Table of Contents |
pt. I | Experimental | |
1. | Experimental Manifestation of Berry Phase in Graphene / Philip Kim | 3 |
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 / Eva Y. Andrei | 29 |
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 / Mildred S. Dresselhaus | 65 |
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 G'-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 / M. Potemski | 113 |
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 / K. Ensslin | 141 |
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 |
pt. II | Theoretical | |
6. | Electronic Properties of Monolayer and Multilayer Graphene / Tsuneya Ando | 173 |
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 / Hideo Aoki | 213 |
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 / Vadim Apalkov | 251 |
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 / Kentaro Nomura | 301 |
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 v = 0 | 311 |
9.4.1. | Field-Induced Dissipative States and Insulating States | 311 |
9.4.2. | Possible Broken Symmetries at v = 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. | v = 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 / Vladimir I. Fal'ko | 327 |
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 |