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

46 Chapter 3: WL/WAL Crossover and Spin Relaxation in Confined Systems
which are persistent for α 2 = α 1 . The first solution is s = s 0 (α 1 ,α 2 ,0) T for Q x = 0 which
is aligned with the effective SO field B SO (k) = −2γ g k x (α 1 ,α 2 ,0) T . In this case, we have
according to Eq.(3.81) H s,RD1 (W) = (1/12)( ˜Q SO W) 2 H s , with ˜Q 2 SO = (2m e(α 2 1 −α2 2 ))2 /α 2
and 1/τ s = 2p 2 F α2 τ, α = √ α 2 1 +α2 2 . As mentioned above by transforming the vector potential
A S , Eq.(3.75), this alignment occurs due to the constraint on the spin-dynamics
imposed by the boundary condition as soon as the wire width W is smaller than the spin
precession length L SO . In addition, we find two spin helix solutions in narrow wires,
⎛ ⎞ ⎛ ⎞
− α 2
α ( ) 0 ( )
s = s 0
⎜ α 1 ⎟ 2π
⎝ α ⎠ sin x +s 0
⎜
L SO
⎝ 0 ⎟ 2π
⎠ cos x , (3.82)
L SO
0
1
and the linearly independent solution, obtained by interchanging cos and sin in Eq.(3.82).
The form of this long persisting spin helix depends therefore on the ratio of linear Rashba
and linear Dresselhaus coupling strength, Fig.3.6, and its spin relaxation rate is diminished
as H s,RD2/3 = (1/2)H s,RD1 .
Figure 3.6: Long persisting spin helix solution of the spin-diffusion equation in a quantum
wire whose width W is smaller than the spin precession length L SO for varying ratio of
linear Rashba α 2 = αsinϕ and linear Dresselhaus coupling, α 1 = αcosϕ, Eq.(3.82), for
fixed α and L SO = π/m e α.
3.4.3 Exact Diagonalization
The exact diagonalization of the inverse Cooperon propagator, as obtained after
the non-Abelian transformation, Eq.(3.73), is performed in the basis of transverse standing
{
waves, satisfyingNeumannboundaryconditions, 1/ √ W, √ 2/ √ }
W cos((nπ/W)(y −W/2))