Hideo Aoki, Mildred S. Dresselhaus, "Physics of Graphene"
2014 | ISBN-10: 3319026321 | 350 pages | PDF | 12,7 MB
This book provides a state of the art report of the knowledge accumulated in graphene research. The fascination with graphene has been growing very rapidly in recent years and the physics of graphene is now becoming one of the most interesting as well as the most fast-moving topics in condensed-matter physics. The Nobel prize in physics awarded in 2010 has given a tremendous impetus to this topic. The horizon of the physics of graphene is ever becoming wider, where physical concepts go hand in hand with advances in experimental techniques. Thus this book is expanding the interests to not only transport but optical and other properties for systems that include multilayer as well as monolayer graphene systems. The book comprises experimental and theoretical knowledge. The book is also accessible to graduate students.
Contents
Part I Experimental
1 Experimental Manifestation of Berry Phase in Graphene . . . . . . 3
Andrea F. Young, Yuanbo Zhang, and Philip Kim
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Pseudospin Chirality in Graphene . . . . . . . . . . . . . . . . . 5
1.3 Berry Phase in Magneto-Oscillations . . . . . . . . . . . . . . . 8
1.4 Pseudospin and Klein Tunneling in Graphene . . . . . . . . . . . 16
1.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2 Probing Dirac Fermions in Graphene by Scanning Tunneling
Microscopy and Spectroscopy . . . . . . . . . . . . . . . . . . . . . 29
Adina Luican-Mayer and Eva Y. Andrei
2.1 Scanning Tunneling Microscopy and Spectroscopy . . . . . . . . 29
2.2 From Disordered Graphene to Ideal Graphene . . . . . . . . . . 31
2.2.1 Surface Topography of Graphene . . . . . . . . . . . . . 33
2.2.2 Tunneling Spectroscopy of Graphene . . . . . . . . . . . 35
2.2.3 Doping and Electron Hole Puddles . . . . . . . . . . . . 36
2.2.4 Landau Levels . . . . . . . . . . . . . . . . . . . . . . . 37
2.2.5 Measuring Small Graphene Devices with Scanning
Probes . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.2.6 Graphene Edges . . . . . . . . . . . . . . . . . . . . . . 49
2.2.7 Strain and Electronic Properties . . . . . . . . . . . . . . 51
2.2.8 Bilayer Graphene . . . . . . . . . . . . . . . . . . . . . 51
2.3 Electronic Properties of Twisted Graphene Layers . . . . . . . . 52
2.3.1 Van Hove Singularities . . . . . . . . . . . . . . . . . . 52
2.3.2 Renormalization of the Fermi Velocity . . . . . . . . . . 55
2.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3 Electron and Phonon Transport in Graphene
in and out of the Bulk . . . . . . . . . . . . . . . . . . . . . . . . . 65
Jean-Paul Issi, Paulo T. Araujo, and Mildred S. Dresselhaus
3.1 General Introduction . . . . . . . . . . . . . . . . . . . . . . . . 66
3.1.1 Graphenes . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.1.2 Transport . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.1.3 Inelastic Scattering of Light . . . . . . . . . . . . . . . . 69
3.1.4 General References and Historical Background . . . . . . 70
3.1.5 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.1.6 Topics Addressed . . . . . . . . . . . . . . . . . . . . . 71
3.2 Electrical Conductivity . . . . . . . . . . . . . . . . . . . . . . 71
3.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 71
3.2.2 Electronic Structure . . . . . . . . . . . . . . . . . . . . 73
3.2.3 Charge Carrier Densities and Scattering . . . . . . . . . . 76
3.2.4 Quantum Effects . . . . . . . . . . . . . . . . . . . . . . 84
3.2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 88
3.3 Thermal Conductivity of Graphene in and out of the Bulk . . . . 88
3.3.1 Preliminary Remarks . . . . . . . . . . . . . . . . . . . 88
3.3.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 89
3.3.3 Comparing the Thermal Conductivity of Graphene
in and out the Bulk . . . . . . . . . . . . . . . . . . . . . 90
