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 Effect In 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. V V Where S is the Seebeck coefficient. It is S = 2,1 = 2,1 defined as the voltage generated per degree − TT ΔT of temperature difference between the two 12 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 Effect The 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 the opposite side. An electrical current is produced, proportional to the temperature gradient between the hot and cold junctions Explanation 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 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‐type material) will produce a negative potential 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, generating electrical power. = = α ⋅ΔTVVoltage L RR == L Thermo σ ⋅ A V 222 σα⋅⋅Δ⋅ AT VIPower ==⋅= RL L
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 Effect In 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 = Π × Iq proportionality constant is the Peltier Coefficient (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. ×=Π TS
Π >0 ; Positive Peltier coefficient. High energy holes move from left to right. Thermal current and electric current flow in same direction.
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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. 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. dQ dT λ I ××= λ is the Thomson coefficient in Volts/Kelvin dx dx Thompson Effect = Seebeck Effect + Peltier Effect
The relationships between the different effects are called the Kelvin relationships. Π First Kelvin relationship: S = T
δS λ Second Kelvin relationship: = δ TT Thermoelectric Figure of Merit (ZT) Coefficient of Performance
Tc + − /1 TTzT chm COPmax = −TT ch zTm ++ 11
where
Seebeck coefficient 2 Electrical conductivity TH = 300 K
max T = 250 K S 2σ 1 C Freon ZT ≡ T Temperature Bi Te κ COP 0 2 3 012345 Thermal conductivity ZT Requirements for a Good Thermoelectric Material
•General considerations for the selection of materials for thermoelectric applications involve: S 2σ –High figure of merit Z = –large Seebeck coefficient α (or S) electrons + κκphonons –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! S 2σ ZT = ×T electrons + κκphonons Minimizing thermal conductivity Thermal conductivity consists of two parts: lattice conductivity (lattice vibrations = phonons), κLattice, and thermal conductivity of charges (electrons and holes), κelectrons:
κ = κ Lattice + κ FreeCarriers Currently, most of the research efforts are devoted to minimizing the lattice conductivity of new phases.
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=1 to 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). •Nanostructured Thermoelectrics Complex Crystal Structures Rattlers: These are weakly bound atoms that fill cages. They have unusually large values of Atomic Displacement Parameters Properties of many clathrate‐like XE24 XE20 compounds can be understood by treating “rattler” atoms as Einstein oscillators and framework atoms as a Debye solid. Skutterudites, LaB6, Tl2SnTe5 A Characteristic Einstein temperature (or frequency) can be assigned to each rattler
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0.00 0 50 100 150 200 250 300 T (K) Advantages of Disadvantages of Thermoelectrics Thermoelectrics •Absence of moving parts •High cost •High reliability •Low efficiency •Quietness •Typically about 3 to 7% •Lack of vibrations •Low maintenance •Simple start up •No pollution •Small •Light weight •No noise •Precise temperature control: within +/‐ 0.1C Applications of Thermoelectric
• Consumer Applications
•Automobile Applications
• Industrial Applications
• Military and Space Applications Consumer Applications
TE Fridge
Beer Cooler
Chocolate Cooler Automobile Applications
Can Cooler Seat Cooler/Warmer Industrial Applications
