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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 converting electricity into a temperature gradient.
The thermoelectric effect is based on a combination of two different effects, namely, the Seebeck effect and the Peltier effect.
Water/Beer Cooler
What is thermoelectricity?Thermoelectricity is the conversion of heat differentials into electricity and viceversa. Thermoelectric energy conversion is one of the direct energy conversion technologies that rely on the electronic properties of the material (semiconductor) for its efficiency. It is based on the Seebeck (Power Generation) and Peltier effects (Heat Pumping).
Seebeck EffectIn 1821, Thomas Seebeck a German Estonian physicist 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. If the temperature gradient is reversed, the direction of the current is reversed.
TV
TTV
SΔ
=−
= 2,1
12
2,1Where S is the Seebeck coefficient. It is defined as the voltage generated per degree of temperature difference between the two points.
S is positive when the direction of the current is the same as the direction of the
The basis of the Seebeck effect is electron mobility in conductors and semiconductors, which is a function of temperature
When two different metals are joined, the relative 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 potential 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 EffectThe Seebeck effect is the conversion of heat differences directly into electricity. When two dissimilar materials with different carrier densities 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 matter releases energy and cools to a lower energy state. The net work is proportional 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 theopposite side. An electrical current is produced, proportional to the temperature gradient between the hot and cold junctions
Explanation of Seebeck EffectIn 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 electrostatic force (and therefore electric potential) to push the charges back to the hot end.
The electric potential produced by a temperature difference is known as the Seebeck effect and the proportionality constant is called the Seebeck coefficient (α or S).If the free charges are positive (the material is p‐type), positive charge will build up on the cold which will have a positive potential. Similarly, negative free charges (n‐typematerial) will produce a negative potential at the cold end.
If the hot ends of the n‐type and p‐typematerial 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, generating electrical power.
LAT
RVVIPower
ALRR
TVVoltage
L
ThermoL
⋅⋅Δ⋅==⋅=
⋅==
Δ⋅==
σασ
α
222
α2σ is the materials property known as the thermoelectric power factor. For efficient operation, high power must be produced with a minimum of heat (Q).κ= Thermal conductivity. The thermal conductivity acts as a thermal short and reduces efficiency.
Peltier EffectIn 1834, a French scientist Jean Peltier found that a thermal difference can be obtained at the junction of two metals, if an electric current is made to flow in them.
Opposite of the Seebeck Effect. The heat current (q) is proportional to the charge current (I) and the proportionality constant is the Peltier Coefficient (Π).
Iq ×Π=
When two materials are joined together, there will be an excess or deficiency in the energy at the junction because the two materials have different Peltier coefficients. The excess energy is released to the lattice at the junction, causing heating, and the deficiency in energy is supplied by the lattice, creating cooling.
The Seebeck and the Peltier coefficients are related to each other through the Kelvin relationship – T is the absolute temperature.
TS ×=ΠΠ >0 ; Positive Peltier coefficient. High energy holes move from left to right. Thermal current and electric current flow in same direction.
Π < 0 ; Negative Peltier coefficient
High energy electrons move from right to left.
Thermal current and electric current flow in opposite directions.
If an electric current is applied to the thermocouple as shown, heat is pumped from the cold junction to the hot junction. The cold junction will rapidly drop below ambient temperature provided heat is removed from the hot side. The temperature gradient will vary according to the magnitude of current applied.
When two dissimilar materials with different carrier densities are connected to each other by an electrical conductor, electrical current (work input), forces the matter to approach a higher energy state and heat is absorbed (cooling). The energy is released (heating) as the matter approaches a lower energy state. The net cooling effect is proportional to the electric current and Peltier Effect coefficient.
The Peltier 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 proportional to the product of the current and the temperature gradient.
dxdTI
dxdQ
××= λ
Thompson Effect = Seebeck Effect + Peltier Effect
λ is the Thomson coefficient in Volts/Kelvin
The relationships between the different effects are called the Kelvin relationships.
