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

3.2 Introduction to quantum dots and various derivations 65
due to the Zeeman effect, since for fixed n + these modes converge into the same
Landau level at large B (Fig. 3.4 and 3.5). Adjacent ω + modes, however, converge
towards different Landau levels, and the Zeeman spin splitting is generally
not large enough to match this energy separation. Thus, to reach coincidence of
spin split ω + modes an additional in-plane magnetic field may be applied, enabling
the independent manipulation of Landau level separation (determined by
the perpendicular field) and Zeeman splitting (given by total field). Such coincidence
technique is utilised experimentally in SO-interacting 2D systems, see
e.g. Ref. [129].
In the following, we concentrate on the low-lying part of the dot spectrum
which is most important for the transport properties in the few-electron regime.
In Fig. 3.1a on page 53, the numerical result of a part of the excitation spectrum
of Hamiltonian (3.23) is shown for typical InGaAs parameters. The energy is
measured is units of ω 0 and the zero-point energy is omitted. Also not shown is
the energy of the ground state |n + =0,n − =0,↓〉. In contrast to the spin degenerate
Fock–Darwin spectrum in GaAs, shown in Fig. 3.4, the spectrum for InGaAs
shows the Zeeman spin splitting and anticrossings as a consequence of SO coupling
when adjacent ω − modes become degenerate. This situation is sketched
schematically in Fig. 3.5. Close to B = B 0 , the SO interaction leads to hybridisation
of |n − + 1,↓〉 and |n − ,↑〉 which results in the anticrossing shown in Fig. 3.1.
The effect of SO coupling on the spectrum is most strikingly seen in this anticrossing.
This marks the different physical relevance of the terms in Eq. (3.24)
and (3.25). The terms on the left correspond to SO-induced transitions where an
up spin is flipped downwards and the index of ω ± modes is excited by one additional
quantum, corresponding to a change of the orbital state in the language
of the quantum dot. Close to resonance B = B 0 these transitions are energy conserving;
the energy which is gained from the spin flip is transferred into a orbital
excitation. On the contrary, the terms on the right of Eqs. (3.24) and (3.25) correspond
to transitions where the spin and orbital degree of freedom are excited
simultaneously, thus being “energy non-conserving” with respect to the total excitation
operator ˆN = ˆn − + ˆn + +J z which counts to the number of boson and spin
excitations in the system.
A similar situation appeared in Sec. 2.2.4 in the high magnetic field limit of
a SO-interacting quantum wire. There, in the context of Eq. (2.26), the notion
of rotating and counter-rotating terms was introduced. Following the same line
of reasoning, we can identify terms in H SO which lead to the left hand side of
Eqs. (3.24) and (3.25) as rotating, whereas terms on the right are counter-rotating.
From quantum optics we know that the rotating-wave approximation (RWA), and
thus the neglect of counter-rotating terms, is well justified if the coupling strength
is weak. Transferring this idea to the SO-coupled QD, we can neglect terms preceeded
by γ + in Eq. (3.23) when the SO coupling is small compared to the con-