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Graphene and Carbon Nanotubes. Ultrafast Optics and Relaxation Dynamics - Product Image

Graphene and Carbon Nanotubes. Ultrafast Optics and Relaxation Dynamics

  • Published: April 2013
  • Region: Global
  • 360 Pages
  • John Wiley and Sons Ltd

A first on ultrafast phenomena in carbon nanostructures like graphene, the most promising candidate for revolutionizing information technology and communication

The book introduces the reader into the ultrafast nanoworld of graphene and carbon nanotubes, including their microscopic tracks and unique optical finger prints. The author reviews the recent progress in this field by combining theoretical and experimental achievements. He offers a clear theoretical foundation by presenting transparently derived equations. Recent experimental breakthroughs are reviewed.

By combining both theory and experiment as well as main results and detailed theoretical derivations, the book turns into an inevitable source for a wider audience from graduate students to researchers in physics, materials science, and electrical engineering who work on optoelectronic devices, renewable energies, or in the semiconductor industry.

Preface XIII

1 Introduction – The Carbon Age 1

1.1 Graphene 3

1.2 Carbon Nanotubes 5

2 Theoretical Framework 9

2.1 Many-Particle Hamilton Operator 10

2.2 Microscopic Bloch Equations 11

2.2.1 Hartree–Fock Approximation 13

2.2.2 Second-Order Born–Markov Approximation 14

2.2.2.1 Detailed Balance 14

2.2.2.2 Markov Approximation 14

2.2.2.3 Coulomb-Induced Relaxation Channels 16

2.2.2.4 Phonon-Induced Scattering Rates 17

2.2.3 Many-Particle Dephasing 18

2.3 Electronic Band Structure of Graphene 19

2.3.1 Structure and Symmetry of Graphene 19

2.3.2 Tight-Binding Approach 21

2.4 Electronic Band Structure of Carbon Nanotubes 24

2.4.1 Structure and Symmetry of CNTs 24

2.4.1.1 Nanotube Symmetry 26

2.4.2 Zone-Folding Approximation 27

2.4.2.1 Helical Quantum Numbers 30

2.4.3 Nanotube Families 31

2.4.4 Trigonal Warping Effect 32

2.4.5 Density of States 33

2.5 Optical Matrix Element 34

2.5.1 Graphene 34

2.5.2 Carbon Nanotubes 36

2.6 Coulomb Matrix Elements 39

2.6.1 Graphene 40

2.6.2 Carbon Nanotubes: Regularized Coulomb Potential 41

2.7 Electron–Phonon Matrix Elements 44

2.7.1 Graphene: Kohn Anomalies 45

2.7.1.1 Optical Phonons 45

2.7.1.2 Acoustic Phonons 46

2.7.2 Carbon Nanotubes 47

2.8 Macroscopic Observables 47

2.8.1 Absorption Coefficient 48

2.8.2 Differential Transmission 49

3 Experimental Techniques for the Study of Ultrafast Nonequilibrium Carrier Dynamics in Graphene 51
Guest article by Stephan Winnerl

