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

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28 Rashba spin-orbit coupling in quantum wires PSfrag replacements atom–light interaction photon number atomic pseudo-spin electric dipole interaction 3 |↑〉 a, a † 2 σ ± |n〉 |↓〉 1 spin-orbit interaction 0 electron spin electron subband number SO-coupled quantum wire Figure 2.5: Analogy between spin-orbit interacting quantum wire and atom-light interaction in quantum optics. mally commuting subsystem with energies of shifted parabolic subbands [69], E 0,n ± = n + 1 ω 0 2 + 1 ( kl 0 ± 1 ) l 2 0 − 1 ( ) 2 l0 , (2.19) 2 2 l SO 8 l SO where ± denotes eigenstates with spin (anti)parallel to σ x . The shift of the dispersion depends on the spin and the strength of the SO coupling k SO = ±1/2l SO . H mix symmetrically couples adjacent subbands of H 0 . In the limit of |kl 0 | ≫ 1 and l 0 /l SO ≪ 1 (corresponding to weak SO coupling) the H mix term can be neglected, leading to the so-called longitudinal-SO approximation [68]. Within this approximation, spinors are given by eigenstates of σ x , where the spin is fixed by the propagation direction of the electron due to k-linear prefactor of σ x in H 0 . Thus, for large momenta, right(left)-moving electrons have approximately spindown(up) polarisation with respect to σ x . The shifted parabolic subbands of the longitudinal-SO approximation are also seen in the result for weak magnetic field, Fig. 2.6 (left panel in upper row). The transversal part of the SO coupling H mix becomes important in the regime where neighbouring subbands of H 0 become degenerate, leading to anticrossings as seen in Fig. 2.6 (central panel in upper row). In general, no common spin quantisation axis can be found. This can also be seen in the spin properties for weak magnetic field, Fig. 2.4a. The zero-B result is given by the envelope of the hysteresis-like solid curves. For finite momenta, the tilt of the spin does depend on k. Only for large momenta, l 0 k ≫ 1, the spin is approximately quantised in σ x direction.
2.2 Spectral properties, various limits 29 increasing spin-orbit coupling PSfrag replacements l B = 4.0 l 0, l so = 4.0 l 0 l B = 4.0 l 0, l so = 1.0 l 0 l B = 4.0 l 0, l so = 0.25 l 0 E E E ω 0 ω 0 ω 0 4 5 5 2 0 B = 0 increasing magnetic field 0 0 -2 -2 0 2 l 0 k -2 0 2 l 0 k -2 0 2 l 0 k l B = 1.0 l 0, l so = 4.0 l 0 l B = 1.0 l 0, l so = 1.0 l 0 l B = 1.0 l 0, l so = 0.25 l 0 E ω 0 8 E ω 0 8 E ω 0 4 4 4 2 0 0 0 -2 -4 0 4 l 0 k -4 0 4 l 0 k -4 0 4 l 0 k l B = 0.25 l 0, l so = 4.0 l 0 l B = 0.25 l 0, l so = 1.0 l 0 l B = 0.25 l 0, l so = 0.25 l 0 E E E ω 0 ω 0 ω 0 80 80 80 40 40 40 0 0 0 -100 0 l 0 k 100 -100 0 l 0 k 100 -100 0 l 0 k 100 Figure 2.6: Low-energy spectra of an SO-interacting quantum wire found by numerical diagonalisation. Various regimes in the interplay of SO coupling and perpendicular magnetic field are shown. For strong magnetic field (lower row) the spectrum of the Jaynes–Cummings model are shown (dashed lines). 2.2.2 Two-band model In the low-energy limit with a strong lateral confinement and a low electron density, Fermi points exist for the lowest subbands, only. Governale and Zülicke introduced a two-band model by truncating the Hilbert space to the two lowest spinsplit subbands [69]. The corresponding matrix representation of dimension 4 × 4 is simple enough to be solved analytically but it goes beyond the longitudinal- SO approximation by showing an anticrossing. This model has been applied to calculate the transmission through an SO-interacting wire of finite length which was attached to perfect leads [69]. If the Fermi energy is placed between the two subband – making the lowest one a transmitting channel whereas the upper
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: 2.2 Spectral properties, various li
Page 37 and 38: 2.2 Spectral properties, various li
Page 39 and 40: 2.3 Electron transport in one-dimen
Page 41 and 42: PSfrag replacements 2.3 Electron tr
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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
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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
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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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