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A practical field guide to thermoelectrics: Fundamentals, synthesis, and characterization Alex Zevalkink, David M. Smiadak, Jeff L. Blackburn, Andrew J. Ferguson, Michael L. Chabinyc, Olivier Delaire, Jian Wang, Kirill Kovnir, Joshua Martin, Laura T. Schelhas, Taylor D. Sparks, Stephen D. Kang, Maxwell T. Dylla, G. Jeffrey Snyder, Brenden R. Ortiz, and Eric S. Toberer
Citation: Applied Physics Reviews 5, 021303 (2018); doi: 10.1063/1.5021094 View online: https://doi.org/10.1063/1.5021094 View Table of Contents: http://aip.scitation.org/toc/are/5/2 Published by the American Institute of Physics
Articles you may be interested in Rocksalt nitride metal/semiconductor superlattices: A new class of artificially structured materials Applied Physics Reviews 5, 021101 (2018); 10.1063/1.5011972 Domain switching in single-phase multiferroics Applied Physics Reviews 5, 021102 (2018); 10.1063/1.5018872 Lightning under water: Diverse reactive environments and evidence of synergistic effects for material treatment and activation Applied Physics Reviews 5, 021103 (2018); 10.1063/1.5024865 Introduction to thermoelectrics APL Materials 4, 104806 (2016); 10.1063/1.4954055 Graphene and related two-dimensional materials: Structure-property relationships for electronics and optoelectronics Applied Physics Reviews 4, 021306 (2017); 10.1063/1.4983646 Tutorial: Defects in semiconductors—Combining experiment and theory Journal of Applied Physics 119, 181101 (2016); 10.1063/1.4948245
APPLIED PHYSICS REVIEWS 5, 021303 (2018)
APPLIED PHYSICS REVIEWS
A practical field guide to thermoelectrics: Fundamentals, synthesis, and characterization Alex Zevalkink,1 David M. Smiadak,1 Jeff L. Blackburn,2 Andrew J. Ferguson,2 Michael L. Chabinyc,3 Olivier Delaire,4,5 Jian Wang,6,7 Kirill Kovnir,6,7 Joshua Martin,8 Laura T. Schelhas,9 Taylor D. Sparks,10 Stephen D. Kang,11 Maxwell T. Dylla,11 G. Jeffrey Snyder,11 Brenden R. Ortiz,12 and Eric S. Toberer12
1 Chemical Engineering and Materials Science Department, Michigan State University, East Lansing, Michigan 48824, USA 2 Chemistry and Nanoscience Center, National Renewable Energy Laboratory, Golden, Colorado 80401, USA 3 Materials Department, University of California Santa Barbara, Santa Barbara, California 93106, USA 4 Department of Mechanical Engineering and Materials Science, Duke University, Durham, North Carolina 27708, USA 5 Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37830, USA 6 Department of Chemistry, Iowa State University, Ames, Iowa 50011, USA 7 Ames Laboratory, US Department of Energy, Ames, Iowa 50011, USA 8 National Institute of Standards and Technology (NIST), Gaithersburg, Maryland 20899, USA 9 Applied Energy Programs, SLAC National Accelerator Laboratory, Menlo Park, California 94025, USA 10 Materials Science and Engineering, University of Utah, Salt Lake City, Utah 84112, USA 11 Department of Materials Science and Engineering, Northwestern University, Evanston, Illinois 60208, USA 12 Department of Physics, Colorado School of Mines, Golden, Colorado 80401, USA
