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

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120 Matrix elements of dot electron-confined phonon interaction induced potential with respect to the confinement direction can be understood as standing waves in the DP interaction potential. The matrix elements of the interaction with the localised dot states |L〉 and |R〉 can be written as Z α n (q ‖ ) = λ ± dp (q ‖) d 3 rρ L (r)tcs(q l,n z) e iq ‖·r ‖ . (D.4) The matrix element β n (q ‖ ) follows from replacing ρ L with ρ R which are the local electron densities in the left and right dot, respectively ρ i (r) = 〈i|Ψ † (r)Ψ(r)|i〉. (D.5) Making the assumption that these densities are smooth functions ρ e centred around the middle of the dots, ρ i (r) ≈ ρ e (r − r i ), we can calculate the interference term |α n (q ‖ )−β n (q ‖ )| 2 =|P e (q ‖ ,q l,n )| 2∣ ∣ ∣λ ± dp (q ‖ )tcs( 1 2 q l,nd sinΘ) ( −1 ± e iq ‖·d )∣ ∣ ∣ 2 .(D.6) The vector d and the angle Θ are related to the dot orientation in the FSQW (see Fig. 6.8 on page 105), and P e (q ‖ ,q l,n ) is a form factor of the electron density, Z P e (q) = d 3 rρ e (r)e iq·r , (D.7) which can be approximated by unity for sharply localised wavefunctions. Piezo-electric coupling The microscopic calculation of the piezo-electric (PZ) potential caused by a confined phonon is more complicated than the previous derivation for the DP case. As seen in chapter 5, the potential V pz is related to the polarisation P, which is induced by the PZ effect, via Poisson’s equation ∇ 2 V pz = 4πe ∇ · P, see Eq. (5.8). In general, all four modes families of confined phonons couple to electrons via PZ interaction. A further complication is the anisotropy of the PZ modulus. Stroscio et al. [176] calculated the interaction potential for Lamb modes [n,q ‖ = (q x ,q y )] by using an averaged expression for the PZ modulus, leading to ( V pz (r) = ±q x q y q l λ ± l (q ‖ )tscq l z + λ t ± (q ‖ )tscq t z ) ] e iq ‖·r ‖ [a n (q ‖ ) + a † n(q ‖ ) (D.8)
121 with λ ± l (q ‖) = F n B pz n (q ‖ ) 3(q2 ‖ − q2 t ) q 2 ‖ + tscq t b , q2 l λ t ± (q ‖ ) = −F n B pz n (q ‖ ) 2(q2 ‖ − 2q2 t ) q 2 ‖ + tscq l b , q2 t (D.9) (D.10) and B pz n (q ‖ ) = (8πeβ/ε)[/2Aρ M ω n (q ‖ )] 1/2 . Here ε is the low frequency permittivity constant. The sign ± and tscx correspond to dilatational and flexural modes as in the previous section. When comparing Eqs. (D.8) and (D.2) the different symmetries of the DP and PZ potentials become apparent. The DP potential for dilatational and flexural modes is shown in Fig. 6.2. By extending the calculation of the matrix elements from the previous section to the anisotropic case we find for the interference term ∣ ∣α n (q ‖ )−β n (q ‖ ) ∣ ∣ 2 = q 2 xq 2 y|q l | 2 ∣ ∣ ∣ ( 1 ± e iq ‖·d )∣ ∣ ∣ 2 [ ) × 1 ∣ λ ± l ‖)tsc( (q 2 q ld sinΘ ) ]∣ 1 ∣∣∣ + λ t ± 2 (q ‖ )tsc( 2 q td sinΘ . (D.11)
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Interaction and confinement in nano
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Abstract iii Abstract It is the pur
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Publications v Publications Some of
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2 Table of Contents 3 Rashba spin-o
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4 Introduction (phonons). Therefore
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6 Introduction Figure 1: Scanning e
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Chapter 1 The Rashba effect Effects
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1.2 The model 11 the position-depen
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1.3 Rashba effect in a perpendicula
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Chapter 2 Rashba spin-orbit couplin
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2.1 Rashba effect and magnetic fiel
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PSfrag replacements 2.1 Rashba effe
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2.2 Spectral properties, various li
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2.3 Electron transport in one-dimen
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PSfrag replacements 2.3 Electron tr
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2.4 Summary 45 For weak SO coupling
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2.4 Summary 47 splitting can become
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Chapter 3 Rashba spin-orbit couplin
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3.1 Spin-orbit driven coherent osci
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3.1 Spin-orbit driven coherent osci
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0 0.5 1.5 3.1 Spin-orbit driven coh
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3.2 Introduction to quantum dots an
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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
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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
Page 96 and 97: 90 Coupled quantum dots in a phonon
Page 102 and 103: ) ) ) ) 3 4 + ) * - 96 Coupled quan
Page 106 and 107: 100 Coupled quantum dots in a phono
Page 110 and 111: PSfrag replacements 1042 Coupled qu
Page 116 and 117: 110 Conclusion due to phonon van-Ho
Page 119 and 120: Appendix A Jaynes-Cummings model Th
Page 121 and 122: Appendix B Fock-Darwin representati
Page 123 and 124: Appendix C Evaluation of the phonon
Page 125: Appendix D Matrix elements of dot e
Page 129 and 130: Bibliography [1] G. Bergmann, Phys.
Page 131 and 132: Bibliography 125 [31] M. Pletyukhov
Page 133 and 134: Bibliography 127 [72] T. Schäpers,
Page 135 and 136: Bibliography 129 [109] U. Merkt, J.
Page 137 and 138: Bibliography 131 [144] B. Auld, Aco
Page 139 and 140: Bibliography 133 [182] A. A. Kisele
Page 141: Acknowledgements Finally, I would l
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