Advanced Solid State Physics — Solutions
Manual
, Advanced Solid State Physics
Version 2009:
Michael Mayrhofer-R., Patrick Kraus, Christoph Heil, Hannes Brandner,
Nicola Schlatter, Immanuel Mayrhuber, Stefan Kirnstötter,
Alexander Volk, Gernot Kapper, Reinhold Hetzel
Version 2010: Martin Nuss
Version 2011: Christopher Albert, Max Sorantin, Benjamin Stickler
Tu Graz, Ss 2011
Lecturer: Univ. Prof. Ph.D. Peter Hadley
,TU Graz, SS 2010 Advanced Solid State Physics
Contents
1 What Is Solid State Physics 1
2 Schrödinger Equation 2
3 Quantization 3
3.1 Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3.1.1 Quantization Of The Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . 6
3.1.2 Quantization Of The Magnetic Flux . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.1.3 Quantisation Of A Charged Particle In A Magnetic Field (With Spin) . . . . . . . . 8
3.1.4 Dissipation ............................................................................................................................ 11
4 Quantization Of The Electromagnetic Field - Quantization Recipe 12
4.1 Thermodynamic Quantities ............................................................................................................. 17
4.1.1 Recipe For The Quantization Of Fields ............................................................................ 18
5 Photonic Crystals 19
5.1 Intruduction: Plane Waves In Crystals ........................................................................................ 19
5.2 Empty Lattice Approximation (Photons) ...................................................................................... 22
5.3 Central Equations ........................................................................................................................... 23
5.4 Estimate The Size Of The Photonic Bandgap ............................................................................ 24
5.5 Density Of States.............................................................................................................................. 25
5.6 Photon Density Of States ................................................................................................................ 26
6 Phonons 27
6.1 Linear Chain ..................................................................................................................................... 27
6.2 Simple Cubic .................................................................................................................................... 30
6.2.1 Raman Spectroscopy ........................................................................................................... 32
7 Electrons 34
7.1 Free Electron Fermi Gas ................................................................................................................. 34
7.1.1 Fermi Energy ....................................................................................................................... 34
7.1.2 Chemical Potential ............................................................................................................... 35
7.1.3 Sommerfeld Expansion ........................................................................................................ 36
7.1.4 Arpes..................................................................................................................................... 39
7.2 Electronic Band Structure Calculations ........................................................................................ 40
7.2.1 Empty Lattice Approximation ............................................................................................ 41
7.2.2 Plane Wave Method - Central Equations ......................................................................... 43
7.2.3 Tight Binding Model .......................................................................................................... 47
7.2.4 Graphene ............................................................................................................................... 55
7.2.5 Carbon Nanotubes ............................................................................................................... 57
7.3 Bandstructure Of Metals And Semiconductors............................................................................. 61
7.4 Direct And Indirect Bandgaps ........................................................................................................ 67
8 Crystal Physics 69
8.1 Stress And Strain ............................................................................................................................. 69
i
, TU Graz, SS 2010 Advanced Solid State Physics
8.2 Statistical Physics ............................................................................................................................ 70
8.3 Crystal Symmetries .......................................................................................................................... 72
8.4 Example - Birefringence .................................................................................................................. 75
9 Magnetism And Response To Electric And Magnetic Fields 76
9.1 Introduction: Electric And Magnetic Fields ............................................................................... 76
9.2 Magnetic Fields ................................................................................................................................ 76
9.3 Magnetic Response Of Atoms And Molecules.............................................................................. 78
9.3.1 Diamagnetic Response ........................................................................................................ 79
9.3.2 Paramagnetic Response ...................................................................................................... 81
9.4 Free Particles In A Weak Magnetic Field ..................................................................................... 81
9.5 Ferromagnetism ................................................................................................................................. 86
10 Transport Regimes 92
10.1 Linear Response Theory .................................................................................................................. 92
10.2 Ballistic Transport ............................................................................................................................ 94
10.3 Drift-Diffusion................................................................................................................................... 95
10.4 Diffusive And Ballistic Transport ................................................................................................... 96
10.5 Skin Depth ........................................................................................................................................ 99
11 Quasiparticles 102
11.1 Fermi Liquid Theory ...................................................................................................................... 102
11.2 Particle Like Quasiparticles ........................................................................................................... 103
11.2.1 Electrons And Holes ......................................................................................................... 103
11.2.2 Bogoliubov Quasiparticles ................................................................................................. 103
11.2.3 Polarons .............................................................................................................................. 103
11.2.4 Bipolarons ........................................................................................................................... 103
11.2.5 Mott Wannier Excitons ..................................................................................................... 103
11.3 Collective Modes ............................................................................................................................ 103
11.3.1 Phonons .............................................................................................................................. 103
11.3.2 Polaritons ............................................................................................................................ 104
11.3.3 Magnons ............................................................................................................................. 106
11.3.4 Plasmons ............................................................................................................................. 110
11.3.5 Surface Plasmons ............................................................................................................... 112
11.3.6 Frenkel Excitations............................................................................................................. 113
11.4 Translational Symmetry................................................................................................................. 113
11.5 Occupation....................................................................................................................................... 114
11.6 Experimental Techniques .............................................................................................................. 115
11.6.1 Raman Spectroscopy ......................................................................................................... 115
11.6.2 Eels ..................................................................................................................................... 115
11.6.3 Inelastic Scattering............................................................................................................. 115
11.6.4 Photoemission ..................................................................................................................... 115
12 Electron-Electron Interactions, Quantum Electronics 116
12.1 Electron Screening ......................................................................................................................... 116
12.2 Single Electron Effects................................................................................................................... 118
ii
Manual
