Page 209 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Transport Theory
,,
(
∂ f E k() x t,,( ) δf kx t) µ ∂ µ ∂ (6.30)
∂
∂

0
= – τ------------------------------------- – E – µ T ------ + vF – ------ – E – µ T 
-------------------
------------------- –
∂ E T t ∂ t ∂  ∂ x T ∂ x
We restrict the situation to a stationary analysis, where the time deriva-
F
tives in (6.30) vanish. Let us assume that the external force is given by
an electric field E , which itself can be expressed as the gradient of an
electrostatic potential (see (4.34)). The force acting on an electron is then
F = – qE = q∇ψ . Note that the minus sign is due to its negative
µ
charge. This allows us to combine the chemical potential and the elec-
ψ
trostatic potential into one term forming the electrochemical potential
–
Electro- η = µ qψ (6.31)
chemical
Potential Then the current densities become
f ∂ 0  ∂ η E – µ T 
∂
∫
3
Current j xt,( ) = – τ v--------v – ------ – ------------------- d k (6.32)
x
E 
n
∂
∂
∂
T
Density x
V k
and
f ∂ 0  ∂ η E – µ T 
∂
∫
3
Energy j xt,( ) = – τ vE--------v – ------ – ------------------- d k (6.33)
u ∂ E  ∂ x T ∂ x
Current
V k
Density
By a simple manipulation we ca see that the latter consists of two terms
f ∂ 0  ∂ η E – µ T 
∂
∫
(
,
(
3
j xt) = µ j – τ v E – µ)--------v – ------ – ------------------- d k
x
E 
u
n
∂
∂
∂
T
x
V k (6.34)
                
j
Q
With (6.34) we see that the current density consists of two parts: one that
is due to particle transport through spatial regions µ j and a second that
n
is due to heat transport j . Since the first part of the energy current is
Q
proportional to the particle current, in the following we focus on j and
Q
j . Both current densities suggest a general structure of the form
n
206 Semiconductors for Micro and Nanosystem Technology