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

Appendix C
Evaluation of the phonon-induced
relaxation rate
In Sec. 3.3.2 the transition rate from the upper to the lower eigenstate of the lowest
JCM-subspace was introduced using Fermi’s golden rule Eq. (3.58),
Γ ep =
V Z
(2π) 2 d 3 q ∣ ∣〈Ψ + 0
|V ep(q)|Ψ − 0 〉∣ ∣ 2 δ(∆ − ω q ), (C.1)
leading [by Eq. (3.56) and (3.59)] to
˜l 4 λ 2 Z
ph
Γ ep =
16(2π) 2 c sin2 θ + sin 2 θ −
= F
Z ∞
0
Z ∞
dq ‖ dq z
0
q 5 ‖
√
q 2 ‖ + q2 z
d 3 q q4 ‖
q e− 2 1 (˜lq ‖ ) 2 δ(∆ − cq),
( √
e − 2 1 (˜lq ‖ ) 2 δ ∆ − c q 2 ‖ + q2 z
(C.2)
)
, (C.3)
= F c
Z ∆/c
0
q 5 ‖
dq ‖ √ ( ∆c
) e − 2 1 (˜lq ‖ ) 2 ,
2
− q
2
‖
(C.4)
with F = sin 2 θ + sin 2 θ − ˜l 4 λ 2 ph
/16πc. Finally, this can be rewritten as
Γ ep = F ( ) ∆ 5 Z 1 t 5
dt √
c c 0 1 −t 2 e−(ξt)2 ,
(C.5)
with the ratio ξ = 2 −1/2 ∆˜l/c = 2 −1/2 (˜l/l 0 )(∆/ω s ), and the time a phonon needs
to propagate through the quantum dot ω −1
s = l 0 /c. Applying Eq. (3.59) we reach
the final result for the phonon induced relaxation rate,
√
Γ ep mP 2l0
=
ω 0 8π(ω s ) 2 sin
l 0 ρ M
˜l
2 θ + sin 2 θ − ξ 5 I(ξ),
(C.6)
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