THERMOELECTRICS

THERMOELECTRICSDESIGN AND MATERIALS

HoSung LeeWestern Michigan University, USA

This edition first published 2017

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Names: Lee, HoSung, author.Title: Thermoelectrics : design and materials / HoSung Lee.Description: Chichester, UK ; Hoboken, NJ : John Wiley & Sons, 2016. |

Includes bibliographical references and index.Identifiers: LCCN 2016025876| ISBN 9781118848951 (cloth) | ISBN 9781118848937

(epub) | ISBN 9781118848920 (epdf)Subjects: LCSH: Thermoelectric apparatus and appliances–Design and

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Cover design by Yujin Lee

ISBN: 9781118848951

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Table of Contents

Preface xiii

1 Introduction 11.1 Introduction 11.2 Thermoelectric Effect 3

1.2.1 Seebeck Effect 31.2.2 Peltier Effect 31.2.3 Thomson Effect 41.2.4 Thomson (or Kelvin) Relationships 4

1.3 The Figure of Merit 41.3.1 New-Generation Thermoelectrics 5Problems 7References 7

2 Thermoelectric Generators 82.1 Ideal Equations 82.2 Performance Parameters of a Thermoelectric Module 112.3 Maximum Parameters for a Thermoelectric Module 122.4 Normalized Parameters 13

Example 2.1 Exhaust Waste Heat Recovery 15

2.5 Effective Material Properties 172.6 Comparison of Calculations with a Commercial Product 18

Problems 19Computer Assignment 21References 22

3 Thermoelectric Coolers 233.1 Ideal Equations 233.2 Maximum Parameters 263.3 Normalized Parameters 27

Example 3.1 Thermoelectric Air Conditioner 29

3.4 Effective Material Properties 333.4.1 Comparison of Calculations with a Commercial Product 34Problems 36Reference 37

viii Table of Contents

4 Optimal Design 384.1 Introduction 384.2 Optimal Design for Thermoelectric Generators 38

Example 4.1 Exhaust Thermoelectric Generators 46

4.3 Optimal Design of Thermoelectric Coolers 49

Example 4.2 Automotive Thermoelectric Air Conditioner 57

Problems 61References 63

5 Thomson Effect, Exact Solution, and Compatibility Factor 645.1 Thermodynamics of Thomson Effect 645.2 Exact Solutions 68

5.2.1 Equations for the Exact Solutions and the Ideal Equation 685.2.2 Thermoelectric Generator 705.2.3 Thermoelectric Coolers 71

5.3 Compatibility Factor 715.4 Thomson Effects 79

5.4.1 Formulation of Basic Equations 795.4.2 Numeric Solutions of Thomson Effect 835.4.3 Comparison between Thomson Effect and Ideal Equation 85Problems 87Projects 88References 88

6 Thermal and Electrical Contact Resistances for Micro and Macro Devices 896.1 Modeling and Validation 896.2 Micro and Macro Thermoelectric Coolers 926.3 Micro and Macro Thermoelectric Generators 94

Problems 97Computer Assignment 97References 98

7 Modeling of Thermoelectric Generators and Coolers With Heat Sinks 997.1 Modeling of Thermoelectric Generators With Heat Sinks 997.2 Plate Fin Heat Sinks 1087.3 Modeling of Thermoelectric Coolers With Heat Sinks 111

Problems 119References 119

8 Applications 1208.1 Exhaust Waste Heat Recovery 120

8.1.1 Recent Studies 1208.1.2 Modeling of Module Tests 1228.1.3 Modeling of a TEG 1268.1.4 New Design of a TEG 133

8.2 Solar Thermoelectric Generators 1388.2.1 Recent Studies 1388.2.2 Modeling of a STEG 1388.2.3 Optimal Design of a STEG (Dimensional Analysis) 1448.2.4 New Design of a STEG 146

ixTable of Contents

8.3 Automotive Thermoelectric Air Conditioner 1498.3.1 Recent Studies 1498.3.2 Modeling of an Air-to-Air TEAC 1508.3.3 Optimal Design of a TEAC 1578.3.4 New Design of a TEAC 160Problems 162References 163

9 Crystal Structure 1649.1 Atomic Mass 164

9.1.1 Avogadro’s Number 164

Example 9.1 Mass of One Atom 164

9.2 Unit Cells of a Crystal 1659.2.1 Bravais Lattices 166

Example 9.2 Lattice Constant of Gold 169

9.3 Crystal Planes 170

Example 9.3 Indices of a Plane 171

Problems 171

10 Physics of Electrons 17210.1 Quantum Mechanics 172

10.1.1 Electromagnetic Wave 17210.1.2 Atomic Structure 17410.1.3 Bohr’s Model 17410.1.4 Line Spectra 17610.1.5 De Broglie Wave 17710.1.6 Heisenberg Uncertainty Principle 17810.1.7 Schrödinger Equation 17810.1.8 A Particle in a One-Dimensional Box 17910.1.9 Quantum Numbers 18110.1.10 Electron Configurations 183

