Page 216 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Local Equilibrium Description
s˙ =
---u˙ –
----n˙
T 1 ∑ µ i i (6.57a)
T
i
1 µ n
j = --- j – ----- j (6.57b)
s u n
T T
where in (6.57b) we are restricted to a single type of particle. It employs
the current densities for entropy, energy, and particles, and is also obtain-
able from the entropic fundamental relation (B 7.2.7). Since energy den-
sity and particle density are both conserved, we have that
⁄
∂u ∂t = – ∇• j , and ∂n ∂t⁄ = – ∇• j . Entropy density is not con-
u n
served, so that
∂s
s˙ = ----- + ∇• j s (6.58)
∂t
From (B 7.2.7) we obtain that

2
1
1 ∂u
n 
∂s ∂x k   µ ∂n   µ n 
----- = ∑ F -------- = --- ------ – ----- ------ = – --- ∇• j + ----- ∇• j n (6.59)
u
k
T
T ∂t
∂t ∂t   T  ∂t   T 
k = 0
⁄
with the entropic intensive parameters F = 1 T , and F = – µ ⁄ T .
0 1 n
The divergence of (6.57b) gives
1
∇• j = ∇•  --- j  – ∇•  µ n 
----- j
s  T   T n 
u
(6.60)
1
 1    µ n   µ n 
= ∇ --- • j + --- ∇• j – ∇ ----- • j – ----- ∇• j
 T  u  u  T  n  T  n
T
Inserting (6.59) and (6.60) into (6.58) results in
 1   µ n 
Entropy s˙ = ∇ --- • j – ∇ ----- • j (6.61)
 T  u  T  n
Production
Defines the
At this point we allow the inclusion of external forces as well. In particu-
Affinities and
Fluxes lar, we have to do so for the charged particles, which are also subject to
the Lorenz force
Semiconductors for Micro and Nanosystem Technology 213