 cm–2. The encapsulated sample also shows the highest intensity ratio I 2D =I G , again consistent with an absence of environmental doping or scattering. Figures 12.10(b) and (c) show electrical transfer curves of encapsulated devices after the same annealing process, and after exposure to oxygen plasma (50 W, 20 s). There is no change in behavior after annealing and only a small (2 V) shift in the Dirac peak position after plasma treatment. Finally, Fig. 12.10(d) shows a long-term study of the stability of an encapsulated device at high bias. In this test, a source–drain voltage of 25 V is applied to a 5 μm long device, inducing a current density of 3:5  108 cm2 . At this current density, the graphene reaches a temperature of 2000
C, and emits bright thermal radiation in the visible range, as shown. Under a mild vacuum of 105 Torr, the current is stable within 0.4% over 106 s, such that the estimated lifetime of this light source is ~5 years. A similar device without h-BN encapsulation shows a lifetime of only a few minutes under similar conditions.

12.3.4

Moiré Superlattice At the atomic scale, coupling between graphene and h-BN results in a periodic moiré pattern due to the isomorphic, but slightly mis-matched, crystal lattices. The moiré wavelength is directly related to the angular rotation between the two lattices [40, 41, 43–46], spanning from approximately 14 nm down to zero between 0
(maximally aligned) and 30
(maximally misaligned), respectively. STM images of the resulting superlattice at different relative twist angles are shown in Fig. 12.11(b) [40]. The moiré pattern acts as a perfectly periodic superlattice potential, superimposed onto the

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229

Fig. 12.10 (a) Raman spectra of graphene on SiO2, on h-BN, and fully h-BN encapsulated [39]. (b) Room temperature resistivity versus gate voltage for h-BN encapsulated graphene before and after baking [28]. (c) Low-temperature resistivity versus gate voltage for h-BN encapsulated graphene before and after exposure to oxygen plasma [28]. (d) Output current stability over a long time period at high bias, sufficient to cause visible light emission from the h-BN encapsulated graphene (inset).

graphene band structure. Moreover, the favorable dielectric characteristics of h-BN maintain high mobility in the graphene channel. This provides unprecedented access to a fully tunable superlattice, without the need for lithographic patterning [27, 47]. Figure 12.11(c) shows transport data measured from a device corresponding to near zero-angle alignment. In addition to the usual resistance peak at the charge neutrality point (CNP), occurring at gate voltage V g e 2 V, two additional satellite resistance peaks appear, symmetrically located at V satl e 30 V away from the CNP. These satellite features are consistent with a depression in the density of states (DOS) at the superlattice Brillouin zone band edge, analogous to spectroscopic measurements by STM [40, 48]. Assuming non-overlapping bands, jV satl j, gives an estimate of the moiré wavelength to be ~14.6 nm, in good agreement with the expected length. The nature of these satellite peaks can be further probed in the semiclassical, low B-field transport regime. In Fig. 12.11(d), longitudinal resistance, Rxx , and transverse Hall resistance, Rxy , are plotted versus gate voltage at B ¼ 1 T. Near the central CNP, the Hall resistance changes sign as the Fermi energy passes from the electron to the hole band. The same trend also appears near V satl , consistent with the Fermi energy passing through a second band edge. This provides further confirmation that the moiré pattern, acting as a periodic potential superlattice, gives rise to a mini-Brillouin zone band [49]. The satellite peaks

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Graphene–BN Heterostructures

Fig. 12.11 Cartoon schematic of graphene on h-BN showing the emergence of a moiré pattern. The moiré wavelength varies with the mismatch angle, θ. (b) STM images of the moiré pattern versus angle (reproduced with permission from [40]. (c) Longitudinal resistance (left axis) and Hall resistance (right axis) versus gate voltage at B ¼ 1 T. The Hall resistance inverts sign and passes through zero at the same gate voltage as the satellite resistance peaks [41]. (d) Hall resistance versus field and density over a large range, revealing the canonical features of the Hofstadter Butterfly spectrum [42].

broaden with increasing temperature, but persist to room temperature (Fig. 3.5(a)) indicating that the coupling between the BLG and h-BN atomic lattices is in excess of ~30 meV. The moiré length scale at zero angle proves to be ideal to study the interplay between the superlattice-induced bloch bands, and magnetic field-induced Landau bands, enabling for the first time confirmation of the predicted fractal band structure known as the Hofstadter Butterfly [41, 45, 46, 50].

12.3.5

Monolayer Band Gap The first studies of moiré-patterned graphene used the layer-by-layer assembly technique. We found that by using our vDW assembly technique, in which the graphene/BN interface remains pristine [20], alignment between the graphene and BN can be achieved by simple application of heat. Figure 12.12(a) inset shows an example of a heterostructure that was assembled with random (and unknown) orientation of each material. After heating the sample to ~350
C, the graphene flake translates and rotates through several microns, despite being fully encapsulated between two BN sheets. This

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231

Fig. 12.12 (a) Zero field resistance versus gate bias measured in a BN encapsulated graphene device with a ~10 nm moiré superlattice. Inset is an optical image showing macroscopic motion of the graphene after heating [50]. (b) Measurements of the gap at the charge neutrality point versus the twist angle (top axis) and corresponding superlattice wavelength (bottom axis). Data points reflect all measurements reported in literature [50].

behavior has been observed in several devices, in each case resulting in a moiré wavelength ranging between 1015 nm (indicating less than 2
angle mismatch [40]). We postulate that the thermally induced motion proceeds until the macro-scale graphene flake finds a local energy minimum, corresponding to crystallographic alignment with one of the BN surfaces [51]. Figure 12.12(a) shows transport of a device fabricated with thermally induced alignment. The three resistance peaks, characteristic of the electronic coupling to a moiré superlattice [40, 41, 45] are evident. The CNP peak resistance exhibits thermally activated behavior, and exceeds 100 kΩ at low temperatures, indicating a moirécoupling-induced band gap [46, 51, 52]. The gap varies continuously with rotation angle, consistent with previous studies of non-encapsulated graphene [46]. At zero angle the energy gap measured by transport is equivalent to the gap energy measured by previous optical studies [53], providing further indication of the high-quality device realized by vdW assembly. This represents the first demonstration of a band-gap opening in monolayer graphene, without significantly degrading the carrier mobility in the on state (the mobility of the moiré-patterned graphene shows the same ballistic transport as the previously discussed devices). Several groups have reported observation of this band gap in moiré-patterned graphene [42, 45, 46, 51, 53, 54]; however, there are some inconsistencies between the studies. Reference [46] finds that the gap decreases continuously as the twist angle increases from 0o to 5o, while reference [51] states that in their studies the gap has never been observed for devices with twist angle larger than 1o. In multiple experiments, it is reported that the gap is only observed for non-encapsualted graphene [45, 51, 54], with the gap suppressed when a BN layer is placed on top of the graphene. In contrast, we routinely observe the existence of a band gap in encapsulated devices fabricated by the

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Graphene–BN Heterostructures

vdW technique [42]. Figure 12.12 summarizes these findings. The difference in the sample preparation methods, and variability in device quality may explain why in some cases a band gap is not observed. Owing to the technological importance of developing a band gap in monolayer graphene, resolving these discrepancies through further experimental and theoretical study is highly desirable.

12.4

Beyond Graphene The mechanical assembly of 2D materials provides several exciting new opportunities in device design that go beyond the individual properties of the constituent layers. For example, vertically layered heterostructures can be made by assembling multiple monolayer graphene crystals seperated by few-layer BN (Fig. 12.13(a)). The BN spacer thickness precisely determines the coulomb interaction and wave function overlap between the graphene layers, providing an unprecedented resolution with which to make vertical superlattices. Tunnel FETs and optoelectronic devices with novel properties have been demonstrated by additionally controlling the rotational lattice orientation between the graphene layers [18]. Graphene represents just one of a larger subset of layered van der Waals materials, including the transition metal dichlcogenides (TMDCs) transition metal oxides (TMOs), and topological insulators (TIs), each of which can also been exfoliated down to a monolayer [18]. For the semiconductor TMDCs, we demonstrated that h-BN as an encapsulating dielctric results in significant enhancment in device mobility [55], much like we observe for graphene devices (Fig. 12.13). In MoS2, mobility of 1,000 to above

Fig. 12.13 (a) Double-layer graphene consisting of two graphene sheets separated by few-layer

BN. Data plot shows measurement of Coulomb drag between the layers. (b). MoS2 device with graphene used to bridge between the metal contact and the MoS2 layer; and cross-section TEM image. Data plot shows mobility versus temperature for different thicknesses of the MoS2 channels [11]. (c) Bilayer NbSe2 device realized by assembly and encapsulation in a glove box enironment to limit oxidation. Superconductivity is recovered (data plot) when devices are prepared in this way [46].

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10, 000 cm2 =V s is achieved utilzing graphene as an improved electrical contact, integrated during the vdW assembly process. This allows a carrier density range of 1012 cm2 to be studied, where strong gate-tunable quantum oscillations are observed [55]. A number of studies by other groups in related materials such phosphorene and WSe2 have found similar results with these techniques [57, 58]. Finally, an engineering challenge facing a number of these materials is their sensitivity to exposure to air. In the case of NbSe2, for example, bulk superconductivity is supressed in the monolayer limit, owing to rapid oxidation. The vdW assembly technqiue, however, allows for isolation of monolayer crystals, followed by encapsulation in h-BN, all within a glove box filled with nitrogen, such that the monolayer flake is never exposed to oxygen. By integrating few-layer graphite as electrodes, the NbSe2 is electrically connected and protected from oxidation in a single step, and superconductivity is recovered [56] (Fig. 12.13). The capability to combine 2D crystals into vertically layered structures, represents a new paradigm in materials engineering, where, instead of atomistic layer-by-layer growth, we can fabricate heterostructures by mechanical assembly of naturally occurring 2D layers. This process is therefore free of many of the constraints of conventional MBE techniques. Owing to the wide range of properties associated with the many available 2D materials, the future of this technique is expected to find many applications in the areas of novel band structure engineering, tunneling devices, tunable optoelectronic response, and hybrid materials for flexible electronics.

12.5

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13

Controlled Growth of Graphene Crystals by Chemical Vapor Deposition: From Solid Metals to Liquid Metals Dechao Geng and Kian Ping Loh

13.1

Introduction Since its debut in 2004 [1], graphene has attracted intense interest because of its extraordinary properties, such as carrier mobility close to 200,000 cm2/Vs [2, 3], long ballistic transport distances at room temperature [4], half-integer quantum Hall effect [5], and broadband light absorption [6]. These unique properties initially raised hopes for creating breakthroughs in electronics, although it was recognized early on that the gapless nature of graphene means it is not going to replace silicon electronics anytime soon. However, graphene still holds great promise for applications such as RF transistors, sensors, nanocomposites, and super-capacitors. To realize the potential applications of graphene in these technologies, the scalable synthesis of graphene is of considerable importance. In the past decade, many graphene fabrication methods have been developed (Fig. 13.1) [7]. Although the mechanical exfoliation of highly oriented pyrolytic graphite has initiated the discovery of graphene and remains the best method to date for studying the physical properties of graphene, it is not a viable method to produce uniform graphene in sizeable amounts due to its poor reproducibility [8–11]. The epitaxial growth of graphene on SiC substrates holds promise for the development of certain graphene-based applications in electronics and optics. Its transfer-free process avoids the effect of residues on the devices. Over the past few years, Walter A. de Heer’s group has intensively explored the growth and applications of graphene and other two-dimensional (2D) materials using the epitaxial growth method [12–15]. Recently, they reported exceptional ballistic transport in epitaxial graphene nanoribbons [16]. It should be noted that the epitaxial method has drawn considerable interest also in the growth of other 2D materials such as silicene [17, 18], germanene [19, 20], and stanene [21]. However, the epitaxial method is restricted to specific substrates with lattice-matched properties, and at present it cannot be readily extended to the industrially important silicon platform. Inspired by mechanical exfoliation, a liquid-phase exfoliation approach has also been developed to generate solution-processible graphene [22–25]. Multilayer graphene flakes can be obtained on a large scale, and the most effective exfoliation methods generally used strong acid and base as the oxidation and reduction agents. Given the

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239

Fig. 13.1 Common graphene production methods that allow a wide range of choices in terms of size, quality and price for any particular application [7]. Reprinted with permission.

destructive process inherent in sonication-assisted exfoliation, the as-synthesized samples are not very uniform, as the size and layer-number are difficult to control [26]. Analysis of commercial liquid-phase exfoliated graphene revealed that the purveyed products contained a significant amount of graphite. It is difficult to achieve precise control over graphene’s size, layer number, and morphology using mechanical or solution-based exfoliation methods. To make highquality, thickness-controllable graphene, which is industrially compatible with regards to either wafer-scale process or roll-to-roll process on metal foil, chemical vapor deposition (CVD) has emerged as the most effective way to advance the industrial development of graphene.

13.2

CVD Method for Graphene Growth

13.2.1

CVD Process It should be noted that the CVD method is a very well-established method in the growth of carbon thin film and layered materials [27–29]. A typical CVD system usually has several components, such as a heating furnace, a gas introduction system, and a reaction chamber [30]. The chemistry of graphene growth is straightforward. The carbon source, usually some carbon-containing hydrocarbon, decompose into carbon clusters on the metal catalyst substrates, followed by diffusion–aggregation of these clusters to form graphene. The complexity lies in how to control the layer uniformity, and how to attain single-crystal domain over macroscopically large areas.

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During the common CVD process for graphene growth, substrate temperature, catalysts, carbon source, and gas ratios are all of considerable importance, and differing growth conditions can lead to considerable differences among the as-grown samples. Among various crucial factors in the CVD process, the metal catalysts have the greatest influence on the as-grown graphene sample, as different catalytic systems can result in completely different growth modes for graphene and other 2D materials. Recent advances in graphene growth using the CVD method have shown that almost all of the transition metals listed in the periodic table can be used as catalysts for graphene growth [31–37]. Depending on the solubility of carbon in different metals and the strength of metal–carbon bonding, different metal catalysts produced graphene films of different thicknesses and crystalline quality. Rational approaches have been applied for the engineering of metal catalyst layers. Among the various metals, Ni and Cu command the most interest for the preparation of uniform, high-quality graphene. Ni-catalyzed, few- or multilayer graphene can be used in stretchable transparent electrodes where the thickness of graphene is needed to ensure sufficient conductivity [38, 39]. In turn, monolayer graphene film grown on Cu is better suited for use in high-mobility field-effect transistors [40]. Metal Ni foil was first introduced for CVD graphene growth in 2008 [38], earlier than the introduction of metal Cu. However, the non-uniform, few-layer characteristics of the as-grown graphene result in comparatively poor electrical performance compared to exfoliated single-layer graphene. Cu was later introduced for the growth of monolayer graphene [40]. Given its ultra-low carbon solubility, a surface catalyzed, self-limiting process can be used to form uniform, monolayer graphene over the whole Cu surface (Fig. 13.2). It was soon realized that by alloying fractions of Ni (~10%) in the Cu

Fig. 13.2 Large-area, high-quality monolayer graphene on solid Cu foil [40]. Reprinted with permission.

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241

substrate, the controlled growth of graphene with different layer numbers can be achieved because of the higher solubility of carbon in Ni [41]. The ratio of Cu and Ni in the alloy catalyst determines the diffusion and precipitation of the carbon in the bulk and on the surface, thus allowing some control over the growth of graphene of different thicknesses. There are ongoing efforts to make bilayer graphene using this approach, by increasing the fraction of nickel to ~20%.

13.2.2

Solid Metals for Use in Graphene Crystal Growth Controlling the Size of Graphene Crystals CVD graphene film is generally polycrystalline and displays electronic properties inferior to the exfoliated counterpart due to the presence of grain boundaries (GBs), which are major sources of intervalley scattering [42]. Thus, the fabrication of single-crystal graphene without structural defects becomes the obvious goal. There are two ways to produce single-crystal graphene: growth from only one nuclei by lateral expansion on the substrate, or growth of a single-crystal domain from the seamless merging of numerous nuclei with aligned orientation. Both approaches have been applied with various degrees of success in the synthesis of single-crystal graphene. In the first approach, the first step is to reduce the number of nucleation sites (Fig.13.3(a)). This can be achieved by a number of methods, such as using annealed copper foil at high temperatures [43, 44], preoxidized copper [45, 46], or polished copper foil [47]. It can be appreciated that growth from a single nuclei may require a long growth time, although the recent breakthrough by Xie suggests that the kinetic limit can be overcome [48]. Alternatively, the stitching of numerous epitaxially aligned graphene grains has the potential to yield uniform, single-crystal graphene film, in a much shorter time compared to growth from a single nuclei (Fig. 13.3(b)). In the single nucleus approach, the key point is the suppression of nucleation density. Ideally, the growth proceeds from only a single seed, thus avoiding grain boundaries. Surface design of the catalytic substrate is shown to be an effective approach for obtaining single-domain growth. The nucleation density of graphene depends on the carbon supply for graphene growth and the active sites for graphene nucleation. Diluting the carbon source proves effective in reducing the number of active sites but may take too long to produce large-sized crystals if the same condition is kept throughout. On the other hand, passivating active sites which have low-energy barriers for nucleation, including atomic steps, grain boundaries, and point defects on the Cu foil, has emerged as a good strategy to control nucleation density. The use of melamine to passivate active sites has been demonstrated as a facile surface engineering method to grow large single-crystalline monolayers which exhibit ultra-high carrier mobilities exceeding 25,000 cm2 V–1 s–1 [50]. Oxidizing the Cu surface to form copper oxide also substantially decreases the graphene nucleation density by passivating the surface active sites [51]; this has now been actively pursued as a method for the growth of centimeter-scale, single-crystal graphene domains (Figs. 13.4(a) and (b)). Alternatively, restricting the delivery of growth flux spatially to a single point was demonstrated to allow a single nucleus to evolve into a monolayer at a fast rate [48]. By locally feeding carbon precursors to a desired position on a substrate composed of

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Controlled Growth of Graphene Crystals

Fig. 13.3 Simplified schemes for growing large-area, single-crystal graphene. (a) Growth mode

from a single nucleus. (b) Growth mode from aligned multiple nuclei [49].

Fig. 13.4 Millimeter-scale single-crystal graphene. Parts a and b, SEM and optical images of centimeter-scale graphene domains on OR-Cu exposed to O2 [51]. Parts c and d, synthesis of ~1.5-inch graphene grains on a Cu85Ni15 alloy substrate from a single nucleus [48]. Reprinted with permission.

optimized Cu–Ni alloy catalyst, a ~1.5-inch-large graphene monolayer was synthesized in 2.5 hours (Figs. 13.4(c) and (d)). Such localized feeding restricts the nucleation zone spatially, and the optimized alloy activates an isothermal segregation mechanism that considerably increases the growth rate.

13.2 CVD Method for Graphene Growth

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Coverage (norm)

1.0 0.8 0.6 0.4 0.2

50 mm

Nucleation regime

1.5 mm 100 mm

Cooperative island growth

0.0 Time

Fig. 13.5 (Dark gray) High nucleation regime leads to high coverage of smaller graphene nuclei at a shorter time while (Light gray) suppressing nucleation density allows cooperative island growth, leading to a large-sized single crystal [52]. Reprinted with permission.

Eres and co-workers [52] proposed a new framework for optimizing the growth of large-area single-crystal graphene. Using a kinetic model, they show that at very low nucleation densities, carbon incorporation occurs by a cooperative island growth mechanism. In order to suppress the nucleation density of graphene, the reactant is cooled transiently by an argon pulse to produce collisional deactivation of the active carbon growth species. This direct intervention of the growth process was found to suppress the nucleation completely. After the argon pulse, the carbon supersaturation was increased (by increasing CH4 flow rate) so that the lateral growth rate, due to the addition of carbon monomers to the edges of existing graphene seeds, could be boosted. The change in growth mechanism from random nucleation to cooperative island growth leads to three orders of magnitude increase in the reactive sticking probability of methane, thus the growth rate of large size single crystal graphene increases rapidly after an induction period (Fig. 13.5). The multiple nuclei approach to the growth of large-scale, single-crystal graphene film is based on the premise of having all the graphene nuclei oriented in the same direction to facilitate seamless merging of the growing grains when they meet. Nonetheless, the question of whether the presence of aligned grains will ensure the absence of all types of grain boundaries is not certain, since translational dislocation can occur even between two parallel hexagons when they impinge, or twin boundaries can occur due to defects on the substrate. It is not easy to observe such grain boundaries with conventional laboratory microscopy techniques, an aberration corrected transmission electron microscopy technique, or a scanning tunneling microscopy technique, is usually needed to identify grain boundaries of very narrow width. To attain alignment of the grains, a growth substrate with an heteroepitaxial relationship to graphene is needed. Based on this concept, the growth of wafer-scale, monolayer, single-crystal graphene film has been realized on a hydrogen-terminated germanium buffer layer of H-terminated Ge (110) crystals (Figs. 13.6(a)–(c)) [49]. The anisotropic, two-fold symmetry of the germanium (110) surface allows the unidirectional alignment of multiple seeds, which were merged to create uniform singlecrystal graphene. Furthermore, the weak interaction between the graphene and the

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(a)

graphene

(c)

(b)

H Ge [110]

(d)

– [110]

(e)

1 mm 111

(f)

On Cu (111) Cu(111)

200 mm

Pristine

Fig. 13.6 Growth of large-area, single-crystal graphene film. (a)–(c) Wafer-scale, monolayer, single-crystal graphene film on a hydrogen-terminated Ge (110) surface [49]. (d)–(f) 6 cm  3 cm single-crystal graphene film on the Cu (111) substrate [53]. Reprinted with permission.

underlying hydrogen-terminated germanium surface enables the facile etch-free dry transfer of graphene and the recycling of the germanium substrate. However, the formation of defects or impurities on the Ge (110) surface makes it difficult to synthesize graphene with a high uniformity and reproducibility. Compared to the H-terminated Ge (110) substrate, the Cu (111) substrate has a hexagonal symmetry (surface nearest-neighbor distance of 0.255 nm) which is lattice matched to graphene (lattice constant of 0.246 nm). In addition, Cu foils are compatible with the industrial roll-to-roll process, thus growers are now developing recipes to make highly textured Cu (111) foil for the heteroepitaxial growth of graphene [53]. Using the concept of seamless stitching, Nguyen and co-workers demonstrated the growth of 6 cm  3 cm monocrystalline graphene without grain boundaries on polished copper (111) foil (Figs. 13.6(d)–(f)) [53]. Other than Cu foil, the lattice matching between Cu and c-plane sapphire allows highly textured Cu (111) film to be grown on sapphire, thus allowing wafer-scale single crystalline graphene to be grown in principle on a Cu (111) film-coated substrate [54]. One question is this: using growth strategy involving the “merging of grains,” how small can the graphene domains (or grain size) be before rotational disorder sets in to make a “mosaic” nanocrystalline graphene. On the (111)-textured Cu film on sapphire, Nai et al. [55] investigated the spectroscopic signatures of graphene film containing highly oriented grains with sizes varying from 20 nm to 150 nm using Angle-resolved Photoemission Spectroscopy (ARPES) and High-Resolution Electron Energy Loss Spectroscopy (HREELS) [55]. The robustness of graphene’s Dirac Cone, as well as dispersion of its phonons, as a function of graphene’s grain size was investigated. At the Κ point, well-defined conical Dirac dispersion can be seen for the nanocrystalline graphene films due to the fact that the majority of the nanosized graphene grains are aligned with the substrate. Not surprisingly, the bands in the Dirac cone are sharpest for films with ~150 nm grains and become

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diffused for films with smaller grain sizes. Although such nanocrystalline graphene films exhibit the spectroscopic fingerprints of near-single crystalline graphene, especially when the grains are larger than 100 nm, the presence of grain boundaries modify the electronic properties of these films, for example, making them n-doped in the vacuum. Due to the reactive nature of grain boundaries, adsorbates such as water and oxygen molecules, which adsorb preferentially on the grain boundaries, often impose a p-doping offset on graphene film. These fluctuations in doping shifted the position of the Dirac point from charge neutrality, and its position is highly sensitive to the environment or heating, which makes device fabrication difficult to control. It can be expected that for a single crystalline graphene film with no grain boundaries (or at least very low density of grain boundaries), the graphene film will be highly stable and the Dirac point should remain close to the charge neutrality point. Thus a good yardstick of a single crystalline graphene film with low density of grain boundaries can be the electronic stability of the film in terms of its Dirac point, besides exhibiting very high mobility. The approach in which commensurate stitching on the Cu (111) surface is used to realize large-area mono-crystal graphene can also be applied to the growth of other layered materials, such as hexagonal boron nitride and transition metal dichalcogenides.

Controlling the Shape of Graphene Crystals The hexagonal shape has been considered to be the most thermodynamically stable morphology of graphene due to its highly symmetrical structures. Warner et al. first reported the growth of hexagonal, few-layer, single-crystal graphene on solid Cu foils using ambient pressure CVD, as shown in Fig. 13.7(a) [56]. Transmission electron microscopy and Raman spectrum measurements confirmed that the as-grown, few-layer graphene mainly comprised approximately 5–10 layers. The selected area electron diffraction of individual crystals suggested that they were single crystals with AB Bernal stacking. It should be noted that the size of the hexagonal graphene crystals was only 2–5 μm. The work has stimulated [56] considerable follow up work on the preparation of hexagonal single-crystal graphene. Almost simultaneously, Yu et al. systematically studied single-crystal graphene growth and showed how the individual grain boundaries between coalescing grains affected graphene’s electronic properties [42]. The edge types and layer numbers of as-grown hexagonal graphene were determined by scanning tunneling microscopy and Raman spectroscopy, in which zig-zag edge types and single-layer properties of the graphene was confirmed, respectively. Notably, in this work, a novel approach was developed to precisely control graphene nucleation using pre-patterned growth seeds. Figure 13.7(b) clearly shows that the pre-seed method leads to well-aligned hexagonal grains. Liu’s group reported the large-scale growth of equiangular or hexagonal singlecrystal graphene on solid Cu foil [57] and fully investigated the effects of the growth parameters, such as temperature, carbon source flow rate, and growth time, on singlecrystal graphene growth, as shown in Figs. 13.7(c) and (d). A qualitative model for hexagonal graphene growth was proposed by combining experimental results and classical nucleation theory. A critical carbon concentration for graphene nucleation was defined, and then the graphene growth stage was studied under varied conditions.

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(b)

(a)

10 mm

2 mm (d) Carbon concentration (a.u.)

(c)

2 mm

50 sccm CH4

Nucleation time

Critical C concentration value for graphene nucleation 30 sccm CH4 20 sccm CH4 1 4

2

10 sccm CH4 7 sccm CH4

3

0

10

20

30

40

50

60

70

Growth time (min) Fig. 13.7 Hexagonal graphene grains on a solid Cu surface. (a) SEM images of few-layer graphene domains on a Cu surface [56]. (b) Seeded growth of graphene grains [42]. (c) AFM amplitude image of hexagonal graphene grains [57]. (d) A plot of C concentrations in the Cu film region as a function of reaction time, showing the relation between experimental conditions and the grown graphene information. Regions 1, 2, 3, and 4, represent the growth, nucleation, no growth of graphene, and growth of regular-shaped graphene, respectively [57]. Reprinted with permission.

Apart from the typical hexagonal shape, other graphene crystal shapes have also drawn intensive interest. Liu grew graphene on cube-textured (100)-oriented Cu (CTO-Cu) foils using CVD [58]. Well-aligned, triangular grains up to several micrometers in size were first self-assembled on CTO-Cu during CVD heating in flowing hydrogen. This work also demonstrated that the shape and possible alignment of the graphene grains can be tuned by changing the properties of the substrate.

13.2.3

Liquid Metals for Growing Graphene Crystals Liquid Cu Solid Cu has long been considered an effective catalyst for the growth of large-area, high-quality graphene, but there are intrinsic defects on the Cu foil. On a solid Cu surface, graphene nucleation preferentially occurs on high surface energy locations,

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such as grain boundaries or defects, resulting in graphene grains of inhomogeneous sizes, and overall difficulty in controlling the nucleation density. Solid Cu foils change into a liquid phase when the growth temperature is higher than the melting point of Cu (1084  C). The introduction of liquid Cu [59–61] provides a particularly effective way to control graphene nuclei, given its complete elimination of grain boundaries on solid Cu foils. Due to the high uniformity of the liquid Cu surface, submillimeter-sized, large-area hexagonal graphene grains (HGFs) could be grown. Considering the fluidity of liquid Cu, Geng et al. used Mo and W foils as supporting substrates for their good wettability with liquid Cu [59, 60]. Remarkably, uniform monolayer graphene flakes and films were prepared on liquid Cu (Fig. 13.8). The size of the hexagonal graphene domains was on average 100 μm, reflecting an advancement in the growth of highly uniformed large-sized graphene. Numerous reasons have been proposed to explain liquid Cu’s success in growing large-sized single-crystal graphene. It is speculated that the liquid phase of Cu allows dynamic restructuring of carbon atoms during crystallization, and largely eliminates the grain boundaries present on a solid Cu surface. It is known that the absence of grain boundaries leads to decreased graphene nucleation and larger graphene crystals.

Fig. 13.8 Uniform hexagonal graphene flakes and films grown on a liquid Cu surface [59, 60]. (a)

Scheme of the CVD process for the synthesis of hexagonal graphene flakes (HFG) on a liquid Cu surface [60]. (b)–(d) SEM images of HFG that are on the verge of merging. (e) SEM image of a sample grown for a long time, showing a continuous graphene film. (f) and (g) Magnified SEM images of HFG, with average sizes ranging from 50 μm to 120 μm [59]. Reprinted with permission.

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A study led by Warner also reported the use of molten Cu in fabricating large-scale, single-crystal graphene [61]. By controlling the amount of H2 present during the growth process, the size of individual graphene crystals grown could reach 200 μm, demonstrating the considerable potential of molten Cu in the growth of large-sized, singlecrystal graphene. Mohsin et al. synthesized millimeter-sized, hexagonal-shaped, singlecrystal graphene on re-solidified Cu foils [62]. A solid Cu foil was first melted and then re-solidified by lowering the temperature. The in-situ processing approach produced a relatively smooth surface that resulted in low graphene nucleation density, enabling the growth of millimeter-sized, single-crystal graphene. Liquid Cu CVD methods are usually operated under ambient conditions, exhibiting significant advantages in scaling up graphene production. It has also been shown that the morphology of single-crystal graphene can be precisely tailored using a liquid Cu surface. Snowflake-like graphene crystals (Fig. 13.9(a)) were grown on liquid Cu surfaces [63], and the manipulation of the Ar: H2 flow rate ratio resulted in the evolution of these graphene flake (GF) shapes. A series of SEM images of typical GFs grown on liquid Cu surfaces under various flow rate ratios of Ar:H2 was obtained (Fig. 13.9(b)–(l)). Using higher ratios of Ar:H2 resulted in the formation of GFs with symmetric dendritic structures. As the Ar:H2 ratio decreased, more compact structures appeared until eventually regular hexagonal GFs with straight edges were obtained. Using pure H2 led to a slight deviation in the perfect hexagonal structures to produce edges with positive curvatures.

Fig. 13.9(b)–(l) SEM images of a series of graphene grains grown on liquid Cu surfaces with different shapes, engineered by varying the Ar/H2 ratio on the liquid Cu surface [63]. Scale bars are 5 μm. Reprinted with permission.

13.2 CVD Method for Graphene Growth

249

Fig. 13.10 Variously shaped GFs formed on a liquid brass surface [63]. Reprinted with permission. Scale bars are 10 μm.

Using liquid brass (30% Zn and 70% Cu) as a catalyst in the approach previously described allowed further fine manipulation of GF morphologies, reflecting the possibility of fine-tuning GF morphology (Fig. 13.10). For example, when the Ar:H2 ratio was low, the GFs assumed a compact hexagonal shape. As the Ar:H2 ratio increased, the vertices of the hexagonal GF crystals became distorted, producing increasingly more concave polyhedral structures. Further increasing this ratio led to the formation of flower-like GFs, similar to the case in which pure Cu was used. To explain the shape evolution of GFs, a diffusion-limited growth mechanism was proposed. The competition between ad-atom diffusion along island edges and ad-atom surface diffusion determines the morphologies of GFs. A geometrically well-defined graphene island is formed when an ad-atom relaxes sufficiently to find an energetically favorable location along island edges or corners; in the absence of such relaxation, diffusion-limited growth leads to dendritic structures.

Liquid Ga In addition to liquid Cu, liquid Ga has also been used as a catalyst to fabricate highquality graphene. Ga is carbon-insoluble, but it is an excellent absorber of carbon, which gives it the potential to act as a good surface catalyst for graphene growth. Jiang’s group studied several molten metals and their alloys for use in graphene growth [64], and liquid Ga exhibits a highly effective catalytic ability. Graphene with a controlled layer number and quality level was prepared on liquid Ga over the course of a few minutes by APCVD. It should be noted that the as-grown graphene was mainly few-layer. Fu et al. further improved CVD growth conditions on liquid Ga, and then prepared uniform, monolayer graphene film and hexagonal single-crystal graphene [65]. The carrier mobility of the as-grown graphene crystals reached a high value of 7400 cm2 V–1 s–1, demonstrating liquid metal’s significant potential in growing highquality graphene. The remarkably high mobility originates from Ga’s characteristics, including low vapor pressure at elevated temperatures, resulting in a homogeneous

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Fig. 13.11 (a)–(f) Schematic illustration of graphene growth over liquid Ga supported on a W substrate. (g), (h) Graphene films transferred from Ga–W substrate onto a SiO2/Si substrate [65]. Reprinted with permission.

surface and low nucleation density (Fig. 13.11). The density was as low as 1/1000 μm2, compared with graphene grown on solid polycrystalline Ga (1/100 μm2). The growth of homogeneous monolayer graphene was attributed to the self-terminating process caused by the high surface energy of liquid Ga. To lower its surface energy, Ga absorbs carbon atoms or clusters on its surface. During growth, the surface-bound carbon species nucleate and form a graphene film. Once the surface of Ga is fully covered by graphene, additional carbon atoms are blocked from absorbing on the Ga surface, thus limiting the growth.

13.3

Prospects This chapter presents the controlled growth of graphene crystals on metal catalysts by CVD. Significant progress has been made in the last few years for the growth of largearea single crystalline graphene, and it is highly plausible that in the near future the size of such single crystalline graphene is limited only by the size of the polished copper foil it is grown on. Two distinct approaches to the growth of single crystalline graphene may converge to allow better control of large-area single crystalline graphene growth. First, polishing the copper foil to obtain a crystallographic textured (111) surface provides an epitaxial substrate for the growth of aligned graphene grains, which can merge seamlessly to produce single crystalline graphene. Next, passivating defects on copper foil to reduce nucleation density allows the growth of much bigger crystalline graphene domains. Future efforts are directed at engineering alloy catalyst based on Ni/Cu to allow the controlled growth of bilayer graphene, which have many interesting properties distinct from monolayer graphene.

13.4 References

251

Compared with the use of solid metals in growing graphene, the use of liquid metals (particularly liquid Cu and Ga) as a catalytic substrate is intriguing as it provides a defect-free surface for the synthesis of large graphene crystals. Large-sized graphene crystals have been successfully synthesized on liquid Cu and Ga using the CVD approach, with excellent tolerance to variations in the growth conditions. However, it is still challenging to make uniform graphene with different layer numbers. A better understanding of the growth mechanisms on various liquid metals is needed. The shortcomings of the liquid metal approach include the much higher temperature needed compared to solid metal CVD, as well as the difficulty in attaining wafer-scale growth. This means that industrialization will be highly challenging. Finally it should also be noted that notwithstanding the rapid progress made in the CVD growth of graphene, there is still room for improvement for the current transfer method of graphene; wet etching or electrochemical bubbling methods usually introduce residues or breakages, compromising the intrinsic properties of graphene. Highfidelity transfer or direct growth of high-quality graphene films on arbitrary substrates is needed to enable wide-ranging applications in photonics or electronics, which includes devices such as optoelectronic modulators, transistors, on-chip biosensors, and tunneling barriers. The direct growth of graphene film on an insulating substrate, such as a SiO2/Si wafer, would be useful for this purpose to enable many applications on the silicon wafer. Recently, Libo Gao et al. developed a face-to-face transfer method for wafer-scale graphene films that accomplished both the growth and transfer steps on one wafer [66]. This spontaneous transfer method relies on nascent gas bubbles and capillary bridges between the graphene film and the underlying substrate during etching of the metal catalyst. In contrast to the previous wet or dry transfer results, the face-toface transfer is non-hand-crafted and is compatible with any size and shape of substrate; this approach also enjoys the benefit of a much reduced density of transfer defects compared with the conventional transfer method. Most importantly, the direct growth and spontaneous attachment of graphene on the underlying substrate is amenable to batch processing in a semiconductor production line, and thus will speed up the technological application of graphene. Thus combining methods to grow large-area single-crystal graphene on copper-coated silicon followed by a face-to-face transfer method, may open up direct deposition of single-crystal graphene on silicon.

13.4

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Part II

14

Electronic Properties and Strain Engineering in Semiconducting Transition Metal Dichalcogenides Rafael Roldán and Francisco Guinea

14.1

Introduction Transition metal dichalcogenides (TMD) form a new family of layered two-dimensional materials that, as graphene, can be easily exfoliated and present promising electrical and optical properties [1]. Among them, semiconducting TMD are of special interest because they present a gap in the optical range of the energy spectrum that makes them candidates for device applications [2–4]. It has been demonstrated experimentally [6–9] that the band structure of those compounds dramatically changes from single-layer to multi-layer samples, involving a transition from an indirect gap for multi-layer samples to a direct gap for single-layer samples [5], pointing out the important role of interlayer hopping of carriers in these compounds [10]. Furthermore, their electronic properties are highly sensitive to external conditions, such as strain, pressure, or temperature. For instance, a semiconducting/ metal transition can be induced under specific conditions [11–23]. Such possibility to tune the gap is very interesting for applications in optoelectronic nanodevices [2]. Semiconducting TMDs present a strong spin–orbit interaction, that lifts the spin degeneracy of the energy bands in single-layer samples due to the absence of inversion symmetry [24]. Since time reversal symmetry must be conserved, the spin splitting in inequivalent valleys must be opposite, leading to the so-called spin–valley coupling [25–27], which has been observed experimentally [28–33]. The coupling of the spin, the valley, and the layer degrees of freedom in semiconducting TMDs opens up the possibility to manipulate them for future applications in spintronics and valleytronics nanodevices [25, 28–31, 34–36]. Another of the fascinating properties of the two-dimensional TMDs is their high stretchability and the possibility to use external strain to manipulate their electronic and optical properties [37]. Strain engineering – which involves the study of how the physical properties of materials can be modified by controlling the external strain fields applied to them – has a perfect platform for its implementation in two-dimensional semiconducting TMDs. Non-uniform strain profiles can be used to create a funnel of excitons, which allows to capture a broad range of the solar spectrum, concentrating carriers in a small region of the samples [11]. Strain engineering can be also used to exploit the piezoelectric properties of atomically thin layers of TMDs, converting mechanical to electrical energy. Molybdenum disulphide, a material which is not 259

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piezoelectric in its bulk configuration, has been shown to become piezoelectric when its thickness is reduced to a single atomic layer (or to an odd number of them) [38]. In this chapter, we describe the electronic properties of 2D semiconducting transition metal dichalcogenides. The corresponding band structure is analyzed in terms of density function theory (DFT) calculations. This information will be used to develop a tightbinding model that contains the relevant orbital contribution to account for the valence and conduction bands. Finally, we will see how the band structure can be modified by means of externally applied strain to the samples.

14.2

Electronic Structure The lattice structure of single and multilayer MX2 (where M ¼ M0 , W and X ¼ S, Se) is schematically shown in Fig. 14.1. The crystal has an inner layer of metal M atoms ordered on a triangular lattice, which is sandwiched between two layers of chalcogen X atoms located on the triangular net of alternating hollow sites. We denote a as the distance between nearest-neighbor in-plane MM and XX atoms, b as the nearestneighbor MX distance and u as the separation between the M and X planes. Therefore, pffiffiffiffiffiffiffiffiffiffi the MX2 crystal constitutes an almost perfect trigonal prism structure with b ’ 7=12a and u ’ a=2. Table 14.1 contains the values of the lattice parameters of the bulk compounds corresponding to the more commonly studied TMDs [40–42]. The inplane Brillouin zone is shown in Fig. 14.2. It ffiffiis ffi a hexagon that contains the high p symmetry points Γ ¼ ð0; 0Þ, M ¼ 4π=3a 0; 3=2 and K ¼ 4π=3að1; 0Þ. The six (a)

X

M

(b)

b

C⬘

2u ª a

a Fig. 14.1 (a) Atomic structure of MX 2 . The bulk configuration has a 2H-MX 2 structure with two MX 2 layers per unit cell, each of them being built up from a trigonal prism coordination unit. The small rectangle represents the unit cell of a monolayer of MX 2 , which is doubled in the bulk crystal. (b) Detail of the trigonal prisms for the two layers in the bulk compound, showing the lattice constants used in the text (adapted with permission from [39]).

14.2 Electronic Structure

261

Table 14.1 Lattice parameters for the TMDs considered in the text. a represents the M M atomic distance, u the internal vertical separation between the M and the X planes, and c 0 the distance between the metal M layers. In bulk systems c ¼ 2c 0 accounts for the z-axis lattice parameter. All values are in Å units.

MoS2 WS2 MoSe2 WSe2

a

u

c0

3:160 3:153 3:288 3:260

1:586 1:571 1:664 1:657

6:140 6:160 6:451 6:422

Fig. 14.2 Two-dimensional Brillouin zone of MX 2 . The high-symmetry points Γ ¼ ð0; 0Þ,  pffiffiffi 

M ¼ 4π=3a 0; 3=2 and K ¼ 4π=3að1; 0Þ are shown. The Q points (which are not highsymmetry points) indicate the position of the conduction band edge in multi-layer samples.

Q valleys that correspond to the minimum of the conduction band for multi-layer samples are also shown. In the following, we study the electronic band structure of single-layer, multi-layer, and bulk TMDs. For this purpose, we use relativistic Density Functional Theory (DFT) calculations, in which the intrinsic spin–orbit interaction term for all atoms is included. Figure 14.3 shows the band structures for single-layer, bilayer, and bulk MX 2 . The most important features of the band structures, such as the band gaps and the energy splitting of the valence and conduction bands at different points of the BZ (due to spin–orbit interaction), are summarized in Table 14.2. Contrary to what happens in other 2D crystals, such as graphene or phosphorene, the valence and conduction bands of TMDs present a very rich orbital contribution. They are made by hybridization of the d orbitals of the transition metal atom M, and the p orbitals of the chalcogen atom X. In particular, the analysis of the orbital contribution for the block of bands containing the first four conduction bands and the first seven valence bands, in an energy window from –7 to 5 eV around the Fermi level, reveals that these bands are mainly constituted by the five 4d orbitals of the metal M and the six

262

Electronic Properties and Strain Engineering

Table 14.2 Band gap and energy splitting of the valence band (VB) and conduction band (CB) at relevant points of the Brillouin zone as obtained from DFT calculations of [39]. The gap for single-layer samples is always direct, whereas for bilayer and bulk compounds we give the direct/indirect band gap for each compound (with the corresponding values separated by a dash as follows: direct–indirect). All the energies are expressed in eV. Band gap

VB(K)

MoS2 WS2 MoSe2 WSe2

1.715 1.659 1.413 1.444

0.153 0.439 0.193 0.439

MoS2 WS2 MoSe2 WSe2

1.710–1.198 1.658–1.338 1.424–1.194 1.442–1.299

0.181 0.451 0.213 0.454

MoS2 WS2 MoSe2 WSe2

1.679–0.788 1.636–0.917 1.393–0.852 1.407–0.910

0.245 0.482 0.267 0.504

VBðΓÞ

CB(K)

CB(Q)

0.0041 0.0333 0.0258 0.0396

0.0832 0.339 0.0 0.275

0.0 0.0357 0.0253 0.0428

0.457 0.528 0.417 0.522

0.0 0.044 0.0228 0.0548

0.874 0.922 0.819 0.919

Monolayer

Bilayer 0.737 0.677 0.649 0.649 Bulk 1.018 1.426 0.695 1.075

Fig. 14.3 Band structure of single layer, (a), bilayer (b), and bulk (c) of MoS2, MoSe2, WS2, and WSe2 obtained from DFT calculations, including spin–orbit coupling. Dashed vertical lines indicate the position of the Q point in the BZ (see text) (adapted with permission from [39]).

(three for each layer) 3p orbitals of the chalcogen X, which sum up to 93% of the total orbital weight of these bands [5]. All the single-layer compounds are direct gap semiconductors, with the gap lying at the two inequivalent K and Kʹ points of the hexagonal Brillouin zone, as shown in Fig. 14.3(a). The most relevant orbital contribution at the edge of the valence band at the K point riginated from a combination of dx2 y2 and dxy orbitals of the metal M, which hybridize with px and py orbitals of the chalcogen atoms X [5]. The edge of the conduction band, on the other hand, has a main contribution due to d3z2 r2 of M, plus a minor contribution of px and py orbitals of X. Multi-layer samples are, on the other hand, indirect gap semiconductors. The maximum of the valence band is placed at the Γ point of the BZ (see Figs. 14.3(b) and

14.2 Electronic Structure

263

(c)), with a major contribution from d 3z2 r2 and pz orbitals of the metal and chalcogen atoms, respectively. The edge of the conduction band in multi-layer samples is at the Q point of the BZ (indicated by a dashed vertical line in Fig. 14.3). The Q point is not a high-symmetry point of the Brillouin zone, and therefore the minima of the conduction band for bilayer and bulk materials do not lie exactly at the same point as for singlelayer samples. The orbital character at this point originates mainly from the d xy and dx2 y2 orbitals of the metal M, and px and py of the chalcogen X, plus a non-negligible contribution of pz and d 3z2 r2 of X and M atoms, respectively. This extremely rich orbital contribution to the relevant bands that occur in semiconducting TMDs makes the construction of a minimum tight-binding model, valid in the whole Brillouin zone, a cumbersome task, as we will see in the next section. Another characteristic of these compounds is the strong spin–orbit coupling, which leads to a large splitting of the valence band at the K and Kʹ points of the BZ. Since spin–orbit interaction is stronger for heavier atoms, the splitting of the valence band for MoX 2 (see top panels of Fig. 14.3) is smaller than the splitting of WX 2 bands (bottom panels of Fig. 14.3). For the Mo compounds, the splitting is of the order of ~150 meV, whereas for the heavier W compounds increases up to ~400 meV. Spin–orbit interaction also leads to a splitting of the conduction band at the K point, as well as at the secondary minimum Q. DFT calculations yield splittings between 3 and 30 meV of the minimum of the conduction band at the K point. In contrast to the valence band (where the splitting is due to the d orbitals of the metal), both atoms give comparable contributions to the conduction band splittings at the K point of the BZ. An interesting aspect of the TMDs is the appearance of a superconducting transition with a critical temperature that strongly depends on the carrier density. A superconducting dome, similar to that observed for the layered cuprate compounds, has been experimentally observed in the temperature-carrier density phase diagram of MoS2 [43–46]. The samples, which were doped by a combination of liquid and solid gating, showed a critical temperature of T c  10:8 K for the optimal doping n  1:2  1014 cm2 . From the theoretical point of view, the origin of superconductivity in heavily doped MoS2 has been studied, by considering the role of both electron–electron and electron–phonon interactions in [47]. The estimations for the strength of the different contributions to the effective coupling suggest that superconductivity in MoS2, under the experimental conditions of [43] and [44], is likely to be induced by the electron–electron interaction. The short-range repulsion between carriers at the conduction band allows for a superconducting phase induced by the electron–electron interaction with a non-trivial structure, where the gap acquires opposite signs in the inequivalent pockets of the conduction band. Functional renormalization group studies suggest that the triplet pairing is driven by ferromagnetic fluctuations [48]. The multi-orbital nature of the conduction band, as well as Hund’s coupling, seems to play an essential role in the physics of MoS2. On the other hand, DFT calculations for the phase diagram of TMDs suggest that phonon mediated superconductivity is also possible for some range of electron doping. A charge-density wave phase is also proposed to exist for higher carrier concentrations [49]. It has also been suggested that spin-triplet p-wave superconductivity can be stabilized by Rashba spin–orbit coupling [50].