Example 10.1 Electronic Configuration of a Silicon Atom 184

10.2 Band Theory and Doping 18510.2.1 Covalent Bonding 18510.2.2 Energy Band 18610.2.3 Pseudo-Potential Well 18610.2.4 Doping, Donors, and Acceptors 187Problems 188References 188

11 Density of States, Fermi Energy, and Energy Bands 18911.1 Current and Energy Transport 18911.2 Electron Density of States 190

11.2.1 Dispersion Relation 19011.2.2 Effective Mass 19011.2.3 Density of States 191

x Table of Contents

11.3 Fermi-Dirac Distribution 19311.4 Electron Concentration 19411.5 Fermi Energy in Metals 195

Example 11.1 Fermi Energy in Gold 196

11.6 Fermi Energy in Semiconductors 197

Example 11.2 Fermi Energy in Doped Semiconductors 198

11.7 Energy Bands 19911.7.1 Multiple Bands 20011.7.2 Direct and Indirect Semiconductors 20011.7.3 Periodic Potential (Kronig-Penney Model) 201Problems 205References 205

12 Thermoelectric Transport Properties for Electrons 20612.1 Boltzmann Transport Equation 20612.2 Simple Model of Metals 208

12.2.1 Electric Current Density 20812.2.2 Electrical Conductivity 208

Example 12.1 Electron Relaxation Time of Gold 210

12.2.3 Seebeck Coefficient 210

Example 12.2 Seebeck Coefficient of Gold 212

12.2.4 Electronic Thermal Conductivity 212

Example 12.3 Electronic Thermal Conductivity of Gold 213

12.3 Power-Law Model for Metals and Semiconductors 21312.3.1 Equipartition Principle 21412.3.2 Parabolic Single-Band Model 215

Example 12.4 Seebeck Coefficient of PbTe 217

Example 12.5 Material Parameter 221

12.4 Electron Relaxation Time 22212.4.1 Acoustic Phonon Scattering 22212.4.2 Polar Optical Phonon Scattering 22212.4.3 Ionized Impurity Scattering 223

Example 12.6 Electron Mobility 223

12.5 Multiband Effects 22412.6 Nonparabolicity 225

Problems 228References 229

13 Phonons 23013.1 Crystal Vibration 230

13.1.1 One Atom in a Primitive Cell 23013.1.2 Two Atoms in a Unit Cell 232

13.2 Specific Heat 23413.2.1 Internal Energy 23413.2.2 Debye Model 235

Example 13.1 Atomic Size and Specific Heat 239

xiTable of Contents

13.3 Lattice Thermal Conductivity 24113.3.1 Klemens-Callaway Model 24113.3.2 Umklapp Processes 24413.3.3 Callaway Model 24413.3.4 Phonon Relaxation Times 245

Example 13.2 Lattice Thermal Conductivity 247

Problems 249References 250

14 Low-Dimensional Nanostructures 25114.1 Low-Dimensional Systems 251

14.1.1 Quantum Well (2D) 251

Example 14.1 Energy Levels of a Quantum Well 255

14.1.2 Quantum Wires (1D) 25614.1.3 Quantum Dots (0D) 25814.1.4 Thermoelectric Transport Properties of Quantum Wells 26014.1.5 Thermoelectric Transport Properties of Quantum Wires 26114.1.6 Proof-of-Principle Studies 26314.1.7 Size Effects of Quantum Well on Lattice Thermal Conductivity 264Problems 267References 267

15 Generic Model of Bulk Silicon and Nanowires 26815.1 Electron Density of States for Bulk and Nanowires 268

15.1.1 Density of States 26815.2 Carrier Concentrations for Two-band Model 269

15.2.1 Bulk 26915.2.2 Nanowires 26915.2.3 Bipolar Effect and Fermi Energy 269

15.3 Electron Transport Properties for Bulk and Nanowires 27015.3.1 Electrical Conductivity 27015.3.2 Seebeck Coefficient 27015.3.3 Electronic Thermal Conductivity 270

15.4 Electron Scattering Mechanisms 27115.4.1 Acoustic-Phonon Scattering 27115.4.2 Ionized Impurity Scattering 27215.4.3 Polar Optical Phonon Scattering 272

15.5 Lattice Thermal Conductivity 27315.6 Phonon Relaxation Time 27315.7 Input Data for Bulk Si and Nanowires 27515.8 Bulk Si 275

15.8.1 Fermi Energy 27515.8.2 Electron Mobility 27515.8.3 Thermoelectric Transport Properties 27515.8.4 Dimensionless Figure of Merit 276

