Preface ...................................................... XIII
Part One. Materials Science and Raman Spectroscopy Background ... 1
1 The sp2 Nanocarbons: Prototypes for Nanoscience and
Nanotechnology ............................................... 3
1.1 Definition of sp2 Nanocarbon Systems .................... 3
1.2 Short Survey from Discovery to Applications ............. 5
1.3 Why sp2 Nanocarbons Are Prototypes for Nanoscience and
Nanotechnology ......................................... 10
1.4 Raman Spectroscopy Applied to sp2 Nanocarbons .......... 11
2 Electrons in sp2 Nanocarbons ................................ 17
2.1 Basic Concepts: from the Electronic Levels in Atoms
and Molecules to Solids ................................ 18
2.1.1 The One-Electron System and the Schrodinger
Equation ........................................ 18
2.1.2 The Schrodinger Equation for the Hydrogen
Molecule ........................................ 20
2.1.3 Many-Electron Systems: the NO Molecule .......... 21
2.1 A Hybridization: the Acetylene C2H2 Molecule .......... 23
2.1.5 Basic Concepts for the Electronic Structure of
Crystals ........................................ 24
2.2 Electrons in Graphene: the Mother of sp2 Nanocarbons ... 27
2.2.1 Crystal Structure of Monolayer Graphene ......... 27
2.2.2 The Π-Bands of Graphene ......................... 28
2.2.3 The σ-Bands of Graphene ......................... 31
2.2.4 N-Layer Graphene Systems ........................ 33
2.2.5 Nanoribbon Structure ............................ 35
2.3 Electrons in Single-Wall Carbon Nanotubes .............. 37
2.3.1 Nanotube Structure .............................. 38
2.3.2 Zone-Folding of Energy Dispersion Relations ..... 40
2.3.3 Density of States ............................... 44
2.3.4 Importance of the Electronic Structure and
Excitation Laser Energy to the Raman Spectra
of SWNTs ........................................ 47
2.4 Beyond the Simple Tight-Binding Approximation and
Zone-Folding Procedure ................................. 48
3 Vibrations in sp Nanocarbons ................................ 53
3.1 Basic Concepts: from the Vibrational Levels in
Molecules to Solids
3.1.1 The Harmonic Oscillator ......................... 55
3.1.2 Normal Vibrational Modes from Molecules to
a Periodic Lattice .............................. 56
3.1.3 The Force Constant Model ........................ 59
3.2 Phonons in Graphene .................................... 61
3.3 Phonons in Nanoribbons ................................. 65
3.4 Phonons in Single-Wall Carbon Nanotubes ................ 66
3.4.1 The Zone-Folding Picture ........................ 66
3.4.2 Beyond the Zone-Folding Picture ................. 67
3.5 Beyond the Force Constant Model and Zone-Folding
Procedure .............................................. 69
4 Raman Spectroscopy: from Graphite to sp2 Nanocarbons ........ 73
4.1 Light Absorption ....................................... 73
4.2 Other Photophysical Phenomena .......................... 75
4.3 Raman Scattering Effect ................................ 78
4.3.1 Light-Matter Interaction and Polarizability:
Classical Description of Raman Effect ........... 79
4.3.2 Characteristics of the Raman Effect ............. 81
4.3.2.1 Stokes and Anti-Stokes Raman
Processes .............................. 81
4.3.2.2 The Raman Spectrum ..................... 82
4.3.2.3 Raman Lineshape and Raman Spectral
Linewidth Гq ........................... 82
4.3.2.4 Energy Units: cm-1 ..................... 84
4.3.2.5 Resonance Raman Scattering and
Resonance Window Linewidth γr ......... 85
4.3.2.6 Momentum Conservation and
Backscattering Configuration of
Light ................................. 86
4.3.2.7 First and Higher-Order Raman
Processes ............................. 86
4.3.2.8 Coherence ............................. 87
4.4 General Overview of the sp2 Carbon Raman Spectra ...... 88
4.4.1 Graphite ....................................... 88
4.4.2 Carbon Nanotubes - Historical Background ....... 92
4.4.3 Graphene ....................................... 96
5 Quantum Description of Raman Scattering ................... 103
5.1 The Fermi Golden Rule ................................ 103
5.2 The Quantum Description of Raman Spectroscopy ........ 108
5.3 Feynman Diagrams for Light Scattering ................ 111
5.4 Interaction Hamiltonians ............................. 114
5.4.1 Electron-Radiation Interaction ................ 114
5.4.2 Electron-Phonon Interaction ................... 115
5.5 Absolute Raman Intensity and the Elaser Dependence ... 116
6 Symmetry Aspects and Selection Rules: Croup Theory ........ 121
6.1 The Basic Concepts of Group Theory ................... 122
6.1.1 Definition of a Group ......................... 122
6.1.2 Representations ............................... 123
6.1.3 Irreducible and Reducible Representations ..... 124
6.1.1 The Character Table ........................... 126
6.1.5 Products and Orthogonality .................... 127
6.1.6 Other Basis Functions ......................... 128