264

Electronic Properties and Strain Engineering

14.3

From Density Functional Theory to Tight-Binding Approximation The low-energy physics of monolayer MoS2 was first described in terms of a massive Dirac Hamiltonian around the K and Kʹ points [25]. More accurate approximations such as k p approximations [51, 52] that account for the presence of trigonal warping and diagonal quadratic terms in momentum, as well as tight-binding approximations [5, 40, 51], were developed later. As we have seen in the previous section, monolayer TMDs are direct band-gap semiconductors with the gap placed at the K and Kʹ points of the Brillouin zone. First-principle calculations show that there are two additional secondary extrema: a local minimum of the conduction band at the Q point, midway between the Γ and K point, and a local maximum of the valence band at the Γ point [53]. These local extrema present in the band structure of single-layer compounds, become absolute extrema in multi-layer samples, for which the minimum of the conduction band is at Q, and the maximum of the valence band is at Γ (see Fig. 14.3). A tight-binding model that aims to be useful for the valence and conduction band in the whole Brillouin zone must capture these features, and this will be the focus of the present section. Furthermore, the orbital content of such a tight-binding model must be appropriate so as to capture the rich orbital contribution that presents the relevant bands above and below the energy gap. The main features of the band structure of TMDs in the whole Brillouin zone, described in the previous section based on first-principle calculations, can be captured by a Slater–Koster tight-binding model that includes 11 bands. These bands corresponds to the five d orbitals of the metal M atom and to the six p orbitals of the two chalcogen X atoms in the unit cell, and the base can be expressed as   ptx ; pty ; ptz ; d3z2 r2 ; d xz ; dyz ; d x2 y2 ; dxy ; pbx ; pby ; pbz , (14.1) where the labels t and b refer to the top and bottom chalcogen planes, respectively. Following the standard Slater–Koster description [54], the model is defined by the hopping integrals between the different orbitals, described in terms of σ, π, and δ ligands. The details of the model can be found in [5, 27]; here we just reproduce the most important results. The hopping processes of the relevant pairs of neighbors are described in terms of the Slater–Koster parameters, namely V pdσ and V pdπ for MX bonds, V ddσ , V ddπ and V ddδ for MM bonds, and V ppσ and V ppπ for XX bonds. Additional parameters of the theory are the crystal fields ϵ 0 , ϵ 1 , ϵ2 , ϵ p , and ϵz , describing,  respectively, the atomic level l ¼ 0 ðd3z2 r2 Þ, the l ¼ 1 dxz ; dyz and l ¼ 2 dx2 y2 , dxy M orbitals, the inplane (px , py ) X orbitals, and the out-of-plane pz X orbitals. In spite of the large Hilbert space considered in this model, a simplification is possible. By performing an unitary transformation that transforms the p orbitals of the top and bottom X layers into their symmetric and antisymmetric combinations with respect to the z-axis, the 11-band model can be decoupled into a six bands block with even symmetry with respect to z ! z inversion, and a five bands block with odd symmetry. Since low-energy excitations belong exclusively to the first block, the relevant bands above and below the gap can be considered by the reduced Hilbert space

14.3 Density Functional Theory to Tight-Binding

  ψ ¼ d 3z2 r2 ; d x2 y2 ; dxy ; pSx ; pSy ; pAz ,

265

(14.2)

where the S and A superscripts refer to symmetric combinations of pffiffiffi and antisymmetric pffiffiffi   the p-orbitals of the chalcogen atom, pSi ¼ 1 2 pti þ pbi and pAi ¼ 1= 2 pti  pbi , where i ¼ x, y, z. A top view of the crystal lattice of MX 2 is sketched in Fig. 14.4. The tight-binding Hamiltonian defined by the base Eq. (14.2), including local spin–orbit coupling, can be expressed in real-space as i X Xh H¼ ϵ μ, v c†i, μ ci, v þ t ij, μv c†i, μ cj, v þ H:c: , (14.3) i, μν ij, μv where c†i, μ creates an electron in the unit cell i in the atomic orbital μ ¼ 1, . . . , 6, belonging to the Hilbert space defined in Eq. (14.2). The Hamiltonian in the k-space can be expressed as [5, 27, 55]   HMM HMX H¼ , HMX † HXX X t MM cos ðk  ai Þ, HMM ¼ ϵ M þ 2 i i¼1, 2, 3 (14.4) X HXX ¼ ϵ X þ 2 t XX i cos ðk  ai Þ, i¼1, 2, 3 X ik  δi t MX , HMX ¼ i e i¼1, 2, 3 where the vectors ðδi Þ and ðai Þ are shown in Fig. 14.4. All the hopping terms t ij, μv are evaluated within the Slater–Koster scheme [5, 27]. The on-site terms of the Hamiltonian can be written in the compact form [27]

Fig. 14.4 A top view of single-layer MX 2 lattice structure. Black (white) circles indicate M ðX Þ atoms. The nearest-neighbors ðδi Þ and the next nearest-neighbors ðai Þ vector are shown in the figure.

266

Electronic Properties and Strain Engineering

 ϵ¼

ϵM 0

 0 , ϵX

(14.5)

where 0

ϵ0

0

0

1

B ϵ2 iλM^sz C ϵM ¼ @ 0 A, ^ 0 iλM sz ϵ2 0 1 λX ⊥ ^ s ϵ þ t i 0 xx B p C 2 z B C B C, ϵ X ¼ B λX ⊥ C ^ i s ϵ þ t 0 p yy @ 2 z A ⊥ 0 0 ϵ z  t zz

(14.6)

where λM and λX are the spin–orbit coupling of the metal ðM Þ and chalcogen atoms ðX Þ, respectively, and ^s z ¼  is the z-component of the spin degree of freedom [27]. The effects of vertical hopping V pp between top and bottom chalcogen atoms is considered ⊥ ⊥ through the terms t ⊥ xx ¼ t yy ¼ V ppπ , and t zz ¼ V ppσ . The nearest-neighbor hopping between M and X atoms are pffiffiffi pffiffiffi pffiffiffi 0 1 12V pdπ þ 3V pdσ pffiffiffi 9V pdπ þ 3V pdσ 3 3V pdπ  V pdσ p ffiffi ffi p ffiffi ffi p ffiffi ffi 2 B C t MX 9V pdπ  3V pdσ 2 3V pdπ þ 3V pdσ A, 1 ¼ pffiffiffi @ 5 3V pdπ þ 3V pdσ 7 7 pffiffiffi pffiffiffi pffiffiffi V pdπ  3 3V pdσ 5 3V pdπ þ 3V pdσ 6V pdπ  3 3V pdσ (14.7) p ffiffi ffi p ffiffi ffi 0 1 0 6 3V pdπ þ 2V pdσ 12V pdπ þ 3V pdσ pffiffiffi pffiffiffi pffiffiffi 2 B C t MX (14.8) 0 6V pdπ  4 3V pdσ 4 3V pdπ  6V pdσ A, 2 ¼ pffiffiffi @ 7 7 14V pdπ 0 0 pffiffiffi pffiffiffi pffiffiffi 0 1 9V pdπ  3V pdσ 3 3V pdπ  V pdσ 12V pdπ þ 3V pdσ pffiffiffi pffiffiffi pffiffiffi 2 B pffiffiffi C t MX 9V pdπ  3V pdσ 2 3V pdπ þ 3V pdσ A, 3 ¼ pffiffiffi @ 5 3V pdπ  3V pdσ 7 7 pffiffiffi pffiffiffi pffiffiffi V pdπ  3 3V pdσ 5 3V pdπ  3V pdσ 6V pdπ þ 3 3V pdσ (14.9) where the direction of the hopping labeled by subindices 1, 2, and 3 is shown in Fig. 14.4. Hopping terms corresponding to processes between the same kind of atoms, MM or XX (see Fig. 14.4), are given by pffiffiffi 0 1 3 3 V þV ð þV Þ  ð V Þ V 3V ddδ ddσ ddδ ddσ ddδ ddσ B C 2 B pffiffiffi C pffiffiffi 2 B C 1 1 3 3 MM B t 1 ¼ B ðV ddδ þV ddσ Þ ðV ddδ þ12V ddπ þ3V ddσ Þ ðV ddδ 4V ddπ þ3V ddσ Þ C C, 4B 2 4 4ffiffiffi C p @ A 3 1 3 ðV ddδ 4V ddπ þ3V ddσ Þ ð3V ddδ þ4V ddπ þ9V ddσ Þ  ðV ddδ V ddσ Þ 4 2 4 (14.10)

14.3 Density Functional Theory to Tight-Binding

267

pffiffiffi 1 3ðV ddδ V ddσ Þ 0 3V ddδ þV ddσ 1 B pffiffiffi C (14.11) t MM V ddδ þ3V ddσ 0 A, 2 ¼ @ 3ðV ddδ V ddσ Þ 4 0 0 4V ddπ 0 1 pffiffiffi 3 3 ðV ddδ þV ddσ Þ ðV ddδ V ddσ Þ B 3V ddδ þV ddσ C 2 2 B pffiffiffi C pffiffiffi B C 1B 3 C 1 3 MM t 3 ¼ B ðV ddδ þV ddσ Þ ðV ddδ þ12V ddπ þ3V ddσ Þ  ðV ddδ 4V ddπ þ3V ddσ Þ C, C 4B 2 4 4 B C pffiffiffi @ 3 A 1 3 ðV ddδ V ddσ Þ  ðV ddδ 4V ddπ þ3V ddσ Þ ð3V ddδ þ4V ddπ þ9V ddσ Þ 4 2 4 (14.12) pffiffiffi  0 1 0 3V ppπ þV ppσ 3 V ppπ V ppσ 1 B pffiffiffi  C XX (14.13) t 1 ¼ @ 3 V ppπ V ppσ V ppπ þ3V ppσ 0 A, 4 0 0 4V ppπ 0 1 0 V ppσ 0 B C t XX (14.14) 2 ¼ @ 0 V ppπ 0 A, 0

0 0

3V ppπ þV ppσ 1 B pffiffiffi  XX t 3 ¼ @  3 V ppπ V ppσ 4 0

0 V ppπ pffiffiffi   3 V ppπ V ppσ V ppπ þ3V ppσ 0

0

1

C 0 A:

(14.15)

4V ppπ

An appropriate set of Slater–Koster parameters for single-layer MoS2 is given in Table 14.3. The band structure calculated with this model is plotted in Fig. 14.5, as compared to DFT calculations. This model can be easily generalized to multi-layer

Table 14.3 Slater–Koster tight-binding parameters for single-layer MoS2. All terms are in units of eV. Crystal fields

ϵ0 ϵ2 ϵp ϵz

–1.094 –1.512 –3.560 –6.886

Intralayer Mo–S

V pdσ V pdπ

3.689 –1.241

Intralayer Mo–Mo

V ddσ V ddπ V ddδ

–0.895 0.252 0.228

Intralayer S–S

V ppσ V ppπ

1.225 –0.467

268

Electronic Properties and Strain Engineering

Fig. 14.5 Tight-binding band structure of single-layer MoS2 (thick black lines) compared with DFT result (thin gray lines). The Slater–Koster parameters are given in Table 14.4.

systems with arbitrary stacking orders. The advantage of the present tight-binding description with respect to first-principles calculations is that it provides a simple starting point for the further inclusion of many-body electron–electron interaction effects by means of quantum field theory techniques, as well as of the dynamical effects of the electron–lattice interaction. Tight-binding approaches are often more convenient than ab initio methods for investigating systems involving a very large number of atoms [23], disordered and inhomogeneous samples [56], and materials nanostructured in large scales (nanoribbons, ripples) or in twisted multilayer materials. In the next section, we will explain how to apply the method to strained samples.

14.4

Including Strain in the Tight-Binding Hamiltonian The use of the above Slater–Koster tight-binding approach is particularly convenient when considering lattice deformations, such as those produced by external strain. If one neglects as a first approximation the corrections to the local atomic potentials due to

14.4 Strain in the Tight-Binding Hamiltonian

269

lattice deformation [57, 58], the effect of strain in the sample is driven by the dependence of the tight-binding parameters on the two-center energy integral elements, which depend on the interatomic distance. At leading order, the modified hopping terms in the presence of strain can be written as [23, 55, 59, 60] 0 1   rij  r0ij   C 0 B t ij, μv rij ¼ t ij, μv rij @1  Λij, μv A, (14.16) 0 rij where r0ij is the distance in the unstrained lattice between two atoms labeled by ði; μÞ and ðj; νÞ. rij is the separation in the presence of strain, which can be calculated from

 rij ðx; y; zÞ ¼ ^I þ ^ε ðx; yÞ  r0ij , (14.17) where ^I is the identity matrix, ^ε ðx; y; zÞ is the strain tensor that accounts for the corresponding profile of strain, and Λij, μv ¼ dln t ij, μv ðr Þ=dlnðr Þ r¼jr0 j is the dimensionij less bond-resolved local electron–phonon coupling. A microscopical estimation of the electron–lattice coupling parameters Λij, μv is in principle possible by means of an accurate analysis based on the direct comparison between ab initio and tight-binding calculations. For instance, the electron–phonon coupling associated with the different interlayer hopping amplitudes in multilayer graphene was estimated in [61]. However, this is a formidable task in transition metal dichalcogenides because of the large number of orbitals/bands that are involved, and because of the lack of a Fermi surface that can be used as a reference. In the absence of any experimental and theoretical estimation for the electron–phonon coupling, it is common to use the Wills–Harrison argument [62], which states that t ij, μv ðr Þ / jrjðlμ þlv þ1Þ , where lμ is the absolute value of the angular momentum of the orbital μ, and lν is the absolute value of the angular momentum of the orbital v. Following this approximation, it can be assumed that the coupling between p orbitals of the chalcogen is Λij, XX  3, the coupling between p orbital of the chalcogen and d orbital of the metal is Λij, MX  4, and the coupling between d orbitals of the metal is Λij, MM  5. This model can be used to simulate the effect of non-uniform strain in the electronic band structure of realistic ribbons of TMDs subject to external deformations. By considering a one-dimensional profile along the y-axis of the strain tensor, and assuming a sinusoidal profile of strain, the deformations along the armchair direction can be described by the strain tensor 0 1 σ 0 0 ^ε ðx; y; zÞ ¼ ^ε ðyÞ ¼ εmax f ðyÞ@ 0 1 0 A, (14.18) 0 0 0 where where L is the characteristic width of the ribbon, εmax the maximum strain attained at the center of the sample y ¼ L=2, σ is the Poisson’s ratio of the membrane and f ðyÞ is a function that accounts for the distribution of strain along the sample. In Fig. 14.6, we show the band structure calculated with this model for a zig-zag MoS2 ribbon (infinite along the x-axis) with periodic boundary conditions and

270

Electronic Properties and Strain Engineering

Fig. 14.6 Calculated electronic band structure for a strained zig-zag MoS2 ribbon. (a) The band structure for the case of an unstrained ribbon. (b) and (c) Band structures calculated for wrinkled ribbons with a 2% and 4% maximum tensile strain, respectively. The vertical arrows indicate the direct band gap, which can be probed in photoluminescence experiments (adapted with permission from [23]).

200 unit cells wide along the y-axis (a is the unit cell size a ¼ 3:16 Å). Each band is composed by 200 subbands that arise from the discrete width of the ribbon in the  y-direction. We have considered a half-sinusoidal distribution of strain f ðyÞ ¼ sin πy L , but other distributions of the strain along the sample, such as Lorentzian distribution, Gaussian distribution, etc., can be considered with this model. Panel (a) corresponds to the band structure for the case of an unstrained ribbon, whereas panels (b) and (c) are the band structures calculated for wrinkled ribbons with a 2% and 4% maximum tensile strain, respectively. The vertical arrows indicate the direct band-gap transitions, and from the calculation, one clearly sees that the corresponding bandgap energy decreases as the uniaxial strain increases. This theoretical result is in agreement with the trend observed experimentally for MoS2 wrinkles subjected to inhomogeneous strain [23].

14.5

Low-Energy Model of Strained Transition Metal Dichalcogenides The Hamiltonian (Eq. (14.3)) includes explicitly the hybridization between the metal and the chalcogen atoms, being therefore the appropriate starting point for a compelling derivation of an effective low-energy model of TMDs in the presence of strain. In this section, we present, following the work of Rostami et al. [55], a simplified model for strained single-layer TMDs valid around the K and Kʹ points of the Brillouin zone.

14.5 Strained Transition Metal Dichalcogenides

271

For this aim, we perform a Taylor expansion in momentum and in strain fields, followed by a canonical projection onto the two (valence and conduction) low-energy bands. From the technical point of view, in order to obtain an effective 2  2 (4  4 including spin) model Hamiltonian, we use the Löwdin partitioning method [63]. Details about the derivation are provided in [55]. Similar to the carbon nanotube and to the graphene cases [59], we first set the momentum coordinates on the relative valley (K-point of the BZ), and we hence derive a strain-dependent Hamiltonian which includes the effect of hopping integrals modification caused by the deformation. The strain-dependent Hamiltonian around K-point, up to second order in strain and momentum, can be written as H ¼ H 0 þ H so , where [55]  2 2    Δ0 þ Δσ z e ħ2 e e þ D þ t 0 a0 q þ τA1  στ þ H0 ¼ q þ τA2 α þ q þ τA3 βσ z , ħ ħ ħ 2 4m0   2 2 λ0 þ λσ z e e 0 0 2 þ δλ þ a0 q þ τA4 λ0 þ q þ τA5 λ σ z H so ¼ τs: (14.19) ħ ħ 2   Here e is the elementary charge, m0 is the free electron mass, στ ¼ τσ x ; σ y are Pauli matrices in the 2  2 “band” space,pand ffiffiffi s ¼  and τ¼  are spin and valley indexes, respectively. Finally a0 ¼ a= 3 and q ¼ qx ; qy is the relative momentum with respect to the K point. The parameters Δ0 , Δ, λ0 , λ, λ00 , λ0 , t 0 , α, and β are strainindependent and they can be obtained directly from the Slater–Koster parameters of the original Hamiltonian (Eq. (14.3)), and they are given in Table 14.4 for the case of MoS2. A detailed derivation of the numerical values of all the parameters of the low-energy model in terms of the original tight-binding parameters can be found in [55]. It is useful to notice that the mass asymmetry parameter, α, and topological term, β, are related to general physical properties of the band structure, like effective mass and energy gap, through the relations α ¼ m0 =mþ and β ¼ m0 =m  4m0 v2 =ðΔ  λ Þ, where v ¼ t 0 a0 =ħ, m ¼ mc mv =ðmv  mc Þ, 2λ ¼ λ0  λ. In addition, mc and mv are the effective masses of the conduction and valence band, and λþ and λ are the spin–orbit coupling of the conduction and valence bands, respectively. The presence of a finite strain induces in the Hamiltonian (Eq. (14.19)) many different terms. The most straightforward are the diagonal ones, i.e. a scalar potential, which contains a spin-independent part, D ¼ diag½Dþ ; D , and a spin-dependent contribution, δλ ¼ diag½δλþ ; δλ . The explicit expressions of D and δλ read   2  2  2 D ¼ α 1 jAj þ α2 V þ ωxy þ α3 V , : (14.20)   2 2 s δλ ¼ αs V þ ω2xy þ αs 1 j Aj þ α 2 3 V Note that the strain fields appear in Eqs. (14.19) and (14.20) onlythrough the represen tative variables A ¼ εxx  εyy  i2εxy , V ¼ εxx þ εyy , and ωxy ¼ ∂uy =∂x  ∂ux =∂y =2. The numerical values of all αi are also reported in Table. 14.4. It should be noticed that the quantitative use of the second-order terms in the scalar potential (D and δλ) should be done with some care because, here, we considered the linear approximation to

272

Electronic Properties and Strain Engineering

Table 14.4 Microscopical parameters of the spinful two-band low-energy model. The upper table describes strain-independent Hamiltonian parameters where t 0 ¼ 2:34 eV, α ¼ 0:01 and β ¼ 1:54; the middle table describes the Hamiltonian parameters related to the strain through a scalar potential (Eq. (14.20)); the lower table describes the Hamiltonian parameters ηi related to the strain-induced coupling to the pseudovector potentials Ai (reproduced from [55] with permission of the American Physical Society). Δ0 ¼ 0:11 eV Δ ¼ 1:82 eV λ00 ¼ 17 meV αþ 1 αþ 2 αþ 3

eV 15.99 ‒3.07 ‒0.17 η1 0:002

α 1 α 2 α 3

eV 15.92 –1.36 0.0 η2 56:551

λ0 ¼ 69 meV λ ¼ 81 meV λ0 ¼ 2 meV αsþ 1 αsþ 2 αsþ 3

η3 1:635

meV ‒61 3.2 3.4 η4 1:362

αs 1 αs 2 αs 3

meV ‒5.7 0.02 0.01

η5 8:180

include deformation in the bond lengths. However, these terms would be negligible for small deformation. In addition to the above discussed diagonal terms, it is interesting to underline the appearance in Eq. (14.19) of five different fictitious gauge fields defined as Ai ¼ ηi A, where Ax ¼ ðħ=ea0 ÞRe½A and Ay ¼ ðħ=ea0 ÞIm½A. The coupling constants ηi are evaluated from the values of the initial Slater–Koster parameters, and their specific value for the case of single-layer MoS2 are reported in Table 14.4. Note that, due to the small value of η1 , the off-diagonal pseudo vector potential (A1 ) results are very weak as compared to the diagonal ones. The opposite happens for the well-known cases of mono- and bilayer graphene, for which the off-diagonal terms are the dominant components of the straindependent Hamiltonian [59, 64]. The weakness of A1 in MoS2 might stem from the large energy gap as compared with graphene, which is a gapless semimetal.

14.6

Strain Engineering in Transition Metal Dichalcogenides In the previous sections, we have summarized some of the theoretical tools that can be used to study the electronic properties of TMDs, in particular when these materials are subjected to external strain. To understand such effects is of fundamental importance in order to exploit the potential of these crystals for their application in strain engineering. Indeed, atomically thin semiconducting transition metal dichalcogenides have shown, apart from their outstanding figures-of-merit and electrical performances, very interesting mechanical properties, unmatched by conventional 3D semiconductors. Application of different kinds of strain is possible in the laboratory with the use of specific experimental techniques, as reviewed in [37]. In particular, nanoindentation experiments on freely suspended single-layer MoS2 have demonstrated mechanical properties approaching those predicted by Griffith for ideal brittle materials in which

14.6 Strain Engineering in Transition Metal Dichalcogenides

273

the fracture point is dominated by the intrinsic strength of the atomic bonds and not by the existence of defects [65]. Contrary to conventional 3D semiconductors like silicon, which typically breaks at strain levels of 1:5%, two-dimensional MoS2 does not break until >10% strain levels, and it can be folded and wrinkled almost at will [37]. Strain engineering involves the study of how the physical properties of materials can be tuned by controlling the external strain fields applied to them. The outstanding stretchability of transition metal dichalcogenides makes them excellent candidates for strain engineering due to the unprecedented tunability levels predicted for these materials that could lead to straintronic devices – devices with electronic and optical properties that are engineered through the application of mechanical deformations. For instance, single-layer MoS2 is expected to undergo a direct-to-indirect gap transition at 2% of tensile uniaxial strain, and a semiconducting-to-metal transition at 1015% of tensile biaxial strain [11, 16, 19]. Such huge tunability of the electronic band gap (from 1:8 eV to 0 eV in single-layer MoS2) can be realistically exploited in thin layers of transition metal dichalcogenides, since these materials can withstand such levels of strain without rupture. It is interesting to compare this behavior with the poor window of tunability of only 0.25 eV achieved for strained silicon at 1.5% biaxial strain, due to the breaking of the bonds beyond that strain value [66]. Another possibility of atomically thin semiconductors is that one can deform them only at a specific region, making it possible to generate localized profiles of strain. A very interesting possibility of local strain engineering is to create a continuous bending of the energy levels of electrons, holes,and excitons when a MoS2 membrane is subjected to a point deformation at its centre. Upon illumination electron–hole pairs are generated and, due to the large exciton binding energy, they are forced to hold together and migrate towards the centre of the membrane, where the strain is maximum and the band gap is minimum [11]. Therefore, strain gradients can be used to concentrate excitons in a small region of the crystal, which can be of interest for fundamental physics (e.g. exciton condensates) and applications (e.g. solar energy funnel). The possibility to create an exciton funnel has been demonstrated experimentally by two different techniques. In [23], atomically thin MoS2 layers are subjected to large local strains of up to 2.5% induced by controlled delamination from an elastomeric substrate. The use of simultaneous scanning Raman and photoluminescence imaging allowed for spatially resolved direct band-gap reduction of up to 90 meV induced by local strain. A funnel effect was observed, in which excitons drift hundreds of nanometers to lower band-gap regions before recombining, demonstrating exciton confinement by local strain. In [67], it has been created an optoelectronic crystal of artificial atoms in strain-induced single-layer MoS2 with a non-uniform strain profile. The sensitivity of transition metal dichalcogenides to mechanical deformations can be seen in the response of the phonon structure to strain. The Raman spectra of these materials contain two main peaks corresponding to the A1g out-of-plane mode, where the top and bottom X atoms move out of plane in opposite directions while M is fixed, and the E12g in-plane mode where the M and X atoms move in-plane in opposite

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Electronic Properties and Strain Engineering

directions [68, 69]. The E12g mode is especially sensitive to applied strain. The degeneracy of this mode is lifted under uniaxial strain, leading to red shifting and splitting into two distinct peaks for strain of just 1% [23, 70, 71]. In Fig. 14.7, we show the correlation between the measured Raman shift of the E12g mode and the strain estimated from the analysis of the AFM topography. The black squares show the results reported in [72] (which follows a linear relationship with a slope 1:7 cm1 per % strain) obtained by applying uniform tensile strain to few-layer MoS2. The change in the size of the direct band gap as a function of the strain is shown in Fig. 14.7(b). The change in the direct band gap is obtained from the shift of the A exciton in the photoluminescence spectra, while the strain can be estimated from the shift of the E12g Raman mode. The modification of the direct gap obtained from the tight-binding model introduced in the previous sections is shown by the solid line in (b). The tightbinding theory describes the order of magnitude of the strain-induced change in the gap, as seen by the coincidence between the calculated and measured extremal points of the distribution; however, there are also many points that fall below the solid line. This is a consequence of the funnel effect [11], which in the present context means that, while the Raman signal represents an average over the whole laser spot area, the photoluminescence gives only information about the minimum gap in the region illuminated by the laser (Fig. 14.7(c)). When including this funnel effect, the local strain tight-binding model fits very well with the experimental results (dashed lines in Fig. 14.7(b)). Apart from the modification of the band structure, strain engineering can exploit the piezoelectric properties of 2D TMDs, converting mechanical to electrical energy. Piezoelectricity is a well-known effect in which stretching or compressing a material causes it to generate an electrical voltage, or the reverse, in which an external voltage leads to contraction or expansion of the crystal membrane. To be piezoelectric, a material must break inversion symmetry. A single atomic layer of MX 2 has inversion symmetry broken, and should be piezoelectric. However, in bulk MX 2 , successive layers are oriented in opposite directions (see Fig. 14.1), and generate positive and negative voltages that cancel each other out and give a zero net piezoelectric effect. This effect has been proven in [38], where strain is induced on exfoliated MoS2 single-layer flakes when the polymer substrate is mechanically bent. The piezoelectric response can be studied by coupling the device to an external load resistor, finding that the straininduced polarization charges at the sample edges can drive the flow of electrons in an external circuit. The electrons flow back in the opposite direction when the substrate is released. When strain is applied in the armchair direction, a positive (negative) voltage and current is generated with increasing (decreasing) strain. This demonstrates the conversion of mechanical energy into electricity [38]. Since one of the necessary conditions for the existence of a piezoelectric effect is the lack of inversion symmetry, when the experiment is done with bilayer or bulk samples, the piezo response is not observed. Therefore, single-layer and few-layer samples of TMDs, with an odd number of layers, present a strong intrinsic piezoelectric response, and are excellent candidates for piezotronics applications in electromechanical sensing, wearable technologies, and implanted devices.

14.6 Strain Engineering in Transition Metal Dichalcogenides

275

Fig. 14.7 Strain tuning the direct band-gap transitions of MoS2. (a) Measured Raman shift of the E12g mode. (b) Change in the energy of the direct band-gap transition as a function of the strain. The solid line in (b) is the band gap versus strain relationship calculated for wrinkled MoS2 ribbons with different levels of maximum strain employing the tight-binding model discussed in the text. The dashed lines show the expected band gap versus strain relationship after accounting for the effect of the finite laser spot size and the funnel effect. (c) Schematic diagram explaining the funnel effect due to the non-homogeneous strain in the wrinkled MoS2 that consists of three processes. (i) Illumination creates excitons in the sample. (ii) The photogenerated excitons drift in the wrinkled sample towards the region with minimal band gap. (iii) The concentrated excitons recombines giving emissions with longer wavelengths (adapted with permission from [23]).

Acknowledgments We acknowledge E. Cappelluti, H. Rostami, J. Silva-Guillén, P. Ordejón, M. P. LópezSancho, H. Ochoa, R. Asgari and A. Castellanos-Gomez for fruitful collaboration in the topics considered in this chapter. Financial support from MINECO (Spain) through

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Electronic Properties and Strain Engineering

grant FIS2014-58445-JIN, and from Project No. PIB2010BZ-00512, the Comunidad Autónoma de Madrid (CAM) MAD2D-CM Program (S2013/MIT-3007), is acknowledged.

14.7

References [1] K. S. Novoselov, D. Jiang, F. Schedin, T. J. Booth, V. V. Khotkevich, S. V. Morozov, and A. K. Geim, Proceedings of the National Academy of Sciences of the United States of America 111, 6198; 102, 10451 (2005). [2] Q. H. Wang, K. Kalantar-Zadeh, A. Kis, J. N. Coleman, and M. S. Strano, Nature Nanotechnology 7, 699 (2012). [3] D. Jariwala, V. K. Sangwan, L. J. Lauhon, T. J. Marks, and M. C. Hersam, ACS Nano 8, 1102 (2014). [4] R. Ganatra and Q. Zhang, ACS Nano 8, 4074 (2014). [5] E. Cappelluti, R. Roldán, J. A. Silva-Guillén, P. Ordejón, and F. Guinea, Physical Review B 88, 075409 (2013). [6] K. F. Mak, C. Lee, J. Hone, J. Shan, and T. F. Heinz, Physical Review Letters 105, 136805 (2010). [7] W. Zhao, R. M. Ribeiro, M. Toh, A. Carvalho, C. Kloc, A. H. Castro Neto, and G. Eda, Nano Letters 13, 5627 (2013). [8] W. Jin, P.-C. Yeh, N. Zaki, D. Zhang, J. T. Sadowski, A. Al-Mahboob, A. M. van der Zande, D. A. Chenet, J. I. Dadap, I. P. Herman, et al., Physical Review Letters 111, 106801 (2013). [9] Y. Zhang, T.-R. Chang, B. Zhou, Y.-T. Cui, H. Yan, Z. Liu, F. Schmitt, J. Lee, R. Moore, Y. Chen, et al., Nature Nanotechnology 9, 111 (2014). [10] A. Castellanos-Gomez, E. Cappelluti, R. Roldán, N. Agraït, F. Guinea, and G. RubioBollinger, Advanced Materials 25, 899 (2013). [11] J. Feng, X. Qian, C.-W. Huang, and J. Li, Nature Photonics 6, 866 (2012). [12] P. Lu, X. Wu, W. Guo, and X. C. Zeng, Physical Chemistry Chemical Physics 14, 13035 (2012). [13] H. Pan and Y.-W. Zhang, The Journal of Physical Chemistry C 116, 11752 (2012). [14] H. Peelaers and C. G. Van de Walle, Physical Review B 86, 241401 (2012). [15] E. Scalise, M. Houssa, G. Pourtois, V. Afanas’ev, and A. Stesmans, Physica E: Lowdimensional Systems and Nanostructures 56, 416 (2014). [16] E. Scalise, M. Houssa, G. Pourtois, V. Afanas’ev, and A. Stesmans, Nano Research 5, 43 (2012). [17] W. S. Yun, S. Han, S. C. Hong, I. G. Kim, and J. Lee, Physical Review B 85, 033305 (2012). [18] Y. Li, Y.-L. Li, C. M. Araujo, W. Luo, and R. Ahuja, Catalysis Science and Technology 3, 2214 (2013). [19] M. Ghorbani-Asl, S. Borini, A. Kuc, and T. Heine, Physical Review B 87, 235434 (2013). [20] H. Shi, H. Pan, Y.-W. Zhang, and B. I. Yakobson, Physical Review B 87, 155304 (2013). [21] L. Hromadová, R. Martoňák, and E. Tosatti, Physical Review B 87, 144105 (2013). [22] S. Horzum, H. Sahin, S. Cahangirov, P. Cudazzo, A. Rubio, T. Serin, and F. M. Peeters, Physical Review B 87, 125415 (2013).

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15

Valley-Spin Physics in 2D Semiconducting Transition Metal Dichalcogenides Hongyi Yu and Wang Yao

15.1

Introduction Monolayer group-VIB transition metal dichalcogenides (TMDs) have recently emerged as a new class of semiconductors with appealing properties. The compounds have the chemical composition of MX2, where M stands for the transition metal element Mo or W, and X for the chalcogen element S or Se. The bulk crystals of the group-VIB TMDs are of the layered structures where the elementary unit, i.e. monolayer, is an X–M–X covalently bonded quasi-2D hexagonal lattice [1]. The monolayers are loosely bound together by the weak van der Waals force to form the bulk crystals. Monolayer TMDs can be extracted from bulk crystals by mechanical exfoliation [2–4], or synthesized using chemical vapor deposition or molecular beam epitaxy [5–10]. When TMDs are thinned down to the monolayer, the band gap crosses from an indirect one to the direct one [2, 3, 6, 11, 12], which is in the visible frequency range. The conduction and valence band edges in the monolayers are both located at the corners of the hexagonal Brillouin zone (BZ), i.e., the K and
K points that are related by time reversal. The two degenerate but inequivalent band extrema constitute a discrete index for low-energy electrons and holes, known as the valley pseudospin. Similar to the use of spin in spintronics, this valley pseudospin of carriers may be utilized to encode information in electronic devices, which has led to the concept of valley-based electronics (or valleytronics) that has been explored in various materials with multivalley band structures [13–22]. Monolayer TMDs have provided a remarkable new laboratory for the exploration of valley physics and valleytronic applications. Key to the utilization of carriers’ internal quantum degree of freedom for information processing is the capability to distinguish between states that represent different pieces of information and to control the dynamics in the Hilbert space spanned by these states. This requires the internal quantum degree of freedom to be associated with measurable physical properties, allowing it to be coupled to external perturbations for measurement and control. In the 2D hexagonal crystals, it was shown earlier using graphene models that, when inversion symmetry is broken, the two degenerate valleys can be distinguished by pseudovector quantities, such as the Berry curvature and the magnetic moment, which must take opposite values at the time-reversal pair of valleys [15, 16]. The valley contrasted Berry curvatures and magnetic moments can couple to external electric and magnetic fields, respectively, 279

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giving rise to the valley Hall effect and valley Zeeman effect. Because of the inversion symmetry breaking, optical interband transitions at the time reversal pair of valleys can also acquire valley-dependent transition selection rules. These valley-dependent phenomena make possible the manipulation of valley pseudospin via electric, magnetic, and optical means, and they have all been discovered in monolayer TMDs where the lattice has structural inversion asymmetry. Moreover, the strong spin–orbit coupling from the transition metal elements gives rise to an effective interaction between the valley pseudospin and the spin, making possible the interplay between these two degrees of freedom, and allowing spin manipulations via the valley phenomena. In this chapter, we give an overview of the valley and spin physics in monolayer TMDs. The content is organized as follows. Section 15.2 briefly introduces the valleydependent electronic structures at the band edges. Section 15.3 presents the various valley and spin-dependent phenomena in monolayer TMDs, including the valley and spin optical selection rules, the valley Zeeman effect, the valley and spin Hall effects, and the non-linear valley and spin currents from triogonal warping. In Section 15.4, we explain how the monolayer valley and spin physics can be inherited in bilayers. Our emphasis is on the basic physics and theoretical framework, while more details on the related experimental discoveries have been covered in other chapters of this collection.

15.2

Electronic Structure at the Band Edges Figures 15.1(a) and (b) illustrate the quasi-2D hexagonal lattice of a TMD monolayer [1]. The M atom is coordinated by the six neighboring X atoms in a trigonal prismatic geometry. With the trigonal prismatic coordination, the structural inversion symmetry is broken in the monolayer, key to the emergence of the valley-dependent physics discussed here. There is mirror symmetry in the out-of-plane ðzÞ direction, which, together with the time reversal symmetry, dictates an effective form of the coupling between the valley and the spin degrees of freedom. First-principles calculations have found that the band edge Bloch wave functions near the conduction band minimum (valence band maximum) at K are predominantly from
 2 2 2 the M dz dx
y  id xy orbital which is the eigenstate of the angular momentum ^ z with magnetic quantum number m ¼ 0 ðm ¼ 2Þ [23], plus a small contrioperator L bution from the X px and py orbitals [1, 24] (Fig. 15.1(d)). The K points are highsymmetry points invariant under the C 3 operation (2π=3 in-plane rotation). Thus the 2 Bloch functions at K are the eigenstates of C 3 with the eigenvalues ei3lπ , where l is an integer that depends on the choice of the rotation center (M, X, or h the hollow center of the hexagon formed by M and X, see Fig. 15.1(a)) [1]. In Table 15.1, the orbital contributions and the quantum number l are given for the conduction and valence band edge Bloch functions at the K point, while their counterparts at the
K point can be obtained by taking time reversal. The C 3 rotational symmetry, of the band edge Bloch functions, b underlies the valley optical selection rule in the monolayer and the interlayer hopping properties in the bilayer, to be discussed in the following sections.

15.2 Electronic Structure at the Band Edges

281

Table 15.1 Rotational symmetry and orbital compositions of conduction and valence band Bloch functions at K point in monolayer TMDs. The Bloch functions are eigenstates of the C 3 rotation with 2 eigenvalue e i 3l π , where l depends on the choice of rotation center that can be an M or X site, or the hollow center h of the M–X hexagon (Fig. 15.1(a)).

CBM VBM

Md

Xp

l (M)

l (X)

l (h)

d z2 d x2
y2 þ idxy

px
ipy px þ ipy

0 +1

+1 –1

–1 0

Fig. 15.1 (a) Monolayer group-VIB transition metal dichalcogenides from the top view. Large (small) spheres denote metal (chalcogen) atoms. (b) Trigonal prismatic coordination around a metal atom. The top view of this triangular prism is denoted by the triangle in (a). (c) The hexagonal first Brillouin zone, and the reciprocal lattice vectors G1 and G2. (d) Band structures of the MoS2 monolayer from first-principles calculations without SOC, with the orbital composition illustrated. The metal d-orbitals and chalcogen p-orbitals are denoted by dots of different symbols, with the dot size proportional to the orbital weight in the corresponding state. (e) Schematics of the SOC splitting at the conduction and valence band edges at K and
K in MoX2 and WX2. Dashed gray (solid black) denotes the spin down (up) bands. ((a)–(c) Reproduced from [1] with permission from the Royal Society of Chemistry; (d) partly adapted with permission from [25]. Copyright 2013, American Physical Society.)

TMDs have a strong spin–orbit-coupling (SOC) originating from the M-d orbitals [1, 24, 26]. The form of the spin–orbit splitting at the band edges is dictated by the symmetries. With the mirror symmetry about the M-atom plane, the Bloch states in the monolayer are invariant under the mirror reflection operation. Considering the fact that mirror reflection of an in-plane spin vector is its opposite, while that of an out-of-plane spin vector is itself, the Bloch states must have their spin either parallel or antiparallel to the out-of-plane ðzÞ direction, i.e. the SOC splitting must be in the z direction. Besides, time reversal symmetry requires the spin splitting to have opposite sign at an arbitrary

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pair of momentum space points k and
k. Thus in the neighborhood of K and
K, the SOC manifests as an effective coupling between the spin component S^z ð¼ 1=2Þ and valley pseudospin component τ z (τ z ¼ 1 for K valley) [1, 24, 26] ^ SOC ¼ λτ z S^z : H

(15.1)

First-principle calculations have found that the valence band at the K point has an SOC splitting of λv e 0:15 eV in the MoX2 monolayer and λv e 0:45 eV in the WX2 monolayer [24, 26–28]. Because of this giant SOC splitting, the valence band edge in monolayer TMDs has the spin index locked with the valley index, i.e. the K ð
KÞ valley has only the spin up (down) low-energy holes, which are formed by removing spin down (up) valence electrons (Fig. 15.1(e)). Compared to the valence band edge from the d x2
y2  id xy orbital that has a finite magnetic quantum number m ¼ 2, the conduction band edge is from the dz2 orbital with m ¼ 0. Thus the on-site spin–orbit coupling vanishes to the leading order. First-principles calculations find that the magnitude of conduction band spin splitting λc is a few meV for MoS2 and tens of meV for MoSe2, WS2, and WSe2 [24, 25, 29–37], which originates from the small X-p orbital compositions, as well as second-order effect mediated by the remote conduction band consisting of M-d xz  id yz orbitals. The two contributions to SOC have opposite signs and their competition leads to the sign difference of λc between MoX2 and WX2 (Fig. 15.1(e)) [24, 25, 30]. The simplest theoretical model that gives a reasonably good description to the top valence band and lowest conduction bands is a three-band tight-binding model [25], constructed with the three M-d-orbitals only, i.e. dz2 and dx2
y2  id xy , which are the major orbital compositions of the band edge Bloch functions in K valleys. Including the M–M nearest-neighbor hopping only, the three-band tight-binding model describes the dispersions and Berry curvatures of conduction and valence bands near the K points [25]. By including up to the third-nearest-neighbor hopping, all three bands agree well with the first-principles ones in the entire BZ [25]. This model has been applied to the study of edge states in TMD nanoribbons [38], quantum dots formed by lateral confinement potential in the extended TMD monolayer [39], intercellular orbital magnetic moment [40, 41], magnetoelectronic and optical properties [42], and magnetoluminescence [43]. Expanding the tight-binding model in the neighborhood of the K points, the kp model for describing the band edge physics can be obtained. By expanding the above three-band model and keeping to the linear order in k, a minimum description of the conduction and valence band edges in τ z K valley has a rather simple form [26]
 ^ kp ¼ at τ z kx σ^ x þ k y σ^ y þ Δ σ^ z
λv τ z S^z : H 2

(15.2)

Here σ^ x=y=z is the Pauli matrix spanned by the conduction and valence states at the τ z K point, S^z is the Pauli matrix for spin, a is the lattice constant, t is the effective hopping integral, k is the wavevector measured from τ z K, and λv is the valence band SOC splitting. This two-band kp model is in fact the massive Dirac fermion model,

15.3 Valley-Spin Physics in Monolayers

283

which captures the low-energy electronic structures in the K valleys of monolayer TMDs, including the band dispersion, the Berry curvatures, and the giant SOC splitting of the valence band. The τ z terms explicitly account for the valley dependence of the electronic structures at the band edges, which lead to the valley Hall effect, valley magnetic moment, and the valley-dependent optical selection rule which will be discussed in the following sections. The two-band kp model has been widely used to study various properties of TMD monolayers because of its simplicity. In the meantime, the simplicity of this model inevitably imposes some limitations on its applications. For example, it cannot account for the electron–hole asymmetry and the trigonal warping of band dispersion. The two effects can be accounted for by including terms quadratic in k [24, 25, 29, 44]. The corrected models with high-order terms have been used to study optical conductivity [45], magneto-optical properties [46], plasmons [47], and spin relaxation [48]. The small conduction band spin splitting at K, absent in the three-band tight-binding model and hence its kp expansion, can also be added as a correction to Eq. (15.2) if needed.

15.3

Valley-Spin Physics in Monolayers

15.3.1

Valley and Spin Optical Selection Rules The optical selection rules for interband transitions at K points are the direct consequence of the C3 rotational symmetry of Bloch functions discussed earlier (cf. Table 15.1) [1, 26, 49]. The interband optical transition matrix element satisfies i3π ðlc
lv ∓1Þ ^  C
1 ^  jψc, K i ¼ hC 3 ψv, K jC 3 P ^  jψc, K i: hψv, K jP hψv, K jP 3 jC 3 ψc, K i ¼ e 2

(15.3) ^  C
1 ^  . A finite ^  P ^ x  iP ^ y is the momentum operator and C 3 P P Here P 3 ¼ e ^ hψv, K jP  jψc, K i then requires lc
lv ∓ 1 ¼ 3N with N being an integer. According to the lc and lv values summarized in Table 15.1, the interband optical transitions have the valley optical selection rules, i.e. coupled to the σ þ ðσ
Þ polarized light only at K ð
KÞ [26] (Fig. 15.2(a)). Owing to the giant SOC splitting in the valence band, the valley optical selection rule also becomes a spin optical selection rule depending on the excitation frequency [26] (Fig. 15.2(a)). Optical excitations of electron–hole pairs with four distinct combinations of their spin and valley indices can be selectively realized through the choice of the polarization and frequency of the excitation light. In principle, such a selection rule is rigorous only at the high-symmetry  K points. Near the K hψv, τ Kþk jP ^  jψc, τ Kþk i2 ¼ points, the kp model in Eq. (15.2)
leads to [26] z z  2 2 m2e aħ2t ð1  τ z cos θÞ2 , where θ ¼ arctan 2kat and Δ0  Δ
λv τ z Sz . With the sizeable Δ0 band gap Δ0 , the valley optical selection rule therefore holds in the neighborhood of K points. First-principles calculations have also shown that the selection rule holds approximately true in a sufficiently large neighborhood of K points [49]. ∓i23π

284

Semiconducting Transition Metal Dichalcogenides

Fig. 15.2 (a) Valley and spin optical transition selection rules. (b) Optical orientation of excitonic valley pseudospin. A bright exciton in the K valley corresponds to valley pseudospin up (τ z ¼ þ1, north pole on the Bloch sphere), which couples exclusively to σ+ circularly polarized photon, while bright exciton in the –K valley corresponds to valley pseudospin down (τ z ¼
1, south pole on the Bloch sphere), which couples to σ
circularly polarized photon. The Bloch sphere equator corresponds to an equal superposition of the two valleys (in-plane valley pseudospin), which couples to linearly polarized photon with the polarization indicated by double arrows (reproduced from [50] with permission from Oxford University Press). (c) Valleydependent Zeeman shifts of the band edges, from three contributions: solid black arrows denote the spin contribution, shaded for the Berry phase contribution, empty for the atomic orbital contribution. See text for explanation. The dashed (solid) curves are the bands under zero (finite) magnetic field. (Adapted from [41]. Copyright 2014, Nature Publishing Group.)

15.3.2

Optical Orientation of Excitonic Valley Pseudospin Through the interband transition, the absorption of a photon excites a valence electron to the conduction band, leaving a vacancy in the filled valence band that is effectively described as a hole. The attractive Coulomb interaction between the negatively charged electron and the positively charged hole leads to their binding into a hydrogen-like bound pair, known as the exciton. Exciton plays a key role in optoelectronic phenomena in monolayer TMDs [50]. It can capture an extra electron or hole to form a negatively or positively charged exciton, also known as trion. The measured trion binding energy, i.e. the difference between the exciton and trion resonances in the absorption or photoluminescence, is of the order of tens of meV, about one order of magnitude larger than that in GaAs quantum wells. This points to an exceptionally large exciton binding energy of the order of hundreds of meV [51–54], which is also predicted by firstprinciples calculations [28, 33, 55–58], and jointly revealed by various measurements including the reflection spectra [59], two-photon absorption [60–63], and scanning tunneling microscopy/spectroscopy [10, 64–66]. Such strong Coulomb binding is due to the large effective masses of both electron and hole and the reduced dielectric screening in the 2D geometry [56, 59, 61, 67]. Meanwhile, the exciton wave function is largely the Wannier type, with electron and hole both well localized near the K points in the momentum space [56, 61]. An exciton can then be classified by the valley

15.3 Valley-Spin Physics in Monolayers

285

configuration of its electron and hole constituents. Due to the momentum conservation, the bright exciton that can radiatively recombine must have its electron and hole in the same valley. The two degenerate valley configurations of bright excitons can then be described by the excitonic valley pseudospin τ, where τ z ¼ þ1 and τ z ¼
1 correspond, respectively, to the exciton being in the K and
K valley (Fig. 15.2(b)). These bright excitons inherit the valley optical selection rules of the band-to-band transitions at K points, allowing their valley specific interconversion with photons of selected helicity: exciton with valley pseudospin up (down) τ z ¼ þ1ð
1Þ can be interconverted with a σ þ ðσ
Þ circularly polarized photon (Fig. 15.2(b)). A linearly polarized photon is a coherent superposition of a σ þ polarized photon and a σ
polarized one, which can then interconvert with an exciton in a coherent superposition of being in the K and
K valley, transferring the optical coherence to valley quantum coherence [51]. Such a bright exciton carries an in-plane valley pseudospin, i.e. on the equator of the Bloch sphere representation of the excitonic valley pseudospin (Fig. 15.2(b)). Different points on the equator then correspond to photon linear polarization along different axis in the monolayer plane. The correspondence between the valley polarization/coherence and the photon polarization has been exploited for optical orientation of the excitonic valley pseudospin using circularly or linearly polarized excitation, as demonstrated in experiments of polarization-resolved photoluminesence [49, 51, 68–72]. The valley selective exciton–photon coupling has also enabled the pump–probe study of valley dynamics [73–77], and the valley selective optical Stark effect [78, 79] at different monolayer TMDs.

