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

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40 Rashba spin-orbit coupling in quantum wires quantum mechanics. Nevertheless, here, we go through the basic calculation to illustrate peculiarities arising from the SO coupling. The solution of the Schrödinger equation can be obtained by using the ansatz I: ψ I (x) =A + ψ k+ (x) + A − ψ k− (x) + B + ψ −k+ (x) + B − ψ −k− (x), (2.47) II: ψ II (x) =C + ψ k ′ + (x) +C − ψ k ′ − (x) + D + ψ −k ′ + (x) + D − ψ −k ′ − (x), (2.48) with the wave vectors of the left and right region, k ± = k ± (κ,k SO ,E 0 ) and k ± ′ = k ± (κ ′ ,k SO ′ ,E′ 0 ), respectively. At the interface the conditions (2.35) and (2.36) have to be satisfied which for a strict-1D system reduce to with the non-diagonal velocity operator ψ I (x = a) = ψ II (x = a), (2.49) ˆv x ψ I (x) ∣ ∣ a = ˆv x ψ II (x) ∣ ∣ a , (2.50) ( ) ˆv x = ∂H = 1 px 2ik SO . (2.51) ∂p x m −2ik SO p x In transport calculations it is customary to normalise the wavefunctions with respect to the probability current. Therefore, we require ∣ ∣〈ψ ks | ˆv x |ψ ks 〉 ∣ ∣ = 1, (2.52) leading to the normalisation constant N ks = √ 1 √ m|k|/2 ( L k 2 + κ 2 + 2k SO κ + s(κ + 2k SO ) √ (2.53) k 2 + κ2), where the length of the wire is denoted by L. Equations (2.49) and (2.50) lead to the transfer matrix M(a) through the magnetic step, ⎛ ⎞ ⎛ ⎞ A + C + N ⎜A − ⎟ ⎝B + ⎠ = Ψ(−a) ˆm−1 ˆm ′ Ψ ′ (a) N ′ ⎜C − ⎟ } {{ } ⎝D + ⎠ , (2.54) B − =:M(a) D −
PSfrag replacements 2.3 Electron transport in one-dimensional systems with Rashba effect 41 κ 1 ,k SO1 ,E 01 κ 2 ,k SO2 ,E 02 κ n ,k SOn ,E 0n a b z x I II XX Figure 2.10: System of stepwise constant external fields with multiple interfaces. with the normalisation matrix ⎛ ⎞ N k+ 0 0 0 N = ⎜ 0 N k− 0 0 ⎟ ⎝ 0 0 N k+ 0 ⎠ , (2.55) 0 0 0 N k− and the matrix of phase coefficients Ψ κ (a) = ⎛ e ik +a 0 0 0 ⎜ ⎝ 0 e ik −a 0 0 0 0 e −ik +a 0 0 0 0 e −ik −a ⎞ ⎟ ⎠ , (2.56) ⎛ ⎞ ξ k+ ξ k− −ξ k+ −ξ k− ˆm = ⎜ 1 1 1 1 ⎟ ⎝φ k+ φ k− φ k+ φ k− ⎠ , (2.57) θ k+ θ k− −θ k+ −θ k− where φ ks = k ξ ks + 2ik SO and θ ks = k − 2ik SO ξ ks . In the case of multiple interfaces between regions of stepwise constant parameters (see Fig. 2.10), one obtains the transfer matrix ˆT of the entire system by connecting the transfer matrices M i (x i ) of the separate interfaces, ⎛ ⎞ ⎛ ⎞ A + X + N ⎜A − ⎟ ⎝B + ⎠ = M 1(a)M 2 (b)...M n (z) N } {{ } n ⎜X − ⎟ ⎝Y + ⎠ . (2.58) B − =: ˆT Y − From this matrix the transmission coefficients for the different spin polarisations are derived easily. We now restrict ourselves to the assumption that all incoming electrons are right-moving in state ψ k+ (i.e. A − = Y + = Y − = 0). Because
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 and 42: PSfrag replacements 2.3 Electron tr
Page 45: 2.3 Electron transport in one-dimen
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
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
Page 92 and 93: 86 Introduction to electron-phonon
Page 94 and 95: 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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