15.9 Si Nanowires 27615.9.1 Electron Properties 27615.9.2 Phonon Properties for Si Nanowires 280Problems 282References 284

xii Table of Contents

16 Theoretical Model of Thermoelectric Transport Properties 28616.1 Introduction 28616.2 Theoretical Equatons 287

16.2.1 Carrier Transport Properties 28716.2.2 Scattering Mechanisms for Electron Relaxation Times 29016.2.3 Lattice Thermal Conductivity 29316.2.4 Phonon Relaxation Times 29316.2.5 Phonon Density of States and Specific Heat 29516.2.6 Dimensionless Figure of Merit 295

16.3 Results and Discussion 29516.3.1 Electron or Hole Scattering Mechanisms 29516.3.2 Transport Properties 299

16.4 Summary 315Problems 316References 316

Appendix A Physical Properties 323

Appendix B Optimal Dimensionless Parameters for TEGs with ZT 2= 1 353

Appendix C ANSYS TEG Tutorial 365

Appendix D Periodic Table 376

Appendix E Thermoelectric Properties 391

Appendix F Fermi Integral 399

Appendix G Hall Factor 402

Appendix H Conversion Factors 405

Index 409

Preface

This book is written as a senior undergraduate or first-year graduate textbook. Thermoelectrics is a study of theenergy conversion between thermal energy and electrical energy in solid state matters. Thermoelectrics is anemerging field with comprehensive applications such as exhaust waste heat recovery, solar energy conversion,automotive air conditioner, deep-space exploration, electronic control and cooling, and medical instrumentation.Thermoelectrics involves multiple interdisciplinary fields: physics, chemistry, electronics, material sciences,nanotechnology, and mechanical engineering. Much of the theories and materials are still under development,mostly on the materials but minimally on the design. The author has taught the thermoelectrics courses in the pastyears with a mind that a textbook is necessary to put a spur on the development. However, the author experiencedconsiderable difficulties, partly because of the need to make a selection from the existing material and partly becausethe customary exposition of many topics to be included does not possess the necessary physical clarity. It is realizedthat the author’s own treatment still has many defects, which are desirable to correct in future editions. The author hasan open mind and appreciation for any comments and defects that may be found in the book. Typically, design andmaterials are separate fields, but in thermoelectrics, the two fields are interrelated particularly when the size is smalland the variation of temperature is large. Hence, this book includes the design and materials for future vigorousengagement.

This book consists of two parts: design (Chapters 1 through 8) and materials (Chapters 9 through 16). The designcovers the theoretical formulation, optimal design, experimental verification, modeling, and applications. Thematerials cover the physics of thermoelectrics for electrons and phonons, experimental verification, modeling,nanostructures, and thermoelectric materials. Each part can be suggestably used for a semester period (usually anintroductory session for the subtle phenomena of thermoelectrics is given in the beginning of class) or two parts in asemester period by skimming some topics when students or readers are familiar with the topics. The author putsignificant effort into managing the contents in Part I with a fundamental heat transfer course as prerequisite and thecontents in Part II with an introductory material science course. The author also attempted to provide detailedderivation of formulas so students or readers can have a conviction on studying thermoelectrics, as well as to providedetailed calculations, so that they can even build their own mathematical programs. Hence, many exercise problemsat the end of chapter ask students or readers to provide Mathcad programs for the problem solutions.

I would like to acknowledge the suggestions and help provided by undergraduate and graduate students throughclasses and research projects. Special thanks are given to Dr. Alaa Attar, who is now professor at King AbdulAzizUniversity, Saudi Arabia, for his help in measurements and computations. I am also indebted to Professor EmeritusHerman Merte Jr. for his lifetime inspiration on the preparation of the book. I am very grateful to Professor EmeritusStanley L. Rajnak, who read the manuscript and made many useful comments.

HOSUNG LEE

KALAMAZOO, MICHIGAN

1Introduction

1.1 Introduction

Thermoelectrics is literally associated with thermal and electrical phenomena. Thermoelectric processes can directly convert thermal energy into electrical energy or vice versa. A thermocouple uses the electrical potential (electro­motive force) generated between two dissimilar wires to measure temperature. Basically, there are two devices: thermoelectric generators and thermoelectric coolers. These devices have no moving parts and require no maintenance. Thermoelectric generators have great potential for waste heat recovery from power plants and automotive vehicles. Such devices can also provide reliable power in remote areas such as in deep space and mountaintop telecommunication sites. Thermoelectric coolers provide refrigeration and temperature control in electronic packages and medical instruments. The science of thermoelectrics has become increasingly important with numerous applications. Since thermoelectricity was discovered in the early nineteenth century, there has not been much improvement in efficiency or materials until the recent development of nanotechnology, which has led to a remarkable improvement in performance. It is, thus, very important to understand the fundamentals of thermo­electrics for the development and the thermal design. We start with a brief history of thermoelectricity.

