Itinerant Spin Dynamics in Structures of ... - Jacobs University
Itinerant Spin Dynamics in Structures of ... - Jacobs University
Itinerant Spin Dynamics in Structures of ... - Jacobs University
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Chapter 3: WL/WAL Crossover and <strong>Sp<strong>in</strong></strong> Relaxation <strong>in</strong> Conf<strong>in</strong>ed Systems 35<br />
detail <strong>in</strong> AppendixC.1. Thus, the problem reduces to the calculation <strong>in</strong> presence <strong>of</strong> SOC<br />
<strong>of</strong> the correlation function<br />
∑<br />
q<br />
E,p + q<br />
E ′ ,p ′ −q<br />
= 1<br />
2πντ<br />
∑<br />
GE,σ R (p+q)G E A ′ ,σ ′(p′ −q), (3.36)<br />
q<br />
which simplifies for weak disorder ǫ F τ ≫ 1 to<br />
∫ dΩ 1<br />
≈<br />
2π 1−iτˆΣ , (3.37)<br />
where<br />
ˆΣ = ǫ p ′ +q,σ ′ −ǫ p−q,σ. (3.38)<br />
For diffusive wires, for which the elastic mean-free path l e is smaller than the wire width<br />
W, the <strong>in</strong>tegral is over all angles <strong>of</strong> velocity v on the Fermi surface. Us<strong>in</strong>g<br />
ǫ p = (p+eA)2 − 1 2m e 2 γ gσ(B+B SO (p)),<br />
v = p−q+eA ,<br />
m e<br />
S = 1 2 (σ +σ′ ),<br />
Q = p+p ′ ,<br />
we obta<strong>in</strong> to lowest order <strong>in</strong> Q,<br />
ˆΣ = −v(Q+2eA+2m e âS)+(Q+2eA)âσ ′ + 1 2 γ g(σ ′ −σ)B. (3.39)<br />
Here, the SO coupl<strong>in</strong>gs are comb<strong>in</strong>ed <strong>in</strong> the matrix<br />
⎛<br />
Thus, the Cooperon becomes<br />
Ĉ(Q) −1 = 1 τ<br />
â =<br />
( ∫ dΩ<br />
1−<br />
2π<br />
⎝ −α 1 +γ D ky 2 −α 2<br />
α 2 α 1 −γ D kx<br />
2<br />
1<br />
1+iτ(v(Q+2eA+2m e âS)+H σ ′ +H Z )<br />
⎞<br />
⎠. (3.40)<br />
)<br />
, (3.41)<br />
where H σ ′ = −(Q+2eA)âσ ′ and the Zeeman coupl<strong>in</strong>g to the external magnetic field yields<br />
H Z = − 1 2 γ g(σ ′ −σ)B. (3.42)