Page 202 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

The Semi-Classical Boltzmann Transport Equation
1
,
--------- v ⊗ f kx t,,(
(
3
Momentum
3
Current ΠΠ Π Π x t) = 4π ∫ v )d k (6.15)
Ω k
Density
Tensor where the tensor product denotes v ⊗( v) = v v . The sum of diagonal
ij i j
elements (6.15) is proportional to the energy density in the harmonic
approximation. The infinite sequence of moments represents the distribu-
tion of the system as well as f kx t,,( ) does. So far there is no advantage
in looking at the moments of the distribution function. The only advan-
tage arises when we are satisfied with the knowledge of a few of
momenta, e.g., the particle density.
To derive equations of motions for the momenta given in (6.10)-(6.13)
we integrate the BTE multiplied with powers of velocity v r with respect
to the wave vector–space. For this purpose we assume that Newton’s law
holds and the force acting on each particle fully determines the time
˙
(
,
⁄
derivative of the wave vector, i.e., we may set k = Fx v) — . Note that
we leave the possibility of the force depending on the particle velocity
open. This will enable us to include interactions with magnetic fields. For
now we assume the force Fx() to depend only on spatial coordinates,
and thus for r = 0 we obtain

C x()
n
∂ Fx() (6.16)
(
,,
(
,,
3
= t ∂ ∫ f kx t) + ------------∇ f kx t,,( ) + v∇ f kx t) d k
k
x
—
Ω k
The first term in (6.16) gives the time derivative of the carrier density
(
n˙ xt) , the second term vanishes due to the Gauss theorem and the fact
,
that f k t() x t() t,( , ) vanishes exponentially as k → ∞
∫ Fx()∇ f kx t,,( )d v = ∫ Fx() f kx t) Γ Ω ) = 0 (6.17)
3
(
,,
(
d
k
v
ΓΩ k )
(
Ω k
The third term gives, by exchanging integration and spatial gradient,
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