In 1821, Thomas J. Seebeck discovered that an electromotive force or a potential difference could be produced by a circuit made from two dissimilar wires when one of the junctions was heated. This is called the Seebeck effect.

Thirteen years later, in 1834, Jean Peltier discovered the reverse process—that the passage of an electric current through a thermocouple produces heating or cooling depending on its direction. This is called the Peltier effect. Although these two effects were demonstrated to exist, it was very difficult to measure each effect as a property of the material because the Seebeck effect is always associated with two dissimilar wires and the Peltier effect is always followed by the additional Joule heating that is heat generation due to the electrical resistance to the passage of a current. Joule heating was discovered in 1841 by James P. Joule.

In 1854, William Thomson (later Lord Kelvin) discovered that if a temperature difference exists between any two points of a current-carrying conductor, heat is either liberated or absorbed depending on the direction of current and material, which is in addition to the Peltier heating. This is called the Thomson effect. He also studied the relationships between these three effects thermodynamically, showing that the electrical Seebeck effect results from a combination of the thermal Peltier and Thomson effects. Although the Thomson effect itself is small compared with the other two, it leads to a very important and useful relationship, which is called the Kelvin relationship.

The mechanisms of thermoelectricity were not understood well until the discovery of electrons at the end of the nineteenth century. Now it is known that solar energy, an electric field, or thermal energy can liberate some electrons from their atomic binding, even at room temperature, moving them (from the valence band to the conduction band of a conductor) where the electrons are free to move. This is the reason why we have electrostatics everywhere. However, when a temperature difference across a conductor is applied as shown in Figure 1.1, the hot region of the conductor produces more free electrons, and diffusion of these electrons (charge carriers including holes) naturally

Thermoelectrics: Design and Materials, First Edition. HoSung Lee. © 2017 John Wiley & Sons, Ltd. Published 2017 by John Wiley & Sons, Ltd.

2 Thermoelectrics

Figure 1.1 Electron concentrations in a thermoelectric material

occurs from the hot region to the cold region. On the other hand, the electron distribution provokes an electric field, which also causes the electrons to move from the hot region to the cold region via the Coulomb forces. Hence, an electromotive force (emf) is generated in a way that an electric current flows against the temperature gradient. As mentioned, the reverse is also true. If a current is applied to the conductor, electrons move and interestingly carry thermal energy. Therefore, a heat flow occurs in the opposite direction of the current, which is also shown in Figure 1.1.

In many applications, a number of thermocouples, each of which consists of p-type and n-type semiconductor elements, are connected electrically in series and thermally in parallel by sandwiching them between two high–thermal conductivity but low–electrical conductivity ceramic plates to form a module, which is shown in Figure 1.2.

Consider two wires made from different metals joined at both ends, as shown in Figure 1.3, forming a close circuit. Ordinarily, nothing will happen. However, when one of the junctions is heated, something interesting happens. Current flows continuously in the circuit. this is the Seebeck effect. The circuit that incorporates both thermal and

Figure 1.2 Cutaway of a typical thermoelectric module

3Introduction

Figure 1.3 Thermocouple

electrical effects is called a thermoelectric circuit. A thermocouple uses the Seebeck effect to measure temperature, and the effect forms the basis of a thermoelectric generator.

In 1834, Jean Peltier discovered the reverse of the Seebeck effect by demonstrating that cooling can take place by applying a current across the junction. The heat pumping is possible without a refrigerator or compressor. The thermal energy can convert to electrical energy without turbine or engines.

There are some advantages of thermoelectric devices despite their low thermal efficiency. There are no moving parts in the device; therefore, there is less potential for failure in operation. Controllability of heating and cooling is very attractive in many applications such as lasers, optical detectors, medical instruments, and microelectronics.

1.2 Thermoelectric Effect

The thermoelectric effect consists of three effects: the Seebeck effect, the Peltier effect, and the Thomson effect.

1.2.1 Seebeck Effect

The Seebeck effect is the conversion of a temperature difference into an electric current. As shown in Figure 1.3, wire A is joined at both ends to wire B and a voltmeter is inserted in wire B. Suppose that a temperature difference is imposed between two junctions; then, it will generally be found that a potential difference or voltage Vwill appear on the voltmeter. The potential difference is proportional to the temperature difference. The potential difference V is

V αABΔT (1.1)

where ΔT=Th Tc and αAB αA αB; αAB is called the Seebeck coefficient (also called the thermopower), which is usually measured in μV/K. The sign of α is positive if the emf tends to drive an electric current through wire A from the hot junction to the cold junction, as shown in Figure 1.3. In practice, one rarely measures the absolute Seebeck coefficient because the voltage meter always reads the relative Seebeck coefficient between wires A and B. The absolute Seebeck coefficient can be calculated from the Thomson coefficient.