Electronic Cooler TE Dehumidifier Military and Space Applications
Night Vision ThermalThermal PropertiesProperties ofof MaterialsMaterials 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 δQ Basic Principles C = δT 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). Fourier’s Law for Heat Conduction
Q (heat flow)
Hot Cold
Th Tc L
T −T dT = kAQ ch = kA L dx
Thermal conductivity Heat Diffusion Equation 1st law (energy conservation) Heat conduction = Rate of change of energy storage
∂T ∂2T C = k 2 Specific heat ∂t ∂ x
•Conditions: t >> t ≡ scattering mean free time of energy carriers L >> 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… 1 km Length Scale Aircraft Automobile 1 m Human
Computer
Butterfly 1 mm Fourier’s law Microprocessor Module MEMS
Blood Cells 1 mm Wavelength of Visible Light Particle MOSFET, NEMS 100 nm l transport Nanotubes, Nanowires Width of DNA 1 nm MeanMean FreeFree PathPath forfor IntermolecularIntermolecular CollisionCollision forfor GasesGases
D D
Total Length Traveled = L
Average Distance between Collisions, A = L/(#of collisions) Total Collision Volume Swept = πD2 L mc Number Density of Molecules = n Mean Free Path Total number of molecules encountered in L 1 swept collision volume = nπD2L A mc == πDn 2L nσ
σ: collision cross‐sectional area MeanMean FreeFree PathPath forfor GasGas MoleculesMolecules
Number Density of kB: Boltzmann constant ‐23 Molecules from Ideal 1.38 x 10 J/K
Gas Law: n = P/kBT 1 k T Mean Free Path: A == B mc nσ Pσ Typical Numbers: Diameter of Molecules, D ≈ 2 Å= 2 x10‐10 m Collision Cross‐section: σ≈1.3 x 10‐19 m
Mean Free Path at Atmospheric Pressure: −23 ×× 3001038.1 −7 A mc ≈ ×≈ m103 μm0.3or 5 × ×103.110 −19
At 1 Torr pressure, Amc ≈ 300 mm; at 1 mTorr, Amc ≈ 30 cm EffectiveEffective MeanMean FreeFree PathPath
Wall
Ab: boundary separation
Wall
Effective Mean Free Path: 111 += AAA bmc KineticKinetic TheoryTheory ofof EnergyEnergy TransportTransport u: energy u(z+A ) Net Energy Flux / # of Molecules z z + A z 1 θ A qz ' zz []()()z zuzuvq +−−= AA z z 2 through Taylor expansion of u u(z-Az) z - A z du du ' vq A −=−= (cos2 θ )vA zzz dz dz Integration over all the solid angles Æ total energy flux 1 du dT 1 dT dT −= vq A CvA −=−= k z 3 dT dz 3 dz dz 1 Thermal conductivity: = Cvk A 3 Specific heat Velocity Mean free path FreeFree ElectronsElectrons inin MetalsMetals atat 00 KK
Fermi Energy –highest occupied energy state: Metal 222 2 kF == 2 3 EF == (3 ηπe ) 22 mm Vacuum 1 F: Work Function Level = 2 3 Fermi Velocity: v = ()3 ηπ E F m e F Energy
EF Fermi Temp: TF = kB
Band Edge
Element Electron Fermi Fermi Fermi Fermi Work Density, ηe Energy Temperature Wavelength Velocity Function 28 -3 4 6 [10 m ] EF [eV] TF [10 K] λF [Å] vF [10 m/s] 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
EffectEffect ofof TemperatureTemperature Fermi‐Dirac equilibrium distribution 1 for the probability of electron ()Ef = occupation of energy ⎛ − EE ⎞ level E at temperature T + exp1 ⎜ F ⎟ ⎝ BTk ⎠ f
kB T 1 T = 0 K Probability, Vacuum Level Increasing T
Occupation 0 E Electron Energy, E F Work Function, F NumberNumber andand EnergyEnergy DensitiesDensities
N ∞ Number density: ηe == ∫ ()e ()dEEDEf ; V 0 ∞ Ee Energy density: ∈e == ∫ ()e ()dEEDEEf V 0
Density of States -- Number of electron states available between energy E and E+dE 2mEm e ()ED = in 3D π == 222 ElectronicElectronic SpecificSpecific HeatHeat andand ThermalThermal ConductivityConductivity
∞ d ∈ df 2 ⎛ ⎞ C = e = E ()dEED π BTk Specific Heat e ∫ C = ⎜ ⎟η k in 3D dT dT e ⎜ ⎟ Be 0 2 ⎝ EF ⎠
1 1 2 Mean free time: Thermal Conductivity = A = vCvCk τ eFeeFee 3 3 te = le / vF
Ae Electron Scattering Mechanisms
Bulk Solids •Defect Scattering • Phonon Scattering Increasing • Boundary Scattering (Film Thickness, Defect Concentration Grain Boundary)
Defect Phonon Scattering Scattering
Temperature, T Grain Grain Boundary ThermalThermal ConductivityConductivity ofof CuCu andand AlAl Matthiessen Rule: 1 1 2 = A = vCvCk τ eFeeFee 1111 3 3 += + e defect boundary ττττphonon