6.1.7 Finding the IRs for Normal Modes
Vibrations .................................... 128
6.1.8 Selection Rules ............................... 130
6.2 First-Order Raman Scattering Selection Rules ......... 130
6.3 Symmetry Aspects of Graphene Systems ................. 132
6.3.1 Group of the Wave Vector ...................... 132
6.3.2 Lattice Vibrations and Π Electrons ............ 135
6.3.3 Selection Rules for the Electron-Photon
Interaction ................................... 138
6.3.4 Selection Rules for First-Order Raman
Scattering .................................... 140
6.3.5 Electron Scattering by q ≠ 0 Phonons .......... 141
6.3.6 Notation Conversion from Space Group to
Point Group Irreducible Representations ....... 141
6.4 Symmetry Aspects of Carbon Nanotubes ................. 142
6.4.1 Compound Operations and Tube Chirality ........ 143
6.4.2 Symmetries for Carbon Nanotubes ............... 145
6.4.3 Electrons in Carbon Nanotubes ................. 151
6.4.4 Phonons in Carbon Nanotubes ................... 151
6.4.5 Selection Rules for First-Order Raman
Scattering .................................... 152
6.4.6 Insights into Selection Rules from Matrix
Elements and Zone Folding ..................... 153
Part Two. Detailed Analysis of Raman Spectroscopy in
Graphene Related Systems ........................... 159
7 The G-band and Time-Independent Perturbations ............. 161
7.1 G-band in Graphene: Double Degeneracy and Strain ..... 162
7.1.1 Strain Dependence of the G-band ............... 163
7.1.2 Application of Strain to Graphene ............. 165
7.2 The G-band in Nanotubes: Curvature Effects on the
Totally Symmetric Phonons ............................ 165
7.2.1 The Eigenvectors .............................. 166
7.2.2 Frequency Dependence on Tube Diameter ......... 168
7.3 The Six G-band Phonons: Confinement Effect ........... 169
7.3.1 Mode Symmetries and Selection Rules in
Carbon Nanotubes .............................. 169
7.3.2 Experimental Observation Through
Polarization Analysis ......................... 170
7.3.3 The Diameter Dependence of ωG ................. 172
7.4 Application of Strain to Nanotubes ................... 174
7.5 Summary .............................................. 175
8 The G-band and the Time-Dependent Perturbations ........... 179
8.1 Adiabatic and Nonadiabatic Approximations ............ 179
8.2 Use of Perturbation Theory for the Phonon Frequency
Shift ................................................ 181
8.2.1 The Effect of Temperature ..................... 181
8.2.2 The Phonon Frequency Renormalization .......... 183
8.3 Experimental Evidence of the Kohn Anomaly on the
G-band of Graphene ................................... 186
8.3.1 Effect of Gate Doping on the G-band of
Single-Layer Graphene ......................... 186
8.3.2 Effect of Gate Doping on the G-band of
Double-Layer Graphene ......................... 286
8.4 Effect of the Kohn Anomaly on the G-band of
M-SWNTs vs. S-SWNTs .................................. 187
8.4.1 The Electron-Phonon Matrix Element: Peierls-
Like Distortion ............................... 188
8.4.2 Effect of Gate Doping on the G-band of
SWNTs: Theory ................................. 191
8.4.3 Comparison with Experiments ................... 194
8.4.4 Chemical Doping of SWNTs ...................... 196
8.5 Summary .............................................. 197
9 Resonance Raman Scattering: Experimental Observations of
the Radial Breathing Mode ................................. 199
9.1 The Diameter and Chiral Angle Dependence of the RBM
Frequency ............................................ 200
9.1.1 Diameter Dependence: Elasticity Theory ........ 200
9.1.2 Environmental Effects on the RBM Frequency .... 202
9.1.3 Frequency Shifts in Double-Wall Carbon
Nanotubes ..................................... 206
9.1 A Linewidths ........................................ 208
9.1.5 Beyond Elasticity Theory: Chiral Angle
Dependence .................................... 209
9.2 Intensity and the Resonance Raman Effect: Isolated
SWNTs ................................................ 211
9.2.1 The Resonance Window .......................... 211
9.2.2 Stokes and Anti-Stokes Spectra with One
Laser Line .................................... 214
9.2.3 Dependence on Light Polarization .............. 215
9.3 Intensity and the Resonance Raman Effect: SWNT
Bundles .............................................. 216
9.3.1 The Spectral Fitting Procedure for an
Ensemble of Large Diameter Tubes .............. 217
9.3.2 The Experimental Kataura Plot ................. 218
9.4 Summary .............................................. 220
10 Theory of Excitons in Carbon Nanotubes .................... 223
10.1 The Extended Tight-Binding Method: σ-Π
Hybridization ........................................ 224
10.2 Overview on the Excitonic Effect ..................... 225
10.2.1 The Hydrogenic Exciton ........................ 226
10.2.2 The Exciton Wave Vector ....................... 227
10.2.3 The Exciton Spin .............................. 228
