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

64 Rashba spin-orbit coupling in quantum dots
PSfrag replacements
|n − + 1〉
|n − + 1,↑〉
E
|n − 〉
|g|µ B B
|n − + 1,↓〉
|n − ,↑〉
|n − ,↓〉
B = 0
B
B ≈ B 0
Figure 3.5: Resonance condition for Zeeman split ω − modes at B = B 0 (n + fixed).
spin sequence of the Zeeman effect where electrons with spin up have a larger
energy than those with spin down. Doing this we arrive at the Hamiltonian
(
H =ω + n + + 1 ) (
+ ω − n − + 1 )
+ 1 2
2 2 |g|µ BBσ z
+ α˜l
) )]
[γ −
(a − σ + + a † −σ − − γ +
(a + σ − + a † +σ + . (3.23)
In analogy to quantum optics we can understand the above Hamiltonian in
terms of a spin which is coupled to two boson modes with energies ω ± . Here,
the coupling is mediated by the SO interaction. The spectral properties of the two
boson modes alone has been discussed in Sec. 3.2.1, leading to the Fock–Darwin
states (Fig. 3.4). In Eq. (3.23) the presence of the spin leads to two effects. (i)
Every state becomes spin split due to the Zeeman effect and (ii) the additional
coupling between boson modes and spin leads to anticrossings in the spectrum. In
the representation in terms of ω ± modes we see that the SO interaction leads to a
coupling between adjacent ω ± modes with opposite spin due to the operators a ±
and a † ± in the last term of the Hamiltonian. There are no direct transitions between
ω + and ω − modes.
From the last term in Eq. (3.23) we see that the SO coupling has only non-zero
matrix elements for
〈n + + 1,n − ,↓|H SO |n + ,n − ,↑〉, 〈n + + 1,n − ,↑|H SO |n + ,n − ,↓〉, (3.24)
〈n + ,n − + 1,↓|H SO |n + ,n − ,↑〉, 〈n + ,n − + 1,↑|H SO |n + ,n − ,↓〉. (3.25)
Therefore, from a perturbative point of view, the SO coupling becomes most
import when adjacent ω ± modes with opposite spin are almost degenerate. The
strategy to reach such a resonance is different for ω + and ω − modes. As a function
of magnetic field, resonances of neighbouring ω − modes can easily occur