Introduction to Graphene-Based Nanomaterials: From Electronic Structure to Quantum Transport

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Graphene is one of the most intensively studied materials, and has unusual electrical, mechanical and thermal properties, which provide almost unlimited potential applications. This book provides an introduction to the electrical and transport properties of graphene and other two dimensional nanomaterials, covering ab-initio to multiscale methods. Updated from the first edition, the authors have added chapters on other two dimensional materials, spin related phenomena, and an improved overview of Berry phase effects. Other topics include powerful order N electronic structure, transport calculations, ac transport and multiscale transport methodologies. Chapters are complemented with concrete examples and case studies, questions and exercises, detailed appendices and computational codes. It is a valuable resource for graduate students and researchers working in physics, materials science or engineering who are interested in the field of graphene-based nanomaterials.

Author(s): Luis E. F. Foa Torres, Stephan Roche, Jean-Christophe Charlier
Edition: 2
Publisher: Cambridge University Press
Year: 2020

Language: English
Pages: 479

Contents......Page 5
Preface to the Second Edition......Page 11
Preface to the First Edition......Page 14
1.1 Carbon Structures and Hybridizations......Page 18
1.2 Carbon Nanostructures......Page 21
1.3 Guide to the Book......Page 25
1.4 Further Reading......Page 27
2.1 Introduction......Page 28
2.2.1 Tight-Binding Description of Graphene......Page 29
2.2.2 Effective Description Close to the Dirac Point and Massless Dirac Fermions......Page 35
2.2.3 Electronic Properties of Graphene beyond the Linear Approximation......Page 37
2.3 Electronic Properties of Few-Layer Graphene......Page 41
2.4 Electronic Properties of Graphene Nanoribbons......Page 46
2.4.1 Electronic Properties of Armchair Nanoribbons (aGNRs)......Page 50
2.4.2 Electronic Properties of Zigzag Nanoribbons (zGNRs)......Page 53
2.5.1 Structural Parameters of CNTs......Page 57
2.5.2 Electronic Structure of CNTs within the Zone-Folding Approximation......Page 58
2.5.3 Curvature Effects: Beyond the Zone-Folding Model......Page 64
2.5.4 Small-Diameter Nanotubes: Beyond the Tight-Binding Approach......Page 65
2.5.5 Nanotubes in Bundles......Page 68
2.5.6 Multiwall Nanotubes......Page 69
2.6.1 Structural Point Defects in Graphene......Page 71
2.6.2 Grain Boundaries and Extended Defects in Graphene......Page 74
2.6.3 Structural Defects at Graphene Edges......Page 79
2.6.4 Defects in Carbon Nanotubes......Page 80
2.7 Further Reading and Problems......Page 83
3.1 Introduction......Page 87
3.2 Hexagonal Boron Nitride Monolayer......Page 89
3.3 Two-Dimensional Transition Metal Dichalcogenides......Page 93
3.4.1 Phosphorene......Page 96
3.4.2 Borophene......Page 98
3.4.3 Silicene, Germanene, and Stanene......Page 100
3.4.4 MXenes......Page 102
3.5 van der Waals Heterostructures......Page 104
3.6 Conclusion......Page 107
3.7 Further Reading......Page 108
4.1.1 Relevant Time and Length Scales......Page 109
4.1.2 Coherent versus Sequential Transport......Page 110
4.2 Landauer–B¨ uttiker Theory......Page 112
4.2.1 Heuristic Derivation of Landauer’s Formula......Page 115
4.3 Boltzmann Semiclassical Transport......Page 116
4.3.1 The Relaxation Time Approximation and the Boltzmann Conductivity......Page 117
4.4 Kubo Formula for the Electronic Conductivity......Page 119
4.4.1 Illustrations for Ballistic and Diffusive Regimes......Page 123
