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

142 Appendix C: Cooperon and Spin Relaxation
duetodephasingc 1 = 1/D e Q 2 SO τ ϕ andelasticscatteringc 2 = 1/D e Q 2 SO τ determinewhether
we have a positive or negative correction. Integrating over all possible wave vectors K =
k/Q SO in the case without boundaries yields
∆σ = − 2e2
2π
∫ √
1 c2
(
(2π) 2 dK(2πK)
0
1
+
E T+ (Q SO K)/Q 2 +
SO
+c 1
1
1
−
E S (Q SO K)/Q 2 +
SO
+c 1 E T0 (Q SO K)/Q 2 SO
+c 1
)
1
E T− (Q SO K)/Q 2 SO
+c 1
(C.38)
= − 2e2
2π
⎧
⎪⎨
arctan
+
⎪⎩
⎧
⎪⎨
arctan
+
⎪⎩
(− 1 2 ln (
1+ c 2
(
(
c 1
)
5 √ 1
4 7
16 +c 1
3 √ 1
4 7
16 +c 1
)
√
)
√
+ 1 2 ln (
1+ c 2
−arctan
7
16 +c 1
+arctan
7
16 +c 1
1+c 1
)
( √
1
16 +c 2+1
√
7
16 +c 1
( √ 1
16 +c 2−1
√
7
16 +c 1
)
)
⎛
− 1 2 ln ⎝
⎛
− 1 2 ln ⎝
2+c 1
√
3
2 +c 1 +c 2 +2
1+c 1
√
3
2 +c 1 +c 2 −2
1
16 +c 2
1
16 +c 2
⎫
⎞
⎪⎬
⎠
⎪⎭
⎫
⎞
⎪⎬
⎠
). (C.39)
⎪⎭
As an example, we choose parameters which have been used in the case of boundaries,
1/D e Q 2 τ SO ϕ = 0.08,1/D e Q 2 τ = 4: SO ∆σ/(2e2 /2π) = −0.29. The exact calculation of wide
wires (Q SO W > 1) approaches this limit as can be seen in Fig.3.11. The weak localization
correction in 2D as function of these
C.5 Exact Diagonalization
We write the inverse Cooperon propagator, the Hamiltonian ˜H c , in the representation
of the longitudinal momentum Q x , the quantized transverse momentum with quantum
numbern ∈ N, and in the representation of singlet and triplet states with quantum numbers
S,m, where we note that ˜H c is diagonal in Q x ,
〈Q x ,n,S,m | ˜H c | Q x ,n ′ ,S ′ ,m ′ 〉. (C.40)