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

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14 The Rashba effect with the magnetic length l B = (/mω c ) 1/2 and the length scale of SO coupling l SO = 2 /2mα. Equation (1.13) shows that the Rashba SO interaction leads to a coupling of adjacent Landau levels with opposite spin. For strong magnetic field, this effect becomes negligible, as seen from the ratio l B /l SO . For typical InAs parameters, we find l SO ∼100nm. Thus, we may expect a significant effect of SO coupling for B ≤ 0.1T, as the prefactor in Eq. (1.13) becomes of order unity. The diagonalisation of the Hamiltonian H = H 0 +H SO is straightforward. The eigenenergies have been given by Bychkov & Rashba [12]. Here, we follow a different approach by noticing [31] that the Hamiltonian is formally identical to the integrable Jaynes–Cummings model (JCM) [32] of quantum optics. There, the JCM is used in the context of atom-light interaction where two atomic levels are described by a pseudo-spin. Transitions between the levels are induced by the electric dipole coupling to a quantised monochromatic radiation field which is modelled by a harmonic oscillator. In this language, H SO describes transitions (σ ± ) between the two atomic levels under absorption/emission of a photon of the radiation field (a, a † ). Conversely, in the SO-interacting system, the roles of atomic pseudo-spin and light field are played by the real spin and the Landau level of the same electron. The properties of the JCM are discussed in appendix A. The above analogy to a quantum optical model will be helpful in elucidating the effect of SO coupling in the following two chapters of the thesis when the models become non-integrable. In section 2.2, the JCM will reappear as a limit for quantum wires with Rashba SO coupling. In chapter 3, the notion of counterrotating coupling terms (see section 2.2.4), which is closely related to definition of the JCM in quantum optics [33], leads us to the derivation of an integrable effective model for SO-interacting quantum dots.
Chapter 2 Rashba spin-orbit coupling in quantum wires In the previous chapter, we have introduced the Rashba effect as a consequence of spin-orbit (SO) interaction in two-dimensional (2D) electron systems with dominating structure-inversion asymmetry. In general, SO interaction couples the spin of a particle to its orbital motion. In mesoscopic systems, the latter can easily be modified by means of geometrical confinement, e. g. due to external gate voltages. In the following, we want to illustrate how effects of SO coupling are modified by further constraining the motion of the electron in a single spatial direction by considering a quasi-one-dimensional system which is defined in the 2DEG. Similar to the Rashba effect, every electric field – and thus also the confining fields – may lead to a coupling between spin and momentum, see Eq. (1.1). These additional contributions to the SO coupling might become important when the lateral confinement is comparable to the vertical constraint that defines the 2DEG. This is the case in wires made by the cleaved-edge overgrowth technique [34] where both lateral and vertical confinement are typically of the order of ∼10nm. A further example where a full three-dimensional description of the SO coupling might be necessary are molecular systems like carbon nanotubes [35, 36]. Throughout this chapter, we restrict ourselves to ballistic quasi-one-dimensional electron systems whose lateral confinement is assumed to be much weaker than the vertical constraint of the 2DEG. This condition is usually fulfilled in quantum wires defined by external gates and etching, leading to a lateral width ≥ 100nm. We describe such a quantum wire by including a parabolic confining potential V c ∝ x 2 into the model for the 2D system [Eq. (1.3)]. In the following section, we present the scientific publication which comprises our main results for the interplay of SO coupling and lateral confinement in quan- 15
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3.2 Introduction to quantum dots an
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3.3 Effects of relaxation 71 page 5
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3.3 Effects of relaxation 73 The re
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3.3 Effects of relaxation 75 with q
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3.3 Effects of relaxation 77 eh 14
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Chapter 4 Conclusion In this part o
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Tuneable quantum dots connected to
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Part II Phonon confinement in nanos
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86 Introduction to electron-phonon
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88 Introduction to electron-phonon
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90 Coupled quantum dots in a phonon
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) ) ) ) 3 4 + ) * - 96 Coupled quan
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100 Coupled quantum dots in a phono
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PSfrag replacements 1042 Coupled qu
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110 Conclusion due to phonon van-Ho
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Appendix A Jaynes-Cummings model Th
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Appendix B Fock-Darwin representati
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Appendix C Evaluation of the phonon
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Appendix D Matrix elements of dot e
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121 with λ ± l (q ‖) = F n B pz
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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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