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

140 Appendix C: Cooperon and Spin Relaxation
where n is the vector normal to the boundary. Noting the relation between the spindiffusion
equation in the s i representation and the triplet components of the Cooperon
density ˜s i ({|⇈〉,|⇉〉,|〉}), Eq.(3.60),
U CD (ǫ ijk B SO,j ) i=1..3,k=1..3 U † CD = −i(〈˜s i |B SO ·S|˜s k 〉) i=1..3,k=1..3 ,
(C.24)
where the matrix U CD is given by Eq.(3.61), we can thereby transform the boundary condition
for the spin-diffusion current, Eq.(C.23), to the triplet components of the Cooperon
density ˜s i ,
0 = n·j˜si | y=±W/2 . (C.25)
Requiring also that the charge density is vanishing normal to the transverse boundaries,
which transforms into the condition −i∂ n˜ρ| Surface = 0 for the singlet component of the
Cooperon density ˜ρ, we finally get the boundary conditions for the Cooperon without external
magnetic field, Eq.(3.70),
(− τ D e
n·〈v F [B SO (k)·S]〉−i∂ n
)
C| Surface = 0. (C.26)
The last expression can be rewritten using the effective vector potential A S , Eq.(3.43),
(n·2eA S −i∂ n )C| Surface = 0.
(C.27)
In the case of Rashba and linear and cubic Dresselhaus SO coupling in (001) systems, we
get
D e
τ 2eA S = −〈v F (B SO (k)·S)〉
= v 2 Fm e
⎛
⎝ −(α 1 − γ D(m ev F ) 2
4
) −α 2
α 2 α 1 − γ D(m ev F ) 2
4
C.3 Relaxation Tensor
⎞
⎠.S.
(C.28)
To connect the effective vector potential A S with the spin relaxation tensor, we
notice that ˆτ can be rewritten in the following way:
1
ˆτ s
= τ(〈B SO (k) 2 〉δ ij −〈B SO (k) i B SO (k) j 〉) i=1..3,j=1..3
(C.29)