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

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38 Chapter 3: WL/WAL Crossover and Spin Relaxation in Confined Systems with Q SO = 2m e α 2 = 2π/L SO , where L SO is the spin precession length. In the representation of the singlet |⇄〉 and triplet modes, {|⇈〉,|⇉〉,|〉} it becomes ⎛ H c = ⎜ ⎝ ⎞ Q 2 0 0 0 √ 0 Q 2 SO +Q2 2Q SO Q + 0 √ √ , (3.49) 0 2Q SO Q − 2Q 2 SO +Q 2 2Q SO Q + ⎟ ⎠ √ 0 0 2Q SO Q − Q 2 SO +Q 2 with Q ± = Q y ±iQ x . Diagonalization yields the gapless singlet eigenvalues and the three triplet Cooperon eigenvalues with a gap due to the SO coupling (see Fig.3.4), (a) (b) Figure 3.4: 2D spectrum of H c , K i = Q i /Q SO . The physical meaning of the gaps in the triplet modes is more comprehensible if H c is related to spin diffusion, where the gaps appear as spin relaxation rates. E S (Q)/D e = Q 2 , (3.50) E T0 (Q)/D e = Q 2 +Q 2 , (3.51) SO √ E T± (Q)/D e = Q 2 + 3 2 Q2 SO ± Q2 SO 2 1+16 Q2 Q 2 , (3.52) SO where E S denotes the singlet eigenvalue and E T0 ,E T± the three triplet eigenvalues. Notice that the two minima of the lowest triplet eigenmode are shifted to Q = ±( √ 15/4)Q SO with a minimal eigenvalue of E/D e = (7/16)Q 2 SO . As we show in the following, this gap in the triplet modes is directly related to the D’yakonov-Perel’ spin relaxation rate 1/τ s .
Chapter 3: WL/WAL Crossover and Spin Relaxation in Confined Systems 39 Spin Diffusion We can get a better understanding of the spin relaxation induced by the SO coupling and impurity scattering by considering directly the spin-diffusion equation for the expectation value of the electron-spin vector [MC00] s(r,t) = 1 2 〈ψ† (r,t)σψ(r,t)〉, (3.53) whereψ † = (ψ † + ,ψ† − )isthetwo-component vector of theup(+), anddown(-) spinfermionic creation operators and ψ the two-component vector of annihilation operators, respectively. In the presence of SO coupling, the spin-diffusion equation becomes for v F | ∇ r s |≪ 1/τ, 0 = ∂ t s+ 1ˆτ s s−D e ∇ 2 s+γ g (B−2τ〈(∇v F )B SO (p)〉)×s (3.54) and we define accordingly the spin-diffusion Hamiltonian H SD 0 = ∂ t s+D e H SD s, (3.55) where the matrix elements of the spin relaxation terms are given by [DP71b, DP71c] (Appendix C.3) 1 = τγ 2 ( g 〈B SO (k) 2 〉δ ij −〈B SO (k) i B SO (k) j 〉 ) . (3.56) (ˆτ s ) ij For pure Rashba SO interaction, the spin-diffusion operator H SD is in momentum representation[SDGR06] ⎛ ⎞ 1 D eτ s +k 2 0 −i2Q SO k x H SD = ⎜ 1 ⎝ 0 D eτ s +k 2 −i2Q SO k y ⎟ ⎠ , (3.57) 2 i2Q SO k x i2Q SO k y D eτ s +k 2 with 1/D e τ s = Q 2 SO. In the 2D case, diagonalization yields the eigenvalues E 0 (k) = k 2 + 1 , (3.58) D e τ s √ E ± (k) = k 2 + 3 1 ± 1 1+16 k2 2D e τ s 2D e τ s Q 2 . (3.59) SO Thus, we find that the spectrum of the spin-diffusion operator and the one of the triplet Cooperon Hamiltonian are identical in 2D (Ref. [MCW97]) as long as time-reversal symmetry is not broken. This confirms that antilocalization in the presence of SO interaction,
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 47: 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
Page 97 and 98: 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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