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

76 Rashba spin-orbit coupling in quantum dots
PSfrag replacements
2.5
2
×100
Γep in 10 −5 ω0
2
1.5
1
1
0
1 2
0.5
0
0
1
B/B 0
2
Figure 3.7: Phonon-induced relaxation rate for InAs quantum dot. Close to B = B 0
the rate is strongly suppressed to Γ ep < 10 −7 ω 0 ≈ 10 4 s −1 .
to a strongly detuned system) the exponential decay of I(ξ) also suppresses the
rate (see appendix C).
The suppression of the relaxation rate close to the resonance can be traced back
to the factor ξ 5 in Eq. (3.61) which in turn originates from the q-dependence of
the matrix element (3.56). In a physical sense, this matrix element is a measure
for how the coupling to phonons modifies the overlap of previously orthogonal
states. Equations (B.4) and (B.5) in App. B show that the interaction with phonons
leads to a displacement of the Fock–Darwin states in the QD. The effectiveness
of this displacement on the orbital overlap manifests itself in the aforementioned
q-dependence of Eq. (3.56), showing the overlap to be maximal at ˜lq ‖ ≈ 1. Thus,
phonons couple most strongly to electrons if their wavelength is comparable to the
size of the QD, or in terms of energy, if the typical phonon energy is of order ω s .
Close to resonance, however, the splitting of eigenstates ∆ is given by the weak
SO coupling, leading to ∆ ≪ ω s . In this regime phonons couple very inefficiently
to the electrons of the QD – thus causing the relaxation rate to be suppressed.
The factor sinθ + sinθ − in Eq. (3.56) corresponds to the misalignment of spin.
Since the electron-phonon interaction (3.51) does not lead to direct spin flips,
transitions between ψ ± are only possible due to the spin admixture of the JCM
eigenstates. This admixture is maximal at resonance with sin 2 θ ± = 1/2. Far from
resonance ψ +(−) are approximately spin-up(down) like, suppressing the transition
rate by the factor sin 2 θ + sin 2 θ − .
For a quantitative analysis we introduce acousto-mechanical parameters of
bulk InAs [142], ρ M = 5.7 · 10 3 Kg/m 3 , c long = 3.8 · 10 3 m/s, c trans = 2.6 · 10 3 m/s,