15.3.3

Valley Magnetic Moment and Valley Zeeman Effect The magnetic moment of a Bloch electron characterizes its response to the magnetic field. In monolayer TMDs, the time reversal symmetry between K and
K requires their magnetic moment to have identical magnitude but opposite signs. An out-of-plane magnetic field then lifts the valley degeneracy, leading to a valley Zeeman effect. For the spin–valley locked band edges, the overall valley Zeeman shift has three contributions [40, 41, 80, 81] (Fig. 15.2(c)). The first is from the spin magnetic moment. The second is the atomic orbital magnetic moment. In monolayer TMDs, the
 conduction (valence) band in the K valley mainly consists of M-d z2 dx2
y2  id xy orbitals with the magnetic quantum m ¼ 0 ðm ¼ 2Þ. This contributes to a Zeeman shift of 0 and 2μB B for the conduction and valence band, respectively. The third is a lattice contribution (i.e. vanishes when the hopping between the lattice sites is turned off), which is also associated with the Berry phase effect of the Bloch electrons [82]. This lattice contribution is given by [16, 83] mn, k ¼
i

^ jui, k i  hui, k jp ^ jun, k i eħ X hun, k jp , 2m2 i6¼n E n, k
E i, k

(15.4)

where un, k is the periodic part of the Bloch function of the nth band at wavevector k, and En, k is the band dispersion.

286

Semiconducting Transition Metal Dichalcogenides

The valley Zeeman shift of the conduction and valence band edges can manifest in the energy cost of generating an electron–hole pair in a specific valley, so the two valley bright exciton states will also have a valley Zeeman splitting, making possible magnetocontrol of the excitonic valley pseudospin dynamics. With the valley optical selection rule, the exciton valley Zeeman splitting may be detected from the polarization-resolved PL measurement, where the magnetic field is expected to shift in opposite directions with the σ þ and σ
polarized PL peaks that correspond to the bright exciton resonances in valley K and
K, respectively [40, 41, 80, 81] (Fig. 15.2(c)). The exciton valley Zeeman splitting measures the difference between the Zeeman shifts of the conduction and valence band edges, so the three contributions to the magnetic moment of the spin–valley locked band edge carriers are partially inherited. Firstly, the spin magnetic moment does not affect the optical resonances, because optical transitions conserve spin so that the shift of the initial and final states due to the spin magnetic moment is identical. Secondly, with
the conduction (valence) band in the K valley  mainly consisting of M-dz2 dx2
y2  id xy orbitals with the magnetic quantum m ¼ 0 ðm ¼ 2Þ, the magnetic moment of the atomic orbital is a major contribution to the valley Zeeman shift of the exciton resonance (Fig. 15.2(c)). Lastly, we note the lattice contribution to the magnetic moment also results in a finite exciton valley Zeeman shift. Within the minimum two-band massive Dirac fermion model of Eq. (15.2), the electron and hole magnetic moment associated with the Berry phase effect exactly cancel because of the particle–hole symmetry. Nevertheless, corrections beyond the two-band model result in a finite difference for the electron and hole magnetic moment, and it is this difference, as well as the atomic orbital contribution, that is measured by the splitting of the σ þ and σ
PL peaks [40, 41].

15.3.4

Berry Phase Effect and Valley Hall Current The Berry phase effect for a particle lies in the dependence of the internal structure on the dynamical parameter [82]. In the context of Bloch electrons, it lies in the dependence of the periodic part of the Bloch function un, k on the wavevector k. Consider a wavepacket of Bloch electrons moving adiabatically in a non-degenerate energy band with band index n. Under smooth perturbations, it is possible to construct the wavepacket with a length scale that is small compared to that of the external perturbation but large compared to the lattice constant. One can then speak simultaneously of the central position of the electron in real and in momentum space, and the semiclassical equation of motion for the Bloch electron is then written [15, 83] r_ ¼

1 ∂E n, k _
k  Ωn, k , ħk_ ¼
eE
er_  B, ħ ∂k

(15.5)

Ω is a pseudovector that captures the Berry phase effect in the Bloch band, known as the Berry curvature [16, 83] Ωn , k ¼ i

^ jui, k i  hui, k jp ^ jun, k i ħ2 X hun, k jp : 2 2 m i6¼n ðEn, k
E i, k Þ

(15.6)

15.3 Valley-Spin Physics in Monolayers

287

Obviously, in an applied electric field the Ω term corresponds to an anomalous velocity perpendicular to the field, i.e. a Hall effect. The Berry curvature has a form similar to the lattice contribution of the magnetic moment m (cf. Eq. (15.4)), so non-zero Berry curvature in general implies a finite magnetic moment, and vice versa. In monolayer TMDs, the K valley conduction and valence bands can be described by the massive Dirac fermion model, which together with the inversion symmetry breaking give rise to sizeable Berry curvatures with opposite signs at the two valleys [15, 16, 26] Ωv, k ¼
Ωc, k ¼ τ z 

2a2 t 2 Δ0 02

Δ þ

4a2 t 2 k2

0 3=2 , Δ  Δ
λv τ z Sz :

(15.7)

,In the presence of an in-plane electric field, the valley contrasting Ω gives rise to a Hall current of the carriers with the sign depending on the valley index [16, 26, 84] (Figs. 15.3(a) and (b)), i.e. a valley Hall effect. The valley Hall effect is an analog of the spin Hall effect [85–87], but with the valley pseudospin playing the role of spin. For hole doped systems, because the spin index is locked with the valley index for the band edge holes, the valley Hall effect is at the same time a spin Hall effect (Fig. 15.3(a)). For n-doped monolayers, the finite conduction band spin splitting can lead to different populations of spin up and down electrons in a given valley (cf. Fig. 15.1(e)), and the spin dependence of Δ0 in Eq. (15.7) also results in different magnitudes of their Berry curvatures. As a result, the valley Hall effect of electrons is also accompanied by the spin Hall effect (Fig. 15.3(b)). Passing a longitudinal current through the monolayer, the valley Hall effect in the 2D bulk will lead to valley polarization on the edges, which can be detected from the Kerr rotation of an incident linearly polarized light based on the valley optical selection rules [88]. The valley contrasting Berry curvature and the valley optical selection rules together can further lead to a Hall effect that can be detected from electrical measurement [16]. Optical field with σ þ ðσ
Þ circular polarization selectively excite electrons and holes in the K ð
KÞ valley. Driven by an in-plane electric field, these photoexcited electrons

Fig. 15.3 (a) Valley Hall effect of holes in TMDs monolayers, which is a spin Hall effect at the same time because of the spin–valley locking of holes. (b) Valley Hall effect of electrons, also accompanied by a spin Hall effect (see text). (c) Valley Hall effect under the excitation by a circularly polarized light. The valley polarized electron and hole tend to move to the opposite edges, contributing with the same sign to the Hall voltage. (Adapted with permission from [26]. Copyright 2013, American Physical Society.)

288

Semiconducting Transition Metal Dichalcogenides

and holes will acquire opposite transverse velocities because of the conduction and valence band Berry curvatures, which then move to the two opposite edges, contributing the same sign to the Hall voltage [16, 26] (Fig. 15.3(c)). Such Hall effects of photo injected carriers have already been observed in monolayer MoS2 and WS2 [89, 90].

15.3.5

Non-Linear Valley-Spin Current Response from Trigonal Warping The valley Hall effect is a linear response to the electric field. The valley current in the transverse direction is always accompanied by the much larger longitudinal charge current, a major cause of dissipation that cannot be removed as it has the same linear dependence on the field. Here we show that there exists another mechanism for generating bulk valley current in monolayer TMDs, which is a quadratic response to the electric field that makes possible current rectification to generate dc valley currents by ac electric field, with the absence of net charge current. The effect arises from the trigonal warping of the band edges at the K valleys, which corrects the parabolic dispersion to [29] Eτz , k ¼

ħ2 k 2 ð1 þ τ z βk cos 3θÞ: 2m∗

(15.8)

Here k  ðk cos θ; k sin θÞ is the wavevector measured from the band edge τ z K point, m∗ is the effective mass, and β characterizes the degree of warping. The valley index τ z in the trigonal warping term ensures the time reversal symmetry between the K and
K valleys, which have opposite trigonal warping (see Figs. 15.4(a) and (b)). The trigonal warping term couples the valley index to the momentum, so the current response to an in-plane electric field becomes valley dependent. This gives rise to a finite valley current jv , defined as the difference between the current responses in the two valleys [91]. For the spin–valley locked band edge carriers in monolayer TMDs, the valley current is also a spin current. Under the relaxation time approximation and in the limit of weak intervalley scattering, such valley current arises in the second order of the applied electric field [91] jv / βjkd j2 ð cos 2θE ;
sin 2θE Þ:

(15.9)

Here kd ¼ eτ p E=ħ is the Fermi surface displacement under the electric field E  ðE cos θE ; E sin θE Þ, with θE the angle between the field direction and the zig-zag crystalline axis, and τ p the momentum relaxation time. Figure 15.4(c) shows the angle dependence of the valley current direction on both the field direction and the crystalline axis. For E in the zig-zag (armchair) direction, the non-linear valley current is along (perpendicular to) the field (see Figs. 15.4(a) and (b)). In an ac electric field, the dc charge current is absent, while there is a dc valley current, as the valley currents in opposite electric fields point in the same direction (Fig. 15.4(c)). From the quadratic dependence of jv on kd in Eq. (15.9), it is clear that the non-linear valley current favors large mobility. Taking a mobility value of e1000 cm2 V
1 s
1 that has been achieved at low temperature [92, 93], we estimate the non-linear valley current magnitude starts to exceed that of the valley Hall current at an electric field of e10 mV μm
1 .

15.4 Valley and Spin Physics in Bilayers

289

Fig. 15.4 (a) Displacements of Fermi pockets at K (right) and –K (left) by an electric field along the

zig-zag direction. The thick arrow on top marked as jþ ðj
Þ denotes the current from the Fermi pocket K (
K). The valley (spin) current jv  jþ
j
is parallel to the field. The thin arrows illustrate the group velocity on the displaced K (
K) valley Fermi surface. (b) The electric field along the armchair direction. The valley (spin) current is perpendicular to the field. (c) The dependence of the spin/valley current direction (large arrow on the circle) on the relative angle θE between the field (small arrow on the circle) and the crystalline axis. (Adapted with permission from [91]. Copyright 2013, American Physical Society.)

We summarize here the key differences between the two mechanisms for bulk valley current generation. The origin of the non-linear valley current is valley-dependent dispersion, arising from the Fermi surface anisotropy for a time-reversal pair of valleys, while the valley Hall current is induced by the finite Berry curvatures of the carriers. The valley Hall effect requires inversion symmetry breaking, which is not necessary for the non-linear valley current studied here. The valley Hall current is always perpendicular to the electric field and is independent of the crystalline axis, while the direction of the non-linear valley current is determined by both the direction of the electric field and the crystalline axis.

15.4

Valley and Spin Physics in Bilayers Natural crystals of the four TMDs of interest (MS2 and MSe2) are mostly the 2H stacked ones, where the adjacent monolayers are rotated 180 to each other, with M sites of one layer sitting right on top of the X sites of the adjacent layer. Figure 15.5(a) shows the unit cell of the 2H stacked bilayer and bulk, where inversion symmetry is restored. Albeit less stable than the 2H stacking, 3R is another possible stacking order for the bulk crystals of the four TMDs of interest. 3R stacking has the same monolayer unit as the 2H stacking, while the neighboring layers are simply translations of each other [94, 95] (Fig. 15.5(b)), so the inversion symmetry is still broken.

290

Semiconducting Transition Metal Dichalcogenides

Fig. 15.5 (a) 2H stacking. Top and side views of a bulk unit cell. (b) 3R stacking. (c) Conduction

and valence band edges of two WX2 monolayers in 2H stacking, where the lower layer is the 180 rotation of the upper layer. The dashed double-headed arrows illustrate the spin-conserving interlayer hoppings. The hopping-induced layer hybridization is substantially quenched by the valley and layered-dependent giant spin splitting. ((a) Reproduced from [1] with permission from the Royal Society of Chemistry.)

The valley and spin physics of the monolayer is inherited in different ways in the 2H and 3R stacking. For the 3R stacking, the interlayer hopping simply vanishes at the K points for both the conduction and valence bands, because of the rotational symmetry of the band edge Bloch functions. The hopping matrix element between two adjacent layers write ^ int C
1 ^ int jψL, K i ¼ hC3 ψU, K jC 3 H hψU, K jH 3 jC 3 ψL, K i i23π ðlU
lL Þ ^ int jψL, K i: hψU, K jH ¼e

(15.10)

A non-zero hopping matrix element therefore requires the two quantum numbers lU and lL to be identical. For the 3R stacking, an X-site in the upper layer sits on top of an M-site in the lower layer, which can be taken as the center of the in-plane C 3 rotations in Eq. (15.10). Thus, the values of lU and lL are to be taken, respectively, from the X and M columns in Table 15.1, which are different for either the conduction or the valence band edge Bloch functions. Thus the interlayer hopping vanishes at K for both the conduction and valence bands. The optical selection rules and valley Hall effect of the monolayer are directly inherited in the 3R bilayer and multilayers [95]. For 2H stacking, similar analysis can show that the interlayer hopping vanishes at K in the conduction band [1, 96, 97]. In the valence band, however, the interlayer hopping is finite at K, and its competition with the large spin–valley coupling has interesting consequences. Figure 15.5(c) schematically shows the band edges of two

15.4 Valley and Spin Physics in Bilayers

291

monolayers in 2H stacking order, with dashed arrows illustrating the interlayer hopping that conserves the spin and momentum. The 180 rotation of the lower layer (relative to the upper layer) switches the K valleys. As a result, the sign of the spin splitting becomes dependent on both the valley index and the layer index. Introducing pseudospin χ to denote the layer degree of freedom (χ z ¼ 1 for upper, and χ z ¼
1 for lower layer), the holes at τ z K can be described by ^ 2H ¼ λv τ z S^z χ z þ t ⊥ χ x : H

(15.11)

The first term characterizes the SOC, i.e. the valley and layered-dependent spin splitting, which manifests as an effective coupling between spin, valley, and layer pseudospins. The second term is the interlayer hopping, where first-principle calculations finds t ⊥ in the order of tens meV. The SOC strength λv then corresponds to the energy cost of the interlayer hopping. In WX2, where λv is much larger than t ⊥ , interlayer hybridization is substantially quenched by the giant SOC splitting [1, 12, 98]. Thus in 2H bilayer WX2, both the conduction and valence Bloch states in the K valleys are predominantly localized in either the upper or the lower layer. The K valley physics is essentially that of the two decoupled monolayers. The valley Hall currents from the two layers average out, but the spin Hall currents add constructively. Under excitation by circularly polarized light, e.g. σ þ , valley
K is excited in the upper layer while valley K is excited in the lower layer, so there is no net valley polarization generated. However, σ þ excites spin up exclusively in both layers, so the spin optical selection rules are still present in the inversion symmetric bilayers. Circular polarized PL can now come from the spin optical selection rule and is not necessarily an indication of inversion symmetry. In MoX2, λv has comparable magnitude to t ⊥ , so the layer hybridization becomes non-negligible. In such a case, an interlayer bias that breaks the inversion symmetry in the bilayer can substantially change the layer polarizations of the Bloch states in the K valley, affecting the valley related phenomena. Experiments have shown that the degree of circular polarization of photoluminescence under a circularly polarized excitation can be continuously tuned between zero and finite as a function of gate voltage in bilayer MoS2 [99]. The tuning of the valley Hall effect by interlayer bias in bilayer MoS2 is also reported recently through the detection of valley polarization accumulated via the Kerr rotation measurements [88, 89]. These experiments indicate a controllable way to tune the valley physical properties by changing the symmetry of the system. Lastly, it is worth noting a unique possibility of controlling the spin and pseudospins in 2H bilayers. With the effective coupling of spin and valley to the layer pseudospin, concerning the carriers in the lower energy spin-split subband, the spin index is now locked to the layer pseudospin polarization in a given valley. As the layer pseudospin couples to the electric field in the perpendicular direction, such spin-layer locking suggests the possibility of controlling these quantum degrees of freedom by various magnetoelectric effects [96, 97, 99]. For example, the electric field can now induce a spin splitting much larger than that possible in a magnetic field [96, 97], while in-plane magnetic field can induce charge oscillations between the layers [96].

292

Semiconducting Transition Metal Dichalcogenides

15.5

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16

Electrical Transport in MoS2: A Prototypical Semiconducting TMDC Andras Kis

16.1

Introduction The sheer number of materials in the layered transition metal dichalcogenide family, around 60 of them already known in late 1960s [1] makes this material class a very interesting playground for exploring mesoscopic physical phenomena at the single- and few-layer limit. Depending on their chemical composition, layered TMDCs can cover a wide range of electrical properties from semiconducting to semimetallic and metallic. Collective phenomena such as charge-density waves and superconductivity have also been observed in the single- and few-layer limit [2] at low temperatures and sometimes high charge densities. This chapter will not be discussing these but will rather focus on electron transport and how it is influenced by disorder in the example of single-layer and ultra-thin MoS2. Since the resurgence of TMDCs in the context of 2D materials, MoS2 has been and remains the most studied material from this class. The easy availability of MoS2 at reasonably high qualities and at low prices in the form of a mineral molybdenite gave rise to a virtuous cycle with a rapid progression of experimental results and theoretical predictions. As a result, we now have the most complete picture of charge transport, and the different scattering mechanisms, that can limit it, built on the example of MoS2. One of the critical advances that brought about the resurgence of layered TMDCs as semiconducting alternatives to graphene was the realization of a field-effect transistor that could be switched on and off, based on single-layer MoS2 [3]. This was the first time a high-quality device on a 2D material other than graphene was demonstrated. Field-effect transistors are not only important as the basic building block used a switch in a digital electronic circuit but is also the basic device used to interface nanoscale semiconductors with the outside world and assess their electrical properties. Crucial for the success of this MoS2 transistor was the use of HfO2 as a top-gate dielectric but also as an encapsulating layer that allowed high-performance levels to be reached. Its role was theoretically explained later by homopolar phonon mode quenching [4, 5] and charged impurity screening by the dielectric and the top-gate [6]. Later experiments showed that even higher mobilities in monolayer MoS2 could be achieved using vacuum annealing that could remove adsorbates [7, 8] or by encapsulating MoS2 between two boron–nitrate (BN) layers [9]. Multi-terminal devices in the dual-gate geometry were also used to demonstrate the metal-insulator transition and band transport in MoS2. Figure 16.2(a) shows the 295

296

Electrical Transport in MoS2

(b)

(a)

10–4

Top gate (tg)

10–5

Vds Ids

Vtg

Source (S) Au

Back gate Silicon

10 mV

10–8 10–9 10–10 10–11 10–13

Back-gate dielectric SiO2

Vbg

10

–7

10–12

Drain (D) Au

Monolayer MoS2

100 mV

10–6

Current Ids (A)

Top-gate dielectric HfO2

Vds = 500 mV

Vbg = 0V

10–14 –4

–2

0

2

4

Top Gate Voltage Vtg (V)

Fig. 16.1 (a) Schematic drawing of a monolayer MoS2-based FET. (b) Current as a function of top-gate voltage. The device showed an on/off current ratio of 108 and n-type transport, typical for MoS2 [3]. (Adapted from [3] with permission © 2011 NPG.)

(a)

(b)

4 3

T = 4.2 K 20 K 30 K 50 K 80 K 120 K 160 K 200 K 240 K

V1

2 µ ~ T–0.78

250 Source

200

Drain V4

150

V3

100

2 1

50

0

0

Mobility µ (cm2/Vs)

5

300

V2 Top gate

Conductance G (µS)

Conductivity s (e2/h)

6 Vds = 500 mV HfO2

100 9

n2D = 0.76 × 1013 cm–2

8

0.96 × 1013 cm–2 1.15 × 1013 cm–2 1.35 × 1013 cm–2

7 6

–4

–2

0

2

Top-gate voltage Vtg (V)

4

1

µ ~ T–0.55

10

100

Temperature (K)

Fig. 16.2 (a) Conductance G and conductivity σ for different top-gate voltages Vtg and temperatures in a dual-gated monolayer MoS2 transistor shown in the inset. (b) Temperature dependence of the mobility. Above ~100 K the mobility decreases due to phonon scattering and follows a Tγ dependence with γ = 0.55–0.78. (Adapted from [11] with permission © 2013 NPG.)

observed sheet conductivity and its dependence on the top-gate voltage. For charge densities above a value of ~1013 cm–2, monolayer MoS2 shows conductivity that increases as the temperature is increased which is a sign of metallic behavior. Electron mobilities extracted from measurements showed an impurity-limited mobility of 120–180 cm2 V–1 s–1. Above 20 K, the mobility starts to decrease and in the 100–300 K range it can be fitted to the power-law dependence μ / Tγ with the exponent γ in the 0.55–0.78 range. This value is much smaller than the experimental values found for bulk crystals (γ = 2.6 [10]) or for bare monolayers (γ = 1.4 [11]), showing that the presence of the top-gate can reduce the rate of mobility degradation with rising temperature.

16.2 Ballistic Transport Simulations

16.2

297

Ballistic Transport Simulations Valuable first insights into the performance limits of MOSFETs based on 2D TMDCs near their scaling limit can be obtained by starting from band structure calculations and using a simple analytical theory for ballistic FETs [12]. Such simulations can then be considered as the best-case scenario for highest achievable current levels and switching slopes. They consider mobile charges at the top of the potential barrier between source and drain electrodes. The barrier is controlled by gate, source and drain potentials through respective capacitors. At zero bias voltage, the equilibrium electron density at the top of the energy barrier is þ∞ ð

N V ds ¼0 ¼

DðE Þf ðE  EF ÞdE,

(16.1)

∞

where D(E) is the density of states at energy E, and f (E – EF) is the Fermi distribution with EF as the Fermi level. The application of gate and drain biases modulates the energy barrier and the electron density is then given by 1 N¼ 2

þ∞ ð

  D E  U scf ½ f ðE  E FS Þ þ f ðE  EFD ÞdE,

(16.2)

∞

where EFS and EFD are the Fermi levels in the source and the drain, respectively, and Uscf is the self-consistent surface potential calculated from the capacitance model describing the transistor characteristics. Figure 16.3 shows the simulations for n-type MOSFETs based on several TMDC monolayers [13]. Mo-based monolayers show similar ON currents due to their effective masses being similar, and they slightly outperform 5 nm thick Si transistors. WSe2 devices outperform reference 5 nm thick Si transistors by ~28% in terms of the ONcurrent.

Fig. 16.3 Ballistic top-of-the-barrier simulations of n-type monolayer MX2 transistor performance. (Adapted with permission from [13]. Copyright 2011 Institute of Electrical and Electronics Engineers.)

298

Electrical Transport in MoS2

Quantum transport models, based on the non-equilibrium Green’s function (NEGF) formalism are the next step up in terms of accuracy [14]. Here, a solution is found by iteratively solving transport equations together with Poisson’s equation until a selfconsistency between charge density and electrostatic potential is achieved. Current is calculated using the expression e I DS ¼ 2 ħ

rffiffiffiffiffiffiffiffiffiffiffiffiffi ð      my kB T E FS  E kx E FD  E kx dE kx F 1=2  F 1=2 T SD ðE kx Þ, (16.3) 2π 3 kB T kB T

where F–1/2 is the Fermi–Dirac integral of order –1/2, TSD the transmission coefficient from source to drain, and EFS and EFD are the Fermi energies in the source and drain electrodes. This results in a device characteristic of ballistic transistors with a 15 nm gate length shown in Fig. 16.4, with a maximum current Imax ~ 1.6 mA/μm. For this simulated gate length, the drain-induced barrier lowering (DIBL) is very small (10 mV/V). Simulations also show that the control of the gate electrode over the device is very good (see Fig. 16.4(b)), due to the small thickness of the semiconducting channel and that devices with a 5 nm gate length show performance close to the ITRS 2026 low-power requirements [15]. A further exploration of device limits was performed using full-band quantum transport simulations [16, 17], which reveal an additional region of negative differential resistance (NDR) in the drain current versus voltage characteristics which are not seen in effective mass-based NEGF calculations. This is attributed to the narrow width in energy of the lowest conduction band but could be suppressed in realistic devices by scattering. Similar to NEGF simulations, full-band simulations show negligibly low DIBL of ~10 mV/V and an ideal subthreshold slope of 60 mV/dec at 300 K, with the best numbers for MoS2. Increasing the mass of the chalcogen atom results in a slight degradation of subthreshold slope and DIBL. The peak transconductance for MoS2 is reached for VDS = 0.2 V and is about 4 mS/μm. Further increase of the bias voltage (a)

(b)

Fig. 16.4 Simulated ID–VG characteristics for a 15 nm gate length monolayer MoS2 transistor [14]. Adapted with permission from [14]. Copyright 2011 American Chemical Society.

16.3 Scattering Mechanisms

299

Fig. 16.5 Result of full-band quantum transport simulations [16, 17]. Adapted with permission from [16, 17]. Copyright 2013 and 2014 AIP Publishing LLC.

reduces the transconductance due to the appearance of NDR. W-based compounds in general show higher transconductance and weaker NDR (Fig. 16.5).

16.3

Scattering Mechanisms While modeling of devices using ballistic simulations can provide upper bounds on their performance which could be relevant for the performance of scaled devices, in order to achieve more realistic results, we also need to consider various sources of scattering. According to sources, these can be divided into intrinsic or extrinsic. Phonon scattering should be the dominant limiting factor in sufficiently clean samples at room temperatures. Kaasbjerg et al. [4] first calculated the strength of electron–phonon interaction in MoS2 from first principles using DFT calculations as the starting point, from which they deduced the acoustic and optical deformation potentials (ADP and ODP) as well as the Fröhlich interaction. They found room temperature phonon-limited mobility for monolayer MoS2 of ~410 cm2 V–1 s–1, determined by zero-order deformation potential scattering and polar optical scattering via the Fröhlich interaction. The mobility drops significantly from ~2500 cm2 V–1 s–1 at T = 100 K to ~400 cm2 V–1 s–1 at T = 300 K. Liu et al. revisited NEGF-based simulations of scaled devices [18] and found that including phonon scattering levels corresponding to this value of room temperature mobility result in a 34.9% reduction in the ON-state current in a device with a gate length of 20 nm and 11.2% in a 5 nm device when compared with results in the ballistic limit. Including piezoelectric interaction [5] lowers the mobility to 320 cm2 V–1 s–1 at room temperature at a change concentration n = 1011 cm–2. The room temperature mobility also follows a dependence on temperature that can be described with a relationship μ / Tγ, where the value of the exponent is γ ~ 1.69 whereas bulk MoS2 is characterized by γ ~ 1.69 [10], which could be an indicator of different electron–phonon coupling in bulk crystals and monolayers. The mean-free path at room temperature should be λ ~ 14 nm.

(b)

(c) 107

Mobility (cm2 V–1 s–1)

(a)

Electrical Transport in MoS2

Mobility (cm2 V–1 s–1)

300

106 105

T=4 K T = 20 K

104 T = 50 K 3

10 1010

1011

1012 n (cm–2)

1013

1010

n = 1 × 1010 cm–2 n = 1 × 1011 cm–2

109

n = 1 × 1012 cm–2 n = 1 × 1013 cm–2 n = 3 × 1013 cm–2

108 107 106 105 104 103

1

10

100

T (K)

Fig. 16.6 (a) Dependence of mobility on carrier concentration [4] (reprinted with permission from [4] ©2012 American Physical Society). (b) Acoustic phonon-limited mobility as a function of carrier density [5]. (c) Acoustic phonon-limited mobility as a function of temperature [5] (reprinted with permission from [5] ©2013 American Physical Society).

For scattering from charge impurities to be dominant, the impurity spacing should be of the order of the phonon mean-free path or smaller, resulting in a lower limit of ~51011 cm–2 for the minimum impurity concentration needed to dominate phonon scattering [4]. At lower temperatures (T < 100 K), scattering from optical phonons is suppressed and acoustic phonons become the dominant source of scattering (Fig. 16.6). Unlike scattering from impurities which can be addressed using for example dielectric engineering [19], scattering from acoustic phonons cannot be eliminated and hence presents an upper limit for mobilities. Such a regime gives rise to a strong change in the temperature-dependence of the carrier mobility as the temperature is lowered under the Bloch–Grüneisen temperature TBG. In the case of monolayer MoS2 with a 2DEG and valley degeneracy of 2, the Bloch–Grüneisen temperature TBG corresponds to qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12 12 (16.4) cm2 K; T LA cm2 K T TA BG  11 n=10 BG  18 n=10 for the transverse (TA) and (LA) acoustic phonons, implying the need to achieve high charge densities, using for example ionic liquid gating. The room temperature phonon-limited mobility could also be strongly influenced by intervalley scattering. In addition to the band gap located at the K point for the monolayer MoS2, DFT calculations also predict that there is a second, local minimum in the conduction band, located at a slightly higher energy than that of the K point. This point is called the Q valley and the energy separation between Q and K valleys EQK is estimated to be between 50 and 200 meV [4, 20]. The emission or absorption of longitudinal acoustic phonons can then mix these two valleys and the relatively large density of states in the Q valley increases the likelihood of scattering. Li et al. [21] found that, for EQK = 70 meV, intervalley scattering can increase the scattering rate for K-valley electrons by almost an order of magnitude at room temperature, resulting in a room temperature mobility of ~130 cm2 V–1 s–1.

16.3 Scattering Mechanisms

301

The dielectric environment can affect the scattering inside 2D semiconducting channels. In the presence of a charged impurity in the semiconducting channel, the mismatch in the dielectric constant between the semiconductor with a dielectric constant εs and the environment (εe) result in the creation of an infinite array of image charges, contributing to the total electrical potential affecting the motion of the electron. This results in an enhancement of Coulomb interaction between the charged impurity and the electron in the case of εe < εs and a suppression for εs < εe. In the presence of a large number of charge carriers inside the 2D semiconductor, the Coulomb potential is further screened. The screening from free carriers has however an opposite dependence on the dielectric environment than in the case of image charges: the free-carrier screening is weaker for εs < εe and stronger for εs < εe. To gain a more complete picture of the influence of the dielectric environment on the electron mobility in a 2D semiconductor, we also need to include another important scattering mechanism that limits the mobility in samples of MoS2 and related materials. It is the remote phonon or surface-optical (SO) phonon scattering, due to electrons in the 2D semiconductor exciting optical phonons in the surrounding dielectric through long-range Coulomb interaction. In a realistic situation, the total of these previously considered contributions – the screening from image charges, screening from free-carriers, SO phonon scattering, and phonon mode quenching – that describe the influence of the dielectric environment on charge transport inside the semiconductor. While at first we could expect that having a high-κ environment would result in the highest mobility, typical high-κ dielectrics also tend to have low optical phonon frequencies, Table 16.1, which makes it easy for charge carriers to excite them. This is clearly illustrated on Fig. 16.7, where the room temperature electron mobility in single-layer MoS2 is plotted for different dielectric environments at 100 K and 300 K. In the absence of SO phonon scattering, the electron mobility is limited by impurities and increases with increased dielectric constant of the environment as the dielectric screening reduces Coulomb scattering. Including SO phonon scattering reduces electron mobility, in some cases to levels under the reference value for SiO2/air interface, corresponding to one of the most common back-gated transistor configurations. The importance of charged impurity and SO phonon scattering in single-layer MoS2 is also shown on Fig. 16.8, where the mobility is plotted as a function of impurity and electron densities. From this, it can be seen that for high impurity concentrations (NI above ~1013 cm–2), the presence of high-κ dielectrics can improve the mobility,

Table 16.1. Surface-optical phonon modes for different dielectrics. Reproduced from [19]. SiO2

AlN

ε0ox

3.9

9.14

ε∞ox

2.5

4.8

ω1SO ðmeVÞ

55.6

81.4

ω2SO ðmeVÞ

138.1

88.5

BN

Al2O3

HfO2

ZrO2

5.09

12.53

23

24

4.1

3.2

93.07

48.18

12.4

16.67

71.41

48.35

57.7

179.1

5.03

4

302

Electrical Transport in MoS2

NI = ns = 1013 cm–2

Fig. 16.7 Electron mobility in MoS2 in environments with different dielectric constants. Dashed lines are related to mobilities without taking into account scattering from surface optical phonons. Reproduced from [19].

(a)

(b) cm2/V

4200 s NI / (1011 cm–2)

T = 300 K NI = 1011 cm–2

(c)

ee

ee A ee

ee T= 300 K ns = 1013 cm–2

T = 300 K NI = 1013 cm–2

Fig. 16.8 Electron mobilities in monolayer MoS2, taking into account different scattering mechanisms as a function of impurity concentration NI with charge density ns = 1013 cm–2 (reproduced from [19]).

16.4 Point Defects

303

especially for low charge carrier concentrations (Fig. 16.8(c)). For NI lower than ~1012 cm–2, low-κ environments show higher electron mobility, with suspended MoS2 showing the potential to reach over 10-000 cm2 V–1 s–1, while BN/BN shows second-best mobilities. Encapsulating MoS2 in BN has indeed resulted in highest lowtemperature mobilities in this material to date, reaching 34,000 cm2 V–1 s–1 for six-layer thick MoS2 [9]. Another promising strategy for achieving high mobilities involves vacuum annealing that can remove adsorbates and related charged scatterers [7, 8]. The fact that the experimentally observed [11] dependence of the electron mobility on temperature showed a power-law behavior (μ / T–γ; γ in the 0.5–0.78 range), which shows that other mechanisms, in addition to scattering from acoustic, optical and remote optical phonons, could be responsible for enhancing the mobility at room temperature. Ong et al. [6] developed a model in which they included the effect of image charges induced in the top-gate by charged impurities in MoS2, Fig. 16.9(a). They find that the Coulomb-impurity limited mobility monotonically increases with charge density n and that the mobility enhancement from the top-gate stack is modest at low temperatures (T < TF, where TF is the Fermi temperature, TF = 29 K for MoS2), since there the screening is dominated by the charge polarizability. At temperatures above TF, the screening by the surroundings of MoS2 become more important. Adding the scattering contributions from phonons, results in the mobilities plotted on Fig. 16.9(b). Phonon scattering from longitudinal acoustic, transverse acoustic, longitudinal optical, and homopolar phonons is included in mobility calculations for bare MoS2 (Fig. 16.9(b)) while for the top-gated MoS2 it is assumed that the homopolar modes are quenched [4, 5]. The γ exponent as a function of temperature is plotted on Fig. 16.9(c), showing that it decreases from 0.75–1.0 to 0.43–0.47 in the presence of the top-gate, indicating that it modifies the Coulomb-impurity scattering.

16.4

Point Defects The main candidates for intrinsic charged impurities are point defects. These are present simply due to entropy and can be introduced in large concentrations during CVD and other growth processes. Figure 16.10(a) shows TEM images and structural models from some of the most common point defects occurring in the CVD material, including sulphur vacancies (VS), bivacancies (VS2), substitutions of S columns by Mo atoms (MoS2) missing MoS3 and MoS6 complexes (VMoS3 and VMoS6), and substitutions of Mo atoms by S columns (S2Mo) [22]. Calculations of their formation energies can give insight into their stability and relative abundance, with point defects having lower formation energies being more stable and appearing more often. Results of such calculations are shown in Fig. 16.10(b) and show that sulphur vacancies have the lowest formation energy and are the most common point defects, with a formation energy of 2.12 eV and observed concentration which could be as high as ~1013 cm–2 [23]. Bivacancies are less likely with a formation energy of 4.14 eV, followed by the SMo antisite defect [23]. These point defects introduce in-gap localized defect states, with Fig. 16.10(d) schematically showing the band structures for VS, VS2, and S2Mo type defects.

Electrical Transport in MoS2

(a)

(b)

mimp + phon (cm2 V−1 s−1)

80

120 100 80

60 40

60 40

20 20

10

101

10 101

102

102 T (K)

T (K) (c)

0 0 mimp + phon, mimp ∝ T−g

1 0.8 g

304

TG TG mimp + phon, mimp ∝ T−g

0.6 0.4 0.2 10

12

14

16

18

20

n (1012 cm−2) Fig. 16.9 (a) Device model used in the calculations of Ong et al. [6]. Single-layer MoS2

is lying on oxide substrate and is covered with a top-gate oxide, which in turn is capped by the metallic top-gate. The charged impurity in the semiconductor induces image charges in the substrate and the top-gate. (b) Plot of total mobility due to charged-impurity and phonon scattering in the absence (left) and presence (right) of the top-gate stack (reprinted with permission from [6] © 2013 American Physical Society).

16.4 Point Defects

305

Fig. 16.10 (a) High-resolution TEM images of most common point defects that can occur in monolayer MoS2. These include sulphur vacancies (VS), bivacancies (VS2), substitutions of S columns by Mo atoms (MoS2) missing MoS3 and MoS6 complexes (VMoS3 and VMoS6) and substitutions of Mo atoms by S columns by (S2Mo) atoms. (b) Structural models of the above-mentioned defects. (c) Sulphur chemical potential. (d) Defect-induced localized energy levels. Additional point defects not considered here include interstitials and Re substitutional defects modeled by Komsa and Krasheninnikov [24]. (Reproduced with permission from [22]. Copyright 2013 American Chemical Society.)

VS and VS2 result in the formation of unoccupied deep levels 0.6 eV below the conduction band minimum, possibly acting as compensation centers in n-type MoS2. They also result in the formation of shallow levels above the valence band, which could explain the difficulty in achieving ambipolar operation in monolayer MoS2. VS2 and S2Mo are also expected to result in the formation of shallow levels below the conduction

Electrical Transport in MoS2

band, which could significantly affect charge carrier transport and introduce disorder. Calculations of the effect of these defects on the phonon-limited mobilities [23] show that the presence of sulphur vacancies (VS and VS2) has a weak effect on the phononlimited electron mobility, which in turn is strongly affected by antisite defects (MoS and MoS2). All these significantly reduce hole mobilities, which is understandable since they all result in shallow levels above the valence band maximum. While these calculations only considered vacancies and substitutions of Mo by S and vice versa, another common impurity in MoS2 samples could be Rhenium substituting Mo atoms, since Re can commonly occur in naturally occurring MoS2 [25]. Its high melting point also makes it difficult to separate from molybdenum and it could be present in molybdenum-based compounds used in CVD synthesis, so we cannot exclude it from being present in CVD materials at this point. Substitutional Re acts as an n-type shallow donor [26] and together with sulphur vacancies could explain why monolayer MoS2 is an n-type semiconductor. Intrinsic point defects, together with defects in the surrounding dielectric or the semiconductor/dielectric interface, can trap charges which can then scatter mobile charge carriers and act as a source of disorder. Temperature-dependent electrical conductivity measurements in the four-probe geometry can very clearly show the effect of point defects and disorder on electrical transport. Figure 16.11 shows the Arrhenius plot of monolayer MoS2 conductance G as a function of inverse temperature and for different charge carrier concentrations. The device in this example consisted of backgated monolayer MoS2 [11]. Two distinct scaling regimes can be distinguished: at relatively high temperatures and charge concentrations, the conductance G follows thermally activated behavior with G / exp(–Ea/kBT), shown as straight lines on Fig. 16.11, and corresponding to nearest-neighbor hopping. At lower temperatures 4

Vbg

G (Vbg) = Goexp(–Ea/kB T )

2

Conductance G (S)

306

2V 5V 9V 13 V 20 V 25 V 30 V 35 V 40 V

10–5

8 6 4

2

10–6

8 6 4

0.004

0.008

0.012

0.016

0.020

1/T (K–1)

Fig. 16.11 Arrhenius plot of four-probe conductance G of back-gated monolayer MoS2. Solid lines are fits to thermally activated behavior (G / exp[–Ea/kBT]) [11]. A departure from this trend and a transition to a weaker temperature dependence can be seen at low temperatures and carrier concentrations (adapted from [11] with permission © 2013 NPG).

307

16.4 Point Defects

10

12

EM E

1011

Type M trap

2

–1.74 V

0.0 10 k

100 k Frequency (Hz)

1M

10–4 10–5 10–6 10–7 10–8 10–9

Type B trap

Type M trap

1

2 3 –2 –1 0 1 Gate voltage V TG (V)

Metal HK MoS2

EF

Emid

0 –1.0

–3

EM Conduction band

0.2 0.1

1013

14

0.3

1014

Valence band

0.4

t it (s)

Cms (pF)

0.5

3

Type B trap –1

0.6

(c) 1015

–2

2.46 V

DOS (10 cm eV )

0.7

Dit (eV–1 cm–2)

(b)

(a)

4

–0.50

0.0

0.5

1.0

Energy E (eV)

Fig. 16.12 (a) Dependence of device capacitance on frequency for different values of the gate voltage. (b) Density and time constant of trap states for different gate voltages. Symbols represent experimental results and lines theoretical models. (c) Density of states for MoS2, including both localized band-tail and mid-gap localized states (adapted from [27] with permission © 2014 NPG).

and charge densities, a crossover to a weaker temperature dependence can be observed, with a conductance that can be fitted by the variable-range hopping model, G / exp(–(T0/T)1/3), typical of hopping transport via localized states in disordered low-dimensional systems. More insight into disorder and localized band tails states in MoS2 can be gained using capacitance and ac conductance measurements as the localized states are characterized with different time constants τ and act as additional capacitors and resistors connected in parallel to the semiconductor. An example of such capacitance measurements as a function of frequency is shown on Fig. 16.12(a) [27]. The presence of a double hump in these measurements indicates the presence of at least two types of traps, with the corresponding density of states and time constants shown in Fig. 16.12(b). These are assigned to mid-gap traps (M) and band edge (B) traps. Their location in the band gap is shown on Fig. 16.12(c). The trap states at the band edge contribute to its smearing that for energies below the band edge energy ED can be described by an exponentially decaying density of states as we move away from the band edge into the band gap

E  ED Dn ðEÞ ¼ αDo exp (16.5) φ with φ = 100 meV the characteristic energy width of the band tail obtained from a fit to data. Such a relatively high value is indicative of relatively large disorder and scattering. Some of the point defects can however be repaired, with the most prominent sulphur vacancy being a prime example. Low-temperature thiol chemistry was successfully used to repair them [28] using (3-mercaptopropyl)trimethoxysilane (MPS) and mild annealing. This was shown to result in a reduction in localized trap states and a very high mobility of 80 cm2 V–1 s–1 in monolayer MoS2 at room temperature. Treating MoS2 with a non-oxidizing organic superacid bis(trifluoromethane) sulfonimide (TFSI) was also recently shown to enhance the photoluminescence and minority carrier lifetime

308

Electrical Transport in MoS2

[29] of MoS2 monolayers by more than two orders of magnitude. While at an early stage such chemistry-based approaches look very promising and could be an interesting path towards realizing high-quality MoS2 and 2D semiconductors with low defect concentration.

16.5

References [1] Wilson JA, Yoffe AD. The transition metal dichalcogenides discussion and interpretation of the observed optical, electrical and structural properties. Adv Phys. 1969 May; 18(73): 193–335. [2] Ye JT, Zhang YJ, Akashi R, Bahramy MS, Arita R, Iwasa Y. Superconducting dome in a gate-tuned band insulator. Science. 2012 November 30; 338(6111): 1193–6. [3] Radisavljevic B, Radenovic A, Brivio J, Giacometti V, Kis A. Single-layer MoS2 transistors. Nat Nanotechnol. 2011; 6(3): 147–50. [4] Kaasbjerg K, Thygesen KS, Jacobsen KW. Phonon-limited mobility in n-type single-layer MoS2 from first principles. Phys Rev B. 2012 March 23; 85(11): 115317. [5] Kaasbjerg K, Thygesen KS, Jauho A-P. Acoustic phonon limited mobility in twodimensional semiconductors: Deformation potential and piezoelectric scattering in monolayer MoS2 from first principles. Phys Rev B. 2013; 87(23): 235312. [6] Ong Z-Y, Fischetti MV. Mobility enhancement and temperature dependence in top-gated single-layer MoS2. Phys Rev B. 2013; 88(16): 165316. [7] Jariwala D, Sangwan VK, Late DJ, Johns JE, Dravid VP, Marks TJ, et al. Band-like transport in high mobility unencapsulated single-layer MoS2 transistors. Appl Phys Lett. 2013; 102(17): 173107. [8] Baugher B, Churchill HOH, Yang Y, Jarillo-Herrero P. Intrinsic electronic transport properties of high quality monolayer and bilayer MoS2. Nano Lett. 2013; 13(9): 4212–16. [9] Cui X, Lee G-H, Kim YD, Arefe G, Huang PY, Lee C-H, et al. Multi-terminal transport measurements of MoS2 using a van der Waals heterostructure device platform. Nat Nanotechnol. 2015 April 27; 10: 534–40. [10] Fivaz R, Mooser E. Mobility of charge carriers in semiconducting layer structures. Phys Rev. 1967; 163(3): 743–55. [11] Radisavljevic B, Kis A. Mobility engineering and a metal–insulator transition in monolayer MoS2. Nat Mater. 2013; 12: 815–20. [12] Rahman A, Guo J, Datta S, Lundstrom MS. Theory of ballistic nanotransistors. IEEE Trans Electron Devices. 2003 September; 50(9): 1853–64. [13] Liu L, Kumar SB, Ouyang Y, Guo J. Performance limits of monolayer transition metal dichalcogenide transistors. IEEE Trans Electron Devices. 2011; 58(9): 3042–7. [14] Yoon Y, Ganapathi K, Salahuddin S. How good can monolayer MoS2 transistors be? Nano Lett. 2011; 11: 3768–73. [15] Alam K, Lake RK. Monolayer transistors beyond the technology road map. IEEE Trans Electron Devices. 2012 December; 59(12): 3250–4. [16] Chang J, Register LF, Banerjee SK. Atomistic full-band simulations of monolayer MoS2 transistors. Appl Phys Lett. 2013 November 25; 103(22): 223509. [17] Chang JW, Register LF, Banerjee SK. Ballistic performance comparison of monolayer transition metal dichalcogenide MX2 (M = Mo, W; X = S, Se, Te) metal–oxide–semiconductor field effect transistors. J Appl Phys. 2014 February; 115(8).

16.5 References

309

[18] Liu L, Lu Y, Guo J. On monolayer MoS2 field-effect transistors at the scaling limit. IEEE Trans Electron Devices. 2013; 60(12): 4133–9. [19] Ma N, Jena D. Charge scattering and mobility in atomically thin semiconductors. Phys Rev X. 2014; 4(1): 011043. [20] Kadantsev ES, Hawrylak P. Electronic structure of a single MoS2 monolayer. Solid State Commun. 2012; 152(10): 909–13. [21] Li X, Mullen JT, Jin Z, Borysenko KM, Buongiorno Nardelli M, Kim KW. Intrinsic electrical transport properties of monolayer silicene and MoS2 from first principles. Phys Rev B. 2013; 87(11): 115418. [22] Zhou W, Zou X, Najmaei S, Liu Z, Shi Y, Kong J, et al. Intrinsic structural defects in monolayer molybdenum disulfide. Nano Lett. 2013; 13: 2615 [23] Hong J, Hu Z, Probert M, Li K, Lv D, Yang X, et al. Exploring atomic defects in molybdenum disulphide monolayers. Nat Commun. 2015 February 19; 6: 6293. [24] Komsa H-P, Krasheninnikov AV. Native defects in bulk and monolayer MoS2 from first principles. Phys Rev B. 2015 March 12; 91(12): 125304. [25] Earnshaw A, Greenwood N. Chemistry of the Elements, 2nd edn. (Elsevier, Amsterdam, 1997). [26] Lin Y-C, Dumcenco DO, Komsa H-P, Niimi Y, Krasheninnikov AV, Huang Y-S, et al. Properties of individual dopant atoms in single-layer MoS2: atomic structure, migration, and enhanced reactivity. Adv Mater. 2014 May 1; 26(18): 2857–61. [27] Zhu W, Low T, Lee Y-H, Wang H, Farmer DB, Kong J, et al. Electronic transport and device prospects of monolayer molybdenum disulphide grown by chemical vapour deposition. Nat Commun. 2014; 5: 3087. [28] Yu Z, Pan Y, Shen Y, Wang Z, Ong Z-Y, Xu T, et al. Towards intrinsic charge transport in monolayer molybdenum disulfide by defect and interface engineering. Nat Commun. 2014 October 20; 5. http://www.nature.com/ncomms/2014/141020/ncomms6290/full/ncomms 6290.html [29] Amani M, Lien D-H, Kiriya D, Xiao J, Azcatl A, Noh J, et al. Near-unity photoluminescence quantum yield in MoS2. Science. 2015 November 27; 350(6264): 1065–8.

