Itinerant Spin Dynamics in Structures of ... - Jacobs University
Itinerant Spin Dynamics in Structures of ... - Jacobs University
Itinerant Spin Dynamics in Structures of ... - Jacobs University
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34 Chapter 3: WL/WAL Crossover and <strong>Sp<strong>in</strong></strong> Relaxation <strong>in</strong> Conf<strong>in</strong>ed Systems<br />
3.2.2 Weak Localization <strong>in</strong> Quantum Wires<br />
If the host lattice <strong>of</strong> the electrons provides SO <strong>in</strong>teraction, quantum corrections to<br />
the conductivity have to be calculated <strong>in</strong> the basis <strong>of</strong> eigenstates <strong>of</strong> the Hamiltonian with<br />
SO <strong>in</strong>teraction<br />
H = 1<br />
2m e<br />
(p+eA) 2 +V(x)− 1 2 γ gσ(B+B SO (p)), (3.32)<br />
where m e is the effective electron mass (see AppendixA for examples <strong>in</strong> semiconductors).<br />
A is the vector potential due to the external magnetic field B. B T SO<br />
= (B SOx ,B SOy ) is<br />
the momentum dependent SO field. σ is a vector, with components σ i , i = x,y,z, the<br />
Pauli matrices, γ g is the gyromagnetic ratio with γ g = gµ B with the effective g factor <strong>of</strong><br />
the material, and µ B = e/2m e is the Bohr magneton constant. In Sec.2.3.3 we presented<br />
the dom<strong>in</strong>ant SO <strong>in</strong>teractions <strong>in</strong> semiconductors: For example, the break<strong>in</strong>g <strong>of</strong> <strong>in</strong>version<br />
symmetry <strong>in</strong> III-V semiconductors causes a SO coupl<strong>in</strong>g, which for quantum wells grown<br />
<strong>in</strong> the [001] direction is given by [Dre55]<br />
− 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)<br />
Here, α 1 = γ D 〈p 2 z〉 is the l<strong>in</strong>ear Dresselhaus parameter, which measures the strength <strong>of</strong> the<br />
term l<strong>in</strong>ear <strong>in</strong> momenta p x ,p y <strong>in</strong> the plane <strong>of</strong> the 2DES. When 〈p 2 z 〉 ∼ 1/a2 z ≥ k2 F (a z is<br />
the thickness <strong>of</strong> the 2DES and k F is the Fermi wavenumber), that term exceeds the cubic<br />
Dresselhaus terms which have coupl<strong>in</strong>g strength γ D . Asymmetric conf<strong>in</strong>ement <strong>of</strong> the 2DES<br />
yields the Rashba term which does not depend on the growth direction<br />
− 1 2 γ gB SO,R = α 2 (ê x p y −ê y p x ), (3.34)<br />
with α 2 the Rashba parameter.[BR84, Ras60] We consider the standard white-noise model<br />
for the impurity potential, V(x), which vanishes on average 〈V(x)〉 = 0, is uncorrelated,<br />
〈V(x)V(x ′ )〉 = δ(x−x ′ )/2πντ, and weak, E F τ ≫ 1. Go<strong>in</strong>g to momentum (Q) and frequency<br />
(ω) representation, and proceed<strong>in</strong>g as presented Sec.3.2.1 but now tak<strong>in</strong>g <strong>in</strong>to account<br />
the sp<strong>in</strong> degree <strong>of</strong> freedom for a two electron <strong>in</strong>terference, we yield the quantum<br />
correction to the static conductivity as [HLN80]<br />
∆σ = −2 e2 D e<br />
∑<br />
2π Vol<br />
Q<br />
∑<br />
α,β=±<br />
C αββα,ω=0 (Q), (3.35)<br />
where α,β = ± are the sp<strong>in</strong> <strong>in</strong>dices, and the Cooperon propagator Ĉ is for E Fτ ≫ 1 (E F ,<br />
Fermi energy), given by Eq.(3.30). The summation over the sp<strong>in</strong>s is described <strong>in</strong> more