10 3 1111 += + e defect AAA boundary A phonon
Copper 10 2 1 1 Electrons dominate k in metals Aluminum 10 1
Defect Scattering Phonon Scattering Thermal Conductivity, k [W/cm-K] Conductivity, k Thermal 10 0 10 0 10 1 10 2 10 3 Temperature, T [K} Crystalline vs. Glasslike Thermal Conductivity
P. W. Anderson, B. I. Halperin, C. M. Varma, Phil. Mag. 25, 1 (1972). CrystalCrystal VibrationVibration
Interatomic Bonding Equation of motion with Energy nearest neighbor interaction
Parabolic Potential of 2 Harmonic Oscillator xd n r m ()−+ 11 −+= 2xxxg nnn o Distance dt 2 Solution
= on exp(− ω )exp(inKatixx )
Eb 1‐D Array of Spring Mass System Spring constant, g Mass, m
Equilibrium Position a
Deformed Position x n-1 xn xn+1 DispersionDispersion RelationRelation
ω 2 gm [ exp2 ( iKa)−−−= exp( )] ( −= cos12 KagiKa )
2g 1 ω ()−= cos1 Ka 2 m
Group Velocity: e od M dω A)
ω (L v = g tic us dK co l A na Speed of Sound: di itu de ng o dω Lo ) M Frequency, A v = lim (T s tic K→0 dK us co A se er sv an Tr
0 Wave vector, K π/a TwoTwo AtomsAtoms PerPer UnitUnit CellCell
Lattice Constant, a
Optical Vibrational y x Modes n‐1 xn yn n+1
LO 2 ω xd n m1 ()−1 −+= 2xyyg nnn TO dt 2 2 yd n Frequency, TA m2 ()+1 −+= 2yxxg nnn LA dt 2
0 Wave vector, K π/a PhononPhonon DispersionDispersion inin GaAsGaAs
LO LO 8 TO TO
6 12 LA LA 4 Frequency (10 Hz) TA 2 TA
0 0.4 0.2 0 0.2 0.4 0.6 0.8 1.0 L (111) Direction Γ Ka/π (100) Direction X EnergyEnergy QuantizationQuantization andand PhononsPhonons
Energy Total Energy of a Quantum Oscillator in a Parabolic Potential ⎛ 1 ⎞ ⎜nu += ⎟=ω Distance ⎝ 2 ⎠ n = 0, 1, 2, 3, 4…; =w/2: zero point energy
Phonon: A quantum of vibrational energy, =w, which travels through the lattice hω Phonons follow Bose‐Einstein statistics. Equilibrium distribution: 1 n = ⎛ =ω ⎞ exp⎜ ⎟ −1 ⎝ BTk ⎠
2π 4π 6π In 3D, allowable wave vector K: , , ,.... LLL LatticeLattice EnergyEnergy ⎡ 1⎤ E = n()ω + =ω p: polarization(LA,TA, LO, TO) l ∑∑ ⎢ , pK ⎥ , pK K: wave vector p K ⎣ 2⎦
Dispersion Relation: = gK (ω)
El ⎡ 1⎤ Energy Density: ∈l == ∑ ∫ ⎢ n()+ ⎥= ()dD ωωωω V p ⎣ 2⎦
Density of States: Number of vibrational states between w and w+dw 2 (ω) dgg D()ω = in 3D 2π 2 dω d ∈ nd Lattice Specific Heat: l Cl = = ∑ ∫ = ()dD ωωω dT p dT DebyeDebye ModelModel w ω = sKv Debye Approximation: ω = vs K 2 2 Debye Density (ω) dgg ω
D()ω = = Frequency, of States: 2 32 2π dω 2π vs
Specific Heat in 3D: 0 Wave vector, K p/a 3 ⎡θ D ⎤ ⎛ ⎞ T x 4dxxeT Debye Temperature [K] = 9ηkC ⎜ ⎟ ⎢ ⎥ Bl ⎜ ⎟ ∫ 2 1 ⎝ θ D ⎠ ⎢ x ⎥ 2 3 ⎣ 0 ()e −1 ⎦ =vs (6 ηπ) θD = kB In 3D, when T << θ , D C(dimnd) 1860 Ga 240 Si 625 NaF 492 4 3 Ge 360 NaCl 321 l ∝∈ , l ∝ TCT B 1250 NaBr 224 Al 394 NaI 164 PhononPhonon SpecificSpecific HeatHeat
10 7 C = 3ηk = 4.7 ×106 J 3ηk T B m3 −K B
6 10 Diamond Each atom has 3 5 Diamond
‐K) 10 3 a thermal energy
of 3KBT (J/m 10 4
Heat 3 C ∝ T3 3 C ∝ T 10 Classical Specific Heat, C (J/m -K) Regime Specific 10 2 θD =1860 K
10 1 10 1 10 2 10 3 10 4 Temperature, T (K) Temperature (K)
In general, when T << qD, d +1 d ∈l ∝ , l ∝ TCT d =1, 2, 3: dimension of the sample PhononPhonon ThermalThermal ConductivityConductivity
1 1 Phonon Scattering Mechanisms Kinetic Theory A == vCvCk 2τ 3 3 lsllsll • Boundary Scattering • Defect & Dislocation Scattering Decreasing Boundary • Phonon‐Phonon Scattering Separation Al
kl Increasing Increasing Defect Defect Concentration Concentration d Phonon ∝ Tk Defect l Boundary Scattering 0.01 0.1 1.0 Phonon Boundary Defect Scattering Temperature, T/qD 0.01 0.1 1.0
Temperature, T/θD ThermalThermal ConductivityConductivity ofof InsulatorsInsulators 10 3 Diamond • Phonons dominate k in insulators
10 2
1 10 Increasing Defect Density
10 0
10 -1 Thermal Conductivity, k [W/cm-K] Thermal Conductivity, Boundary Defect Scattering Scattering 10 -2 10 0 10 1 10 2 10 3 Temperature, T [K]