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

138 Appendix C: Cooperon and Spin Relaxation
For αβ =↑↓
〈↑↓ |⇄〉〈⇄ |↓↑〉 = − 1 2 ,
〈↑↓ |⇄〉〈⇉ |↓↑〉 = + 1 2 ,
〈↑↓ |⇉〉〈⇄ |↓↑〉 = − 1 2 ,
〈↑↓ |⇉〉〈⇉ |↓↑〉 = + 1 2 ,
(C.5)
(C.6)
(C.7)
(C.8)
For αβ =↓↑
〈↓↑ |⇄〉〈⇄ |↑↓〉 = − 1 2 ,
〈↓↑ |⇄〉〈⇉ |↑↓〉 = − 1 2 ,
〈↓↑ |⇉〉〈⇄ |↑↓〉 = + 1 2 ,
〈↓↑ |⇉〉〈⇉ |↑↓〉 = + 1 2 .
(C.9)
(C.10)
(C.11)
(C.12)
Inserting the eigenvectors,
∑
C αββα = ∑
αβ αβ
we end up with
∑
〈αβ|mS〉 〈 m ′ S ′ |βα 〉 〈mS|i〉 〈 i|C|m ′ S ′〉 ,
∑
mS i
m ′ S ′
(C.13)
= ∑ i
(〈⇈ |i〉〈i|⇈〉+〈⇉ |i〉〈i|⇉〉+〈 |i〉〈i|〉−〈⇄ |i〉〈i|⇄〉)λ −1
i
. (C.14)
Writing
∑
C αββα = C ↑↑,↑↑ +C ↑↓,↓↑ +C ↓↑,↑↓ +C ↓↓,↓↓
αβ
⎡⎛
⎞ ⎤
1 0 0 0
0 0 1 0
= Tr
C
⎢⎜
⎣⎝
0 1 0 0 ⎟ ⎥
⎠ ⎦
0 0 0 1
≡ Tr[ΛC]
(C.15)
(C.16)
(C.17)