1.2.2 Peltier Effect

When current flows across a junction between two different wires, it is found that heat must be continuously added or subtracted at the junction in order to keep its temperature constant, which is illustrated in Figure 1.4. The heat is

Figure 1.4 Schematic for the Peltier effect and the Thomson effect

4 Thermoelectrics

proportional to the current flow and changes sign when the current is reversed. Thus, the Peltier heat absorbed or liberated is

_QPeltier πABI (1.2)

where πAB is the Peltier coefficient and the sign of πAB is positive if the junction at which the current enters wire A is heated and the junction at which the current leaves wire A is cooled. The Peltier heating or cooling is reversiblebetween heat and electricity. This means that heating (or cooling) will produce electricity and electricity will produce heating (or cooling) without a loss of energy.

1.2.3 Thomson Effect

When current flows as shown in Figure 1.4, heat is absorbed in wire A due to the negative temperature gradient and liberated in wire B due to the positive temperature gradient, which is experimental observation [1], depending on the material. The Thomson heat is proportional to both the electric current and the temperature gradient, which is schematically shown in Figure 1.4. Thus, the Thomson heat absorbed or liberated across a wire is

_ T (1.3)QThomson τABI

where τ is the Thomson coefficient. The Thomson coefficient is unique among the three thermoelectric coefficients because it is the only thermoelectric coefficient directly measurable for individual materials. There is other form of heat, called Joule heating (I2R), which is irreversible and is always generated as current flows in a wire. The Thomson heat is reversible between heat and electricity.

1.2.4 Thomson (or Kelvin) Relationships

The interrelationships between the three thermoelectric effects are important in order to understand the basic phenomena. In 1854, Thomson [2] studied the relationships thermodynamically and provided two relationships as shown in Equations (1.4) and (1.5) by applying the first and second laws of thermodynamics with the assumption that the reversible and irreversible processes in thermoelectricity are separable. The necessity for the assumption remained an objection to the theory until the advent of the new thermodynamics. The Thomson effect is relatively small compared with the Peltier effect, but it plays an important role in deducing the Thomson relationships. These relationships were later completely confirmed by experiments (See Chapter 5 for details).

πAB αABT (1.4)

dαABτAB T (1.5)

dT

Equation (1.4) leads to the very useful Peltier cooling in Equation (1.2) as

_QPeltier αABTI (1.6)

where T is the temperature at a junction between two different materials and the dot above the heat Q indicates the amount of heat transported per unit time.

1.3 The Figure of Merit

The performance of thermoelectric devices is measured by the figure of merit (Z), with units 1/K:

α2 α2σZ (1.7)

ρk k

where

α = Seebeck coefficient, μV/K ρ = electrical resistivity, Ωcm

5Introduction

σ = 1/ρ= electrical conductivity (Ωcm) 1

k = thermal conductivity, W/mK

The dimensionless figure of merit is defined by ZT, where T is the absolute temperature. There is no fundamental limit on ZT, but for decades it was limited to values around ZT 1 in existing devices. The larger the value of ZT, the greater is the energy conversion efficiency of the material. The quantity of α2σ is defined as the power factor. Therefore, both the Seebeck coefficient α and electrical conductivity σ must be large, while the thermal conductivity k must be minimized. This well-known interdependence among the physical properties makes it challenging to develop strategies for improving a material’s ZT.

1.3.1 New-Generation Thermoelectrics

Although Seebeck observed thermoelectric phenomena in 1821 and Altenkirch defined Equation (1.7) in 1911, it took several decades to develop the first functioning devices in the 1950s and 1960s. They are now called the first-generation thermoelectrics with an average of Z∼ 1.0. Devices made of them can operate at ∼5% conversion efficiency. After several more decades of stagnancy, new theoretical ideas relating to size effects on thermoelectric properties in the 1990s stimulated new experimental research that eventually led to significant advances in the following decade. Although the theoretical ideas were originally about prediction on raising the power factor, the experimental breakthroughs were achieved by significantly decreasing the lattice thermal conductivity. Among a wide variety of research approaches, one has emerged, which has led to a near doubling of ZT at high temperatures and defines the second generation of bulk thermoelectric materials with ZT in the range of 1.3–1.7. This approach uses nanoscale precipitates and composition inhomogeneities to dramatically suppress the lattice thermal conduc­tivity. These second-generation materials are expected to eventually produce power-generation devices with conversion efficiencies of 11–15% [3].

Third-generation bulk thermoelectrics have been under development recently, which integrate many cutting-edge ZT-enhancing approaches simultaneously, namely, enhancement of Seebeck coefficients through valence band convergence, retention of the carrier mobility through band energy offset minimization between matrix and precipitates, and reduction of the lattice thermal conductivity through all length-scale lattice disorder and nanoscale endotaxial precipitates to mesoscale grain boundaries and interfaces. This third generation of bulk thermoelectrics exhibits high ZT, ranging from 1.8 to 2.2, depending on the temperature difference, and a consequent predicted device conversion efficiency increase to ∼15–20% [3].

