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

Chapter 3: WL/WAL Crossover and Spin Relaxation in Confined Systems 39
Spin Diffusion
We can get a better understanding of the spin relaxation induced by the SO
coupling and impurity scattering by considering directly the spin-diffusion equation for the
expectation value of the electron-spin vector [MC00]
s(r,t) = 1 2 〈ψ† (r,t)σψ(r,t)〉, (3.53)
whereψ † = (ψ † + ,ψ† − )isthetwo-component vector of theup(+), anddown(-) spinfermionic
creation operators and ψ the two-component vector of annihilation operators, respectively.
In the presence of SO coupling, the spin-diffusion equation becomes for v F | ∇ r s |≪ 1/τ,
0 = ∂ t s+ 1ˆτ s
s−D e ∇ 2 s+γ g (B−2τ〈(∇v F )B SO (p)〉)×s (3.54)
and we define accordingly the spin-diffusion Hamiltonian H SD
0 = ∂ t s+D e H SD s, (3.55)
where the matrix elements of the spin relaxation terms are given by [DP71b, DP71c]
(Appendix C.3)
1
= τγ 2 (
g 〈B SO (k) 2 〉δ ij −〈B SO (k) i B SO (k) j 〉 ) . (3.56)
(ˆτ s ) ij
For pure Rashba SO interaction, the spin-diffusion operator H SD is in momentum representation[SDGR06]
⎛
⎞
1
D eτ s
+k 2 0 −i2Q SO k x
H SD = ⎜ 1
⎝ 0
D eτ s
+k 2 −i2Q SO k y
⎟
⎠ , (3.57)
2
i2Q SO k x i2Q SO k y D eτ s
+k 2
with 1/D e τ s = Q 2 SO. In the 2D case, diagonalization yields the eigenvalues
E 0 (k) = k 2 + 1 , (3.58)
D e τ s
√
E ± (k) = k 2 + 3 1
± 1 1+16 k2
2D e τ s 2D e τ s Q 2 . (3.59)
SO
Thus, we find that the spectrum of the spin-diffusion operator and the one of the triplet
Cooperon Hamiltonian are identical in 2D (Ref. [MCW97]) as long as time-reversal symmetry
is not broken. This confirms that antilocalization in the presence of SO interaction,