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

68 Chapter 4: Direction Dependence of Spin Relaxation and Diffusive-Ballistic Crossover
[AOMO01, DHR + 04] The properties of spin relaxation in systems with this growth direction
have also been related to WL measurements, Ref.[HPB + 97]. We present analytical
explanations for dimensional spin relaxation reduction and discuss the crossover from WL
to WAL, Sec.4.4.
It was shown in the previous chapter that the cubic Dresselhaus SOC leads to a term,
Eq.(3.45), which hinders the spin relaxation to vanish for small wire widths Q SO W ≪ 1
but W ≫ l e , with the wire width W and the elastic mean free path l e . As we will show
in the following sections, this term is width dependent but the finite spin relaxation rate
is not reduced if the wire is rotated in the (001) plane. However some of the experiments
are done on ballistic wires, i.e. in the regime where W ≫ l e does not hold, and we need
to modify the theory used in Ref.[Ket07] and presented in the previous chapter to enable
us to study the crossover from diffusive to ballistic wires. In Sec.4.5 we show how the spin
relaxation which is due to cubic Dresselhaus SOC reduces with the number of channels in
the quantum wire.
We consider again the Hamiltonian with SOC, Eq.(3.32)
H = 1
2m e
(p+eA) 2 +V(x)− 1 2 γ gσ(B+B SO (p)), (4.1)
where m e is the effective electron mass. A is the vector potential due to the external
magnetic field B. B T SO
= (B SOx ,B SOy ) is the momentum dependent SO field. σ is a
vector, with components σ i , i = x,y,z, the Pauli matrices, γ g is the gyromagnetic ratio
with γ g = gµ B with the effective g factor of the material, and µ B = e/2m e is the Bohr
magneton constant. To analyze the spin relaxation for different wire directions we use for
the SO interaction which is caused by BIA to lowest order in the wave vector k the general
form, Eq.(2.13),
− 1 2 γ ∑
gB SO,D = γ D ê i p i (p 2 i+1 −p 2 i+2) (4.2)
where the principal crystal axes are given by i ∈ {x,y,z},i → ((i − 1) mod 3) + 1 and
the spin-orbit coefficient for the bulk semiconductor γ D . We consider the standard whitenoise
model for the impurity potential as in the previous chapter, V(x), which vanishes on
average 〈V(x)〉 = 0, is uncorrelated, 〈V(x)V(x ′ )〉 = δ(x−x ′ )/2πντ, and weak, E F τ ≫ 1.
To address both, the WL corrections as well as the spin relaxation rates in the system, our
starting point is also here the Cooperon[HLN80]
Ĉ(Q) −1 = 1 ( ∫ dϕ
1−
τ 2π
i
1
1+iτ(v(Q+2eA+2m e âS)+H σ ′ +H Z )
)
, (4.3)