First Kelvin relationship: T
S Π=
Second Kelvin relationship: TTS λ
δδ
=
11/1
COPmax ++−+
−=
m
chm
ch
czT
TTzTTT
T
Coefficient of Performance
where
Thermoelectric Figure of Merit (ZT)
TSZTκσ2
≡
Seebeck coefficientElectrical conductivity
Thermal conductivity
Temperature0
1
2
0 1 2 3 4 5ZT
CO
P max
Bi2Te3
FreonTH = 300 KTC = 250 K
Requirements for a Good Thermoelectric Material
• General considerations for the selection of materials for thermoelectric applications involve:– High figure of merit – large Seebeck coefficient α (or S)– high electrical conductivity σ– low thermal conductivity κLattice+κelectrons
– Possibility of obtaining both n‐type and p‐type thermoelements.– No viable superconducting passive legs developed yet
• Good mechanical, metallurgical and thermal characteristics– Capable of operating over a wide temperature range. Especially true for high temperature applications.
– To allow their use in practical thermoelectric devices– Materials cost can be an important issue!
phononselectrons
Sκκ
σ+
=2
Z
TSphononselectrons
×+
=κκ
σ2
ZT
Thermal conductivity consists of two parts: lattice conductivity (lattice vibrations = phonons), κLattice, and thermal conductivity of charges (electrons and holes), κelectrons:
Currently, most of the research efforts are devoted to minimizing the lattice conductivity of new phases.
Minimizing thermal conductivity
rsFreeCarrieLattice κκκ +=
Some ways to reduce the lattice conductivity:(1) use of heavy elements, e.g. Bi2Te3, Sb2Te3 and PbTe; (2) a large number N of atoms in the unit cell: the fraction of vibrational modes (phonons) that carry heat efficiently to 1/N;(3) rattling of the atoms, e.g. filled skutterudite CeFe4Sb12; disorder in atomic structure: random atomic distribution 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 conductivity typical for heavily doped semiconductors and thermal conductivity typical for amorphous solids. In fact, its thermal conductivity is the lowest among state‐of‐the art thermoelectric materials:
Minimize thermal conductivity and maximize electrical conductivity has been the biggest dilemma for the last 40 years.�
Bismuth telluride is the standard with ZT=1to match a refrigerator you need ZT= 4 ‐ 5 to recover waste heat from car ZT = 2
Can the conflicting requirements be met by nano‐scale material design?
Reduce the lattice thermal conductivity by:
•Complex crystal structure of high atomic number materials.
•Rattlers in the structure (Atomic Displacement Parameter – ADP).
•NanostructuredThermoelectrics
Complex Crystal Structures
Rattlers:These are weakly bound atoms that
fill cages.They have unusually large values of
Atomic Displacement ParametersProperties of many clathrate‐like
compounds can be understood by treating “rattler” atoms as Einstein oscillators and framework atoms as a Debye solid.Skutterudites, LaB6, Tl2SnTe5A Characteristic Einstein temperature
(or frequency) can be assigned to each rattler
XE20XE24
Eu8‐eGa16Ge30 Phase With the Ba8Ga16Sn30Clathrate 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 (<u2> �) From 6d Site
0.00
0.05
0.10
0.15
0 50 100 150 200 250 300
X8Ga16Ge30U eq
(Å2 )
T (K)
Eu
Sr
Ba
Advantages of Thermoelectrics•Absence of moving parts•High reliability•Quietness•Lack of vibrations•Low maintenance•Simple start up•No pollution•Small•Light weight•No noise•Precise temperature control: within +/‐ 0.1C
Disadvantages of Thermoelectrics•High cost•Low efficiency•Typically about 3 to 7%
Applications of Thermoelectric
• Consumer Applications
• Automobile Applications
• Industrial Applications
• Military and Space Applications
Consumer Applications
Beer Cooler
TE Fridge
Chocolate Cooler
Automobile Applications
Seat Cooler/WarmerCan Cooler
Industrial Applications
Electronic Cooler TE Dehumidifier
Military and Space Applications
Night Vision
Thermal Properties of MaterialsThermal Properties of Materials
Basic Principles•Macroscopic Thermal Transport Theory– Diffusion
‐‐ Fourier’s Law‐‐ Diffusion Equation
•Microscale Thermal Transport Theory – Particle Transport
‐‐ Kinetic Theory of Gases‐‐ Electrons in Metals‐‐ Phonons in Insulators‐‐ Boltzmann Transport Theory
Basic PrinciplesTQC
δδ
=
Heat is a form of energy. The thermal properties describe how a solid responds to changes in its thermal energy.