3.3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 101
3.4 Inelastic Scattering of Light—Raman Scattering . . . . . . . . . 101
3.4.1 A Brief Overview of Inelastic Scattering of Light . . . . 101
3.4.2 The G-Band Mode . . . . . . . . . . . . . . . . . . . . . 103
3.4.3 The G1 -Band (or 2D) Mode . . . . . . . . . . . . . . . . 104
3.4.4 The Disorder-Induced D-Band Mode . . . . . . . . . . . 105
3.4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 108
3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
4 Optical Magneto-Spectroscopy of Graphene-Based Systems . . . . 113
C. Faugeras, M. Orlita, and M. Potemski
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.2 Magneto-Spectroscopy of Graphene . . . . . . . . . . . . . . . . 115
4.2.1 Classical Cyclotron Resonance of Dirac Fermions . . . . 115
4.2.2 Magneto-Optical Response of Graphene:
Quantum Regime . . . . . . . . . . . . . . . . . . . . . 117
4.2.3 Landau Level Fan Charts and Fermi Velocity . . . . . . . 120
4.2.4 Beyond Simple Band Models . . . . . . . . . . . . . . . 121
4.2.5 Scattering/Disorder . . . . . . . . . . . . . . . . . . . . 121
4.2.6 Electron-Electron Interaction . . . . . . . . . . . . . . . 122
4.2.7 Effects of Electron-Phonon Interaction . . . . . . . . . . 123
4.3 Magneto-Spectroscopy of Bilayer Graphene . . . . . . . . . . . 124
4.4 Graphite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
4.4.1 Simplified Models for the Band Structure . . . . . . . . . 126
4.4.2 Full Slonczewski-Weiss-McClure Model . . . . . . . . . 128
4.4.3 Band Structure Close to the Neutrality Point: Proximity
to Lifshitz Transition . . . . . . . . . . . . . . . . . . . 129
4.4.4 Scattering Efficiency . . . . . . . . . . . . . . . . . . . . 131
4.4.5 Electron-Phonon Coupling . . . . . . . . . . . . . . . . 132
4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
5 Graphene Constrictions . . . . . . . . . . . . . . . . . . . . . . . . 141
S. Dr?scher, F. Molitor, T. Ihn, and K. Ensslin
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
5.1.1 Graphene Electronics . . . . . . . . . . . . . . . . . . . 141
5.1.2 Graphene Nanostructures . . . . . . . . . . . . . . . . . 142
5.2 Constrictions in Conventional Semiconductors . . . . . . . . . . 143
5.3 Conductance in Graphene Constrictions . . . . . . . . . . . . . . 144
5.3.1 Nanoribbons with Ideal Edges . . . . . . . . . . . . . . . 144
5.3.2 Extension to Disordered Edges . . . . . . . . . . . . . . 146
5.4 Experimental Observations and Microscopic Pictures . . . . . . 146
5.4.1 Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . 146
5.4.2 Dependence of Transport on the Charge Carrier
Density . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
5.4.3 Dependence of Transport on the Applied Voltage Bias . . 148
5.4.4 Microscopic Pictures . . . . . . . . . . . . . . . . . . . 151
5.4.5 Geometry Dependence . . . . . . . . . . . . . . . . . . . 152
5.5 Further Experiments for More Detailed Understanding . . . . . . 153
5.5.1 Temperature Dependence . . . . . . . . . . . . . . . . . 153
5.5.2 Magnetic Field Dependence . . . . . . . . . . . . . . . . 156
5.5.3 Side-Gate Influence . . . . . . . . . . . . . . . . . . . . 158
5.5.4 Thermal Cycling . . . . . . . . . . . . . . . . . . . . . . 160
5.5.5 Tunneling Coupling in a Double Quantum Dot . . . . . . 161
5.6 Recent Advances and Outlook . . . . . . . . . . . . . . . . . . . 164
5.6.1 Bottom-Up Growth of Nanoribbons . . . . . . . . . . . . 164
5.6.2 Quantized Conductance in Suspended Nanoribbons . . . 165
5.6.3 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . 166
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
Part II Theoretical
6 Electronic Properties of Monolayer and Multilayer Graphene . . . 173
Mikito Koshino and Tsuneya Ando
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
6.2 Electronic Structure of Graphene . . . . . . . . . . . . . . . . . 174
