THERMOELECTRICS DESIGN AND MATERIALS

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5 THERMOELECTRICS DESIGN AND MATERIALS HoSung Lee Western Michigan University, USA

6 This edition first published John Wiley & Sons Ltd Registered office John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom For details of our global editorial offices, for customer services and for information about how to apply for permission to reuse the copyright material in this book please see our website at The right of the author to be identified as the author of this work has been asserted in accordance with the Copyright, Designs and Patents Act All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by the UK Copyright, Designs and Patents Act 1988, without the prior permission of the publisher. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. Designations used by companies to distinguish their products are often claimed as trademarks. All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners. The publisher is not associated with any product or vendor mentioned in this book. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. It is sold on the understanding that the publisher is not engaged in rendering professional services and neither the publisher nor the author shall be liable for damages arising herefrom. If professional advice or other expert assistance is required, the services of a competent professional should be sought. Library of Congress Cataloging-in-Publication Data Names: Lee, HoSung, author. Title: Thermoelectrics : design and materials / HoSung Lee. Description: Chichester, UK ; Hoboken, NJ : John Wiley & Sons, Includes bibliographical references and index. Identifiers: LCCN ISBN (cloth) ISBN (epub) ISBN (epdf) Subjects: LCSH: Thermoelectric apparatus and appliances Design and construction. Thermoelectric materials. Classification: LCC TK2950.L DDC /243 dc23 LC record available at A catalogue record for this book is available from the British Library. Cover design by Yujin Lee ISBN: Set in 9.5/11.5 TimesLTStd-Roman by Thomson Digital, Noida, India

7 For Young-Ae and Yujin

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9 Table of Contents Preface xiii 1 Introduction Introduction Thermoelectric Effect Seebeck Effect Peltier Effect Thomson Effect Thomson (or Kelvin) Relationships The Figure of Merit New-Generation Thermoelectrics 5 Problems 7 References 7 2 Thermoelectric Generators Ideal Equations Performance Parameters of a Thermoelectric Module Maximum Parameters for a Thermoelectric Module Normalized Parameters 13 Example 2.1 Exhaust Waste Heat Recovery Effective Material Properties Comparison of Calculations with a Commercial Product 18 Problems 19 Computer Assignment 21 References 22 3 Thermoelectric Coolers Ideal Equations Maximum Parameters Normalized Parameters 27 Example 3.1 Thermoelectric Air Conditioner Effective Material Properties Comparison of Calculations with a Commercial Product 34 Problems 36 Reference 37

10 viii Table of Contents 4 Optimal Design Introduction Optimal Design for Thermoelectric Generators 38 Example 4.1 Exhaust Thermoelectric Generators Optimal Design of Thermoelectric Coolers 49 Example 4.2 Automotive Thermoelectric Air Conditioner 57 Problems 61 References 63 5 Thomson Effect, Exact Solution, and Compatibility Factor Thermodynamics of Thomson Effect Exact Solutions Equations for the Exact Solutions and the Ideal Equation Thermoelectric Generator Thermoelectric Coolers Compatibility Factor Thomson Effects Formulation of Basic Equations Numeric Solutions of Thomson Effect Comparison between Thomson Effect and Ideal Equation 85 Problems 87 Projects 88 References 88 6 Thermal and Electrical Contact Resistances for Micro and Macro Devices Modeling and Validation Micro and Macro Thermoelectric Coolers Micro and Macro Thermoelectric Generators 94 Problems 97 Computer Assignment 97 References 98 7 Modeling of Thermoelectric Generators and Coolers With Heat Sinks Modeling of Thermoelectric Generators With Heat Sinks Plate Fin Heat Sinks Modeling of Thermoelectric Coolers With Heat Sinks 111 Problems 119 References Applications Exhaust Waste Heat Recovery Recent Studies Modeling of Module Tests Modeling of a TEG New Design of a TEG Solar Thermoelectric Generators Recent Studies Modeling of a STEG Optimal Design of a STEG (Dimensional Analysis) New Design of a STEG 146