4.4.2 Kubo versus Landauer......Page 126
4.4.4 The Kubo Formalism in Real Space......Page 127
4.4.5 Scaling Theory of Localization......Page 130
4.5 Quantum Transport beyond the Fully Coherent or Decoherent Limits......Page 134
4.6 Further Reading and Problems......Page 135
5.1 The Klein Tunneling Mechanism......Page 137
5.1.1 Klein Tunneling through Monolayer Graphene with a Single (Impurity) Potential Barrier......Page 138
5.1.2 Klein Tunneling through Bilayer Graphene with a Single (Impurity) Potential Barrier......Page 143
5.2 Ballistic Transport in Carbon Nanotubes and Graphene......Page 145
5.2.1 Ballistic Motion and Conductance Quantization......Page 146
5.2.2 Mode Decomposition in Real Space......Page 147
5.2.3 Fabry–P ´ erot Conductance Oscillations......Page 151
5.2.4 Contact Effects: SWNT-Based Heterojunctions and the Role of Contacts between Metals and Carbon-Based Devices......Page 154
5.3 Ballistic Motion through a Graphene Constriction: The 2D Limit and the Minimum Conductivity......Page 159
5.4 Further Reading and Problems......Page 160
6.1 Elastic Mean Free Path......Page 162
6.1.1 Temperature Dependence of the Mean Free Path......Page 165
6.1.2 Inelastic Mean Free Path in the High-Bias Regime......Page 167
6.1.3 Quantum Interference Effects and Localization Phenomena in Disordered Graphene-Based Materials......Page 169
6.1.4 Edge Disorder and Transport Gaps in Graphene Nanoribbons......Page 171
6.2.1 Two-Dimensional Disordered Graphene: Experimental and Theoretical Overview......Page 173
6.2.2 Metallic versus Insulating State and Minimum Conductivity......Page 177
6.2.3 Boltzmann Transport in Two-Dimensional Graphene......Page 178
6.2.4 Kubo Transport: Graphene with Anderson Disorder......Page 185
6.2.5 Kubo Transport: Graphene with Gaussian Impurities......Page 187
6.2.6 Weak Localization Phenonema in Disordered Graphene......Page 192
6.2.7 Strong Localization in Disordered Graphene......Page 200
6.3 Graphene with Monovacancies......Page 202
6.3.1 Electronic Structure of Graphene with Monovacancies......Page 204
6.3.2 Transport Features of Graphene with Monovacancies......Page 206
6.4.1 Motivation and Structural Models......Page 211
6.4.2 Electronic Properties of Polycrystalline Graphene......Page 215
6.4.3 Mean Free Path, Conductivity and Charge Mobility......Page 216
6.5 Graphene Quantum Dots......Page 219
6.5.1 Generalities on Coulomb Blockade......Page 220
6.5.2 Confining Charges in Graphene Devices......Page 222
6.6 Further Reading and Problems......Page 225
7.1 Berry Phase......Page 227
7.2 Graphene’s Berry Phase and Its Observation in ARPES Experiments......Page 230
7.3 Anomalous Velocity and Valley Hall Effect......Page 231
7.4 The Peierls Substitution......Page 233
7.5 Aharonov–Bohm Gap Opening and Orbital Degeneracy Splitting in Carbon Nanotubes......Page 234
7.6 Landau Levels in Graphene......Page 239
7.7 Quantum Hall Effect in Graphene......Page 242
7.7.1 Experimental Observation of Hall Quantization in Graphene......Page 243
7.7.2 Remarks for the Numerical Investigation of the Hall Response......Page 244
7.7.3 The Mystery of the Zero-Energy Landau Level Splitting......Page 245
7.7.4 Universal Longitudinal Conductivity at the Dirac Point......Page 246
7.8 The Haldane Model......Page 249
7.9 Further Reading and Problems......Page 252
8.1 Introduction......Page 254
8.2 Spin–Orbit Coupling in Graphene......Page 256
8.2.1 Derivation from the Dirac Equation......Page 257
8.2.2 Theoretical Estimation of the SOC Terms Magnitude......Page 262
8.3 Spin Transport Measurements and Spin Lifetime......Page 263