3.1 The Principle of Pump-Probe Experiments 51

3.1.1 Introduction to the Technique 52

3.1.2 Technical Realization of Pump-Probe Experiments 54

3.1.3 Temporal Resolution 56

3.1.4 Artifacts in Pump-Probe Signals 56

3.2 Characteristics of Short Radiation Pulses 58

3.2.1 The Fourier Limit 58

3.2.2 Auto-Correlation as a Technique to Characterize Short Radiation Pulses 59

3.2.3 Chirped Pulses 61

3.3 Sources of Short Infrared and Terahertz Radiation Pulses 63

3.3.1 The Titanium-Sapphire Laser 63

3.3.2 Optical Parametric Generation and Amplification 65

3.3.3 Difference-Frequency Generation 66

3.3.4 Generation of Single-Cycle Terahertz Radiation with Photoconductive Antennas 67

3.3.5 The Free-Electron Laser 69

3.3.6 Generation of a Femtosecond White-Light Continuum 71

3.4 Single-Color and Two-Color Pump-Probe Experiments on Graphene 72

3.4.1 Graphene Samples 72

3.4.2 Review of Single-Color Experiments 75

3.4.3 Review of Two-Color Experiments 76

Part One Electronic Properties – Carrier Relaxation Dynamics 83

4 Relaxation Dynamics in Graphene 85

4.1 Experimental Studies 85

4.1.1 High-Resolution Experiment in the Infrared 86

4.1.2 Pump-Probe Experiment Close to the Dirac Point 88

4.2 Relaxation Channels in Graphene 90

4.2.1 Coulomb-Induced Relaxation Channels 90

4.2.2 Auger Scattering Channels 91

4.2.3 Phonon-Induced Relaxation Channels 92

4.3 Optically Induced Nonequilibrium Carrier Distribution 93

4.4 Carrier Dynamics 96

4.4.1 Orientational Relaxation toward an Isotropic Carrier Distribution 98

4.4.2 Thermalization of the Excited Carrier System 102

4.4.3 Energy Dissipation and Carrier Cooling 103

4.4.3.1 Energy Dissipation 104

4.4.3.2 Temperature and Chemical Potential 105

4.4.4 Time- and Momentum-Resolved Relaxation Dynamics 107

4.4.5 Differential Transmission Spectra 108

4.5 Phonon Dynamics 110

4.5.1 Momentum-Resolved Dynamics 110

4.5.2 Time-Resolved Dynamics 113

4.5.3 Momentum- and Time-Resolved Dynamics 115

4.6 Pump Fluence Dependence 116

4.6.1 Thermalization 116

4.6.2 Isotropic Carrier Distribution 118

4.6.3 Phonon Occupation 119

4.6.4 Microscopic Polarization 120

4.6.5 Differential Transmission Spectra 121

4.6.6 Saturation Behavior 122

4.6.7 Temperature and Chemical Potential 125

4.6.8 Energy Density 126

4.7 Influence of the Substrate 127

4.8 Auger-Induced Carrier Multiplication 130

4.8.1 Coulomb-Induced Relaxation Dynamics 132

4.8.2 Influence of Phonon-Induced Recombination Processes 133

4.8.3 Pump-Fluence Dependence 134

4.8.4 Analytic Description of the Carrier Multiplication 135

4.9 Optical Gain 137

4.10 Relaxation Dynamics near the Dirac Point 141

5 Carrier Dynamics in Carbon Nanotubes 145

5.1 Experimental Studies 145

5.2 Phonon-Induced Relaxation Dynamics 148

5.2.1 Scattering via Optical Phonons 149

5.2.2 Scattering with Acoustic Phonons 152

5.2.3 Scattering Driven by Both Optical and Acoustic Phonons 152

5.2.4 Dependence on the Excitation Energy 154

5.2.5 Diameter and Chirality Dependence 155

5.2.6 Intersubband Relaxation Channels 156

5.3 Coulomb-Induced Quantum-Kinetic Carrier Dynamics 158

5.3.1 Non-Markov Relaxation Dynamics 159

5.3.2 Influence of the SurroundingMedium 162

5.3.3 Excitation-Induced Dephasing 162

Part Two Optical Properties – Absorption Spectra 165

6 Absorption Spectra of Carbon Nanotubes 167

6.1 Experimental Studies 167

6.1.1 Excitonic Absorption Spectra 168

6.1.2 Optical Assignment of Nanotubes 170

6.1.3 Functionalized Carbon Nanotubes 171

6.2 Absorption of Semiconducting Carbon Nanotubes 173

6.2.1 Free-Particle Spectra 173

6.2.2 Coulomb-Renormalized Spectra 176

6.2.3 Excitonic Spectra 178

6.2.4 Diameter and Chirality Dependence 180

6.2.4.1 Transition Energy 180

6.2.4.2 Excitonic Binding Energy 182

6.2.4.3 Oscillator Strength (Transition Intensity) 183

6.2.5 Excited Excitonic Transitions 187

6.2.6 Influence of the SurroundingMedium 189

6.3 Absorption of Metallic Carbon Nanotubes 192

6.3.1 Free-Particle Spectra 193

6.3.2 Excitonic Spectra 194

6.3.2.1 Comparison to Semiconducting Nanotubes 195

6.3.2.2 Trigonal Warping Effect 196

6.3.3 Diameter and Chirality Dependence 196

6.3.3.1 Transition Energy 197

6.3.3.2 Excitonic Binding Energy 198

6.3.3.3 Oscillator Strength 199

6.3.4 Rayleigh Scattering Spectra 201

6.3.5 Phonon-Induced Side-Peaks 204

6.4 Absorption of Functionalized Carbon Nanotubes 208

6.4.1 Spiropyran-Functionalized Nanotubes 209

7 Absorption Spectrum of Graphene 215

7.1 Experimental Studies 215

7.2 Absorbance and Conductivity in Graphene 216

7.2.1 Free-Particle Absorbance 218

7.2.2 Excitonic Absorbance 220

Appendix A Introduction to the Appendices 223

A.1 Microscopic Processes in Carbon Nanostructures 223

A.2 Outline of the Theoretical Description 225

Appendix B Observables in Optical Experiments 227

B.1 Temporal and Spectral Information in Measurements 227

B.2 Intensity-Related Optical Observables 230

B.2.1 Linear Optics 230

B.2.2 Nonlinear Signals 231

B.3 Specific Solutions of the Wave Equation for Graphene and Carbon Nanotubes 231

