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

6.2 Details 105
PSfrag replacements
b
d
Θ
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Figure 6.8: Orientation of the quantum dots in the phonon cavity.
In bulk GaAs systems without phonon confinement, PZ interaction normally
dominates the coupling to long wavelength acoustic phonons because typical matrix
elements relate to the DP coupling as |V pz /V dp | 2 ∝ (eβ/Ξ) 2 q −2 , with the DP
constant Ξ and the PZ modulus β, see chapter 5. Thus, for long wavelength
phonons (q → 0) the PZ coupling prevails. In a phonon cavity, however, one
can argue that the confinement-induced quantisation of the transverse wave vector
gives a lower bound for the wavelength q min ∝ 1/b. Therefore, for small FSQWs
the DP coupling dominates.
We now turn to the discussion of the orientation dependence of the DP induced
inelastic current for the examples of lateral and vertical configurations. We start
with the vertical configuration (Θ = π/2) in which the dot vector is orthogonal
to the propagation of the phonons, q ‖· d = 0. Thus, from Eq. (6.24) follows that
only flexural modes contribute to the inelastic current through the DQD via DP
scattering. This can be understood by a simple symmetry consideration. Dilatational
modes induce a symmetrical DP interaction potential (see Fig. 6.2) which
affects both dots equally, hence not changing the electron transport. On the other
hand, the antisymmetric potential induced by the flexural modes strongly alters
the dot levels. A numerical evaluation of Eq. (6.15) leads to the inelastic scattering
rate which is shown in Fig. 6.3 (top). In spite of the modifications due to the
electron-phonon interaction, the main features of phonon confinement, subband
quantisation and van-Hove singularities, clearly remain visible.
On the contrary, for the lateral configuration (Θ = 0), Eq. (6.24) predicts that
the effective DOS (6.16) vanishes for flexural modes. Thus, all phonons which are
spontaneously emitted from the DQD correspond to dilatational modes. Flexural
modes induce an antisymmetric interaction potential (Fig. 6.2) which vanishes
at the location of the dots in the mid-plane of the FSQW (z = 0). On the other
hand, dilatational modes strongly couple to the DQD since they induce a non-zero
potential and lead to a phase shift in the renormalisation of the dot levels due to the
finite time a phonon needs to propagate from one dot to the next. The calculated
inelastic scattering rate for the lateral configuration is shown in Fig. 6.3 (bottom).
As for the lateral case, the van-Hove singularities remain visible in the transport.
When comparing the inelastic rates with the thermodynamic DOS, the most
obvious difference between the two configurations is seen in the behaviour at the