Spin-orbit coupling and electron-phonon scattering - Fachbereich ...

2.3 Electron transport in one-dimensional systems with Rashba effect 33
SO coupling and perpendicular magnetic field act simultaneously. This energy
splitting at k = 0 can also be calculated perturbatively.
We start from the unperturbed spectrum at k = 0 of the Hamiltonian (2.26),
which is equivalent to Eq. (2.5), E (0)
n,↑↓
= Ω(n + 1/2) ± δ/2. The perturbation is
given by
H 1 = 1 l
[( 0
γ r a + γ c a †) σ + +
(γ c a + γ r a †) ]
σ − . (2.30)
2 l SO
Since H 1 is non-diagonal in spin, first order corrections vanish. Second order nondegenerate
perturbation theory straightforwardly gives for the limit |Ω ± δ| > 1
(far from resonance)
E (2)
n,↑ = 1 ( ) 2 (
l0 γ
2
)
c n
4 l SO Ω + δ − γ2 r(n + 1)
, (2.31)
Ω − δ
E (2)
n,↓ = 1 ( ) 2 (
l0 γ
2
)
r n
4 l SO Ω − δ − γ2 c(n + 1)
. (2.32)
Ω + δ
This leads to the k = 0 energy splitting ∆ n := E n,↑ − E n,↓ given by Eq. (2.13)
which is discussed on page 21.
2.3 Electron transport in one-dimensional systems
with Rashba effect
In low-dimensional systems which arise from quantum confinement, the theoretical
treatment of transport depends on how it is driven through a system. When
describing transport parallel to the barriers of the nanostructure (e.g. along the
axis of a quantum wire), in many cases, kinetic Boltzmann equation formalisms
can be used, thus, ignoring the phase information of the particles [3]. Effects
of quantum mechanics solely enter through the appropriate quantum states in the
presence of confinement and by transitions between them induced by scattering
potentials. On the contrary, if transport is driven through barriers, quantum interference
can be expected to become important when the particle traverses regions
in which the medium changes on a length scale comparable to the phase coherence
length of the particle. This condition is typically fulfilled in mesoscopic systems.
In the following, we are interested in transport through open ballistic systems
where particles are injected from leads and where the transport properties
of the system are determined by the transmission and reflection amplitudes. The
quantum mechanical derivation of the transport properties of such systems generally
involves the solution of the Schrödinger equation. Numerical approaches for