17

Optical Properties of TMD Heterostructures Pasqual Rivera, Wang Yao, and Xiaodong Xu

Heterostructures formed by 3D semiconductors have led to repeated breakthroughs in both condensed matter physics and modern optoelectronic technologies. In analogy to these 3D heterostructures, a natural extension of the studies of 2D semiconductors is the investigation of van der Waals bound heterostructures built from them [1–3]. The techniques developed during the nanofabrication of graphene-based electronic devices have been adapted to allow for the assembly of complex layered heterostructures. In this section, we will consider those made with monolayer semiconductors from the family of transition metal dichalcogenides (TMDs). When these 2D semiconductors are placed in contact with one another, an atomically sharp interface is formed. The close proximity causes interlayer coupling, which can dramatically affect the electronic and optical properties of the layers. Distinct from double quantum well structures formed by 3D semiconductors, such 2D heterostructures inherit the valley physical properties from the constituent monolayers. For instance, a relative rotation between the crystal axes of the monolayers will introduce a twist angle between their valleys in momentum space, which may allow for engineering the valley-optoelectronic properties. Here, we will first present the contemporary experimental results and understanding of 2D heterostructures without their valley functionalities. Afterwards, we will discuss the theoretical background of the valley optoelectronic properties and present the current outlook with recent experimental progress.

17.1

Fundamentals of 2D TMD Heterostructures

17.1.1

Band Alignment of MX2 The electronic properties of 2D semiconducting heterobilayers follow from their relative band alignment. Calculations of the band gaps and work functions of the monolayer TMDs predict a staggered band alignment for these heterostructures [4–12], as is qualitatively shown in Fig. 17.1(a). The MoX2/WX2 heterostructure (Fig. 17.1(b)) therefore manifests a type-II band alignment, with the conduction band minimum in the monolayer MX2 and the valence band maximum in the monolayer WX2. Density functional theory calculations show the full band structure (Fig. 17.1(c)) of the MoX2/WX2 heterostructure with AA-type stacking and idealized lattice matching [13]. The results indicate that the band edges remain at the K points in the first Brillouin zone,

310

17.1 Fundamentals of 2D TMD Heterostructures

311

(b)

(a) MoS2

WS2 MoSe2 WSe2 MoX2

–4

Band Energy (eV)

WX2 (c) –5

–6

Fig. 17.1 Monolayer TMD band alignment and heterobilayer stacking. (a) Qualitative band alignment for the MX2 and WX2 monolayer TMD. (b) Illustration of the MoX2/WX2 heterobilayer. (c) Calculated band structure of the MoX2/WX2 heterobilayer. The inset shows the band edge electronic wave functions at the K points, with conduction electrons localized predominantly in the MoX2 (top layer) and valence electrons in the WX2 (bottom layer). (Reprinted with permission from [13]. Copyright (2015) by the American Physical Society.)

just as in the isolated monolayers. In this idealized heterobilayer, the K points in the MoX2 conduction band are primarily composed of the dz2 orbitals of the Mo atoms (top layer  of inset), and  the valence band edges in the WX2 are composed mainly of the p1ffiffi d x2 y2  id xy orbitals of the W atom (bottom layer of inset) [17, 18]. Given the 2 spatial separation, the interlayer coupling at the K points is not expected to cause significant hybridization of the band edges [5, 9, 10, 16]. This is different than that seen in the multilayers of MX2, where the band edges shift away from the K points to produce an indirect band gap [10, 12, 14–16]. Recent micro-ARPES (angle resolved photoemission spectroscopy) measurements on monolayer MoSe2/WSe2 heterostructures have confirmed the above understanding on band parameters and hybridization, and determined the valence band offset to be e300 meV [19]. In addition, scanning tunneling spectroscopy and X-ray photoelectron spectroscopy have been used to determine the conduction and valence band offsets of ~800 meV in MoS2/WSe2 heterostructures [5].

17.1.2

Heterostructure Fabrication 2D heterostructures can be readily synthesized by manually stacking different monolayers. Currently, polymer stamp-based transfer techniques are the most suitable method

312

Optical Properties of TMD Heterostructures

(a) External heater

Opcal microscope

Three axis micromanipulator

Sample on substrate

Heang stage Stamp mounted on glass slide Rotaon manipulator

(b) 1. PDMS PC Heang stage

2.

3.

4.

5.

Sample Substrate Fig. 17.2 Transfer stage and process. (a) Typical transfer stage for constructing van der Waals bound vertical heterostructures. The setup allows for optical imaging through the clear polymer stamp, as well as individual positioning controls for both sample and stamp. The heating stage provides fine control of the vertical position of the sample on substrate, and the rotation manipulator allows for control of the relative rotation. The entire bottom stage is also controlled by micromanipulators (not shown). (b) Flow chart of the polymer stamp pick-up technique outlined in the text.

for fabricating vertically stacked 2D heterostructures [20, 21]. A typical transfer stage for heterostructure fabrication is shown in Fig. 17.2(a). The combination of the critical components identified in this graphic allow for precision positioning and control during the transfer of the sample from a substrate to a polymer stamp in the following manner (see Fig. 17.2(b)). 1.

2.

3.

4. 5.

The stamp is prepared by attaching a thin layer of polymer – such as polycarbonate (PC) – and the identified sample on substrate is oriented and mounted to the heating stage. The stamp is brought to within a few microns of the sample surface with the aid of the optical microscope and manipulators. Thin film interference allows one to identify the lowest spot on the stamp as it nears the substrate, and the sample on the substrate is positioned nearby. The external heater slowly raises the temperature of the heating stage. Thermal expansion brings the sample in contact with the stamp surface in a controlled manner. The heating stage is slowly cooled so that the polymer stamp peels away from the substrate, taking the sample with it. The sample has now been transferred to the stamp and is ready for successive fabrication steps.

17.1 Fundamentals of 2D TMD Heterostructures

(a)

(b)

313

(c)

MoSe2

WSe2

Fig. 17.3 Monolayer TMD characterization. (a) Optical micrographs of monolayer MoSe2 (top) and WSe2 (bottom). Scale bar is 5 μm. (b) Atomic-force microscopy characterization of the surface of each sample. (c) Linear-polarization resolved second harmonic generation (SHG) intensity (circles) of the samples oriented as in (a). This SHG signal is strongest along the armchair axes of the crystal. The zig-zag axes of the crystal are located at 30 rotation (solid lines).

Using this transfer technique, it is possible to assemble the MX2 heterobilayer with control over the twisting angle between the constituent layers. In the following example, the monolayers are obtained by exfoliation of bulk crystals onto ~280 nm SiO2 on Si substrates. This allows for visual identification of the monolayer samples by their contrast using a high magnification optical microscope [22]. The optical micrographs of a monolayer MoSe2 (top) and a monolayer WSe2 (bottom) are shown in Fig. 17.3(a). The monolayer character of these samples can be readily verified by the room temperature photoluminescence (PL) spectrum. The sample surfaces are then characterized by atomic-force microscopy to ensure that they are free of significant surface contamination (Fig. 17.3(b)). Having large areas free from surface residues, these samples are suitable for use in assembling a heterobilayer. The crystal axes of the monolayers are then determined by analyzing the intensity of SHG that is co-linearly polarized with the excitation source [23–25]. An example of the polarization resolved SHG intensity as a function of the angle of the excitation polarization is shown in Fig. 17.3(c). The peaks of the SHG intensity correspond to the armchair axes of the crystals [23–25], and the zig-zag axes are at 30 rotation (solid lines in Fig. 17.3(c)). Comparing the axial SHG intensity maps with in-situ optical micrographs clearly shows that the straight edges of these two samples correspond to the zig-zag axes of the crystal. One should be aware, however, that this technique is not phase sensitive, so the crystal axes are only resolved to an arbitrary 60 rotation. To assemble the heterobilayer, the top monolayer (MoSe2) is first transferred from the substrate (SiO2 on Si) to a PC on PDMS stamp. The monolayer orientation on the stamp is imaged to aid in aligning the bottom layer (WSe2 on SiO2 on Si). The bottom layer is then manipulated to the desired relative rotation and position with respect to the top monolayer on the stamp. Computer software allows one to overlay the images to ensure

314

Optical Properties of TMD Heterostructures

1.

Monolayer MoSe2

2.

3.

Monolayer MSe2

Fig. 17.4 Heterobilayer fabrication. 1. The top layer of the heterobilayer is transferred from

the substrate to the stamp using the procedure outlined in Fig. 17.2(b). The image shows the transferred sample attached to the polymer stamp. 2. Using the crystal axes previously determined, the bottom monolayer is rotated and positioned to achieve the desired twisting angle and stacking configuration, as shown [3]. The polymer stamp is melted onto the substrate and the PDMS is lifted away. The good agreement of the sample edges with the dashed lines (60 angle) demonstrates the accuracy of the stacking procedure.

the desired positioning prior to bringing the monolayers in contact with one another. The accuracy of this technique can be considerably enhanced by simultaneously triangulating the positions of bulk pieces on the substrate and on the stamp. This process is shown in Fig. 17.4, where the monolayers shown in Fig. 17.3 have been assembled into a MoSe2/WSe2 heterobilayer with the crystal axes aligned (modulo 60 ). A solvent rinse of the PC leaves the heterostructure on the SiO2, which is ideal for the fabrication of electrical contacts using standard electron beam lithography. Considering the high sensitivity of 2D material properties to surface contaminants, the transfer process will continue to evolve towards heterostructures with pristine interfaces. The realization of commercially viable electronic and optoelectronic heterostructure devices currently hinges on the scalable growth of high-quality, large area, monolayer crystals, as well as the ability to process them into heterostructures with high-quality interfaces. Further, the ability to control the crystal axes of each layer may be an opportunity to engineer the optoelectronic response for tailored end-use applications. Towards this end, researchers are developing scalable growth of 2D heterostructures by vapor transport techniques [26–28], which is a promising approach for routine synthesis for experimental studies and potential applications.

17.1.3

Ultra-fast Interlayer Charge Transfer and Photovoltaic Response The type II band alignment of the MoX2/WX2 heterostructure (Fig. 17.5(a)) is useful for many optoelectronic devices. For example, under photoexcitation, electrons and holes will relax to different layers without an applied bias. This intrinsic charge separation facilitates subsequent carrier collection, which is ideal for many photovoltaic applications. Notably, the heterostructure has an atomically sharp interface, so there is a vanishingly small depletion region between the two layers. The significant conduction and valence band offsets drive ultra-fast ð< 1 psÞ charge transport between the layers of

17.2 Interlayer Exciton Properties

315

(a)

WX2

MoX2

(b) MoX2

WX2

Fig. 17.5 Type II band alignment of the MoX2/WX2 heterobilayer. (a) Type II alignment and interlayer charge transfer in the MoX2/WX2 heterobilayer. Γe ðΓh Þ represents the interlayer charge transfer rate for electrons (holes) and Δc ðΔv Þ is the conduction (valence) band offset between the two monolayers. (b) Illustration of the interlayer exciton ðXI Þ, the Coulomb bound electron and hole from different layers of the heterobilayer.

the MoSe2/MoS2 heterobilayer [29]. Further, interlayer hole transfer in the MoS2/WS2 heterobilayer was determined to be less than 50 fs using non-degenerate ultra-fast pump–probe measurements [30]. Presently, charge transport rates have not shown dependence on the twist angle between the layers – this possibility will be interesting to explore in the future. Since electrons and holes are localized in opposite layers, the 2D heterobilayer manifests a vertical p–n junction [26, 28, 31–36]. The formation of vertical p–n junction has been confirmed experimentally by electron transport measurements showing diode-like behavior [32, 34–36]. Optical absorption and PL measurements, which will be discussed in detail in the following sections, further support this conclusion [28, 31–33, 37, 38]. Since the p–n junction is the back-bone of modern electronic and photonic devices, such as light-emitting diodes and photodetectors, 2D semiconductor heterostructures are naturally suitable for nanoscale optoelectronic and photovoltaic applications. Already, tunable photocurrent generation has been demonstrated in 2D MX2/WX2 heterostructures [34–37].

17.2

Interlayer Exciton Properties

17.2.1

Observation of Interlayer Exciton The strong Coulomb interaction in 2D allows for the binding of the electrons and holes in the opposite layers of the heterobilayer, i.e. formation of interlayer excitons

316

Optical Properties of TMD Heterostructures

(XI in Fig. 17.5(b)) [29, 31, 32, 37]. When this exciton recombines, the emitted photon energy corresponds to the reduced band gap between the conduction band minimum in the MoX2 and the valence band maximum in the WX2, less the interlayer exciton binding energy. Therefore, the underlying type-II alignment makes the interlayer exciton the lowest energy excitonic state in the system, which is readily seen from the heterobilayer PL spectrum [26, 31, 32, 37, 38]. Here, we again use the MoSe2/WSe2 heterobilayer as an example. Figure 17.6(a) shows the optical image of such a heterostructure in which the staggered stacking configuration allows the direct comparison of the PL from the MoSe2/WSe2 heterobilayer to that from isolated monolayers of WSe2 and MoSe2. Low temperature PL spectra from these three regions are shown in Fig. 17.6(b). The PL spectra from the WSe2 and MoSe2 monolayers, labeled as XW and XM, respectively, are typical for monolayers of these materials [31]. The high-energy PL from the heterobilayer resembles the summation of the PL from the MoSe2 and WSe2 monolayer regions (Fig. 17.6(c)). This implies the band hybridization at the K point is weak, consistent with the μ-ARPES measurement [19]. (a)

(b) WSe2 XW MoSe2

5 m

XI

MoSe2/WSe2

XM

XW

(c)

Y ( m)

6 (i)

(ii)

(c)

(iii) XM

4 2 0

1

3

5

1

3

5

1

3

5

X (mm) Fig. 17.6 Interlayer exciton photoluminescence. (a) Optical micrograph of a MoSe2/WSe2

heterobilayer. The white line borders the heterostructure region. (b) Photoluminescence spectrum from the monolayer WSe2 (top), heterobilayer (middle), and monolayer MoSe2 (bottom) at T = 20 K. Here, the heterobilayer region shows interlayer exciton ðXI Þ photoluminescence peak at 1.4 eV, as well as quenching of the intralayer excitons in the monolayers. (c) Spatial distribution of the integrated PL from intralayer excitons in (i) monolayer WSe2  XW , (iii) monolayer MoSe2  XM , and (ii) the interlayer excitons – XI – showing that the interlayer exciton is located only at the heterobilayer region. (Reprinted by permission from Macmillan Publishers Ltd: Nature Communications [31]. Copyright (2015).)

17.2 Interlayer Exciton Properties

317

The spectrum from the heterobilayer also shows the emergence of pronounced low-energy emission (XI at ~1.4 eV). Spatially resolved measurements show that this low-energy emission is isolated entirely to the heterostructure region [31, 32] (Fig. 17.6(c)). The reduced energy PL is a consequence of the reduced band gap of the heterobilayer (cf. Fig. 17.5(a)), and is a hallmark of the interlayer exciton. The relative energy position between interlayer and intralayer exciton implies a large binding energy for XI, which is consistent with the small interlayer spacing (0 are the nth-order Umklapp-assisted light cones which are defined by the momentum 0 mismatch of τκ on Cn and on C 0n (see Fig. 17.10(d)). From the crystal symmetry,  τκ E  ð0Þ ^ 3 Q0 , ^ 3 rotational symmetry: Q0 , C the main light cones for X Q, τ 0 , τ are related by the C

322

Optical Properties of TMD Heterostructures

(b)

(d)

(a)

(c)

Fig. 17.10 Finite momentum light cones of interlayer exciton. (a) The light coupling of the

interlayer exciton with momentum mismatch Q0 . Positions of the valleys in the top (dots) and bottom (circles) layers in the extended zone scheme for both (b) MoSe2/WSe2 and (c) MoS2/WS2 heterobilayers. (d) The positions of the Q0 , Q1 , and Q2 light cones in the phase space of Q. The brightness of the light cones indicates the relative strength of the optical dipole. (Reprinted with permission from [13]. Copyright (2015) by the American Physical Society.)

^ 2 Q0 . The light cones at Q0 , C ^ Q , and C ^ 2 Q0 are found from time reversal and C 3 3  E 3 0  ð0Þ and correspond to the state X Q, τ 0 , τ .

17.3.3

Interlayer Exciton Recombination Pathways Having the interlayer exciton wave function in the absenceof interlayer coupling, we  ^T ^ now introduce the interlayer hopping of electrons and holes H T as a perturbation. H can hybridize the Bloch functions in the two layers, and the corrected interlayer exciton eigenstate, to first order, is D D   ð0Þ E   ð 0Þ E H ^ T X 0  ^ T X 0  E  E X X τ 0 , q0 H X X τ , q  , τ , τ Q Q, τ , τ  ð1Þ  ð0Þ X τ 0 , q0 þ jX τ , q i: X Q, τ0 , τ ¼ X Q, τ0 , τ þ E I ðQ Þ  E t E I ðQÞ  E b q q0 (17.2)     Here, X τ0 , q0 X τ , q represents the intralayer exciton in the top (bottom) layer with

energy Et ðEb Þ and center-of-mass wavevector q0 ðqÞ defined in the individual monolayer. The second term on the right-hand side corresponds to the interlayer hopping of the hole from the τK valley in the bottom layer to the τ 0 K 0 valley in the top layer to form the intralayer exciton. The third term similarly corresponds to the interlayer hopping of the electron from the τK valley of the top layer to the τ 0 K 0 valley of the bottom layer to form the intralayer exciton. These interlayer hopping processes are shown in Fig. 17.11.

17.3 Valley Optoelectronic Properties of 2D Heterostructure

(a)

323

(b)

MoX2

WX2

MoX2

WX2

Fig. 17.11 Light coupling mediated by interlayer hopping. Intralayer hopping of the hole (gray dashed line) and the electron (black dashed line) couples the interlayer exciton to the intralayer exciton in MoX2 and WX2, respectively, for heterobilayer with twisting angle near (a) θ ¼ 60 and (b) θ ¼ 0 . (Adapted with permission from [13]. Copyright (2015) by the American Physical Society.)

The optical coupling of the interlayer exciton is of primary interest here as this determines its impact potential for end use optoelectronic applications and provides the means to probe the unique physics found in this new system. At the light cones Q ¼ τκ  τ 0 κ 0 , the transition matrix elements for the optical dipole of the interlayer exciton in the presence of the interlayer hopping perturbation are

Dτ 0 , τ , Q

D   ð0Þ E ^ T X 0    D   E X τ0 , q0 ¼0 H  Q, τ , τ ^ X ð0Þ 0 þ ^ X τ 0 , q0 ¼0 ¼ 0D 0D Q, τ , τ E I ðQÞ  E t D   ð0Þ E ^ T X 0    X τ , q¼0 H  Q, τ , τ ^ X τ , q¼0 , þ 0 D EI ðQÞ  Eb

(17.3)

^ is the electric dipole operator and j0i is the vacuum state. The first term on the where D     right represents the direct recombination of ek0 , τ 0 and hk, τ , as these basis functions for electron and hole, respectively, in the two layers can have finite overlap. The second and third terms represent the optical coupling mediated by the interlayer hopping to the bright intralayer exciton states in the top and bottom monolayers.

17.3.4

Optical Selection Rules of Interlayer Excitons The intralayer exciton transition dipoles in the top with  and bottom layers are associated       ^ ^ the valley optical selection rules, respectively, h0 D X τ 0 , q0 ¼0 i ¼ Deτ0 , and h0 D X τ , q¼0 i ¼ pffiffi represents the circular polarization vector of the emitted photon. Deτ , where e  xiy 2 D is the interband transition dipole for the intralayer exciton. Using the two-center approximation [13], the three terms in Eq. (17.3) are, respectively,

324

Optical Properties of TMD Heterostructures

D   E X iτκr ^ Xð00 Þ 0 D δQ, τκτ 0 κ0 D2τ , 0 ðτκ Þe τ , τ, Q / 0 κ, κ D   ð0Þ E X iτκr ^ T X 0 X τ 0 , q0 ¼0 H δQ, τκτ0 κ0 t 2τ , 2τ 0 ðτκ Þe Q, τ , τ  κ, κ0 D E X   ð0Þ ^ T X 0 X τ , q¼0 H δQ, τκτ0 κ0 t 00 ðτκ Þeiτκr , Q, τ , τ  0 κ, κ (Dm ) are the Fourier components of the hopping integral (transition dipole) where t m m0 m0 between a W and a Mo d-orbital with magnetic quantum numbers m and m0 , respectively. t m and Dm decay fast with the increase of jτκ j, thus in practice only the main m0 m0 ^ 3 Q0 and C ^ 2 Q0 ) need to be considered for optoelectronic light cones (Q0 , C 3

responses as the optical dipole strength becomes very weak in the Umklapp light cones (Qn>0 ) where jτκ j is large. The overall transition dipole for the interlayer exciton can be expressed as ! X t 2τ   0 ðτκ Þeτ 0 t 00 ðτκ Þeτ iτκr 2τ 2τ Dτ 0 , τ , Q ¼ e δQ, τκτ0 κ0 D0 ðτκ Þ αeτ0 þ βeτ þ D þD , E I ðQ Þ  E t EI ðQÞ  E b κ, κ0 where α and β are the weighted coefficients of the direct interlayer exciton emission polarization (see supplementary material of [42]).   The essential result now is that X Q, τ 0 , τ may couple with photons with general elliptical polarization, without the assistance of phonon or impurity scattering at the ^ 3 rotational symmetry of the finite velocity light cones Q. Considering the underlying C ^ 3 Q0 , and C ^ 2 Q0 are also C ^3 interlayer exciton states, the elliptical polarization at Q0 , C 3

rotations of each other. Therefore, without mutual quantum coherence, the sum of emission from these three light cones combines to produce light with a finite degree ^ 3 Q0 , and of circular polarization but no linear polarization. The light cones Q0 , C 2 ^ Q0 are the time reversal counterparts of the above three where the excitons have the C 3 opposite valley configuration, and the elliptical polarization also has the opposite helicity.

17.3.5

Valley Functionalities of Interlayer Excitons At each light cone, we see that the major axis of the elliptical polarization is locked to Q (see Fig. 17.12(a) for an illustration), which also corresponds to the kinematic momentum of the interlayer exciton at the light cone. This means that the finite-velocity light cones not only allow for the optical injection of valley polarization, but this absorption will also produce a valley-dependent exciton current. This results from the fact that light cones for different valley configurations not only correspond to different polarizations, but also different in-plane velocity of the interlayer exciton. For example, under excitation by a linearly polarized light, each of the light cones will absorb at different rates depending on their elliptical polarization, which results in a net valley current, as shown in Fig. 17.12(a). The long lifetime and long-range drift-diffusion of the interlayer exciton may further allow for demonstrations of valley Hall effects through the observation of the spatial profile of the polarization (cf. Fig. 17.12(b)). Additionally, the

17.4 Outlook

325

Fig. 17.12 Valley injection and valley current in heterobilayers. (a) Injection of valley current by excitation with linearly polarized light. (b) Valley Hall effect of interlayer excitons in the heterobilayer. The spatial distribution of emission polarization depends on the distribution of interlayer exciton in the Q-space. (c) The net valley current induces a perpendicular number current due to the Berry curvature. (Reprinted with permission from [13]. Copyright (2015) by the American Physical Society.)

optical injection of valley current induces a number current in the perpendicular direction due to Berry curvature effects, i.e. the inverse valley Hall effect shown in Fig. 17.12(c).

17.4

Outlook This theoretical framework illustrates the unique valley-dependent physical properties of interlayer excitons in TMD heterobilayers. This points to exciting opportunities based on 2D heterostructures for optoelectronics with valley functionalities. Notably, the most recent measurements in MoSe2/WSe2 heterostructures with nearly aligned crystal axes show the promise of this direction. Circularly polarized interlayer exciton PL is observed by circularly polarized optical pumping, demonstrating the optical generation of valley-polarized interlayer exciton. Time and polarization resolved PL reveals the valley polarization life time of ~40 ns [44], many orders of magnitude longer than that in monolayers. This long valley lifetime is a consequence of the spatially indirect nature of XI, which suppresses the intravalley electron–hole exchange interaction, a main cause of the valley depolarization of intralayer excitons. The long lifetime also enables the observation of drift-diffusion of valley degrees of freedom in the 2D plane, with the spatial pattern of polarization evolving from a disk to a ring pattern as a function of optical excitation intensity, implying fascinating valleydependent many-body interaction effects [44]. Based on the theoretical predictions, we anticipate future work on interlayer valley exciton properties as a function of twist angle of fundamental scientific interest. These properties include the evolution of optical selection rules, the exciton and valley polarization life times, interlayer charge/spin transport dynamics, and in-plane exciton transport. Realization of valley functionalities, such as interlayer valley exciton Hall effects and optical injection of valley current, would be challenging but intriguing. The understanding of the fundamental properties will be important for studying exciton

326

Optical Properties of TMD Heterostructures

quantum phenomena, such as excitonic superfluidity, condensation, and exciton droplet/liquid. Considering the possibilities presented above, the interlayer exciton in the TMD heterobilayer presents a promising platform for studies of 2D heterostructure optoelectronics, including LEDs, nanolasers, and photovoltaics.

17.5

References [1] Geim, A. K. and Grigorieva, I. V. Van der Waals heterostructures. Nature 499, 419–425 (2013). [2] Britnell, L. et al. Strong light–matter interactions in heterostructures of atomically thin films. Science 340, 1311–1314 (2013). [3] Novoselov, K. S. and Castro Neto, A. H. Two-dimensional crystals-based heterostructures: materials with tailored properties. Physica Scripta T146, 014006 (2012). [4] Amin, B., Singh, N. and Schwingenschlögl, U. Heterostructures of transition metal dichalcogenides. Physical Review B 92, 075439 (2015). [5] Chiu, M. H. et al. Determination of band alignment in the single-layer MoS2/WSe2 heterojunction. Nature Communications 6, 7666 (2015). [6] Constantinescu, G. C. and Hine, N. D. M. Energy landscape and band-structure tuning in realistic MoS2/MoSe2 heterostructures. Physical Review B 91, 195416 (2015). [7] Gong, C. et al. Band alignment of two-dimensional transition metal dichalcogenides: Application in tunnel field effect transistors. Applied Physics Letters 103, 053513 (2013). [8] Kang, J., Tongay, S., Zhou, J., Li, J., and Wu, J. Band offsets and heterostructures of twodimensional semiconductors. Applied Physics Letters 102, 012111–012114 (2013). [9] Komsa, H.-P. and Krasheninnikov, A. V. Electronic structures and optical properties of realistic transition metal dichalcogenide heterostructures from first principles. Physical Review B 88, 085315 (2013). [10] Kośmider, K. and Fernández-Rossier, J. Electronic properties of the MoS2–WS2 heterojunction. Physical Review B 87, 075451 (2013). [11] Terrones, H., Lopez-Urias, F., and Terrones, M. Novel hetero-layered materials with tunable direct band gaps by sandwiching different metal disulfides and diselenides. Scientific Reports 3, 1549 (2013). [12] Debbichi, L., Eriksson, O., and Lebègue, S. Electronic structure of two-dimensional transition metal dichalcogenide bilayers from ab initio theory. Physical Review B 89, 205311 (2014). [13] Yu, H., Wang, Y., Tong, Q., Xu, X., and Yao, W. Anomalous light cones and valley optical selection rules of interlayer excitons in twisted heterobilayers. Physical Review Letters 115, 187002 (2015). [14] Mak, K. F., Lee, C., Hone, J., Shan, J., and Heinz, T. F. Atomically thin MoS2: a new directgap semiconductor. Physical Review Letters 105, 136805 (2010). [15] Splendiani, A. et al. Emerging photoluminescence in monolayer MoS2. Nano Letters 10, 1271–1275 (2010). [16] Liu, G. B., Xiao, D., Yao, Y., Xu, X., and Yao, W. Electronic structures and theoretical modelling of two-dimensional group-VIB transition metal dichalcogenides. Chemical Society Reviews 44, 2643–2663 (2015). [17] Mattheiss, L. F. Band Structures of Transition-Metal-Dichalcogenide Layer Compounds. Physical Review B 8, 3719–3740 (1973).

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[18] Xiao, D., Liu, G. B., Feng, W., Xu, X., and Yao, W. Coupled spin and valley physics in monolayers of MoS2 and other group-VI dichalcogenides. Physical Review Letters 108, 196802 (2012). [19] Wilson, N. R. et al. Band parameters and hybridization in 2D semiconductor heterostructures from photoemission spectroscopy. arXiv:1601.05865 (2016). [20] Wang, L. et al. One-dimensional electrical contact to a two-dimensional material. Science 342, 614–617 (2013). [21] Zomer, P. J., Guimarães, M. H. D., Brant, J. C., Tombros, N., and van Wees, B. J. Fast pick up technique for high quality heterostructures of bilayer graphene and hexagonal boron nitride. Applied Physics Letters 105, 013101 (2014). [22] Wang, Q. H., Kalantar-Zadeh, K., Kis, A., Coleman, J. N., and Strano, M. S. Electronics and optoelectronics of two-dimensional transition metal dichalcogenides. Nature Nanotechnology 7, 699–712 (2012). [23] Kumar, N. et al. Second harmonic microscopy of monolayer MoS2. Physical Review B 87 (2013). [24] Li, Y. et al. Probing symmetry properties of few-layer MoS2 and h-BN by optical secondharmonic generation. Nano Letters 13, 3329–3333 (2013). [25] Malard, L. M., Alencar, T. V., Barboza, A. P. M., Mak, K. F., and de Paula, A. M. Observation of intense second harmonic generation from MoS2 atomic crystals. Physical Review B 87 (2013). [26] Lin, Y. C. et al. Atomically thin resonant tunnel diodes built from synthetic van der Waals heterostructures. Nature Communications 6, 7311 (2015). [27] Yu, Y. et al. Equally efficient interlayer exciton relaxation and improved absorption in epitaxial and nonepitaxial MoS2/WS2 heterostructures. Nano Letters 15, 486–491 (2015). [28] Gong, Y. et al. Vertical and in-plane heterostructures from WS2/MoS2 monolayers. Nature Materials 13, 1135–1142 (2014). [29] Ceballos, F., Bellus, M. Z., Chiu, H. Y., and Zhao, H. Ultrafast charge separation and indirect exciton formation in a MoS2–MoSe2 van der Waals heterostructure. ACS Nano 8, 12717–12724 (2014). [30] Hong, X. et al. Ultrafast charge transfer in atomically thin MoS2/WS2 heterostructures. Nature Nanotechnology 9, 682–686 (2014). [31] Rivera, P. et al. Observation of long-lived interlayer excitons in monolayer MoSe2–WSe2 heterostructures. Nature Communications 6, 6242 (2015). [32] Fang, H. et al. Strong interlayer coupling in van der Waals heterostructures built from single-layer chalcogenides. Proceedings of the National Academy of Sciences of the United States of America 111, 6198–6202 (2014). [33] Chiu, M. H. et al. Spectroscopic signatures for interlayer coupling in MoS2–WSe2 van der Waals stacking. ACS Nano 8, 9649–9656 (2014). [34] Lee, C. H. et al. Atomically thin p–n junctions with van der Waals heterointerfaces. Nature Nanotechnology 9, 676–681 (2014). [35] Furchi, M. M., Pospischil, A., Libisch, F., Burgdorfer, J., and Mueller, T. Photovoltaic effect in an electrically tunable van der Waals heterojunction. Nano Letters 14, 4785–4791 (2014). [36] Cheng, R. et al. Electroluminescence and photocurrent generation from atomically sharp WSe2/MoS2 heterojunction p–n diodes. Nano Letters 14, 5590–5597 (2014). [37] Heo, H. et al. Interlayer orientation-dependent light absorption and emission in monolayer semiconductor stacks. Nature Communications 6, 7372 (2015).

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[38] Ceballos, F., Bellus, M. Z., Chiu, H. Y., and Zhao, H. Probing charge transfer excitons in a MoSe2–WS2 van der Waals heterostructure. Nanoscale 7, 17523–17528 (2015). [39] Butov, L. V., Gossard, A. C., and Chemla, D. S. Macroscopically ordered state in an exciton system. Nature 418, 751–754 (2002). [40] Snoke, D., Denev, S., Liu, Y., Pfeiffer, L., and West, K. Long-range transport in excitonic dark states in coupled quantum wells. Nature 418, 754–757 (2002). [41] Leonard, J. R. et al. Spin transport of excitons. Nano Letters 9, 4204–4208 (2009). [42] Schuller, J. A. et al. Orientation of luminescent excitons in layered nanomaterials. Nature Nanotechnology 8, 271–276 (2013). [43] Xu, X., Yao, W., Xiao, D., and Heinz, T. F. Spin and pseudospins in layered transition metal dichalcogenides. Nature Physics 10, 343–350 (2014). [44] Rivera, P. R. et al. Valley-polarized exciton dynamics in a 2D semiconductor heterostructure. Science 351 (6274), 688–691 (2015). [45] Dufferwiel, S. et al. Exciton-polaritons in van der Waals heterostructures embedded in tunable microcavities. Nature Communications 6 (2015). [46] Palummo, M., Bernardi, M., and Grossman, J. C. Exciton radiative lifetimes in twodimensional transition metal dichalcogenides. Nano Letters 15, 2794–2800 (2015). [47] Moody, G. et al. Intrinsic homogeneous linewidth and broadening mechanisms of excitons in monolayer transition metal dichalcogenides. Nature Communications 6 (2015). [48] Korn, T., Heydrich, S., Hirmer, M., Schmutzler, J., and Schüller, C. Low-temperature photocarrier dynamics in monolayer MoS2. Applied Physics Letters 99, 102109 (2011). [49] Fogler, M. M., Butov, L. V., and Novoselov, K. S. High-temperature superfluidity with indirect excitons in van der Waals heterostructures. Nature Communications 5, 4555 (2014).

18

TMDs – Optoelectronic Devices Thomas Mueller

18.1

Introduction Transition metal dichalcogenides (TMDs) offer properties that complement those of graphene. Graphene is a semimetal with high carrier mobility and peculiar electrical transport behavior, but the lack of a band gap hampers its use in many optoelectronic applications such as lasers or photovoltaics. In contrast, semiconducting TMDs (in particular, MoS2, MoSe2, WS2, WSe2, MoTe2, and WTe2) possess a sizeable band gap of the order of 1–2.5 eV. A layer-dependent indirect-to-direct band-gap transition was observed in these materials, which resulted in bright photoluminescence from monolayers, whereas photoluminescence from bulk was found to be practically absent [1, 2]. The reported photoluminescence quantum yields are still poor. Likely, unintentional doping and crystal defects play a major role in suppressing the luminescence. The room temperature yield of monolayer MoS2, for example, was found to be [1] 4
10–3. In transition metal selenides, yields an order of magnitude better were obtained [3, 4]. The optical properties of ultra-thin TMDs are dominated by excitonic rather than band-to-band transitions. Excitons are strongly bound in two-dimensional (2D) materials due to reduced dielectric screening and enhanced electron–hole interactions. Several excitonic transitions can be observed, where the two lowest energy peaks, known as A and B-excitons, arise from optical transitions between the valence bands with strong spin–orbit splitting and the weakly split conduction bands. [1] The exciton binding energies of mono- and few-layer TMDs have been determined by various experimental and theoretical techniques [5–11]. The binding energies increase with decreasing material thickness and are typically of the order of ~0.5 eV in monolayers. Excitons in 2D semiconductors are thus stable at room temperature and dominate the optical and optoelectronic properties. The strong exciton confinement in TMDs further results in short radiative lifetimes, of the order of tens of picoseconds, which should enable light emission with high modulation bandwidth. The band structure of TMD monolayers exhibits two non-equivalent valleys, K and K0 , located at the corners of the first Brillouin zone. The broken inversion symmetry in TMD monolayers, together with time reversal symmetry, gives rise to spin–valley coupling, i.e. the sign of the hole spin is opposed in the K and K0 valleys. This leads to valley-dependent optical selection rules for interband transitions: right-handed circularly polarized light only couples to the K-valley; left-handed circularly polarized 329

330

TMDs – Optoelectronic Devices

light couples to the K 0 -valley [12, 13]. The ability to control and detect the valley polarization, not only optically but also electrically, could provide a basis for constructing a new generation of valley-dependent optoelectronic devices for information processing.

18.2

Light-Emitting Diodes and Lasers Electroluminescence is the generation of light by a current passing through an electronic device. It is the result of radiative recombination of electron–hole pairs (or excitons) in a semiconductor and forms the basis of light-emitting diodes (LEDs) and semiconductor lasers. Radiative recombination occurs much more efficiently in direct band-gap semiconductors than in indirect gap materials, which is why TMD monolayers are usually preferred over multi-layers or bulk. Nevertheless, some works have reported strong electroluminescence in multi-layer TMDs, either by carrier redistribution from the indirect to the direct valleys in a high electric field [14] or by inducing a very high carrier concentration to fill the indirect valleys in the conduction and valence bands [15].

18.2.1

Hot-Carrier Electroluminescence Electroluminescence from a 2D semiconductor was first reported by Sundaram et al. [16] by driving a unipolar (electron) current through a MoS2 monolayer and exploiting hot-carrier processes for electron–hole pair (exciton) generation. Under high drainsource bias, strong band bending occurs at the metal contacts of a TMD field-effect transistor. Electrons that are injected from the contact into the semiconductor are accelerated by the electric field. If these electrons acquire a kinetic energy that is larger than the MoS2 band gap before being scattered by phonons, electron–hole pairs can be generated via impact excitation. This process is known to be very efficient in lowdimensional systems due to the strong electron–hole interactions [17, 18]. The large exciton binding energy in low-dimensional materials prevents the excitons from being dissociated by the strong electric field. Characteristic features of impact excitation are a threshold behavior of the electroluminescence (electrons need to acquire a kinetic energy of at least e1:5E g , where E g is the semiconductor band gap) and the exponential increase of the emission above threshold [17]. Spatial mapping of the light emission confirmed that the electroluminescence is localized near the metal-semiconductor interface where the electric field is highest. The electroluminescence efficiency was estimated to be 105, and the emission peaked at the MoS2 A-exciton wavelength in accordance with photoluminescence measurements.

18.2.2

Light-Emitting Diodes LEDs are based on semiconductor p–n junctions, in which both electrons and holes are injected. Both types of carriers recombine radiatively, releasing energy in the form of photons. This process is thresholdless and more efficient than electron–hole pair generation by impact excitation. In a traditional LED, the semiconductor material is

18.2 Light-Emitting Diodes and Lasers

331

doped with impurities to create the p- and n-regions. As stable chemical doping is currently difficult to achieve in TMDs, electrostatic doping was used by Pospischil et al. [19], Baugher et al. [20], and Ross et al. [21] to realize TMD monolayer p–n junctions. In these devices, split-gate electrodes couple to two different regions of a TMD, as illustrated in Fig. 18.1(a). By biasing one gate electrode with a positive voltage and the other with a negative voltage, electrons and holes, respectively, are drawn into the transistor channel and a lateral junction is formed. As ambipolar electrical transport behavior is difficult to achieve in MoS2 monolayers, and has as yet only been accomplished using ionic liquid gating [22], WSe2 monolayers were employed in [19–21]. Applying voltages of opposite polarities to the gate electrodes led to diode-like current rectification, whereas applying voltages of same polarity resulted in resistive behavior. The devices also showed a photovoltaic effect under illumination. By driving a forward current through the electrostatically defined junction, electroluminescence emission was obtained. The emission occurred at the same wavelength as the photoluminescence in the near infrared spectral regime (~750 nm), indicating that electrons and holes form excitons before radiative recombination. Electroluminescence in such devices was measureable at injection currents as low as [21] 200 pA and with efficiencies of 0.1%, mainly limited by resistive losses at the contacts and non-radiative recombination. The spectral weight of emission from the neutral and charged excitons was found to be tunable by the bias current [21]. Similar studies, performed with ionic liquid gated WS2, yielded visible light emission at ~630 nm wavelength [23].

18.2.3

3D/2D van der Waals Heterojunctions Devices based on vertical structures, in which the current flow is perpendicular to the 2D layers, have also been demonstrated. The vertical heterostructure design allows the performance to be improved in several respects: (i) reduced contact resistance due to the larger contact area, (ii) higher current densities which allows for brighter emission, (iii) luminescence from the whole device area, and (iv) easier scalability. Vertical structures can be fabricated by stacking of 2D materials in a layered configuration on top of each other, or on top of a traditional 3D semiconductor such as silicon. The van der Waals interaction between the 2D (or 2D and 3D) materials keeps the heterostack together. Electrically driven light emission from vertical 3D/2D heterojunctions (Fig. 18.1(b)) was demonstrated by Ye et al. [24] and Lopez-Sanchez et al. [25]. Their devices were composed of a MoS2 monolayer, which is intrinsically n-doped, and a p-doped silicon wafer. The p-doped silicon served as a hole injection layer into MoS2, whereas electrons were injected into MoS2 through a standard metal contact. Light emission occurred across the entire face of the junction [25]. The electroluminescence originated from excitons related to the optical transitions between the conduction and valence bands in MoS2. Both A- and B-excitons were observed, with the B-exciton likely being excited by impact excitation. Radiative emission from silicon was not observed due to its indirect band gap. In addition, bound-exciton related emission features were identified and, at a high injection rates, exciton–exciton annihilation [26] of the bound excitons was studied [24]. Another work utilized p-doped GaN to inject holes into n-type MoS2 for light emission [14].

332

TMDs – Optoelectronic Devices

Fig. 18.1 (a) Schematic drawing of a TMD light-emitting diode. Split-gate electrodes couple to two

different regions of a WSe2 monolayer so that electrons/holes are drawn into the WSe2 channel and a lateral junction is formed. Reproduced with permission from [19]. (b) Schematic drawing of a bulk silicon/MoS2 monolayer heterojunction device and corresponding band diagram. Reproduced with permission from [25]. (c) Schematic of a Si/SiO2/h-BN/Gr/h-BN/MoS2/h-BN/ Gr/h-BN heterostructure LED (Gr, graphene; h-BN, hexagonal boron nitride) and band diagram of the biased heterostructure. Reproduced with permission from [27].

18.2.4

2D van der Waals Heterostructures Figure 18.1(c) schematically depicts the architecture of a van der Waals heterojunction that is entirely composed of 2D materials [27]. It was made by stacking of metallic (graphene, Gr), insulating (hexagonal boron nitride, h-BN) and semiconducting (TMD monolayers, WS2 or MoS2) 2D crystals, to realize a vertical LED. In this device, electrons and holes are injected into a TMD monolayer from two graphene electrodes. Thin boron nitride layers to both sides of the TMD serve to reduce the direct tunneling between the graphene sheets, resulting in electron and hole accumulation in the semiconductor. Electroluminescence emission was obtained above a threshold that can be associated with the alignment of the top and bottom graphene Fermi levels with the TMD conduction and valence band edges, respectively. The electroluminescence emission wavelength was found to be similar to that of the photoluminescence emission from negatively charged excitons, suggesting more efficient electron than hole

18.3 Photovoltaic Devices

333

injection. An external quantum efficiency of more than 1% was obtained – ten times larger than that of the planar p–n diodes. This improvement most likely is due to the lower contact resistance. The quantum efficiency was further improved by stacking of multiple devices in the series. At low temperature, efficiencies of up to 8.4% were reported, comparable to those of state-of-the-art organic LEDs. Also samples on elastic and transparent substrates were prepared, demonstrating the capability of such devices for flexible and semitransparent optoelectronics.

18.2.5

Lasing in TMDs It has been shown that coupling a TMD light emitter to an optical cavity can significantly enhance the spontaneous emission rate due to the Purcell effect [28, 29]. Using cavities with a high-quality factor, Wu et al. [30], Salehzadeh et al. [31], and Ye et al. [32] demonstrated that sufficient optical gain can be provided in a TMD by optical pumping to compensate for the cavity losses and to achieve lasing. A photonic crystal cavity (Fig. 18.2) was utilized in [30], whereas microdisk cavities were employed in [31] and [32]. Both cavity designs offer the potential for ultra-low threshold lasing, and continuous-wave operation, and an optical pumping threshold as low as 27 nW at 130 K was reported [30]. Hallmark features of a laser are the abrupt changes of the slope of the output light intensity and the emission linewidth around the lasing threshold. Both features were indeed observed, but photon statistics measurements still need to be demonstrated.

18.3

Photovoltaic Devices Photovoltaic cells allow the conversion of sun energy into electricity. Typically, inorganic or organic semiconductors are used, but also TMDs have long been considered

Fig. 18.2 Monolayer WSe2 nanolaser; device architecture with electric-field profile of the cavity mode (left), and polarization-resolved emission spectrum (right). An optical image of the device is shown in the inset (reproduced with permission from [30]).

334

TMDs – Optoelectronic Devices

for photovoltaics, because of their band gaps in the visible part of the electromagnetic spectrum and their strong optical absorption [33]. From a technological standpoint, abundance of source materials, chemical stability, environmental sustainability, and low cost of production are essential requirements. All these demands can potentially be fulfilled by TMDs, making them interesting candidates for flexible thin film solar cells. The electronic band gap in 2D semiconductors is higher than the optical gap due to the weak screening and large exciton binding energies. As in organic solar cells, photogenerated excitons must thus disassociate into free carriers to generate a current. Theoretical studies of graphene/MoS2 [34] and WS2/MoS2 [35] (all monolayers) heterojunction solar cells have suggested achievable power conversion efficiencies of ~1%, limited by the low optical absorption of the ultra-thin materials. Absorption enhancement using plasmonics or vertical stacking of 2D layers will thus be necessary for these devices to become competitive with other technologies.

18.3.1

Lateral p–n and Schottky Junctions The photovoltaic properties of bulk TMDs have been studied experimentally since the 1980s [33]. The photovoltaic properties of monolayers were studied only recently using p–n and Schottky junctions. When biased in diode configuration, the device presented in Fig. 18.1(a) showed a photovoltaic response [19, 20]. As expected, the electrical device characteristics were not affected by light when the device was operated as resistor. Despite the low optical absorption of the monolayer material, power conversion efficiencies (the percentage of the incident light energy that is converted into electrical energy) of ~0.5% were obtained, comparable to those reported for the early bulk devices [33]. For a similar device layout with multi-layer MoSe2, power conversion efficiencies surpassing 5% and fill factors of 70% were reported [36]. Similar experiments were performed using other 2D semiconductors, such as black phosphorus [37]. Multijunction solar cells could hence potentially be constructed by stacking of 2D materials with different band gaps, which would allow for more efficient absorption of the solar spectrum without losses due to carrier thermalization. Lateral built-in fields in 2D semiconductors also occur at Schottky junctions. Fontana et al. [38] studied MoS2 field-effect transistors with two different contact metals – palladium and gold. Upon illumination, photogenerated electron–hole pairs were separated by the built-in potential from the space charge regions at the contacts. Electrons accumulated on the gold contacts and holes on the palladium side, which gave rise to an open circuit voltage and a sizeable photovoltaic effect.

18.3.2

2D van der Waals Heterostructures Although experimental studies of lateral junctions have shown encouraging results, the complex electrode arrangement does not allow for easy scalability for which a vertical geometry would be desirable. Such vertical heterostructures can be obtained by stacking of 2D materials in a layered configuration to realize a van der Waals heterostructure. Photovoltaic effects in TMD van der Waals heterojunctions were first demonstrated by Furchi et al. [39], Lee et al. [40], and Cheng et al. [41]. The device structure is depicted in Fig. 18.3(a). The electron affinity (the energy required to excite an electron from the

18.3 Photovoltaic Devices

335

Fig. 18.3 (a) Left: Schematic drawing and energy band diagram of a WSe2/MoS2 heterojunction solar cell. The lowest energy electron states are spatially located in the MoS2 layer and the highest energy hole states are located in the WSe2 sheet. Right: Electrical characteristics of the device under optical illumination with 180–6400 W/m2. Inset: Electrical power that is extracted from the device (reproduced with permission from [39]). (b) Schematic and band diagram of a WSe2/MoS2 heterojunction embedded between two graphene layers for vertical carrier extraction (reproduced with permission from [40]).

bottom of the conduction band into the vacuum) of MoS2 is larger than that of WSe2. The lowest energy electron states are thus spatially located in the MoS2 layer and the highest-energy hole states lie in the WSe2, forming a type-II heterojunction (schematic energy band diagram in the inset of Fig. 18.3(a)). Photoluminescence studies and optical pump/probe measurements confirmed efficient and ultra-fast charge transfer between TMDs with almost 100% efficiency and 6.6 MV/cm).

