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

Appendix C
Cooperon and Spin Relaxation
C.1 Sum Formula for the Cooperon
Writing the Cooperon (Eq.3.43)
Ĉ(Q) =
1
D e (Q+2eA+2eA S ) 2 +H γD
.
(C.1)
in singlet, |S = 0;m = 0〉 = (|↑↓〉 − |↓↑〉)/ √ 2 ≡ |⇄〉 and triplet |S = 1;m = 0〉 = (|↑↓〉 +
|↓↑〉)/ √ 2 ≡ |⇉〉,|S = 1;m = 1〉 ≡ |⇈〉,|S = 1;m = −1〉 ≡ |〉 representation, without
magnetic field the singlet sector is decoupled from the triplet one. To sum over C αββα in
the case of a finite magnetic field and having calculated the eigenvectors |i〉 and eigenvalue
λ i for the Cooperon Hamiltonian, we can use the following simplification:
∑
C αββα = ∑ ∑
〈αβ|mS〉 〈 m ′ S ′ |βα 〉〈 mS|C|m ′ S ′〉 .
αβ αβ mS
m ′ S ′
Only several of the prefactors 〈αβ|mS〉〈m ′ S ′ |βα〉 are non-zero:
(C.2)
For αβ =⇈
〈⇈ |⇈〉〈⇈ |⇈〉 = 1, (C.3)
For αβ =
〈 |〉〈 |〉 = 1, (C.4)
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