Thermoelectric
Materials

Thermoelectric
devices
are
based
on
a
phenomenon
known
as
the
thermoelectric
effect
which
is
the
direct
conversion
of
a
temperature
gradient
across
two
dissimilar
materials
into
electricity.
The
materials
used
are
known
as
thermoelectric
materials.
The
thermoelectric
effect
is
reversible
i.e.
it
directly
conver;ng
electricity
into
a
temperature
gradient.


The
thermoelectric
effect
is
based
on
a
combina;on
of
two
different
effects,
namely,
the
Seebeck
effect
and
the
Pel4er
effect.


Water/Beer
Cooler


What
is
thermoelectricity?
Thermoelectricity
is
the
conversion
of
heat
differen;als
into
electricity
and
viceversa.
Thermoelectric
energy
conversion
is
one
of
the
direct
energy
conversion
technologies
that
rely
on
the
electronic
proper;es
of
the
material
(semiconductor)
for
its
efficiency.
It
is
based
on
the
Seebeck
(Power
Genera;on)
and
Pel;er
effects
(Heat
Pumping).


Seebeck
Effect
In
1821,
Thomas
Seebeck
a
German
Estonian
physicist
found
that
an
electric
current
would
flow
con;nuously
in
a
closed
circuit
made
up
of
two
dissimilar
metals,
if
the
 junc;ons
 of
 the
metals
 were
maintained
 at
 two
different
temperatures.

If
 the
 temperature
gradient
 is
 reversed,
 the
direc;on
of
the
current
is
reversed.


Where
S
is
the
Seebeck
coefficient.
It
is
defined
as
the
voltage
generated
per
degree
of
temperature
difference
between
the
two
points.


S
is
posi;ve
when
the
direc;on
of
the
current
is
the
same
as
the
direc;on
of
the
voltage


The
basis
of
the
Seebeck
effect
is
electron
mobility
in
conductors
and
semiconductors,
which
is
a
func;on
of
temperature


When
two
different
metals
are
joined,
the
rela;ve
difference
in
electron
mobility
in
each
of
the
metals
will
make
that
the
electrons
from
the
more
“mobile”
metal
jump
to
the
less
mobile
metal.


A
poten4al
difference
is
created
between
the
two
conductors.

In
the
absence
of
a
circuit,
this
causes
charge
to
accumulate
in
one

conductor,
and
charge
to
be
depleted
in
the
other
conductor.


Example:
Type
K
thermocouple


Measure ?

The
Seebeck
Effect
The
Seebeck
effect
is
the
conversion
of
heat
differences
directly
into
electricity.
When
two
dissimilar
materials
with
different
carrier
densi;es
are
connected
to
each
other
by
an
electrical
conductor
and
heat
is
applied
to
one
side
of
the
connectors,
some
of
the
heat
input
is
converted
to
electrical
current,
as
the
higher
energy

maUer
releases
energy
and
cools
to
a
lower
energy
state.
The
net
work
is
propor;onal
to
the
temperature
difference
and
Seebeck
coefficient.



The
simplest
thermoelectric
generator
consists
of
a
thermocouple,
comprising
a
p‐type
and
n‐type
thermo‐element
connected
electrically
in
series
and
thermally
in
parallel.



Heat
is
pumped
into
one
side
of
the
couple
and
rejected
from
the
opposite
side.
An
electrical
current
is
produced,
propor;onal
to
the
temperature
gradient
between
the
hot
and
cold
junc;ons



Explana1on
of
Seebeck
Effect
In
a
thermoelectric
material
there
are
free
carriers
which
carry
both
charge
and
heat.



If
a
material
is
placed
in
a
temperature
gradient,
where
one
side
is
cold
and
the
other
is
hot,
the
carriers
at
the
hot
end
will
move
faster
than
those
at
the
cold
end.

The
faster
hot
carriers
will
diffuse
further
than
the
cold
carriers
and
so
there
will
be
a
net
build
up
of
carriers
(higher
density)
at
the
cold
end.