11 Table of Contents ix 8.3 Automotive Thermoelectric Air Conditioner Recent Studies Modeling of an Air-to-Air TEAC Optimal Design of a TEAC New Design of a TEAC 160 Problems 162 References Crystal Structure Atomic Mass Avogadro s Number 164 Example 9.1 Mass of One Atom Unit Cells of a Crystal Bravais Lattices 166 Example 9.2 Lattice Constant of Gold Crystal Planes 170 Example 9.3 Indices of a Plane 171 Problems Physics of Electrons Quantum Mechanics Electromagnetic Wave Atomic Structure Bohr s Model Line Spectra De Broglie Wave Heisenberg Uncertainty Principle Schrödinger Equation A Particle in a One-Dimensional Box Quantum Numbers Electron Configurations 183 Example 10.1 Electronic Configuration of a Silicon Atom Band Theory and Doping Covalent Bonding Energy Band Pseudo-Potential Well Doping, Donors, and Acceptors 187 Problems 188 References Density of States, Fermi Energy, and Energy Bands Current and Energy Transport Electron Density of States Dispersion Relation Effective Mass Density of States 191

12 x Table of Contents 11.3 Fermi-Dirac Distribution Electron Concentration Fermi Energy in Metals 195 Example 11.1 Fermi Energy in Gold Fermi Energy in Semiconductors 197 Example 11.2 Fermi Energy in Doped Semiconductors Energy Bands Multiple Bands Direct and Indirect Semiconductors Periodic Potential (Kronig-Penney Model) 201 Problems 205 References Thermoelectric Transport Properties for Electrons Boltzmann Transport Equation Simple Model of Metals Electric Current Density Electrical Conductivity 208 Example 12.1 Electron Relaxation Time of Gold Seebeck Coefficient 210 Example 12.2 Seebeck Coefficient of Gold Electronic Thermal Conductivity 212 Example 12.3 Electronic Thermal Conductivity of Gold Power-Law Model for Metals and Semiconductors Equipartition Principle Parabolic Single-Band Model 215 Example 12.4 Seebeck Coefficient of PbTe 217 Example 12.5 Material Parameter Electron Relaxation Time Acoustic Phonon Scattering Polar Optical Phonon Scattering Ionized Impurity Scattering 223 Example 12.6 Electron Mobility Multiband Effects Nonparabolicity 225 Problems 228 References Phonons Crystal Vibration One Atom in a Primitive Cell Two Atoms in a Unit Cell Specific Heat Internal Energy Debye Model 235 Example 13.1 Atomic Size and Specific Heat 239

13 Table of Contents xi 13.3 Lattice Thermal Conductivity Klemens-Callaway Model Umklapp Processes Callaway Model Phonon Relaxation Times 245 Example 13.2 Lattice Thermal Conductivity 247 Problems 249 References Low-Dimensional Nanostructures Low-Dimensional Systems Quantum Well (2D) 251 Example 14.1 Energy Levels of a Quantum Well Quantum Wires (1D) Quantum Dots (0D) Thermoelectric Transport Properties of Quantum Wells Thermoelectric Transport Properties of Quantum Wires Proof-of-Principle Studies Size Effects of Quantum Well on Lattice Thermal Conductivity 264 Problems 267 References Generic Model of Bulk Silicon and Nanowires Electron Density of States for Bulk and Nanowires Density of States Carrier Concentrations for Two-band Model Bulk Nanowires Bipolar Effect and Fermi Energy Electron Transport Properties for Bulk and Nanowires Electrical Conductivity Seebeck Coefficient Electronic Thermal Conductivity Electron Scattering Mechanisms Acoustic-Phonon Scattering Ionized Impurity Scattering Polar Optical Phonon Scattering Lattice Thermal Conductivity Phonon Relaxation Time Input Data for Bulk Si and Nanowires Bulk Si Fermi Energy Electron Mobility Thermoelectric Transport Properties Dimensionless Figure of Merit Si Nanowires Electron Properties Phonon Properties for Si Nanowires 280 Problems 282 References 284

