Page 197 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Transport Theory
particle are known. If there are no particles created or destroyed in a
given volume element of the phase space the total time derivative of the
phase-space density vanishes
d ∂ ˙
(
(
,,
,,
(
,,
f kx t) = f kx t) + k∇ f kx t,,( ) + x ˙∇ f kx t) (6.3)
x
k
t d t ∂
where ∇ and ∇ denote the gradients with respect to the wave vector
k x
and the position. A volume in phase-space evolves in time according to
the trajectories that its member particles follow. Assuming that
⁄
x ˙ = v = —k m and combine position and wave vector into a single
,
vector ξ = { xk} in the six-dimensional phase-space, we rewrite (6.3)
as a continuity equation like that for current continuity in Section 4.1.2
∂ ( , ∇ f ξ t() t,(( ˙
t ∂ f ξ t() t) + ξ )ξ) = 0 (6.4)
˙
⁄
If we write k = F — considering the Newton law, where F does not
depend on the wave vector, and take into account that does not explic-
k
˙
x
itly depend on , ∇ ξ = 0 holds and (6.4) corresponds to (6.3). The
ξ
term f ξ t() t,( )ξ ˙ we interpret as a probability current density in phase-
space, i.e., the probability density that flows out of or into a six-dimen-
3
3
sional phase-space volume element d xd k = d 6 . ξ
For a constant external force along the x-direction the deformation of
F
a unit circle in a two dimensional phase space xk,( x ) is shown in
Figure 6.1. Without any applied force the phase-space density changes
a) k b) k
x x

Figure 6.1. Phase-space density
evolving according to (6.3), a)
x x
without applied force and b) with
constant applied force in x-direc-
tion.

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