10.2.4 Localization of Wavefunctions in Real Space ... 229
10.2.5 Uniqueness of the Exciton in Graphite, SWNTs
and C60 ....................................... 230
10.3 Exciton Symmetry .................................... 231
10.3.1 The Symmetry of Excitons ..................... 231
10.3.2 Selection Rules for Optical Absorption ....... 234
10.4 Exciton Calculations for Carbon Nanotubes ............ 234
10.4.1 Bethe-Salpeter Equation ....................... 235
10.4.2 Exciton Energy Dispersion ..................... 236
10.4.3 Exciton Wavefunctions ......................... 237
10.4.4 Family Patterns in Exciton Photophysics ....... 241
10.5 Exciton Size Effect: the Importance of Dielectric
Screening ............................................ 243
10.5.1 Coulomb Interaction by the 2s and σ
Electrons ..................................... 243
10.5.2 The Effect of the Environmental Dielectric
Constant kenv Term ............................ 245
10.5.3 Further Theoretical Considerations about
Screening ..................................... 246
10.6 Summary .............................................. 248
11 Tight-Binding Method for Calculating Raman Spectra ........ 251
11.1 General Considerations for Calculating Raman
Spectra .............................................. 252
11.2 The (n, m) Dependence of the RBM Intensity:
Experiment ........................................... 253
11.3 Simple Tight-Binding Calculation for the Electronic
Structure ............................................ 255
11.4 Extended Tight-Binding Calculation for Electronic
Structures ........................................... 258
11.5 Tight-Binding Calculation for Phonons ................ 259
11.5.1 Bond Polarization Theory for the Raman
Spectra ....................................... 260
11.5.2 Non-Linear Fitting of Force Constant Sets ..... 261
11.6 Calculation of the Electron-Photon Matrix Element .... 263
11.6.1 Electric Dipole Vector for Graphene ........... 264
11.7 Calculation of the Electron-Phonon Interaction ....... 266
11.8 Extension to Exciton States .......................... 269
11.8.1 Exciton-Photon Matrix Element ................. 270
11.8.2 The Exciton-Phonon Interaction ................ 271
11.9 Matrix Elements for the Resonance Raman Process ...... 272
11.10 Calculating the Resonance Window Width .............. 273
11.11 Summary ............................................. 274
12 Dispersive G'-band and Higher-Order Processes: the
Double Resonance Process .................................. 277
12.1 General Aspects of Higher-Order Raman Processes ...... 278
12.2 The Double Resonance Process in Graphene ............. 280
12.2.1 The Double Resonance Process .................. 280
12.2.2 The Dependence of the ωG' Frequency on the
Excitation Laser Energy ....................... 284
12.2.3 The Dependence of the G'-band on the Number
of Graphene Layers ............................ 286
12.2.4 Characterization of the Graphene Stacking
Order by the G' Spectra ....................... 288
12.3 Generalizing the Double Resonance Process to Other
Raman Modes .......................................... 289
12.4 The Double Resonance Process in Carbon Nanotubes ..... 290
12.4.1 The G'-band in SWNTs Bundles .................. 292
12.4.2 The (n, m) Dependence of the G'-band .......... 294
12.5 Summary .............................................. 296
13 Disorder Effects in the Raman Spectra of sp2 Carbons ..... 299
13.1 Quantum Modeling of the Elastic Scattering Event ..... 301
13.2 The Frequency of the Defect-Induced Peaks: the
Double Resonance Process ............................. 304
13.3 Quantifying Disorder in Graphene and Nanographite
from Raman Intensity Analysis ........................ 307
13.3.1 Zero-Dimensional Defects Induced by Ion
Bombardment ................................... 308
13.3.2 The Local Activation Model .................... 310
13.3.3 One-Dimensional Defects Represented by the
Boundaries of Nanocrystallites ................ 313
13.3.4 Absolute Raman Cross-Section .................. 317
13.4 Defect-Induced Selection Rules: Dependence on Edge
Atomic Structure ..................................... 317
13.5 Specificities of Disorder in the Raman Spectra of
Carbon Nanotubes ..................................... 320
13.6 Local Effects Revealed by Near-Field Measurements .... 321
13.7 Summary .............................................. 323
14 Summary of Raman Spectroscopy on sp2 Nanocarbons .......... 327
14.1 Mode Assignments, Electron, and Phonon Dispersions ... 327
14.2 The G-band ........................................... 328
14.3 The Radial Breathing Mode (RBM) ...................... 330
14.4 G'-band .............................................. 332
14.5 D-band ............................................... 333
14.6 Perspectives ......................................... 334
References ................................................ 335
Index ........................................................ 351
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