Table 1.1 shows the thermoelectric properties of bulk nanocomposite semiconductors. Figure 1.5 shows the dimensionless figures of merit for the materials in Table 1.1.

Figure 1.5 Dimensionless figures of merit for various nanocomposite thermoelectric materials

6 Thermoelectrics

Table 1.1 Thermoelectric Properties of Single Crystal and Bulk Nanocomposite Semiconductors

Type Temperature (K) α (μV/Κ) σ (Ωcm) 1 ke (W/mK) k (W/mK) ZT Authors

Bi2Te3 p-type single crystals 300 230 500 0.6 2.0 0.5 Jeon et al. (1991) [4] BiSbTe p-type, nanocomposites 400 220 700 0.6 1.0 1.4 Poudel et al. (2008) [5] Bi2Te2.7Se0.3 n-type nanocomposites 400 210 700 0.6 1.2 1.0 Yan et al. (2010) [6] PbTe-SrTe p-type nanocomposites 900 270 300 0.4 1.1 2.2 Biswas et al. (2012) [7] Si70Ge30 n-type single crystals 1000 350 320 0.5 4.0 0.8 Dismukes et al. (1964) [8] Si80Ge20 n-type nanocomposites 1200 250 400 0.5 2.8 1.3 Wang et al. (2008) [9] CoSb3 n-type single crystals 800 240 800 0.5 4.0 0.6 Caillat et al. (1996) [10] Yb-CoSb3 n-type, Yb-filled skutterudites 800 200 1600 2.0 3.2 1.3 Tang et al. (2015) [11] Yb14MnSb11 p-type, Zintl compound 1200 190 200 0.7 1.1 Brown et al. (2006) [12] La3Te4 n-type single crystals 1200 280 80 0.3 0.7 1.1 May et al. (2010) [13]

7Introduction

Problems

1.1 Briefly describe the thermoelectric effect.

1.2 Describe the dimensionless figure of merit and why it is important in thermoelectric design.

1.3 Describe the Thomson relations.

References

1. Nettleton, H.R. The Thomson effect. Proceedings of the Physical Society of London. 1922. 2. Thomson, W., Account of researches in thermo-electricity. Proceedings of the Royal Society of London, 1854. 7: p. 49–58. 3. Zhao, L.-D., V.P. Dravid, and M.G. Kanatzidis, The panoscopic approach to high performance thermoelectrics. Energy &

Environmental Science, 2014. 7(1): p. 251. 4. Jeon, H.-W., et al., Electrical and thermoelectrical properties of undoped Bi2Te3-SbeTe3 and Bi2Te3-Sb2Te3-Sb2Se3 single

crystals. Journal of Physics and Chemistry of Solids, 1991. 52(4): p. 579–585. 5. Poudel, B., et al., High-thermoelectric performance of nanostructured bismuth antimony telluride bulk alloys. Science, 2008.

320(5876): p. 634–8. 6. Yan, X., et al., Experimental studies on anisotropic thermoelectric properties and structures of n-type Bi2Te2.7Se0.3. Nano

Letters, 2010. 10(9): p. 3373–8. 7. Biswas, K., et al., High-performance bulk thermoelectrics with all-scale hierarchical architectures. Nature, 2012. 489(7416):

p. 414–8. 8. Dismukes, J.P., et al., Thermal and electrical properties of heavily doped Ge-Si alloys up to 1300°K. Journal of Applied

Physics, 1964. 35(10): p. 2899. 9. Wang, X.W., et al., Enhanced thermoelectric figure of merit in nanostructured n-type silicon germanium bulk alloy. Applied

Physics Letters, 2008. 93(19): p. 193121. 10. Caillat, T., A. Borshchevsky, and J.P. Fleurial, Properties of single crystalline semiconducting CoSb3. Journal of Applied

Physics, 1996. 80(8): p. 4442. 11. Tang, X., et al., Synthesis and thermoelectric properties of p-type- and n-type-filled skutterudite R[sub y]M[sub x]Co[sub

4 x]Sb[sub 12](R:Ce,Ba,Y;M:Fe, Ni). Journal of Applied Physics, 2005. 97(9): p. 093712. 12. Brown, S.R., et al., Yb14MnSb11: New high efficiency thermoelectric material for power generation. Chemistry of Materials,

2006. 18: p. 1873–1877. 13. May, A.F., J.-P. Fleurial, and G.J. Snyder, Optimizing thermoelectric efficiency in La3 xTe4via Yb substitution. Chemistry of

Materials, 2010. 22(9): p. 2995–2999.

