Page 196 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

The Semi-Classical Boltzmann Transport Equation
reason is that position and momentum are canonically conjugated vari-
ables. According to (5.6) the density of states is 1 ( ⁄ 2π) 3 . Note that for
two independent spin directions an electron may assume this value twice.
(
3
3
3 –
Thus 4π ) f kx t,,( )d kd x gives the number of particles which
1
3
x
reside in a small volume element d x around the position vector and
3
k
in a small volume element d k around the wave vector . We think of
this phase space simply as the vector space formed by the three positions
and the three momenta of a carrier. f kx t,,( ) is a particle density in
phase space similar to the charge density and total charge in (4.33). Per-
forming the integration over the entire phase space Ω the system
assumes
(
( 4π ) 1 ∫ f kx t)d kd x = N (6.1)
3 –
3
3
,,
Ω
gives us the number N of particles present in the system.The BTE is a
law of motion for a single particle density to evolve in this space. A gen-
eral treatment of the BTE can be found in 6.1 a more semiconductor spe-
cific description is given in [6.2] and [6.3]
∂ ( ,, k∇ f kx t,,( ( ,, Cf kx t))
˙
(
(
,,
t ∂ f kx t) + k ) + x ˙∇ f kx t) = (6.2)
x
The l.h.s. of (6.2) is called the streaming motion term or convective term.
The path a single particle follows we call a phase-space trajectory.
Knowing this trajectory of a single particle with arbitrary initial condi-
tion the streaming motion term is determined. This will be discussed in
the next section. The r.h.s. of (6.2) is called the collision term or scatter-
ing term and is discussed in 6.1.2.

6.1.1 The Streaming Motion

,,
(
(
In the case where Cf kx t)) = 0 there is only streaming motion
along the individual particle trajectories. These trajectories are given
once the temporal evolution of position x t() and velocity k t() of each

Semiconductors for Micro and Nanosystem Technology 193