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

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36 Chapter 3: WL/WAL Crossover and Spin Relaxation in Confined Systems It follows that for weak disorder and without Zeeman coupling, the Cooperon depends only on the total momentum Q and the total spin S. Expanding the Cooperon to second order in (Q+2eA+2m e âS) and performing the angular integral which is for 2D diffusion (elastic mean-free path l e smaller than wire width W) continuous from 0 to 2π and yields Ĉ(Q) = 1 D e (Q+2eA+2eA S ) 2 +H γD . (3.43) The effective vector potential due to SO interaction, A S = m eˆαS/e (where ˆα = 〈â〉 denotes the matrix Eq.(3.40), as averaged over angle), couples to total spin vector S whose components are four by four matrices. The cubic Dresselhaus coupling is found to reduce the effect of the linear one to ˜α 1 := α 1 −m e γ D E F /2. (3.44) Furthermore, it gives rise to the spin relaxation term in Eq.(3.43), H γD = D e (m 2 e E Fγ D ) 2 (Sx 2 +S2 y ). (3.45) In the representation of the singlet, |⇄〉 and triplet states |⇉〉,|⇈〉,|〉 (Tab.3.1), Ĉ destate (index: electron-number) m s S |⇄〉 := 1 √ 2 (|↑〉 1 |↓〉 2 −|↑〉 2 |↓〉 1 ) 0 0 |⇈〉 := |↑〉 1 |↑〉 2 1 1 |⇉〉 := 1 √ 2 (|↑〉 1 |↓〉 2 +|↑〉 2 |↓〉 1 ) 0 1 |〉 := |↓〉 1 |↓〉 2 −1 1 Table 3.1: Singlet and triplet states couples into a singlet and a triplet sector. Thus, the quantum conductivity is a sum of singlet and triplet terms ⎛ ∆σ = −2 e2 D e 2π Vol ∑ 1 − ⎜ D Q ⎝ e (Q+2eA) } {{ 2 + ∑ } singlet contribution 〈 ∣ 〉 ∣ ∣∣S S = 1,m ∣Ĉ(Q) = 1,m . (3.46) ⎟ m=0,±1 ⎠ } {{ } triplet contribution ⎞
Chapter 3: WL/WAL Crossover and Spin Relaxation in Confined Systems 37 With the cutoffs due to dephasing 1/τ ϕ and elastic scattering 1/τ, we can integrate over all possible wave vectors Q in the 2D case analytically (Appendix C.4). In 2D, one can treat the magnetic field nonperturbatively using the basis of Landau bands.- [HLN80, KSZ + 96, MZM + 03, AF01, LG98, Gol05] In wires with widths smaller than cyclotron length k F l 2 B (l B, the magnetic length, defined by Bl 2 B = 1/e), the Landau basis is not suitable. There is another way to treat magnetic fields: quantum corrections are due to the interference between closed time-reversed paths. In magnetic fields, the electrons acquire a magnetic phase, which breaks time-reversal invariance. Averaging over all closed paths, one obtains a rate with which the magnetic field breaks the time-reversal invariance, 1/τ B . Like the dephasing rate 1/τ ϕ , it cuts off the divergence arising from quantum corrections with small wave vectors Q 2 < 1/D e τ B . In 2D systems, τ B is the time an electron diffuses along a closed path enclosing one magnetic flux quantum, D e τ B = lB 2 . In wires of finite width W the area which the electron path encloses in a time τ B is W √ D e τ B . Requiring that this encloses one flux quantum gives 1/τ B = D e e 2 W 2 B 2 /3. For arbitrary magnetic field, the relation 1 τ B = D e (2e) 2 B 2 〈y 2 〉, (3.47) with the expectation value of the square of the transverse position 〈y 2 〉, yields 1/τ B = ( 1−1/(1+W 2 /3lB 2 )) D e /lB 2 . Thus, it is sufficient to diagonalize the Cooperon propagator as given by Eq.(3.43) without magnetic field, as we will do in the next chapters, and to add the magnetic rate 1/τ B together with dephasing rate 1/τ ϕ to the denominator of Ĉ(Q) when calculating the conductivity correction, Eq.(3.46). 3.3 The Cooperon and Spin Diffusion in 2D The Cooperon can be diagonalized analytically in 2D for pure Rashba coupling, α 1 = 0,γ D = 0. For this case, we define the Cooperon Hamilton operator as H c := Ĉ−1 = Q 2 +2Q SO (Q y S x −Q x S y )+Q 2 SO D (S2 y +S2 x ), (3.48) e
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 and 36: Chapter 3 WL/WAL Crossover and Spin
Page 37 and 38: Chapter 3: WL/WAL Crossover and Spi
Page 45: Chapter 3: WL/WAL Crossover and Spi
Page 77 and 78: Chapter 4 Direction Dependence of S
Page 79 and 80: Chapter 4: Direction Dependence of
Page 93 and 94: Chapter 5 Spin Hall Effect 5.1 Intr
Page 95 and 96: 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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