22.2 Anisotropic Response of Black Phosphorus

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Fig. 22.3 (a) Schematic image of a back-gate BP field-effect transistor. (b) Output and (c) Transfer

characteristics of a BP field-effect transistor. (Adapted with permission from [19]. Copyright 2014 American Chemical Society.)

Fig. 22.4 Field-effect mobility (red open circles) and Hall mobility (filled squares, three different values of n) as a function of temperature on a logarithmic scale. A power-law dependence μ  T –0.5 (black dashed line) is plotted in the high-temperature region. (Adapted with permission from [20]. Copyright 2014 Nature Publisher.)

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Anisotropic Properties of Black Phosphorus

This is attributed to the narrow band-gap nature of BP, where the Fermi-level can be tuned either near the valence band, leading to the hole conduction, or the conduction band, leading to the electron conduction. Many layered materials can be exfoliated from bulk down to one single atomic layer, but only a small number of them are stable in an ambient atmosphere. The reduced chemical stability of 2D materials is associated with the energy needed to maintain stable bonding configurations, which is affected by electrostatics and structural buckling. Thus, ambient stability is likely to be a big concern for BP since phosphorus atoms have free lone pairs and valence bond angles of 102 [61]. The surface instability has been observed in several studies regarding the isolation of BP thin layers [22, 23], but the mechanism of surface degradation is not well understood. Researchers have suggested that O2 saturated H2O irreversibly reacts with BP to form oxidized phosphorus species, with a significant cost of degraded electronic characteristics [62–64]. Certain passivation techniques have been proposed recently, such as atomic layer deposited AlxOy [65–67], hydrophobic polymer [68], and multi-layer h-BN [69–73]; thus the BP transistors can sustain good electrical performance for weeks and months. Li et al. recently achieved enhanced Hall mobility in BP at low temperatures [69]. The device is accomplished by constructing a van der Waals heterostructure with few-layer BP sandwiched between two h-BN flakes, and being placed on the graphite back-gate. The top h-BN flake not only provides as effective passivation layer to avoid flake degradation from ambient exposure it also prevents surface scattering from neighboring adsorbates. More importantly, the bottom h-BN allows the electrons in the graphite to screen the impurity potential at the BP/h-BN interface, as the impurities at the interface are mainly responsible for limited low-temperature Hall mobility of BP. High-quality h-BN-encapsulated BP has brought arrier Hall mobility up to 6000 cm2/V s at low temperature. Shubnikov–de Haas oscillations and quantum Hall effects are observed in BP [69–73]. Contact to 2D materials is another grand challenge in 2D electronics in general. The demonstrated TMDs and BP transistors are Schottky-barrier transistors with significant large contact resistance due to metal/2D Schottky barriers. The contact resistance on BP transistors is around 1.0–2.0 Ωmm [38], smaller than the contact resistance in MoS2 in general, due to its narrower band-gap. Here, we discuss the anisotropic contact resistance on BP due to its anisotropic atomic structures and electrical transport properties. When current flows from the semiconductor to the metal, it encounters the resistances at the metal contacts. Contact resistance measurement determines the specific contact resistivity which is not the resistance of the metal-semiconductor interface alone, but it is a practical quantity describing the total real contact [74]. Therefore, the semiconductor/metal contact should be more fairly represented by the equivalent circuit with the current choosing the path of least resistance, and the potential distribution under the contact is determined by both resistors and can be written as follows [74]: pffiffiffiffiffiffiffiffiffiffiffi I Rsh ρc cosh ½ðL  xÞ=LT  V ð xÞ ¼ , (22.1) sinh ðL=LT Þ W where ρc is semiconductor/metal interface resistance, Rsh is sheet resistance underneath the metal contact, x is the lateral distance from the contact edge, L and W are the contact

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Table 22.1 Contact resistance, and corresponding field-effect mobility for BP transistors. Device Set

Mobility (cm2/V s)

Contact Resistance (Ωmm)

1 2 3 4

151.1 108.7 92.9 76.8

1.7 2.4 3.4 4.6

Fig. 22.5 Resistor network of BP contact structure, ρc is semiconductor/metal interface resistance, Rsh is sheet resistance underneath the metal contact.

length and width, and I is the current flowing into the contact. The voltage is highest near the contact edge and drops nearly exponentially with distance. Usually, the “1/e” distance of the voltage drop is defined as the transfer length LT and can be expressed as LT = (ρc/Rsh)1/2. In conventional Si MOSFETs, both ρc and Rsh are almost fixed numbers for implanted regions, while in BP transistors, they are modulated by gate voltage. In the resistor network of the BP contact structure, as shown in Fig. 22.5, contact resistance depends on the semiconductor/metal interface resistance ρc, as well as the sheet resistance Rsh. More importantly, the anisotropic characteristic-induced deviation in mobility would lead to different sheet resistances, such that the BP contact resistance should experience anisotropic property as well. The interpretation is further supported by the observation of anisotropic contact resistance in electric measurements. Through studying transistor behaviors with various channel lengths, the contact resistance on BP can be extracted from the transfer length method (TLM). Flakes with similar thicknesses have been patterned upon the TLM structure with 3 μm, 2 μm, 1.5 μm, 1 μm, and 500 nm channel lengths, and source–drain metals are kept as Ni/Au. Contact resistance, and corresponding field-effect mobility at a channel length of 1 μm for each independent sets were summarized in Table 22.1. It is evident that the BP flake with highest mobility generates the lowest contact resistance, which successfully support our previous theoretical prediction about the anisotropic characteristic of BP contact resistance.

22.2.2

Thermal Anisotropic Behavior In order to study the thermal anisotropic behavior, we need to identify the atomic structure and crystal orientation of BP films first. It has been shown experimentally by

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Anisotropic Properties of Black Phosphorus

many groups that the photonic method could also examine the BP axes [21, 75–79]. In fact, the polarization-resolved Raman spectroscopy offers an important method to accurately determine the crystalline orientation of thin film BP, which is especially useful for such a relatively small flake due to the small laser spot size. In Raman spectroscopy, the laser light interacts with molecular vibrations, phonons, or other excitations in the system, resulting in the energy of the laser photons being shifted up or down. The shift in energy gives sufficient information about the vibrational modes in the system, so as to examine the atomic bond structure. Previous studies of Raman spectra have shown that there are three prominent active modes in BP, and they can be observed at around 365, 440, and 470 cm–1, corresponding to three Ag1, B2g, and Ag2 vibration modes, respectively [19, 21, 77]. The out-of-plane Ag1 mode occurs due to opposing vibrations of top and bottom phosphorus atoms with respect to each other. The B2g mode that describes the bond movement along the in-plane zig-zag direction is associated with the Ag2 mode, which has the dominate component along the in-plane armchair vibration, as depicted in Fig. 22.6(a) [75]. To observe the polarized Raman scattering, a linear polarizer is placed at the spectrometer entrance. With the detection polarization perpendicular (VH configuration) or parallel (VV configuration) to the incoming laser polarization, the optical phonon modes of different symmetries can be selected or eliminated when lattice principal axes are aligned with the laser polarization. Specifically in BP, the Ag modes and the B2g mode can be filtered out in VH and VV configurations, respectively, when either the armchair or zig-zag axis is aligned with the laser polarization [75, 76]. An experimental demonstration of a polarization-resolved Raman spectrum was conducted by Luo et al. [75], shown in Fig. 22.6(b). The solid lines represent the two principal axes under the VV configuration, where the B2g mode is eliminated. Meanwhile, VH configurations of the zig-zag and the armchair axis are elucidated in dashed lines with a single peak located at the B2g mode. To further distinguish the two principal axes, we looked into the Ag2/Ag1 Raman intensity ratio in the VV configuration. The armchair-oriented atomic vibrations of Ag2 phonons lead to maximized Ag2 Raman intensity when laser polarization is along the armchair direction, while the Ag1 Raman intensity remains unchanged because the Ag1 phonon vibrations are out-of-plane. Therefore, the Ag2/Ag1 intensity ratio becomes larger (~2) with armchair-polarized laser excitation, and smaller (~1) with zig-zag-polarized laser excitation, which serves as Raman signatures of armchair and zig-zag lattice axes. We should notice that, it is insufficient to claim BP Raman spectroscopy without understanding the angular-dependent Raman spectrum. In this case, the peak positions of modes do not change as the excitation light polarization varies with flake rotation; however, the relative intensities of these three peaks do change significantly with the polarization direction. Mathematical calculation of Raman modes has been introduced in the first place to help us understand angular-dependent polarization-resolved Raman measurement [75]. Even though the electronic and photonic anisotropic properties of BP have been widely reported, thermal transport of BP is still in its infancy. Recently, the thermoelectric power of bulk BP has been investigated, indicating BP as an efficient thermoelectric material with a relatively high temperature of 380 K [80]. First-principles calculations

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Fig. 22.6 (a) Atom vibration modes in BP Raman spectra system. (b) Polarized-Raman spectra with respect to VV and VH measurements. (Reproduced with permission from [75]. Copyright 2015 Nature Publishing Group.)

showed that the existence of anisotropy was related to asymmetric phonon dispersion, whereas the intrinsic phonon scattering rates were found to be similar along the armchair and zig-zag directions. Surface scattering in the phosphorene thin films was shown to strongly suppress the contribution of long-mean-free-path acoustic phonons. Moreover, calculations also raised interest in BP in thermoelectric applications, claiming that because of the anisotropic lattice structure, the armchair direction of BP possesses high electrical conductivity but low lattice thermal conductivity. On the other hand, the zig-zag direction might not be a favorite choice for electronic transport of BP due to relatively small carrier mobility; instead, it accelerated thermal transport in the channel [81–84]. Experimental demonstration of anisotropic in-plane thermal conductivity of suspended BP measured by micro-Raman spectroscopy was reported recently by Luo et al. at Purdue University [75]. The candidate BP flakes were firstly transferred onto 3 μm wide slits fabricated on 200 nm thick free-standing SiN membranes. Meanwhile, two slits were patterned to be mutually perpendicular to form a “T” shape, so as to investigate separately the armchair and zig-zag thermal transport on the same flake. It should be emphasized that the rectangular geometry along with a nearly one-dimensional laser heater source in the center guarantees that the heat conduction in the BP sample was most sensitive to the thermal conductivity perpendicular to the slit. Raman thermometer calibration was first conducted on a 9.5 nm-thick BP sample,

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Anisotropic Properties of Black Phosphorus

Fig. 22.7 Raman thermometer calibration results of the 9.5 nm-thick BP film. (a) Four sample Raman spectra taken at 24, 42, 57, and 72  C with armchair-polarized laser. The dashed lines correspond to the peak positions at 24  C. (b) The Ag2 Raman shift as a function of temperature for both armchair- and zig-zag-polarized laser. The dashed lines show linear fit results. (Reproduced with permission from [75]. Copyright 2015 Nature Publishing Group.)

with laser temperature rises from a room temperature of 24 up to 72  C. As shown in Fig. 22.7(a), the zig-zag-polarized excitation yielded temperature coefficients of larger absolute values, which might be caused by anisotropic thermal expansion during the laser heating process. In the Raman spectroscopy, the Ag2 mode was found to be the most sensitive to temperature changing among a total of three modes, as well as showing the highest Raman intensity. Therefore, Ag2 mode has been selected and plotted in Fig. 22.7(b) to evaluate anisotropic thermal expansion of BP, indicating a significant anisotropic thermometer along these two principle axes. Micro-Raman thermal conductivity experiments were performed at room temperature to investigate detailed anisotropic in-plane anisotropic behavior. To best achieve the desired onedimensional heat transfer, rectangular aperture was placed in front of the objective lens to produce a laser focal line instead of a circular spot. In the direction where the aperture cut the laser beam, the partially filled objective lens aperture produced a larger width at the focal point, yielding a stretched line-shaped focal spot. Figure 22.8(a) plots the Raman-measured temperature rise θRaman versus the absorbed laser power PA for the suspended 9.5 nm thick BP film in which the data show a linear correlation similar to other micro-Raman experiments. The absorbed laser power was determined by PA = AP = (1 – R – T)P, where P is the incident laser power, A is the absorptivity, R is the reflectivity, and T is the transmissivity. The reflectivity of BP films was measured using a beam splitter in the incident laser path, which deviated from the reflected light to a separate path where its intensity was measured; by comparing the reflected light intensity of the BP films and a silvercoated mirror as a reference, the reflectivity of BP films could be calculated. The transmissivity of the BP films was measured under the slit, by dividing the transmitted laser intensity on the BP-covered slit by that at the adjacent empty slit. Note that

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Fig. 22.8 Thermal conductivity measurements of BP using the micro-Raman technique. (a) Laser-power dependent temperature rise (θRaman) of the 16.1 nm-thick BP film determined by the micro-Raman spectroscopy along armchair and zig-zag transport directions. The dashed lines are linear fits. (b) Extracted armchair and zig-zag in-plane thermal conductivities of multiple BP films. Dashed lines are the results of theoretical modeling. The gray error bars account for the uncertainty of SiN substrate thermal conductivity kSiN, while the black error bars do not. (c) The anisotropic ratio kzig-zag/karmchair at different BP thicknesses. The ratio at 12 nm is calculated using linearly interpolated armchair thermal conductivity from adjacent thicknesses. (Reproduced with permission from [75]. Copyright 2015 Nature Publishing Group.)

A, R, and T are all anisotropic quantities due to anisotropic optical conductivity, and the measured absorptivity of the 9.5 nm-thick suspended film (~12.0% for armchair polarization and ~2.9% for zig-zag polarization) agrees well with the theoretical predictions. Due to the much higher absorption of armchair-polarized light in BP films, all the thermal conductivity measurements were carried out with an armchair-polarized laser beam in order to reduce the uncertainty of PA. The measured anisotropic in-plane thermal conductivity values of BP are summarized in Fig. 22.8(b). As expected, the BP has demonstrated significant anisotropic thermal transport behavior in the channel, where armchair and zig-zag thermal conductivities are ~20 and ~40 W m–1 K–1 for BP films thicker than 15 nm, respectively. In addition, how thermal conductivity varies with flake thickness is also included in the study and is shown in Fig. 22.8(c). The

426

Anisotropic Properties of Black Phosphorus

thermal conductivity anisotropic ratio kzig-zag/karmchair, which drops from ~2 to ~1.5 as BP thickness decreases to less than 10 nm. Large anisotropy in-plane thermal conductivity of BP nanoribbons at temperatures higher than 100 K was experimentally observed very recently [85]. It was evident that zig-zag and armchair had similar thermal conductivity at lower temperatures between 30 and 100 K. However, zig-zag nanoribbon had a higher κ than armchair nanoribbon, by as much as ~7 W m–1 K–1 at temperatures above ~100 K. Furthermore, the ratio of κ along zig-zag and armchair directions, increased with temperature, reaching up to ~2 at ~300 K. The increasing anisotropy ratio was attributed to the increased contribution from phonon–phonon scattering, which became the dominating scattering mechanism at high temperatures. At a constant temperature of 300 K, the thermal conductivity decreased as the nanoribbon thickness was reduced from ~300 nm to ~50 nm, but the anisotropy ratio stayed around 2 within the thickness range. Similar results were also obtained on 138–552 nm thick BP by the Northwestern group using conventional timedomain thermos-reflectance and the beam-offset method [86].

22.2.3

Mechanical Anisotropic Behavior Here, we focus on another property of BP: its unique puckered structure allows BP to exhibit substantial anisotropy in the mechanical properties with respect to strains [87–95]. In this study, Du et al. from Purdue University employed Raman spectroscopy to experimentally demonstrate the strong anisotropic mechanical properties in BP [96]. The geometric anisotropy introduced by the pucker implies that BP would exhibit significant anisotropic lattice vibrations in response to uniaxial strain along armchair or zig-zag directions, which can be observed directly from Raman spectroscopy. Previous studies of Raman spectra have shown that there are three prominent active modes in BP. The cross-plane Ag1 mode occurs due to opposing vibrations of top and bottom phosphorus atoms with respect to each other. The B2g mode describes the bond movement along the in-plane zig-zag direction. The Ag2 mode has a dominate component along the in-plane armchair direction. To start our experiment, few-layer BP was exfoliated from the bulk crystal by the standard scotch tape method and transferred to a conducting Si substrate with a 300 nm SiO2 capping layer. Polarized Raman spectroscopy was utilized to determine the flake orientation. With the detection polarization parallel to the incident laser polarization, the active phonon mode of B2g is not detected due to matrix cancellation when the two principle lattice axes are aligned with the laser polarization. The Ag2/Ag1 Raman intensity ratio can further be used to distinguish the specific armchair or zig-zag axis. The armchair-oriented Ag2 mode intensity is maximized and is about two times higher than the intensity of the Ag1 mode when the laser polarization is along the armchair direction. The intensity of Ag2 is comparable to Ag1 when the laser is aligned along the zigzag direction. The candidate BP flakes were all pre-characterized by the polarized Raman system. A representative 7.3 nm-thick BP flake was used in this study. The armchair and zig-zag lattice axes were determined by the Ag2/Ag1 intensity ratio. We first investigate the evolution of the Raman spectra of BP with uniaxial tensile and compressive strains, summarized in Fig. 22.9(a). The laser polarization is

22.2 Anisotropic Response of Black Phosphorus

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Fig. 22.9 Raman evolution of uniaxial armchair strained BP. (a) Raman spectra of BP for both stretch and compress armchair strains. The dashed lines are here to guide the Raman peak position shift. Raman shift of (b) Ag1, (c) B2g modes in armchair strained BP. The dashed lines show linear fit results, and error bars are determined from Lorentzian peak fittings. (Reproduced with permission from [96]. Copyright American Chemical Society.)

aligned along the zig-zag axis of BP flake, and the strain direction is along armchair direction based on our apparatus set-up [96]. In our work, we used Lorentzian functions to fit the Raman spectra and obtained the peak frequency of each mode at different strains. For unstrained BP, consistent with previous reports [19, 21, 75], we observe the cross-plane vibration mode of Ag1 at ~362 cm–1, and in-plane vibration modes of B2g and Ag2 at ~439 cm–1 and ~467 cm–1, respectively. The Ag1 and B2g modes show the same linear trend of Raman frequency shift with respect to the applied strain, while the rate of frequency shift is different in these two modes. Both Ag1 and B2g modes experience a red-shift when BP is under tensile strain along the armchair direction, with a slope of 1.37 cm–1 %–1 and 1.07 cm–1 %–1, respectively. On the other hand, Ag1 and B2g have a blue-shift at a rate of 1.34 cm–1 %–1 and 0.70 cm–1 %–1 under uniaxial compressive strain, as shown in Figs. 22.9(b) and (c). It is worth mentioning that we did not observe a measurable Raman shift in the Ag2 mode with various armchair strains, and we believe this can be attributed to the fact that the BP structure is anisotropic and is much softer along the armchair direction, compared to the zig-zag direction. The strain sensitivity of BP from the Raman peak positions is greater than that of molybdenum disulfide (MoS2), but slightly smaller than those of carbon

428

Anisotropic Properties of Black Phosphorus

Fig. 22.10 Raman evolution of uniaxial zig-zag strained BP. (a) Raman spectra of BP for both

stretch and compress zig-zag strains. The dashed lines are here to guide the Raman peak position shift. Raman shift of (b) Ag1, (c) Ag2, and (d) B2g modes in zig-zag strained BP. The dashed lines show linear fit results, and error bars are determined from Lorentzian peak fitting. (Reproduced with permission from [96]. Copyright American Chemical Society.)

nanotubes and graphene [97, 98]. The error bars in the figures are extracted from the Lorentzian peak fittings, which are significantly smaller than the strain-induced frequency shift. For the applied strain less than 0.2%, the Raman peak position remains the same at each strain level after multiple loading and unloading cycles, indicating our experiments are highly reliable. In addition, the absence of discrete jumps of any of the three Raman modes under monotonically varied strains assures that the BP flake does not slip against the substrate during strain experiments. To apply strain along the zig-zag direction, the BP sample substrate was rotated by 90 , and the polarized Raman system was introduced again to verify the BP orientation. The corresponding Raman spectra of zig-zag strained BP are presented in Fig. 22.10(a). Figure 22.10(b) illustrates the Ag1 peak position as a function of tensile and compressive zig-zag strain. The dominate Ag1 mode peak overlaps with another peak appearing as a weak shoulder. In this case, we fitted the data by two Lorentzian functions and treated the main peak as the peak for the Ag1 mode. This phenomenon only happens along the armchair lattice of BP flakes, and it has been observed in our previous studies as well [75]. The Ag1 mode shows monotonic behavior for the full range of zig-zag strain with a slope of 0.56 cm–1 %–1 under tensile strain and 1.07 cm–1 %–1 under

22.4 References

429

compressive strain. The Ag2 mode and B2g mode also show monotonic behavior for the full range of zig-zag strain with a slope of 1.98 cm–1 %–1 and 4.69 cm–1 %–1 under tensile strain, 2.73 cm–1 %–1 and 5.46 cm–1 %–1 under compressive strain. In general, the Raman peak shifts are a factor of 5–7 more sensitive under zig-zag strains compared to those under armchair strains. This strong anisotropic mechanical property can be ascribed simply to the softer bonds in armchair direction in BP films.

22.3

Conclusion We have presented here an overview of BP synthesis and its anisotropic property studies. Progress in BP science and technology have been built upon the success in bulk crystal synthesis and layer exfoliation. It is demonstrated that both high pressure and catalyst involved procedures are reliable; however, there still exists an urgency to develop large-area wafer-scale BP thin film synthesis to satisfy future technology development. Moreover, in this chapter, we have carefully reviewed anisotropic properties of BP with the focus on its electrical properties. The anisotropic characteristics of BP, with respect to its electronic, thermal, and mechanic behaviors, have distinguished BP as a unique 2D material among other layer structured crystals.

Acknowledgments This material is based upon work supported by US National Science Foundation (ECCS-1449270), NSF/AFOSR under EFRI 2-DARE Grant EFMA-1433459, and Army Research Office (W911NF-14-1-0572). The authors would like to thank our collaborators A. Neal, J. Maassen, M.S. Lundstrom, Z. Zhu, D. Tomanek, J.C.M. Hwang, Y. Deng, G. Qiu, Y.Q. Wu, X.F. Li, and A.R. Charnas for their contributions on this phosphorene journey.

22.4

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[47] Krebs, H., Weitz, H., and Worms, K. H. Über die Struktur und Eigenschaften der Halbmetalle. VIII. Die katalytische Darstellung des schwarzen Phosphors. Anorg. Allg. Chem., 280, 119–33, 1955. [48] Brown, A. and Rundqvist, S. Refinement of the crystal structure of black phosphorus. Acta Cryst., 19, 684–5, 1965. [49] Maruyama, Y., Suzuki, S., Kobayashi, K., and Tanuma, S. Synthesis and some properties of black phosphorus single crystal. Physica B, 105, 99–102, 1981. [50] Maruyama, Y., Inabe, T., Nishii, T., He, L., Dann, A. J., Shirotani, I., Fahy, M. R., and Willis, M. R. Electrical conductivity of black phosphorus–silicon compound. Synthetic Met., 29, 213–18, 1989. [51] Maruyama, Y., Inabe, T., He, L., and Oshima, K. Electrical conductivity of black phosphorous–germanium compound. Synthetic Met., 43, 4067–70, 1991. [52] Lange, S., Schmidt, P., and Nilges, T. Au3SnP7@black phosphorus: an easy access to black phosphorus. Inorg. Chem., 46, 4028–35, 2007. [53] Nilges, T., Kersting, M., and Pfeifer, T. A fast low-pressure transport route to large black phosphorus single crystals. J. Solid State Chem., 181, 1707–11, 2008. [54] Köpf, M., Eckstein, N., Pfister, D., Grotz, C., Krüger, I., Greiwe, M., Hansen, T., Kohlmann, H., and Nilges, T. Access and in situ growth of phosphorene-precursor black phosphorus. J. Crystal Growth, 405, 6–10, 2014. [55] Yang, Z., Hao, J., Yuan, S., Lin, S., Yau, H. M., Dai, J., and Lau, S. P. Field-effect transistors based on amorphous black phosphorus ultrathin films by pulsed laser deposition. Adv. Mater., 27, 3748–54, 2015. [56] Qiu, G., Nian, Q., Deng, Y., Jin, S., Charnas, A. R., Cheng, G., Ye, P. D. Synthesis of black phosphorus films by ultra-fast laser exfoliation. In preparation. [57] Liu, H., Du, Y., Deng, Y., and Ye, P. D. Semiconducting black phosphorus: synthesis, transport properties and electronic applications. Chem. Soc. Rev., 44, 2732–43, 2015. [58] Takao, Y., Asahina, H., and Morita, A. Electronic structure of black phosphorus in tight binding approach. J. Phys. Soc. Jpn, 50, 3362–9, 1981. [59] Asahina, H., Shindo, K., and Morita, A. Electronic structure of black phosphorus in selfconsistent pseudopotential approach. J. Phys. Soc. Jpn, 51, 1192–9, 1982. [60] Goodman, N. B., Ley, L., and Bullett, D. W. Valence-band structures of phosphorus allotropes. Phys. Rev. B, 27, 7440–50, 1983. [61] Rodin, A. S., Carbalho, A., and Neto, A. H. C. Strain-induced gap modification in black phosphorus. Phys. Rev. Lett., 112, 176801, 2014. [62] Favron, A, Gaufres, E., Fossard, F., Levesque, P. L., Heureux, A. P., Tang, N. Y.-W., Loiseau, A., Leonelli, R., Francoeur, R. S., and Martel, R. arXiv:1408.0345, 2014. [63] Wood, J. D., Wells, S. A., Jariwala, D., Chen, K. S., Cho, E., Sangwan, V. K., Liu, X., Lauhon, L. J., Marks, T. J., and Hersam, M. C. Effective passivation of exfoliated black phosphorus transistors against ambient degradation. Nano Lett., 14, 6964–70, 2014. [64] Molle, A., Grazianetti, C., Chiappe, D., Cinquanta, E., Cianci, E., Tallarida, G., and Fanciulli, M. Hindering the oxidation of silicene with non-reactive encapsulation. Adv. Funct. Mater., 23, 4340–4, 2013. [65] Liu, H., Neal, A. T., Si, M., Du, Y., and Ye, P. D. The effect of dielectric capping on fewlayer phosphorene transistors: Tuning the Schottky barrier heights. IEEE Electron Device Lett., 35, 795–7, 2014.

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23

Optical Properties and Optoelectronic Applications of Black Phosphorus Andres Castellanos-Gomez and Mo Li

23.1

Introduction The recent isolation of black phosphorus (a layered material composed of phosphorus atoms held together by strong in-plane bonds forming layers that interact between them through weak van der Waals forces) has unleashed the interest of the community working on 2D materials because of its interesting and attractive electronic properties including: narrow intrinsic gap, ambipolar field effect, and high carrier mobility [1–11]. The outstanding electrical properties of black phosphorus, discussed in detail in the previous chapter, has already motivated the fabrication of black phosphorus-based devices such as field-effect transistors [1, 3, 4, 12–21], inverter amplifiers [22]. Apart from its electrical properties, black phosphorus also displays remarkable optical properties [23–27], unmatched by any other 2D materials that have been isolated to date, further motivating the recent surge of works on this novel material. This chapter reviews the main optical properties of black phosphorus, focusing on its thickness-dependent band gap and its unusual in-plane anisotropy. Then, recent works demonstrating black phosphorus optoelectronic devices (phototransistors, high-speed photodetectors and solar) are summarized. Finally, the future perspectives of black phosphorus’ application in optoelectronic are discussed.

23.2

Optical Properties In this section, the main optical properties of black phosphorus will be reviewed, stressing the thickness dependence of the band gap and the characteristic anisotropic optical properties of black phosphorus.

23.2.1

Thickness-Dependent Band Gap In spite of its morphological similarity to graphene, black phosphorus differs considerably from graphene in its electronic properties. While graphene is a zero-gap semiconductor, black phosphorus presents a sizeable gap which makes it attractive for optoelectronic applications [28–30]. Moreover, similarly to other 2D semiconducting materials, the band-gap value strongly depends on the number of layers due to the quantum confinement effect in the out-of-plane direction [31–33]. In order to illustrate 435

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Fig. 23.1 (a) Calculated band structure of monolayer, bilayer, and trilayer black phosphorus (data extracted from [34]). (b) Thickness dependence of the black phosphorus band gap, calculated with different approaches (data extracted from [2, 3, 21, 24, 26, 35–38]). (Adapted with permission from [39]. Copyright 2015 American Chemical Society.)

this effect, Fig. 23.1(a) compares the band structures of monolayer, bilayer, and trilayer black phosphorus obtained through ab initio calculations with the GW approximations [34]. A general feature of all the calculated band structures is that the band gap remains direct at the Γ point of the Brillouin zone. In semiconducting transition, metal dichalcogenides, on the other hand, the gap is at the K point and it is only direct for singlelayers (becoming and an indirect gap semiconductor for multilayer samples) [31, 32]. Another feature, clearly visible in Fig. 23.1(a), is the strong dependence on the number of layers (much stronger than that reported for other 2D semiconductor materials). Figure 23.1(b) summarizes the thickness dependence of the black phosphorus band-gap value. The figure compares the values calculated through different ab initio methods [3, 26, 35], and experimental values measured by photoluminescence [2, 24, 36], infrared spectroscopy [21], and scanning tunneling spectroscopy [37, 38] techniques. Although the magnitude of the theoretical band-gap value strongly depends on the approximation employed to calculate the band structure, in all of the cases there is a marked thickness dependence: from a large gap (close to 2 eV) for a single-layer that monotonically decreases to a narrow band-gap value (about 0.3 eV) for bulk black phosphorus. This thickness-dependent band gap, due to quantum confinement of the charge carriers in the out-of-plane direction, is much stronger that that observed in other 2D semiconductor materials [31, 32] and it provides a remarkable way of tuning the optical and electrical properties of black phosphorus. Therefore, one can select the thickness of black phosphorus material to design a nanodevice with a band-gap value that is optimized for a certain application. It is important to highlight that, due to the quantum confinement effect, the black phosphorus band-gap value spans over a wide energy range (0.3 eV to 2.0 eV) that was not covered by any of the other 2D materials isolated to date. This range can be increased even further by alloying black phosphorus with arsenic (b-AsxP1–x) covering the range between 0.15 eV and 0.30 eV, as recently reported by Liu et al. [40]. Figure 23.2 compares the band-gap values spanned by black phosphorus and black arsenic-phosphorus with other 2D materials. According to Fig. 23.2, graphene can cover

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437

Fig. 23.2 Comparison of the band-gap values for different 2D semiconductor materials. The band-gap values for conventional semiconductors have been included also for comparison. The horizontal bars spanning a range of band-gap values indicate that the band gap can be tuned over that range by changing the number of layers, straining or alloying. In conventional semiconductors, the bar indicates that the band gap can be continuously tuned by alloying the semiconductors (e.g. Si1–xGex or In1–xGaxAs). The range of band-gap values required for certain applications has been highlighted in the bottom part of the figure to illustrate the potential applications of the different semiconductors. (Reproduced with permission from [39]. Copyright 2015 American Chemical Society.)

the range from 0 eV to 0.2 eV. In fact, although pristine graphene is a zero-gap semiconductor, one can open a band gap in graphene using different methods such as patterning graphene into nanoribbon shape or by applying a perpendicular electric field to bilayer graphene [41–43]. Other isolated 2D materials are semiconductors; they present an intrinsic band gap. Among these materials transition metal dichalcogenides are the most studied so far and they present a band gap in the range of 1.0–2.0 eV [44, 45]. As illustrated in Fig. 23.2, each material covers a relatively wide range of band gaps as the gap depends on the thickness of the material. The band gap can be tuned further by applying strain or by alloying with other materials (e.g. MoS2 xSex presents a band-gap value in between those of MoS2 and MoSe2). Interestingly, there is a big empty gap between the values covered by graphene and those covered by other 2D materials. Black phosphorus bridges that gap between graphene and transition metal dichalcogenides, and most importantly a lot of applications require semiconductor materials with a band-gap value in that range of energies

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(that have remained elusive so far). For example, although photovoltaic energy harvesting and photocatalysis applications are optimized for semiconductors with a 1.2–1.6 eV band gap [46], other applications such as fiber optic telecommunications [47], thermal imaging, and thermoelectric power generation require semiconductors with band-gap values in the range of 0.3 eV and 1.0 eV, and are thus out of reach of the twodimensional semiconductors isolated to date. Figure 23.2 also displays the band-gap values for conventional semiconductors to facilitate a direct comparison with 2D semiconductors. The horizontal bars spanning a range of band-gap values indicate that the band gap can be continuously tuned by alloying different semiconducting materials (e.g. Si1–xGex or In1–xGaxAs). By taking a look at conventional semiconductors, black phosphorus has the potential to substitute and/or complement Pb-chalcogenides, In1–xGaxAs, In1–xGaxSb and Si1–xGex semiconductors, with the difference being that atomically thin black phosphorus is very transparent and flexible.

23.2.2

Anisotropic Optical Properties Although, like graphite, black phosphorus is a single elemental material with a layered honeycomb lattice structure, each layer of phosphorus is puckered along the armchair directions due to sp3 hybridization. This broken crystalline symmetry leads to an anisotropic band structure, which has been theoretically studied as early as 1950s [28, 48] and reviewed by Morita in 1980s [30]. As a consequence of the anisotropic band structure, black phosphorus has high anisotropy in its electrical, thermal, mechanical, and optical properties which assume high variation between the two in-plane crystallographic directions along the armchair (x-axis) and zig-zag (y-axis) lattice axes. As the first three anisotropic properties have been discussed in other chapters, here we focus on black phosphorus anisotropic optical properties, including optical absorption and photoluminescence. In experiments, the crystalline orientation of black phosphorus can be determined with polarization-resolved Raman spectroscopy by comparing the relative intensity of different modes. Because the A2g and B2g modes in the Raman spectrum originate from the in-plane vibrational mode along the armchair and zig-zag axes, respectively, their intensity varies sinusoidally with the linear polarization of the normally incident excitation light: the A2g mode intensity reaches maximum when the light polarization is aligned with the armchair axis [49, 50, 24].

23.2.3

Linear Dichroism Linear dichroism is the difference in optical absorption for linearly polarized light with the polarization direction aligned to different crystallographic axes in an anisotropic material. Normally, for light incident to the layer plane of black phosphorus, the dichroism is the different absorption for light polarized along the zig-zag or armchair axes. The frequency-dependent dielectric function of black phosphorus was calculated based, historically, on the interband transition matrix for bulk black phosphorus by

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439

Fig. 23.3 Polarization resolved Raman spectroscopy study of black phosphorus. The intensity of the A2g and B2g modes depends sinusoidally on the angle between excitation light polarization and the armchair direction. (Adapted with permission from [50]. Copyright 2014 American Chemical Society.)

Morita [30] and more recently for few-layer black phosphorus by Low et al. [51] and Qiao et al. [35]. The results indicate that in line with the calculated band structure, the imaginary part of the complex dielectric function and correspondingly the absorption coefficient is highly different for light polarized along the armchair (x-polarized) and zig-zag (y-polarized) directions, and therefore reveals large linear dichroism in black phosphorus. Particularly, because of the forbidden dipole transition along directions other than the Γ–Z direction in the k-space, x-polarized light shows a peak near the absorption threshold that is absent for the y-polarized light. The predicted linear dichroism property of black phosphorus has been measured and confirmed in cleaved bulk samples. In few-layer black phosphorus that was exfoliated from bulk crystal, the infrared spectroscopy method was first measured by Xia et al. to determine anisotropy absorption [21]. The results (Fig. 23.4) showed a cos2(θ) dependence of the extinction spectrum near the absorption threshold around a wavelength of 4.17 μm (wavenumber 2400 cm–1). Consistent with the result from bulk samples, an absorption peak can be observed for x-polarized light but not for y-polarized light at a wavenumber of 2700 cm–1, where a maximal contrast ratio of 6.6 between the extinction coefficients (23.2% and 3.5%, respectively) for the two orthogonal polarization directions was measured. Xia et al. also measured the anisotropy of DC conductivity in their samples and compared with optical absorption (i.e. AC conductivity); the two properties show the same angular dependence with respect to the lattice axes. Similar measurements of the linear dichroism were carried out by several other groups in optical spectrum spanning from infrared to visible bands and the results show the anisotropy is also strongly dependent on the optical wavelength, in agreement with band theory [52]. The linear dichroism gets even stronger in monolayer black phosphorus due to the strong quantum confinement and very distinct energy dispersion for carriers along different lattice axes [24, 53]. The high dichroism in black phosphorus thin films has important implications for their optoelectronic applications. For photodetection, to optimize the detection

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Fig. 23.4 Linear dichroism in black phosphorus. ((a) Reproduced with permission from [21]. Copyright 2014 Macmillan Publishers Ltd. (b) Reproduced with permission from [51]. Copyright 2014 American Chemical Society.)

efficiency, it is important to align the input light polarization in the armchair direction to maximize absorption.This was confirmed in the results by Yuan et al. [52], showing that the photoresponsivity at certain wavelengths can vary by a factor of 4 when the light polarization is rotated by 90 . The high linear dichroism of few-layer black phosphorus can also possibly be utilized to implement an optical polarizer using a fiber or waveguide integrated configuration similar to the graphene-based polarizer [54], while the difference is that the polarizing effect of graphene is not due to dichroism but the different dispersion of TE and TM modes. More interestingly, since dichroism is strongest near the absorption threshold, which is tunable in black phosphorus by modifying the Fermi level through electrostatic gating, it is thus possible to control electrical optical polarization in black phosphorus integrated devices.

23.2.4

Linearly Polarized Photoluminescence (PL) The asymmetric lattice structure also leads to anisotropic photoluminescence in black phosphorus, which is found to be strong because of the direct band gap of black phosphorus and because it mainly originates from the excitonic states. Photoluminescence (PL) measurement is a powerful tool to probe the band structure and excitonic states in semiconductor materials. It has been used to study monolayer and few-layer black phosphorus with varying thicknesses [2, 50] as shown in Figs. 23.5(a) and (b). The measured energy spectrum of the emitted photons provides a lower bound estimate of the band gap of black phosphorus, which corroborates the predicted value of the band gap that is much larger in few-layer than in bulk black phosphorus. The anisotropy in black phosphorus’ photoluminescence was first revealed by Wang et al. [24] in monolayer black phosphorus using a polarization resolved photoluminescence measurement (Figs. 23.5(c) and (d)). Their results show that the PL intensity is highly dependent on the polarization of excitation light and the emitted light is also polarized highly linearly; the strongest photoluminescence intensity was measured when both the polarization of the excitation light and the orientation of the polarizer before the detector are aligned with the armchair direction (x-axis). The dependence of

23.2 Optical Properties

441

Fig. 23.5 PL and anisotropy in black phosphorus. ((a) Reproduced with permission from [2]. Copyright 2014 American Chemical Society. (b) Reproduced with permission from [50]. Copyright 2014 American Chemical Society. (c) and (d) Reproduced with permission from [24]. Copyright 2015 Macmillan Publishers Ltd.)

the excitation light can be attributed to the linear dichroism of black phosphorus discussed above, such that armchair direction polarized light is more strongly absorbed, leading to more efficiently excitation. Remarkably, regardless of the excitation polarization, the photoluminescent emission is highly polarized with a very high extinction ratio: the intensity of the y-polarized emission is less than 3% that of the xpolarized emission. The strongly polarized photoluminescent emission is consistent with the anisotropic exciton states due to the asymmetric lattice. First-principles calculation shows that the electron wave function of the exciton states extends along the armchair direction, in which the photoexcited carriers are more mobile and less bounded than in the zig-zag direction. This anisotropic exciton emission makes black phosphorous unique compared to other two-dimensional materials. Its study can provide important insights into carrier dynamics and band structures. Unlike transition metal dichalcogenides, where the thickness is varied, the band gap in monolayer to multilayer black phosphorus remains direct at the Γ point with varying band-gap sizes. Thus, the anisotropic excitonic photoluminescence in black phosphorus with different numbers of layers should have common features but also be tunable by the layer number. It is also possible to electrically tune the excitonic states by electrostatic gating as has been achieved in transition metal dichalcogenides monolayers and junctions [55, 56]. Efficient photoluminescence is often a prerequisite for

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electro-luminescence and the results obtained so far are very promising for achieving black phosphorus electroluminescent devices.

23.3

Optoelectronic Devices In this section, the recent advances in applying black phosphorus in optoelectronic devices will be reviewed. Special focus will be paid to photodetectors and solar cells because of their potential impact on society and industry.

23.3.1

Phototransistors The lack of an intrinsic band gap in graphene, which strongly reduces its applicability in optoelectronic devices, motivated the research on other 2D materials with an intrinsic band gap [44]. Although silicene, a 2D layer of silicon atoms [57], can be considered as a semiconducting counterpart of graphene, its environmental instability hampers its applicability and therefore further studies. Among the studied 2D semiconductors, the transition metal dichalcogenides (TMDCs) show promising properties for optoelectronic applications but they typically suffer from slow photoresponse, and they are only suited for applications in part of the visible range of the electromagnetic spectrum (because of the large band gap of Mo- and W-based compounds) [44, 58–62]. Black phosphorus owning a small direct band gap together with a fast photoresponse is a promising alternative to other 2D semiconductor materials in photodetection [63]. After the first works demonstrating the isolation of black phosphorus, simple photodetector devices (based in the phototransistor geometry) were rapidly reported [52, 64–67]. The simplest phototransistor device is based on a field-effect transistor with a back-gate electrode in which the semiconducting channel can be directly illuminated (see Fig. 23.6(a)). Upon illumination with photons with energy higher than the bandgap, electron–hole pairs are generated. When a voltage bias is applied across the device, the electron–hole pairs can be separated, thus increasing the current flowing through the device. Therefore, the light can be detected by the difference between the current flowing through the device in the dark state and upon illumination. This would be the simplest scenario where the photocurrent generation is only due to photoconductance but other photocurrent generation mechanisms (photogating, photovoltaic, photothermoelectric, or photobolometric) might also play a relevant role on black phosphorus photodetectors (see [63] for further discussions about the differences between the different photocurrent generation mechanisms). Research on black phosphorus phototodetectors has been extensive since the first works isolating and characterizing black phosphorus nanosheets. Phototransistors based on ultrathin (3 to 8 nm thick) black phosphorus have been fabricated on SiO2/Si substrates [4]. In the dark state, black phosphorus phototransistors display ambipolar transport with hole mobilities of the order of 100 cm2 V–1 s–1 and an ON/OFF ratio >1  103. Figure 23.6(e) shows an example of photodetection in one of these ultra-thin black phosphorus phototransistor devices with 940 nm, 640 nm, and 532 nm illumination wavelengths.

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Fig. 23.6 (a) Schematic diagram of a phototransistor based on black phosphorus. (b) to (d) Different

examples of phototransistor devices based on mechanically exfoliated black phosphorus. (e) Field-effect transistor characteristics of the device shown in (b) in dark state and upon illumination with different wavelengths. (f) Scanning photocurrent map acquired on the device shown in (c). (d) Spatially resolved photocurrent measured along the dashed line in (d) ((a), (b), and (e) Adapted with permission from [4]. Copyright 2014 American Chemical Society. (c) and (f) Adapted with permission from [66]. Copyright 2014 American Chemical Society. (d) and (g) Adapted from [67] with permission of The Royal Society of Chemistry.)

The phototransistor can detect excitation wavelengths up to 940 nm, demonstrating that black phosphorus-based devices can be operated in the near-infrared (NIR) part of the spectrum, unlike photodetectors based on other layered materials that are usually operated in the visible or ultra-violet region due to their larger band gap. The responsivity of these devices can reach 4.8 mA W–1, and the rise and fall times are 1 and 4 ms, respectively. In the work by Hong et al., a deeper insight into the photocurrent generation mechanisms in black phosphorus phototransistors was achieved by scanning photocurrent microscopy (SPCM) on a 8 nm thick black phosphorus transistor (see Figs. 23.6(c) and (f)) [67]. The authors found that both the photovoltaic effect (due to the presence of Schottky barriers at the metal/semiconductor interfaces) and the photothermoelectric effect can contribute to photocurrent generation. In the OFF state, the authors found that the photocurrent generation is dominated by the electron–hole separation induced by the electric field at the Schottky barriers. In the ON state, the contact resistance is drastically reduced (the Schottky barrier decreases and thus the photovoltaic effect weakens), allowing the photothermoelectric effect to contribute to the photocurrent (in good agreement with the analysis by Low et al. on doped black phosphorus phototransistors [65]). Moreover Hong et al. also studied the role of the strong in-plane anisotropy of black phosphorus in photocurrent generation. When the incident light is linearly polarized along the armchair direction, the photocurrent is enhanced; conversely, when aligning the polarization with the zig-zag direction the photocurrent is reduced. The polarizationdependent photocurrent is a direct result of the strongly asymmetric band structure of fewlayer black phosphorus, highlighting the difference from other 2D semiconductors, such as transition metal dichalcogenides, which are rather isotropic within the plane.

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Table 23.1 Summary of the figures-of-merit of different few-layer black phosphorus photodetectors (PNJ stands for p–n junction). Measurement conditions

Thickness (nm)

Vds (V) Vg (V) λ (nm)

Responsivity P (mW cm–2) R (mA W–1)

Rise time (ms)

Spectral range

8 6 6 120 120 30–50 11.5

0.2 0 0.5 0 0 0–0.1 0.4

16 1900 1900 3106 107 N/A 1.91

1 1.5 N/A N/A N/A 0.04 No data

VIS–NIR 4 VIS–NIR 64 VIS–IR 64 VIS–IR 65, 66 VIS–NIR 65, 66 VIS–NIR 52 VIS–NIR 68

0 PNJ PNJ 0 0 0–2.5 8

640 532 532 1550 532 1100–1700 1550

4.8 0.50 0.28 5.0 0.20 0.25–1.5 130

Reference

Fig. 23.7 Summary of the responsivity and time response of photodetectors based on graphene, transition metal dichalcogenides (TMDCs), and black phosphorus. (Adapted with permission from [39]. Copyright 2015 American Chemical Society.)

Table 23.1 summarizes the figures-of-merit reported for different black phosphorus phototransistors and Fig. 23.7 visually compares the photoresponse and the response time measured for photodetectors based on graphene (data extracted from [68–72]), transition metal dichalcogenides (data extracted from [58, 60–62, 73–83]), and black phosphorus (data from [4, 52, 64, 66, 84]). While graphene detectors typically display a fast but rather weak photoresponse, transition metal dichalcogenides can have photoresponse values as high as 103 A/W (104 times larger than that of pristine graphene-based photodetectors) but they typically suffer from very low response time, hampering their use in applications such as video-rate imaging. The first black phosphorus photodetectors (mainly based on phototransistor geometry) showed photoresponse times faster than most transition metal dichalcogenide-based photodetectors [4, 52, 64, 66]. Note that the

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RC time of the experimental setup (~1 ms) is very close to the time response values reported for these early black phosphorus photodetectors and thus it is very probable that these values are limited by the experimental setup and that they should be considered as lower bounds for the black phosphorus time response. In fact, more sophisticated device engineering (integrating a black phosphorus detector on a silicon photonic waveguide to overcome the RC time limitation in the photodetection setup) allowed for a time response of a black phosphorus photodetector < 1 ns to be demonstrated [84]. The next section gives further details on high-speed black phosphorus photodetectors.

23.3.2

High Speed Photodetectors The narrow and direct band gap of black phosphorus, along with its excellent transport performance, makes it particularly promising for the photodetection of infrared light, especially for optical communication, which predominantly uses the near-infrared band in fiber optics and silicon photonics. Compared with germanium and compound semiconductors such as indium gallium arsenide (InGaAs), black phosphorus boasts the potential of low-cost growth and epitaxy-free integration with silicon photonics. To satisfy the bandwidth requirement of practical optical communication, however, the photoresponse speed of black phosphorus photodetectors needs to be dramatically improved to rival the state-of-the-art germanium photodetectors that can be monolithically integrated into silicon photonics, and have reached a speed beyond 40 GHz [85, 86]. Therefore, to fulfill the potential of black phosphorus photodetectors in

Fig. 23.8 Waveguide integrated black phosphorus photodetector in near-infrared light. Low dark current and high operation speed at 3 Gbits/sec are demonstrated.

