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

116 Fock–Darwin representation of electron-phonon interaction
Here, we have omitted the effect of the operator exp(iq z z) because it solely leads
to a form factor (basically the Fourier transform of the z-component of the wavefunction).
For 2D systems with strong confinement in the growth direction this
factor is of order unity.
The matrix elements of the displacement operator with number states of a
harmonic oscillator have been derived (see e.g. Eq. (32) in Ref. [184]),
√
n!
〈n ′ |D(α)|n〉 =
n ′ ! αn′ −n (
L n′ −n
n |α|
2 ) , n ′ > n, (B.6)
with the generalised Laguerre polynomial L m n . This leads to the matrix elements
of the electron-phonon interaction in the Fock–Darwin basis,
√
〈n ′ +,n ′ −,σ ′ n + !n − !( )
|V ep (q)|n + ,n − ,σ〉 = δ σσ ′λ q α
n ′ +!n ′ + n ′ + −n +
( )
q α
− n ′ − −n −
q
−!
× e − 2(|α 1 + q | 2 +|α − q | 2 ) n L ′ +−n +
(
n + |α
+
q | 2) L n′ −−n −
(
n − |α
−
q | 2) . (B.7)
The introduction of displacement operators in Eq. (B.4) provides an illustrative
sight on the effect of electron-phonon interaction in the quantum dot. In
the Fock–Darwin basis, the scattering with phonons leads to a displacement of
the electron wavefunction. The extend of this displacement is determined by the
phonon wavelength.