The heat capacity (C) of a solid quantifies the relationship 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 conductivity
HotTh
ColdTc
L
Q (heat flow)
Fourier’s Law for Heat Conduction
dxdTkA
LTTkAQ ch =
−=
Heat Diffusion Equation
xTk
tTC 2
2
∂∂
=∂∂
Specific heat
Heat conduction = Rate of change of energy storage
1st law (energy conservation)
•Conditions: t >> t ≡ scattering mean free time of energy carriersL >> l ≡ scattering mean free path of energy carriers
Breaks down for applications involving thermal transport in small length/ time scales, e.g. nanoelectronics, nanostructures, NEMS,ultrafast laser materials processing…
Length Scale
1 m
1 mm
1 mm
1 nm
Human
Automobile
Butterfly
1 km
Aircraft
Computer
Wavelength of Visible Light
MEMS
Width of DNA
MOSFET, NEMS
Blood Cells
Microprocessor Module
Nanotubes, Nanowires
Particle transport100 nm
Fourier’s law
l
D
D
Mean Free Path for Intermolecular Mean Free Path for Intermolecular Collision for GasesCollision for Gases
Total Length Traveled = L
Total Collision Volume Swept = πD2 L
Number Density of Molecules = n
Total number of molecules encountered inswept collision volume = nπD2L
Average Distance betweenCollisions, mc = L/(#of collisions)
Mean Free Path
σπ nLDnL
mc1
2 ==
σ: collision cross‐sectional area
Mean Free Path for Gas MoleculesMean Free Path for Gas Molecules
Number Density ofMolecules from IdealGas Law: n = P/kBT
kB: Boltzmann constant1.38 x 10‐23 J/K
Mean Free Path:
σσ PTk
nB
mc ==1
Typical Numbers:
Diameter of Molecules, D ≈ 2 Å = 2 x10‐10 mCollision Cross‐section: σ ≈ 1.3 x 10‐19 m
Mean Free Path at Atmospheric Pressure:
m0.3or m103103.110
3001038.1 7195
23μ−
−
−×≈
××××
≈mc
At 1 Torr pressure, mc ≈ 300 mm; at 1 mTorr, mc ≈ 30 cm
Wall
Wall
b: boundary separation
Effective Mean Free Path:
Effective Mean Free PathEffective Mean Free Path
bmc
111+=
Kinetic Theory of Energy TransportKinetic Theory of Energy Transport
z
z - z
z + z
u(z- z)
u(z+ z)
θ ( ) ( )[ ]zzzz zuzuvq +−−=21'qz
Net Energy Flux / # of Molecules
( )dzduv
dzduvq zzz θ2cos' −=−=
through Taylor expansion of u
u: energy
dzdTk
dzdTCv
dzdT
dTduvqz −=−=−=
31
31
Integration over all the solid angles total energy flux
Cvk31
=
Specific heat Velocity Mean free path
Thermal conductivity:
Free Electrons in Metals at 0 KFree Electrons in Metals at 0 K
EF
F: Work Function
Energy
Fermi Energy – highest occupied energy state:
Fermi Velocity:
( )
( ) 312
322
222
3
322
eF
eF
F
mv
mmkE
ηπ
ηπ
=
==
Element Electron Density, ηe [1028 m-3]
Fermi Energy EF [eV]
Fermi Temperature TF [104 K]
Fermi Wavelength λF [Å]
Fermi Velocity vF [106 m/s]
Work Function Φ [eV]
Cu 8.47 7.00 8.16 4.65 1.57 4.44 Au 5.90 5.53 6.42 5.22 1.40 4.3 Fe 17.0 11.1 13.0 2.67 1.98 4.31 Al 18.1 11.7 13.6 3.59 2.03 4.25
VacuumLevel
Band Edge
Fermi Temp:B
FF k
ET =
Metal
Effect of TemperatureEffect of Temperature