In
the
steady
state,
the
effect
of
the
density
gradient
will
exactly
counteract
the
effect
of
the
temperature
gradient
so
there
is
no
net
flow
of
carriers.

The
buildup
of
charge
at
the
cold
end
will
also
produce
a
repulsive
electrosta;c
force
(and
therefore
electric
poten;al)
to
push
the
charges
back
to
the
hot
end.


The
electric
poten;al
produced
by
a
temperature
difference
is
known
as
the
Seebeck
effect
and
the
propor;onality
constant
is
called
the
Seebeck
coefficient
(α or
S).

If
the
free
charges
are
posi;ve
(the
material
is
p‐type),
posi;ve
charge
will
build
up
on
the
cold
which
will
have
a
posi;ve
poten;al.
Similarly,
nega;ve
free
charges
(n‐type
material)
will
produce
a
nega;ve
poten;al
at
the
cold
end.
If
the
hot
ends
of
the
n‐type
and
p‐type
material
are
electrically
connected,
and
a
load
connected
across
the
cold
ends,
the
voltage
produced
by
the
Seebeck
effect
will
cause
current
to
flow
through
the
load,
genera;ng
electrical
power.



α2σ is
the
materials
property
known
as
the
thermoelectric
power
factor.

For
efficient
opera;on,
high
power
must
be
produced
with
a
minimum
of
heat
(Q). κ= Thermal
conduc;vity.
The
thermal
conduc;vity
acts
as
a
thermal
short
and
reduces
efficiency.


Pel4er
Effect
In
1834,
a
French
scien;st
Jean
Pel;er
found
that

a
thermal
difference
can
be
obtained
at
the
junc;on
of
two
metals,
if
an
electric
current
is
made
to
flow
in
them.



Opposite
of
the
Seebeck
Effect.
The
heat
current
(q)
is
propor;onal
to
the
charge
current
(I)
and
the
propor;onality
constant
is
the
Pel;er
Coefficient
(Π).


When
two
materials
are
joined
together,
there
will
be
an
excess
or
deficiency
in
the
energy
at
the
junc;on
because
the
two
materials
have
different
Pel;er
coefficients.
The
excess
energy
is
released
to
the
la^ce
at
the
junc;on,
causing
hea;ng,
and
the
deficiency
in
energy
is
supplied
by
the
la^ce,
crea;ng
cooling.



The
Seebeck
and
the
Pel;er
coefficients
are
related
to
each
other
through
the
Kelvin
rela;onship
–
T
is
the
absolute
temperature.


Π >0
; Posi;ve
Pel;er
coefficient.
High
energy
holes
move
from
lea
to
right.
Thermal
current
and
electric
current
flow
in
same
direc;on.


Π

When
two
dissimilar
materials
with
different
carrier
densi;es
are
connected
to
each
other
by
an
electrical
conductor,
electrical
current
(work
input),
forces
the
maUer
to
approach
a
higher
energy
state
and
heat
is
absorbed
(cooling).
The
energy
is
released
(hea;ng)
as
the
maUer
approaches
a
lower
energy
state.
The
net
cooling
effect
is
propor;onal
to
the
electric
current
and
Pel;er
Effect
coefficient.


The
Pel1er
Effect


Thompson

Effect


William
Thompson
(1824‐1907)
also
known
as
Lord
Kelvin.
He
observed
that
when
an
electric
current
flows
through
a
conductor,
the
ends
of
which
are
maintained
at
different
temperatures
(gradient
temperature),
heat
is
evolved
at
a
rate
approximately
propor;onal
to
the
product
of
the
current
and
the
temperature
gradient.


Thompson
Effect
=
Seebeck
Effect
+
Pel;er
Effect


is
the
Thomson
coefficient
in
Volts/Kelvin


The
rela;onships
between
the
different
effects
are
called
the
Kelvin
rela;onships.


