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

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76 Rashba spin-orbit coupling in quantum dots PSfrag replacements 2.5 2 ×100 Γep in 10 −5 ω0 2 1.5 1 1 0 1 2 0.5 0 0 1 B/B 0 2 Figure 3.7: Phonon-induced relaxation rate for InAs quantum dot. Close to B = B 0 the rate is strongly suppressed to Γ ep < 10 −7 ω 0 ≈ 10 4 s −1 . to a strongly detuned system) the exponential decay of I(ξ) also suppresses the rate (see appendix C). The suppression of the relaxation rate close to the resonance can be traced back to the factor ξ 5 in Eq. (3.61) which in turn originates from the q-dependence of the matrix element (3.56). In a physical sense, this matrix element is a measure for how the coupling to phonons modifies the overlap of previously orthogonal states. Equations (B.4) and (B.5) in App. B show that the interaction with phonons leads to a displacement of the Fock–Darwin states in the QD. The effectiveness of this displacement on the orbital overlap manifests itself in the aforementioned q-dependence of Eq. (3.56), showing the overlap to be maximal at ˜lq ‖ ≈ 1. Thus, phonons couple most strongly to electrons if their wavelength is comparable to the size of the QD, or in terms of energy, if the typical phonon energy is of order ω s . Close to resonance, however, the splitting of eigenstates ∆ is given by the weak SO coupling, leading to ∆ ≪ ω s . In this regime phonons couple very inefficiently to the electrons of the QD – thus causing the relaxation rate to be suppressed. The factor sinθ + sinθ − in Eq. (3.56) corresponds to the misalignment of spin. Since the electron-phonon interaction (3.51) does not lead to direct spin flips, transitions between ψ ± are only possible due to the spin admixture of the JCM eigenstates. This admixture is maximal at resonance with sin 2 θ ± = 1/2. Far from resonance ψ +(−) are approximately spin-up(down) like, suppressing the transition rate by the factor sin 2 θ + sin 2 θ − . For a quantitative analysis we introduce acousto-mechanical parameters of bulk InAs [142], ρ M = 5.7 · 10 3 Kg/m 3 , c long = 3.8 · 10 3 m/s, c trans = 2.6 · 10 3 m/s,
3.3 Effects of relaxation 77 eh 14 = 3.5 · 10 6 eV/cm, leading to a piezo-electric coupling P = 3.0 · 10 −21 J 2 /m 2 by Eq. (3.60). For our QD size of l 0 = 150nm (corresponding to ω 0 = 10 11 Hz) the characteristic acoustic frequency is ω s = c/l 0 = 2.5·10 10 Hz. The dimensionless prefactor in Eq. (3.61) can be evaluated to mP/8π(ω s ) 2 l 0 ρ M ≈ 10 −3 . The magnetic field dependence of the phonon-induced relaxation rate is shown in Fig. 3.7 (see Fig. 3.3c for log-scale plot) for a SO coupling strength α = 1.5 × 10 −12 eVm. Close to the resonance the rate is strongly suppressed, Γ ep < 10 −7 ω 0 ≈ 10 4 s −1 . Comparison with the Rabi frequency of 2 GHz (see Sec. 3.2.4) shows that the robustness of pure spin qubits is not significantly weakened by the SO-induced hybridisation to the orbital degree of freedom. There are, of course, further sources of decoherence in any experimental realisation. For example, background charge fluctuations and noise in the gate voltages of the device affect the electrostatic definition of the quantum dot and hence the detuning δ. In addition, higher order (co-tunnelling) processes may scatter the states during the coherent evolution period. These additional mechanisms were also present in the experiment of Hayashi et al. [7], which demonstrated the general feasibility of observing coherent oscillations in quantum dots on a nanosecond time scale. The feasibility of our proposal is enhanced over and above this by the following considerations. Firstly, we work with a perpendicular magnetic field, and thus the confinement of the electron is not wholly electrostatic and thus more robust against charge/voltage fluctuations. Also, in the Hayashi experiment, a rather strong coupling to the leads is applied. Since we use weak coupling, the effects of co-tunnelling are further reduced. Finally, we point out that the counter-rotating terms which are neglected in the rotating-wave approximation in section 3.2.3, do not induce additional first order relaxation transitions. This is because counter-rotating terms couple to states which are beyond the two levels in a JCM-subspace. For comparison, the value for bulk GaAs is P GaAs = 5.4 · 10 −20 J 2 /m 2 .
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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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Page 31 and 32: PSfrag replacements 2.1 Rashba effe
Page 33 and 34: 2.2 Spectral properties, various li
Page 39 and 40: 2.3 Electron transport in one-dimen
Page 41 and 42: 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
Page 59 and 60: 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
Page 75 and 76: PSfrag replacements 3.2 Introductio
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: 3.3 Effects of relaxation 75 with q
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
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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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