Page 208 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Local Equilibrium Description
6.2.1 Irreversible Fluxes and Thermodynamic Forces
The distribution function that locally deviates from its equilibrium value
,,
(
,,
is given by f kx t,,( ) = f ( kx t) + δf kx t) . Once we have
0
,,
δf kx t) it is possible to calculate all the current densities introduced
(
in 6.1.4 as moments of δf kx t,,( ) . We insert the expansion of the distri-
bution function into the BTE (6.2) with the scattering term given by a
relaxation time approximation (6.23) and obtain
∂ ( ,, F ( ,, δf kx t)
,,
(
----∇ f kx t,,(
τ
t ∂ f kx t) + — k ) + v∇ f kx t) = – ------------------------- . (6.25)
x
The deviation is assumed to be small and thus to a first approximation its
contribution to the streaming motion term is neglected. We obtain
(
∂ f ( ,, ----∇ f ( ,, v∇ f ( ,, δf kx t)
,,
F
τ
t ∂ 0 kx t) + — k 0 kx t) + x 0 kx t) = – ------------------------- (6.26)
Inserting (6.24) in (6.26) we can immediately solve for δf kx t,,( ) . To
this end we calculate the effect of the streaming motion operator on the
Fermi distribution as given by (6.24), where the gradient in k-space gives
(
∂ f E k() x t)
,,
0
,,
∇ f ( kx t) = -------------------------------------∇ E k() . (6.27)
k 0
k
∂
E
The gradient in real space is
∂ f E k() x t,,( ) E – µ T∂ 1 ∂ µ
0
∇ f ( kx t) = -------------------------------------k T – ------------------- – --------------- (6.28)
,,
x 0
k T ∂
B
∂
k T x∂
E
2
x
B
B
The partial derivative with respect to time gives
(
∂ f E k() x t) ∂ f E k() x t,,( ) E – µ T ∂ µ
,,
∂
0
0
------------------------------------- = ------------------------------------- – ------------------- – ------ (6.29)
t ∂ ∂ E T t ∂ t ∂
Solving (6.26) for δf we obtain
Semiconductors for Micro and Nanosystem Technology 205