Page 215 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Transport Theory
∫ b ∇η s = ∫ b qε∇Ts (6.53)
d
d
a a
This yields
T 1 T 2 T 0
η – η a
b
(
d
----------------- = ∫ ε T + ∫ ε T + ∫ ε T = ( – ε – ε ) T – T ) (6.54)
d
d
2
1
A
B
q
T 0 T 1 T 2
b
B
a
Since and are located in the same material , µ = µ b holds
a
and (6.54) gives us the so–called absolute differential thermovoltage
Absolute δψ = ( – ε – ε )δT (6.55)
B
A
Differential
which is the inverse of the Peltier effect, i.e., an electrostatic potential
Thermo-
voltage difference builds up in a system of two different materials where no
current is flowing due to a temperature gradient.
6.2.2 Formal Transport Theory
Our analysis here will be strictly macroscopic, and for this we invoke
equilibrium thermodynamics. Assuming local equilibrium we might still
say that the temperature and the electrochemical potential are “good”
quantities, at least locally. We allow them to vary in space. In particular,
we start with the fundamental energy relation (B 7.2.3) which we write
using density extensive variables and for multiple component particle
systems with densities n i and their respective intensive chemical poten-
tials µ i

du = Tds + ∑ µ dn i (6.56a)
i
i
1 µ i
ds = ---du – ∑ ----dn i (6.56b)
T T
i
When this expression is taken per unit time, we obtain the rate equations

212 Semiconductors for Micro and Nanosystem Technology