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

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100 Coupled quantum dots in a phonon cavity Following Ref. [149], for T = 0 the stationary inelastic current in lowest order in T c is given by the frequency-depending inelastic scattering rate, I inel ≈ −eγ, with ρ eff (ω) = ∑ q γ(ω) ≈ 2πT 2 c ρ eff (ω), (6.15) |α q − β q | 2 2 ω 2 δ(ω − ω q ), (6.16) where ρ eff is an effective density of states (DOS) of the phonon environment. Every state is weighted by the electron-phonon interaction through the interference term |α q − β q | 2 / 2 ω 2 . Brandes & Kramer [149] utilised the interference in the coupling of the individual dots in Eq. (6.15) to explain the oscillatory behaviour of the inelastic current which was found in Ref. [147]. Equation (6.15) shows the direct relation between inelastic current and DOS of the phonon environment. Therefore, a DQD is an ideal tool to investigate peculiar phonon systems like e.g. nanomechanical resonators by means of electron transport. In this section we investigate the influence of a mechanical confinement of the elastic medium on the transport characteristics of an embedded DQD. Scattering rates for electron-phonon interaction in phonon cavities have been investigated by Stroscio et al. [175, 176, 182]. One of the systems they considered was a free-standing quantum well (FSQW) in a slab geometry (see Fig. 6.4) which is confined in the z-direction and widely extended in x- and y-directions. In the following, we use a FSQW with area A ≫b 2 as a model for a phonon cavity. We describe vibrations by a displacement field u(r,t). Assuming a homogenous, isotropic and linear medium of the FSQW, the general solution for the displacement field u(r,t) is determined by four families of modes – two families of shear waves (vertically and longitudinally polarised), and two families of Lamb waves ‡ (dilatational and flexural waves). Stroscio et al. showed that electron scattering by shear waves can be neglected since these modes do not interact with electrons via the deformation potential interaction which is the dominant coupling mechanism in small FSQW [176]. Therefore, in the following, we restrict ourselves to the coupling to dilatational and flexural modes. A further effect of the phonon confinement is the spectral quantisation in subbands familiar from other confined systems. For each fixed in-plane wave vector q ‖ there are infinitely many modes related to a discrete set of transversal wave vectors pointing in the direction of the confinement. Since there are two velocities of sound in the elastic medium, related to longitudinal and transversal wave propagation, c l and c t , there are also two transversal wave vectors which we will denote as q l and q t . This is in contrast to the unconfined bulk case where one can ‡ See Ref. [176] and references therein.
6.2 Details 101 r z r PSfrag replacements r ‖ y x z 2b Figure 6.4: Free-standing quantum well (FSQW) as a model for a phonon cavity. The slab is confined in z-direction and widely extended in x- and y-direction. separate both polarisations. In the confined geometry of the FSQW, the boundary conditions at the surface of the slab lead to a coupling between longitudinal and transversal propagation [172]. The relation between the in-plane wave vector component q ‖ and the transversal components q l and q t for Lamb waves is determined by the so-called Rayleigh– Lamb equations, tanq t,n b tanq l,n b = − [ 4q 2 ‖ q l,n q t,n (q 2 ‖ − q2 t,n) 2 ] α , (6.17) [ω n (q ‖ )] 2 = c 2 l (q2 ‖ + q2 l,n ) = c2 t (q 2 ‖ + q2 t,n), (6.18) where α = +1 for dilatational modes and α = −1 for flexural modes. The integer n = 0,1,2,... denotes the phonon subband index. The second Rayleigh–Lamb equation is the dispersion relation of confined phonons. An analytical solution of the Eqs. (6.17) and (6.18) is in general not feasible, thus a numerical approach has to be applied [172, 175, 182, 183]. Once the solutions q l,n (q ‖ ) and q t,n (q ‖ ) are known, one can calculate the displacement field associated with a confined phonon in a Lamb mode (n,q ‖ ) that propagates in the x-direction [176], u n (q ‖ ,z) = (u n,x ,0,u n,z ), (6.19)
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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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PSfrag replacements 2.3 Electron tr
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2.4 Summary 45 For weak SO coupling
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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 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 108 and 109: 102 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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