252
8 Transport
Fig. 8.28 Thermal
conductivity of Ge vs.
temperature. The enriched
Ge consists of 96% 74 Ge
while the natural isotope
mix is 20% 70 Ge, 27%
72 Ge, 8% 73 Ge, 27% 74 Ge
and 8% 76 Ge. The dashed
line shows a κ ∝ T 3
dependence at low
temperatures. Adapted
from [815]
100
50
20
10
5
2
1
0.5
0.2
1
5
enriched Ge
74
0
0
2
0
0
5
0
0
1
0
2
0
5
0
1
1
-
1
-
2
T
3
j = σ ˆ
E + L ∇T
(8.70)
q = M ˆ
E + N ∇T ,
(8.71)
where ˆ
E and ∇T are the stimulators for the currents. From the experimental point of view there is
interest to express the equations in j and ∇T since these quantities are measurable. With new coefficients
they read
ˆ
E = ρ j + S ∇T
(8.72)
q = j − κ ∇T ,
(8.73)
where ρ, S and are the specific resistance, thermoelectric power and Peltier coefficient (transported
energy per unit charge), respectively. The relations with the coefficients σ, L, M, and N are given by
ρ =
1
σ
(8.74a)
S = −
L
σ
(8.74b)
=
M
σ
(8.74c)
κ =
M L
σ
− N .
(8.74d)
8.14.1 Thermopower and Seebeck Effect
A semiconductor shall have two ends at different temperatures T 2 and T 1 and a temperature gradient
in between in an open circuit, i.e. j = 0. Then a field ˆ
E = S ∇T and a voltage U = S/(T 2 − T 1 ) will
arise. This effect is called the thermoelectric or Seebeck effect. S is termed the Seebeck coefficient or
the thermoelectric power, often also denoted as Q in the literature. The voltage can be measured and
used to determine the temperature at one end if the temperature at the other end is known, forming