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