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

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92 Coupled quantum dots in a phonon cavity Figure 6.1: Scheme for lateral (L) and vertical (V) configurations of a double quantum dot in a phonon nano-cavity. states, typical properties [144, 172] of a nano-size slab such as phonon-subband quantisation can be detected in the staircase-like electronic current I(ε) though the dots. In addition, and very strikingly, for certain wave vectors we find negative phonon group velocities and phonon van Hove singularities close to which one can strongly excite characteristic phonon modes with specific emission patterns. The controlled enhancement or reduction of spontaneous light emission from atoms is well-known in cavity QED [173]. Here a single, confined photon mode can be tuned on or off resonance with an atomic transition frequency. In contrast to cavity QED, the vanishing of spontaneous emission in phonon cavities is due to real zeros in the phonon deformation potential or polarisation fields rather than gaps in the density of states. This is a peculiar consequence of the boundary conditions for vibration modes which lead to complicated nonlinear dispersions even for homogeneous slabs. As a result, phonon cavities confined in only one spatial direction support a continuous spectrum, and yet suppression of spontaneous emission is possible. As a model, we consider two tunnel-coupled quantum dots embedded in an infinite semiconductor slab of thickness 2b in a vertical or a lateral configuration, Fig. 6.1. The dots are weakly coupled to external leads, and we assume that both the energy difference ε and the coupling strength T c between the dots can be externally tuned by gate voltages. In the Coulomb blockade regime, we adopt the usual description in terms of three many body ground states [149, 174] |0〉, |L〉, and |R〉 that have no or one additional electron in either of the dots, respectively. The coupling to the phonon environment of the slab is described by an effective spin-boson Hamiltonian H = ε 2 σ z + T c σ x +∑ω q a † qa q q + ∑(α q n L + β q n R ) q ( ) a q + a † −q , (6.1)
# $ ( ' 6.1 Control of dephasing and phonon emission in coupled quantum dots 93 "! %$& Figure 6.2: Deformation potential induced by dilatational (left) and flexural modes (centre) at q ‖ b = π/2 (n = 2 subbands). Right: displacement field u(x,z) of n = 0 dilatational mode at ∆ = ω 0 . Greyscale: moduli of deformation potentials (left) and displacement fields (right) (arb. units). with σ z = |L〉〈L|−|R〉〈R|, σ x = |L〉〈R|+|R〉〈L|, n i = |i〉〈i| and α q (β q ) the coupling matrix element between electrons in dot L(R) and phonons with dispersion ω q . The stationary current can be calculated by using a master equation [149] and considering T c as a perturbation. We consider weak electron-phonon coupling and calculate the inelastic scattering rate γ(ω) = 2π∑ q Tc 2 |α q − β q | 2 2 ω 2 δ(ω − ω q ). (6.2) For ω = (ε 2 + 4T 2 c ) 1/2 this is the rate for spontaneous emission at zero temperature due to electron transitions from the upper to the lower hybridised dot level. On the other hand, in lowest order in T c , the total current I(ε) can be decomposed into an elastic Breit-Wigner type resonance and an inelastic component I in (ε) ≈ −eγ(ε), where −e is the electron charge. The double dot, supporting an inelastic current I in (ε), therefore can be regarded as an analyser of the phonon system [147, 148]. One can also consider the double dot as an emitter of phonons of energy ω at a tunable rate γ(ω). We show below how the phonon confinement within the slab leads to steps in I in (ε) and tunable strong enhancement or nearly complete suppression of the electron-phonon coupling. We describe phonons by a displacement field u(r) which is determined by the vibrational modes of the slab [143]. For the following, it is sufficient to consider dilatational and flexural modes (Lamb waves). The symmetries of their displacement fields differ with respect to the slab’s mid-plane. They either yield a
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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 63 and 64: 0 0.5 1.5 3.1 Spin-orbit driven coh
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 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
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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