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

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98 Coupled quantum dots in a phonon cavity mechanisms such as coupling to electronic excitations in the leads. In contrast to cavity QED, where a single, confined photon mode can be tuned on or off resonance with an atomic transition frequency, the vanishing of spontaneous emission in phonon cavities is due to zeros in the phonon deformation potential or polarisation fields rather than gaps in the density of states. In addition, we found that phonon emission into characteristic modes can be enormously enhanced due to van Hove singularities that could act as strong fingerprints of the phonon confinement if experimentally detected. This work was supported by the EU via TMR and RTN projects FMRX- CT98-0180 and HPRN-CT2000-0144, DFG projects Kr 627/9-1, Br 1528/4-1, and project EPSRC R44690/01. Discussions with R. H. Blick, T. Fujisawa, W. G. van der Wiel, and L. P. Kouwenhoven are acknowledged. 6.2 Details ∗ 6.2.1 Model In this section, we derive the low-temperature current-voltage characteristic of a double quantum dot (DQD) which interacts with a bath of confined phonons in the non-linear transport regime. We assume that the DQD is weakly coupled to leads. The parameters are chosen such that the transport is dominated by Coulomb blockade. Following Ref. [149], we describe the DQD as a two-level system consisting of one additional electron in either the left, |L〉 = |N +1,M〉, or the right dot, |R〉 = |N,M +1〉, associated with the energies E L and E R . The state |0〉 = |N,M〉 denotes the N + M electrons many body ground state of the double dot. Using this basis set for the electron states and defining the operators n L = |L〉〈L|, n R = |R〉〈R|, p = |L〉〈R|, (6.6) s L = |0〉〈L|, s R = |0〉〈R|, (6.7) the Hamiltonian is H = H 0 + H T + H ld + H ep , (6.8) ∗ Parts of the following results have been published in: S. Debald, T. Vorrath, T. Brandes, and B. Kramer, Phonons and Phonon Confinement in Transport through Double Quantum Dots, Proc. 25th Int. Conf. Semicond., Osaka (2000); T. Vorrath, S. Debald, B. Kramer, and T. Brandes, Phonon Cavity Models for Quantum Dot Based Qubits, Proc. 26th Int. Conf. Semicond., Edinburgh (2002); S. Debald, T. Brandes, and B. Kramer, Nonlinear Electron Transport through Double Quantum Dots Coupled to Confined Phonons, Int. Journal of Modern Physics B 17, 5471 (2003).
6.2 Details 99 with the Hamiltonian of the decoupled components H 0 = E L n L + E R n R + H p + H leads , (6.9) the tunnelling Hamiltonian between the dots H T = T c (p + p † ), (6.10) the phonon system the Hamiltonian of the leads H p = ∑ω q a † qa q , (6.11) q the electron-phonon coupling H leads = ∑(ε L k c† k c k + ε R k d† k d k), (6.12) k H ep = ∑(α q n L + β q n R )(a q + a † −q), (6.13) q and the coupling between the leads and the dots H ld = ∑(V k c † k s L +W k d † k s R + h.c.). (6.14) k The coupling between the two dots is parametrised with the tunnel matrix element T c . The electronic reservoirs of the leads (H leads ) and the phonon environment (H p ) are assumed to be in thermal equilibrium. H ep describes the interaction † of the electrons in the dots with acoustic phonons, α q and β q are the matrix elements of the interaction potential in the representation of |L〉 and |R〉 (see appendix D). Introduction of pseudo-spin operators σ x and σ z leads to the effective spin-boson Hamiltonian (6.1) with ε = E L − E R . Applying a dc bias between source and drain, the current through the DQD is determined by the expectation value of the current operator Î = ieT c (p − p † ). The current as a function of the energy difference of the dot levels consists of two parts [147, 148], I = I el + I inel . The elastic contribution I el which arises from resonant tunnelling [181] is dominating the transport around ε = 0. For ε > 0, the current is determined by inelastic contributions due to spontaneous emission of acoustic phonons [147, 148]. † The influence of the non-diagonal part of the electron-phonon interaction has been shown to be negligible [149, 180].
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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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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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2.4 Summary 45 For weak SO coupling
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Page 55 and 56: Chapter 3 Rashba spin-orbit couplin
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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
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Page 89: Part II Phonon confinement in nanos
Page 92 and 93: 86 Introduction to electron-phonon
Page 94 and 95: 88 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
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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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