, Advanced Solid State Physics
Version 2009:
Michael Mayrhofer-R., Patrick Kraus, Christoph Heil, Hannes Brandner,
Nicola Schlatter, Immanuel Mayrhuber, Stefan Kirnstötter,
Alexander Volk, Gernot Kapper, Reinhold Hetzel
Version 2010: Martin Nuss
Version 2011: Christopher Albert, Max Sorantin, Benjamin Stickler
Tu Graz, Ss 2011
Lecturer: Univ. Prof. Ph.D. Peter Hadley
,TU Graz, SS 2010 Advanced Solid State Physics
Contents
1 What Is Solid State Physics 1
2 Schrödinger Equation 2
3 Quantization 3
3.1 Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3.1.1 Quantization Of The Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . 6
3.1.2 Quantization Of The Magnetic Flux . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.1.3 Quantisation Of A Charged Particle In A Magnetic Field (With Spin) . . . . . . . . 8
3.1.4 Dissipation ............................................................................................................................ 11
4 Quantization Of The Electromagnetic Field - Quantization Recipe 12
4.1 Thermodynamic Quantities ............................................................................................................. 17
4.1.1 Recipe For The Quantization Of Fields ............................................................................ 18
5 Photonic Crystals 19
5.1 Intruduction: Plane Waves In Crystals ........................................................................................ 19
5.2 Empty Lattice Approximation (Photons) ...................................................................................... 22
5.3 Central Equations ........................................................................................................................... 23
5.4 Estimate The Size Of The Photonic Bandgap ............................................................................ 24
5.5 Density Of States.............................................................................................................................. 25
5.6 Photon Density Of States ................................................................................................................ 26
6 Phonons 27
6.1 Linear Chain ..................................................................................................................................... 27
6.2 Simple Cubic .................................................................................................................................... 30
6.2.1 Raman Spectroscopy ........................................................................................................... 32
7 Electrons 34
7.1 Free Electron Fermi Gas ................................................................................................................. 34
7.1.1 Fermi Energy ....................................................................................................................... 34
7.1.2 Chemical Potential ............................................................................................................... 35
7.1.3 Sommerfeld Expansion ........................................................................................................ 36
7.1.4 Arpes..................................................................................................................................... 39
7.2 Electronic Band Structure Calculations ........................................................................................ 40
7.2.1 Empty Lattice Approximation ............................................................................................ 41
7.2.2 Plane Wave Method - Central Equations ......................................................................... 43
7.2.3 Tight Binding Model .......................................................................................................... 47
7.2.4 Graphene ............................................................................................................................... 55
7.2.5 Carbon Nanotubes ............................................................................................................... 57
7.3 Bandstructure Of Metals And Semiconductors............................................................................. 61
7.4 Direct And Indirect Bandgaps ........................................................................................................ 67
8 Crystal Physics 69
8.1 Stress And Strain ............................................................................................................................. 69
i
, TU Graz, SS 2010 Advanced Solid State Physics
8.2 Statistical Physics ............................................................................................................................ 70
8.3 Crystal Symmetries .......................................................................................................................... 72
8.4 Example - Birefringence .................................................................................................................. 75
9 Magnetism And Response To Electric And Magnetic Fields 76
9.1 Introduction: Electric And Magnetic Fields ............................................................................... 76
9.2 Magnetic Fields ................................................................................................................................ 76
9.3 Magnetic Response Of Atoms And Molecules.............................................................................. 78
9.3.1 Diamagnetic Response ........................................................................................................ 79
9.3.2 Paramagnetic Response ...................................................................................................... 81
9.4 Free Particles In A Weak Magnetic Field ..................................................................................... 81
9.5 Ferromagnetism ................................................................................................................................. 86
10 Transport Regimes 92
10.1 Linear Response Theory .................................................................................................................. 92
10.2 Ballistic Transport ............................................................................................................................ 94
10.3 Drift-Diffusion................................................................................................................................... 95
10.4 Diffusive And Ballistic Transport ................................................................................................... 96
10.5 Skin Depth ........................................................................................................................................ 99
11 Quasiparticles 102
11.1 Fermi Liquid Theory ...................................................................................................................... 102
11.2 Particle Like Quasiparticles ........................................................................................................... 103
11.2.1 Electrons And Holes ......................................................................................................... 103
11.2.2 Bogoliubov Quasiparticles ................................................................................................. 103
11.2.3 Polarons .............................................................................................................................. 103
11.2.4 Bipolarons ........................................................................................................................... 103
11.2.5 Mott Wannier Excitons ..................................................................................................... 103
11.3 Collective Modes ............................................................................................................................ 103
11.3.1 Phonons .............................................................................................................................. 103
11.3.2 Polaritons ............................................................................................................................ 104
11.3.3 Magnons ............................................................................................................................. 106
11.3.4 Plasmons ............................................................................................................................. 110
11.3.5 Surface Plasmons ............................................................................................................... 112
11.3.6 Frenkel Excitations............................................................................................................. 113
11.4 Translational Symmetry................................................................................................................. 113
11.5 Occupation....................................................................................................................................... 114
11.6 Experimental Techniques .............................................................................................................. 115
11.6.1 Raman Spectroscopy ......................................................................................................... 115
11.6.2 Eels ..................................................................................................................................... 115
11.6.3 Inelastic Scattering............................................................................................................. 115
11.6.4 Photoemission ..................................................................................................................... 115
12 Electron-Electron Interactions, Quantum Electronics 116
12.1 Electron Screening ......................................................................................................................... 116
12.2 Single Electron Effects................................................................................................................... 118
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