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

Chapter 4: Direction Dependence of Spin Relaxation and Diffusive-Ballistic Crossover 71
functions (with k x ,k y ) to H c , Eq.(4.10), we end up with (singlet part left out)
H c
q 2 2
=
⎛
⎜
⎝
⎞
2+Q 2 f θφ −2i exp(2iθ)
4+Q 2 f ⎟ θφ ⎠ (4.16)
c.c. 2+Q 2
with k x /q 2 = Qcos(φ), k y /q 2 = Qsin(φ) and
f θφ = (i −1) √ 2exp(iθ)(cos(φ+θ)−sin(φ+θ))Q. (4.17)
Vanishing spinrelaxation is foundat Q = 0 for arbitrary values of θ (the spinwith vanishing
spin relaxation is pointing along the [110] direction[SEL03]). Another solution is found at
Q = 2 with the condition θ + φ = 3π/4, which is equivalent to the [110] crystallographic
direction.[CWd07]
4.2.2 Quasi-1D wire
In the following we consider spin and charge conserving boundaries. Due to the
SOC we have, as in the previous chapter, Eq.(3.70), modified Neumann condition with the
slight change which is the angle dependency of the vector potential:
(− τ D e
n·〈v F [γ g B SO (k)·S]〉−i∂ n
)
C| ∂S = 0, (4.18)
where 〈...〉 denotes the average over the direction of v F and k which we rewrite for the
rotated x-y system
(−i∂ y +2e(A S ) y
)C
(
x,y = ± W )
= 0, ∀x, (4.19)
2
where n is the unit vector normal to the boundary ∂S and x is the coordinate along the
wire. In order to do a diagonalization taking only the zero-mode into account, we have
to simplify the boundary condition. A transformation acting in the transverse direction is
needed according to Eq.(4.10): Ĉ → ˜Ĉ = U A ĈU † A
, by using the transformation
U =½4 −i sin(q s y) 1 q s
A y +(cos(q s y)−1) 1 q 2 sA 2 y (4.20)
with A y = (α x2 +q 2 )S x −α x1 S y and q s = √ (α x2 +q 2 ) 2 +α 2 x1 .