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

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74 Rashba spin-orbit coupling in quantum dots The potential which is induced by a bulk acoustic phonon with (three-dimensional) wave vector q is given by [139] V ep (q) = λ q e iq·r( ) b q + b † −q , (3.51) with the phonon operators b q and b † q, and the coupling parameter λ q depending on the mechanism of electron-phonon interaction, e.g. deformation potential or piezo-electric coupling. § In the following, we are interested in phonon-induced transitions from the upper to the lower eigenstate within a JCM subspace, representing excited and ground state of the qubit under consideration, Ψ + → Ψ − , with Ψ ± given by Eq. (3.29). Since the electron-phonon interaction does not depend on the spin, only transition matrix elements which are diagonal in spin space contribute, 〈Ψ + |V ep (q)|Ψ − 〉 =cosθ + cosθ − 〈n + ,n − ,↑ |V ep (q)|n + ,n − ,↑〉 + sinθ + sinθ − 〈n + ,n − + 1,↓ |V ep (q)|n + ,n − + 1,↓〉. (3.52) In appendix B, matrix elements of the electron-phonon interaction are calculated in the Fock–Darwin basis {|n + ,n − ,σ〉}, leading to 〈Ψ + |V ep (q)|Ψ − ( 〉 = λ q L n+ |α + q | 2) e − 2(|α 1 + q | 2 +|α − q | 2 ) [ ( × cosθ + cosθ − L n− |α − q | 2) + sinθ + sinθ − L n− +1( |α − q | 2)] , (3.53) with the Laguerre polynomials L n and complex phonon wave vectors α ± q := ±˜l(q y ± iq x )/2. For the ease of computation we concentrate on the lowest qubit, n + = n − = 0. Generalisation to higher qubits is straightforward. With L 0 (x) = 1 and L 1 (x) = 1 − x we have 〈Ψ + 0 |V ep(q)|Ψ − 0 〉 = λ q e − 1 2(|α + q | 2 +|α − q | 2 ) [ cosθ + cosθ − + sinθ + sinθ − − |α − q | 2 sinθ + sinθ − ]. (3.54) The q-independent summand on the right vanishes because of the orthogonality of Ψ ± 0 , cosθ + cosθ − + sinθ + sinθ − = 〈Ψ + 0 |Ψ− 0 〉 = 0, (3.55) leading to 〈Ψ + 0 |V ep(q)|Ψ − 0 〉 = −λ (˜lq ‖ ) 2 q sinθ + sinθ − e − 4(˜lq 1 ‖ ) 2 , (3.56) 4 § An introduction to electron-phonon interaction is given in chapter 5.
3.3 Effects of relaxation 75 with q ‖ = (q 2 x + q 2 y) 1/2 . The phonon-induced transition rate can be calculated in first order by Fermi’s golden rule, Γ ep = 2π ∑ q ∣ ∣〈Ψ + 0 |V ep(q)|Ψ − 0 〉∣ ∣ 2 δ(E + − E − − ω q ) (3.57) = V Z (2π) 2 d 3 q ∣ ∣〈Ψ + 0 |V ep(q)|Ψ − 0 〉∣ ∣ 2 δ(∆ − ω q ), (3.58) with ∆ = E + − E − , the volume of the system V , and the phonon frequency ω q . For low temperatures the electron-phonon interaction is dominated by the piezo-electric coupling to long-wavelength bulk (3D) acoustic phonons, see chapter 5. We follow Ref. [111, 140] and apply an angular average of the anisotropic piezo-electric modulus for zinc blende crystal structures, leading to the coupling parameter [141] |λ q | 2 = 1 V λ 2 ph cq , λ2 ph = P 2ρ M , (3.59) where c is the longitudinal velocity of sound, ω q = cq = c(q 2 ‖ + q2 z ) 1/2 is the phonon dispersion, ρ M the mass density of the semiconductor, and P the averaged piezo-electric coupling [140] ( P =(eh 14 ) 2 12 35 + 1 ) 16 , (3.60) x 35 with the piezo-electric constant eh 14 and the sound velocity ratio x = c trans /c long . With this model of the electron-phonon interaction, the phonon-induced relaxation rate Eq. (3.58) can be written as (see appendix C) √ Γ ep mP 2l0 = ω 0 8π(ω s ) 2 sin l 0 ρ M ˜l 2 θ + sin 2 θ − ξ 5 I(ξ), (3.61) with the characteristic acoustic frequency ω s = c/l 0 and the ratio ξ = 2 −1/2 (˜l/l 0 )(∆/ω s ). The function I(ξ) = R 1 0 dt exp[−(ξt) 2 ]t 5 /(1 − t 2 ) 1/2 is evaluated in appendix C and can be estimated by I(ξ) ≤ 8/15 for ξ < 1. The rotating-wave approximation (RWA) which is applied to derive the effective model in Sec. 3.2.3 requires the SO coupling to be small compared to the confinement. In addition, to observe coherent oscillations the system needs to be driven close to resonance (see Sec. 3.2.4). In this regime, the energy ratio is ∆/ω s ≪ 1, leading to ξ ≪ 1, and thus suppressing the phonon-induced relaxation due to Γ ep ∝ ξ 5 , see Eq. (3.61). On the contrary, for ξ ≫ 1 (corresponding
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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 33 and 34: 2.2 Spectral properties, various li
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Page 51 and 52: 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 65 and 66: 3.2 Introduction to quantum dots an
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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.
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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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