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

2.2 Spectral properties, various limits 27
2.2 Spectral properties, various limits
Geometrical confinement leads to quantisation of the orbital motion. Therefore,
the natural language to describe the wavefunction in a quasi-1D electron system is
in terms of confined transverse subbands. In Sec. 2.1.2, the effective mass Hamiltonian
of the spin-orbit (SO) interacting quantum wire (QWR) [Eq. (2.1) on page
18] is rewritten in the basis of transverse subbands corresponding to parabolic
confinement. Formally, this leads to a bosonic representation [Eq. (2.5) on page
19] in which the SO interaction leads to a coupling between nearest-neighbouring
subbands of opposite spin, see Fig. 2.2a. In Eq. (2.5) the operators a † k and a k
act on the transverse eigenstates such that a † k a k gives the subband number of the
electron that propagates with longitudinal momentum k. Clearly, electrons are
fermions. We use the notion “bosonic” because the oscillator operators have a
commutator [a k ,a † k ] = 1. Here, a† k and a k describe transitions between transverse
subbands – not particle creation and annihilation. In this form, the Hamiltonian
resembles models of matter-light interaction of quantum optics like the Rabi
Hamiltonian [82], where, in the simplest case, two atomic levels – representing a
pseudo-spin – are coupled to a monochromatic radiation mode, see Fig. 2.5. In
quantum optics such pseudo-spin boson models consist of distinct physical subsystems
(atom and light) whereas in the case of QWR two degrees of freedom of
the same particle are coupled.
Due to the complexity of the coupling between spin and orbitals in Eq. (2.5),
in general, no analytical solution of the Hamiltonian is feasible. We apply an exact
numerical diagonalisation to find the spectral properties of the wire. Figure 2.6
shows the low-energy spectra of the QWR for various strengths of confinement,
SO coupling and perpendicular magnetic field. For a better physical understanding
we summarise analytical limits and approximations in the following.
2.2.1 Zero magnetic field
For B = 0 the Hamiltonian (2.5) reduces to
H(k)
= a †
ω
k a k + 1
0 2 + 1 2 (kl 0) 2 + 1 (
l 0
kl 0 σ x +
i
)
√
(a k − a †
2 l SO 2
k
)σ y , (2.17)
=: H 0 + H mix , (2.18)
with H mix = 2 −3/2 i(l 0 /l SO )(a k − a † k )σ y. This limit was studied in detail by
Moroz & Barnes [67], and Governale & Zülicke [69,70]. H 0 represents the maxi-