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

144 Appendix C: Cooperon and Spin Relaxation
C n = 1 ( n
) 2
2 +K2 x + 2P 2 −n 2 sin(Pπ)
− ,
P 2(n+P)(n−P) Pπ
(C.49)
D n = − 1 4 − n 2 −2P 2 sin(Pπ)
, (C.50)
4(n−P)(n+P) Pπ
√
2(2n 2 −P 2 )
F n =
2 ( n− P 2)(
n+
P
) sin(Pπ 2 ) . (C.51)
Pπ
2 2
For n ≠ n ′ , the spin matrices have the form
⎛
〈Q x ,n | ˜H c | Q x ,n ′ 〉 = Q2 SO
π P ⎜
⎝
0 0 0 0
0 a ig d
0 −ig b if
0 d −if c
⎞
. (C.52)
⎟
⎠
Calculating the matrix elements for n = 0,n ′ > 0, we get
⎛(
a = √ 1 1+(−1) n′) (
sin(Pπ) 4 −1+(−1) n′) K x cos ( ) ⎞
Pπ
2
⎝
2 (n ′ −2P)(n ′ −
⎠,
+2P) (n ′ −P)(n ′ (C.53)
+P)
√ (
2 1+(−1) n′) sin(Pπ)
b = −
(n ′ −2P)(n ′ , (C.54)
+2P)
⎛ (
c = √ 1 ⎝ 4 −1+(−1) n′) K x cos ( ) (
Pπ
2
1+(−1) n′) ⎞
sin(Pπ)
2 (n ′ −P)(n ′ + ⎠,
+P) (n ′ −2P)(n ′ (C.55)
+2P)
d =
(
1+(−1) n′) sin(Pπ)
√
2(n ′ −2P)(n ′ +2P) ,
⎛(
−1+(−1) n′) (
cos(Pπ)
f = 2⎝
2(n ′ −2P)(n ′ −
+2P)
⎛(
−1+(−1) n′) (
cos(Pπ)
g = −2⎝
2(n ′ −2P)(n ′ +
+2P)
1+(−1) n′) K x sin ( Pπ
2
(n ′ −P)(n ′ +P)
1+(−1) n′) K x sin ( Pπ
2
(n ′ −P)(n ′ +P)
) ⎞
(C.56)
⎠, (C.57)
) ⎞
⎠. (C.58)