# Thermoelectric Effects Seebeck and Peltier Seebeck Effect in a

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```					                     CALIFORNIA INSTITUTE OF TECHNOLOGY
APh 114b Solid State Physics Lecture 15
H. Atwater Winter, 2009

Thermoelectric Effects: Seebeck and Peltier

Last time, we developed constitutive relations for the Seebeck coefficient Q and the Peltier
coefficient . Let’s now consider some simple important examples of these phenomena.

Seebeck Effect in a Thermocouple

In a metal subject to a thermal gradient ∇T there exists an effective electric field under
conditions of zero net current flow
E  Q|∇T|
j  e 2 K 0 E − e K 1 ∇T
T
where j  0,
Q  1 K1
eT K 0

In practice we typically measure the difference in thermopower Q A − Q B between two
different metals A and B by measuring the open circuit voltage V AB

V AB  −  E  dr  −  Q ∂T dr
∂r
T1                 T2       T0
V AB    T       Q B dT   Q A dT   Q B dT
T1       T2
0

T2
Note that at open circuit, and for V AB   Q A − Q B  dT.
T     1
T 1 ≠ T 2 , a potential V AB is developed, and it is independent of T 0 . For a metal, we note
Q  k E T
2
kT
2 e F

Peltier Heat Pump

In a metal subject to an electric field but no temperature gradient, there will be an effective
thermal current
u  eK 1 E
and
j  K20 E
e
for ∇T  0. So
u  K21  j  K −1
0
e
scalar:
u  K 1  j  j
eK 0

The Peltier effect is measured by the heat absorbed or generated at the junction between
two dissimilar materials. Sending an electric current around a circuit of two dissimilar
materials cools one juction and heats the other.

Given the close correspondence between the Seebeck and Peltier effects, it is perhaps not
surprising that materials with large Seebeck coefficients Q have larger Peltier coefficient .

Optical Properties of Solids
We now embark on a discussion of the optical properties of solids. In optical experiments
in which we measure reflectivity, transmission coefficient, luminescence, or light scattering,
we can learn a great deal about energy band structure, impurities, phonons, defects and
magnetic excitations. In particular, optical measurements are used to determine the complex
dielectric or complex conductivity, which is directly related to bandstructure. We start with
Maxwell’s Equations:
∇  H − 1 ∂D  4 j
c ∂t        c

∇  E  1 ∂B  0
c ∂t
∇D  0
(assume zero net charge density)
∇B  0
and the constitutive relations:
D  E
B  H
j  E
These yields a wave equation:
 ∂ 2 E   4 ∂E
∇2E             
c ∂t
2     2
c 2 ∂t

For optical fields the solution is of form
E  E 0 e ikr−t
where
k ≡ complex propagation constant
realk  wavevector
imaginaryk  absorption
substitution yields:
 2   4i
− k2       2
−
c        c2

We can write
k   c
c
 c ≡ complex dielectric constant
 c    4i   1  i 2

 c ≡ complex conductivity
By definition
 c  4i   
                   4i  c

4i
 c    
4i

Having defined  c and  c , we will relate these to
1) laboratory observables (reflectivity, etc.)
2) properties of solids (carrier density, band structure, etc.)

More definitions that you need to know:
n 2   c  n  i 2
c
n c ≡ complex refractive index.
Taking   1 in nonmagnetic materials,
 c  n 2   1  i 2  n  i 2
c
so (real part)
1  n2 − 2
and (imaginary part)
 2  2n

For a beam incident on a solid, we can develop the picture below:


The wave inside the solid is E x  E 0 e ikz−t with k     c   nc

The boundary condition E 0  E 1  E 2 gives the wave in free space as
z                 z
E x E 1 e i c −t  E 2 e i− c −t
incident beam   reflected beam

From

∇  E − c ∂H  c H
i
∂t
we get
∂E x  i H
c    y
∂z
so
E0k  E1  − E2   E0  nc
c      c      c

The normal incidence reflectivity is
2
R       E2
E1

Noting
E 2  1 E 0 1 − n c 
2
E 1  1 E 0 1  n c 
2

so
1 − nc     2
1 − n 2  k 2
R                  
1  nc             1  n 2  k 2
where R  1.

Power continuity dictates:
1  RAT
R  reflectivity, A  absorbance, T  transmittance.

We have now related an observable, R, to properties of solids such as n c .

If we can measure reflectivity at normal incidence over all frequencies, we can obtain both
n and k. This is because n and k are not independent, but are related through the
Kramers-Kronig relations. In assessing the optical response of a solid, we will examine both:
i) intraband transitions (e.g. Free carrier transport)
ii) interband transitions

Free Carrier Contribution to Optical Properties: Drude Theory
For free carriers, we can apply the semi-classical equations developed for transport:
∗
m ∗ dv  m v  eE 0 e −it
dt
where the time harmonic carrier motion is describable as v  v 0 e −it . So
∗
−m ∗ i  m v 0  eE 0


The current j  nev 0   c E 0 giving
v0            eE 0
m∗
   − im ∗ 
and
c           ne 2 
∗
m 1 − i

We recall that j  E where j     E  with
1               ∂ 2 Ek
m           12
         ∂k 2

Suppose the only conduction mechanism we treat explicitly is that due to free carriers.
Then all other mechanisms can be included in a core dielectric constant,  core ,
   core  4i


Without free carrier absorption,   0 and    core . In the Drude theory,

   core  4i         ne 2 
        m1 − i

There are two limiting cases:
  1 (low frequencies)
  1 (high frequencies)

Low Frequencies   1

Here we obtain
   core  4ine 
2
m
Since the free carrier term is proportional to 1/, this term dominates at low frequencies.
For Si,   11. 5, in Ge,   16 and   100 in PbTe. To find n and k, take the square root of 

 ≃    4ne 2  i
m
n.b.:
i  e i/4  1  i
2

So in the low frequency limit, n ≃ k and both are large. The reflectivity
n − 1 2  k 2
≃ n2  k2 ≃ 1
2    2
R
n  1 2  k 2   n k
(  1).

So at low frequencies, materials with free carriers are perfect reflectors.

High Frequencies   1

In this limit,
2
   core − 4ne2
m
and
 ≃     core  real
Thus n  0 and k ≃ 0, so
n − 1 2
R
n  1 2
where n         core .

Thus the Drude theory predicts that free carrier materials become dielectrics at high
frequencies. The characteristic frequency at which the metallic-to-dielectric transition occurs
is called the plasma frequency (the frequency at which the real part of  vanishes).

Drude theory:
1  i
   1  i 2   core  4i ∗ ne 
2
 m 1 − i 1  i
so
 1   core −     4ne 2  2
∗
m 1   2  2 

 1  p   0   core −      4ne 2  2
m ∗ 1   2  2 
p

so
2
 2  4ne − 12
∗
p
m  core 
2
 2 ≃ 4ne
∗
p
m  core

At low frequencies, free carrier conduction dominates and the reflectivity is R ≃ 1. At
n−1 2
high frequencies, R ≃       2 which is large is n  1. Near the plasma frequency, , is small.
n1
Moreover,  2 is also small since at  p ,
2      4      ne 2 
∗
m  p 1   p  2
and if  p   1,
 2 ≃  core / p   small

At  p ,  1  0 and n ≃ k and  2 ≃ 2nk ≃ 2n 2 . So at  p , n tends to be small, and thus R is
small.

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