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

100 Coupled quantum dots in a phonon cavity
Following Ref. [149], for T = 0 the stationary inelastic current in lowest order
in T c is given by the frequency-depending inelastic scattering rate, I inel ≈ −eγ,
with
ρ eff (ω) = ∑
q
γ(ω) ≈ 2πT 2
c ρ eff (ω), (6.15)
|α q − β q | 2
2 ω 2 δ(ω − ω q ), (6.16)
where ρ eff is an effective density of states (DOS) of the phonon environment. Every
state is weighted by the electron-phonon interaction through the interference
term |α q − β q | 2 / 2 ω 2 . Brandes & Kramer [149] utilised the interference in the
coupling of the individual dots in Eq. (6.15) to explain the oscillatory behaviour
of the inelastic current which was found in Ref. [147].
Equation (6.15) shows the direct relation between inelastic current and DOS
of the phonon environment. Therefore, a DQD is an ideal tool to investigate peculiar
phonon systems like e.g. nanomechanical resonators by means of electron
transport. In this section we investigate the influence of a mechanical confinement
of the elastic medium on the transport characteristics of an embedded DQD.
Scattering rates for electron-phonon interaction in phonon cavities have been
investigated by Stroscio et al. [175, 176, 182]. One of the systems they considered
was a free-standing quantum well (FSQW) in a slab geometry (see Fig. 6.4)
which is confined in the z-direction and widely extended in x- and y-directions. In
the following, we use a FSQW with area A ≫b 2 as a model for a phonon cavity.
We describe vibrations by a displacement field u(r,t). Assuming a homogenous,
isotropic and linear medium of the FSQW, the general solution for the displacement
field u(r,t) is determined by four families of modes – two families of shear
waves (vertically and longitudinally polarised), and two families of Lamb waves ‡
(dilatational and flexural waves). Stroscio et al. showed that electron scattering by
shear waves can be neglected since these modes do not interact with electrons via
the deformation potential interaction which is the dominant coupling mechanism
in small FSQW [176]. Therefore, in the following, we restrict ourselves to the
coupling to dilatational and flexural modes.
A further effect of the phonon confinement is the spectral quantisation in subbands
familiar from other confined systems. For each fixed in-plane wave vector
q ‖ there are infinitely many modes related to a discrete set of transversal wave
vectors pointing in the direction of the confinement. Since there are two velocities
of sound in the elastic medium, related to longitudinal and transversal wave
propagation, c l and c t , there are also two transversal wave vectors which we will
denote as q l and q t . This is in contrast to the unconfined bulk case where one can
‡ See Ref. [176] and references therein.