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

80 Chapter 4: Direction Dependence of Spin Relaxation and Diffusive-Ballistic Crossover
free path l e is comparable to the wire width W, we cannot perform the step from
to
∑
q
E,p + q
E ′ ,p ′ −q
∫ 2π
0
= 1
2πντ
∑
GE,σ R (p+q)G E A ′ ,σ ′(p′ −q) (4.47)
q
dϕ
2π (1+iτve ϕ(Q+2m e a(ϕ).S)) −1 (4.48)
and integrate in Eq.(4.3) over the Fermi surface in a continuous way. Instead, we assume
k F /W to be finite and sum over the number of discrete channels N = [k F W/π], where [...]
is the integer part. Because H γD ∼ E 2 F
this constant term due to cubic Dresselhaus should
reduce if we reduce the number of channels. If we expand the Cooperon to second order in
(Q+2eA+2m e âS) before averaging over the Fermi surface, 〈...〉, and use the Matsubara
trick, we get
C −1 ( ( ) )
= 2f 1 Q y +2α 2 S x +2 α 1 −γ D v 2f 2
3
S y
D e f 1
( )
+2f 2
(Q x −2α 2 S y −2 α 1 −γ D v 2f 2
3
S x
)
f 2
[( ) ( )
+8γDv 2 4 f 4 − f2 3
Sx 2 + f 5 − f2 3
Sy
2 f 2 f 1
]
, (4.49)
with m e = 1 and functions f i (ϕ) (App.E) which depend on the number of transverse modes
N. In the diffusive case we can perform the continuous sum over the angle ϕ in Eq.(E.3)-
(E.7), and we receive the old result with f 1 = f 2 = 1/2, f 3 = 1/8 and f 4 = f 5 = 1/16:
H c = (Q y +2α 2 S x +2
(α 1 − 1 )
2 γ DE F S y ) 2 +(Q x −2α 2 S y −2
(α 1 − 1 )
2 γ DE F S x ) 2
+(γ D E F ) 2 (S 2 x +S 2 y). (4.50)
4.5.1 Spin Relaxation at Q SO W ≪ 1
In the first section we analyzed the lowest spin relaxation in wires of different
direction ina(001) system. We have shown, that forevery direction thereis still afinitespin
relaxation at wire width which fulfill the condition Q SO W ≪ 1 due to cubic Dresselhaus
SOC. It is clear that this finite spin relaxation vanishes when the width is equal to the
Fermi wave length λ F . In the following we show how this finite spin relaxation depends
on the number of transverse channels N. We show in Ref.[WK] that the findings are
consistent with calculations going beyond the perturbative ansatz. This is possible in a