14 xii Table of Contents 16 Theoretical Model of Thermoelectric Transport Properties Introduction Theoretical Equatons Carrier Transport Properties Scattering Mechanisms for Electron Relaxation Times Lattice Thermal Conductivity Phonon Relaxation Times Phonon Density of States and Specific Heat Dimensionless Figure of Merit Results and Discussion Electron or Hole Scattering Mechanisms Transport Properties Summary 315 Problems 316 References 316 Appendix A Physical Properties 323 Appendix B Optimal Dimensionless Parameters for TEGs with ZT 2 = 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

15 Preface This book is written as a senior undergraduate or first-year graduate textbook. Thermoelectrics is a study of the energy conversion between thermal energy and electrical energy in solid state matters. Thermoelectrics is an emerging 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 past years with a mind that a textbook is necessary to put a spur on the development. However, the author experienced considerable difficulties, partly because of the need to make a selection from the existing material and partly because the customary exposition of many topics to be included does not possess the necessary physical clarity. It is realized that the author s own treatment still has many defects, which are desirable to correct in future editions. The author has an open mind and appreciation for any comments and defects that may be found in the book. Typically, design and materials are separate fields, but in thermoelectrics, the two fields are interrelated particularly when the size is small and the variation of temperature is large. Hence, this book includes the design and materials for future vigorous engagement. This book consists of two parts: design (Chapters 1 through 8) and materials (Chapters 9 through 16). The design covers the theoretical formulation, optimal design, experimental verification, modeling, and applications. The materials 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 an introductory session for the subtle phenomena of thermoelectrics is given in the beginning of class) or two parts in a semester period by skimming some topics when students or readers are familiar with the topics. The author put significant effort into managing the contents in Part I with a fundamental heat transfer course as prerequisite and the contents in Part II with an introductory material science course. The author also attempted to provide detailed derivation of formulas so students or readers can have a conviction on studying thermoelectrics, as well as to provide detailed calculations, so that they can even build their own mathematical programs. Hence, many exercise problems at 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 through classes and research projects. Special thanks are given to Dr. Alaa Attar, who is now professor at King AbdulAziz University, Saudi Arabia, for his help in measurements and computations. I am also indebted to Professor Emeritus Herman Merte Jr. for his lifetime inspiration on the preparation of the book. I am very grateful to Professor Emeritus Stanley L. Rajnak, who read the manuscript and made many useful comments. HOSUNG LEE KALAMAZOO, MICHIGAN

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17 1 Introduction 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 John Wiley & Sons, Ltd. Published 2017 by John Wiley & Sons, Ltd.

18 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

19 Introduction 3 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 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 V will appear on the voltmeter. The potential difference is proportional to the temperature difference. The potential difference V is V α AB ΔT (1.1) where ΔT = T h T c and α AB α A α B ; α AB 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 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

20 4 Thermoelectrics proportional to the current flow and changes sign when the current is reversed. Thus, the Peltier heat absorbed or liberated is _Q Peltier π AB I (1.2) where π AB is the Peltier coefficient and the sign of π AB 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 reversible between heat and electricity. This means that heating (or cooling) will produce electricity and electricity will produce heating (or cooling) without a loss of energy 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 Q_ Thomson τ AB I T (1.3) 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 (I 2 R), which is irreversible and is always generated as current flows in a wire. The Thomson heat is reversible between heat and electricity 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). π AB α AB T (1.4) dα AB τ AB T (1.5) dt Equation (1.4) leads to the very useful Peltier cooling in Equation (1.2) as _Q Peltier α AB TI (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: where α = Seebeck coefficient, μv/k ρ = electrical resistivity, Ωcm α 2 σ Z (1.7) ρk k α 2

21 Introduction 5 σ = 1/ρ = electrical conductivity (Ωcm) k = thermal conductivity, W/mK 1 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 α 2 σ 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 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 firstgeneration 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 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