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optoelectronics, it is imperative to improve the response speed, together with the responsivity and noise performance. The first GHz black phosphorus photodetector was demonstrated by Youngblood et al. [84] by integration with a silicon photonic waveguide. In their device, 10 to 100 nm thick black phosphorus thin film was exfoliated and transferred onto a single mode waveguide with source and drain contacts. To investigate the photocurrent generation mechanism and to optimize device performance, a graphene gate is also fabricated on top of the black phosphorus to electrostatically tune the Fermi level. The waveguide integrated configuration is more advantageous than normal incident devices in that the absorption length is determined by the width rather than the thickness of the black phosphorus, and thus can be long enough to obtain nearly 100% absorption. In contrast, for normal incident light, on average each layer of black phosphorus absorbs less than 0.4% of light, with a factor of 8 less than graphene. In the device by Youngblood et al., the black phosphorus channel is 6.5 μm wide, contributing 17.5% of the total absorption of light in the waveguide. By measuring the photoresponsivity while changing the applied gate voltage, it was found that the responsivity is maximized when the black phosphorus channel is gated to be in the off-state with low carrier concentration. More interestingly, when the black phosphorus channel is gated to be more ntype doped, the sign of the photocurrent changes from positive to negative with a much lower responsivity, indicating that the photocurrent generation mechanism has changed. The generation of positive photocurrent when the black phosphorus channel has low carrier concentration is attributed to the photovoltaic effect, while the negative photocurrent in the high carrier concentration regime is attributed to the bolometric effect. The maximal responsivity obtained in devices using 11.5 nm thick black phosphorus is 135 mA/W under 0.4 V bias, and reaches more than 650 mA/W in 100 nm thick phosphorus under 2.0 V bias. As stated, the responsivity varies with the gate voltage and the photocurrent mechanism changes from photovoltaic to bolometric, but the response speed is also strongly dependent on the generation mechanism. Because the bolometric effect stems from optical heating of the waveguide, the thermal time constant of the device limits the response speed, which was found to be less than 200 kHz. In stark contrast, in the photovoltaic regime, the response speed is measured to approach 3 GHz. The authors further demonstrated digital communication at a date rate of 3 Gbits/s with an open eye diagram. Therefore, with a high responsivity, high response speed, and low dark current, the low doping regime should be the optimal condition to operate the black phosphorus photodetector. The response speed of a photodetector is intrinsically limited by the carrier recombination time τr in the photosensitive material and the transit time τt for the photocarriers to travel out of the detection region, and externally by the RC time constant of the whole device τRC in which the total capacitance and resistance of the contacting leads and pads have to be accounted for. While measurement of the carrier recombination kinetics has not been reported, it can be expected from the 2D nature and high trap state density in black phosphorus that the recombination rate is high and the lifetime of carriers is very short. For reference, the recombination lifetime of carriers in graphene is tens of picoseconds. Therefore, the response speed of a black phosphorus

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photodetector is unlikely to be limited by the carrier recombination rate but more by the transit time and RC time constant. The photodetectors as reviewed in this section all use a lateral configuration in which the carriers have to travel in the plane of black phosphorus. For the generated photocarriers to be extracted by the source and drain contacts, they must travel across the BP channel of width L. This channel transit time in the drift-diffusion regime is given by τt = (L/2)2/μ
Vb, where μ is the carrier mobility and Vb is the bias voltage. For example, assuming L = 1 μm, typical carrier (mostly holes) mobility of μ = 100 cm2/V
s and bias voltage Vb = 1 V, τt will be 0.1 ns. Such a transport time will limit the response speed of the BP photodetector to 0.6 ns; therefore, the reported 3 GHz bandwidth is indeed limited by the transit time. Furthermore, in order to achieve high photodetection quantum efficiency, it is ideal to have τt shorter than the carrier recombination time to reduce loss of photocarriers. The recombination rate in BP has not been reported but can be expected to be high given the narrow, direct gap of BP. Therefore, to further improve the performance of BP photodetectors, in terms of internal quantum efficiency and response speed, it is critical to reduce the transport time τt. While increasing mobility μ is important, the channel length L plays a more important role in τt because of the quadratic dependence. However, in a laterally configured device, reducing τt simply by decreasing channel length L faces the trade-off of diminishing active detector area. Particularly, in waveguide integrated devices, the minimal channel length L is bounded by the waveguide width because the source and drain contacts have to be placed far enough from the waveguide to reduce optical absorption. This limitation becomes even more severe for mid-infrared devices because the waveguide width increases proportionally with the operation wavelength so midinfrared waveguides are much larger than those for shorter wavelength operation. Consequently, both the photodetection efficiency and photoresponse speed of laterally configured BP photodetectors are limited by the channel width. To overcome this limitation, the photodetector can be constructed using a vertical configuration with BP as the photoactive layer sandwiched vertically between two contacts. In such a configuration, the channel length is reduced to the thickness of the BP layer, and the total active area is not reduced. Photocarriers generated in the middle of the BP layers only need to travel through the layers to reach the contacts. Therefore, the transport time τt is drastically reduced even accounting for the fact that the out-ofplane mobility is likely much lower than in-plane mobility in BP. Even assuming a through-layer mobility of μ = 1 cm2/V
s and bias voltage of 0.1 V, the transport time in BP of 10 nm thick is only 10 ps. However, for such a vertical device to work, the contacts must be made of transparent material. For this purpose, graphene serves as an ideal material to be used as the electrical contact because if affords both low optical absorption and high electrical conductivity. Similar vertical heterostructures of monolayer transition-metal dichalcogenides and graphene have already been reported by many groups [59, 87–89] and picosecond photoresponse time has been demonstrated [90].

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In a graphene–BP–graphene vertical heterostructure, a bias voltage is applied to facilitate the separation and transport of electron and hole photocarriers to be collected, respectively, by the top and bottom graphene contacts. They will then transport in the graphene layers laterally with high mobility, for both electrons and holes, to generate photocurrent. By optimizing the initial doping in the top and bottom graphene contact, an ideal band alignment within the heterostructure can be achieved with the Fermi levels of both layers of graphene lying within the band gap of BP. With such a band alignment, the dark current associated with thermal injection carriers from the graphene into the BP can be minimized. Additionally, the majority of photocarriers generated in the graphene layers have sufficient energy to overcome the barrier to enter the BP and be collected. Overall, the photodetection internal quantum efficiency of such a vertical graphene–BP–graphene heterostructure will be very close to 100%, and its response speed will reach above 100 GHz. The heterostructure is also suitable for both normal incidence and waveguide integrated photodetectors based on different application needs.

23.3.3

Solar cells Electrostatically gated Building on the ambipolar field-effect of black phosphorus (described in detail in the previous chapter, which focused on the electronic properties of black phosphorus), Buscema and Groenendijk et al. have fabricated p–n junctions via local electrostatic gating. The devices are based on two local split gates [64]. An h-BN layer, used as an atomically flat and disorder-free gate dielectric, is transferred on top of two electrodes that will be used as local gates. Then a few-layer black phosphorus flake is transferred as the channel material. The devices have been fabricated by exploiting a recently developed all-dry deterministic transfer method. Electrical contacts are defined in a final step by e-beam lithography. Figure 23.9 shows one black phosphorus p–n junction device based on this approach and an example of Ids–Vds characteristics of a locally gated device in different gate configurations. When the two gates are biased with the same polarity, the device shows an almost metallic behavior while for the PN or NP configuration, the Ids–Vds curves become rectifying. Upon illumination, the Ids–Vds curves display a net current at zero-bias (so-called short circuit current Isc) and the reverse current increases, as shown in Fig. 23.9. This is a feature of the photovoltaic effect where the generated photocurrent adds to the reverse-bias current. The inset in Fig. 23.9 shows the electrical power (Pel = Vds
Ids) generated by the device, which reaches about 13 pW under the largest illumination power. Even near-infrared (NIR) photons give rise to photocurrent, opening up the possibility to harvest NIR photons in photovoltaic applications. Compared to electrostatically gated solar cells realized with other layered materials such as WSe2 [56, 73, 91, 92], the black phosphorus solar cells present a more extended wavelength operation range, a similar external quantum efficient EQE, but a lower open-circuit voltage (Voc). The lower open-circuit voltage is consistent with the smaller band gap of the black phosphorus compared with that of WSe2.

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Fig. 23.9 (a) Optical images of the steps employed to fabricate black phosphorus devices with two

local gates. Underneath the optical images there are cross-sectional diagrams of the device at the different fabrication steps. (b) Different device configurations using the two local gates and the corresponding current versus voltage characteristics. (c) Current versus voltage characteristics upon illumination showing the photovoltaic effect (generation of current at zero bias and flow of current at zero bias voltage). (Inset) Photogenerated electrical power through photovoltaic effect (adapted from [64]).

Van der Waals Heterostructures Another concept of the black phosphorus-based solar cell device, alternative to the splitgate architecture, relies on artificial vertical stacking of two nanosheets with n-type and p-type doping to form a vertical p–n junction with a built-in electric field due to the difference in doping between the two stacked layers [93]. In [93], Deng et al. fabricated a black phosphorus-based solar cell by exploiting the intrinsic p-type doping of exfoliated black phosphorus and the strong n-type behavior of CVD grown MoS2 in a vertically stacked geometry (see Fig. 23.10(a) for a sketch and an optical image of the fabricated device). Due to the different doping levels of the two nanolayers, the device showed a strong rectifying (diode-like) behavior (see Fig. 23.10(b)), and upon illumination the built-in electric field at the interface between the two materials was strong enough to separate the photogenerated electron–hole pairs. Figure 23.10(c) shows the photogenerated electrical power in the solar cell. Similar concepts of p–n junction-based solar cells with black phosphorus have been fabricated by combining black phosphorus with conventional n-type semiconductors. Figure 23.10(d) shows a sketch illustrating the fabrication of a p–n junction based on GaAs and black phosphorus reported by Gehring et al. [94] and an optical image of the resulting device. Similar to the MoS2/black phosphorus p–n junction devices, these GaAs/black phosphorus p–n junctions show marked rectifying behavior (due to the difference in the doping level between the two materials), and upon illumination the

450

Optoelectronic Applications of Black Phosphorus

Fig. 23.10 (a) Schematic diagram (top) and optical image (bottom) of a p–n junction based on

vertical stacking of black phosphorus and MoS2. (b) Current versus voltage characteristics of the p–n junction device shown in (a) upon illumination and (c) the electrical power photogenerated by the photovoltaic effect. (d) Schematic diagram (top) and optical image (bottom) of a p–n junction based on vertical stacking of black phosphorus on top of an n-type GaAs substrate. ((9a), (b), and (c) Adapted with permission from [93]. Copyright 2014 American Chemical Society. (d) Adapted with permission from [94]. Copyright 2015 AIP Publishing LLC.)

electric field at the heterojunction also separates the photogenerated carriers, reaching external quantum efficiencies (EQE) values of up to 10%.

23.4

Outlook and Remarks The prospects for the applications of black phosphorus in optoelectronic devices are very bright. Compared with other 2D materials, including graphene and transition metal dichalcogenides, the direct and narrow band gap of black phosphorus is a significant advantage for infrared light detection and modulation. Its band gap ranges, from ~0.3 eV in many-layer films to 1.5–1.8 eV in monolayers, cover the mid-infrared spectral range up to ~4.1 μm in wavelength, where important applications, including free-space and fiber optics-based communication and remote spectroscopic sensing, are emerging, as mid-infrared laser sources are becoming more available [95, 96]. The current technology of mid-infrared detection is based on mercury cadmium telluride (MCT) and InSb photodetectors or quantum well infrared photodetectors (QWIP). Both use expensive materials that have to be epitaxially grown and are not suitable for integration with silicon technology. In contrast, 2D materials, such as black phosphorus, can be directly transferred onto silicon and bonded with van der Waals force. Given the already demonstrated performance in the near-infrared and the excellent transport properties of black phosphorus devices, the development of black phosphorus

23.4 Outlook and Remarks

451

mid-infrared photodetectors is very promising, with high performance to be integrated with silicon mid-infrared photonics [96]. To achieve performance rivaling the current technology, black phosphorus can benefit by forming heterostructures with other 2D materials, such as graphene and boron nitride. Particularly, since the band-gap size is determined by the number of layers in black phosphorus, it is desirable to control the layer numbers in different regions of a device in order to obtain the desired band alignment to facilitate the efficiency of, for example, photocarrier generation and transportation. Although black phosphorus photodetector devices have already been demonstrated, only relatively simple architectures have been studied so far and more complex device architectures and combinations of black phosphorus with other two-dimensional materials or other semiconductor materials need to be explored [93, 94, 97–99]. For example, the combination of quantum dots with graphene have dramatically improved the detectivity of graphene-based photodetectors (up to a 109 time improvement) [100]. Another topic that is attracting interest in the field of optoelectronics, is the use of two-dimensional semiconductors for light emission by exploiting the electroluminescence [56, 91, 92, 101–103]. Black phosphorus light-emitting devices hold the promise to emit photons in the NIR part of the spectrum while maintaining high flexibility and transparency. Despite of all the promises, however, there are two major roadblocks that must be solved before black phosphorus optoelectronic devices can achieve wide applications. The first is the lack of method to grow large-area black phosphorus films. While other 2D materials, including graphene and transition metal dichalcogenides, have been successfully grown with chemical vapor deposition (CVD) methods in many areas, with high-quality, black phosphorus, currently they can only be exfoliated from bulk crystals in small flakes. Although millimeter sized single crystals can now be grown of high quality and purity [104], and larger flakes can be exfoliated from those, it is still not sufficient for large-scale integration in black phosphorus devices. Therefore, development of a large-area growth method is imperative for research into black phosphorus. So far, there are preliminary results demonstrating large-area growth of nanocrystalline [105] or amorphous black phosphorus [12]. It seems CVD methods, with the use of proper precursors that inhibit 3D growth but promote 2D growth of thin film on a substrate, is one of the promising ways to get highly crystalline large-area films. The second roadblock is the chemical stability of black phosphorus. It is widely known that after being exfoliated from bulk crystal, black phosphorus thin film degrades, if exposed in ambient air, within hours through photoassisted reaction with water and oxygen to form oxidized phosphorus species [17, 106]. It was later found that oxide overlayers, such as aluminum oxide and hafnium oxide, deposited by atomic layer deposition (ALD) methods, provide an effective passivation layer that can preserve encapsulated black phosphorus for weeks [107]. Sandwiching black phosphorus between graphene or hexagonal boron nitride layers can also passivate black phosphorus [108]. However, if the fabrication process is not carried out in an oxygen and water free environment, such as a dry glovebox, the black phosphorus can still be contaminated. Many non-ideal performances reported in the literature were attributed to

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contamination and degradation of black phosphorus. Therefore, methods providing long-term and stable passivation of black phosphorus, and a fabrication process that does not cause further exposure to oxygen and water are utterly important to achieve the promised performance of black phosphorus devices.

23.5

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24

Silicene, Germanene, and Stanene Guy Le Lay, Eric Salomon, and Thierry Angot

24.1

Introduction Recently, group IV honeycomb monolayer materials, that is, the silicon, germanium, and tin-based analogues of graphene, namely, silicene, germanene, and stanene, have attracted considerable interest. These novel synthetic two-dimensional (2D) Si, Ge, and Sn allotropes are artificially created, since, at variance with graphene, which inherits from graphite, they have no parent crystal in nature. Due to the strong effective spin–orbit coupling, these analogues are predicted to be robust two-dimensional (2D) topological insulators and ideal candidates for the quantum spin Hall effect at accessible temperatures, even at room temperature (RT). Exotic high temperature superconductivity is also theoretically suggested. Furthermore, these emerging elemental 2D materials are expected to be directly compatible with the current Si-based device technologies. In this respect, the first silicene field-effect transistors operating at room temperature were fabricated in early 2015. This already demonstrates the potential of these novel materials for future applications in nanoelectronics. As we will see in this chapter, a cornucopia of unprecedented exotic properties, which appear out of reach of graphene, are envisaged for silicene, graphene, stanene, and their derivatives. Many of them are likely to be exploited in spintronics, valleytronics, or quantum computing.

24.2

The Advent of Silicene Graphene (Part I of this book), which stems from graphite, its mother natural crystal, came to the forefront of nanoscience in 2004 with its outstanding properties; it has further initiated the search for other atom-thin 2D materials. In the wake of graphene, researchers first looked for existing semiconducting or insulating layered crystals that could possibly be peeled down to single monolayer flakes using the now famous scotch tape method. Success came with transition metal dichalcogenides (TMDs), which could be easily exfoliated (see Part II of this book). Typically, molybdenum disulfide, MoS2, permitted the fabrication of the first TMD single-layer transistor in 2011 [1]. More recently, the (re)-discovery in 2014 of black

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459

Fig. 24.1 Illustration of the buckled honeycomb lattice of standalone silicene: (a) perspective view,

(b) top view.

phosphorous, an already existing elemental direct gap semiconductor, has led to the first few-layer phosphorene transistors [2]. In the realm of 2D materials, besides the ones initially peeled from lamellar crystals, the synthetic emerging elemental ones appear as strong contenders to graphene in a booming new field. The main reason is that the opening of a sufficient band gap for current semiconductor electronic applications does not appear possible in practice for graphene. As a consequence, the question raised by Ross Kozarsky, senior analyst at Lux Research, at the Washington DC, Nanotech Conference and Expo 2013: “Is Graphene the Next Silicon . . . Or Just the Next Carbon Nanotube?” was answered without appeal last July: “Just don’t expect graphene to live up to the untenable hype, or become the next silicon.” The first suggestion of a Si-based 2D material can be traced back to an article published in 1994 [3], but essentially unnoticed for 17 years. Its two authors K. Takeda and K. Shiraishi had theoretically predicted the possible existence of a novel 2D hexagonal silicon allotrope with a corrugated stage, i.e., a mixed sp2/sp3 hybridization of its Si atoms, yielding a ~0.4 nm buckling between the A and B sublattices, as shown in Fig. 24.1, at variance with graphene, which is flat because of the pure sp2 hybridization of its C atoms.

24.3

Epitaxial Silicene

24.3.1

The Archetype Silicene Phase In a creative endeavor, a few groups tried to synthesize silicene, as the hypothesized 2D material was named in 2007 [4], and further confirmed to be dynamically stable in a low buckled configuration with a lattice parameter of 0.383 nm, Si—Si bond distance of 0.225 nm, and buckling height of 0.044 nm in 2009 [5]. The first compelling evidence of the realization of silicene was published in 2012 [6]: graphene’s silicon cousin was created in situ, under ultra-high vacuum (UHV), by molecular beam epitaxy (MBE) as an artificial atom-thin lattice of Si atoms epitaxially grown on top of the (111) surface of a silver single crystal template held at ~200–220
C. This growth mirrored extensive ones of silver onto the Si(111) surface, where an atomically abrupt interface was formed

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Fig. 24.2 (a) Filled states STM image (10 nm  10 nm, tunnel current 0.55 nA, sample bias –520 mV) of the archetype silicene phase on silver (111). The observed “flower pattern” results from the protruding Si atoms within the 3  3 reconstructed silicene layer which coincides with a 4  4 Ag(111) supercell; the 3  3Si/4  4Ag supercell is indicated. (b) Perspective view of the atomic structure of the archetype 3  3 reconstructed silicene layer, as extracted from the Ag(111) surface.

with a magic coincidence between four Ag(111) unit vectors and three Si(111) ones perfectly aligned in a parallel arrangement [7]. As seen in Fig. 24.2, the unique “flower pattern” observed in scanning tunneling microscopy (STM) imaging does not directly reveal the structure of the silicene monolayer. The characteristic underlying honeycomb arrangement of the Si atoms was inferred from the Ag(111) 4  4 low-energy electron diffraction pattern pointing to the epitaxial relationship mentioned above, the measurement of a new linear band at the common. The KSi/KAg point of the surface Brillouin zone (non-dispersing with the k⊥ vector in angle-resolved photoemission measurements) and thorough density functional theory calculations within the general gradient approximation (DFT-GGA calculations) enable the exact simulation of the STM images. All this confirmed the initial guess of a 3  3 reconstructed silicene sheet in perfect match with a 4  4 coincidence supercell on Ag(111) yielding a nominal coverage of 18 Si atoms per 16 Ag surface atoms, i.e., θSi = 1.125. The prototype 3  3 silicene phase was thus created but not a primitive one that could have been expected from theoretical studies of standalone silicene for a non-interacting couple. As a matter of fact, all silicene phases synthesized until now on silver [8–12] or other metallic substrates, namely, ZrB2(0001) [13] and Ir (111) [14] surfaces are reconstructed because the silicene/metal interaction is rather strong. As demonstrated through fundamental surface crystallography arguments and DFT calculations, a pristine, unreconstructed, silicene sheet on Ag(111) has not been made [15]. The 3  3 silicene phase on Ag(111) is recognized as the archetype silicene phase by all experimental and theoretical groups worldwide. It has passed crucial diffraction tests, which have revealed its detailed atomic structure. Both reflection high-energy positron diffraction (RHEPD) [16] and low-energy electron diffraction (LEED) [17] experiments have confirmed that the “flower pattern” seen in STM imaging originates from Si atoms within the honeycomb array, which markedly protrude because they

24.3 Epitaxial Silicene

461

mainly sit on top of Ag atoms. The intrinsic buckling and inherent adaptability of silicene facilitates the emergence of the 3  3 puckered structure which conforms to the substrate in a compliant fashion and can cover 95% of the surface [16]. The puckering of 0.083 nm derived from these RHEPD measurements and all other geometrical quantities are in very good agreement with the values calculated previously [6].

24.3.2

Sister Silicene Phases The adaptive character of silicene favors the appearance of other rotational epitaxial phases on Ag(111), observed in LEED patterns. Differently buckled rotated phases in coincidence with √13  √13R(13.9
) Ag(111) supercells may cover large areas of the substrate’s surface at slightly higher preparation temperatures (~250
C). We have observed new STM structures in four different domains supporting an interpretation in terms of √7  √7R(19.1
) reconstructed silicene in small extension (by ~2%) with a slightly reduced coverage of θSi = 1.077 [18, 15]. Instead, others favor a 3  3 silicene reconstruction with a higher nominal coverage of θSi = 1.385 [19]; however, we note that it could exist only in two different domains. As yet no structural determination has solved the issue. In STM imaging at 4.2 K, this phase reveals a surprising vortex-like arrangement as shown in Fig. 24.3. We assign this to compressive strain, a consequence of the negative thermal expansion coefficient of silicene [21]: when the silicene/Ag(111) system prepared at ~210
C is cooled down, silicene expands, while silver shrinks. The stress is relieved through a complex buckling re-arrangement into large triangular domains and vortices, which still maintain, because of its adaptability, the continuity of the silicene layer, as shown in Figs. 24.3(b) and (c). A few other phases have also been described in the literature, e.g., by Arafune et al. [22], but identification as real silicene phases is not fully ascertained.

Fig. 24.3 (a) High-resolution topography STM image taken on a √13  √13 domain showing a large hexagonal pattern (U = 3 mV; I = 0.2 nA). (b) Enlarged STM image cut from the square indicated in (a). (c) Ball structural model for the vortex pattern shown in (b). It is based on the models of 20 (adapted from figure 4 of [8]).

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Silicene, Germanene, and Stanene

24.3.3

The Fate of the 2√3  2√3 Superstructure A 2√3  2√3R(30
) superstructure (noted in terms of Ag(111) basis vectors) was initially claimed in 2010, just by observation of STM topographs, to reveal directly a highly perfect, nearly flat, honeycomb silicene arrangement with a very small in-plane Si—Si distance of dSi–Si ~ 0.19 nm that would be formed only between 220
C and 250
C [23]. It turns out that this was just a false start: such data have never been reproduced, and, actually, the apparent honeycomb structure was shown to be merely a contrast reversed image of the bare Ag(111) surface [24]. Later some of the previous authors presented the 2√3  2√3R(30
) superstructure grown, instead, at 300
C; however, this superstructure had in this case a highly defective appearance in STM imaging [25]. Further, the real nature of the corresponding hypothetical √7  √7R(19.1
) silicene structure was shown to be inherently defective and inhomogeneous, a mere patchwork of fragmented pieces [10, 15, 26]. In addition, several observations and measurements have revealed its sudden death, to end, in a dynamic fading process at 300
C, on the one hand, in multilayer islands through a dewetting mechanism [27, 28], and, on the other hand, in the burial of an alloy [29], most probably confined below the surface.

24.4

Electronic Structure of Silicene

24.4.1

Electronic Structure of Standalone Silicene and Germanene Most studies of the electronic structure of silicene and germanene concern their freestanding forms. As mentioned above and illustrated in Fig. 24.4, Cahangirov et al. [5] demonstrated the existence of Dirac cones at the corners of the Brillouin zone (BZ),

Fig. 24.4 Band structures of free-standing low-buckled silicene (Si-LB) and germanene (Ge-LB) (adapted with permission from figure 2 of [5]).

24.5 Functionalization of Silicene

463

as in in graphene and with comparable Fermi velocities for the so-called lowbuckled structures. Silicene and germanene are thus analogues of graphene, keeping many of its properties despite their intrinsic bucklings. However, the heavier Si and Ge elements and the significant bucklings confer to silicene and germanene specific key intrinsic properties that are out of reach of graphene. Typically, application of an electric field perpendicular to the 2D lattices breaks the symmetry between the two A and B sublattices, which can open a band gap [30]. Furthermore, the significant non-trivial band gaps related to a much stronger effective spin–orbital coupling than in graphene, would make silicene and germanene potential 2D topological insulators (TIs), possibly yielding a quantum spin Hall (QSH) effect at easily accessible temperatures [31, 32]. Such QSH states in TIs provide new opportunities for low-dissipation spintronic devices due to the absence of backscattering and possible superconductivity, as recalled by Kou et al. [33].

24.4.2

Electronic Structure of Epitaxial 3  3 Silicene on Ag(111) Studies on the electronic structure of epitaxial monolayer silicene have mainly focused on the archetype 3  3 phase, prompted by the initial observation of a new linearly dispersing band, seemingly gapped by ~0.6 eV at the Ksilicene point of the 1  1 silicene BZ, due to interaction with the Ag(111) surface [6]. The same gap was also measured at the zone centre Γ00 and at the MAg points, which are Γ points of the 3  3 BZ of reconstructed epitaxial silicene [34]. Interestingly, when extracted from the Ag(111) surface, the 3  3 reconstructed silicene monolayer possesses a direct band gap, as demonstrated in state-of-the-art first-principles calculations, including many-body effects by the GW approximation and also incorporating the van der Waals interaction with the Ag substrate [35]. However, the assignment to pure silicene-related π bands has been questioned [36]. Instead, the new band SP has been assigned to a strong hybridization with the silver bands [37]. Nevertheless, Fig. 24.5 shows that silicene on Ag(111) still forms graphene-like bands featuring a steep cone-like dispersion around KAg with effective Fermi velocity 1.3  106 m s–1 and saddle points around MAg. As underlined by Houssa et al., this could open new areas of research focusing on new hybrid semiconductor-metallic systems with graphene-like electronic band structures [35]. The strong hybridization could also explain why Landau levels were not measured in STS experiments under a high magnetic field [38, 39].

24.5

Functionalization of Silicene As opposed to graphene, which is essentially inert, silicene and germanene are chemically reactive, thus enabling rather easy functionalization to tune their properties and further expand their palette of applications. In the following, we will just consider hydrogenation to illustrate promising possibilities.

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Silicene, Germanene, and Stanene

Fig. 24.5 Band dispersions of (a) clean Ag(111) (SSS: Shockley surface state, Ag sp: bulk Ag sp band) and (b) 3  3 epitaxial silicene on Ag(111) (SB is the new hybrid surface band). (c) Constant energy contour at EB = –0.45 eV at the MAg point. (d) Energy dispersion along the KAg–MAg–KAg direction (perpendicular to Γ–MAg). Bottom left: scheme of the Brillouin zones. Bottom right: illustration of the saddle point formed at MAg (adapted from figure 26 of [36]).

24.5.1

Hydrogenation of Free-Standing Silicene: Theoretical Studies Different hydrogenation adsorption configurations have been treated by several authors and recently re-examined by Yu et al. [40]. To summarize, full hydrogenation on both sides gives non-magnetic semiconductors, namely silicane and germanane, for which the chair configuration is found to have the lowest energy. For silicane, the gap of ~2.21 eV is indirect, while for germanane it is direct with a value of ~1.33 eV. One-side fully hydrogenated silicene and germanene have indirect band gaps. Regardless of the arrangement of hydrogen atoms, band-gap opening due to semihydrogenation is predicted both for silicene and germanene. Remarkably, chair configurations in this semihydrogenation case are ferromagnetic, the spin moments being mainly carried by the Si or Ge atoms that are not bonded with H.

24.5.2

Hydrogenation of 3  3 Epitaxial Silicene on Ag(111): Experimental Results Wu’s group was the first to report on the ordered and reversible partial hydrogenation of the epitaxial 3  3 silicene phase [41]. Amazingly, the 3  3 supercell is maintained,

24.6 Multilayer Silicene

465

Fig. 24.6 STM image (8 nm  8 nm, 0.33 nA, –200 mV = filled states, 3D rendering) of ordered partial hydrogenation of the epitaxial 3  3 silicene phase (the 3  3 supercell is indicated in top left corner); the protrusions correspond to H atoms bonded to Si atoms.

but, indeed, with a very different internal structure, as seen in the STM image of Fig. 24.6 obtained in Marseille, where similar observations have been made [42]. Here, the H atoms saturate the Si dangling bonds in a surprising manner: they favor one of the sublattices (six H atoms on one sublattice on the right half of the supercell) over the other (a single H atom on the other sublattice on the left half of the supercell). Obviously, this imbalance is likely to favor magnetic ordering, if not weird topological properties. At higher doses, H atoms may even bond below the silicene layer and full hydrogenation may be reached, as strong resistance to oxygen, testified in high resolution electron energy loss spectroscopy (HREELS) measurements, seem to indicate that despite over night exposure in a poor vacuum no OH related vibration loss was detected [42]. The strange aspect is the preservation of a 3  3 supercell, possibly pointing to reconstructed epitaxial “silicane” on Ag(111), while, instead, H intercalation could have been anticipated to favor lifting off from the Ag(111) surface, as in the case of graphene on silicon carbide. Obviously, all these intriguing results require more work, especially serious theoretical studies.

24.6

Multilayer Silicene Upon continuing to deposit Si onto Ag(111) beyond the first silicene monolayer, multilayer silicene grows in a terrace fashion, like rice fields in mountainous regions [43]. Multilayer silicene has just one unique structure: a √3  √3R(30
) reconstruction with respect to primitive silicene. However, depending on the first silicene layer, whether showing the sole 3  3 silicene phase on 4  4 domains or, instead, also √13  √13R(13.9
) reconstructed domains with respect to Ag(111), as discussed

466

Silicene, Germanene, and Stanene

Fig. 24.7 (a) Terrace growth of multilayer silicene with domains oriented at 30
, 33
, and 5.2


with respect to the in-plane Ag [10] direction (35 nm  35 nm; It = 0.27 nA; bias Vt = –1.1 V (filled states)), step height: 3.1 Å. (b) Zoom-in showing the honeycomb arrangement of the √3  √3R(30
) silicene reconstruction (5.5 nm  5.5 nm; It = 0.16 nA; Vt = –0.56 V (filled states); the √3  √3 cell is indicated).

above in Section 24.3, the √3  √3 structure growing on top can have orientations, respectively, at 30
, 33
and 5.2
[18, 44], as illustrated in Fig. 24.7. STS measurements showing linear dispersions [45] and synchrotron radiation photoemission measurements have demonstrated that √3  √3R(30
) multilayer silicene hosts massless Dirac fermions, evidenced in the cone sections recorded in ARPES around the zone centre Γ (see Fig. 24.8), with a very high Fermi velocity of ~0.3  106 m s–1, about one third of that of free-standing graphene [46]. The Dirac point is located at ~0.3 eV below the Fermi level because of charge transfer from the metallic silver substrate yielding partial filling of the π* upper cone. Transport measurements performed in situ using a four-probe STM, free of shortcircuit paths through the Ag support, have revealed a sheet resistance of silicene, which is comparable to the one found for graphite in nanograins [43]. Furthermore, multi-layer silicene is self-protected for at least 24 hours in ambient air by an ultra-thin native oxide skin, which preserves the integrity of the silicene sheets underneath [47]. Clearly, both results are highly promising for electronic applications. Nevertheless, the true nature of multilayer silicene is debated. Despite Raman signatures differing from those of silicon [47, 48] and a measured in-plane lattice parameter shrunk by ~4% with respect to that of the (111) plane of bulk silicon [43, 49], several authors argue that it simply does not exist: it would be cubic diamondlike silicon (111) layers either terminated by the Si(111)√3  √3R(30
)-Ag reconstruction due to the one monolayer of silver segregating at the surface [50], or, instead, having just an intrinsic weak buckling forming a honeycomb structure with Dirac surface states without invoking Ag surfactants [51, 52].

24.7 Germanene and Stanene

467

Fig. 24.8 Section of a Dirac cone (left) measured by SR-ARPES along the Γ–K direction at the center of the Brillouin zones (right); dotted line: Ag(111) surface BZ; dashed line: BZ of primitive silicene; full line: BZ of √3  √3R(30
) multilayer silicene.

We hope that such controversial issues will be solved soon and multilayer silicene will be confirmed with its anticipated outstanding properties. Typically, itinerant ferromagnetism and p + ip0 superconductivity is predicted in doped bilayer silicene [53]. As seen above, bilayers grown on Ag(111) are effectively doped due to charge transfer from the substrate. This could be a reason of the indication, although not presently confirmed, for high-Tc superconductivity up to nearly 40 K for the silicene √3  √3R(30
) phase [54].

24.7

Germanene and Stanene 24.7.1 Single-layer and Multilayer Germanene Single-layer germanene has been synthesized recently on gold (see Fig. 24.9) [55], aluminum [56], and platinum [57] (111) surfaces (although this has been disputed in the latter case because of the likely formation, instead, of a surface alloy [58]), as well as on Ge2Pt clusters [59–61]. Just like silicene, free-standing germanene does not exist, but it is predicted to be a buckled honeycomb 2D Dirac material [5] with extremely high mobilities of its charge carriers [62]. The spin–orbit coupling along with the ~0.64 Å buckling opens up a ~24 meV band gap at the Dirac points significantly higher than in silicene (1.55 meV); this, together with the non-trivial topological properties, might result in a quantum spin

468

Silicene, Germanene, and Stanene

Fig. 24.9 (a) STM topograph (–1.12 V; 1.58 nA) and (b) calculated atomic geometry of the

weakly buckled (by ~0.2 Å) √3  √3 reconstructed phase of germanene in coincidence with a √7  √7 supercell on gold (111).

Hall effect detectable at RT [63]. Electrostatic fields externally applied can permit tuning the band gap and can generate topological phase transitions and helical zeroenergy modes [32]. Proximity with an s-wave superconductor and photoirradiation could create a photoinduced topological superconductor hosting controllable Majorana fermions [64]. These fascinating properties offer tantalizing prospects for future applications in electronics, spintronics, and quantum computing. Since Ge is currently used as a performance booster in thin FET channels, perspectives of using germanene for scaling down beyond the 5 nm node while increasing the speed and lowering the energy consumption of electronic devices are very promising. However, single-layer germanene created directly on metal surfaces is likely to lose the massless Dirac fermion character and other unique physical properties of freestanding germanene because of the interfacial coupling. This issue has just been overcome by synthesizing multilayer epitaxial germanene and demonstrating that it possesses Dirac cones [65], as does its sibling, multilayer silicene [43, 46, 47]. Indeed, this will facilitate transfer to semiconducting or insulating substrates and make germanene more practical for use in devices.

24.7.2

Single-Layer Stanene Stanene, sometimes written stannene (from stannum, latin) and also sometimes coined tinene, has just been realized on Bi2Te3(111) substrates [66]. The sizeable spin–orbit coupling in stanene could permit exploiting its topological insulator phases and lead to the development of a new class of low-energy consumption nanoelectronic devices. Especially, as theoretically described, but not yet realized, a germanium (111) substrate could be very convenient to render stanene amenable for implementation in current semiconductor technology as a topological transistor. The transistor channel would be replaced by the 2D TI, the conductivity switched on and off at high speed by tuning the gate voltage with low-energy consumption [67].

24.9 References

24.8

469

Summary We have described the birth of single-layer silicene, germanene, and stanene, graphene’s group IV elemental cousins, respectively, in 2012, 2014, and 2015, and, further, the debut of multilayer silicene and germanene. These novel synthetic twodimensional Si, Ge, and Sn allotropes are artificially created in situ under ultra-high vacuum, since, at variance with graphene, which descends from graphite, they have no parent crystal in nature. Indeed, this is an extraordinary development. We have also surveyed some of their fascinating properties, notably, their easy functionalization, which make them promising candidates and rivals of graphene for ultimate scaling of nanoelectronic devices. The recent fabrication of the first silicene field-effect transistors with ambipolar characteristics operating at room temperature demonstrates their potential as emerging 2D electronic materials. Since germanene and stanene, with near room temperature 2D topological insulators, are also expected to be easily incorporated into the existing silicon-based industry, their advent, which could lead to the development of a new class of low-energy consumption nanoelectronic devices, excites a rapidly expanding community. Still, the properties and potential applications of these emerging 2D materials are far from being fully explored, but prospects for nanoelectronics, spintronics, and quantum computing are highly promising.

24.9

References [1] Radisavljevic B, Radnovic A, Brivio J, Giacometti V and Kis A, Nature Nanotechnol. 6 147 (2011). [2] Li L, Yu Y, Ye G-J, Ge Q, Ou X, Wu H, Feng D, Chen X-H and Zhang Y, Nature Nanotechnol. 9 372 (2014). [3] Takeda K and Shiraishi K, Phys. Rev. B 50 14916 (1994). [4] Guzman-Verri G and Lew Yan Voon L, Phys. Rev. B 76 075131 (2007). [5] Cahangirov S, Topsakal M, Aktürk E, Sahin H and Ciraci S, Phys. Rev. Lett. 102 236804 (2009). [6] Vogt P, De Padova P, Quaresima C, Avila J, Frantzeskakis E, Asensio M C, Resta A, Ealet, B and Le Lay G, Phys. Rev. Lett. 108 155501 (2012). [7] Le Lay G, Surface Sci. 132 169 (1983). [8] Liu Z-L, Wang M-X, Xu J-P, Ge J-F, Le Lay G, Vogt P, Qian D, Gao C-L, Liu C and Jia J-F, New J. Phys. 16 075006 (2014). [9] Wu K-H Chin. Phys. B 24 086802 (2015). [10] Liu Z-L, Wang M-X, Liu C, Jia J-F, Vogt P, Quaresima C, Ottaviani C, Olivieri, B, De Padova P and Le Lay L, APL Mater. 2 092513 (2014). [11] Li H, Fu H-X and Meng S, Chin. Phys. B 24 086102 (2015). [12] Le Lay G, Salomon E, De Padova P, Layet J-M and Angot T, Aust. J. Chem. 67 1370 (2014). [13] Fleurence A, Friedlein R, Ozaki T, Kawai H, Wang Y, and Yamada-Takamura Y, Phys. Rev. Lett. 108 245501 (2012).

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25

Predictions of Single-Layer Honeycomb Structures from First Principles S. Ciraci and S. Cahangirov

25.1

Motivation and Methodology Finding a contender for graphene in the field of 2D electronics and in other possible potential applications of nanotechnology has derived active search for graphene like novel structures, which do not exist in nature. As a matter of fact, the types of 3D layered materials, which make the exfoliation of their single-layer (SL) structures possible, are limited only to graphite, 2h-BN, 2h-MoS2, 2h-WS2, black phosphorus etc. However, most of desired electronic and magnetic properties demand materials that do not have layered allotropes. In view of the location of C, B, and N elements in the periodic table, which constitute SL graphene and BN, questions have been raised as to whether other group IV elements, group III–V and II–VI compounds may also form SL structures. The theoretical methods have provided for quick answers to guide further experiments. These methods, based on the quantum theory, have now reached now a level of providing accurate predictions for chemical, mechanical, electronic, magnetic, and optical properties of matter. In our group, we have carried out studies to explore novel materials in SL structure constituted by group IV elements, group III–V and II–VI, group V elements, transition metal oxides, and dichalcogenides, MX2 in h- and t-structures. We also consider their functionalization by decoration of ad-atoms, by creation of the mesh of vacancies and voids, by formation of nanoribbons or in-plane heterostructures. Most of the elements which construct SL materials have valence orbitals similar to carbon. These are atoms having s2 and pm valence orbitals, which can allow three folded, planar sp2 hybrid orbitals to form σ-bonds between two atoms located at the corners of hexagons. This way a three-fold coordinated honeycomb structure can be constructed. Remaining p orbitals form bonding (antibonding) π- (π*-) bonds with nearest neighbors. While the σ-bonds between atoms maintain the mechanical strength, π–π*-bonds assure the planar geometry and dominate the electronic energy structure near the Fermi level. SL structures including at least one element from the first row of the periodic table, prefer a planar structure such as graphene, h-BN and SiC, since the π-bond is strong enough to maintain the planar geometry. However, the situation is different for SL structures constructed by elements from rows lying below the first one, where nearest-neighbor distance is relatively longer and hence a weaker π-bond cannot maintain the planar geometry. At the end, the structure is stabilized by dehybridization of planar sp2

472

25.1 Motivation and Methodology

473

orbitals, and eventually rehybridization of sp3-like orbitals. Accordingly, the structure is buckled, where alternating atoms located at the corners of the hexagon are displaced in opposite and perpendicular directions. In this structural transformation, the projection of atoms continue to form again a honeycomb structure with a 2D hexagonal lattice. Minimizing the calculated total energy and also atomic forces at each atomic site attains a theoretical prediction of a structure or its functional form. Once the structure optimization resulted in a new SL honeycomb structure, the main issue is whether this structure is stable. Especially, the stability of a structure above room temperature is necessary for technological applications. First, ab-initio phonon calculations are carried out to check whether the SL structure remains stable after small displacements of atoms. The structure is viewed as stable when all the frequencies of phonon modes are positive and hence the SL structure corresponds to a local minimum on the Born–Oppenheimer (BO) surface. Imaginary frequencies indicate that the displacements of corresponding modes cannot be restored, and then the structure eventually dissassociates. Even if positive phonon frequencies indicate stability, it cannot be assumed that the structure corresponds to a deep local minimum on the BO surface and will remain stable under thermal excitations at high temperature. The stability at high temperatures is then investigated by performing ab-initio, finite temperature molecular dynamics (MD) calculations using two different approaches. Either the Nosè thermostat is used and Newton’s equations are integrated through the Verlet algorithm with a time step of 1–2 femtoseconds, or the velocities of atoms were scaled at each time step to keep the temperature constant. MD simulations carried out for several picoseconds at temperatures as high as 1000 K to ensure that the SL structure does not dissociate and hence can remain stable at least above room temperature. Notably, some of the honeycomb structures deduced by the total energy and force calculations were dissociated already at low temperatures after a few time steps, since they were actually unstable. In addition to phonon frequency and high temperature MD calculations, the stability of optimized structures are subjected to further tests. For example, the possibility that the optimized structure can undergo reconstruction covering several primitive unit cells is tested by optimization in large n  n supercells. Another possibility that the optimized structure may dissociate or change into clusters is examined by the adsorption of specific ad-atoms or by the formation of defects. Positive cohesive and formation energies are indicative of stability. High mechanical strength suggests robustness. An optimized SL structure, which passes all these stringent tests, is then considered to be stable in the freestanding state even if its parent 3D crystal is not layered like graphite. It should be noted that the stability of an SL structure does not mean that it can be synthesized; rather it means that this structure remains stable once it is synthesized in freestanding form. Since certain SL structures can be synthesized only by growing them on specific inert substrates, SL structure–substrate interaction may modify the properties calculated for the freestanding form. Therefore, the properties of the SL structure grown on substrates are calculated to see whether they are affected by the substrate. Single-layer, bilayer, multilayer, and layered periodic structures derived from freestanding SL structures may be stable and display properties gradually

474

Single-Layer Honeycomb Structures from First Principles

changing with the number of layers. These multilayers can be considered as new polymorphisms of a given SL structure. The stable SL honeycomb structures are characterized by calculating their equilibrium optimized structural parameters: total energy ET; cohesive energy EC relative to constituent free atoms; formation energy Ef relative to the allotrope having lowest total energy (in the global minimum); elastic, electronic, magnetic, optical properties, etc. In-plane stiffness, C = A o 1 ∂2ET/∂ε2 (Ao being the equilibrium area of the unit cell) and the Poisson’s ratio ν = –εy /εx are relevant quantities to quantify the strength and elastic properties of an SL structure. Because of their dimensionality, these structures attain high Poisson ratio and high uniaxial strain under uniaxial stress. In this respect, monitoring of the electronic structure – in particular, of the fundamental band gap with applied strain – is crucial for SL materials. The total charge density ρ(r), charge transfer between constituent atoms are also calculated to provide further information about the character of the binding and bond formation. The total energies of structures and atomic forces are calculated from the firstprinciples pseudopotential calculations based on the spin-polarized density functional theory (DFT) using the Vienna ab-initio simulation package (VASP) [1]. Since the fundamental band gaps are underestimated by standard DFT, calculations are carried out using the HSE06 hybrid functional [2] and quasi-particle GWo corrections [3].

25.2

Group IV Elements: Silicene, Germanene Even before synthesis of isolated graphene, theoretical studies based on the minimization of the total energy have pointed out that the single layer of Si in a buckled honeycomb structure can exists [4, 5]. However, freestanding silicene and germanene, together with their signature of massless Dirac fermion, ambipolar effects and nanoribbons showing familiar behavior, were first predicted after an extensive stability analysis [6, 7]. The need to unravel the exotic electronic structure and its integrability into the well-established silicon technology has placed silicene at the forefront of intensive theoretical and experimental research. Since Si and Ge do not have any 3D layered allotropes like graphite with weak interlayer van der Waals interaction, silicene cannot be exfoliated and hence freestanding silicene cannot exist in nature. The stable structure of silicene (germanene) has the following calculated values: the 2D hexagonal lattice constant is a = 3.83 Å (3.97 Å); the buckling height is Δ = 0.44 Å (0.64 Å) [6]. The phonon dispersion curves of silicene and germanene calculated for optimized buckled structures are shown in Fig. 25.1(c). Their stability continues to exist even above room temperature as revealed from ab-initio MD calculations performed at 1000 K for 10 picoseconds; similar tests have been done for finite size flakes, indicating stability above room temperature. The electronic energy band structures of silicene and germanene presented in Fig. 25.1(d) show the π- and π*-bands linearly crossing at the Fermi level. Spin–orbit coupling included later brought about other features such as topological insulator behavior [8]. Not only the physical properties of 2D periodically repeating silicene and germanene are similar to graphene, but also those of their

475

25.2 Group IV Elements: Silicene, Germanene

(c)

Planar Silicene

a2

Wavenumber (cm–1)

a1

120º

Buckled Silicene a1

a2

(d)

Silicene Phonons

600

Silicene Electrons 3

500

2

400

1

Energy (eV)

(a)

300 200 100

0 –1 –2

0 Г

K

M

–3

Г

Г

K

M

Г

116.2º

Germanene Phonons

300

Germanene Electrons

3

(b)

K Г

250

2

200

1

Energy (eV)

M

Wavenumber (cm–1)

First Brillouin Zone

150 100 50

0 –1 –2

0

–3 Г

K

M

Г

Г

K

M

Г

Fig. 25.1 (a) Ball and stick model of planar and buckled silicene. (b) The first Brillouin zone of a

honeycomb structure and corresponding symmetry points. (c) Calculated phonon dispersion curves of SL silicene and germanene in optimized, buckled honeycomb structure. (d) Electronic energy band structure of buckled silicene and germanene.

nanoribbons are reminiscent of graphene nanoribbons. For example, the armchair nanoribbons of silicene display the behavior similar to graphene, except that the edge atoms of the former are 2  1 reconstructed [6, 7]. Prediction of SL silicene boosted efforts to grow silicene on a substrate. Silicene was synthesized for the first time on an Ag(111) substrate [9]. It was shown that silicene acquires a 3  3 reconstruction, which is in perfect match with the 4  4 supercell of the Ag(111) surface. Moreover, linear bands near the Fermi level revealed by ARPES performed on the 3  3 silicene grown on Ag(111) are attributed to the significant hybridization between silicene and Ag sp bands [10]. The √3  √3 reconstruction is also frequently observed when silicene is deposited on an Ag(111) surface. Here two bright spots are formed in each √3  √3 supercell of silicone, making a honeycomb STM pattern [11, 12]. In contrast to 3  3 and √7  √7 reconstructions, the √3  √3 reconstruction in silicene is not matched by any lattice vector on the Ag(111) surface. Furthermore, it was found that the in-plane lattice constant of √3  √3 silicene is 5% smaller than the corresponding value in freestanding silicene. A model was proposed to explain the spontaneous formation of these 5% contracted √3  √3 silicene structures [13]. According to this model, adding more Si

476

Single-Layer Honeycomb Structures from First Principles

atoms on top of already formed silicene creates so-called dumbbell units [14, 15]. As the number of dumbbells increase, they organize themselves in such a way that there are two dumbbell units in each √3  √3 supercell. The resulting structure is spontaneously contracted to a lattice constant of 6.4 Å, which is what was measured in the experiments [11,12]. Later it was shown that even more layers with the √3  √3 reconstruction grow as silicon continues to be deposited [16]. Interestingly, the multilayer silicene grown in this fashion was shown to have a metallic character. On the theoretical side, it has been possible to extend the aforementioned √3  √3 dumbbell monolayer of silicene into a layered dumbbell structure called silicite [17]. This new allotrope of silicon is only 0.17 eV/atom less favorable than cubic diamond silicon and has an enhanced absorption in the visible range. However, it does not reproduce ~3.0 Å interlayer separation observed in multilayer silicene experiments. Recently, multilayer silicene reaching 40 layers was reported [18]. This structure was exposed to ambient air for 24 hours and survived by creating a thin oxide layer on the surface. More recently, a transistor made of silicene was shown to operate at room temperature [19]. Ambipolar Dirac charge transport with a room temperature mobility of 100 cm2/V s was measured in this system. Germanene was also synthesized by depositing germanium atoms on an Au(111) substrate [20]. The resulting structure was complex with coexisting phases, one of which was shown to be √3  √3 germanene matched by √7  √7 Au(111). Notably, adsorption of additional Ge ad-atoms on germanene creates dumbbell units such as silicene [21]. Germanene was also synthesized on an Al(111) surface [22]. In this case, the observed structure was uniform consisting of 2  2 germanene matched by a 3  3 Al(111) surface. Finally, stanene was also synthesized by depositing tin atoms on Bi2Te3(111) surface [23].

