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

34 Chapter 3: WL/WAL Crossover and Spin Relaxation in Confined Systems
3.2.2 Weak Localization in Quantum Wires
If the host lattice of the electrons provides SO interaction, quantum corrections to
the conductivity have to be calculated in the basis of eigenstates of the Hamiltonian with
SO interaction
H = 1
2m e
(p+eA) 2 +V(x)− 1 2 γ gσ(B+B SO (p)), (3.32)
where m e is the effective electron mass (see AppendixA for examples in semiconductors).
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. In Sec.2.3.3 we presented
the dominant SO interactions in semiconductors: For example, the breaking of inversion
symmetry in III-V semiconductors causes a SO coupling, which for quantum wells grown
in the [001] direction is given by [Dre55]
− 1 2 γ gB SO,D = α 1 (−ê x p x +ê y p y )+γ D (ê x p x p 2 y −ê yp y p 2 x ). (3.33)
Here, α 1 = γ D 〈p 2 z〉 is the linear Dresselhaus parameter, which measures the strength of the
term linear in momenta p x ,p y in the plane of the 2DES. When 〈p 2 z 〉 ∼ 1/a2 z ≥ k2 F (a z is
the thickness of the 2DES and k F is the Fermi wavenumber), that term exceeds the cubic
Dresselhaus terms which have coupling strength γ D . Asymmetric confinement of the 2DES
yields the Rashba term which does not depend on the growth direction
− 1 2 γ gB SO,R = α 2 (ê x p y −ê y p x ), (3.34)
with α 2 the Rashba parameter.[BR84, Ras60] We consider the standard white-noise model
for the impurity potential, 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. Going to momentum (Q) and frequency
(ω) representation, and proceeding as presented Sec.3.2.1 but now taking into account
the spin degree of freedom for a two electron interference, we yield the quantum
correction to the static conductivity as [HLN80]
∆σ = −2 e2 D e
∑
2π Vol
Q
∑
α,β=±
C αββα,ω=0 (Q), (3.35)
where α,β = ± are the spin indices, and the Cooperon propagator Ĉ is for E Fτ ≫ 1 (E F ,
Fermi energy), given by Eq.(3.30). The summation over the spins is described in more