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

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22 Chapter 2: Spin Dynamics: Overview and Analysis of 2D Systems Perturbation theory yields then directly the corresponding spin relaxation rate 1 ∑ ∫ dθ = πνn i τ s 2π (1−cosθ) | V SO(k,k ′ ) αβ | 2 , (2.34) α,β proportional to the concentration of impurities n i . Here α,β = ± denotes the spin indices. Sincethespin-orbitinteraction increases withtheatomic numberZ oftheimpurityelement, this spin relaxation increases as Z 2 , being stronger for heavier element impurities. 2.4.5 Bir-Aronov-Pikus Spin Relaxation Theexchangeinteraction J betweenelectrons andholesinp-dopedsemiconductors results in spin relaxation, as well.[BAP76] Its strength is proportional to the density of holes p and depends on their itinerancy. If the holes are localized they act like magnetic impurities. If they are itinerant, the spin of the conduction electrons is transfered by the exchange interaction to the holes, where the spin-orbit splitting of the valence bands results in fast spin relaxation of the hole spin due to the Elliott-Yafet, or the D’yakonov-Perel’ mechanism. 2.4.6 Magnetic Impurities Magnetic impurities have a spin S which interacts with the spin of the conduction electrons by the exchange interaction J, resulting in a spatially and temporarily fluctuating local magnetic field B MI (r) = − ∑ i Jδ(r−R i )S, (2.35) where the sum is over the position of the magnetic impurities R i . This gives rise to spin relaxation of the conduction electrons, with a rate given by 1 τ Ms = 2πn M νJ 2 S(S +1), (2.36) where n M is the density of magnetic impurities, and ν is the DOS at the Fermi energy. Here, S is the spin quantum number of the magnetic impurity, which can take the values S = 1/2,1,3/2,2.... Antiferromagnetic exchange interaction between the magnetic impurity spin and the conduction electrons results in a competition between the conduction electrons to form a singlet with the impurity spin, which results in enhanced nonmagnetic and magnetic scattering. At low temperatures the magnetic impurity spin is screened by
Chapter 2: Spin Dynamics: Overview and Analysis of 2D Systems 23 the conduction electrons resulting in a vanishing of the magnetic scattering rate. Thus, the spin scattering from magnetic impurities has a maximum at a temperature of the order of the Kondo temperature T K ∼ E F exp(−1/νJ)[MACR06, MHEZ71, ZBvDA04]. In semiconductors T K is exponentially small due to the small effective mass m e (see Appendix A) and the resulting small DOS ν. Therefore, the magnetic moments remain free at the experimentally achievable temperatures. At large concentration of magnetic impurities, the RKKY-exchange interaction between the magnetic impurities quenches however the spin quantum dynamics, so that S(S +1) is replaced by its classical value S 2 . In Mn-p-doped GaAs, the exchange interaction between the Mn dopants and the holes can result in compensation of the hole spins and therefore a suppression of the Bir-Aronov-Pikus (BAP) spin relaxation[ADK + 08]. 2.4.7 Nuclear Spins Nuclear spins interact by the hyperfine interaction with conduction electrons. The hyperfine interaction between nuclear spins Î and the conduction electron spin, ŝ, results in a local Zeeman field given by [OW53] ˆB N (r) = − 8π 3 g 0 µ B ∑ γ n Îδ(r−R n ), (2.37) γ g whereγ n isthegyromagneticratioofthenuclearspin. Thespatialandtemporalfluctuations ofthishyperfineinteraction fieldresultinspinrelaxationproportionaltoitsvariance, similar to the spin relaxation by magnetic impurities. n 2.4.8 Magnetic Field Dependence of Spin Relaxation The magnetic field changes the electron momentum due to the Lorentz force, resulting in a continuous change of the spin-orbit field, which similar to the momentum scattering results in motional narrowing and thereby a reduction of DP spin relaxation: [Ivc73, PT84, BB04], 1 τ s ∼ τ 1+ωcτ 2 2. (2.38) Another source of a magnetic field dependence is the precession around the external magnetic field. In bulk semiconductors and for magnetic fields perpendicularto a quantum well, the orbital mechanism is dominating, however. This magnetic field dependence can be used to identify the spin relaxation mechanism, since the EYS does have only a weak magnetic
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 31: 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 77 and 78: Chapter 4 Direction Dependence of S
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Page 81 and 82: Chapter 4: Direction Dependence of
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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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