Page 245 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Interacting Subsystems
A Fourier transform yields
e
,
Ψ eff ( p ω) = Ψ ext ( p ω) – -----δn p ω,( ) (7.14)
,
2
p
Using (7.11) we have
e 0 () eff ext
,
,
,
1 + -----χ ( p ω) Ψ ( p ω) = Ψ ( p ω) (7.15)
2 n
p
which gives us the relation between the external potential and the effec-
tive potential and which we call the dynamic dielectric function
,
Ψ ext ( p ω) e 0 ()
,
(
,
ε p ω) = -------------------------- = 1 + -----χ ( p ω) (7.16)
2 n
,
Ψ eff ( p ω) p
Random in random phase approximation which means that only the internal Cou-
Phase lomb potential in addition to an external potential is taken into account
approx-
and all other possible interactions are neglected.
imation
7.2.3 Screening
The term in the definition of the dynamic dielectric function (7.16) that
arises from the density susceptibility acts as a screening for the external
potential. Let us, for this purpose, analyze the static dielectric function
,
(
ε p 0) , which, taking into account (7.11), reads
2 ∂f λ 2
e
2 ∫
(
3
,
ε p 0) = 1 – --------------- 3 0 d k = 1 + ----- 2 (7.17)
p 4π ∂ E k() p
Ω k
We defined the screening parameter λ
2
2 e ∂f 0
3
λ = – -------- 3∫ d k (7.18)
4π ∂ E k()
Ω k
Screening which is a wavelength the so called screening wavelength. Its inverse
Length may interpreted as the screening length. This can be easily seen for the

242 Semiconductors for Micro and Nanosystem Technology