Page 217 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Transport Theory
B)
v ×
(
q E +
(6.62)
=
f
n
L
n
that is caused by an externally applied electromagnetic field. This means
that we have to modify the particle affinities to yield
 1   f L µ n 
s˙ = ∇ --- • j + ------ – ∇ ----- • j (6.63)
 T  u  T T  n
However, since by definition the particle current densities and the particle
velocity vectors are parallel, the cross products drop out of the result and
we obtain
 1   q E µ n 
n
s˙ = ∇ --- • j + --------- – ∇ ----- • j , or (6.64a)
 T  u  T T  n
 1   q ∇ ψ µ n 
n
s˙ = ∇ --- • j + – -------------- – ∇ ----- • j (6.64b)
 T  u  T T  n
if we make use of the relation between the electric field and the electro-
static potential ψ . We see that entropy production by the electrons or
holes is naturally “driven” by two forces, their concentration gradient (or
chemical potential gradient) represented by ∇ µ n , and the externally
applied field (or electrical potential gradient) represented by ψ∇ . We can
now simplify matters by introducing the electrochemical potential η n for
the charge carriers to obtain

 1   η n 
s˙ = ∇ --- • j + – ∇ ----- • j (6.64c)
 T  u  T  n
Noting that

∇ ( 1 T) = – ∇ T T 2 (6.65)
⁄
⁄
and

∇ ( η ⁄ ) = ∇ η ⁄ T – η ∇ T T 2 (6.66)
⁄
T
n
n
n
we obtain
214 Semiconductors for Micro and Nanosystem Technology