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

Appendix C: Cooperon and Spin Relaxation 141
using U CD , Eq.(3.60),
= τU † CD{〈B SO (k) x 〉S 2 x +〈B SO(k) y 〉S 2 y
+〈B SO (k) x B SO (k) y 〉(S x S y +S y S x )}U CD
(C.30)
= τU † CD〈(B SO (k).S) 2 〉U CD (C.31)
= τ
vF
2 U † CD〈(v F B SO (k).S) 2 〉U CD .
(C.32)
Because
τ
vF
2 〈(v F [B SO (k).S]) 2 〉 = 2τ
vF
2 〈(v F [B SO (k).S])〉 2 (C.33)
is true for linear Rashba and linear Dresselhaus SO coupling but, in general, false if cubicin-k
terms are included in the SO field, we have to write
τ〈(B SO (k).S) 2 〉 = 2τ
vF
2 〈(v F B SO (k).S)〉 2 +ct
(C.34)
so that we conclude
1
ˆτ s
= U † CD(D e (2eA S ) 2 +ct)U CD
(C.35)
with the separated cubic part ct = D e m 2 eE 2 F γ2 D (S2 x+S 2 y). This reflects nothing but the fact
that the effective SO Zeeman term in Eq.(3.32) can only be rewritten as a vector potential
A S when the SO coupling is linear in momentum.
As an example we assume a very general Cooperon that means the growth direction of the
material and the SO coupling should be very general. We set
(2eA S ) 2 +H γ = (α 11 S x +α 21 S y +α 31 S z ) 2 +(α 12 S x +α 22 S y +α 32 S z ) 2 +α 4 S 2 z
(C.36)
After the transformation we get
⎛
⎞
α 2 21
1
ˆτ = 22 +α2 31 +α2 32 +α 4 −α 11 α 21 −α 12 α 22 −α 11 α 31 −α 12 α 32
⎜
⎝ −α 11 α 21 −α 12 α 22 α 2 11 +α2 12 +α2 31 +α2 32 +α 4 −α 21 α 31 −α 22 α 32
⎟
⎠ .
−α 11 α 31 −α 12 α 32 −α 21 α 31 −α 22 α 32 α 2 11 +α2 12 +α2 21 +α2 22
(C.37)
C.4 Weak Localization Correction in 2D
In contrast to the case where we have a wire with a finite width, we can calculate
the weak localization correction to the conductivity analytically in the 2D case. The cutoffs