8.4 Spin Dynamics and Relaxation Mechanisms......Page 265
8.4.1 Dyakonov–Perel Mechanism......Page 267
8.4.2 Elliot–Yafet Mechanism for Graphene......Page 270
8.4.3 Spin–Pseudopsin Entanglement and Spin Relaxation......Page 272
8.5 Manipulating Spin by Proximity Effects......Page 276
8.5.2 Magnetic Proximity Effects in Vertical Spin Devices......Page 277
8.5.3 Weak Antilocalization in Graphene/TMD Heterostructures......Page 279
8.5.4 Spin Transport Anisotropy......Page 280
8.6.1 Introductory Picture and Basics......Page 286
8.6.2 Enhanced SHE in Graphene?......Page 288
8.7 Spin Transport Formalism and Computational Methodologies......Page 291
8.8 Further Reading......Page 294
9.1 Introduction: Why AC Fields?......Page 295
9.2 Adiabatic Approximation......Page 296
9.3 Floquet Theory......Page 297
9.3.1 Average Current and Density of States......Page 298
9.3.2 Homogeneous Driving and the Tien–Gordon Model......Page 300
9.4 Overview of AC Transport in Carbon-Based Devices......Page 301
9.5 AC Transport and Laser-Induced Effects on the Electronic Properties of Graphene......Page 303
9.6 Further Reading and Problems......Page 307
10.1 Introduction......Page 310
10.2.2 Boron-Doped Metallic Carbon Nanotubes......Page 311
10.2.3 Nitrogen-Doped Metallic Carbon Nanotubes......Page 314
10.3.1 Monatomic Oxygen Defects......Page 321
10.3.2 Atomic Hydrogen Defects......Page 324
10.4 Structural Point Defects Embedded in Graphene......Page 327
10.5.1 Introduction......Page 330
10.5.2 Carbon Nanotubes......Page 331
10.5.3 Defective Carbon Nanotubes......Page 333
10.5.4 Doped Carbon Nanotubes......Page 338
10.5.5 Functionalized Carbon Nanotubes......Page 340
10.5.6 Carbon Nanotubes Decorated with Metal Clusters......Page 347
10.5.7 Graphene Nanoribbons......Page 348
10.5.8 Graphene Nanoribbons with Point Defects......Page 349
10.5.9 Graphene Nanoribbons with Edge Reconstruction......Page 351
10.5.10 Graphene Nanoribbons with Edge Disorder......Page 352
10.5.11 Doped Graphene Nanoribbons......Page 359
10.5.12 GNR-Based Networks......Page 364
10.6 Conclusion......Page 369
10.7 Further Reading......Page 370
A.2.1 The Schr ¨ odinger Equation......Page 371
A.2.2 The Born–Oppenheimer Approximation......Page 372
A.2.3 The Hartree Approximation......Page 373
A.2.4 The Hartree–Fock Approximation......Page 374
A.3.1 The Thomas–Fermi Model......Page 375
A.3.2 The Hohenberg–Kohn Theorem......Page 376
A.3.3 The Kohn–Sham Equations......Page 377
A.3.4 The Exchange–Correlation Functionals......Page 379
A.4.1 Crystal Lattice and Reciprocal Space......Page 381
A.4.2 The Plane Wave Representation......Page 382
A.4.3 k-Point Grids and Band Structures......Page 383
A.4.4 The Pseudopotential Approximation......Page 384
A.4.5 Available DFT Codes......Page 387
B.1 Introduction......Page 390
B.2.1 Hedin’s Equations......Page 391
B.2.2 GW Approximation......Page 392
B.3.1 Perturbative Approach......Page 393
B.3.2 Plasmon Pole......Page 394
C.1 Phase-Coherent Quantum Transport and the Green’s Function Formalism......Page 396
C.2 Self-Energy Corrections and Recursive Green’s Functions Techniques......Page 404
C.3 Dyson’s Equation and an Application to Treatment of Disordered Systems......Page 406
C.4 Computing Transport Properties within Ab Initio Simulations......Page 410
D.1 Lanczos Method for the Density of States......Page 418
D.1.1 Termination of the Continued Fraction......Page 421
D.2 Wavepacket Propagation Method......Page 422
D.3 Lanczos Method for Computing Off-Diagonal Green’s Functions......Page 428
References......Page 430
Index......Page 474