(Received 31 December 2017; accepted 2 April 2018; published online 27 June 2018) The study of thermoelectric materials spans condensed matter physics, materials science and engineering, and solid-state chemistry. The diversity of the participants and the inherent complexity of the topic mean that it is difficult, if not impossible, for a researcher to be fluent in all aspects of the field. This review, which grew out of a one-week summer school for graduate students, aims to provide an introduction and practical guidance for selected conceptual, synthetic, and characterization approaches and to craft a common umbrella of language, theory, and experimental practice for those engaged in the field of thermoelectric materials. This review does not attempt to cover all major aspects of thermoelectric materials research or review state-of-the-art thermoelectric materials. Rather, the topics discussed herein reflect the expertise and experience of the authors. We begin by discussing a universal approach to modeling electronic transport using Landauer theory. The core sections of the review are focused on bulk inorganic materials and include a discussion of effective strategies for powder and single crystal synthesis, the use of national synchrotron sources to characterize crystalline materials, error analysis, and modeling of transport data using an effective mass model, and characterization of phonon behavior using inelastic neutron scattering and ultrasonic speed of sound measurements. The final core section discusses the challenges faced when synthesizing carbon-based samples and the measuring or interpretation of their transport properties. We conclude this review with a brief discussion of some of the grand challenges and opportunities that remain to be addressed in the study of thermoelectrics. Published by AIP Publishing. https://doi.org/10.1063/1.5021094
TABLE OF CONTENTS
I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . II. THERMOELECTRIC TRANSPORT THEORY . . A. Introduction to Landauer theory . . . . . . . . . . . B. Charge transport in an electric field. . . . . . . . C. Charge transport in a temperature gradient . . 1. Short-circuit condition . . . . . . . . . . . . . . . . 2. Open-circuit condition . . . . . . . . . . . . . . . . 1931-9401/2018/5(2)/021303/50/$30.00
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D. Heat transport . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Heat carried by a particle. . . . . . . . . . . . . . 2. Electronic thermal conductivity. . . . . . . . . 3. Lattice thermal conductivity . . . . . . . . . . . E. The energy-dependent transport function. . . . 1. Role in maximizing power factor . . . . . . . 2. The ideal G(E) for electrons . . . . . . . . . . . F. Power from a thermoelectric device . . . . . . . . III. INORGANIC SYNTHESIS METHODS . . . . . . . . A. General guidelines . . . . . . . . . . . . . . . . . . . . . .
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1. Reporting and literature . . . . . . . . . . . . . . . 2. Preparation of reagents . . . . . . . . . . . . . . . . B. Basic powder synthesis routes. . . . . . . . . . . . . 1. Ball milling. . . . . . . . . . . . . . . . . . . . . . . . . . 2. Case study: Incremental ball-milling synthesis of KAlSb4 and KGaSb4 . . . . . . . 3. High-temperature reactions . . . . . . . . . . . . 4. Arc melting. . . . . . . . . . . . . . . . . . . . . . . . . . 5. Flux synthesis. . . . . . . . . . . . . . . . . . . . . . . . C. Advanced methods: Exploiting reaction pathways . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Decomposition . . . . . . . . . . . . . . . . . . . . . . . 2. Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Multi-step reactions. . . . . . . . . . . . . . . . . . . D. Sample consolidation . . . . . . . . . . . . . . . . . . . . 1. Cold press/annealing . . . . . . . . . . . . . . . . . . 2. Hot-pressing or spark plasma sintering (SPS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E. Beyond powder synthesis: Crystal growth. . . 