22 6 Thermoelectrics Table 1.1 Thermoelectric Properties of Single Crystal and Bulk Nanocomposite Semiconductors Type Temperature (K) α (μv/κ) σ (Ωcm) k e (W/mK) k (W/mK) ZT Authors Bi 2 Te 3 p-type single crystals Jeon et al. (1991) [4] BiSbTe p-type, nanocomposites Poudel et al. (2008) [5] Bi 2 Te 2.7 Se 0.3 n-type nanocomposites Yan et al. (2010) [6] PbTe-SrTe p-type nanocomposites Biswas et al. (2012) [7] Si 70 Ge 30 n-type single crystals Dismukes et al. (1964) [8] Si 80 Ge 20 n-type nanocomposites Wang et al. (2008) [9] CoSb 3 n-type single crystals Caillat et al. (1996) [10] Yb-CoSb 3 n-type, Yb-filled skutterudites Tang et al. (2015) [11] Yb 14 MnSb 11 p-type, Zintl compound Brown et al. (2006) [12] La 3 Te 4 n-type single crystals May et al. (2010) [13] 1

23 Introduction 7 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 Thomson, W., Account of researches in thermo-electricity. Proceedings of the Royal Society of London, : p Zhao, L.-D., V.P. Dravid, and M.G. Kanatzidis, The panoscopic approach to high performance thermoelectrics. Energy & Environmental Science, (1): p 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, (4): p Poudel, B., et al., High-thermoelectric performance of nanostructured bismuth antimony telluride bulk alloys. Science, (5876): p Yan, X., et al., Experimental studies on anisotropic thermoelectric properties and structures of n-type Bi2Te2.7Se0.3. Nano Letters, (9): p Biswas, K., et al., High-performance bulk thermoelectrics with all-scale hierarchical architectures. Nature, (7416): p Dismukes, J.P., et al., Thermal and electrical properties of heavily doped Ge-Si alloys up to 1300 K. Journal of Applied Physics, (10): p Wang, X.W., et al., Enhanced thermoelectric figure of merit in nanostructured n-type silicon germanium bulk alloy. Applied Physics Letters, (19): p Caillat, T., A. Borshchevsky, and J.P. Fleurial, Properties of single crystalline semiconducting CoSb3. Journal of Applied Physics, (8): p 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, (9): p Brown, S.R., et al., Yb14MnSb11: New high efficiency thermoelectric material for power generation. Chemistry of Materials, : p May, A.F., J.-P. Fleurial, and G.J. Snyder, Optimizing thermoelectric efficiency in La3 xte4via Yb substitution. Chemistry of Materials, (9): p

24 2 Thermoelectric 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 (2.4) Thermoelectrics: Design and Materials, First Edition. HoSung Lee John Wiley & Sons, Ltd. Published 2017 by John Wiley & Sons, Ltd.

25 Thermoelectric Generators 9 For steady state, we have ~?~q q_ 0 (2.5) where _q is expressed by [3] Substituting Equations (2.3) and (2.6) into Equation (2.5) yields _q ~E?~j j 2 ρ ~j? α~t (2.6) ~? k ~T j 2 ρ T dα j? ~ dt ~ T 0 (2.7) The Thomson coefficient τ, originally obtained from the Thomson relations, is defined by τ T dα 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 τ becomes 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

26 10 Thermoelectrics 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 d dt I 2 ρ ka 0 (2.9) dx dx A The solution for the temperature gradient with two boundary conditions (T x 0 T h and T x L T c ) in Figure 2.2 is dt I 2 ρl T h T c dx x 0 2A 2 k L (2.10) Equation (2.3) is expressed in terms of p-type and n-type thermoelements. dt dt _Q h n α p α n T c I ka ka (2.11) dx x 0 p dx x 0 n _ where Q h 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 ρ p L _ p ρ n L n k p A p k n A n n α T I 2 p α n h I T h T c 2 A p A n L p L n Q h (2.12)

