ABSTRACT
In the context of quantum wires and nanostructures, in general, such a symmetry-lowering perturbation is provided by the ever-present spin-orbital (SO) coupling.
Spin-orbital interaction originates from relativistic correction to electron’s motion. Textbook examples of this consists in considering electrons moving with velocity v→ in an external electric eld E
s − o. In electron’s reference frame that eld produces magnetic eld B
s − o × v →/c2. en, spin-orbital coupling emerges
as an interaction of electron’s spin S→ with that magnetic eld:
μ = ⋅ ×
(30.1)
s − o ∝ Ze/r2 is the electric eld of the nuclei, while in fabricated nanostructures, E
s − o = −∇
Vconf(r) is associated with the structural asymmetry of the con- nement potential. SO coupling due to structure inversion asymmetry (SIA) is known as the Rashba coupling (Bychkov and Rashba 1984). For the two-dimensional electron gas, where E
s − o ∝ zˆ∂z Vconf(r) is related to the normal to the plane of motion gradient of the potential, it reads
R ( )x y y xH p p α
= σ − σ
(30.2)
e Rashba constant αR is a phenomenological parameter that describes the magnitude of E
s − o. e magnitude of asymmetry and hence the Rashba coupling strength can be controlled by the external gate voltage. In addition to the noted asymmetry of con ning potentials (which include quantum-well potential that
30.1 Introduction ...........................................................................................................................30-1 30.2 Spin-Orbit-Mediated Interaction between Spins of Localized Electrons ....................30-2
Calculation of the vdW Coupling • E ect of the Magnetic Field 30.3 Magnetized Quantum Wire with Spin-Orbit Interaction ............................................. 30-4
30.4 Conclusions...........................................................................................................................30-10 Appendix 30.A: Bosonization Basics ............................................................................................30-11 Acknowledgments ...........................................................................................................................30-13 References .........................................................................................................................................30-13
con nes electrons to a two-dimensional layer as well as transverse [in-plane] potential that forms the one-dimensional channel (Moroz and Barnes 1999, 2000), spin-orbit interaction is inherent to semiconductors of either zinc-blende or wurtzite lattice structures lacking bulk inversion symmetry (Dresselhaus 1955).