First
Kelvin
rela;onship:



Second
Kelvin
rela;onship:


Coefficient
of
Performance


where



Thermoelectric
Figure
of
Merit
(ZT)


Seebeck coefficient Electrical conductivity

Thermal conductivity

Temperature Bi2Te3

Freon

TH = 300 K TC = 250 K

Requirements
for
a
Good
Thermoelectric
Material


•  General
considera;ons
for
the
selec;on
of
materials
for
thermoelectric
applica;ons
involve:
–  High
figure
of
merit



–  large
Seebeck
coefficient
α
(or
S)
–  high
electrical
conduc;vity
σ–  low
thermal
conduc;vity
κLaDce+κelectrons–  Possibility
of
obtaining
both
n‐type
and
p‐type
thermoelements.
–  No
viable
superconduc;ng
passive
legs
developed
yet


•  Good
mechanical,
metallurgical
and
thermal
characteris;cs
–  Capable
of
opera;ng
over
a
wide
temperature
range.
Especially
true
for
high
temperature
applica;ons.


–  To
allow
their
use
in
prac;cal
thermoelectric
devices
–  Materials
cost
can
be
an
important
issue!


Thermal
conduc;vity
consists
of
two
parts:

la^ce
conduc;vity
(la^ce
vibra;ons
=
phonons),
κLa,ce,

and
thermal
conduc;vity
of
charges
(electrons
and
holes),
κelectrons:


Currently,
most
of
the
research
efforts
are
devoted
to
minimizing
the
la^ce
conduc;vity
of
new
phases.


Minimizing
thermal
conduc1vity


Some
ways
to
reduce
the
la^ce
conduc;vity:
(1)
use
of
heavy
elements,
e.g.
Bi2Te3,
Sb2Te3
and
PbTe;

(2)
a
large
number
N
of
atoms
in
the
unit
cell:
the
frac;on
of
vibra;onal
modes
(phonons)
that
carry
heat
efficiently
to
1/N;
(3)
raUling
of
the
atoms,
e.g.
filled
skuUerudite
CeFe4Sb12;

disorder
in
atomic
structure:
random
atomic
distribu;on
and
deficiencies.



The
last
approach
is
nicely
realized
in
"Zn4Sb3",
which
can
be
called
an
"electron‐crystal
and
phonon‐glass"
according
to
Slack.
This
material
has
electrical
conduc;vity
typical
for
heavily
doped
semiconductors
and
thermal
conduc;vity
typical
for
amorphous
solids.
In
fact,
its
thermal
conduc;vity
is
the
lowest
among
state‐of‐the
art
thermoelectric
materials:


Minimize
thermal
conduc4vity
and
maximize
electrical
conduc4vity
has
been
the
biggest
dilemma
for
the
last
40
years.!



Bismuth
telluride
is
the
standard
with
 
 
ZT=1
to
match
a
refrigerator
you
need
 
 
 
ZT=
4
‐
5

to
recover
waste
heat
from
car
 
 
 
ZT
=
2


Can
the
conflic4ng
requirements
be
met
by
nano‐scale
material
design?


Reduce
the
la^ce
thermal
conduc;vity
by:


• Complex
crystal
structure
of
high
atomic
number
materials.


• RaUlers
in
the
structure
(Atomic
Displacement
Parameter
–
ADP).


• Nanostructured
Thermoelectrics


Complex
Crystal
Structures


RaMlers:


 These
are
weakly
bound
atoms
that
fill
cages.


 They
have
unusually
large
values
of
Atomic
Displacement
Parameters


 Proper1es
of
many
clathrate‐like
compounds
can
be
understood
by
trea1ng
“raSler”
atoms
as
Einstein
oscillators
and
framework
atoms
as
a
Debye
solid.


 SkuSerudites,
LaB6,
Tl2SnTe5


 A
Characteris1c
Einstein
temperature
(or
frequency)
can
be
assigned
to
each
raSler


Eu8‐eGa16Ge30
Phase
With
the
Ba8Ga16Sn30

Clathrate
Structure
Type:
a
=
10.62
Å


Eu
Nuclear
Density
Map
at
Center
of
Large
Cage


Tunneling
States
!


