Itinerant Spin Dynamics in Structures of ... - Jacobs University

More documents

Recommendations

Info

10 Chapter 2: Spin Dynamics: Overview and Analysis of 2D Systems Injecting an electron at position r 0 into a conductor with previously constant electrondensity ρ 0 , thesolutionof thediffusionequation yieldsthat theelectron density spreads in space according to ρ(r,t) = ρ 0 +exp(−(r−r 0 ) 2 /4D e t)/(4πD e t) d D/2 . The dimension d D is equal to the kinetic dimension d, d D = d, if the elastic mean free path l e is smaller than the size of the sample in all directions. If the elastic mean free path is larger than the sample size in one direction the diffusion dimension reduces by one, accordingly. Thus, on average the variance of the distance the electron moves after time t is 〈(r−r 0 ) 2 〉 = 2d D D e t. This introduces a new length scale, the diffusion length L D (t) = √ D e t. We can rewrite the density as ρ = 〈ψ † (r,t)ψ(r,t)〉, where ψ † = (ψ † + ,ψ† − ) is the two-component vector of the up (+), and down (-) spin fermionic creation operators, and ψ the 2-component vector of annihilation operators, respectively, 〈...〉 denotes the expectation value. Accordingly, the spin density s(r,t) is expected to satisfy a diffusion equation, as well. The spin density is defined by where σ is the vector of Pauli matrices, ⎛ ⎞ ⎛ σ x = ⎝ 0 1 ⎠,σ y = ⎝ 0 −i 1 0 i 0 s(r,t) = 1 2 〈ψ† (r,t)σψ(r,t)〉, (2.9) ⎞ ⎠, and σ z = ⎛ ⎝ 1 0 0 −1 Thus the z-component of the spin density is half the difference between the density of spin up and down electrons, s z = (ρ + −ρ − )/2, which is the local spin polarization of the electron system. Thus, we can directly infer the diffusion equation for s z , and, similarly, for the other components of the spin density, yielding, without magnetic field and spin-orbit interaction,[Tor56] ∂s ∂t = D e∇ 2 s− sˆτ . (2.10) s Here, in the spin relaxation term we introduced the tensor ˆτ s , which can have non-diagonal matrix elements. In the case of a diagonal matrix, τ sxx = τ syy = τ 2 , is the spin dephasing time, and τ szz = τ 1 the spin relaxation time. The spin diffusion equation can be written as a continuity equation for the spin density vector, by defining the spin diffusion current of the spin components s i , ⎞ ⎠. J si = −D e ∇s i . (2.11) Thus, we get the continuity equation for the spin density components s i , ∂s i ∂t +∇J s i = − ∑ j s j (ˆτ s ) ij . (2.12)
Chapter 2: Spin Dynamics: Overview and Analysis of 2D Systems 11 2.3.3 Spin Orbit Interaction in Semiconductors SOI in semiconductors is closely related to breaking of symmetries which lift spin degeneracy: In the case without magnetic field B we start with a twofold degeneracy: Time inversion symmetry, E ↑ (k) = E ↓ (−k) and space inversion symmetry, E ↑ (k) = E ↑ (−k). As a consequence we have E ↑ (k) = E ↓ (k). In the following we show how the degeneracy is lifted in semiconductor devices in case of B = 0. While silicon and germanium have in their diamond structure an inversion symmetry around every midpoint on each line connecting nearest neighbor atoms, this is not the case for III-V-semiconductors like GaAs, InAs, InSb, or ZnS. These have a zinc-blende structure which can be obtained from a diamond structure with neighbored sites occupied by the two different elements. Therefore the inversion symmetry is broken, which results in spin-orbit coupling. This can be understood by noticing that pairs like Ga-As are local dipoles whose electric field is responsible for SOC if inversion symmetry is broken 2 . Similarly, that symmetry is broken in II-VI-semiconductors. This bulk inversion asymmetry (BIA) coupling, or often so called Dresselhaus-coupling, is anisotropic, as given by [Dre55] [ H D = γ D σx k x (ky 2 −k2 z )+σ yk y (kz 2 −k2 x )+σ zk z (kx 2 −k2 y )] , (2.13) where γ D is the Dresselhaus-spin-orbit coefficient. Band structure calculations yield the following values: γ D = 27.6 eVÅ(GaAs), = 27.2 eVÅ(InAs), = 760.1 eVÅ(InSb) [Win03]. Some values extracted in experiments are listed in Tab.A. Confinement in quantum wells with width a z on the order of the Fermi wave length λ F yields accordingly a spin-orbit interaction where the momentum in growth direction is of the order of 1/a z . Because of the anisotropy of the Dresselhaus term, the spin-orbit interaction depends strongly on the growth direction of the quantum well. Grown in [001] direction, one gets, taking the expectation value of Eq.