( )⎟⎠
⎞⎜⎝
⎛ −+=
TkEE
Ef
B
Fexp1
1Fermi‐Dirac equilibrium distributionfor the probability of electron occupation of energy level E at temperature T
0
1
EFElectron Energy, E
Occup
ation Prob
ability, f
Work Function, F
Increasing T
T = 0 K
k TB
Vacuum Level
( ) ( )
( ) ( )dEEDEEfVE
dEEDEfVN
ee
e
ee
∫==∈
∫==
∞
∞
0
0;η
Number and Energy DensitiesNumber and Energy Densities
Density of States -- Number of electron states available betweenenergy E and E+dE
( ) 2222mEmEDe π
=
Number density:
Energy density:
in 3D
Electronic Specific Heat and Thermal ConductivityElectronic Specific Heat and Thermal Conductivity
( )dEEDdTdfE
dTdC e
e ∫∞
=∈
=0
BeF
Be k
ETk
C ηπ⎟⎟⎠
⎞⎜⎜⎝
⎛=
2
2
eFeeFee vCvCk τ231
31
==
Specific Heat
Thermal Conductivity
Electron Scattering Mechanisms• Defect Scattering• Phonon Scattering• Boundary Scattering (Film Thickness,
Grain Boundary)
Grain Grain Boundary
e
Temperature, T
Defect Scattering
PhononScattering
IncreasingDefect Concentration
Bulk Solids
Mean free time:te = le / vF
in 3D
10 310 210 110 010 0
10 1
10 2
10 3
Temperature, T [K}
Ther
mal
Con
duct
ivity
, k [
W/c
m-K
]
Copper
Aluminum
Defect Scattering Phonon Scattering
11
eFeeFee vCvCk τ231
31
==Matthiessen Rule:
Thermal Conductivity of Cu and AlThermal Conductivity of Cu and Al
phononboundarydefecte
phononboundarydefecte
1111
1111
++=
++=ττττ
Electrons dominate k in metals
Crystalline vs. Glasslike Thermal Conductivity
P. W. Anderson, B. I. Halperin, C. M. Varma, Phil. Mag. 25, 1 (1972).
Crystal VibrationCrystal Vibration
Energy
Distancero
Parabolic Potential of Harmonic Oscillator
Eb
Interatomic Bonding
a
Spring constant, g Mass, m
xn xn+1xn-1
Equilibrium Position
Deformed Position
1‐D Array of Spring Mass System
( )nnnn xxxg
dtxdm 2112
2−+= −+
Equation of motion withnearest neighbor interaction
( ) ( )inKatixx on expexp ω−=
Solution
Dispersion RelationDispersion Relation
( ) ( )[ ] ( )
( ) 21
cos12
cos12expexp22
Kamg
KagiKaiKagm
−=
−=−−−=
ω
ω
Freq
uency, ω
Wave vector, K0 π/a
Longitudina
l Acou
stic (LA
) Mode
Transv
erse A
coustic
(TA) Mo
de
Group Velocity:
dKdvg
ω=
Speed of Sound:
dKdv
Ks
ω0
lim→
=
Lattice Constant, a
xn ynyn‐1 xn+1
( )
( )nnnn
nnnn
yxxgdt
ydm
xyygdt
xdm
2
2
12
2
2
12
2
1
−+=
−+=
+
−
Two Atoms Per Unit CellTwo Atoms Per Unit Cell
Freq
uency, ω
LATA
Wave vector, K0 π/a
LO
TO
OpticalVibrationalModes
0 0.2 0.4 0.6 0.8 1.00.20.40
2
4
6
8
(111) Direction (100) DirectionΓ XL Ka/π
LA
TATA
LA
LO
TO
LO
TO
Freq
uenc
y (1
0 H
z)12
Phonon Dispersion in GaAsPhonon Dispersion in GaAs
Energy Quantization and PhononsEnergy Quantization and Phonons
hω
Energy
Distance
Total Energy of a QuantumOscillator in a Parabolic Potential
ω⎟⎠⎞
⎜⎝⎛ +=
21nu
n = 0, 1, 2, 3, 4…; w/2: zero point energy
Phonon: A quantum of vibrational energy, w, which travels through the lattice
Phonons follow Bose‐Einstein statistics.