1. Which material types can be grown by Bridgman method? . . . . . . . . . . . . . . . . . . . 2. What should be considered for successful Bridgman growth? . . . . . . . . . . IV. STRUCTURAL CHARACTERIZATION. . . . . . . A. In-situ X-ray diffraction . . . . . . . . . . . . . . . . . . B. Resonant X-ray diffraction (REXD). . . . . . . . C. X-ray absorption spectroscopy (XAS) . . . . . . D. Practical advice . . . . . . . . . . . . . . . . . . . . . . . . . V. ELECTRONIC TRANSPORT MEASUREMENT AND UNCERTAINTY . . . . . . . . . . . . . . . . . . . . . . . A. Definitions and resources. . . . . . . . . . . . . . . . . B. Measurement of the Seebeck coefficient . . . . 1. Seebeck coefficient measurement conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. 2-probe vs. 4-probe arrangements. . . . . . . C. Measurement of the electrical resistivity . . . . VI. ANALYSIS OF ELECTRONIC TRANSPORT DATA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Material quality factor analysis using only a, r, and j. . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Bipolar effects . . . . . . . . . . . . . . . . . . . . . . . . . . C. Modeling complex electronic structures . . . . VII. MEASURING PHONON BEHAVIOR IN CRYSTALLINE MATERIALS. . . . . . . . . . . . . . . A. Basics of lattice dynamics . . . . . . . . . . . . . . . . 1. Interatomic force-constants and dynamical matrix . . . . . . . . . . . . . . . . . . . . . 2. Phonon density of states and dispersions. B. Debye model . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Characterization of the speed of sound . . . . . 1. Ultrasonic methods . . . . . . . . . . . . . . . . . . . 2. Considerations for porous samples . . . . . . 3. Temperature-dependent speed of sound measurements . . . . . . . . . . . . . . . . . . . . . . . . D. Anharmonicity and the quasiharmonic approximation . . . . . . . . . . . . . . . . . . . . . . . . . . E. Neutron scattering measurements of phonons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1. Powder or incoherently scattering samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Single-crystals . . . . . . . . . . . . . . . . . . . . . . . 3. Practical considerations . . . . . . . . . . . . . . . VIII. INTRODUCTION TO ORGANIC THERMOELECTRICS . . . . . . . . . . . . . . . . . . . . . A. Chemical and electronic structure of carbon-based materials . . . . . . . . . . . . . . . . . . . B. The problem of polydispersity . . . . . . . . . . . . 1. Electronic character of SWCNTs . . . . . . . 2. Electronic bandgap of SWCNTs . . . . . . . . C. Processing organic thermoelectric materials . D. Charge carrier doping. . . . . . . . . . . . . . . . . . . . 1. Measuring charge carrier concentrations . 2. Dopants and doping stability . . . . . . . . . . . E. Thermoelectric properties. . . . . . . . . . . . . . . . . 1. Thermoelectric power factor of organic semiconductors. . . . . . . . . . . . . . . . . . . . . . . 2. Ionic transport in semiconducting polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Thermal transport . . . . . . . . . . . . . . . . . . . . 4. Consequences for anisotropic transport . . IX. GRAND CHALLENGES AND OPPORTUNITIES . . . . . . . . . . . . . . . . . . . . . . . . . . A. Mysteries of quasiparticles . . . . . . . . . . . . . . . B. Large-scale calculations . . . . . . . . . . . . . . . . . . C. Synthetic chemistry. . . . . . . . . . . . . . . . . . . . . . D. Measurement techniques . . . . . . . . . . . . . . . . . E. Thermoelectric devices . . . . . . . . . . . . . . . . . . . F. Uncharted terrain in organic materials . . . . . .
