Page 244 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Electron-Electron
e
,,
(
ip---Ψ p z)∇ f E k()(
(
,,
izδf kp z) +
–
k
— ( , k 0 ) + ipv δf kp z) = 0 (7.9)
(
which we may solve for δf kp z) to give
,,
∂f 0
,
(
epv Ψ p z)
k ∂ E k()
,,
(
δf kp z) = ---------------------------------------------- (7.10)
ω + iη – pv
k
(7.10) must be interpreted as the variation or response of the distribution
function due to an perturbation. From (6.10) we calculate the density
response function by integration of (7.10) over k-space and thus obtain
∂f 0
(
,
epv Ψ p ω)
1 k ∂ E k() 0 ()
3∫
3
,
,
(
,
(
δn p ω) = -------- ------------------------------------------------d k = χ n ( p ω)Ψ p ω) (7.11)
4π ω + iη – pv k
Ω k
χ 0 () ( p ω) is the zero order density susceptibility which gives us the spa-
,
n
tial variations in carrier density due to lowest order approximation. This
function is essential for calculating the dielectric properties of the semi-
conductor material.
7.2.2 The Dielectric Function
We now turn to the initial point where electrons as charged particles are
interacting. This interaction appears to create an internal potential
int
,
,
Ψ ( x t) in addition to the external potential Ψ ext ( x t) so that we have
,
to calculate with an effective potential Ψ eff ( x t) of the form
int
,
,
,
Effective Ψ eff ( x t) = Ψ ext ( x t) + Ψ ( x t) (7.12)
Potential
The additional internal potential is due to charge density variations and
according to (4.37) may be written as
e
int
Ψ ( x t) = – 4πε ∫ δn x'() 3 (7.13)
,
----------- ----------------d x'
x'
x –
0
Ω r
Semiconductors for Micro and Nanosystem Technology 241