2Thermoelectric Generators

2.1 Ideal Equations

In 1821, Thomas J. Seebeck discovered that an electromotive force or potential difference could be produced by a circuit made from two dissimilar wires when one junction was heated. This is called the Seebeck effect. In 1834, Jean Peltier discovered the reverse process—that the passage of an electric current through a thermocouple produces heating or cooling depending on its direction [1]. This is called the Peltier effect (or Peltier cooling). In 1854, William Thomson discovered that if a temperature difference exists between any two points of a current-carrying conductor, heat is either absorbed or liberated depending on the direction of current and material [2]. This is called the Thomson effect (or Thomson heat). These three effects are called the thermoelectric effects.

Let us consider a non–uniformly heated thermoelectric material. For an isotropic substance, the continuity equation for a constant current gives

)

?

)

j 0 (2.1)

~

~The electric field E is affected by the current density~j and the temperature gradient ~T . The relationships are known as Ohm’s law and the Seebeck effect [3]. The electric field is then expressed as

~E ~jρ α~T (2.2)

where ρ is the electrical resistivity. The heat flux ~q is also affected by both the field ~E and the temperature gradient T . However, their coefficients were not readily attainable at that time. Thomson in 1854 arrived at the relationship

assuming that thermoelectric phenomena and thermal conduction are independent [2]. Later, Onsager [4] supported that relationship by presenting the reciprocal principle, which was experimentally proved. The Thomson relationship and the Onsager’s principle yielded a formula for the heat flow density vector (heat flux),

~q αT~j k~T (2.3)

This is the most important equation in thermoelectrics (will be discussed later in detail). The general heat diffusion equation is given by

~ ?~q _q ρcp@T

@t(2.4)

Thermoelectrics: Design and Materials, First Edition. HoSung Lee. © 2017 John Wiley & Sons, Ltd. Published 2017 by John Wiley & Sons, Ltd.

9Thermoelectric Generators

For steady state, we have

~ ?~q q_ 0 (2.5)

where _q is expressed by [3]

_q ~E ?~j j2ρ ~j ? α~T (2.6)

Substituting Equations (2.3) and (2.6) into Equation (2.5) yields

~ ? k~T j2ρ j ?~Tdα

dT~ T 0 (2.7)

The Thomson coefficient τ, originally obtained from the Thomson relations, is defined by

τ Tdα

dT(2.8)

In Equation (2.7), the first term is the thermal conduction, the second term is the Joule heating, and the third term is the Thomson heat. Note that if the Seebeck coefficient α is independent of temperature, the Thomson coefficient τbecomes 0 and then the Thomson heat is absent. The two equations, (2.3) and (2.7), govern thermoelectric phenomena.

Consider a steady-state one-dimensional thermoelectric generator module as shown in Figure 2.1. The module consists of many p-type and n-type thermocouples, where one thermocouple (unicouple) with a circuit is shown in

Figure 2.1 Cutaway of a thermoelectric generator module

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Figure 2.2 The p- and n-type unit thermocouple for a thermoelectric generator

Figure 2.2. We assume that the electrical and thermal contact resistances are negligible, the Seebeck coefficient is independent of temperature, and the radiation and convection at the surfaces of the elements are negligible. Then, Equation (2.7) reduces to

I2ρd dTkA 0 (2.9)

dx dx A

The solution for the temperature gradient with two boundary conditions (Tx 0 Th and Tx L Tc) in Figure 2.2 is

I2ρL Th TcdT(2.10)

dx x 0 2A2k L

Equation (2.3) is expressed in terms of p-type and n-type thermoelements.

dT dT_Qh n TcI kA kA (2.11)αp αn dx dxx 0 x 0p n

_where Qh is the rate of heat absorbed at the hot junction in Figure 2.2 and n is the number of thermocouples. Substituting Equation (2.10) into Equation (2.11) gives

1 ρpLp ρ Ln kpAp knAnn_ I2Qh n (2.12)ThIαp αn Th Tc2 Ap An Lp Ln

_ __

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Finally, the heat absorbed at the hot junction with temperature Th is expressed as

_Qh n αThI1 I2R K Th Tc (2.13)

2

where

α αp αn (2.14)

ρ Lp LnρnpR (2.15)Ap An

kpAp knAnK (2.16)Lp Ln

R is the electrical resistance and K is the thermal conductance. If we assume that the p-type and n-type thermocouples are similar, we have that R= ρL/A and K= kA/L, where ρ= ρp+ ρn and k= kp+ kn. Equation (2.13) is called the ideal equation and has been widely used in science and industry. The rate of heat liberated at the cold junction is given by

1 _ I2RQc n αTcI K Th Tc (2.17)2

From the first law of thermodynamics for the thermoelectric module, the power output is Wn Qh Qc. The total power output is then expressed in terms of the internal properties as

_ I2RWn n αI Th Tc (2.18)

However, the total power output in Figure 2.2 can be defined by an external load resistance as

