Page 213 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Transport Theory
Suppose we have a constant current flowing from material to , as
indicated in Figure 6.4. A B
j
A z B
Figure 6.4. Contact between two Π Π
A B
different materials showing the
z
Peltier effect. −ε 0 ε

Integrate (6.47) over small volume around the interface between A
and B
ε ε ε
∂  1 
∫
(
,
lim t ∂ ∫ ux t) z = lim –  i ∇Π z + i 2 σ ∫ ---dz
d
d
ε → 0 ε → 0   (6.48)
ε – ε – ε –
= i – ( Π – Π )
z B A
Peltier Effect Integration and taking the limit in (6.48) are equivalent to asking for
the amount of heat produced in a very small region around the inter-
face. We see that in the limit ε → 0 the second integral vanishes, i.e.,
the Joule heating is a volume effect, whereas the first integral gives a
finite contribution in this limit if Π ≠ Π A , i.e., it is a surface or inter-
B
face effect. This means that we have a local heating or cooling at the
interface between material A and B, which is called the Peltier effect.

2. If there is no particle current present, i.e., j = 0 , then, according to
n
(6.41a), the electrochemical potential gradient is completely deter-
mined by the temperature gradient
N 12
∇η = – -------------∇T (6.49)
T N 11
Their constant of proportionality divided by the unit charge

210 Semiconductors for Micro and Nanosystem Technology