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

72 Chapter 4: Direction Dependence of Spin Relaxation and Diffusive-Ballistic Crossover
Spin relaxation
We diagonalize the Hamiltonian, Eq.(4.10), after applying the transformation U,
taking only the lowest mode into account similar to the calculation presented in Sec.3.4.2.
The spectrum of the Hamiltonian for small wire width, Wq s < 1, is given by
(
α
E 1/2 (k x > 0) = kx 2 2
±k x
(2q sm − x1 +α 2 2
)
x2 2) −q2 W 2
12q sm
+ 3 (
qs3
2
2 2 +q2 sm ∓ q2 s3 α
2
x1 +α 2 ) 2W
x2 −q2 2
2
2k x 96q
( sm
q 2
s3
(α
2 sm) +q2 2
x1 +α 2 ) 2
x2 −q2 2
−
W 2 , (4.21)
24q 2 sm
E 1 (k x = 0) = q 2 s3 +q2 sm − (α2 x1 +α2 x2 −q2 2 )2 + q2 s3
E 2 (k x = 0) = q2 s3
2 +q2 sm + q2 s3
2
2 q2 s
W 2 , (4.22)
12
q2 2α2 x1
3qsm
2 W 2 , (4.23)
( q 2
s3
(α
E 3 = kx 2 + q2 s3
2 + 2 sm) +q2 2
x1 +α 2 x2 −q2 2
12q 2 sm
) 2
W 2 , (4.24)
with q sm = √ (α x2 −q 2 ) 2 +α 2 x1 . First we notice that the only θ dependence is in the
term q sm , which disappears if the Dresselhaus SOC strength ˜α 1 , which is shifted due to
the cubic term, equals the Rashba SOC strength α 2 and the angle of the boundary is
θ = (1/4 + n)π, n ∈. Assuming the term proportional to W 2 /k x to be small, the
absolute minimum can be found at
( ) (α
E 1/2,min = 3 qs3
2 q 2
2 2 + sm − q2 s3 2
2 x1 +α 2 ) 2
x2 −q2 2
W 2 (4.25)
24q 2 sm
which is independent of the width W if α x1 (θ = 0) = −q 2 and/or the direction of the wire
is pointing in
θ = 1 ( 2〈k
2
2 arcsin z 〉(m e γ D ) 2 ((m e v) 2 −2〈kz〉)−q 2 2
2 )
(m 3 ev 2 γ D −4〈kz〉m 2 . (4.26)
e γ D )q 2
The second possible absolute minimum, which dominates for sufficient small width W and
q sm ≠ 0 (compare with E 2 (k x = 0)), is found at
( q 2 (α
s3
E 3,min = q2 s3
2 + 2 sm) +q2 2
x1 +α 2 ) 2
x2 −q2 2
12qsm
2 W 2 . (4.27)