Equilibrium distribution:
1exp
1
−⎟⎟⎠
⎞⎜⎜⎝
⎛=
Tk
n
B
ω
In 3D, allowable wave vector K: ,....6,4,2LLLπππ
Lattice EnergyLattice Energy
( ) pKpKp
l nE ,, 21 ωω∑∑ ⎥⎦
⎤⎢⎣⎡ +=
K
p: polarization(LA,TA, LO, TO)K: wave vector
Dispersion Relation: ( )ωgK =
Energy Density: ( ) ( ) ωωωω dDnVE
p
ll ∫ ⎥⎦
⎤⎢⎣⎡ +∑==∈
21
( ) ( )ωπ
ωωddggD 2
2
2=
Density of States: Number of vibrational states between w and w+dw
Lattice Specific Heat: ( ) ωωω dDdT
nddTdC
p
ll ∫∑=
∈=
in 3D
Debye ModelDebye Model
Freq
uency, w
Wave vector, K0 p/a
Kvs=ωKvs=ωDebye Approximation:
( ) ( )32
2
2
2
22 svddggD
πω
ωπωω ==Debye Density
of States:
( )B
sD k
v 3126 ηπθ =
C(dimnd) 1860 Ga 240Si 625 NaF 492Ge 360 NaCl 321B 1250 NaBr 224Al 394 NaI 164
Debye Temperature [K]
Specific Heat in 3D:
( ) ⎥⎥
⎦
⎤
⎢⎢
⎣
⎡
−⎟⎟⎠
⎞⎜⎜⎝
⎛= ∫
TD
x
x
DBl
e
dxxeTkC
θ
θη
02
43
19
In 3D, when T << θD,
34, TCT ll ∝∝∈
Phonon Specific HeatPhonon Specific Heat
10 410 310 210 110 1
10 2
10 3
10 4
10 5
10 6
10 7
Temperature, T (K)
Spec
ific
Hea
t, C
(J/
m -
K)3
C ∝ T 3
C = 3ηkB = 4.7 ×106 Jm3 −K
θD =1860 K
Diamond
ClassicalRegime
In general, when T << qD,
dl
dl TCT ∝∝∈ + ,1
d =1, 2, 3: dimension of the sample
Each atom has a thermal energy of 3KBT
Specific Heat (J/m
3 ‐K)
Temperature (K)
C ∝ T3
3ηkBT
Diamond
Phonon Thermal ConductivityPhonon Thermal Conductivity
lsllsll vCvCk τ231
31
==Kinetic Theory
l
Temperature, T/qD
BoundaryPhononScatteringDefect
Decreasing BoundarySeparation
IncreasingDefectConcentration
Phonon Scattering Mechanisms
• Boundary Scattering
• Defect & Dislocation Scattering
• Phonon‐Phonon Scattering
0.01 0.1 1.0
Temperature, T/θD
0.01 0.1 1.00.01 0.1 1.0
kl
dl Tk ∝
BoundaryPhononScatteringDefect
Increasing DefectConcentration
10 310 210 110 010 -2
10 -1
10 0
10 1
10 2
10 3
Temperature, T [K]
Ther
mal
Con
duct
ivity
, k [W
/cm
-K]
Diamond
BoundaryScattering
DefectScattering
IncreasingDefect Density
• Phonons dominate k in insulators
Thermal Conductivity of InsulatorsThermal Conductivity of Insulators