Page 201 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Transport Theory
the velocity of the respective particles and retain only information about
the density in positional space
1
4π ∫
(
,,
,
(
3
Density n x t) = --------- f kx t)d k (6.10)
3
Ω k
This is equivalent to the zeroth order moment of f kx t,,( ) with respect
to the wave vector and we call n x t,( ) the spatial density of the system.
The first-order moment
1
1
,,
(
(
,
Current j x t,( ) = --------- --- ∇ E k()( ) f kx t)d k = v n x t) (6.11)
3
n 4π ∫ — k a
3
Density
Ω k
represents the particle current density, where we know that
∇ E k() —⁄ = v is the group velocity of a particle with a specific k. v is
k a
the average velocity of a particle. This velocity times the particle density
results in the current density as given in (6.11). The energy density
1
,
--------- E k() f vx t,,(
(
3
Energy u x t) = 4π ∫ )d v (6.12)
3
Density
Ω k
is a second order moment if the energy given in the harmonic approxima-
2 2
tion E = — k ⁄ ( 2m∗ ) . The same approximation yields a third order
moment
1
,
(
--------- E k()v f kx t,,(
3
Energy j x t) = 4π ∫ )d k (6.13)
u
3
Current Ω k
Density
that is called the energy current density. In general we have the r-th order
moment m r given by
a r
(
--------- v f kx t,,(
,
3
m x t) = 4π ∫ r )d k (6.14)
r 3
Ω k
A special second-order moment not contained in (6.12) is the momentum
current density, a tensorial quantity given by
198 Semiconductors for Micro and Nanosystem Technology