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

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12 The Rashba effect (a) 1 (b) E±(k) in a.u. 0.8 0.6 0.4 0.2 k y E − E − E + E + ∆k E F k x 0 −1 −0.5 0 0.5 1 k in a.u. Figure 1.1: Effect of Rashba spin-orbit coupling on dispersion (a) and spin (b). where k = k(cosϕ,sinϕ,0). The dispersion (1.4) consists of two branches which are non-degenerate for k > 0, see Fig. 1.1a. The spin state of the electron which follows from Eq. (1.5) is given by 〈σ〉 = ±(sinϕ,−cosϕ,0) and is shown in Fig. 1.1b. These profound effects of Rashba SO coupling can be understood by rewriting Eq. (1.2) as the Zeeman effect of a momentum dependent effective magnetic field B SO (p) · σ. The amplitude of B SO is proportional to the momentum while its orientation in the 2D plane is orthogonal to the direction of propagation. Thus, for fixed propagation direction, the spin is quantised perpendicular to p, see Fig. 1.1b. This shows that there is no common axis of spin quantisation in 2D with Rashba SO coupling. Thus, the subscript ± only corresponds to a spin frame which is local in k space. In experiment, the coupling parameter α is commonly determined by Shubnikov–de Haas (SdH) measurements [28] or weak antilocalisation analysis [1, 29] in weak magnetic fields. In 2D, the frequency of SdH oscillations in the magnetotransport is proportional to the electron density in the system [3]. If the spin degeneracy is broken by Rashba SO coupling the two branches of the dispersion (1.4) have different wave vectors at a given Fermi energy E F = 2 2m k2 F± ± αk F± . (1.6) As a consequence, the difference in Fermi wave vector is proportional to the strength of SO coupling, ∆k = |k F+ − k F− | = 2mα/ 2 . The electron density depends parabolically on the Fermi wave vector in 2D. Thus, SO coupling leads to different electron populations N ± ∝ k 2 F± of the two branches E ±(k). In the context of SdH measurements, this leads to two different frequencies which result in a beating pattern in the SdH oscillations, see e.g. [30]. From the analysis of the beating frequency, the strength of SO coupling α can be determined [18]. We remark that beating patterns in magneto-oscillations are not restricted to the Rashba
1.3 Rashba effect in a perpendicular magnetic field 13 effect. Any effect which leads to a B = 0 spin splitting (like BIA) may also result in a beating pattern. 1.3 Rashba effect in a perpendicular magnetic field In the following, we present the effect of an additional perpendicular magnetic field which is a common tool to introduce a tuneable energy scale, i.e. the cyclotron energy ω c = eB/mc. In addition to this energy scale, which finally leads to the quantisation of the system into Landau levels, the Zeeman effect is also expected to alter the spin state of the electrons. This system serves as an example for the interplay of the Rashba effect with a further energy scale. Furthermore, it shows an illustrative analogy to a quantum optical model which will be useful in the following chapters of the thesis when dealing with non-integrable models. We extend the Hamiltonian (1.3) of the previous section by including a perpendicular magnetic field (B = B ê z ), H = H 0 + H SO , (1.7) H 0 = 1 ( p + e ) 2 2m c A 1 + 2 gµ BBσ z , (1.8) H SO = − α [(p + e ) ] c A × σ , p = (p x, p y ). (1.9) z We follow the standard derivation of quantised Landau levels in symmetric gauge A = (−y,x)B/2, by defining x ± = 1 √ 2 (y ± ix), p ± = 1 √ 2 (p y ∓ ip x ), (1.10) and creation and annihilation operators ( 1 a = √ p − − i mωc 2 mω cx + ), ( a † 1 = √ p + + i mωc 2 mω cx − ), (1.11) leading to the representation in terms of Landau levels, ( H 0 = a † a + 1 ) + 1 ω c 2 2 δσ z, (1.12) with the dimensionless Zeeman splitting δ = mg/2m 0 , (m 0 : bare mass of electron). Expressing the SO coupling in the same representation gives H SO = 1 l ( √ B aσ + + a † σ − ), σ ± = 1 ω c 2 l SO 2 (σ x ± iσ y ), (1.13)
Page 1 and 2: Interaction and confinement in nano
Page 3 and 4: Abstract iii Abstract It is the pur
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Page 8 and 9: 2 Table of Contents 3 Rashba spin-o
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Page 12 and 13: 6 Introduction Figure 1: Scanning e
Page 15 and 16: Chapter 1 The Rashba effect Effects
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3.2 Introduction to quantum dots an
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PSfrag replacements 3.2 Introductio
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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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