8.14 Coupled Heat and Charge Transport
253
(a)
0.9
0.6
0.3
0
-0.3
-0.6
-0.9
Ge
1000
Thermoelectric force S (mV/K)
Temperature (K)
100 200 300 400 500 600 700 800 900
(b)
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
200 300
Si
0
0
9
0
0
0
1
0
0
8
0
0
7
0
0
6
0
0
5
Thermoelectric force S T (V)
Temperature (K)
400
intri nsic
n-type
p-type
Fig. 8.29 a Seebeck coefficient S for n- and p-doped germanium. Experimental data (symbols) and theory (lines). N A −
N D is 5.7 × 10 15 cm −3 (white circles), 1.7 × 10 17 cm −3 (grey) and 7.2 × 10 18 cm −3 (black); N D − N A is 3.3 × 10 15 cm −3
(white squares), 1.1 × 10 17 cm −3 (grey) and 6.2 × 10 17 cm −3 (black). Adapted from [821]. b Thermoelectric force
of lowly doped n- and p-silicon as a function of temperature. Solid line is from simple model calculation and symbols
represent data from silicon samples with the approximate doping of circles: 1 × 10 15 cm −3 B, 2 × 10 14 cm −3 donors,
squares: 4 × 10 14 cm −3 P, 9 × 10 13 cm −3 acceptors. Adapted from [822]
a thermometer. The Seebeck coefficient is positive if the electric field is in the same direction as the
temperature gradient.
A famous relation from irreversible thermodynamics connects it to the Peltier coefficient via
S =
T
.
(8.75)
The Seebeck coefficient is related to the energy transport by charge carriers. The heat (energy) flow is
obviously from the hot to the cold end (assuming here T 2 > T 1 ), so is the flow of charge carriers. In a
simple picture, if the energy is carried by (hot) holes, the current (by definition the direction of positive
charge carriers) is from the hot to the cold end (2 → 1); if the energy flow is carried by electrons, the
current flows from the cold to the hot end (1 → 2). Accordingly, energy transport by electrons and
holes gives rise to different signs of the thermoelectric coefficient (Fig. 8.29). If the cold (unheated)
substrate is grounded, the sign of the voltage at a hot solder tip pressed (carefully) on the surface of
the semiconductor yields the conductivity type, n-type (p-type) for a negative (positive) voltage.
However, the semiconductor should not be heated so strongly that intrinsic conduction arises. In this
case the conductivity and the thermoelectric power is determined by the carrier type with the higher
mobility; typically, and for the case of silicon shown in Fig. 8.29, these are the electrons thus yielding
a negative Seebeck coefficient in the intrinsic regime.
For band conduction the thermopower (J.29) is given for electrons (S n ) and holes (S p ) by [823] (for
a derivation see Appendix J.4)
S n = −
k
e
E C − E F
kT
+ A C
(8.76a)
S p =
k
e
E F − E V
kT
+ A V
,
(8.76b)
where A i are constants (J.31a) depending on the energy dependence of the density of states and the
mobility. The sign of the thermopower tells whether conduction takes place above (negative sign) or
below (positive sign) the Fermi level.
253
(a)
0.9
0.6
0.3
0
-0.3
-0.6
-0.9
Ge
1000
Thermoelectric force S (mV/K)
Temperature (K)
100 200 300 400 500 600 700 800 900
(b)
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
200 300
Si
0
0
9
0
0
0
1
0
0
8
0
0
7
0
0
6
0
0
5
Thermoelectric force S T (V)
Temperature (K)
400
intri nsic
n-type
p-type
Fig. 8.29 a Seebeck coefficient S for n- and p-doped germanium. Experimental data (symbols) and theory (lines). N A −
N D is 5.7 × 10 15 cm −3 (white circles), 1.7 × 10 17 cm −3 (grey) and 7.2 × 10 18 cm −3 (black); N D − N A is 3.3 × 10 15 cm −3
(white squares), 1.1 × 10 17 cm −3 (grey) and 6.2 × 10 17 cm −3 (black). Adapted from [821]. b Thermoelectric force
of lowly doped n- and p-silicon as a function of temperature. Solid line is from simple model calculation and symbols
represent data from silicon samples with the approximate doping of circles: 1 × 10 15 cm −3 B, 2 × 10 14 cm −3 donors,
squares: 4 × 10 14 cm −3 P, 9 × 10 13 cm −3 acceptors. Adapted from [822]
a thermometer. The Seebeck coefficient is positive if the electric field is in the same direction as the
temperature gradient.
A famous relation from irreversible thermodynamics connects it to the Peltier coefficient via
S =
T
.
(8.75)
The Seebeck coefficient is related to the energy transport by charge carriers. The heat (energy) flow is
obviously from the hot to the cold end (assuming here T 2 > T 1 ), so is the flow of charge carriers. In a
simple picture, if the energy is carried by (hot) holes, the current (by definition the direction of positive
charge carriers) is from the hot to the cold end (2 → 1); if the energy flow is carried by electrons, the
current flows from the cold to the hot end (1 → 2). Accordingly, energy transport by electrons and
holes gives rise to different signs of the thermoelectric coefficient (Fig. 8.29). If the cold (unheated)
substrate is grounded, the sign of the voltage at a hot solder tip pressed (carefully) on the surface of
the semiconductor yields the conductivity type, n-type (p-type) for a negative (positive) voltage.
However, the semiconductor should not be heated so strongly that intrinsic conduction arises. In this
case the conductivity and the thermoelectric power is determined by the carrier type with the higher
mobility; typically, and for the case of silicon shown in Fig. 8.29, these are the electrons thus yielding
a negative Seebeck coefficient in the intrinsic regime.
For band conduction the thermopower (J.29) is given for electrons (S n ) and holes (S p ) by [823] (for
a derivation see Appendix J.4)
S n = −
k
e
E C − E F
kT
+ A C
(8.76a)
S p =
k
e
E F − E V
kT
+ A V
,
(8.76b)
where A i are constants (J.31a) depending on the energy dependence of the density of states and the
mobility. The sign of the thermopower tells whether conduction takes place above (negative sign) or
below (positive sign) the Fermi level.