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

106 Coupled quantum dots in a phonon cavity
onset of higher subbands. In the vertical configuration the jumps of the DOS also
appear in the inelastic rate whereas for the lateral orientation continuous cusps
are seen only. Although the microscopic coupling between electrons and confined
phonons is the same in both configurations, the effect on the DQD depends on
the orientation. Apart from the modes that lead to van–Hove singularities, the
subband minimum is at q ‖ = 0. These modes correspond to non-propagating excitations
of the elastic medium which nevertheless lead to displacement patterns
with a non-vanishing local change in volume, and thus induce a DP potential. In
contrast to the vertical configuration, the lateral setup with both dots in the middle
of the cavity (Θ = 0) is not sensitive to non-propagating excitations because the
displacement for such modes is constant at the locations of the dots; only propagating
modes lead to a finite phase shift in the interaction potentials between the
dots and thus affect the transport.
Although being of minor relevance in small FSQWs, the PZ coupling also
contributes to the inelastic scattering rate. The orientation dependence and the
corresponding decoupling of individual Lamb modes is reversed as compared to
the DP case, see Eq. (6.24).
6.2.3 Discussion
From the inelastic rates, which are shown in Fig. 6.3, for the example of DP
electron-confined phonon interaction, we can see that in addition to the generic
subband quantisation also the van–Hove singularities of the phonon DOS show
up in the current through the DQD, acting as clear fingerprints of phonon confinement.
Close to the singularity, the application of the perturbative derivation which
led to the inelastic rate for weak electron-phonon coupling, Eqs. (6.15) and (6.16),
is questionable. However, led by the numerical results for the phonon DOS, we
can investigate the influence of the van–Hove singularity in a non-perturbative
way by decomposing the phonon spectral density (6.16) to find an approximate
analytical model
ρ eff (ω) = ρ Ohm + ρ 0 δ(ω − ω v ), (6.25)
consisting of a van–Hove singularity at ω v and a background of lower order
subbands which are assumed to form an ohmic bath. Following Ref. [149], the
inelastic current at zero temperature can be written as
I inel ≈ 2πT 2
c ∑
n
w n δ(ω − nω v ). (6.26)
Thus, in addition to the main peak a ω v , the van–Hove singularity leads to
non-perturbative satellite peaks at harmonics of the frequency ω v with oscillator