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

120 Matrix elements of dot electron-confined phonon interaction
induced potential with respect to the confinement direction can be understood as
standing waves in the DP interaction potential.
The matrix elements of the interaction with the localised dot states |L〉 and |R〉
can be written as
Z
α n (q ‖ ) = λ ± dp (q ‖) d 3 rρ L (r)tcs(q l,n z) e iq ‖·r ‖
.
(D.4)
The matrix element β n (q ‖ ) follows from replacing ρ L with ρ R which are the local
electron densities in the left and right dot, respectively
ρ i (r) = 〈i|Ψ † (r)Ψ(r)|i〉.
(D.5)
Making the assumption that these densities are smooth functions ρ e centred around
the middle of the dots, ρ i (r) ≈ ρ e (r − r i ), we can calculate the interference term
|α n (q ‖ )−β n (q ‖ )| 2 =|P e (q ‖ ,q l,n )| 2∣ ∣ ∣λ
±
dp
(q ‖ )tcs( 1
2 q l,nd sinΘ) (
−1 ± e iq ‖·d )∣ ∣ ∣
2
.(D.6)
The vector d and the angle Θ are related to the dot orientation in the FSQW (see
Fig. 6.8 on page 105), and P e (q ‖ ,q l,n ) is a form factor of the electron density,
Z
P e (q) = d 3 rρ e (r)e iq·r ,
(D.7)
which can be approximated by unity for sharply localised wavefunctions.
Piezo-electric coupling
The microscopic calculation of the piezo-electric (PZ) potential caused by a confined
phonon is more complicated than the previous derivation for the DP case.
As seen in chapter 5, the potential V pz is related to the polarisation P, which is
induced by the PZ effect, via Poisson’s equation ∇ 2 V pz = 4πe ∇ · P, see Eq. (5.8).
In general, all four modes families of confined phonons couple to electrons
via PZ interaction. A further complication is the anisotropy of the PZ modulus.
Stroscio et al. [176] calculated the interaction potential for Lamb modes [n,q ‖ =
(q x ,q y )] by using an averaged expression for the PZ modulus, leading to
(
V pz (r) = ±q x q y q l λ
±
l
(q ‖ )tscq l z + λ t ± (q ‖ )tscq t z ) ]
e iq ‖·r ‖
[a n (q ‖ ) + a † n(q ‖ ) (D.8)