Page 218 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Local Equilibrium Description
1 

-------- •
s˙ = --- –   ∇ T  ( j – η • j ) + – ( ∇ η ) • j  (6.66a)

n
n
n
u
n
T  T  
Using dQ = TdS in the fundamental energy relation (B 7.2.4) we
obtain j = j – µ ⋅ j n , which, when inserted into (6.66a), gives
q
n
u
1   ∇ T  
s˙ = --- – -------- • j + – ( ∇ η ) • j  (6.66b)

q
n
n
T   T  
and which defines the affinities associated with the fluxes (see Box 7.2)
of this system. Equation (6.66a) also tells us that the heat flux is com-
prised of energy transport as well as particle energy transport, and that
the affinities needed for the following steps are the gradients of tempera-
ture and particle electrochemical potentials.
Assumptions We now summarize the assumptions underlying the previous analysis for
Needed to transport coefficients
Define the
Transport • The system has no “memory” and is purely resistive. The fluxes
Coefficients depend only on the local values of the intensive parameters and on the
affinities, and do so “instantaneously”.
• Each flux is zero for zero affinity. If we add the assumption of linear
dependence, we can truncate a series expansion of the fluxes w.r.t. the
affinities after the first nonzero term, i.e.,
nc
k ∑ L •
j = kl F l (6.67)
l = 1
for nc transport entities, and assuming that the transport coefficients
L are of tensor nature.
kl

The remarkable Onsager theory proves that, in the presence of a mag-
T
B
netic field L () = L ( – B) , and hence that, in the absence of a mag-
kl kl
T
netic field L = L . The approach in the above section was to write
kl kl
down formal expression for the transport components, and then to associ-
Semiconductors for Micro and Nanosystem Technology 215