27 Thermoelectric Generators 11 Finally, the heat absorbed at the hot junction with temperature T h is expressed as _Q h n αt h I 1 I 2 R KT h T c (2.13) 2 where α α p α n (2.14) ρ p L p ρ n L n R (2.15) A p From the first law of thermodynamics for the thermoelectric module, the power output is W_ n Q_ h Q_ c. The total power output is then expressed in terms of the internal properties as A n k p A p k n A n K (2.16) L p 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 = ρ p + ρ n and k = k p + k n. 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 L n _ 1 Q I 2 c n αt c I R KT h T c (2.17) 2 W_ I 2 n n αi T h T c R (2.18) However, the total power output in Figure 2.2 can be defined by an external load resistance as _W n ni 2 R L (2.19) _ Equating Equations (2.18) and (2.19) with W n IV n gives the total voltage as V n nir L n α T h T c IR (2.20) 2.2 Performance Parameters of a Thermoelectric Module From Equation (2.20), the electrical current for the module is obtained as T c α T h I (2.21) R R L 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 T h T c R L (2.22) V n RL R 1 R

28 12 Thermoelectrics Inserting Equation (2.21) in Equation (2.19) gives the power output as 2 T c R L nα 2 _ T h W R n R R L 1 R 2 (2.23) The conversion (or thermal) efficiency is defined as the ratio of the power output over the heat absorbed at the hot junction: _W n (2.24) η th _ Q h Inserting Equations (2.13) and (2.23) into Equation (2.24) gives an expression for the conversion efficiency: T c R L 1 T h R (2.25) R L 1 T c 1 R L T c R 2 2ZT R η th 2 where the average temperature is defined as T ηc 1 T c =T h. T h T h T c Maximum Parameters for a Thermoelectric Module Because the maximum current inherently occurs in a short circuit where R L current for the module is T h. It is noted that the Carnot cycle efficiency is 0 in Equation (2.21), the maximum I max α T h T c (2.26) R The maximum voltage inherently occurs in an open circuit where I = 0 in Equation (2.20). The maximum voltage is V max nα T h T c (2.27) 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 R L =R 1, which leads to the maximum power output as 2 T c nα 2 _ T h (2.28) 4R W max 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 R L =R 1 ZT. Then, the maximum conversion efficiency η is max T c 1 ZT 1 ηmax 1 (2.29) T h T c 1 ZT T h There are a total of four essential maximum parameters: I max, V max, W_ max, and η max. However, there is also the maximum power efficiency. The maximum power efficiency is obtained by letting R L =R 1 in Equation (2.25). The

29 Thermoelectric Generators 13 maximum power efficiency η mp is T c 1 T η h mp (2.30) 1 T c 2 T c ZT T h T h Note there are two thermal efficiencies: the maximum power efficiency η mp efficiency η max. and the maximum conversion 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 R L _W 4 R W_ max R L 1 R 2 (2.31) Equations (2.21) and (2.26) give the normalized currents as I 1 I max R L 1 R (2.32) Equations (2.22) and (2.27) give the normalized voltage as R L V n R (2.33) V max R L 1 R Equations (2.25) and (2.29) give the normalized thermal efficiency as R L 1 ZT η th R T h η 2 max R L 1 T c 1 R L T c ZT R 2 T h 2ZT R T c T h 1 (2.34) Note that the normalized values in Equations (2.31) through (2.33) are a function only of R L =R, while Equation (2.34) is a function of three parameters: T c =T h, R L =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 R L =R. The maximum power W_ max occurs at R L =R 1, while η max occurs approximately at R L =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 η max is

30 14 Thermoelectrics Figure 2.3 Normalized chart I for thermoelectric generators, where T c /T h = 0.7 and ZT 1 are used presented in Figure 2.5 as a function of both the dimensionless figure of merit (ZT) and T c /T h. 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 T c /T h = 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 T c /T h = 0.7 and ZT 1 are used