B.3.1 Normal Incidence on a Stack of Graphene Layers 232

B.3.1.1 Absorption in a Single Graphene Layer 234

B.3.1.2 Enhancement of the Radiation Coupling in a Stack of Many Graphene Layers 236

B.3.2 Light Propagation through Dispersed Carbon Structures 236

B.3.3 Light Scattering from Single Carbon Structures 238

B.4 Differential Transmission 239

Appendix C Second Quantization 241

C.1 Lagrange Formalism for Particles 242

C.1.1 Law of Least Action for Particle Dynamics 242

C.1.2 Lagrange Equations for Many Particles 244

C.2 Lagrange Formalism for Fields 246

C.2.1 Law of Least Action for Fields 246

C.2.2 Euler–Lagrange Equations for Fields 247

C.2.3 Schrödinger Field in a Potential 248

C.2.4 Maxwell Field 249

C.2.4.1 Free Static Electric Field 249

C.2.4.2 Free Electromagnetic Field in Vacuum 249

C.3 Quantization of Free Fields 251

C.3.1 General Scheme for Field Quantization 251

C.3.2 Quantization of the Free Schrödinger Field 255

C.3.3 Quantization of the Free Electromagnetic Field 257

C.3.4 Eigenvalue Problem for the Field Modes 259

C.4 Quantization of Interacting Fields 261

C.4.1 Classical Particles in the Maxwell Field 261

C.4.2 Interaction of Schrödinger and Maxwell Field 263

C.4.3 Interaction of Different Schrödinger Fields 265

C.5 Electron–Phonon Interaction in Second Quantization 266

C.5.1 Born–Oppenheimer Scheme 266

C.5.2 Electron–Phonon Coupling 268

C.6 Many-Particle Hamilton Operator 271

C.7 Electron–Light Interaction 272

C.8 Electrons and Phonons in Periodic Solid-State Structures 273

C.8.1 Electrons 273

C.8.2 Phonons 275

Appendix D Equations of Motion 279

D.1 Hierarchy Problem 279

D.2 Macroscopic Observables 281

D.3 The Relevant Density Operator 284

D.4 Treatment of a Bath 288

Appendix E Mean-Field and Correlation Effects 291

E.1 Mean-Field Contributions (Hartree–Fock) 291

E.2 Coulomb Correlations in an Equation of Motion Approach 294

E.2.1 Hartree–Fock Level 295

E.2.1.1 Free Particles 296

E.2.1.2 Coulomb Interaction 296

E.2.2 Second-Order Born–Markov Level 297

E.2.3 Screened Bloch Equations 301

E.3 Correlation Contributions: Electron–Phonon Interaction 303

E.4 A More Systematic Way to Correlation Effects: Screened Electron–Electron Interaction 306

References 313

Index 327

Ermin Malic graduated in Physics from Technical University (TU) Berlin. During his PhD thesis, he was a visiting researcher at the MIT and the University of Modena, Italy. From 2003 to 2008, he was a fellow of the Studienstiftung des Deutschen Volkes and the Friedrich-Ebert Stiftung. He received the DAAD and the Chorofas award for outstanding scientific research. After a post-doctoral stay at CIN2 in Barcelona, he is now leading the Einstein Junior Research Group on Microscopic Study of Carbon-based Hybrid Nanostructures at TU Berlin.. . Professor Andreas Knorr works in the field of nonlinear optics and quantum electronics of nanostructured solids. His research is focused on the interaction of light and matter, self-consistent solutions of Maxwell- and material equations and many body effects in open quantum systems. Since 2000 Andreas Knorr has a professorship at the Technical University of Berlin. His scientific career, which started at the Friedrich-Schiller-University Jena led him to the Universities of New Mexico, Arizona (College of Optical Sciences), Marburg, Göttingen and to Sandia National Labs Albuquerque and NTT Tokio.

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