Sr
Nuclear
Density
Map
at
Center
of
Large
Cage
Tunneling
States?


Ba
Nuclear
Density
Map
at
Center
of
Large
Cage
(
6d
site
of


clathrate
structure)


X8Ga16Ge30
(X=
Ba,
Sr,
Eu)


ADP
Data
(
)
From
6d
Site


Advantages
of
Thermoelectrics
• Absence
of
moving
parts
• High
reliability
• Quietness
• Lack
of
vibra;ons
• Low
maintenance
• Simple
start
up
• No
pollu;on
• Small
• Light
weight
• No
noise
• Precise
temperature
control:
within
+/‐
0.1C


Disadvantages
of
Thermoelectrics
• High
cost
• Low
efficiency
• Typically
about
3
to
7%


Applica1ons
of
Thermoelectric


•  Consumer
Applica;ons


•  Automobile
Applica;ons


•  Industrial
Applica;ons


•  Military
and
Space
Applica;ons


Consumer
Applica1ons


Beer
Cooler


TE
Fridge


Chocolate
Cooler


Automobile
Applica1ons


Seat
Cooler/Warmer
Can
Cooler


Industrial
Applica1ons


Electronic
Cooler
 TE
Dehumidifier


Military
and
Space
Applica;ons


Night
Vision


Basic
Principles
• 
Macroscopic
Thermal
Transport
Theory–
Diffusion




‐‐
Fourier’s
Law

‐‐
Diffusion
Equa1on


• 
Microscale
Thermal
Transport
Theory
–
Par1cle
Transport



‐‐
Kine1c
Theory
of
Gases

‐‐
Electrons
in
Metals

‐‐
Phonons
in
Insulators

‐‐
Boltzmann
Transport
Theory


Basic
Principles


Heat
is
a
form
of
energy.
The
thermal
proper;es
describe
how
a
solid
responds
to
changes
in
its
thermal
energy.


The
heat
capacity
(C)
of
a
solid
quan;fies
the
rela;onship
between
the
temperature
of
the
body
(T)
and
the
energy
(Q)
supplied
to
it.


The
measured
value
of
the
heat
capacity
is
found
to
depend
on
whether
the
measurement
is
made
at
constant
volume
(CV)
or
at
constant
pressure
(CP).


Thermal
conduc1vity


Hot
Th


Cold
Tc


L


Q
(heat
flow)


Fourier’s
Law
for
Heat
Conduc1on


Heat
Diffusion
Equa1on


Specific
heat


Heat
conduc;on
=
Rate
of
change
of
energy
storage


1st
law
(energy
conserva;on)


• Condi;ons:


t
>>
t
≡
scaUering
mean
free
;me
of
energy
carriers

L
>>
l
≡
scaUering
mean
free
path
of
energy
carriers


Breaks
down
for
applica;ons
involving
thermal
transport
in
small
length/
;me
scales,
e.g.
nanoelectronics,
nanostructures,
NEMS,
ultrafast
laser
materials
processing…




Length
Scale


1
m


1
mm


1
mm


1
nm


Human


Automobile


BuSerfly


1
km



Aircraf


Computer


Wavelength
of
Visible
Light


MEMS


Width
of
DNA


MOSFET,
NEMS


Blood
Cells


Microprocessor
Module


Nanotubes,
Nanowires


Par1cle
transport
100
nm


Fourier’s
law


l


D


D


Total
Length
Traveled
=
L


Total
Collision
Volume
Swept
=
πD2
L


Number
Density
of
Molecules
=
n



Total
number
of
molecules
encountered
in
swept
collision
volume
=
nπD2L


Average
Distance
between
Collisions,
mc
=
L/(#of
collisions)


Mean
Free
Path


σ:
collision
cross‐sec;onal
area


Number
Density
of
Molecules
from
Ideal
Gas
Law:



















n
=
P/kBT


kB:
Boltzmann
constant







1.38
x
10‐23
J/K


Mean
Free
Path:




Typical
Numbers:


