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

14 Chapter 2: Spin Dynamics: Overview and Analysis of 2D Systems
close to the Fermi energy is in general anisotropic as given by
E ± (k) = 1 √
k 2 ±αk 1−4 α 1α 2
2m e α 2 cosθsinθ, (2.19)
where k =| k |, α = √ α 2 1 +α2 2 , and k x = kcosθ. Thus, when an electron is injected with
energy E, with momentum k along the [100]-direction, k x = k,k y = 0, its wave function is a
superposition of plain waves with the positive momenta k ± = ∓αm e +m e (α 2 +2E/m e ) 1/2 .
The momentum difference k − − k + = 2m e α causes a rotation of the electron eigenstate
in the spin subspace. When at x = 0 the electron spin was polarized up spin, with the
eigenvector
⎛
⎞
ψ(x = 0) = ⎝ 1 0
⎠,
then, when its momentum points in x-direction, at a distance x, it will have rotated the
spin as described by the eigenvector
⎛ ⎞ ⎛
ψ(x) = 1 ⎝ 1 ⎠e ik +x + 1 ⎝
2 α 1 +iα 2 2
α
1
− α 1+iα 2
α
⎞
⎠e ik −x . (2.20)
In Fig.2.3 we plot the corresponding spin density as defined in Eq.(2.9) for pure Rashba
coupling, α 1 = 0. Thespinwill pointagain intheinitial direction, whenthephasedifference
1
sz
0
1
0 L SO 2 L SO
x
Figure 2.3: Precession of a spin injected at x = 0, polarized in z-direction, as it moves by
one spin precession length L SO = π/m e α through the wire with linear Rashba spin-orbit
coupling α 2 .
between the two plain waves is 2π, which gives the condition for spin precession length as