Spin-orbit coupling and electron-phonon scattering - Fachbereich ...

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2 Table of Contents 3 Rashba spin-orbit coupling in quantum dots 49 3.1 Spin-orbit driven coherent oscillations in a few-electron quantum dot (Publication) . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.2 Introduction to quantum dots and various derivations . . . . . . . 58 3.2.1 Introduction to few-electron quantum dots . . . . . . . . . 58 3.2.2 Spin-orbit effects in quantum dots . . . . . . . . . . . . . 62 3.2.3 Derivation of the effective model . . . . . . . . . . . . . . 63 3.2.4 Coherent oscillations . . . . . . . . . . . . . . . . . . . . 66 3.2.5 The current . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.3 Effects of relaxation . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.3.1 Effects of relaxation on coherent oscillations . . . . . . . 72 3.3.2 Phonon induced relaxation rates . . . . . . . . . . . . . . 73 4 Conclusion 79 II Phonon confinement in nanostructures 83 5 Introduction to electron-phonon interaction 85 6 Coupled quantum dots in a phonon cavity 89 6.1 Control of dephasing and phonon emission in coupled quantum dots (Publication) . . . . . . . . . . . . . . . . . . . . . . . . . . 90 6.2 Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 6.2.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 6.2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 6.2.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 106 7 Conclusion 109 Appendices 113 A Jaynes–Cummings model 113 B Fock–Darwin representation of electron-phonon interaction 115 C Evaluation of the phonon-induced relaxation rate 117 D Matrix elements of dot electron-confined phonon interaction 119 Bibliography 123
Introduction With the immense technological progress in the field of nanoprocessing in the last two decades, it is feasible to fabricate high-precision nanostructured electronic devices in semiconductors. In such artificial structures, the length scales of the system may become comparable or even smaller than the dephasing distance. The latter is the average length an electron can propagate before its quantum mechanical phase becomes destroyed by some process. Therefore, the quantum mechanical behaviour of the electrons manifests itself in striking quantum interference phenomena in the properties of the nanostructures. Examples are the weak localisation quantum corrections to the conductance of disordered films [1] and the Aharonov–Bohm oscillations in the magneto-transport of tiny ring structures [2]. In addition, in clean samples electrons can propagate large distances without being scattered at imperfections. In GaAs/GaAlAs semiconductor heterostructures, a mean free path (average distance between successive scattering events) of several µm can be reached at low temperatures [3]. Thus, in such nanostructures, electron propagation is often well described in a ballistic picture. A prominent example for ballistic transport in nanostructures is the quantisation of conductance through small constrictions (quantum point contacts) [4, 5]. Recently, the observation of coherent oscillations in the time evolution of a quantum state in a Josephson junction [6] or coupled semiconductor quantum dots [7] has been achieved. These oscillations – the back and forth flopping between two states in a quantum mechanical superposition – directly show quantum mechanics at work. The observation of coherent oscillations in solid-state systems represents the frontier in our ability to control nature at a microscopic level. This effort goes hand-in-hand with the search for workable quantum bits (qubits) and dream of quantum computation. The above examples are well described in an effective single-particle model which treats the electrons (or Cooper pairs as in case of the Josephson junction) as non-interacting particles. However, this picture has some limitations. For instance, the coherent oscillations can only be traced for times smaller than the dephasing time. In general, the quantum phase of an electron will be randomised by inelastic scattering events with e.g. other electrons or with lattice vibrations 3
Page 1 and 2: Interaction and confinement in nano
Page 3 and 4: Abstract iii Abstract It is the pur
Page 5: Publications v Publications Some of
Page 10 and 11: 4 Introduction (phonons). Therefore
Page 12 and 13: 6 Introduction Figure 1: Scanning e
Page 15 and 16: Chapter 1 The Rashba effect Effects
Page 17 and 18: 1.2 The model 11 the position-depen
Page 19 and 20: 1.3 Rashba effect in a perpendicula
Page 21 and 22: Chapter 2 Rashba spin-orbit couplin
Page 23 and 24: 2.1 Rashba effect and magnetic fiel
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Page 33 and 34: 2.2 Spectral properties, various li
Page 39 and 40: 2.3 Electron transport in one-dimen
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Page 51 and 52: 2.4 Summary 45 For weak SO coupling
Page 53 and 54: 2.4 Summary 47 splitting can become
Page 55 and 56: Chapter 3 Rashba spin-orbit couplin
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3.1 Spin-orbit driven coherent osci
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0 0.5 1.5 3.1 Spin-orbit driven coh
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3.2 Introduction to quantum dots an
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PSfrag replacements 3.2 Introductio
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3.3 Effects of relaxation 71 page 5
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3.3 Effects of relaxation 73 The re
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3.3 Effects of relaxation 75 with q
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3.3 Effects of relaxation 77 eh 14
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Chapter 4 Conclusion In this part o
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Tuneable quantum dots connected to
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Part II Phonon confinement in nanos
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86 Introduction to electron-phonon
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88 Introduction to electron-phonon
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90 Coupled quantum dots in a phonon
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) ) ) ) 3 4 + ) * - 96 Coupled quan
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100 Coupled quantum dots in a phono
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PSfrag replacements 1042 Coupled qu
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110 Conclusion due to phonon van-Ho
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Appendix A Jaynes-Cummings model Th
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Appendix B Fock-Darwin representati
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Appendix C Evaluation of the phonon
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Appendix D Matrix elements of dot e
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121 with λ ± l (q ‖) = F n B pz
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Bibliography [1] G. Bergmann, Phys.
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Bibliography 125 [31] M. Pletyukhov
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Bibliography 127 [72] T. Schäpers,
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Bibliography 129 [109] U. Merkt, J.
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Bibliography 131 [144] B. Auld, Aco
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Bibliography 133 [182] A. A. Kisele
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Acknowledgements Finally, I would l
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