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

16 Chapter 2: Spin Dynamics: Overview and Analysis of 2D Systems
averaging their product yields a finite value, since ∆x depends on the momentum at time t,
k(t), yielding 〈∆xB SOi (k(t))〉 = 2∆t〈v F B SOi (k(t))〉, where 〈...〉 denotes the average over
the Fermi surface. This way, we can also evaluate the average of the spin-orbit term in
Eq.(2.22), expanded to first order in ∆x, and get, substituting ∆t → τ the spin diffusion
equation,
∂s
∂t = −B×s+D e∇ 2 s+2τ〈(∇v F )B SO (p)〉×s− 1ˆτ s
s, (2.23)
Spin polarized electrons injected into the sample spread diffusively, and their spin polarization,
while spreading diffusively as well, decays in amplitude exponentially in time. Since,
between scattering events the spins precess around the spin-orbit fields, one expects also an
oscillation of the polarization amplitude in space. One can find the spatial distribution of
the spin density which is the solution of Eq.(2.23) with the smallest decay rate Γ s . As an
example, the solution for linear Rashba coupling is, [SDGR06]
s(x,t) = (ê q cos(qx)+Aê z sin(qx))e −t/τs , (2.24)
with 1/τ s = 7/16τ s0 where 1/τ s0 = 2τkF 2α2 2 and where the amplitude of the momentum q is
determined by D e q 2 = 15/16τ s0 , and A = 3/ √ 15, and ê q = q/q. This solution is plotted in
Fig.2.4 for ê q = (1,1,0)/ √ 2. We will derive this solution in the context of local corrections
to the static conductivity, Chapter3. Thereby we show that by adding Dresselhaus SOC it
will be even possible to create persistent solutions.
InFig.2.5 weplotthelinearly independentsolution obtained by interchanging cos andsinin
Eq.(2.24), with the spin pointing in z-direction, initially. We choose ê q = ê x . Comparison
with the ballistic precession of the spin, Fig 2.5 shows that the period of precession is
enhanced by the factor 4/ √ 15 in the diffusive wire, and that the amplitude of the spin
density is modulated, changing from 1 to A = 3/ √ 15.
Injecting a spin-polarized electron at one point, say x = 0, its density spreads
the same way it does without spin-orbit interaction, ρ(r,t) = exp(−r 2 /4D e t)/(4πD e t) dD/2 ,
where r is the distance to the injection point. However, the decay of the spin density is
periodically modulated as a function of 2π √ 15/16r/L SO .[Fro01] The spin-orbit interaction
together with the scattering from impurities is also a source of spin relaxation, as we discuss
in the next Section together with other mechanisms of spin relaxation. We can find the
classical spin diffusion current in the presence of spin-orbit interaction, in a similar way
as one can derive the classical diffusion current: The current at the position r is a sum