6.2.1 Effective Hamiltonian . . . . . . . . . . . . . . . . . . . 174
6.2.2 Landau Levels . . . . . . . . . . . . . . . . . . . . . . . 177
6.2.3 Band Gap in Graphene . . . . . . . . . . . . . . . . . . . 179
6.3 Orbital Diamagnetism . . . . . . . . . . . . . . . . . . . . . . . 181
6.3.1 The Susceptibility Singularity . . . . . . . . . . . . . . . 181
6.3.2 Response to a Non-uniform Magnetic Field . . . . . . . . 183
6.4 Transport Properties . . . . . . . . . . . . . . . . . . . . . . . . 184
6.4.1 Boltzmann Conductivity . . . . . . . . . . . . . . . . . . 185
6.4.2 Self-consistent Born Approximation . . . . . . . . . . . 187
6.5 Optical Properties . . . . . . . . . . . . . . . . . . . . . . . . . 189
6.6 Bilayer Graphene . . . . . . . . . . . . . . . . . . . . . . . . . 191
6.6.1 Electronic Structure . . . . . . . . . . . . . . . . . . . . 191
6.6.2 Landau Levels . . . . . . . . . . . . . . . . . . . . . . . 193
6.6.3 Gapped Bilayer Graphene . . . . . . . . . . . . . . . . . 194
6.6.4 Orbital Diamagnetism . . . . . . . . . . . . . . . . . . . 196
6.6.5 Transport Properties . . . . . . . . . . . . . . . . . . . . 198
6.6.6 Optical Properties . . . . . . . . . . . . . . . . . . . . . 200
6.7 Multilayer Graphenes . . . . . . . . . . . . . . . . . . . . . . . 202
6.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
7 Graphene: Topological Properties, Chiral Symmetry
and Their Manipulation . . . . . . . . . . . . . . . . . . . . . . . . 213
Yasuhiro Hatsugai and Hideo Aoki
7.1 Chiral Symmetry as a Generic Symmetry in Graphene . . . . . . 213
7.2 Chiral Symmetry, Dirac Cones and Fermion Doubling . . . . . . 215
7.2.1 Chiral Symmetry for Lattice Systems . . . . . . . . . . . 215
7.2.2 Fermion Doubling for Chiral Symmetric Lattice
Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . 218
7.2.3 When and How Dirac Cones Appear?—Generalised
Chiral Symmetry . . . . . . . . . . . . . . . . . . . . . . 221
7.3 Hall Conductivity of Dirac Fermions in Magnetic Fields . . . . . 223
7.3.1 Landau Level of the Dirac Fermions . . . . . . . . . . . 223
7.3.2 Stability of the n = 0 Landau Level . . . . . . . . . . . . 224
7.3.3 Massless vs Massive Dirac Fermions . . . . . . . . . . . 226
7.3.4 Chern Number for Many-Particle Configurations . . . . . 228
7.3.5 Quantum Hall Effect in Graphene . . . . . . . . . . . . . 231
7.4 Bulk-Edge Correspondence for the Chiral-Symmetric Dirac
Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
7.4.1 Boundary Physics of Graphene . . . . . . . . . . . . . . 233
7.4.2 Types of Edges and Zero-Energy Edge States . . . . . . . 234
7.4.3 Edge States and Chiral Symmetry . . . . . . . . . . . . . 235
7.4.4 Quantum Hall Edge States of Graphene . . . . . . . . . . 238
7.4.5 n = 0 Landau Level and the Zero Modes . . . . . . . . . 239
7.5 Optical Hall Effect in Graphene . . . . . . . . . . . . . . . . . . 239
7.6 Nonequilibrium Control of Topological Property . . . . . . . . . 241
7.7 Chiral Symmetry for Interacting Electrons . . . . . . . . . . . . 245
7.8 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . 247
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
8 Aspects of the Fractional Quantum Hall Effect in Graphene . . . . 251
Tapash Chakraborty and Vadim Apalkov
8.1 A Brief History of the Fractional Quantum Hall Effect . . . . . . 251
8.1.1 A Novel Many-Body Incompressible State . . . . . . . . 253
8.1.2 Pseudopotential Description of Interacting Electrons . . . 254
8.1.3 Composite Fermions and the Fermion-Chern-Simons
Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
8.2 The Advent of Graphene . . . . . . . . . . . . . . . . . . . . . . 256
8.2.1 Massless Dirac Fermions . . . . . . . . . . . . . . . . . 257
8.2.2 Landau Levels in Graphene . . . . . . . . . . . . . . . . 258
8.2.3 Pseudopotentials in Graphene . . . . . . . . . . . . . . . 260
8.2.4 Nature of the Incompressible States in Graphene . . . . . 262
