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

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36 Rashba spin-orbit coupling in quantum wires with the velocity operator ˆv x = ∂H/∂p x which becomes non-diagonal in systems with SO coupling [87]. In a quasi-1D system the continuity conditions (2.35) and (2.36) – which finally give the solution of the Schrödinger equation of the whole system – cannot be fulfilled for all y when the set of transverse modes is restricted to propagating ones only (clearly seen for the two modes sketched in Fig. 2.7). Evanescent modes (EM) have to be included to obtain a sufficiently complete set of transverse wave functions. § Once the solution is found, meaning {A n ,B n ,C n ,D n } in Eq. (2.33) and (2.34) are known, the transmission properties of the system can be derived from the transfer-matrix ˆT , which relates amplitudes on the left to those on right, ( A = ˆT B) ( C D) , or equivalently ( ( B A = Ŝ , (2.38) C) D) defining the scattering or Ŝ–matrix which connects incoming and outgoing fluxes. When addressing the effect of SO coupling on the transmission properties of quasi-1D systems, like the parabolically confined quantum wire in the previous sections, the ansatz for the wavefunctions in (2.33) has to be generalised (i) to include the spin, and (ii) to allow for the peculiarities due the SO coupling like transverse modes depending explicitely on the longitudinal momentum k, making φ k,n (y) ≠ φ −k,n (y), and the multitude of possible wave vectors for a given energy due to non-parabolicities of the dispersion (Fig. 2.6). The real inconvenience, however, is the determination of EM. The parabolic QWR with SO coupling in the previous sections exemplifies the general non-integrability of the transverse Schrödinger equation, H(k)φ k = E k φ k , see Eq. (2.5). Therefore, the determination of propagating modes for energy E F corresponds to numerically finding those real k n such that E F is contained in the spectrum of H(k n ). On the contrary, EM have in general complex wave vectors, thus drastically complicating the numerical evaluation of the spectrum of H(k). Although challenging, but technically still feasible, in the context of spintronics to our knowledge no exact investigation of the properties of EM has been performed within the above scheme yet. Evanescent modes are localised close to the interface between adjacent sections of the quasi-1D system due to their exponential character. Therefore they do not extend over the whole system, leading one to naively believe they have no influence on the transmission characteristics. However, in confined systems the boundary conditions couple propagating and EM which aquire a non-zero amplitude in the superposition (2.33) at the expense of the propagating ones; thus they do affect the transmission properties indirectly. § We remark that the diagonalisation of a tight-binding model is exact within the dimension of the lattice, the solution naturally includes evanescent modes.
2.3 Electron transport in one-dimensional systems with Rashba effect 37 This influence of EM has been highlighted in quasi-1D systems without SO coupling by e.g. showing that the localisation of EM around defects may lead to perfect transparency or opaqueness as a function of Fermi energy [85]. Since EM are a basic wave phenomenon, their influence also appears in other fields of physics like microwave waveguides and near-field optics [88, 89]. Because of the complications in finding EM in SO-coupled systems, normally approximations are being applied to make the Hamiltonian integrable. In Sec. 2.2.1 the longitudinal-SO approximation and the two-band model are presented as examples. Within the latter approximation, Governale & Zülicke [69] showed that in a quasi-1D hybrid system of waveguides with different strengths of SO coupling, scattering to EM leads to spin accumulation close to the interface, thus highlighting the general importance of EM. The results which are derived by the mode matching analysis so far depend very much on the applied approximation. For instance, Wang et al. studied periodically stubbed waveguides as an extension of the Datta & Das [42] spin-FET design. They applied the longitudinal-SO approximation for single [44] and multisubband [90] transmission, and found a tuneable spin current modulation similar to the original spin-FET. Later, Wang & Vasilopoulos [45] extended the calculation by applying the two-band model and found that the previously neglected subband mixing leads to drastic changes of the transmission properties. 2.3.2 Strict-1D limit of a quantum wire In this section, we treat the strict-1D limit of a ballistic QWR (1D-QWR) by truncating the Hilbert space to its lowest transverse subband. This is a reasonable approximation for the case of strong lateral confinement where the subband separation is the dominating energy scale and the SO coupling is weak. In this limit, the Hamiltonian of a 1D-QWR is given by ⎛ 1 2m p2 x + 1 2 gµ ⎞ i BB αp x H 1D − E 0 = ⎝ − i αp x 1 2m p2 x − 1 2 gµ BB ⎠, (2.39) where the effect of a constant electrostatic background potential is absorbed into the energy offset E 0 . In the following, we will demonstrate for the basic example of the 1D-QWR how the electronic transport is influenced when magnetic field and SO coupling act simultaneously in the wire. In this strict-1D limit there is no problem associated with the finding of EM. For the case of a periodic magnetic modulation, we show that a commensurability effect appears in the spin-dependent conductance when the modulation period becomes comparable with the SO-induced spin precession length.
Page 1 and 2: Interaction and confinement in nano
Page 3 and 4: Abstract iii Abstract It is the pur
Page 5: Publications v Publications Some of
Page 8 and 9: 2 Table of Contents 3 Rashba spin-o
Page 10 and 11: 4 Introduction (phonons). Therefore
Page 12 and 13: 6 Introduction Figure 1: Scanning e
Page 15 and 16: Chapter 1 The Rashba effect Effects
Page 17 and 18: 1.2 The model 11 the position-depen
Page 19 and 20: 1.3 Rashba effect in a perpendicula
Page 21 and 22: Chapter 2 Rashba spin-orbit couplin
Page 23 and 24: 2.1 Rashba effect and magnetic fiel
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: PSfrag replacements 2.3 Electron tr
Page 47 and 48: PSfrag replacements 2.3 Electron tr
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
Page 61 and 62: £££££££££££££¢¢¢¢
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
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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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