Page 243 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Interacting Subsystems
electronic system as non-interacting particles for which only an effective
interaction due to density fluctuations is put into the streaming motion
term of the Boltzmann transport equation.
7.2.1 The Coulomb Potential (Poisson Equation)
˙
⁄
In 6.1.1 we wrote k = F — considering Newton’s law. Here F was
supposed to be an external force field. Electrons are charged particles and
thus even at low densities will interact with each other through the elec-
trostatic potential.

Let us assume that the force term is due to an externally applied electro-
static potential, which gives an electric field of the form E = – ∇Ψ x()
(see (4.34)) and the force term gets F = e∇Ψ x() . For the streaming
motion term of the Boltzmann transport equation (6.2) we therefore write

∂ ( ,, e 1
---∇ E k()∇ f kx t,,(
---∇Ψ x()∇ f kx t,,(
t ∂ f kx t) + — k ) + — k x ) = 0 (7.6)
in the absence of scattering, where we introduced a real band structure
⁄
E k() through the group velocity v = ∇ E k() — . Let us further
k
g
assume that we are near equilibrium and the distribution function
(
f kx t) is given by a well known equilibrium distribution f E k()( . )
,,
0
(
,
Ψ x t) is a weak perturbation that might also depend on time and we
(
,,
want to calculate δf kx t,,( ) = f kx t) – f E k()) the variation in
(
0
the distribution function due to this perturbational potential. For this pur-
pose we write both Ψ x t,( ) and δf kx t,,( ) as Fourier transforms in the
spatial coordinate and Laplace transforms in time. This yields
[
,
(
,
(
Ψ x t) = Re Ψ p z)e px – zt ] (7.7)
[
(
,,
(
,,
,,
(
δf kx t) = f kx t) – f E k()( ) = Re δf kp z)e px – zt ] (7.8)
0
where z = ω + iη and η introduces an adiabatic switching on of the
perturbation in time. We insert (7.7) and (7.8) in (7.6) and neglect the
,
(
second order term δf kp z,,( )Ψ p z) arising due to the gradient term
∇ f kx t,,( ) in (7.6). This yields
k
240 Semiconductors for Micro and Nanosystem Technology