_Wn nI2RL (2.19)

_Equating Equations (2.18) and (2.19) with Wn IVn gives the total voltage as

Vn nIRL n α Th Tc IR (2.20)

2.2 Performance Parameters of a Thermoelectric Module

From Equation (2.20), the electrical current for the module is obtained as

α Th TcI (2.21)RRL

Note that the current I is independent of the number of thermocouples. Inserting this into Equation (2.20) gives the voltage across the module by

nα Th Tc RL (2.22)Vn RL R1 R

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Inserting Equation (2.21) in Equation (2.19) gives the power output as

RL2nα2 Th Tc_ R

2 (2.23)Wn R RL1 R

The conversion (or thermal) efficiency is defined as the ratio of the power output over the heat absorbed at the hot junction:

_Wn (2.24)ηth _Qh

Inserting Equations (2.13) and (2.23) into Equation (2.24) gives an expression for the conversion efficiency:

Tc RL1 RTh (2.25)ηth 21 1RL Tc RL Tc1 1 1 1

R 2 R2ZTTh Th

Tcwhere the average temperature is defined as T Th2 . It is noted that the Carnot cycle efficiency is

η 1 Tc=Th .c

2.3 Maximum Parameters for a Thermoelectric Module

Because the maximum current inherently occurs in a short circuit where RL 0 in Equation (2.21), the maximum current for the module is

α Th Tc (2.26)Imax R

The maximum voltage inherently occurs in an open circuit where I= 0 in Equation (2.20). The maximum voltage is

(2.27)Vmax nα Th Tc

_The maximum power output is attained by differentiating the power output W in Equation (2.23) with respect to the ratio of the load resistance to the internal resistance and setting it to 0. The result yields a relationship of RL=R 1, which leads to the maximum power output as

2nα2 Th Tc_ (2.28)Wmax 4R

The maximum conversion efficiency can be obtained by differentiating the conversion efficiency in Equation (2.25) with respect to the ratio of the load resistance to the internal resistance and setting it to zero. The result yields a relationship of RL=R 1 ZT . Then, the maximum conversion efficiency η ismax

Tc 1 ZT 1 η 1 (2.29)max TcTh 1 ZT

Th

_There are a total of four essential maximum parameters: Imax, Vmax, Wmax, and ηmax. However, there is also the maximum power efficiency. The maximum power efficiency is obtained by letting RL=R 1 in Equation (2.25). The

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maximum power efficiency ηmp is

Tc1 Th (2.30)ηmp 1 2Tc Tc2 1 1

2 ZTTh Th

Note there are two thermal efficiencies: the maximum power efficiency ηmp and the maximum conversion efficiency ηmax.

2.4 Normalized Parameters

If we divide the actual values by the maximum values, we can normalize the characteristics of a thermoelectric generator. The normalized power output can be obtained by dividing Equation (2.23) by Equation (2.28), which leads to

RL4 _W R2 (2.31)

_ RLWmax 1

R

Equations (2.21) and (2.26) give the normalized currents as

I 1 (2.32)RLImax 1

R

Equations (2.22) and (2.27) give the normalized voltage as

RLVn R (2.33)RLVmax 1

R

Equations (2.25) and (2.29) give the normalized thermal efficiency as

RL Tc1 ZTηth R Th

2ηmax RL 1 Tc 1 RL Tc1 1 1 1 1 ZTR 2 Th R2ZT Th

(2.34)

Note that the normalized values in Equations (2.31) through (2.33) are a function only of RL=R, while Equation (2.34) is a function of three parameters: Tc=Th, RL=R, and ZT .

It is noted, as shown in Figure 2.3, that the maximum power output and the maximum conversion efficiency appear close each other with respect to RL=R. The maximum power Wmax occurs at RL=R 1, while ηmax occurs _

approximately at RL=R 1:5. The various parameters are presented against the normalized current, which is shown in Figure 2.4. This plot is often used in the specification of commercial modules. Note that the current indicates half the maximum current for the maximum power output. The maximum conversion efficiency ηmax is

1

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Figure 2.3 Normalized chart I for thermoelectric generators, where Tc /Th= 0.7 and ZT 1 are used

presented in Figure 2.5 as a function of both the dimensionless figure of merit (ZT) and Tc / Th. Considering a conventional combustion process (where the thermal efficiency is about 30%) where the high and low junction temperatures would be at 1500 K and 500 K leads to Tc / Th= 0.3. Therefore, to compete with the conventional way of the thermal efficiency (30%), the thermoelectric material should be at least ZT 3, which has been the goal in this field. Much development is needed when considering the current technology of thermoelectric material of ZT 1. However, it is thought that there is a strong potential that nanotechnology would contribute to ZT 3 in near future.

Figure 2.4 Normalized chart II for thermoelectric generators, where Tc /Th= 0.7 and ZT 1 are used