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

40 Chapter 3: WL/WAL Crossover and Spin Relaxation in Confined Systems
which has its cause in the suppression of the triplet modes in Eq.(3.46), is indeed a direct
measure of the spin relaxation. Mathematically, there exists a unitary transformation
H c = U CD H SD U † CD, (3.60)
⎛ ⎞
U CD =
⎜
⎝
− 1 √
2
i √
2
0
0 0 1
1√
2
i √
2
0
⎟
⎠ , (3.61)
with the according transformation between spin-density components s i and the triplet components
of the Cooperon density ˜s,
1
√
2
(−s x +is y ) = ˜s ⇈ , (3.62)
s z = ˜s ⇉ , (3.63)
1
√ (s x +is y ) = ˜s . 2
(3.64)
This is a consequence of the fact that the four-component vector of charge density ρ =
(ρ + + ρ − )/2 and spin-density vector S are related to the density vector ˆρ with the four
components 〈ψ † αψ β
〉/ √ 2, where α,β = ±, by a unitary transformation.
Relation to the Diffuson
The classical evolution of the four-component density vector ˆρ is by definition
governed by the diffusion operator, the Diffuson. The Diffuson is related to the Cooperon
in momentum space by substituting Q → p−p ′ and the sum of the spins of the retarded
and advanced parts, σ and σ ′ , by their difference. Using this substitution, Eq.(3.48) leads
thus to the inverse of the Diffuson propagator
H d :=
ˆD
−1
= Q 2 +2Q SO (Q y˜Sx −Q x˜Sy )+Q 2
D (˜S SO y 2 + ˜S x 2 ), (3.65)
e
with ˜S = (σ ′ −σ)/2, which has the same spectrum as the Cooperon, as long as the timereversal
symmetry is not broken. In the representation of singlet and triplet modes the
diffusion Hamiltonian becomes
⎛ √
2Q 2 SO
+Q 2 2Q SO Q − 0 − √ ⎞
2Q SO Q +
√ 2Q SO Q + Q 2 SO
H d =
+Q2 0 0
. (3.66)
⎜
⎝ 0 0 Q 2 0 ⎟
− √ ⎠
2Q SO Q − 0 0 Q 2 SO
+Q 2