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

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88 Introduction to electron-phonon interaction with the transversal and longitudinal velocities of sound c t and c l , respectively. For bulk media follows that the eigenmodes are given by plane waves (sq), u(r,t) = ∑û s (q)e i(q·r−ω sq t) , (5.13) s,q where s = {l,t} denotes the different longitudinal and transversal polarisations, and û s (q) is the corresponding polarisation vector. With proper normalisation one can show that the bulk matrix elements (5.6) for the absorption of a phonon (sq) via deformation and piezo-electric potentials are given by [139] M def lk→l ′ k ′ = δ k ′ ,k+q M pz lk→l ′ k ′ = δ k ′ ,k+q [ ] 1 1 2 √ √ iΞ q, (5.14) V 2ρ M c s [ ] 1 1 2 √ V 2ρ M c s eβ √ q . (5.15) We mention two important points. (i) The phase difference of π/2 between the matrix elements shows that the piezo-electric and deformation potentials are indeed independent scattering mechanisms, hence they do not interfere. (ii) Comparison of the amplitudes of the matrix elements yields, |M pz | 2 ( ) eβ 2 |M def | 2 = . (5.16) Ξq Thus, in piezo-electric bulk materials, the scattering of electrons by long wavelength (q → 0) acoustic phonons is typically dominated by the piezo-electric coupling potential.
Chapter 6 Coupled quantum dots in a phonon cavity Quantum coherence has become a major issue in the study of electronic transport of low-dimensional artificial structures. It can be observed on length scales of the order of or smaller than the dephasing distance. The latter is the mean distance an electron can propagate within the dephasing time which is determined by scattering processes that destroy the quantum phase. Generally, the dephasing time depends on temperature. For high temperatures, phase breaking scatterings are so frequent that quantum signatures are completely absent in the electron transport. Only at sufficiently low temperature, scattering events become rare. Then, quantum coherence is maintained over long periods of time such that interference effects can be observed in the current transport. An important fundamental question is whether or not one can control the coupling to phase breaking modes “coherently” such that interference effects due to the coupling itself become experimentally accessible. Recently, the coupling of semiconductor quantum dots [145–148] has turned out to be a promising method for preparing and controlling superpositions of electronic states. If two dots are coupled to each other and to external leads, Coulomb blockade guarantees that only one additional electron at a time can tunnel between the dots and the leads. Dephasing at very low temperature then is energetically possible only by bosonic low-energy excitations of the environment. It has turned out that the coupling to low-energetic phonons governs the dynamics of double quantum dots (DQDs) [147, 148] since they are essentially realisations of twolevel systems within a semiconductor host material [149]. Therefore, a logical step towards the control of dephasing in DQDs is the control of the vibrational properties of such structures. Enormous progress has been made in the fabrication of partly suspended or free-standing nanostructures (“phonon cavities”) [150–153]. These considerably differ in their mechanical 89
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Interaction and confinement in nano
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Abstract iii Abstract It is the pur
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Publications v Publications Some of
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2 Table of Contents 3 Rashba spin-o
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4 Introduction (phonons). Therefore
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6 Introduction Figure 1: Scanning e
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Chapter 1 The Rashba effect Effects
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1.2 The model 11 the position-depen
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1.3 Rashba effect in a perpendicula
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Chapter 2 Rashba spin-orbit couplin
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2.1 Rashba effect and magnetic fiel
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PSfrag replacements 2.1 Rashba effe
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2.2 Spectral properties, various li
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2.3 Electron transport in one-dimen
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PSfrag replacements 2.3 Electron tr
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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
Page 57 and 58: 3.1 Spin-orbit driven coherent osci
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Page 65 and 66: 3.2 Introduction to quantum dots an
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Page 77 and 78: 3.3 Effects of relaxation 71 page 5
Page 79 and 80: 3.3 Effects of relaxation 73 The re
Page 81 and 82: 3.3 Effects of relaxation 75 with q
Page 83 and 84: 3.3 Effects of relaxation 77 eh 14
Page 85 and 86: Chapter 4 Conclusion In this part o
Page 87 and 88: Tuneable quantum dots connected to
Page 89: Part II Phonon confinement in nanos
Page 92 and 93: 86 Introduction to electron-phonon
Page 96 and 97: 90 Coupled quantum dots in a phonon
Page 102 and 103: ) ) ) ) 3 4 + ) * - 96 Coupled quan
Page 106 and 107: 100 Coupled quantum dots in a phono
Page 110 and 111: PSfrag replacements 1042 Coupled qu
Page 116 and 117: 110 Conclusion due to phonon van-Ho
Page 119 and 120: Appendix A Jaynes-Cummings model Th
Page 121 and 122: Appendix B Fock-Darwin representati
Page 123 and 124: Appendix C Evaluation of the phonon
Page 125 and 126: Appendix D Matrix elements of dot e
Page 127 and 128: 121 with λ ± l (q ‖) = F n B pz
Page 129 and 130: Bibliography [1] G. Bergmann, Phys.
Page 131 and 132: Bibliography 125 [31] M. Pletyukhov
Page 133 and 134: Bibliography 127 [72] T. Schäpers,
Page 135 and 136: Bibliography 129 [109] U. Merkt, J.
Page 137 and 138: Bibliography 131 [144] B. Auld, Aco
Page 139 and 140: Bibliography 133 [182] A. A. Kisele
Page 141: Acknowledgements Finally, I would l
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