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

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26 Chapter 3: WL/WAL Crossover and Spin Relaxation in Confined Systems precession axis and the amplitude of the total precession field is changing, since | B+B SO (−p) |=| B−B SO (p) |≠| B+B SO (p) |, resulting in spin dephasing and relaxation, as the sign of the momentum changes randomly. 3.1.2 Wires with W > λ F In this Chapter, we show, however, that the condition for a coherent spin precession is not only the 1d wire, 1/τ s is already strongly reduced in much wider wires: as soon as the wire width W is smaller than bulk spin precession length L SO , which is the length on which the electron spin precesses a full cycle. This explains the reduction of the spin relaxation rate in quantum wires for widths exceeding both the elastic mean-free path l e and λ F , as observed with optical[HSM + 06] as well as with WL measurements[DLS + 05, LSK + 07, SGP + 06, WGZ + 06, KKN09]. As an example we show two experiments in Fig.3.1, where the significant dimensional reduction has been observed. Since L SO can be several µm and is not changed significantly as the wire width W is reduced, such a reduction of spin relaxation can be very useful for applications: the spin of conduction electrons precesses coherently as it moves along the wire on length scale L SO . It becomes randomized and relaxes on the longer length scale L s (W) = √ D e τ s only [D e = v F l e /2 (v F , Fermi velocity) is the 2D diffusion constant]. To understand the connection between the conductivity measurements and spin relaxation we recall that quantum interference of electrons in low-dimensional, disordered conductors is known to result in corrections to the electrical conductivity ∆σ. This quantum correction, the WL effect, is a very sensitive tool to study dephasing and symmetry-breaking mechanisms in conductors.[AAKL82, Ber84, CS86] The entanglement of spin and charge by SO interaction reverses the effect of WL and thereby enhances the conductivity. This WAL effect was predicted by Hikami et al.[HLN80] for conductors with impurities of heavy elements. As conduction electrons scatter from such impurities, the SO interaction randomizes their spin. The resulting spin relaxation suppresses interference in spin triplet configurations. Sincethetime-reversal operation changes not only thesign of momentum but also the sign of the spin, the interference in singlet configuration remains unaffected. Since singlet interference reduces the electron’s return probability, it enhances the conductivity, which is named the WAL effect. In weak magnetic fields, the singlet contributions are suppressed. Thereby, the conductivity is reduced and the magnetoconductivity becomes negative. The
Chapter 3: WL/WAL Crossover and Spin Relaxation in Confined Systems 27 magnetoconductivity of wires is thus related to the magnitude of the spin relaxation rate. InSec.3.2, we firstderive thequantumcorrections to theconductivity for wires with general bulk SO interaction and relate it to the Cooperon propagator. In Sec.3.3, we diagonalize the Cooperon for two-dimensional (2D) electron systems with Rashba SO interaction. We compare the spectrum of the triplet Cooperon with the one of the spin-diffusion equation. InSec.3.4, we presentthesolution of the Cooperonequation for awiregeometry. We review the solutions of the spin-diffusion equation in the wire geometry and compare the resulting spin relaxation rate with the one extracted from the Cooperon equation. Then we proceed to calculate the quantum corrections to the conductivity using the exact diagonalization of the Cooperon propagator. In the last part of this section, we consider two other kinds of boundary conditions. We calculate the spin relaxation rate in narrow wires with adiabatic boundaries, which arise in wires with smooth lateral confinement and regard also tubular wires. In Sec.3.5, we study the influence of the Zeeman coupling to a magnetic field perpendicular to the quantum well in a system with sharp boundaries and analyze how the magnetoconductivity is modified. In Sec.3.6, we draw the conclusions and compare with experimental results. In AppendixC.2, we give the derivation of the non-Abelian Neumann boundary conditions for the Cooperon propagator. In AppendixC.3, we show the connection between the effective vector potential A S due to SO coupling and the spin relaxation tensor. InAppendixC.4, we give theexact quantumcorrection totheelectrical conductivity in 2D. In AppendixC.5, we detail the diagonalization of the Cooperon propagator. 3.2 Quantum Transport Corrections 3.2.1 Diagrammatic Approach As the temperature is lowered, we expect quantum mechanical coherence to be more important: The phase coherence length l ϕ increases with decreasing temperature. If l ϕ is much larger then the elastic scattering length but smaller then the sample size one would expect that all interference effects disappear due to self-averaging. However, it was found that one process seems to survive, as measurements show in logarithmically increasing resistance as temperature decreases, Fig.3.2. In order to introduce the problem of dephasing and WL, we begin with a semiclassical picture of how an electron propagates from a point r to r ′ : The corresponding probability amplitude P is given as the sum over
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 35: Chapter 3 WL/WAL Crossover and Spin
Page 39 and 40: Chapter 3: WL/WAL Crossover and Spi
Page 77 and 78: Chapter 4 Direction Dependence of S
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Chapter 4: Direction Dependence of
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Chapter 5 Spin Hall Effect 5.1 Intr
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Chapter 5: Spin Hall Effect 85 curr
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Chapter 5: Spin Hall Effect 87 with
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Chapter 5: Spin Hall Effect 89 1.0
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Chapter 5: Spin Hall Effect 91 as s
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Chapter 5: Spin Hall Effect 93 V⩵
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Chapter 5: Spin Hall Effect 95 0.4
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Chapter 5: Spin Hall Effect 97 oper
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Chapter 5: Spin Hall Effect 99 the
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Chapter 5: Spin Hall Effect 101 str
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Chapter 5: Spin Hall Effect 103 Com
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Chapter 5: Spin Hall Effect 105 Fin
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Chapter 6: Critical Discussion and
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Chapter 6: Critical Discussion and
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List of Figures 111 3.3 Exemplifica
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List of Figures 113 4.2 The spin de
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List of Tables 3.1 Singlet and trip
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Bibliography 117 [BB04] [BBF + 88]
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Bibliography 119 [DP71b] [DP71c] [D
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Bibliography 121 [HSM + 06] [HSM +
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Bibliography 123 [MACR06] T. Mickli
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Bibliography 125 [PMT88] [PP95] [PT
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Bibliography 127 [Tor56] H. C. Torr
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Appendix A SOC Strength in the Expe
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Appendix A: SOC Strength in the Exp
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Appendix B: Linear Response 133 in
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Appendix B: Linear Response 135 Pro
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Appendix C Cooperon and Spin Relaxa
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Appendix C: Cooperon and Spin Relax
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Appendix D: Hamiltonian in [110] gr
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Appendix E: Summation over the Ferm
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Appendix F: KPM 151 integer running
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