25.2.1.

Silicon Carbide Bulk SiC is a material, which is convenient for high temperature and high power devices. One expects that SL SiC can be synthesized, since graphene and silicene are already synthesized, and it may exhibit physical properties which are desired for specific applications in 2D electronics. First-principles calculations have predicted that SL SiC is stable in a honeycomb structure [24]. It is an ionic compound semiconductor with significant charge transfer from the Si to C atom and has a fundamental band gap of EG = 2.53 eV obtained using GGA, which increases to 3.90 eV after GoWo corrections. Other relevant properties, i.e. bond length, cohesive energy, and in-plane stiffness, are calculated to be d = 1.79 Å, EC = 11.94 eV/per SiC and C = 166 J/m2, respectively. When compared with the calculated values of 3D bulk SiC in zincblende or wurtzite structure and 1D chain structures, those of SL SiC in a honeycomb structure display intermediate values, except that the band gap is largest in the SL honeycomb structure [24].

25.2.2

Silicatene None of the allotropes of silica (i.e. amorphous or crystalline quartz) is known to have a graphite-like layered structure. Despite that, efforts have been devoted to

25.2 Group IV Elements: Silicene, Germanene

(a)

(b)

sp3

sp2

β

sp2

α

d2

sp3

sp3 10 9°

d1

2.6 eV

120°

Silicatene

°

0

14

d0 aβ = 5.39 Å hβ-silica

Si2O5

ay

220°

0.7 eV

96 °

sp3

sp3

477

aα = 5.18 Å hα-silica

ax

Fig. 25.2 (a) hα-silica derived from hβ-silica. (b) Silicatene derived from hα-silica

(adapted with permission from [26]).

grow a 2D ultra-thin polymorph of silica on substrates [25]. Recently, stable SL allotropes of silica, named as hα-silica and silicatene have been predicted [26]. The optimized structure of hα-silica, which is derived from ideal hβ-silica by lowering energy by 0.7 eV is described in Fig. 25.2(a). Remarkably, hα-silica is predicted to have a negative Poisson’s ν = 0.21. That is, as hα-silica is stretched along the x-direction, it also expands in the y-direction, owing to its squeezed structure consisting of twisted and bent Si—O—Si bonds resulting in a reentrant structure as described in Fig. 25.2(a). The negative Poisson’s ratio is a rare situation and those extreme materials having this property are called auxetic or metamaterials. The semiconductor hα-silica is non-magnetic with a direct band gap of 2.2 eV. Moreover, it is a rather rare situation that the variation in the calculated band gap is strain specific; it increases with increasing uniaxial strain εx, but it decreases with increasing εy. Owing to the dangling bonds oozing from Si atoms, hα-silica is rather reactive; through the saturation of Si dangling bonds upon oxidation it transforms to Si2O5 and the band gap of hα-silica increases from 2.2 eV to 6 eV, attributing a high insulating character and inertness like 3D silica. While the hexagon-like 2D geometry in Fig. 25.2(a) is maintained, sp2 bonded Si atoms change to sp3 bonded Si atoms and hence restore the rotary reflection symmetry. This way, Si atoms acquire the fourfold coordination of oxygen atoms as shown in Fig. 25.2(b) as in 3D silica. Upon heating, Si2O5 undergoes a structural transformation by further lowering (i.e. becoming more energetic) its total energy by 2.63 eV. In this transformation, the first half of the dangling Si—O bonds rotate from top to bottom so that all are relocated at the bottom side. Eventually, they are paired to form O—O bonds. The optimized structure predicted in Fig. 25.2(b) replicates the structure of the SL silica in a honeycomb structure named silicatene, the growth of which was achieved recently on a Ru(0001) surface [28].

478

Single-Layer Honeycomb Structures from First Principles

25.3

Group III–V and II–VI Compounds Groups III–V and II–VI compound semiconductors in zincblende or wurtzite structures dominate electronics, and optoelectronics. The question whether GaAs can form single wall nanotubes and SL honeycomb structures like graphene was addressed in 2005 [5]. Motivated by these early results, a comprehensive study has been carried out to explore SL structures of group IV elements and group III–V compounds [27]. Ab-initio phonon frequency calculations, which resulted in positive frequencies, have demonstrated that 17 new group IV–IV and group III–V compounds can remain stable in a SL honeycomb structure once they are synthesized. The equilibrium structure parameters, cohesive energy, energy band gap, the ratio of effective charges, Poisson’s ratio, and in-plane stiffness calculated using LDA approximation are presented in Table 25.1. Using the calculated values from Table 25.1, interesting correlations between cohesive energy and the lattice constant, and between in-plane stiffness and cohesive energy were deduced as shown in Fig. 25.3. For example, as EC decreases, the lattice constant a increases with increasing average row number of constituents. Similarly, C and EC are correlated and both increase with decreasing average row number of constituent elements. Additionally, the commensurate 1D heterostructures of these materials constructed from their nanoribbons, such as GaN/AlN having multiple quantum well structures with their band-lineups, have been proposed as an extension to 2D SL honeycomb structures [27]. It should be noted that by increasing the widths of nanoribbons and constructing these heterostructures, one can attain in-plane heterostructures or composite structures [29].

25.3.1

Group II–VI Compound: ZnO Bulk ZnO is an important optoelectronic material, because of its wide band gap of 3.3 eV and large exiton binding energy of 60 meV leading to LED and solar cell applications. Two-monolayer-thick ZnO(0001) films have been grown on an Ag(111) surface [30]. Based on first-principles calculations, an SL ZnO in planar honeycomb structure has been found to be stable [31]. It is a non-magnetic semiconductor and has a lattice constant of a = 1.89 Å and a direct band gap (calculated by GGA and corrected by GoWo) EG = 5.64 eV. In the bilayer of ZnO, the band gap decreases to 5.10 eV and saturates at 3.32 eV in the graphitic h-ZnO structure. Zig-zag nanoribbons of ZnO are ferromagnetic metals due to spins localized in oxygen atoms at the edges. Whereas bare and H saturated armchair nanoribbons of ZnO are a non-magnetic semiconductor, energy band gap saturates at 1.75 eV as their widths increase.

25.3.2

α-Graphyne and α-BNyne The stable SL structures α-graphyne and α-BNyne are derived from a honeycomb lattice with additional n atoms between the atoms placed at the corners of hexagon [32]. The

479

Geometry

Planar Buckled Buckled Planar Planar Buckled Buckled Buckled Planar

Geometry

Planar Planar Planar Planar Buckled Buckled Buckled Buckled Planar Planar Buckled Buckled Planar

Material

Graphene Silicene Germanene SiC GeC SnGe SiGe SnSi SnC

Material

BN AlN GaN InN InP InAs InSb GaAs BP Bas GaP AlSb BSb

0.00 0.00 0.00 0.00 0.51 0.62 0.73 0.55 0.00 0.00 0.40 0.60 0.00

Δ (Å)

θ (deg)

120.0 120.0 120.0 120.0 115.8 114.1 113.2 114.7 120.0 120.0 116.6 114.8 120.0

0.00 0.44 0.64 0.00 0.00 0.73 0.55 0.67 0.00

Δ (Å)

120.0 116.4 113.0 120.0 120.0 112.3 114.5 113.3 120.0

θ (deg)

1.45 1.79 1.85 2.06 2.46 2.55 2.74 2.36 1.83 1.93 2.25 2.57 2.12

d (Å)

1.42 2.25 2.38 1.77 1.86 2.57 2.31 2.52 2.05

d (Å) 20.08 10.32 8.30 15.25 13.23 8.30 9.62 8.72 11.63

EC (eV)

2.51 3.09 3.20 3.57 4.17 4.28 4.57 3.97 3.18 3.35 3.84 4.33 3.68

a (Å) 17.65 14.30 12.74 10.93 8.37 7.85 7.11 8.48 13.26 11.02 8.49 8.04 10.27

EC (eV)

GROUP III–V

2.46 3.83 3.97 3.07 3.22 4.27 3.89 4.21 3.55

a (Å)

GROUP IV

4.61/KK-6.86/ΓK 3.08/ΓM-5.57/ΓM 2.27/ΓK-5.00/ΓK 0.62/ΓK-5.76/ΓΓ 1.18/ΓK-2.88/ΓK 0.86/ΓΓ-2.07/ΓΓ 0.68/ΓΓ-1.84/ΓΓ 1.29/ΓK-2.96/ΓK 0.82/KK-1.81/KK 0.71/KK-1.24/KK 1.92/ΓK-3.80/KM 1.49/KM-2.16/KK 0.39/KK-0.23/KK

EG (eV) LDA-GW0

Semimetal Semimetal Semimetal 2.52/KM-4.19/KM 2.09/KK-3.83/KK 0.23/KK-0.40/KK 0.02/KK-0.00/KK 0.23/KK-0.68/KK 1.18/ΓK-6.18/ΓK

EG (eV) LDA-GW0

0.85/7.15 0.73/7.27 1.70/6.30 1.80/6.20 2.36/5.64 2.47/5.53 2.70/5.30 2.47/5.53 2.49/5.51 2.82/5.18 2.32/5.68 1.58/6.42 3.39/4.61

Zc/Za

0.0/0.0 0.0/0.0 0.0/0.0 1.53/6.47 2.82/5.18 3.80/4.20 3.66/4.34 3.89/4.11 2.85/5.15

Zc*/Za*

0.21 0.46 0.48 0.59 0.43 0.43 0.43 0.35 0.28 0.29 0.35 0.37 0.34

ν

0.16 0.30 0.33 0.29 0.33 0.38 0.32 0.37 0.41

ν

267 116 110 67 39 33 27 48 135 119 59 35 91

C (J/m2)

335 62 48 166 142 35 57 40 98

C (J/m2)

Table 25.1 Calculated values for group IV elements, their binary compounds, and group III–V compounds forming a stable SL honeycomb structure. These are angled between neighboring bonds θ; buckling parameter Δ; bond length d; 2D hexagonal lattice constant a; cohesive energy EC; fundamental band gap EG calculated by LDA and corrected by GWo with symmetry points indicating where the minimum (maximum) of conduction (valence) band occurs; calculated effective charges on the constituent cation/anion Zc*/Za*; Poisson’s ratio ν; in-plane stiffness C (this table is taken with permission from [27]).

Single-Layer Honeycomb Structures from First Principles

20

350

Graphene

Cohesive Energy (eV)

18 BN 16

SiC

14

AlN BP GeC SnC GaN

12

Si SiGe SnSi Alsb GaP InAs Insb 4.0 3.5 4.5

BAs InN BSb

10 8 2.5

3.0

In-plane Stiffness (J/m2)

480

Graphene

300

BN

250 200 150 100 50 0

GeC SiC BAs BP AlN BSb GaN SnC GaP Si InN SiGe Insb

o

Lattice Constant (A)

10 15 Cohesive Energy (eV)

20

Fig. 25.3 Correlations between the cohesive energy EC and lattice constant a, and between inplane stiffness C and cohesive energy EC among stable SL honeycomb structures. Squares and circles are for planar and buckled structures, respectively (adapted with permission from [27]).

SL structure α-graphyne is stable for n = even and exhibits Dirac cones similar to graphene. However, for n = odd, it is unstable and undergoes a structural transformation by breaking hexagonal symmetry and opening a band gap. The SL structure α-BNyne is a semiconductor with the band gap decreasing with increasing n; EG = 4.13 eV for n = 2, but it decreases to EG = 3.46 eV. Both α-graphyne and α-BNyne form stable bilayers with AB stacking.

25.4

Group V Elements: Nitrogene and Antimonene More recently, the fabrication of a field-effect transistors, using micrometer sized flakes consisting of two to three layers of black phosphorus [33] and theoretical analysis [34], revealing the stability of its single-layer allotropes, i.e. blue and black phospherenes, brought group V elements into focus. Recent theoretical analysis exploring the idea of whether Sb and N can form SL structures have concluded that these two elements can also form stable, SL buckled honeycomb structures, called nitrogene and antimonene, respectively [35, 36]. Notably, while strong the N2 molecule is triple bonded, nitrogene is constructed from threefold coordinated and single-bonded N atoms similar to the 3D cg-N crystalline phase. However, unlike semimetallic graphene or silicene which have perfect electron– hole symmetry, nitrogene is a wide band-gap insulator with a DFT band gap of EG = 3.96 eV (EG = 5.96 eV after HSE correction). The buckling distance is Δ = 0.7 Å and cohesive energy is EC = 3.67 eV/atom [35, 36]. Moreover, nitrogene can form stable nanoribbons with band gaps in the range of 0.6 eV < EG < 2.2 eV, bilayer and 3D graphitic structure named nitrogenite. Antimonene has a stable SL buckled honeycomb (h-Sb) structure, as well as an asymmetric washboard (aW-Sb) structure; the latter has slightly higher cohesive energy.

25.5 Transition Metal Oxides and Dichalcogenides

481

Here we consider only h-Sb, which has cohesive energy EC = 2.87 eV and is a nonmagnetic semiconductor with an indirect band gap of 1.04 eV, calculated within PBE approximation, which occurs between the minimum of the conduction band along the Γ–M direction and a maximum of the valance band at the Γ point. Upon HSE correction the indirect band gap increases to 1.55 eV. Apparently, the band gap of h-Sb lies in the range, which is convenient for several 2D electronic applications. Free-standing SL antimonene is metallized when grown on substrates such as a Ge(111) surface or germanene. Also interlayer coupling is significant and attributes metallicity to bilayer and multilayer antimonene [35, 36].

25.5

Transition Metal Oxides and Dichalcogenides Three-dimensional transition metal oxides or dichacogenides MX2 (M, transition metal; X, oxygen or chalcogen atoms) compounds constitute one of the most interesting classes of crystals; their wide range of properties have been investigated since 1960. Some of these compounds have D6h-point group symmetry and are layered structures formed by the stacking of weakly (vdW) interacting 2D MX2 layers and are specified as 2h-MX2 like layered MoS2 crystals. Another type of layered structure is specified as a 2t-structure (centered honeycomb) and has D3d-point-group symmetry. Some 3D MX2 structures are known to be stable in rutile, 3R, marcasite, anatase, pyrite, and tetragonal structures. Interest in 2D materials has led to the synthesis of SL MoS2 (38), WS2 (39) with honeycomb structure and NBSe2 only on SiO2. Coleman et al. reported liquid exfoliation of MoS2, WS2, MoSe2, TaS2, NbS2, NiTe2, and MoTe2 (40). In both h and t structures, instead of forming covalent sp2-bonding with three neighboring atoms as in graphene, each M atom has the six nearest X atoms and each X atom has the three nearest M atoms forming p–d hybridized ionic M-X bonds. These 2D materials have

Table 25.2 Calculated values of stable, SL, MX2 in h- and t-structures: Lattice constants, a = b; bond lengths, dM–X, dX–X; X—M—X bond angle, θ; cohesive energy per MX2 unit, EC; energy band gap, EG; total magnetic moment in the unit cell, μ; in-plane stiffness, C. (The full version of this table can be found in [37]. Reproduced with permission.)

Material

Geometry

a (Å)

dM–X (Å)

dX–X (Å)

θ (deg)

EC (eV)

EG (eV) LDA-GW0

μ (μB)

C (N/m)

MnS2 MnSe2 MnTe2 MoS2 MoSe2 MoTe2 WO2 WS2 WSe2 WTe2

t t t h h h h h h h

3.12 3.27 3.54 3.11 3.24 3.46 2.80 3.13 3.25 3.47

2.27 2.39 2.59 2.37 2.50 2.69 2.03 2.39 2.51 2.70

3.29 3.50 3.77 3.11 3.32 3.59 2.45 3.13 3.34 3.61

93.08 93.78 93.56 81.62 83.05 83.88 74.12 81.74 83.24 83.96

14.82 13.61 12.27 19.05 17.47 15.65 24.56 20.81 19.07 17.05

Metal Metal Metal 1.87–2.57 1.62–2.31 1.25–1.85 1.37–2.87 1.98–2.84 1.68–2.38 1.24–1.85

2.38 2.35 2.29 NM NM NM NM NM NM NM

66.87 56.61 44.77 138.12 118.37 92.78 250.00 151.48 130.04 99.17

482

Single-Layer Honeycomb Structures from First Principles

shown exceptional physical and chemical properties. For example, transistors fabricated from a SL MoS2 presented features, which are superior to those of graphene [41]. Also SL MoS2 appears to be promising for optoelectronic devices, solar cells, LEDs, and HER (hydrogen evaluation reactions). In an extensive theoretical study exploring other possible SL structures, out of 88 different combinations (M = Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Nb, Mo, W and X = O, S, Se, Te) 52 freestanding, stable h-MX2 and t-MX2 structures have been predicted [37]. The optimized lattice constants and values calculated using LDA for selected h-MX2 and t-MX2 are presented in Table 25.2.

25.6

Conclusions Theoretical studies outlined in this chapter predicted 79 new stable, SL honeycomb structures of different elements with electronic and magnetic properties, which may be utilized in the emerging field of nanotechnology. Some of these theoretical predictions have been realized by synthesizing novel SL materials, which are now subjects of active research.

Acknowledgments Authors acknowledge valuable contributions of their collaborators E. Aktürk, O. Üzengi Aktürk, C. Ataca, E. Durgun, V. O. Özturk, H. Sevinçli, H. Şahin and M. Topsakal to various studies and papers, on which this review is based.

25.7

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Index

0D defects, 56 13C isotope, 93 2D band, graphene, 80, 90 2D density of states, 400 2D material defects, 359 2D peak, graphene, 228 2D quantum spin Hall effect, 458 2D van der Waals heterostructures, 332 2DEG, plasmons, 44, 108–109 2DEGs, 45, 122 3
3 silicene phase, 460 3-band tight binding model, 282 3D resistivity, 225 3D van der Waals heterojunctions, 331 4
4 matrices, graphene, 15 α-BNyne, 480 α-graphynes, 480 π band, 474 π* band, 474 π-bond, 472 π-states, graphene, 12, 16 σ-states, graphene, 12 A and B excitons, 329, 331 AA stacking, graphene, 8 AB Bernal stacking, graphene growth, 245 AB bilayer graphene, 47, 78–79 ab-initio phonon calculations, 473 ab-initio phonon frequency calculations, 478 absorbance, 42 absorbance universal value, graphene, 40 absorbance, graphene, 39, 180 absorption, 30, 39, 41, 44–45, 135, 180, 192 absorption cross-section, graphene, 120, 122 absorption of phonons, 26 absorption resonance, graphene, 131 absorption/reflection spectroscopy, graphene, 38 access resistance, 163 acoustic (A) phonon, 34, 71 acoustic deformation potential, 34 acoustic modes, 10 acoustic phonon scattering, 34, 117, 121, 183, 225

acoustic phonon, graphene, 34, 77, 92, 96 acoustic phonons, dispersion, graphene, 96 admittance, 106–107 AFM topography analysis, 34, 274 all-dry deterministic transfer method, 448 ambipolar operation, 305 Ampère’s law, 105 amplifier, graphene, 164 amplitude attenuation length, plasmons, 115 amplitude lifetime, plasmons, 115 angular momentum conservation, 43 angular-dependent polarization-resolved Raman measurement, 422 anomalous quantum Hall effect, 207 anti-bonding states, 388 antimonene, 480 antimonene asymmetric washboard structure (aW-Sb), 480 antimonene buckled honeycomb (B-Sb), 480 anti-Stokes (aS), Raman spectroscopy, 77–78 anti-Stokes scattering, 78 anti-Stokes scattering, Raman spectroscopy, 78 archetype silicene phase, 460 armchair direction, 57, 59 armchair edges, graphene, 81 ARPES, 48, 118 ARPES, 46 asymmetric metal contacts, 186–187 atomic force microscopy (AFM), 53, 167 atomic hydrogen, 32 atomic layer deposition (ALD), 451 atomic orbital magnetic moment, 285 atomic vacancy, graphene, 32, 56 atomically sharp defects, 21 atomic-scale inhomogeneities, 15 back gated electric field, 36 back-gated BP transistor, 418 back-gating graphene, 145 background permittivity, 109 ballistic graphene devices, 148 ballistic limit, 299 ballistic spin propagation, 202

485

486

Index

ballistic transport, 226–227, 297 ballistic transport, epitaxial graphene nanoribbons, 238 band counting algorithm, 410 band gap, 12, 47, 160, 168, 175, 230–231, 234 band gap engineering, graphene, 160, 171 band gap opening, 167 band gap opening, bilayer graphene, 170, 187 band gap, bilayer graphene, 19, 47–48, 164, 170 bandgap opening, graphene, 231 bandstructure, graphene, 180 basal plane, graphene, 55 basic transport properties, graphene, 25, 145 Becke three parameters Lee–Yang–Par (B3LYP), 382 bending stress, graphene, 61 benzyl viologen, 170–171 Bernal AB stacking, graphene, 7, 170 Bernal stacked, bilayer graphene, 94 Bernal stacking, bilayer graphene, 17 Bernal stacking, graphene, 8, 46 Bernal structure, 219 Berry curvature, 286–287 Berry phase effect, 207, 286 Bethe–Salpeter equations, 398 beyond RPA, 114 Bi2Sr2Co2O8 layer, 172 bilayer, 18, 20, 48, 74, 84, 125, 134 bilayer and multilayer graphene, 46 bilayer graphene, 46–47, 47, 74–75, 77–78, 91, 93–94, 126, 159, 170, 172, 175, 185 bilayer graphene bolometer, 185 bilayer graphene devices, 170 bilayer graphene transistors, 169 bilayer graphene, FET, 170 bilayer optical transitions, 38 birefringence, 110 birefringent dielectric, 111 birefringent dielectric, h-BN, 110 birefringent material h-BN, 110 bismuth selenide, 214 bismuth-flux method, 415 black arsenic–phosphorus band gap, 436 black phospherene, 72, 480 black phosporus, 74, 78 BLG, 230 Bloch electron Berry phase effect, 285 Bloch electron magnetic moment, 285 Bloch electrons, 286 Bloch function, 286 Bloch functions rotational symmetry, 283 Bloch wave functions, 280 Bloch–Grüneisen temperature, 300 block copolymer lithography, 168 blue phospherene, 72, 74, 480 blue phosphorene in-plane deformation, 405 BN encapsulated graphene, 231

bolometers, 185 bolometric effect, 135, 183, 187 Boltzmann equation, 26 Boltzmann transport theory, 25, 27 Born approximation, 26, 29 Born–Oppenheimer (BO) surface, 473 Bose–Einstein distribution, phonons, 77 bound excitons, 331 BP ab-initio methods, 436 BP absorption spectra, 394 BP acoustic modes, 399 BP ambient stability, 420 BP ambipolar field effect, 448 BP angle-dependent electrical conductivity, 381 BP angle-dependent electronic mobility, 404 BP angle-dependent thermal conductivity, 381 BP angular dependence drain current, 418 BP anisotropic carrier effective masses, 391 BP anisotropic lattice vibration, 426 BP anisotropic medium, 389 BP anisotropic optical conductivity, 425 BP anisotropic optical properties, 438 BP anisotropic photoluminescence, 440 BP anisotropic Poisson’s ratio, 406 BP anisotropic properties, 381, 414 BP anisotropic sound velocity, 399 BP anisotropic thermal expansion, 424 BP anisotropic thermal transport behavior, 425 BP anisotropy absorption, 439 BP anisotropy ratio, 404 BP armchair, 381, 399, 416 BP armchair mobility, 417 BP armchair phonon mode, 400 BP armchair tensile strains, 381 BP asymmetric lattice structure, 440 BP asymmetric phonon dispersion, 423 BP atomic orbital wave functions, 383 BP ballistic transport limit, 401 BP band structure calculation, 417 BP bending stiffness, 407 BP binding energies, 398 BP bolometric effect, 446 BP Boltzmann transport equation, 400 BP Bridgman method synthesis, 415 BP Brillouin zone, 399 BP bulk band structures, 382 BP bulk optical bandgap, 409 BP carrier effective mass, 417 BP carrier mobility, 175, 391 BP catalyst-based synthesis, 415 BP catalyst developed synthesis, 415 BP charge doping, 393 BP chemical stability, 451 BP compressive strains, 426 BP compressive zigzag strain, 428 BP contact resistance transfer length method (TLM), 421

Index

BP coupled hinge-like bonding configurations, 414 BP crystal structure, 381, 414 BP crystalline orientation, 438 BP direct optical transition, 394 BP doped phototransistors, 443 BP Drude model, 391 BP effective Hamiltonian, 383 BP effective mass, 386, 417 BP effective mass for conduction band state, 386 BP effective mass for valence band state, 386 BP elastic acoustic phonon-carrier scattering, 391 BP electrical properties, 429 BP electrical transport, 391 BP electroluminescence, 442, 451 BP electron mobility, 391 BP electron transport, 402 BP electron–hole Coulomb interaction, 394 BP electronic anisotropic behavior, 417 BP electronic bandgap, 387 BP electronic hybridization, 388 BP electronic properties, 389 BP electronic transport, 423 BP electrostatically gate, 448 BP energy dispersion, 439 BP enhanced Hall mobility, 420 BP exciton eigenfunctions, 397 BP exciton eigenstates, 397 BP exciton spectrum, 397 BP excitonic polarization, 381 BP excitons, 394 BP external quantum efficiencies (EQE), 450 BP extinction ratio, 441 BP few layer band structures, 382 BP few layer bandgap, 387 BP few layer electronic band structure, 382 BP few layer electronic bandgap, 388 BP few layer optical absorption spectra, 393 BP fiber optic telecommunications, 438 BP field-effect mobility, 418 BP field-effect transistors, 416–417, 480 BP flake mobility, 421 BP fracture strain, 407 BP G-point phonon mode, 400 BP ground state exciton energy, 397 BP GW approximation, 436 BP Hall mobility, 418 BP Hamiltonian matrix, 383 BP high hydrostatic pressure synthesis, 414 BP high-speed photodetectors, 445 BP hole concentration, 404 BP hole effective masses, 391 BP impurity scattering, 391 BP infrared photodetection, 445 BP in-plane anisotropy, 435 BP in-plane transport, 402 BP inter-band transition matrix, 438

487

BP interfacial Coulomb scattering, 391 BP Landauer formalism, 400 BP large-area synthesis, 415, 451 BP laser exfoliation technique, 416 BP light polarization, 393 BP linear dichroism, 438–439 BP linearly polarized photoluminescence, 440 BP local-orbital method, 417 BP longitudinal acoustic phonon dispersion, 399 BP low-energy acoustic modes, 400 BP mechanical anisotropic behavior, 426 BP monolayer GGA functionals, 382 BP monolayer GW methods, 382 BP monolayer k. p approximation, 383 BP monolayer meta-GGA functionals, 382 BP monolayer optical bandgap, 381 BP nanoribbons, 426 BP near-infrared phototransistor, 443 BP non-hydrogenic Rydberg series, 397 BP non-toxic catalyst developed process, 415 BP non-toxic reaction method, 415 BP on SiO2/Si substrates, 442 BP optical absorption, 392, 439 BP optical phonon modes symmetries, 422 BP optical properties, 414, 435 BP orthotropic plate model, 406 BP oscillator strength, 398 BP oscillator strength ground exciton state, 398 BP oscillator strength second excited exciton state, 398 BP Pauli blocking, 394 BP phase transition process, 414 BP phonon backscattering, 400 BP phonon dispersion, 399–400 BP phonon energies, 399 BP phonon mean-free-path, 401 BP phonon relaxation time, 400 BP phonon transport, 399 BP phonon transport properties, 400 BP phonon–phonon scattering, 426 BP photocurrent generation, 443 BP photocurrent generation mechanisms, 442 BP photodetection, 439 BP photodetection efficiency, 447 BP photodetection quantum efficiency, 447 BP photodetector, 442, 447 BP photodetector dark current, 446 BP photodetector devices, 442 BP photodetector response speed, 446 BP photoluminescence, 394 BP photoresponse speed, 447 BP photo-responsivity, 446 BP photothermoelectric effect, 443 BP phototransistor, 442, 444 BP phototransistors display ambipolar transport, 442 BP photovoltaic effect, 443, 449 BP plasmons dispersion, 390

488

Index

BP p–n junctions, 448 BP polarizability, 389 BP polarized light absorption, 393 BP P—P bond elongation, 407 BP P—P bond length, 407 BP p-type doping, 449 BP pulsed laser deposition, 416 BP quantum confinement, 435, 439 BP quantum confinement effect, 436 BP quantum Hall effects, 420 BP Raman modes, 428 BP recombination rate, 447 BP Schottky-barrier transistors, 420 BP Schrödinger equation, 383 BP scotch tape exfoliation method, 426 BP self-consistent pseudopotential method, 417 BP shear modulus, 406 BP Shubnikov–de Haas oscillations, 420 BP solar cell device, 449 BP solar cells, 448 BP strain-induced frequency shift, 428 BP surface roughness scattering, 402 BP synthesis, 414 BP tensile strain, 428 BP thermal anisotropic behavior, 421 BP thermal conductivity, 423 BP thermal properties, 399 BP thermal transport, 399, 422 BP thermoelectric conversion, 399 BP thermoelectric Landauer approach, 402 BP thermoelectric performance, 402 BP thermoelectricity, 402 BP thickness dependent band gap, 435–436 BP tight-bonding method, 417 BP transformation from white/red phosphorus, 415 BP transistor contact resistance, 420 BP transport distribution, 400 BP ultra-fast laser direct writing, 416 BP Umklapp phonon–phonon scattering, 402 BP uniaxial tensile, 426 BP van der Waals heterostructures, 449 BP vertical configuration photodetector, 447 BP Young’s modulus, 405 BP zigzag, 381, 399, 416 BP zigzag mobility, 417 BP zigzag phonon mode, 400 BP zigzag tensile strains, 406 BP zigzag-polarized laser excitation, 422 BP/h-BN interface, 420 BP/MoS2 p–n junction, 449 BP/silicon photonic waveguide, 446 BPA armchair-polarized laser excitation, 422 BP-GaAs p–n junction, 449 BP-MoS2 rectifying behavior, 449 Bravais lattice, 11 breaking load, graphene, 64 breaking strength, mechanical properties, 57

breathing mode, bilayer graphene, 75 Brillouin zone, 80, 94, 142, 198, 209, 229 Brillouin zone, graphene, 12, 71, 94 Brillouin zone, phonon, 72–73, 81 Brillouin zone, superlattice, 229 broadband light absorption, 238 bulk semiconductor flexible membranes, 340 Burgers vector, 364 BurtonCabreraFrank theory, 370 Bychkov–Rashba splitting, 199 C=O vibrational absorption, plasmon enhancement, 134 carbon nanotubes, 53, 56, 160, 167 carbon nanotubes (CNTs), 98 carrier confinement and guiding, 150 carrier density, 27, 29–35, 42, 46, 104, 117, 130–131, 180, 224–227, 225 carrier density control, 181 carrier density, electrical control, 27, 146 carrier density, graphene, 27, 41, 45 carrier density, MoS2, 233 carrier mobility, 25, 30, 32–35, 175, 183, 200, 219, 231 carrier mobility degradation, 170 carrier mobility GNRs, 168 carrier mobility, graphene, 25, 30, 32–35, 131, 146, 159, 225, 238 carrier mobility, nanoribbons, 159 carrier–carrier scattering, 191 carrier–carrier scattering, plasmons, 132 Cauchy stress, mechanical properties, 58, 60–61, 63 cavity-integrated graphene photodetectors, 184 CdTe/HgTe/CdTe quantum wells, 214 channel length, graphene transistors, 161 charge carrier scattering, graphene, 25 charge neutrality point (CNP), 27, 171, 229 charge transport rates, 315 charged impurities, 25, 28, 29–31, 33, 199 charged impurities, graphene, 25, 32, 81 charged impurity scattering, graphene, 28, 34 charged surface states, scattering, 219 chemical doping, graphene, 30, 170 chemical vapor deposition, 39, 42, 65, 344–345, 359 chemical vapor deposition process, graphene, 239 chiral, 133 chiral (helical) states, graphene, 16 chiral components, few-layer graphene, 46 chiral currents, 215 chiral nature, electrons, 12 chiral properties, 18 chiraledge channels, 214 circular dichroism, graphene, 44 circular-toothed antenna, 187 classical forbidden region, 144 coherence length LC, phonons, 79

Index

coherence length, La, graphene, 83 coherent phonon spectroscopy, graphene, 75 collective excitations (plasmons), 180 combination D+D0 mode, graphene, 80–81 complementary inverters, 170 composites, thermal conductivity, 99 compression factor, plasmons, 128 computer CPU temperature, 100 conductance oscillations, 155 conduction band, 12 conductive coatings, graphene, 96–97 conductivity, 27–31, 34, 39, 106, 114 conductivity losses, graphene, 116 conductivity mismatch, 201, 213 conductivity relaxation time, plasmons, 116 conductivity, Drude model, 108 conductivity, graphene, 25–34 conductivity, in-plane thermal, 91 conductivity, local, 112 conductivity, microscopic approach, 111 conductivity, non-local, 117 conductivity, non-Drude, 108 confinement factor, graphene plasmons, 134 conical Dirac dispersion, graphene, 244 conical points, 12, 14–15, 17 contact resistance, 201, 224 contact resistance, graphene, 175, 223–224 cooperative island growth, graphene, 243 copper oxide, 30 Coulomb potential, 28 coupled harmonic oscillators, 124 coupled oscillator model, 76 coupled plasmon–phonon dispersion, 122 coupled two harmonic oscillators, 125 coupling parameter, coupled oscillators, 125 covalent bonds, 7 Cr adhesion layer, 224 cracks, graphene, 52 crystal structure of bilayer graphene, 17 crystallite size La, graphene, 79, 82 Cu and Ni ratio in alloy catalyst, graphene growth, 241 current saturation, 164 current saturation velocity, 160 cut-off frequency, 161, 163 cut-off frequency, fT, 160 CVD, 45, 67–68, 90, 93, 133, 163, 193–194 cyclotron, 38 cyclotron frequency, 43, 133 cyclotron mass, graphene, 132 D and G band ratio, graphene, 82 D band scattering, graphene, 81–82 D band, graphene, 80–82 D peak, Raman spectroscopy, 64 Dʹ peak, Raman spectroscopy, 64 damped harmonic oscillator, 122, 124

489

dark current, 184 dark modes, 120 dc conductivity, 41 deep electron traps, 360 defect control engineering, 359 defective graphene, 31–33, 64 defects, 21, 31–34, 48, 52–53, 56–57, 63–64, 70, 80–83, 92–93, 128, 180, 207 defects equilibrium concentration, 360 defects in h-BN, 362 deformation mechanisms, graphene, 57 degenerate ultrafast pump-probe measurements, 315 density functional theory (DFT), 60, 260–261, 267, 474 density of states (DOS), 27, 120, 172 density-density response function, plasmons, 112–114 density-dependent resistivity, 28–35 dephasing, spin, 209–210 diamond, 7–9, 54, 78 diamond-like carbon (DLC), 161, 164 dielectric, 105, 108, 116, 129, 162 dielectric breakdown, graphene, 131 dielectric constant, 30 dielectric dispersion, 108 dielectric environment, 30–31, 110, 112, 301 dielectric environment, plasmons, 104, 106, 110, 134 dielectric response, 112 dielectric screening, 30, 301 diethyl sulphide ((C2H5)2S), 351 differential reflectivity, 76 diffusion-limited growth, graphene flake, 249 dipolar emitter, plasmons, 122, 126 Dirac cone, graphene, 39, 244 Dirac electrons, graphene, 44–45 Dirac fermions, 29 Dirac Hamiltonian, 16, 199 Dirac plasmons, 114 Dirac point, graphene, 29–32, 39 Dirac spinors, 16 directional, thermal emitters, 192 dislocation edge type, 363 dislocation screw type, 363 Dislocations in h-BN, 363 Dislocations, topological defects, 363 disorder amount, graphene, 80, 82–83, 244 disorder induced bands, graphene, 81 disorder induced-Raman, 79 disorder scattering, graphene, 25–36, 92 disorder, graphene, 116, 141, 145, 210 dispersion, 10, 40, 96, 108, 111, 116–117, 123, 128, 133, 144, 170 dispersion law, 108 dispersion ZA phonons, 95 dispersion, graphene, 180

490

Index

dispersion, graphene plasmons, 106, 115 displacement field, 48, 170–171 dopant diffusion, 361 doped graphene absorption, 180 doping levels, graphene, 30, 45 double-resonance mechanism, Raman spectroscopy, 80 double-resonance, Raman scattering, 73, 75 driven oscillator, 125 Drude conductivity, graphene, 42, 116 Drude formula, 116 Drude model, graphene, 41–42, 104, 108, 189 Drude scattering, graphene, 42 Drude transport time, 117 Drude weight, 42, 43, 104, 105, 108, 131, 180 dwell time, 207 Dyakonov–Perel mechanism, 204, 206, 212 Dyakonov–Shur mechanism, 187 Dyakonov–Shur mechanism, THz generation, 183 dynamical dielectric function, phonon, 123 dynamical dielectric function, plasmons, 113, 116 E2g phonon, 71 edge contact geometry, 223 edge contact resistance, 224 edge contacts, 223, 226 edge defects, 372 edge magnetoplasmons, graphene, 133 edge modes, 121 edge scattering, 121, 169–170, 227 edge states, 146–147 edges/grain boundaries, 79 edges/grain boundaries, graphene, 81 EELS, 118, 223 effective Hamiltonian matrix, 198, 384 effective hopping integral, 282 effective k.p Hamiltonian matrix, 384 effective magnetic field, 199, 208 eigenfunctions, graphene, 16 elastic constants, 59 elastic energy density, graphene, 57 elastic instability, mechanical properties, 63 elastic response, non-linear, graphene, 57 elastic stiffness, graphene, 55 elastic strain energy density, mechanical properties, 57–58, 60 elastic strain, graphene, 56–57 elastic structural instability, 64 elastomer stamp, 221 electric field, 25, 104–105, 127, 129, 160, 187 electric gating, 20 electrical doping, graphene, 40 electrical gating, 48 electrical gating, graphene, 38 electrical transport in graphene, 25 electro-absorption modulation, 187–188 electrodynamic modes, graphene, 105

electrolyte gating, graphene, 40, 131 electrolyte layer gating, 40 electromagnetic modes, 104 electron beam irradiation, 367, 374 electron beam lithography, 167 electron doping, graphene, 48 electron drift velocity, 160 electron hole finite kinetic momentum, 321 electron hole interlayer hopping, 322 electron mobility, 160, 225 electron mobility, graphene, 141, 226 electron resonant scattering, 207 electron velocity, graphene, 14 electron–electron interactions, 46, 112, 118 electron–hole pairs, 38, 78, 114, 120, 185 electron–hole puddles, 29, 208, 211–212, 220 electron–hole symmetry, 12, 14 electronic energy structure, 472 electronic gating, 48 electronic structure control, bilayer graphene, 39, 48 electronic structure of bilayer graphene, 17, 46, 47 electronic structure of single-layer graphene, 11 electronic structure, few layer graphene, 46 electronic structure, graphene, 39 electron–lattice coupling parameters, 269 electron–phonon coupling, 34–36, 269 electron–phonon matrix element, 34 electron–phonon scattering, 34–36, 403 electrostatic doping, 45, 131, 184, 194, 224 electrostatic gating, graphene, 40, 42 electrostatic transparency, 172 electrostatic/ chemical doping, 181 elementary excitations, graphene, 38–39 Elliott–Yafet mechanism, 204, 207 Elliot–Yafet spin relaxation mechanism, 200 emission of phonons, 26 emitter and graphene plasmons, graphene, 126 encapsulation, 233 energy bands, solids, 7 energy lifetime, plasmons, 115 energy loss spectrum, 118 energy spectrum, graphene, 13 enhancement of the near-field, graphene ribbons, 129 environmental insensitivity, graphene, 227 epitaxial 3
3 silicene electronic structure, 463 epitaxial graphene, 48, 128 epitaxial graphene transitors, 163 epitaxial growth method, germanene, 238 epitaxial growth method, graphene, 238 equibiaxial strain, 58 equibiaxial strain, mechanical properties, 59, 61, 63 erbium-doped fiber, 191 europium oxide, 213 evanescent modes, 12, 120, 191 ex-situ experiments, 33 ex situ molecular doping, 361 exchange interaction, spin flip, 206

Index

exciting plasmons, 126 exciton condensation, 273 exciton funnel, 372 exciton Hamiltonian, 396 exciton wave function, 396 exciton–exciton annihilation, 331 exciton–exciton separation, 318 excitonic condensation, 319 excitonic superfluidity, 319 excitonic valley pseudospin, 285 excitonic valley pseudospin, 249 exciton–photon coupling, 285 exfoliated graphene, 38, 46, 67 exfoliated graphene, microscopy, 39 experimental validation, graphene, mechanical properties, 61 external quantum efficiency, 194 extinction spectra, bilayer graphene, 125 extinction spectra, doping, 125 extinction spectra, graphene, 120 extinction spectra, ribbon array, 125 extrinsic plasmon damping, graphene, 117 extrinsic spin splitting, 200 Fabry–Pérot oscillations, 149 Fabry–Pérot resonances, 146–147 face-to-face transfer method, graphene, 251 failure, graphene, defects, 52 failure, probability, graphene, 55 Fano structure, graphene, 126 far-infrared absorption, graphene, 180 far-infrared spectroscopy, graphene, 42 Faraday rotation, graphene, 44 femtosecond optical pulses, 189 Fermat’s principle, 149 Fermi energy, 12, 15, 39–43, 48, 105, 109, 113, 118, 120, 127, 131, 135, 141, 229 Fermi velocity, 21, 26, 34, 40, 41, 43, 105, 108, 113, 121, 180 Fermi’s golden rule, 26 ferromagnetic electrodes, 197 ferromagnetic metallic contacts, 213 few-layer graphene, 46, 47, 75–76 few-layer graphene growth, 25, 46, 92, 245 few-layer graphene, spectra, 46 field effect devices, 25, 30–31 field effect mobilities, 25, 27 field-effect transistor (FET), 40, 183 filling factor, Landau level, 43 fine structure constant, 21, 39, 128, 180 finite exciton valley Zeeman shift, 286 finite velocity light cones, 321 first-principles pseudopotential calculations, 474 flexible electronics, 169, 174, 233 flexible metal–chalcogenide ligand bonding, 364 flexible optoelectronics, 333 flexible RF electronics, 164

491

flexible thin-film solar cells, 334 flexural phonons, graphene, 36 fluorinated graphene, 33 Fourier spectrum, 34 Fourier transform infrared spectrometer (FTIR), 45, 120 fracture, graphene, 53, 54 Frank van der Merwe mode, 345 Frank’s rule, 372 Franz–Keldysh effect, 187 free carrier absorption, graphene, 42–43 free carrier density, 181 free carriers, graphene, 41 free carriers, intraband transitions, 180 free standing germanene, 467 freely suspended single-layer MoS2, 272 freestanding single layer structure, 473 frequency mixer, graphene, 164 frequency multiplier, graphene, 164 frequency-dependent sheet conductivity, graphene, 41 Fresnel reflection coefficient, 105, 126 Friedel virtual bound state, 207 Frohlich coupling, 123 Frohlich interaction, 299 Frohlich oscillator strength, 123 full-band quantum transport simulations, 298 fullerenes, 8, 22, 101 funnel effect, 273 G band, 74 G band phonon, graphene, 71 G band, graphene, 73–74, 77, 79–80, 82–83, 90 G peak, graphene, 64, 90, 228 GaAs metamorphic HEMT, 163 gain medium, 191 GaN/AlN quantum well, 478 gap localized defect states, 303 gap opening, 16 gapless semiconductor, graphene, 12 gated interband transitions, graphene, 39 gate-induced absorption spectra, 171 gate-tunable interband transitions, graphene, 40 gate-tunable intraband absorption, graphene, 41 gating suspended graphene, 146 Gaussian distribution, 209 germanene, 8, 74, 458 germanene band gap, 463–464 germanene charge carrier mobilities, 467 germanene Dirac cones, 468 germanene electronic structure, 462 germanene hydrogenated indirect band gap, 464 germanene interfacial coupling, 468 germanene massless Dirac fermion, 468, 474 germanene non-trivial topological properties, 467 germanene phonon dispersion curves, 474

492

Index

germanene quantum spin Hall effect, 468 germanene spin–orbit coupling, 467 germanene stable structure, 474 germanene topological insulators, 463, 474 giant magnetoresistance, 197 GL films, graphene, 97 GNR band gap, 167 grain boundaries, 65, 67, 92, 130 grain boundaries topological defects, 363 grain boundaries, graphene, 65, 241 grain boundary electrical transport, 368 grain boundary in electronic transport, 368 grain boundary in metallicity, 368 grain boundary magnetism, 368 grain boundary S vacancies migration, 368 grain boundary spintronics applications, 368 grain boundary topological defects, 368 graphene, 7, 11, 14, 16, 18, 20, 30, 32, 35, 43, 45, 50, 57, 63, 66, 90, 93–94, 100, 105–106, 108, 113, 123, 127, 129–130, 132–134, 159–160, 163, 171, 192, 210, 219–220, 228 graphene, optoelectronic devices, 180 graphene absorbance, 38 graphene absorption spectrum, 180 graphene anti-dots, plasmons, 119 graphene as a bridge between condensed matter and high energy physics, 20 graphene band structure, 12 graphene barristors, 172 graphene carrier density, 27–35, 122 graphene chemical bonding, 7 graphene corrugation, 33 graphene crystal growth, 241 graphene crystals growth on liquid Cu, 246 graphene crystals growth on liquid Ga, 249 graphene crystals shape controlling, 245 graphene crystals size controlling, 241 graphene dielectric interaction, 163 graphene disc plasmons, 119–120 graphene dots array, plasmons, 119 graphene edge dislocation, 364 graphene electronics, 159 graphene fillers, 99–100 graphene flake morphology, 249 graphene in-plane deformation, 405 graphene laminate, 97–98 graphene layers, 73 graphene magnetic impurities, 33 graphene magnetism, 368 graphene mechanical properties, 52 graphene microdisks, plasmons, 46 graphene micro-ribbons, graphene, 46 graphene nano discs, plasmons, 122 graphene nanomesh (GNM) device, 168 graphene nanomesh transistors, 168 graphene nanomesh, 169 graphene nanoribbon (GNR) transistors, 176

graphene nanoribbon array photodetector, 184 graphene nanoribbon array, plasmons, 119 graphene nanoribbons, 159, 166, 181 graphene nanoribbons, plasmons, 120 Graphene nanostructures, 166 graphene OLEDs, 194 graphene on Si/SiO2 substrate, 92 graphene–P3HT–Au organic VTFT, 173 graphene pentagon–heptagon pair, 364 graphene phonons, 34 graphene photoconductive photodetector, 186 graphene photodetector, 184 graphene plasmon excitation, 119 graphene plasmonics, 104 graphene plasmons, 45, 104, 118, 126, 129, 131, 134 graphene plasmons dispersion, 118 graphene plasmons, experiments, 117 graphene plasmons, square root dispersion, 108 graphene p–n junctions, 144 graphene p–n junctions, photoresponse, 184 graphene polarizer, 192–193 graphene RF circuits, 164–165 graphene RF transistors, 160 graphene ribbons, graphene, 45, 124 graphene ribbons, plasmons, 45 graphene Riemann surfaces, 370 graphene saturable absorber, 191 graphene spin valves, 204 graphene spintronics, 197–198 graphene StoneWales (SW) rotation, 367 graphene sublattices, 113 graphene terahertz detector, 186 graphene transistor, 161 graphene transistor with lead sulfide (PbS) quantum dots, 339 graphene zero bandgap, 413 graphene, atomic structure, 72 graphene, elastic constants, 61 graphene, many layers, 74 graphene, nano-structured, 83 graphene/MgO/Co spin-valve, 204 graphene/MoS2 extremely high photo gain, 348 graphene/MoS2 ultrasensitive detection of DNA hybridization, 348 graphene–BN heterostructures, 219 graphene–BP–graphene vertical heterostructure, 447 graphene–light analogs, 148 graphene–MoS2 hybrids, 348 graphene–MoS2–metal FETs, 173 graphene–MoS2–metal heterostructure, 172 graphene–Poly(3-hexylthiophene-2,5-diyl) heterostructure, 174 graphene–SiC substrate, 201 graphene–TMD–graphene junction, 339 graphene–TMD–graphene photodetector, 339 graphite, 7–9, 17, 30, 35, 55–56, 61, 75, 77, 90, 92 grating coupling, plasmons, 45

