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

Appendix C: Cooperon and Spin Relaxation 139
in singlet-triplet representation by using the transformation U ST ,
⎛ ⎞ ⎛ ⎞ ⎛ ⎞
|⇈〉 |⇄〉 0 √2 1
−√ 1 2
0
|↑↓〉
|⇈〉
1 0 0 0
U ST ≡
⊗
=
, (C.18)
⎜
⎝ |↓↑〉 ⎟ ⎜
⎠ ⎝ |⇉〉 ⎟ ⎜ 1
⎠ ⎝ 0 √2 √2 1
0 ⎟
⎠
|〉 |〉 0 0 0 1
it can be seen that the singlet term has positive contribution to the conductivity in contrast
to the triplet terms which have a different sign: Transforming Λ using U ST we get
⎛ ⎞
−1 0 0 0
U ST ΛU −1
ST = 0 1 0 0
, (C.19)
⎜
⎝ 0 0 1 0 ⎟
⎠
0 0 0 1
where we can immediately extract the signs.
C.2 Spin-Conserving Boundary
In the following we set γ g = 1. In order to generate a finite system, we need to
specify the boundary conditions. These can be different for the spin and charge current.
Here we derive the spin-conserving boundary conditions. Let us first recall the diffusion
current density j at position r as derived from a classical picture in Sec.2.3.4, which is given
by
j si (r,t) = 〈vs k i (r,t)〉−D e ∇s i (r,t),
(C.20)
where s k i is the part of the spin-density which evolved from the spin-density at r − ∆x
moving with velocity v and momentum k. Using the Bloch equation
∂ŝ
∂t = ŝ×B SO(k)− 1ˆτ s
ŝ,
we rewrite the first term in Eq.(C.20) yielding the total spin-diffusion current as
(C.21)
j si = −τ〈v F [B SO (k)×S] i
〉−D e ∇s i .
(C.22)
In Sec.3.4.1 we consider specular scattering from the boundary with the condition that the
spin is conserved, so that the spin current density normal to the boundary must vanish
n·j si | ±W/2 = 0,
(C.23)