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

More documents

Recommendations

Info

34 Rashba spin-orbit coupling in quantum wires one-dimensional systems often employ discretised lattice models. In such singleparticle tight-binding formulations, numerical techniques like the transfer-matrix or the recursive Green’s function method are used to calculate the exact quantum coherent solution of the system for a given energy. Then, for instance, the conductance can be calculated after coherently connecting the system to leads [3]. These methods have extensively been applied to study localisation problems in disordered systems [84]. Recently, with increasing interest in spin-dependent transport, various spintronic devices have been addressed with tight-binding approaches. For instance, Bulgakov et al. [56] studied the spin-dependent transmission properties of a ballistic cross-junction structure. They found a SO-induced Hall-like effect when driving a spin polarised current through a four-terminal device (without magnetic field). Kiselev & Kim [57, 58] investigated ballistic T-shaped structures with SO interacting intersection region. They found that the device may redirect electrons with opposite spins from an unpolarised current, thus acting as a spin filter. Such behaviour is similar to the Stern-Gerlach experiment where the motion of paramagnetic atoms in a inhomogeneous magnetic field is altered depending on the spin. In addition to such atomistic lattice calculations, continuum formulations of the transfer-matrix method (mode matching analysis) have also been applied to study e.g. the influence of scattering at impurities [85] and conductance fluctuations [86] in quasi-1D systems. The generalisation of this approach to include the effect of spin-orbit (SO) coupling, whilst straightforward in principle, is technically rather difficult because of the imperative determination of evanescent modes. In the following we briefly formulate the necessity to treat evanescent modes in quasi-1D transport and the associated technical complications caused by SO coupling. As an example of the mode matching analysis, in Sec. 2.3.2, we discuss the interplay of SO coupling and an external magnetic modulation in the integrable strict-1D limit of a quantum wire. We find a commensurability effect in the spindependent transport characteristics when the modulation period is comparable to the SO-induced spin precession length. 2.3.1 Transmission in quasi-1D systems When applying numerical tight-binding models to the problem of transmission through quasi-1D systems (e.g. in quantum waveguide structures as shown in Fig. 2.7) the discretisation of the lattice sets the dimension of the Hilbert space. Within this space the numerical solution of the Schrödiger equation is exact. On the contrary, when utilising a continuum formulation of the transfer-matrix method, the mode matching analysis [3] may be used to derive the solution. The basic idea is to split a given geometry into sections and to calculate the solution of the Schrödinger equation by connecting the solutions of the individual sections.
PSfrag replacements 2.3 Electron transport in one-dimensional systems with Rashba effect 35 E x = 0 E E F I II E F y x Figure 2.7: Sketch of interface between two quasi-1D wires with different widths (centre) and their dispersion relations (left and right). At the Fermi energy (dashed) only the lowest two subbands are propagating. This is done by “matching” the transverse modes (and their probability flux ‡ ) at the boundaries between the sections. We illustrate this with the basic example of a discontinuity of a quantum waveguide which is shown in Fig. 2.7. In contrast to Sec. 2.1 and 2.2, in the following, we change the geometry such that the quasi-1D system is parallel to the x-axis. We are looking for the stationary-state solution for the energy E F . Expansion into transverse modes yields ψ I (x,y) = [1ex]ψ II (x,y) = ∞ ∑ n=0 ∞ ∑ n=0 ( ) A n e ikI nx + B n e −ikI nx φ I n(y), (2.33) ( ) C n e ikII n x + D n e −ikII n x φ II n (y), (2.34) with modes {φ I(I) n } of region I(I), resp. The wave vectors follow from the energy dispersions, En(k I n) I = En II (kn II ) = E F . Depending on E F , modes are either propagating if min(E n ) < E F with real k n , or evanescent if min(E n ) > E F , leading to imaginary k n and thus to exponentially decaying (or growing) wavefunctions. At the interface (x=0) the usual boundary conditions have to be satisfied for all y, ψ I (x = 0,y) = ψ II (x = 0,y), (2.35) ˆv x ψ I (x,y) ∣ ∣ x=0 = ˆv x ψ II (x,y) ∣ ∣ x=0 , (2.36) requiring a continuous wavefunction and the continuity equation for the probability density, ∂ t ρ=−∂ x J x , to be fulfilled. The probability current is given by J x = 1 2[ ψ∗ ˆv x ψ + ψ( ˆv x ψ) ∗] , (2.37) ‡ Note that for SO-interacting systems the continuity condition at the interface affects the probability flux; only without SO coupling this reduces to the continuity of the derivative of the wavefunction.
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: 2.3 Electron transport in one-dimen
Page 43 and 44: 2.3 Electron transport in one-dimen
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
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 98 and 99:
Page 100 and 101:
Page 102 and 103:
) ) ) ) 3 4 + ) * - 96 Coupled quan
Page 104 and 105:
Page 106 and 107:
100 Coupled quantum dots in a phono
Page 108 and 109:
Page 110 and 111:
PSfrag replacements 1042 Coupled qu
Page 112 and 113:
Page 114 and 115:
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 and 126:
Appendix D Matrix elements of dot e
Page 127 and 128:
121 with λ ± l (q ‖) = F n B pz
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
show all

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

Create successful ePaper yourself

Delete template?

Save as template?