Diameter
of
Molecules,
D
≈
2
Å
=
2
x10‐10
m
Collision
Cross‐sec;on:
σ
≈
1.3
x
10‐19
m


Mean
Free
Path
at
Atmospheric
Pressure:




At
1
Torr
pressure,
mc
≈
300
mm;

at
1
mTorr,
mc
≈
30
cm


Wall


Wall


b:
boundary
separa;on


Effec;ve
Mean
Free
Path:




z

z - z

z + z

u(z-z)

u(z+z)

θ qz

Net Energy Flux / # of Molecules

through Taylor expansion of u

u: energy

Integration over all the solid angles total energy flux

Thermal conductivity:

Specific heat Velocity Mean free path


EF
F:

Work
Func;on


Energy


Fermi
Energy
–
highest
occupied
energy
state:


Fermi
Velocity:


Vacuum
Level


Band
Edge


Fermi
Temp:


Metal


Fermi‐Dirac
equilibrium
distribu;on

for
the
probability
of
electron

occupa;on
of
energy

level
E
at
temperature
T


0


1


E
F
Electron
Energy,

E


Occup

a;on


Probability,

f


Work
Func;on,
F


Increasing

T


T

=
0
K
k
T
B


Vacuum

Level


Density of States -- Number of electron states available between energy E and E+dE

Number density:

Energy density:

in 3D

Specific
Heat


Thermal
Conduc;vity


Electron
ScaSering
Mechanisms
• 

Defect
ScaUering
• 

Phonon
ScaUering
• 

Boundary
ScaUering
(Film
Thickness,






Grain
Boundary)


e


Temperature,
T


Defect

ScaUering


Phonon
ScaUering


Increasing
Defect
Concentra;on





Bulk
Solids


Mean
free
;me:

te
=
le
/
vF



in
3D


MaUhiessen
Rule:


Electrons
dominate
k
in
metals



Crystalline
vs.
Glasslike
Thermal
Conduc;vity


P.
W.
Anderson,
B.
I.
Halperin,
C.
M.
Varma,
Phil.
Mag.
25,
1
(1972).


Interatomic
Bonding


1‐D
Array
of
Spring
Mass
System


Equa;on
of
mo;on
with
nearest
neighbor
interac;on


Solu;on


Freq

uency,
ω

Wave
vector,
K
0
 π/a


Group
Velocity:


Speed
of
Sound:


La^ce
Constant,
a


xn
 yn
yn‐1
 xn+1


Freq

uency,
ω

Wave
vector,
K
0
 π/a


LA
TA


LO


TO


Op;cal
Vibra;onal
Modes


Total
Energy
of
a
Quantum
Oscillator
in
a
Parabolic
Poten;al


n
=
0,
1,
2,
3,
4…;

w/2:

zero
point
energy


Phonon:

A
quantum
of
vibra;onal
energy,


w,
which
travels
through
the
la^ce


Phonons
follow
Bose‐Einstein
sta1s1cs.


Equilibrium
distribu;on:



In
3D,
allowable
wave
vector
K:


p:
polariza;on(LA,TA,
LO,
TO)
K:
wave
vector


Dispersion
Rela;on:



Energy
Density:



Density
of
States:
 Number
of
vibra;onal
states
between
w
and
w+dw


La^ce
Specific
Heat:


in
3D


Freq

uency,
w


Wave
vector,
K
0
 p/a


Debye
Approxima;on:


Debye
Density

of
States:


Debye
Temperature
[K]


Specific
Heat
in
3D:


In
3D,
when
T


Classical
Regime


In
general,
when
T


Kine;c
Theory


l


Temperature,
T/qD


Boundary
Phonon
ScaUering
Defect


Decreasing
Boundary
Separa;on


Increasing
Defect
Concentra;on


Phonon
ScaUering
Mechanisms


• 
Boundary
ScaUering
• 
Defect
&
Disloca;on
ScaUering
• 
Phonon‐Phonon
ScaUering


0.01
 0.1
 1.0


• 
Phonons
dominate
k
in
insulators