(2.13) in the direction normal to the plane, noting that 〈k z 〉 = 〈kz〉 3 = 0, [Dre55] H D[001] = α 1 (−σ x k x +σ y k y )+γ D (σ x k x ky 2 −σ y k y kx). 2 (2.14) where α 1 = γ D 〈kz〉 2 is the linear Dresselhaus parameter. Thus, inserting an electron with momentum along the x-direction, with its spin initially polarized in z-direction, it will 2 Starting from an extended Kane model where p-like higher energy bands are included one gets a Hamiltonian with matrix elements which are only nonzero if the crystal has no center of inversion.[FMAE + 07]
Page 1 and 2: Itinerant Spin Dynamics in Structur
Page 3 and 4: Itinerant Spin Dynamics in Structur
Page 5 and 6: Contents v 3.2.2 Weak Localization
Page 7 and 8: Citations to Previously Published W
Page 9 and 10: Dedicated to Linda, my parents and
Page 11 and 12: Chapter 1 Introduction Structure of
Page 13 and 14: Chapter 1: Introduction 3 the coupl
Page 15 and 16: Chapter 1: Introduction 5 Throughou
Page 17 and 18: Chapter 2: Spin Dynamics: Overview
Page 19: Chapter 2: Spin Dynamics: Overview
Page 35 and 36: Chapter 3 WL/WAL Crossover and Spin
Page 37 and 38: Chapter 3: WL/WAL Crossover and Spi
Page 71 and 72:
Chapter 3: WL/WAL Crossover and Spi
Page 73 and 74:
Page 75 and 76:
Page 77 and 78:
Chapter 4 Direction Dependence of S
Page 79 and 80:
Chapter 4: Direction Dependence of
Page 81 and 82:
Page 83 and 84:
Page 85 and 86:
Page 87 and 88:
Page 89 and 90:
Page 91 and 92:
Page 93 and 94:
Chapter 5 Spin Hall Effect 5.1 Intr
Page 95 and 96:
Chapter 5: Spin Hall Effect 85 curr
Page 97 and 98:
Chapter 5: Spin Hall Effect 87 with
Page 99 and 100:
Chapter 5: Spin Hall Effect 89 1.0
Page 101 and 102:
Chapter 5: Spin Hall Effect 91 as s
Page 103 and 104:
Chapter 5: Spin Hall Effect 93 V⩵
Page 105 and 106:
Chapter 5: Spin Hall Effect 95 0.4
Page 107 and 108:
Chapter 5: Spin Hall Effect 97 oper
Page 109 and 110:
Chapter 5: Spin Hall Effect 99 the
Page 111 and 112:
Chapter 5: Spin Hall Effect 101 str
Page 113 and 114:
Chapter 5: Spin Hall Effect 103 Com
Page 115 and 116:
Chapter 5: Spin Hall Effect 105 Fin
Page 117 and 118:
Chapter 6: Critical Discussion and
Page 119 and 120:
Chapter 6: Critical Discussion and
Page 121 and 122:
List of Figures 111 3.3 Exemplifica
Page 123 and 124:
List of Figures 113 4.2 The spin de
Page 125 and 126:
List of Tables 3.1 Singlet and trip
Page 127 and 128:
Bibliography 117 [BB04] [BBF + 88]
Page 129 and 130:
Bibliography 119 [DP71b] [DP71c] [D
Page 131 and 132:
Bibliography 121 [HSM + 06] [HSM +
Page 133 and 134:
Bibliography 123 [MACR06] T. Mickli
Page 135 and 136:
Bibliography 125 [PMT88] [PP95] [PT
Page 137 and 138:
Bibliography 127 [Tor56] H. C. Torr
Page 139 and 140:
Appendix A SOC Strength in the Expe
Page 141 and 142:
Appendix A: SOC Strength in the Exp
Page 143 and 144:
Appendix B: Linear Response 133 in
Page 145 and 146:
Appendix B: Linear Response 135 Pro
Page 147 and 148:
Appendix C Cooperon and Spin Relaxa
Page 149 and 150:
Appendix C: Cooperon and Spin Relax
Page 151 and 152:
Page 153 and 154:
Page 155 and 156:
Page 157 and 158:
Appendix D: Hamiltonian in [110] gr
Page 159 and 160:
Appendix E: Summation over the Ferm
Page 161:
Appendix F: KPM 151 integer running
show all

Itinerant Spin Dynamics in Structures of ... - Jacobs University

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?