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

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PSfrag replacements 1042 Coupled quantum dots in a phonon cavity 6 10 0 ω in µeV 60 120 14 Phonon DOS ν d/f in a.u. 18 600 400 200 ν d (ω) ν f (ω) 0 0 4 8 12 16 ω in ω b Figure 6.7: Thermodynamical density of states ν d/f of dilatational and flexural phonons. The arrows mark van–Hove singularities. Upper energy scale corresponds to a GaAs cavity of width 2b = 1µm, leading to ω b = c l /b = 7.5µeV. for electron-confined phonon interaction via deformation potential (DP) and piezoelectric (PZ) coupling. The interference is influenced by the orientation of the DQD in the phonon cavity. From Eqs. (D.6) and (D.11) follows the orientation dependence for coupling to Lamb modes ⎧ ⎪⎨ |α n (q ‖ ) − β n (q ‖ )| 2 ∝ ⎪⎩ ∣ ∣tcs ( 1 2 q l,n d sinΘ ) ( 1 ∓ e iq ‖·d )∣ ∣ ∣ 2 ∣ ∣tsc ( 1 2 q l/t,n d sinΘ ) ( 1 ± e iq ‖·d )∣ ∣ ∣ 2 for DP, for PZ, (6.24) where the upper (lower) sign, and tcsx = cosx (or sinx), holds for dilatational (flexural) modes; tscx follows by exchanging sinx and cosx. The vector d connects both dots (see Fig. 6.8), Θ gives the orientation of the dots in the cavity with Θ = 0,(π/2) corresponding to the lateral (vertical) configurations in Fig. 6.1. The different symmetries of the DP and PZ potentials with respect to the mid-plane of the cavity (appendix D) manifest themselves in the tcsx and tscx terms in Eq. (6.24). For instance, dilatational modes lead to a symmetric DP potential (shown in Fig. 6.2) while inducing an antisymmetric PZ potential.
6.2 Details 105 PSfrag replacements b d Θ z y Figure 6.8: Orientation of the quantum dots in the phonon cavity. In bulk GaAs systems without phonon confinement, PZ interaction normally dominates the coupling to long wavelength acoustic phonons because typical matrix elements relate to the DP coupling as |V pz /V dp | 2 ∝ (eβ/Ξ) 2 q −2 , with the DP constant Ξ and the PZ modulus β, see chapter 5. Thus, for long wavelength phonons (q → 0) the PZ coupling prevails. In a phonon cavity, however, one can argue that the confinement-induced quantisation of the transverse wave vector gives a lower bound for the wavelength q min ∝ 1/b. Therefore, for small FSQWs the DP coupling dominates. We now turn to the discussion of the orientation dependence of the DP induced inelastic current for the examples of lateral and vertical configurations. We start with the vertical configuration (Θ = π/2) in which the dot vector is orthogonal to the propagation of the phonons, q ‖· d = 0. Thus, from Eq. (6.24) follows that only flexural modes contribute to the inelastic current through the DQD via DP scattering. This can be understood by a simple symmetry consideration. Dilatational modes induce a symmetrical DP interaction potential (see Fig. 6.2) which affects both dots equally, hence not changing the electron transport. On the other hand, the antisymmetric potential induced by the flexural modes strongly alters the dot levels. A numerical evaluation of Eq. (6.15) leads to the inelastic scattering rate which is shown in Fig. 6.3 (top). In spite of the modifications due to the electron-phonon interaction, the main features of phonon confinement, subband quantisation and van-Hove singularities, clearly remain visible. On the contrary, for the lateral configuration (Θ = 0), Eq. (6.24) predicts that the effective DOS (6.16) vanishes for flexural modes. Thus, all phonons which are spontaneously emitted from the DQD correspond to dilatational modes. Flexural modes induce an antisymmetric interaction potential (Fig. 6.2) which vanishes at the location of the dots in the mid-plane of the FSQW (z = 0). On the other hand, dilatational modes strongly couple to the DQD since they induce a non-zero potential and lead to a phase shift in the renormalisation of the dot levels due to the finite time a phonon needs to propagate from one dot to the next. The calculated inelastic scattering rate for the lateral configuration is shown in Fig. 6.3 (bottom). As for the lateral case, the van-Hove singularities remain visible in the transport. When comparing the inelastic rates with the thermodynamic DOS, the most obvious difference between the two configurations is seen in the behaviour at the
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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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2.4 Summary 45 For weak SO coupling
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2.4 Summary 47 splitting can become
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Chapter 3 Rashba spin-orbit couplin
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3.1 Spin-orbit driven coherent osci
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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
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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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