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

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74 Chapter 4: Direction Dependence of Spin Relaxation and Diffusive-Ballistic Crossover At definite angles the dephasing time diverges - as for the in-plane polarized states with eigenvalue E 2 (k x = 0) - , 1 τ s (W) = 2D eq 2 2(1−sin(2θ)) (4.28) whichisplottedinFig.(4.2). Wehavelongest spindephasingtimeatθ = (1/4+n)π, n ∈. For θ = (3/4+n)π, n ∈we get the 2D result T 2 = 1/(4q 2 2 D e). We notice that this is the eigenvalue to the triplet state |S = 1;m = 0〉 of the spin relaxation tensor [DP71b, DP71c], which we derived in Sec.3.3. 1 = τγ 2 ( g 〈B SO (k) 2 〉δ ij −〈B SO (k) i B SO (k) j 〉 ) (4.29) (ˆτ s ) ij This gives an analytical description of numerical calculation done by J.Liu et al., Ref.- [LLK + 10]. Switching on cubic Dresselhaus SOC leads to finite spin dephasing time for all angles θ. In addition T 2 is then width dependent. In the case of strong cubic Dresselhaus SOC where q 2 s3 /2 = q2 1 = q2 2 , the dephasing time T 2 is angle independent and for q 2 s3 /2 > q2 1 = q2 2 the minima in T 2 (θ) change to maxima and vice versa. 50.0 0. 0.1 T2Deq 2 2 10.0 5.0 1.0 0.2 0.3 0.4 0.5 ∈ 0 Π Π 3Π Π 5Π 3Π 7Π 2Π 4 2 4 4 2 4 θ Figure4.2: ThespindephasingtimeT 2 ofaspininitially orientedalongthe[001] directionin units of (D e q2 2 ) for the special case of equal Rashba and lin. Dresselhaus SOC. The different curves show different strength of cubic Dresselhaus in units of q s3 /q 2 . In the case of finite cubic Dresselhaus SOC we set W = 0.4/q 2 . If q s3 = 0: T 2 diverges at θ = (1/4+n)π, n (dashed vertical lines). The horizontal dashed line indicated the 2D spin dephasing time, T 2 = 1/(4q 2 2 D e).
Chapter 4: Direction Dependence of Spin Relaxation and Diffusive-Ballistic Crossover 75 Special case: θ = 0 In this case the longitudinal direction of the wire is [100]. If we neglect the term proportional to W 2 /k x in Eq.(4.21) the lowest spin relaxation is found to be or depending whether ( 1 α = q2 2 s3 x1 −q 2 ) ( 2 D e τ s 2 + 2 12q 2 s q 2 s + q2 s3 2 ( 1 α = 3q2 2 s3 x1 −q 2 ) ( ) 2 D e τ s 4 + 2 qs 2 − q2 s3 2 24qs 2 ( α − q2 2 s3 x1 −q 2 ) ( 2 4 + 2 q 2 s +3 q2 s3 2 ) W 2 ) W 2 W 2 (4.30) , (4.31) 24qs 2 (4.32) is negative or positive. This shows that the cubic Dresselhaus term adds not only to the relaxation rate by a constant term but is also width dependent. However, this width dependence does not reduce the spin relaxation rate below q 2 s3 /2. 4.3 Spin relaxation in quasi-1D wire with [110] growth direction To get the spin-relaxation in a [110] quantum wire with Rashba and Dresselhaus SOC again we have to rotate the spacial coordinate system of the Dresselhaus Hamiltonian Eq.(4.2) but now with the rotation matrix We get R = ⎛ ⎜ ⎝ √1 1 2 0 √2 − 1 √ 2 0 1 √2 0 1 0 ⎞ H D[110] γ D = σ x (−k 2 x k z −2k 2 y k z +k 3 z ) +σ y (4k x k y k z ) ⎟ ⎠ . (4.33) +σ z (kx 3 −2k xky 2 −k xkz 2 ). (4.34) The confinement in z-direction (z ≡[110]) leads to 〈k z 〉 = 〈k 3 z 〉 = 0, and 〈k2 z 〉 = ∫ |∇φ| 2 dz. The Hamiltonian for the quantum wire in [110] direction has then the following form
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Itinerant Spin Dynamics in Structur
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Itinerant Spin Dynamics in Structur
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Contents v 3.2.2 Weak Localization
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Citations to Previously Published W
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Dedicated to Linda, my parents and
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Chapter 1 Introduction Structure of
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Chapter 1: Introduction 3 the coupl
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Chapter 1: Introduction 5 Throughou
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Chapter 2: Spin Dynamics: Overview
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Page 33 and 34: 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 93 and 94: Chapter 5 Spin Hall Effect 5.1 Intr
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Page 101 and 102: Chapter 5: Spin Hall Effect 91 as s
Page 103 and 104: Chapter 5: Spin Hall Effect 93 V⩵
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Page 107 and 108: Chapter 5: Spin Hall Effect 97 oper
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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
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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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