Index

grating patterning, 191 Green–Lagrange strain, mechanical properties, 58, 60 group III–V, 472, 478 group II–VI, 472, 478 group IV, 472 group IV honeycomb monolayer materials, 458 group V, 472 group velocity, 149 growth, bilayer graphene, 241 gyromagnetic ratio, 202 half-integer quantum Hall effect, 43 Hall resistance, graphene, 229–230 Hall resistivity, 213 Hamiltonian, 13, 15–19, 26, 114, 142–143, 199 Hanle spin precession, 202 hard disk drives, 197 harmonic approximation, 10 harmonic oscillator, phonon, 123 harmonic oscillator, phonons, 77, 135 Hartree potential, 112 h-BN, 9, 30, 111, 117, 128, 130, 203, 210–211, 221, 230, 230, 232 h-BN, 219 h-BN, 221 h-BN/graphene/h-BN, 222 h-BN/graphene/h-BN devices, 220 h-BN/graphene/h-BN heterostructure, 221 h-BN alpha-B phase, 362 h-BN asymmetric grain boundary, 364 h-BN asymmetry doping, 362 h-BN B vacancies defects, 362 h-BN B vacancy formation energy, 362 h-BN B-centered point defects, 362 h-BN birefringent dielectric, 110 h-BN disorder-free gate dielectric, 448 h-BN encapsulated graphene, 200, 227 h-BN encapsulation, 110, 227 h-BN gate dielectric, 150 h-BN heteroelemental composition, 364 h-BN migration mechanisms, 370 h-BN optical phonon scattering, 117 h-BN strain energy, 364 h-BN substrate, graphene, 25, 146–147 h-BN surface optical phonons, 219 h-BN symmetric grain boundary, 364 h-BN tunneling barrier, 149 h-BN zigzag-type edges, 362 h-BN–few-layer BP–hBN heterostructure, 420 H—C bond, graphene, 32 heat conduction, graphene, 90, 92–93, 96–99, 101 heat dissipation, 192 heating or sputtering methods, 352 heteroepitaxial growth, graphene, 243–244 Heyd–Scuseria–Ernzerhof (HSE06), 382 HfO2 top gate dielectric, 295 high-performance graphene, 225

493

high-electron-mobility transistor (HEMT), 187 Hofstadter butterfly spectrum, 230 honeycomb lattice, 11, 113 Hooke’s Law, mechanical properties, 58 hopping parameter, 17, 20 hot band luminescence, graphene, 181 hot carrier dynamics, graphene, 190 hot carriers, 183 hot electron–hole plasma, 191 hot electrons, 204 hot-carrier electroluminescence, 330 HREELS, 118 HSE06 hybrid functional, 474 hybrid photodetectors, 339 hybrid phototransistor, 339 hybridization, light-plasmon, 109 hybridization, plasmons, 123, 127 hybridized sp states, graphene, 7 hydrogen impurities, 33, 207 hydrogenated graphene, 33, 214 hydrogenation, 33 hydrogen-silsesquioxane (HSQ) resist, 223 hyperbolic bands, graphene, 46 hyperbolic materials, 111 hyperbolic polaritons, 129 hyperfine coupling, 198 h-ZnO structure, 478 hα-silica, 477 hβ-silica, 477 IaS/IS intensity ratio, 78 ice layers, 30 ID/IG ratio, 82 ideal graphene, 15 IdsVds curves, BP, 448 III–IV semiconductors, 225 III–V coupled quantum wells, 317 iLO phonon, graphene, 80 imaging graphene plasmons, 46 impact excitation, 330 impedance, 106 impedance function, 116 impurities, 25, 28, 30–31, 33, 36, 48, 204, 207, 219 impurity scattering, 25–36, 403 in situ direct growth, 361 in-situ experiments, 30, 32, 35 indefinite materials, 111 indentation, graphene, 53, 66 indenter tip, 53–56, 61–62, 67–68 indium tin oxide (ITO), 193 induced bandgap, 43 induced transparency phenomenon, graphene, 126 inductance, 105, 109, 161 inductive sheet conductivity, plasmons, 108 inelastic scattering, phonons, 26, 77 infrared, 39, 43, 124, 127, 135, 184, 187 infrared absorption, 46

494

Index

infrared absorption spectroscopy, graphene, 47, 72 infrared absorption, graphene, 72, 135 infrared conductivity, 47 infrared light sources, 191 infrared nanoscopy, 46 infrared photons, 39 infrared spectra, 47, 135 infrared spectroscopy, graphene, 43 infrared transmission spectra, graphene, 43 inhomogeneous strain, 270 InP high-electron-mobility transistor (HEMT), 163 in-plane longitudinal optical (iLO) phonons, 71 in-plane stress, graphene, 61–62 in-plane transversal phonons, 71 intrinsic lifetime, plasmons, 114 interband, 39–40, 42–43, 127, 132, 181 interband excitations, 180 interband optical transition matrix, 283 interband transitions, 116 interband transitions control, graphene, 187–188 interband transitions, graphene, 39–40, 42–43, 180, 180, 187, 189 interband, graphene, 39 intercellular orbital magnetic moment, 282 interference fringes, plasmons, 128–129, 132 interlayer exciton Berry curvature effects, 325 interlayer exciton light cones, 321 interlayer exciton long-range drift-diffusion, 324 interlayer exciton photoluminescence, 316 interlayer exciton recombination pathways, 322 interlayer exciton valley Hall effects, 324–325 interlayer vibrations, graphene, 75, 77 internal electric field, 181 intervalley resonance, 73 intervalley scattering, 15, 18, 21, 33, 80, 143, 212, 242 intraband absorption, graphene, 42–43 intraband graphene transitions, graphene, 39 intraband transitions, graphene, 39, 116, 189 intralayer exciton, 322 intralayer exciton transition dipoles, 323 intravalley electron–hole exchange interaction, 325 intravalley resonance, 73 intravalley, Raman scattering, 80 intrinsic bandwidth, graphene photodetector, 187 intrinsic cut-off frequency, 164 intrinsic doping, graphene, 131 intrinsic point defects, 306 intrinsic rippling, graphene, 34 intrinsic strength, graphene, 52–53, 56, 57, 61, 63, 67–68 inverse damping ratio, plasmons, 115, 117 inversion symmetry breaking, bilayer graphene, 48, 170 inversion symmetry, graphene, 32, 48 ion-bombarded graphene, 80 ion-implantation, 361 irradiation induced heating, 183

isotopes, thermal conductivity, 93 iTO phonon, graphene, 80 joint density of states, graphene, 40 K, Kʹ points, 12 Kerr rotation measurements, 287, 291 Kerr-lensing, 189 kinematic momentum, 321 kinetic inductance, graphene, 104–106, 108 Kirchhoff’s law, 192 Klein paradox, graphene, 31, 146 Klein tunneling, graphene, 17, 21, 143, 146, 207 Kondo effect, graphene, 33 K-point states, graphene, 48 Kramers degeneracy, 199 Kramers–Kronig relations, 44 Kubo formalism, 42 LA phonon, 34 Lagrangian strain, 58 Landau bands, 230 Landau damping, plasmons, 114, 116, 120 Landau level (LL) structures, graphene, 43 Landau level energies, graphene, 43 Landau level transitions, graphene, 39, 43–44 Landau levels, 147 Landau quantization, 18 LanTraP simulation tool, 400 large-scale growth heterostructures, 344 large-scale growth single crystalline monolayers, 344 large-area single crystal growth, graphene, 242 large-scale growth graphene on solid Cu, 245 Larmor frequency Ω, 202 lateral metal–TMD–metal detector, 337 lateral WSe2–MoS2 monolayer heterojunction controlled growth, 350 lattice scattering, 116 lattice unharmonicity, 92 layer-by-layer growth mode, 233, 345 layered dependent spin splitting, 291 layered materials, 173–174 layered van der Waals materials, 232 layer-plus-island growth, 350 lead sulfide PbS quantum dots, 339 lifetime quality factor, plasmons, 115–116 lifetime, phonons, 77–78, 125, 185 lifetime, plasmons, 115–116 light absorption, 45, 92 light electric field, plasmon excitation, 120 light modulators, 187 light switching, 181 light to current conversion, 181 light-coupling of interlayer exciton, 319 light-matter interaction, graphene, 38, 122 line defects, graphene, 65 linear conductivity vs. carrier density, 32

Index

linear dispersion, graphene, 10, 39, 40, 43, 108, 111, 198 liquid phase exfoliation, graphene, 98, 239 LL transitions selection rule, 44 local density of optical states (LDOS), graphene, 128, 130 local electron–phonon coupling, 269 local exciton density, 318 local hot spots, 192 localized edge modes, graphene, 121 localized plasmon, 45, 119 localized plasmon modes, graphene ribbons, 129 localized plasmon resonance, graphene, 45, 192 localized plasmon, graphene ribbons, 122 localized plasmon, plasmons, 45 localized plasmons, 119, 181 localized plasmons, graphene, 45, 181 localized plasmons, graphene nanoribbons, 181 localized Wannier functions, 384 localized plasmon resonances, graphene, 45 logic electronics, 170 logic OR gate, 168 longitudinal acoustic (LA) phonons, graphene, 34, 36, 92–93 longitudinal resistance, 229 long-range coupling, 123 long-range defects, 81 long-range scatterers, 28–31, 117 long-range, defects, 29, 72, 81 Lorentz force, 227 Lorentzian profile, 121, 208 low-buckled structures, 463 Löwdin partition method, 384 lumped element, 106 magnetic field, 25–27, 33, 43, 47, 132–133, 157, 202, 204, 210, 213, 214, 230 magnetic field sensors, 197 magnetic field, graphene, 43, 142 magnetic field, transverse, 227 magnetic proximity effects, spin, 213 magnetoelectric effects, 291 magnetoelectronic, 282 magnetoluminescence, 282 magnetoplasmons, graphene, 132 magnetoplasmons, graphene, 132 magnetoresistance, 197, 201 magnetoresistive random-access memory (MRAM), 197–198 magneto-transport, 33 massive Dirac fermion model, 287 massless Dirac electrons, 43 massless Dirac fermions, 20, 198 massless electrons, graphene, 42 maximum oscillation frequency fMAX, 162 maximum strength, graphene, 67 mean free path, graphene, 224, 226–227

495

mechanical, 69 mechanical assembly, 233 mechanical assembly of graphene–BN heterostructures, 220 mechanical assembly, 2D materials, 232 mechanical exfoliation, graphite, 55, 68 mechanical instabilities, 63 mechanical properties, graphene, 55, 65, 68 Mermin–Wagner theorem, 10 metal catalysts, graphene growth, 240 metal dichalcogenides (TMD), 90 metal organic chemical vapor deposition, 351 metal particles, plasmons, 121 metal thin film plasmons, 109 metal–graphene junctions, 181 metal–graphene microcavity, 192 metal–graphene–metal photodetector, 184 metallic bonding, 7 metallic ferromagnet, 201 metallic octahedral 1T phase, 374 mexican hat dispersion, 19 mica, 30 mica, substrate, 9 microelectromechanical systems (MEMS), 197 micromanipulator, 221 micro-Raman thermal conductivity experiments, 424 mid-gap states, 31 mid-gap states, graphene, 36 mid-infrared resonances, plasmon, 120 minimum conductivity, 27, 29–32 mismatch angle, bilayer, 74 mobility, 21, 27–28, 30–33, 35, 163, 169, 226–227 mobility, GNRs, 166 mode volume, plasmons, 122, 130 mode-locked laser, 190 mode-locking, 191 modulation depth, 188 moiré pattern, 230 moiré superlattice, 228, 231 moiré-coupling induced bandgap, 234 moiré-patterned graphene, 230–231 molecular dynamics (MD), 92 molecular dynamics (MD) calculations, 473 molecular orbitals, 7 molybdenum hexacarbonyl (Mo(CO)6), 351 momentum conservation, 41, 72, 80–81, 191 momentum relaxation lifetime, graphene, 43 momentum scattering, 211 monolayer, 75, 134, 231 monolayer bandgap, 234 monolayer graphene, 46, 73 monolayer graphene growth, 240 monolayer graphene/MoS2 power conversion efficiencies, 334

496

Index

monolayer WS2/MoS2 power conversion efficiencies, 38, 334 MoS2, 2, 68, 172, 232, 262, 481 MoS2 2DEG, 300 MoS2 anti-sites, 360 MoS2 atomically thin semiconductors, 273 MoS2 chemical vapor transport methods, 361 MoS2 circularly polarized light, 341 MoS2 configuration entropy, 360 MoS2 Coulomb-impurity scattering, 303 MoS2 dark current, 339 MoS2 disorder, 306 MoS2 double S vacancy, 360 MoS2 electron mobility, 359 MoS2 electron transport, 295 MoS2 electron–electron interaction, 263 MoS2 electron–phonon interaction, 263, 299 MoS2 ferromagnetic fluctuations, 263 MoS2 field-effect transistor, 295, 334 MoS2 flexible electronics, 375 MoS2 Hall voltage, 341 MoS2 helicoid geometry, 370 MoS2 homopolar phonon mode quenching, 295 MoS2 hydrodesulfurization (HDS), 373 MoS2 hydrogen evolution reaction (HER), 373 MoS2 interstitials, 360 MoS2 intervalley scattering, 300 MoS2 intrinsic strength, 273 MoS2 magnetization density, 368 MoS2 massive Dirac Hamiltonian, 264 MoS2 mechanical properties, 68 MoS2 nanoscale transistor, 374 MoS2 nucleation energy barrier, 346 MoS2 photoactive material, 339 MoS2 photovoltaic effect, 334 MoS2 piezoelectric interaction, 299 MoS2 quantum dot, 374 MoS2 quantum lead, 374 MoS2 S coverage, 372 MoS2 S vacancy, 372 MoS2 Schottky diode, 374 MoS2 screw dislocations, 370 MoS2 sheet conductivity, 296 MoS2 short-range repulsion, 263 MoS2 single S vacancy, 360 MoS2 single vacancy formation energy, 360 MoS2 spintronics devices, 361 MoS2 superconductivity, 263 MoS2 synthesis, 481 MoS2 tellurium-assisted low-temperature synthesis, 347 MoS2 triangular crystals, 346 MoS2 triangular monolayer flakes, 346 MoS2 trigonal warping, 264 MoS2 vacancies, 360 MoS2 valleytronic devices, 361 MoS2 vapor phase growth, 346

MoS2 α boundary, 374 MoS2 β boundary, 374 MoS2 γ boundary, 374 mosaic nanocrystalline, graphene, 244 MoSe2, 262, 481 MoSe2 midgap-state-induced metallicity, 368 MoSe2 molecular beam epitaxy growth, 367 MoSe2 optical micrograph, 313 MoSe2 strain energy, 367 MoSe2 vacancy lines, 367 MoSe2/WSe2 heterobilayer, 314 MoX2/WX2 AA-type stacking, 310 MoX2/WX2 full band structure, 310 MoX2/WX2 hybridization, 311 MoX2/WX2 idealized lattice matching, 310 MoX2/WX2 interlayer hole transfer, 315 MoX2/WX2 type II band alignment, 314 multilayer antimonene, 481 multilayer graphene, 39, 46, 100 multilayer graphene/insulator stacks, 122 multilayer silicene, 476 multiple nuclei method approach, graphene, 243 MX2, 472 MX2, asymmetric Z-grain boundary, 365 MX2, band alignment, 310 MX2, dislocation diversity, 364 MX2, dislocations, 365 MX2, point defects, 365 MX2, small tilt grain boundary, 365 MX2, symmetric A-grain boundary, 365 MX2/WX2, tunable photocurrent generation, 315 nanoindentation, 53, 55, 56, 61–62, 67 nanoribbon arrays, graphene, 120, 134 nanoribbon extinction spectra, 124 nanoribbon plasmon, damping, 121 nanoribbons, 20, 124, 134–135, 181, 192 nanosolenoids, 370 nanotube devices, 35 nanotubes, 8 NbSe2, 233 NbSe2 superconductivity, 233 nearest-neighbor hopping parameter, 13 near-field microscopy (s-SNOM), graphene, 127 near-infrared photons, 448 negative differential resistance (NDR), 298 negative GB/interface energy, 367 negative index, 150 negative index materials, 149 negative index refraction, 149 negative magnetoresistance, 33 negative permittivity, 109 negative refractive index materials, 149 nitrogene, 480 nitrogenite, 480 noise-equivalent-power, graphene photodetector, 187

Index

497

nonequilibrium Green’s function (NEGF) formalism, 298 nonlinear and anisotropic response of graphene, 57 nonlinear valley current, 288 nonlinear valley spin current, 288 non-local resistance, graphene, 150, 153 non-relativistic electrons, 31 non-symorphic space group, 75 non-zero bandgap, bilayer graphene, 48 Nosè thermostat, 473 nucleation density, graphene, 241

orthorhombic lattice, graphene, 74 orthotropic model, 405–406 oscillator, 164 oscillator strength, 35, 42, 123 oscillator strength, graphene, 43, 122, 131, 180 oscillator strength, graphene plasmons, 45 oscillator strength, phonon, 123 out-of-plane (ZA) phonons, graphene, 92 out-of-plane acoustic modes, graphene, 10 oxidation, graphene, 64 oxygen plasma, 64

on/off ratio, 167–168, 170–171, 173 one-dimensional edge-contact, 222 one-dimensional plasmons, graphene, 121 on/off ratio, transistor, 159 OPG carrier guiding, 153 opposite valley configuration, 324 optical (O) phonon, 33, 71 optical absorption control, graphene, 187 optical absorption modulation, graphene, 180 optical absorption of graphene, effect of doping, 41 optical absorption spectroscopy, 118 optical absorption, graphene, 38–39, 40, 187, 190 optical cavity, 184 optical communication networks, 184 optical communications, 184 optical conductivity, 189 optical conductivity, few-layer graphene, 46 optical conductivity, graphene, 39, 104, 120, 180 optical coupling, interlayer exciton, 323 optical desert, 191 optical fiber waveguides, 152 optical phonon, graphene, 35, 71, 77–79, 96, 124, 126 optical phonons, plasmon decay, 121 optical phonons, SiO2, 34 optical properties control, graphene, 181 optical properties of interlayer exciton, 320 optical properties THz, graphene, 189 optical properties, graphene, 38, 77, 180 optical selection rules, 283 optical sheet conductivity, graphene, 42 optical spectroscopy, graphene, 38 optical telecommunication, 185 optical transitions, graphene, 48 optical transmission/absorption, graphene, 21 optical transmission/reflection spectroscopy, graphene, 38 optical waveguide, 188 optoelectronics, 38, 181 optoelectronics, bilayer graphene, 170 optothermal Raman spectroscopy, 90–91, 93–94 organic field-effect transistors, 338 oriented graphene fillers, 99

parabolic band structure, bilayer graphene, 170 parabolic dispersion, 20, 288 parabolic dispersion, graphene bilayer, 18 parasitic capacitance, 162–163 particle–hole symmetry, 286 passive mode-locking, 189 passive optical elements, graphene, 192 patterned graphene, plasmons, 119, 121 Pauli blocking, 42–43, 114, 191 Pauli blocking, graphene, 40–41, 187 Pauli blocking, interband excitations, 181 Pauli matrices, 15, 198, 199, 271, 282 Pauli matrices, graphene, 15 PbS quantum dots, 185, 339 PbS–MoS2 detector, 339 PbS–MoS2 interface, 339 p–d hybridized ionic M—X bonds, 481 PDMS, 221 peak responsivity, 183 PEDOT:PSS films, 193 Perdew–Burke–Ernzerhof (PBE), 382 perfect transmission, graphene, 144 periodic table, 472 permittivity, 106 permittivity dispersion, 110 phase 1T–2H interfaces, 374 phase velocity, 107–109, 111, 116, 149 phonon, 34–36, 63, 75, 77, 79, 81, 83, 92, 94, 122–123, 127 phonon anharmonicities, phonons, 77 phonon branches, graphene, 71, 75, 93–94 phonon coherence, graphene, 79 phonon density of states, graphene, 95 phonon dispersion, graphene, 71–73, 92, 96 phonon engineering, graphene, 96 phonon group velocity, 96 phonon mean free path, graphene, 90, 141 phonon mode frequencies, 473 phonon mode quenching, 301 phonon modes, out of plane, graphene, 96 phonon modes, thermal properties, 81, 92 phonon polaritons, 191 phonon scattering, graphene, 34–36, 72–73, 75, 77, 79–80, 93 phonon spectrum, 10

498

Index

phonon temperature, phonons, 77, 79 phonon thermal conductivity, 96 phonon wavevector, 81 phonon resonance, 129 phonons, 25–26, 28, 35, 38, 71–73, 77–78, 83, 90, 94, 96, 108, 110, 112, 117–118, 121–122, 124, 180, 204 phosphorene, 233, 413 phosphorene few-layer transistors, 459 phosphorene optical properties, 414 photoconductive effect, 338 photoconductive gain, 183, 185, 339 photocurrent, 182 photocurrent generation mechanisms in graphene, 182 photocurrent sign, 182 photodetector responsivity, graphene, 184 photodetectors, 135, 184, 185 photodetectors, graphene, 184 photoelectric (PE) effect, 181 photo-gate, 183 photogating effect, 338 photogenerated carrier relaxation, 336 photoluminescence excitation spectroscopy, 317 photonic applications, graphene, 181 photoresponse time, 183 photo-thermoelectric effect, 181, 338 photovoltage, 181–182 photovoltaic effect, 338 photovoltaic response, 314 physical vapor deposition method, 351 physical vapor phase transport, 344–345 pinholes, 204 Piola–Kirchhoff stress, mechanical properties, 58, 60–61 pi-orbital, graphene, 38 pi-plasmon, 118 Planck’s law, 191 plasma frequency, 42, 181 plasma-bombardment-induced defects, 362 plasmaron, graphene, 118 plasmon absorption, 45 plasmon damping, graphene, 115–116 plasmon decay, graphene, 130 plasmon dispersion, 107, 109, 114, 118 plasmon dispersion, graphene, 106–107, 119–120 plasmon enhancement, 184 plasmon excitation, graphene, 39, 44, 46, 119 plasmon frequency, graphene, 109 plasmon lifetime, 117 plasmon lifetime, graphene, 117, 121–122, 134 plasmon oscillator strength, graphene, 121 plasmon resonance, 105, 119, 131 plasmon resonance frequency, 122, 181 plasmon resonance, graphene, 45, 80, 105, 126–127, 134, 181 plasmon tunability, graphene, 130, 132

plasmon wavelength, graphene, 109 plasmon wavevector, 104, 389 plasmon, graphene, 46, 104, 127 plasmon–optical phonon coupling, 119 plasmon–phonon coupling, graphene, 122 plasmon–phonon hybridization, graphene, 108, 119, 123, 127 plasmon–phonon mode energies, 123 plasmon–phonon modes, graphene, 124 plasmon–phonon polaritons, graphene, 117 plasmons, 38, 44, 105, 124, 126–133, 137, 180, 191 plasmons, 2D systems, 108 plasmons, graphene nanoribbons, 121, 124 plasmons, graphene ribbons, 119 plasmons, patterned graphene, 119 p–n junction, graphene, 141–142, 182 PNG carrier guiding, 153 p–n–p junctions, 21 point defect scattering, graphene, 25–36, 93 point defects, 25–36, 53, 303 point defects repair, 307 point defects, graphene, 56, 64, 67, 82 Poisson’s ratio, 60, 474 Poisson’s ratio, graphene, 55 Poisson’s ratio, mechanical properties, 55, 59 polar optical phonons, 30, 35, 36 polar optical phonons, SiO2, phonons, 34 polar phonon scattering, 34 polar substrate phonon scattering, 30, 35, 36, 121 polar substrates, 123 polarization vector P, 81 polycrystalline graphene, graphene, 79 polymer stamp, 314 polymer-stamp-based transfer techniques, 311 polymer transfer technique, graphene, 220 polymethylmethacrylate, 30 potassium, 48 potassium, adsorption, 30 pristine graphene, 35 probing phonons, 79 probing plasmons, EELS, 118 propagating plasmons, graphene, 45, 90, 111, 115, 121, 128 propagating plasmons, thermal excitation, 191 propagation factor, plasmons, 117 propagation length, plasmons, 115–116 propagation losses, plasmons, 128 propagation quality factor, plasmons, 115 proteins, vibrational enahncement, 134 pseudospin, 16, 142–144, 157, 198, 199, 206–207, 207, 210, 212 pseudospin conservation, 143 PTE effect, 184 Purcell effect, 192, 333 Purcell enhancement, plasmons, 122 Purcell-factor, 122

Index

quality factor, plasmons, 117, 122 quantum confinement, 79, 167–168 quantum Goos–Hänchen, graphene, 149 quantum Hall effect, graphene, 146, 148, 238 quantum hall regime, 133 quantum interference, 215 quantum memory, phonons, 78 quantum relativistic effects, 12, 21 quantum resistance, 224 quantum spin Hall effect, 214, 463 quantum transport models, 298 quantum-confined Stark effect, 187 quasi-particle GWo corrections, 474 quasi-degenerate perturbation theory, 384 quasi-electrostatic limit, 107–110 quasiparticles, 118 radiative modes, 191 radio frequency (RF) transistors, 160 Raman optothermal technique, 97 Raman optothermal, graphene, 90 Raman scattering, graphene, 73, 77, 82 Raman spectra, graphene on h-BN, 229 Raman spectroscopy, 35, 71, 74–75, 79, 82 Raman spectroscopy, graphene, 38, 64 random phase approximation (RPA), 29, 111, 113, 389 Rashba field, 212 Rashba SOC Hamiltonian, 199 Rashba SOC magnetic field, 199 Rashba spin–orbit coupling, 203, 207, 209 Rashba SOC field, 208 reactance, 106 real-space imaging, plasmons, 127 reciprocal lattice vector, 12, 14 red phosphorus, 414 reflection phase, plasmon, 120 relativistic atomic collapse, 21 relativistic Dirac equation, 142 relativistic spin–orbit coupling, 213 relaxation time, 27, 183, 211 relaxation time approximation, 288 relaxation time, phonons, 96 relaxation time, plasmons, 104, 109, 134 resistance, 106 resistance, negative, 226 resistance, graphene, 105, 146, 161, 187, 230 resistivity, 26, 29, 33, 34, 224 resistivity, graphene, 34–36 resonance, 32, 43, 45, 78, 127, 129, 132, 181, 193 resonance scatterers, graphene, 25, 32 resonant cavity, 183 resonant scatterers, 31–32, 206 reststrahlen frequency, 108 retardation effects, 108 RF frequency doubler, 165 rhombohedral graphite, 7

499

ribbon array, bilayer graphene, 45, 126 ribbon arrays, graphene, 45 ribbon arrays, optical response, graphene, 45 ripples, graphene, 11, 34 ripples, 11, 22, 25, 33, 212 ripplocations, 372 room temperature photoluminescence spectrum, 313 Ross Kozarsky, 459 RPA, 29, 31 RPA conductivity, 107, 114 rupture, graphene, 52, 54–56, 63 Rydberg series of excited excitonic states, 394 saddle-point singularity, 180 Salisbury screen, graphene, 122 saturable absorber, 189, 191 saturable absorber mirrors, SESAMs, 189 saturable absorption, 189–190 saturation velocity, 159 scale-invariant quantum conductivity, 215 scaling law, plasmons, 45 scanning photocurrent microscopy (SPCM), 443 scattering, 18, 25–28, 30–31, 34–36, 34, 72, 78, 79, 81–82, 92, 96, 117, 121, 127–128, 135, 163, 205, 210, 219 scattering by resonant scatterers, graphene, 31 scattering by ripples, 34 scattering from ripples, 116 scattering from substrate, 34–36, 116 scattering losses, 116 scattering matrix elements, 26 scattering mechanisms, 25–36, 299 scattering, phonons, 34–36, 180 scattering, plasmon, 115 scattering, thermal vibrations, 116 Schottky contact, 172 Schottky diodes, 187 screened Coulomb potential, 28, 210, 396 screened electron–hole interaction potential, 395 screened potential, plasmons, 112–113 second-order effective Hamiltonian, 384 second-order G0 mode, graphene, 80–81 Seebeck coefficient, 181–182 seeing graphene monolayer, 38 selection rules, graphene, 44 self-aligned gate, 161 self-sustaining oscillation, plasmon, 106 semiconductor/dielectric interface, 306 semiconductors, 15, 21, 27, 42, 122, 141, 149, 160, 168–169, 187, 200, 201, 204 semi-transparent optoelectronics, 333 sensitized graphene photodetectors, 185 shear mode, bilayer graphene, 75–76 sheet current, 105 sheet current density, 104 sheet inductance, graphene, 105

500

Index

sheet resistance, graphene, 194 sheet resistivity, 225 sheet-plasmon, graphene, 118 short-range defect scattering, 121 short-range defects, graphene, 29, 31, 34, 81 short-range scatterers, graphene, 36 Shubnikov–de Haas oscillations, 156 Si sp2/sp3 hybridization, 459 SiC band gap, 476 SiC honeycomb structure, 476 SiC substrate, 123 SiC substrate, graphene, 47, 119, 129–130 SiC synthesis, 476 silica honeycomb structure, 477 silicane band gap, 464 silicatene, 476 silicatene auxetic, 477 silicatene metamaterials, 477 silicene, 8, 74, 458 silicene √3
√3, 475 silicene 2√3
2√3R(30 ) superstructure, 462 silicene ambipolar Dirac charge transport, 476 silicene armchair nanoribbons, 475 silicene band gap, 463 silicene Dirac cone, 467 silicene electronic structure, 462 silicene ferromagnetism, 467 silicene field effect transistors, 458, 469 silicene flower pattern, 460 silicene functionalization, 463 silicene GW approximation, 463 silicene honeycomb, 462 silicene hybridization, 463 silicene hydrogenated indirect bad gap, 464 silicene hydrogenation, 465 silicene linear dispersions, 466 silicene massless Dirac fermion, 466, 474 silicene molecular beam epitaxy (MBE), 459 silicene negative thermal expansion coefficient, 461 silicene phonon dispersion curves, 474 silicene rotational epitaxial phases, 461 silicene sheet resistance, 466 silicene stable structure, 474 silicene synthesis, 475 silicene topological insulator, 463, 474 silicene/metal interaction, 460 silicite, 476 silicon carbide, 476 silicone, epitaxial growth method, 238 single layer honeycomb structure, 473 single layer materials band gap, 474 single layer polymorphism, 474 single layer structure uniaxial strain, 474 single nucleus growth approach, graphene, 241 single-particle excitations, 118 SiO2, 30, 91, 189, 191, 203, 210–211, 219–221, 301 SiO2 dielectric, 40

SiO2 substrate, 34 SiO2 surface modes, 34–36, 124 SiO2, substrate, 9 sister silicene phases, 461 skipping tranjectories, 154 Slater–Koster parameters, 264, 268, 271–272 Slater–Koster scheme, 265 Slater–Koster tight-binding model, 264, 268 SLZnO structure, 478 snake state trajectories, 149, 153–156 Snell’s law, 135, 144, 148–149, 155 snowflake-like graphene crystals, 248 SO phonon frequency, 123 SO phonon modes, 124 SOC in graphene, 198 soft phonon mode instability, 63 soft phonon mode, mechanical properties, 68 soft phonon, graphene, 63 solid state metal oxide precursor vaporization, 351 sp2 bonding, graphene, 8 sp2–sp3 bonding mixture, 8 sp3 bonding, 8 space group, 72 spatial coherence, phonons, 79 spatially indirect effects, 317 spin accumulation, 202 spin-Bloch diffusion equation, 202 spin current, 213 spin diffusion length, 200 spin dynamics, graphene, 208–209 spin gating, 212 spin Hall currents, 291 spin Hall effect, 200, 212, 287 spin Hall effect, 213 spin injection, 200 spin lifetime, 198, 200, 203, 207, 210–211 spin lifetime, 2D materials, 200 spin lifetime, graphene, 200 spin optical selection rules, 291 spin–orbit, 205, 207, 212, 213 spin–orbit coupling, 7, 9, 16, 197–199, 208, 214 spin–orbit enhancement, 213 spin orientation manipulation, 202 spin polarizers, 202 spin precession, 202, 205 spin precession frequency, 208 spin precession time, 205, 207, 211 spin relaxation, 201, 211 spin relaxation length, 200, 212 spin relaxation mechanism, 204, 206 spin relaxation time, 198, 201–202, 205, 210 spin resistance, 201 spin scattering, 197 spin transfer torque effect, 197 spin–valley coupling, 329 spin–valley-polarized LED, 340

Index

spin–valley-polarized photovoltage/current generation, 336 spinors, 113, 144 spin-polarized electron injection, 197 spin-polarized electrons, 213 spin-polarized transport, graphene, 33 spin-torque, 213 spin-torque applications, 204 spin-value, 197, 201 spintronics, 9, 197, 200 stable single layer honeycomb structures, 474 stacking domain-walls solitons, 370 stacking orders, graphene, 46 stanene, 74, 458 stanene spin–orbit coupling, 468 stanene topological insulator phases, 468 stanene topological transistor, 468 stanine, epitaxial growth method, 238 Stark effect, 199, 318 Stark shift, 317 Stokes scattering, 77–79 Stokes scattering (S), Raman spectroscopy, 77–78 Stokes/anti-Stokes scattering ratio, 191 Stoner model, 368 strain, 20, 54, 57–60, 62–63, 68, 71, 92, 212 strain anisotropic, graphene, 74 strain-dependent Hamiltonian, 271, 272 strain included in tight-binding Hamiltonian, 272 strain tensor, 269 strain uniaxial, 59 strain, compressive, 72 strain, equibiaxial, 59 strain, graphene, 36, 58, 60–61, 60, 68, 74 strained TMD, 270 straintronic devices, 273 Stranski–Krastanov growth, 350 Stranski–Krastanov mode, 345 stress–strain, 405 stress–strain relationship, graphene, 58 structural defects, graphene, 25, 241 structural engineering, 38 structure stability, 473 sublattice, 13, 15, 18, 142 sublattice, graphene, 8, 11 sublattices A and B, 13 substrate phonons, 25, 192 substrate surface roughness, scattering, 219 substrate-induced disorder, 219 sulfur vacancies, 306 sulfurization of metal thin films, 344 sulfurization/selenization of transition metal oxides, 345–346 sum rule, graphene, 43 suppression of backscattering, graphene, 198 surface admittance, 106 surface corrugations, graphene, 36 surface optical phonons, 123, 219

501

surface phonon mode (SO), 123 surface plasmon polariton (SPP), 109 surface plasmons, dispersion, 109 surface transfer doping, graphene, 131 surface-optical (SO) phonon scattering, 301 suspended graphene, 10–11, 25, 36, 54, 66, 68, 90–91, 93–94, 96, 146, 153 symmetry arguments, 383 symmetry breaking, bilayer graphene, 171 symmetry points, graphene, 12 tangent modulus, mechanical properties, 58, 63 temperature gradient, 182 temperature-dependent resistivity, 27, 35 temperature-dependent resistivity, 35 terahertz (THz) radiation detection, graphene, 183 terahertz and infrared detection, graphene, 184 terahertz detectors, 187 terahertz modulator, graphene, 189 terahertz plasmon resonances, 120 terahertz range, graphene, 180 ternary van der Waals alloys, 348 thallium ad-atoms, graphene, 215 thallium islands, 94, 215 thermal (in)stability of 2D crystals, 9 thermal coatings, graphene, 97 thermal compression, 11 thermal conductivity, 77, 92–96, 99–100 thermal conductivity of plastics, 97 thermal conductivity, few layer graphene, 97 thermal conductivity, graphene, 90, 92–93, 98 thermal conductivity, suspended graphene, 91, 94 thermal contact resistance, graphene, 99 thermal emission, plasmon enhancement, 135, 192 thermal instability, graphene, 10 thermal interface materials (TIM), 96 thermal light emission, graphene, 192 thermal management, graphene, 96, 98–100 thermal pastes, 98 thermal properties, graphene, 90, 93 thermal radiation emitter, graphene, 228 thermal radiation sources, 191 thermal radiation, graphene on SiO2, 191 thermal resistance, graphene, 185, 200 thermoelectric effects, 204 thermolysis of thiosalts, 344 thin films, plasmons, 110 THz detector, 187 THz light sources, 191 THz plasmon resonance, graphene, 133 THz radiation, 183, 187 tight-binding approximation, 267 tight-binding Hamiltonian, 13, 17, 265 tight-binding Hamiltonian energy bands, 387 tight-binding model, 12, 384, 386 tight-binding model, graphene, 39 tilt grain boundary, 368

502

Index

TIM efficiency, graphene, 100 TIMs electronics, 98 TMD barriers for dislocations, 370 TMD berry curvatures, 279 TMD bilayered interlayer hopping properties, 280 TMD bilayered valley and spin physics, 289 TMD bottom-up synthesis, 345 TMD bottom-up synthesis, 344 TMD carrier density, 263 TMD catalysts for hydrogen evolution reaction, 374 TMD channel transistors, 418 TMD chemical vapor deposition, 279 TMD chemical vapor deposition grown device, 338 TMD circularly polarized electroluminescence, 340 TMD circularly polarized emission, 340 TMD circularly polarized luminescence, 340 TMD conduction band (CB), 262 TMD controllable synthesis, 345 TMD edge-epitaxy, 350 TMD electrical output current, 337 TMD electroluminescence, 330–331, 340 TMD electron–hole pairs, 340 TMD electronic properties, 259, 344 TMD electronic structure, 260 TMD electronic structure at the band edges, 280 TMD electrostatic doping, 335 TMD energy splitting, 262 TMD equilibrium concentration of defects, 360 TMD excitonic transitions, 340 TMD external deformations, 269 TMD extrinsic defects, 344 TMD field effect transistor photocurrent generation, 338 TMD field-effect transistor, 330, 374 TMD first principle calculations, 264 TMD grain boundary polar discontinuity, 368 TMD growth on graphene, 346 TMD growth on h-BN, 346 TMD heterostructure fabrication, 311 TMD heterostructures vertically stacked, 348 TMD hexagonal Brillouin zone, 279 TMD hybrid detector, 337 TMD hybridization, 261, 270 TMD hydrophilicity, 374 TMD interlayer exciton, 315, 317 TMD intralayer exciton, 317 TMD intrinsic p–n junction, 350 TMD inversion symmetry, 259 TMD island growth, 345 TMD isomorphism, 347 TMD kp model, 283 TMD large scale growth nanosheets, 344 TMD laser, 330, 333 TMD lateral heteroepitaxy growth, 345 TMD layer-plus-island growth, 345 TMD light emitter, 333

TMD light emitting diodes, 330 TMD liquid phase exfoliation, 344 TMD many-body interactions, 318 TMD mass production, 354 TMD mechanical deformations, 273 TMD mechanical exfoliation, 279 TMD mechanical properties, 272 TMD migration mechanisms, 370 TMD molecular beam epitaxy, 279 TMD monolayered valley-selective optical Stark effect, 285 TMD monolayered valley-spin physics, 283 TMD MOSFETs, 297 TMD nanoribbons, 282 TMD non-uniform strain, 269 TMD nucleation density, 346 TMD nucleation energy barrier, 345 TMD optical properties, 310 TMD oscillator strength, 318 TMD phonon Brillouin zone, 260, 264, 271 TMD photocurrent, 336 TMD photodetectors, 336 TMD photoelectric, 338 TMD photoluminescence, 331 TMD photon absorption, 339 TMD photo-thermoelectric effect, 338 TMD photovoltaic device, 334 TMD photovoltaic effect, 331, 334 TMD physical vapor deposition method, 353 TMD piezoelectric properties, 274 TMD p–n junctions, 331, 334 TMD point defects, 360 TMD power conversion efficiency, 336 TMD quantum dots, 282 TMD Rashba spin–orbit coupling, 263 TMD Schottky barrier, 338 TMD Schottky junctions, 334 TMD semiconducting, 269, 329 TMD short radiative lifetimes, 329 TMD spin relaxation, 283 TMD spin–orbit coupling, 262, 281 TMD spin–orbit interaction, 259 TMD spintronics, 259 TMD spin–valley coupling, 259 TMD strain engineering, 259, 272–273 TMD stretchability, 273 TMD strong light–matter interactions, 339 TMD supercapacitor electrodes, 374 TMD synthesis, 344 TMD synthesized on fluorophlogopite mica (KMg3AlSi3O10F2), 350 TMD ternary alloy synthesis, 344, 347–348 TMD top-down synthesis, 344 TMD twin grain boundary metallicity, 368 TMD two-step growth method, 350 TMD ultrafast charge transfer, 335 TMD unit cell, 260

Index

TMD valence band (VB), 262 TMD valley Hall effect, 341 TMD valley polarization detection, 340 TMD valley pseudospin, 280 TMD valleytronics nano-devices, 259 TMD van der Waals epitaxy, 350 TMD van der Waals force, 279 TMD van der Waals heterostructure device, 337 TMD van der Waals stacking, 344 TMD vertical p–n junctions, 335–336 TMDs vertically stacked, 350 top-gating graphene, 145 topological defects, 363 topological insulators, 20, 214, 232 topological insulators, 232 topological state, 214 touch screens, graphene, 194 transconductance, 162 transfer characteristics, 166 transfer characteristics, bilayer FET, 171 transfer technique fabrication, 313 transit time, 183 transition metal dichalcogenides (TMD), 472 transition metal oxides, 233, 472 transmission of carriers, p–n interface, 143 transmission probability, angle dependence, graphene, 145 transparency dip, nanoribbons, 134 transparent conductive electrodes, 193 transport bandgap, graphene, 48 transport gap, bilayer, 170 transport in magnetic field, 146 transversal (T) phonons, 71 transverse acoustic (TA) phonons, graphene, 92–93 transverse magnetic field, 105 trap state density, 446 trapping lifetime, 183 trigonal warping, 180, 288, 340 trigonal warping, graphene, 17 trilayer graphene, 46, 74, 91 trion, 284 trion first-principles calculations, 284 tunable bandgap, bilayer, 48 tunability of Fermi level, 200 tunability of the electron system, graphene, 25 tunability, carrier density, 127, 130 tunable band gap, 170 tunable bandgap, bilayer graphene, 46 tunable interband transitions, 41 tunable metasurfaces, graphene, 135 tunable THz filter, graphene, 193 tungsten hexacarbonyl (W(CO)6), 351 twin boundary grain, 368 twin grain boundaries conductivity, 368 twisted bilayer, graphene, 73–74, 93–94 two-band massive Dirac fermion model, 286 two-center approximation, 323

503

two-dimensional electron systems (2DES), 28 type-II band alignment, 310, 335 ultrafast carrier dynamics, 184 ultrafast interlayer charge transfer, 314, 317 ultrafast lasers, graphene, 189 ultrahigh speed transistors, graphene, 160 ultrarelativistic regime, 20 ultra-short channel transistors, 174 Umklapp scattering, 92, 94, 401 Umklapp-assisted light cones, 321 Umklapp-assisted momentum conservation, 321 uniaxial stress, 61 unilateral gain (U), 162 unipolar resistance graphene, 146 unit cell, graphene, 11, 14, 58, 60, 71, 74–75, 94 universal conductance, graphene, 39 universal value, graphene, 31 utrafast fiber laser, 190 vacancy defects, graphene, 31–33 valence band, 12 valley and spin degrees of freedom coupling, 280 valley circular dichroism, 340 valley degeneracy, 285 valley dependent spin splitting, 291 valley functionalities of interlayer excitons, 324 valley Hall current, 289, 291 valley Hall current, 286 valley Hall effect, 280, 287 valley Hall effect, inversion symmetry breaking, 289 valley magnetic moment, 285 valley optical selection rule, 280, 283 valley optoelectronic, 310 valley polarization, 287 valley polarization detection, 330 valley polarization/coherence, 285 valley pseudospin, 279, 287 valley pseudospin component, 282 valley quantum coherence, 285 valley Zeeman effect, 280, 285 valley Zeeman shift, 285 valley Zeeman splitting, 286 valley-dependent dispersion, 289 valley-dependent exciton current, 324 valley-dependent optical selection rules, 329 valley-dependent optoelectronic devices, 330 valley-dependent optoelectronic devices, 340 valleytronics, 279 van der Pauw measurement, 227 van der Waals (vdW) transfer, 220 van der Waals adhesion, 53, 220 van der Waals assembly, 220–221, 226, 233 van der Waals bonds, 75 van der Waals epitaxy, 1 van der Waals heterojunction diode, 335 van der Waals heterostructures photodetectors, 339

504

Index

van der Waals interactions, 7 vapor transport techniques, 314 vapor–solid growth method, 352 variable range hopping, 33 Verlet algorithm, 473 vertical electric field, 170 vertical field-effect transistors, 172 vertical graphene transistors, 171 vertical structures fabrication, 331 vertical transistor (VFET), 173 vertical tunneling transistors, 172 vertically stacked 2D heterostructures fabrication, 312 Veselago device, 151 Veselago lensing, 149 vibrational absorption, graphene, 134–135 vibrational modes, 75 vibrational spectroscopy, graphene, 134 vibrations out of plane, graphene, 96 vibrations, graphene, 71–72, 74–75, 79, 134–135 vibrations, interlayer, graphene, 74–75 vibrations, quantum nature, graphene, 77 video and biomedical imaging, 184 Vienna ab-initio simulation package (VASP), 474 Volmer–Weber mode, 345 Wannier type exciton wave function, 284 Wannier–Mott exciton Hamiltonian, 396 wave function, 14–15, 157 wave guides, graphene, 149 waveguide-integrated photodetectors, 185 wavepackets, 211 wavevector, 71, 73, 75, 120, 126–127 wavevector, modulation, 73 wavevector, phonon, 10 weak intervalley scattering, 288

Weibull modulus, mechanical properties, 55 white noise disorder, 31 white phosphorus, 414 Wills–Harrison argument, 269 WS2, 262 WS2 photoluminescence intensity, 373 WS2 synthesis, 481 WS2 tellurium-assisted low-temperature synthesis, 347 WSe2, 233, 262 WSe2 helicoid geometry, 370 WSe2 optical micrograph, 313 WSe2 p–n junctions, 340 WSe2 quantum photonics, 362 WSe2 quantum-information, 362 WSe2 screw dislocations, 370 WSe2 single photon emission, 362 WSe2 three-fold rotational defect, 367 Young’s modulus, 52–53, 55–56, 59–60, 65–66, 193 yttrium iron garnet (YIG), 213 ZA phonons, 94, 96 Zeeman Hall effect, 213 Zeeman term, 199 zero bandgap, graphene, 48 zeroth-order Neumann functions, 396 zeroth-order Struve function, 396 zigzag direction, 57 zigzag edges, graphene, 81, 245 zigzag MoS2 band structure, 269 ZnO armchair nanoribbon, 478 ZnO ferromagnetic, 478 ZnO optoelectronic material, 478 ZnO zigzag naoribbons, 478 zone-folding, graphite, 46