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

118 Evaluation of the phonon-induced relaxation rate
PSfrag replacements
0.5
0.4
I(ξ)
0.3
0.2
0.1
0
0
1 2 3 4
ξ
Figure C.1: Numerical evaluation of the integral I(ξ).
with the characteristic frequency ω 0 = /ml0 2 and the integral
I(ξ) =
Z 1
0
t 5
dt √
1 −t 2 e−(ξt)2 .
(C.7)
For the numerical evaluation of the integral (C.7) a quickly converging series expression
is derived. The integral can be rewritten as
I(ξ) = 1 ∂ 2 Z 1
ξ 4 ∂ε 2 dt
0
= 1 ∂ 2 [
ξ 4 ∂ε 2
t
√
1 −t 2 e−ε(ξt)2 ∣ ∣∣∣ε→1
,
1 − 2εξ 2 Z 1
0
(C.8)
√
]
dt t 1 −t 2 e −ε(ξt)2 , (C.9)
ε→1
by partial integration. Successive N-fold partial integration leads to
[
I(ξ) = 1 ∂ 2 N∑
(
−2εξ
2 ) n (
−2εξ
2 ) ]
N+1 Z 1
ξ 4 ∂ε 2 n=0
(2n + 1)!! + dt t(1 −t 2 ) N+ 2 1 e
−ε(ξt) 2 (C.10)
(2N + 1)!! 0
ε→1
with the double factorial (2n + 1)!! = 1 · 3 · 5 · ... · (2n − 1) · (2n + 1). In the limit
N → ∞ the last summand in Eq. (C.10) vanishes, leading to
I(ξ) = 1 ∞
ξ ∑ 4
n=2
√ π
=
n(n − 1) (
−2ξ
2 ) n
, (C.11)
(2n + 1)!!
∞
2ξ ∑ 4 (−1)
n=2
n n(n − 1)
Γ ( n + 3 ) ξ 2n , (C.12)
2
with the Gamma function Γ. This, for reasonably small ξ, quickly converging
result for I(ξ) is shown in Fig. C.1.