Thermopower (Seebeck Effect)
• In 1821, Thomas Seebeck found that an electric current would
flow continuously in a closed circuit made up of two dissimilar
metals, if the junctions of the metals were maintained at two
different temperatures.
r
∇T
+
Free electron gas
+
+
+
-e
-e
-e
-e
-e
r
E
Origines of thermopower :
diffusion, « phonon-drag »,
« magnon-drag »
S= dV / dT;
S is the Seebeck Coefficient with units of Volts per Kelvin
S is positive when the direction of electric current is same as the
direction of thermal current
Peltier Effect
• In 1834, a French watchmaker and part time
physicist, Jean Peltier found that an electrical
current would produce a temperature gradient at the
junction of two dissimilar metals.
П < 0 ; Negative Peltier coefficient
High energy electrons move from
right to left.
Thermal current and electric current
flow in opposite directions.
(electronic)
Explanation of Thermopower in metals
(Band Structure View Point)
Key role of Fermi-Dirac equilibrium
distribution
for the probability of electron
occupation of energy
level E at temperature T
1
f (E ) =
 E − EF
1 + exp
 k BT



Energy
EF
Occupation Probability, f
Effect of Temperature
k BT
1
DOS
T= 0 K
Vacuum
Level
Increasing T
0
Electron Energy,
E
EF
Φ
Work Function,
Thermopower in metals (Band Structure View Point)
m
h 2π 2
2mE
h2
α <0
in 3D
α =0
Energy
De ( E ) =
EF
Energy
EF
• From energy dependent
conductivity.
• Mott formula:
N-type
Energy
EF
DOS
Thermopower:
α >0
Metal
P-type
π 2 k B2T ∂σ
s=(
)
|E = E
3eσ ∂E
F
• Note log derivative (not
an extensive quantity –
multiplicative factors in
density of states (specific
heat, entropy) or in σ do
not change S.
Density of States
-- Number of electron states available between energy E and E+dE
Thermopower in non-metals
EF
n
N-type
EFp P-type
Es
EF
Valence
Band
Valence
Band
DOS
Band gap
Semiconductor
kB
Sd =
e
Conduction
Band
Energy
Energy
Conduction
Band
 Es

 T + B 
 kB

DOS
Hopping
semiconductor
n

1
−
k
N
S d = B ln βsβo
n
e 
N




{ }
•B is configurational entropy term
Energy
Thermopower-absolute value at 300 K
10-1
5
Insulators
10-2
DOS
Conduction
Band
Energy
Germanium, Silicium
Bi2Te3
5
Bismuth
DOS
Semiconductor
Semi-metals
DOS
10-5
5
10-6
5
Semimetal
Nickel
Metal
Ag, Cu, Au
EF
DOS
10-7
EF
Constantan (Cu – Ni)
Energy
IαI (V/K)
10-4
E Fp
Valence
Band
Semiconductors
5
Insulator
EFn
Energy
5
10-3
EF
Metal
Thermopower-temperature dependence
0
200
400
600
800
1000
430
Thermoelectric power (kB/e)
5.0
-1
metal S~γ~ 50 mJ mol K
4.0
-1
344
gap-SC-S~300/T
3.0
258
hopping-SC-S~lnΣconfig (10%)
2.0
172
hopping-SC-S~lnΣconfig (20%)
1.0
86
-1
-1
metal S~γ~ 5 mJ mol K
0.0
0
200
400
T (K)
600
800
0
1000