Preface ........................................................ xi
1 The electronic structure of ideal graphene ................... 1
1.1 The carbon atom ......................................... 1
1.2 л States in graphene .................................... 5
1.3 Massless Dirac fermions in graphene .................... 10
1.4 The electronic structure of bilayer graphene ........... 14
1.5 Multilayer graphene .................................... 19
2 Electron states in a magnetic field ......................... 23
2.1 The effective Hamiltonian .............................. 23
2.2 Landau quantization for massless Dirac fermions ........ 27
2.3 Topological protection of the zero-energy states ....... 31
2.4 Semiclassical quantization conditions and Berry's
phase .................................................. 34
2.5 Landau levels in bilayer graphene ...................... 39
2.6 The case of bilayer graphene: trigonal warping
effects ................................................ 42
2.7 A unified description of single-layer and bilayer
graphene ............................................... 45
2.8 Magnetic oscillations in single-layer graphene ......... 47
2.9 The anomalous quantum Hall effect in single-layer
and bilayer graphene ................................... 53
2.10 Effects of smooth disorder and an external electric
field on the Landau levels ............................. 58
3 Quantum transport via evanescent waves ...................... 63
3.1 Zitterbewegung as an intrinsic disorder ................ 63
3.2 The Landauer-formula approach .......................... 67
3.3 Conformal mapping and Corbino geometry ................. 70
3.4 The Aharonov-Bohm effect in undoped graphene ........... 73
4 The Klein paradox and chiral tunnelling ..................... 77
4.1 The Klein paradox ...................................... 77
4.2 The massless case: the role of chirality ............... 84
4.3 Klein tunnelling in single-layer graphene .............. 86
4.4 Klein tunnelling for a smooth potential barrier and
the effect of magnetic fields .......................... 90
4.5 Negative refraction coefficient and Veselago lenses
for electrons in graphene .............................. 93
4.6 Klein tunnelling and minimal conductivity .............. 94
4.7 Chiral tunnelling in bilayer graphene .................. 97
5 Edges, nanoribbons and quantum dots ........................ 103
5.1 The neutrino billiard model ........................... 103
5.2 A generic boundary condition: valley mixing ........... 108
5.3 Boundary conditions for a terminated honeycomb
lattice ............................................... 112
5.4 Electronic states of graphene nanoribbons ............. 116
5.5 Conductance quantization in graphene nanoribbons ...... 120
5.6 The band gap in graphene nanoribbons with generic
boundary conditions ................................... 124
5.7 Energy levels in graphene quantum dots ................ 126
5.8 Edge states in magnetic fields and the anomalous
quantum Hall effect ................................... 129
6 Point defects .............................................. 134
6.1 Scattering theory for Dirac electrons ................. 134
6.2 Scattering by a region of constant potential .......... 138
6.3 Scattering theory for bilayer graphene in the
parabolic-band approximation .......................... 141
6.4 General theory of defects in a honeycomb lattice ...... 145
6.5 The case of vacancies ................................. 150
6.6 Adsorbates on graphene ................................ 152
6.7 Scanning tunnelling microscopy of point defects on
graphene .............................................. 156
6.8 Long-range interaction between adatoms on graphene .... 158
7 Optics and response functions .............................. 161
7.1 Light absorption by Dirac fermions: visualization of
the fine-structure constant ........................... 161
7.2 The optics of Dirac fermions: the pseudospin
precession formalism .................................. 163
7.3 The absence of many-body corrections to the
universal optical conductivity ........................ 167
7.4 The magneto-optics of Dirac fermions .................. 169
7.5 Optical properties of graphene beyond the Dirac
approximation ......................................... 171
7.6 The dielectric function of Dirac fermions ............. 174
7.7 Static screening ...................................... 179
7.8 Plasmons .............................................. 181
7.9 Transverse response functions and diamagnetic
susceptibility ........................................ 182
8 The Coulomb problem ........................................ 185
8.1 Scattering of Dirac fermions by point charges ......... 185
8.2 Relativistic collapse for supercritical charges ....... 191
8.3 Nonlinear screening of charge impurities .............. 195
8.4 Inter-electron Coulomb interaction and
renormalization of the Fermi velocity ................. 201
9 Crystal lattice dynamics, structure and thermodynamics ..... 205
9.1 Phonon spectra of graphene ............................ 205
9.2 The theory of elasticity for thin plates .............. 210
9.3 The statistical mechanics of flexible membranes ....... 219
9.4 Scaling properties of membranes and intrinsic
ripples in graphene ................................... 222
9.5 The self-consistent screening approximation ........... 230
9.6 Thermodynamic and other thermal properties of
graphene .............................................. 234
9.7 Raman spectra of graphene ............................. 238
10 Gauge fields and strain engineering ........................ 243
10.1 Strain-induced pseudomagnetic fields .................. 243
10.2 Pseudomagnetic fields of frozen ripples ............... 246
10.3 Pseudomagnetic fields of ripples: the effect of
in-plane relaxation ................................... 251
10.4 The zero-field quantum Hall effect by strain
engineering ........................................... 254
10.5 The pseudo-Aharonov-Bohm effect and transport gap in
suspended graphene .................................... 258
10.6 Gap opening by combination of strain and electric
field ................................................. 261
11 Scattering mechanisms and transport properties ............. 266
11.1 The semiclassical Boltzmann equation and limits of
its applicability ..................................... 266
11.2 The Kubo-Nakano-Mori formula for resistivity .......... 274
11.3 Scattering mechanisms in graphene on substrate ........ 279
11.4 Intrinsic mobility and transport properties of
suspended graphene flakes ............................. 287
11.5 Nonlocal transport in magnetic fields ................. 293
11.6 Beyond the Boltzmann equation: localization and
antilocalization ...................................... 298
12 Spin effects and magnetism ................................. 301
12.1 General remarks on itinerant-electron magnetism ....... 301
12.2 Defect-induced magnetism in graphene .................. 309
12.3 Magnetic edges ........................................ 313
12.4 Spin-orbit coupling ................................... 316
References .................................................... 322
Index ......................................................... 338
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