254
8 Transport
(a)
500
0
-500
-1000
-1500
-2000
Thermoelectric power S (mV/K)
Temperature (K)
Si
0
10
20
30
40
50
60
(b)
0
-500
-1000
10
14
10
15
10
16
10
17
10
18
10
19
N -N (cm )
D
A
-3
Si
Thermoelectric power S ( V/K)
Fig. 8.30 a Thermoelectric power S of highly doped n-type silicon as a function of temperature. Circles are experimental
data and dashed lines guides to the eye. The approximate doping of the samples is white: 2.7 × 10 19 cm −3 As, grey:
2.2 × 10 18 cm −3 As, black: 1.1 × 10 18 cm −3 As and 1.0 × 10 18 cm −3 B with N D − N D = 1.25 × 10 17 cm −3 at room
temperature. Adapted from [822]. b Thermopower of doped n-type silicon at room temperature as a function of doping
concentration. Experimental data (symbols) from [822] and theory (solid line) from [824]
If the Fermi level is fixed and both electrons and holes contribute (two-band conduction), the
thermopower is (evaluating (J.32), b = σ n /σ p and the gap center energy E M = (E C − E V )/2)
S =
k
e
1 − b
1 + b
E g
2 kT
+
E F − E M
kT
+
A V − b A C
1 + b
.
(8.77)
In the case of intrinsic conduction from (7.18) E F − E M = (kT /2) ln(N V /N C ).
The thermoelectric power from some highly doped n-type silicon samples is depicted in Fig. 8.30a.
At low temperature the (low) conductivity is due to conduction in a donor impurity band (cmp.
Sect. 7.5.7). At high compensation of about 90% (grey data points in Fig. 8.30a), the band is only 10%
filled and acts like a valence band with positive thermopower at sufficiently low temperature when the
free carrier density is small. Without compensation, the thermopower remains negative since the almost
completely filled impurity band acts conduction band like. The dependence of thermopower on doping
has been simulated in [824] (Fig. 8.30); the decrease with increasing doping is mostly attributed to the
reduced mobility due to ionized impurity scattering. The increase of thermopower at low temperatures
is due to the phonon-drag effect which is discussed for the samples from [822] in [825].
As a figure of merit for the production of thermoelectric power the Z T -value is used, Z T =
σ S
2 T /κ.
8.14.2 Peltier Effect
In a semiconductor with a temperature difference at its ends a current flow will be allowed now (short
circuit). The current leads via the charge transport also to a heat (or energy) transport. This effect is
called the Peltier effect. The Peltier coefficient is negative (positive) for electrons (holes). The total
amount of energy P that is transported consists of the generation term and the loss due to transport:
P = j · ˆ
E − ∇ · q .
(8.78)
With (8.72) and (8.73) we find
8 Transport
(a)
500
0
-500
-1000
-1500
-2000
Thermoelectric power S (mV/K)
Temperature (K)
Si
0
10
20
30
40
50
60
(b)
0
-500
-1000
10
14
10
15
10
16
10
17
10
18
10
19
N -N (cm )
D
A
-3
Si
Thermoelectric power S ( V/K)
Fig. 8.30 a Thermoelectric power S of highly doped n-type silicon as a function of temperature. Circles are experimental
data and dashed lines guides to the eye. The approximate doping of the samples is white: 2.7 × 10 19 cm −3 As, grey:
2.2 × 10 18 cm −3 As, black: 1.1 × 10 18 cm −3 As and 1.0 × 10 18 cm −3 B with N D − N D = 1.25 × 10 17 cm −3 at room
temperature. Adapted from [822]. b Thermopower of doped n-type silicon at room temperature as a function of doping
concentration. Experimental data (symbols) from [822] and theory (solid line) from [824]
If the Fermi level is fixed and both electrons and holes contribute (two-band conduction), the
thermopower is (evaluating (J.32), b = σ n /σ p and the gap center energy E M = (E C − E V )/2)
S =
k
e
1 − b
1 + b
E g
2 kT
+
E F − E M
kT
+
A V − b A C
1 + b
.
(8.77)
In the case of intrinsic conduction from (7.18) E F − E M = (kT /2) ln(N V /N C ).
The thermoelectric power from some highly doped n-type silicon samples is depicted in Fig. 8.30a.
At low temperature the (low) conductivity is due to conduction in a donor impurity band (cmp.
Sect. 7.5.7). At high compensation of about 90% (grey data points in Fig. 8.30a), the band is only 10%
filled and acts like a valence band with positive thermopower at sufficiently low temperature when the
free carrier density is small. Without compensation, the thermopower remains negative since the almost
completely filled impurity band acts conduction band like. The dependence of thermopower on doping
has been simulated in [824] (Fig. 8.30); the decrease with increasing doping is mostly attributed to the
reduced mobility due to ionized impurity scattering. The increase of thermopower at low temperatures
is due to the phonon-drag effect which is discussed for the samples from [822] in [825].
As a figure of merit for the production of thermoelectric power the Z T -value is used, Z T =
σ S
2 T /κ.
8.14.2 Peltier Effect
In a semiconductor with a temperature difference at its ends a current flow will be allowed now (short
circuit). The current leads via the charge transport also to a heat (or energy) transport. This effect is
called the Peltier effect. The Peltier coefficient is negative (positive) for electrons (holes). The total
amount of energy P that is transported consists of the generation term and the loss due to transport:
P = j · ˆ
E − ∇ · q .
(8.78)
With (8.72) and (8.73) we find