8.2.5 Experimental Observations of the Incompressible
States . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
8.3 Bilayer Graphene . . . . . . . . . . . . . . . . . . . . . . . . . 267
8.3.1 Magnetic Field Effects . . . . . . . . . . . . . . . . . . . 268
8.3.2 Biased Bilayer Graphene . . . . . . . . . . . . . . . . . 269
8.3.3 Pseudopotentials in Bilayer Graphene . . . . . . . . . . . 271
8.3.4 Novel Effects from Electron-Electron Interactions . . . . 272
8.3.5 Interacting Electrons in Rotated Bilayer Graphene . . . . 277
8.4 Fractional Quantum Hall Effect in Trilayer Graphene . . . . . . 279
8.5 Some Unique Properties of Interacting Dirac Fermions . . . . . . 283
8.5.1 The Pfaffians in Condensed Matter . . . . . . . . . . . . 283
8.5.2 The Pfaffians in Graphene . . . . . . . . . . . . . . . . . 285
8.5.3 Interacting Dirac Fermions on the Surface
of a Topological Insulator . . . . . . . . . . . . . . . . . 290
8.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
9 Symmetry Breaking in Graphene’s Quantum Hall Regime:
The Competition Between Interactions and Disorder . . . . . . . . 301
Yafis Barlas, A.H. MacDonald, and Kentaro Nomura
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
9.2 The Quantum Hall Effect of Massless Dirac Fermions . . . . . . 303
9.2.1 Landau Levels and Quantized Hall Conductivities . . . . 303
9.2.2 Zero-Field Mobility and Charged Impurities . . . . . . . 305
9.2.3 Self-consistent Treatment of Screened Impurities
in a Magnetic Field . . . . . . . . . . . . . . . . . . . . 306
9.3 Spontaneous Breaking of Spin and Valley Symmetry . . . . . . . 307
9.3.1 Exchange Interactions . . . . . . . . . . . . . . . . . . . 307
9.3.2 Phase Diagram: Disorder vs Exchange . . . . . . . . . . 309
9.4 Field-Induced Insulator at ν = 0 . . . . . . . . . . . . . . . . . . 311
9.4.1 Field-Induced Dissipative States and Insulating States . . 311
9.4.2 Possible Broken Symmetries at ν = 0 . . . . . . . . . . . 312
9.4.3 Field-Induced Transition and Divergence
of Resistance . . . . . . . . . . . . . . . . . . . . . . . . 314
9.5 Quantum Hall Ferromagnetism in Bilayer Graphene . . . . . . . 316
9.5.1 Bilayer Graphene . . . . . . . . . . . . . . . . . . . . . 316
9.5.2 Octet Hund’s Rules . . . . . . . . . . . . . . . . . . . . 317
9.5.3 Collective Modes of Landau Level Pseudospins . . . . . 319
9.5.4 Instabilities, Ordering and Topological Excitations
of LL-Pseudospins . . . . . . . . . . . . . . . . . . . . . 320
9.5.5 ν = 0 QH Plateaus in Bilayer Graphene . . . . . . . . . . 321
9.6 Quantum Hall Ferromagnetism at Fractional Fillings . . . . . . . 322
9.7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . 323
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325
10 Weak Localization and Spin-Orbit Coupling in Monolayer
and Bilayer Graphene . . . . . . . . . . . . . . . . . . . . . . . . . 327
Edward McCann and Vladimir I. Fal’ko
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327
10.2 The Low-Energy Hamiltonian of Monolayer Graphene . . . . . . 328
10.2.1 Massless Dirac-Like Quasiparticles in Monolayer
Graphene . . . . . . . . . . . . . . . . . . . . . . . . . . 328
10.2.2 Model of Disorder in Monolayer Graphene . . . . . . . . 330
10.2.3 Spin-Orbit Coupling in Monolayer Graphene . . . . . . . 332
10.3 Weak Localization vs Antilocalization in Monolayer
Graphene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332
10.4 The Low-Energy Hamiltonian of Bilayer Graphene . . . . . . . . 337
10.4.1 Massive Chiral Quasiparticles in Bilayer Graphene . . . . 337
10.4.2 Model of Disorder in Bilayer Graphene . . . . . . . . . . 339
10.4.3 Spin-Orbit Coupling in Bilayer Graphene . . . . . . . . . 339
10.5 Weak Localization in Bilayer Graphene . . . . . . . . . . . . . . 340
10.6 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . 344
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
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