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I. INTRODUCTION
The study of thermoelectric materials, which convert thermal energy to electrical energy in the solid state, has grown immensely in the past decade. What was once a relatively niche area has transformed into a research area spanning condensed matter physics, materials science and engineering, and solid-state chemistry. The diversity of the participants and the inherent complexity of the topic—fundamentally, the study of electronic and thermal transport in a broadly defined group of materials—means that it is difficult, if not impossible, for one person to be fluent in all aspects of the field. Further, synthesis and measurement techniques for structure and transport pose inherent difficulties in both their execution and interpretation. The present review article grew out of a one-week thermoelectric summer school held for graduate students at the Colorado School of Mines. While the summer school began with the basics of thermoelectric transport, our primary aim was to discuss commonly misunderstood conceptual approaches, to raise awareness of powerful synthesis and characterization techniques, and to encourage discussions of the best practices and pitfalls associated with each approach. During the final day of the school, significant discussion was devoted to future directions within the study of thermoelectric materials. Mirroring the organization of the workshop, we begin this review with an introduction to transport theory using
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Landauer’s approach. Landauer theory is a universal description of charge and heat transport and is therefore equally applicable to inorganic materials, nano-materials, polymers, or even walruses, if you so choose; the first chapter will derive each of the transport coefficients present in the thermoelectric material figure-of-merit zT, which determines the maximum efficiency of a thermoelectric material. We do not assume any prior knowledge of thermoelectric transport. The following chapters are intended primarily for those studying inorganic materials. We first discuss effective strategies for the synthesis of bulk inorganic samples (single and polycrystalline) and encourage readers to take advantage of national synchrotron sources to characterize the structure of their materials, highlighting methods that remain under-utilized in our field. We then turn our attention to electronic and thermal transport properties, discussing first the fundamentals of uncertainty and error analysis, followed by a tutorial for modeling electronic transport within an effective mass model. This analysis is a valuable exercise for those who have measured the Seebeck coefficient and electrical conductivity of one or more samples and are interested in further optimizing the performance through doping. The next chapter focuses on methods for characterizing phonon behavior in crystalline materials using speed of sound measurements and inelastic neutron scattering. The final core chapter of this review provides an overview of the obstacles faced when synthesizing carbon-based samples and the measurement or interpretation of their transport properties. Although such materials do not exhibit performance comparable with inorganic materials and have not been as extensively developed, they have some attractive characteristics, and it is worthwhile to delve into the unique challenges faced when working with non-crystalline materials. This review does neither attempt to cover all aspects of thermoelectric materials research nor is it a review of stateof-the-art thermoelectric materials. Rather, the authors hope to provide an introduction and practical guidance for selected conceptual, synthetic, and characterization approaches, as well as to craft a common umbrella of language, theory, and experimental practice for those engaged in the complex and diverse field of thermoelectric materials. The organization of this review does not require a linear approach to read it. Rather, treat this as a resource for understanding different aspects of the field. Each section is self-contained and designed to be accessible to those with little or no experience with the approaches described. II. THERMOELECTRIC TRANSPORT THEORY
Thermoelectric researchers need a universal language to discuss charge and heat transport in their materials. Landauer theory is ideal for this purpose; it is approachable, intuitive, and applicable to both charge and heat transport. The theory developed in this section yields the same results as historically dominant transport theories (for example, the Boltzmann transport equation within the relaxation time approximation). However, Landauer theory will unite electronic and thermal transports, which are often viewed as disparate sub-fields of thermoelectrics.
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In Sec. II A, we introduce the core of Landauer theory before discussing electronic transport induced from electric fields and temperature gradients—two processes integral to thermoelectrics. The theory developed for charge transport is then adapted to discuss heat carried by electrons and phonons. This microscopic theory is general and applies to inorganic and organic materials, crystalline and amorphous. In the final subsections, we discuss specific metrics for power factor and lattice thermal conductivity in crystalline materials and examine how these properties determine the material figure-of-merit zT and the performance of a thermoelectric device. A. Introduction to Landauer theory
In 1957, Landauer sought to explain the diffusive transport of charges in disordered metals containing polarizable point defect scattering sources.1–6 At the time, Boltzmann transport theory was the dominant approach to describe transport in crystalline solids, and Landauer’s developments were largely ignored. However, the emergence of nanostructured and molecular electronics in the mid-1980s rekindled interest in Landauer’s transport framework.7–10 The Landauer approach is easily understood by considering a quantum system where we track the transport of individual charge carriers; do not worry, we would not need to solve Schr€odinger’s equation. As an example, consider a benzene ring attached between two gold electrodes. The Landauer approach specifies boundary conditions at the gold electrodes on either side of the benzene ring: fixed electron chemical potentials, l, and temperatures, T. The boundary conditions set the electron occupation statistics, f, of the gold electrodes, which are a function of energy, E. As an aside, the Fermi-level, Ef, is used interchangeably with the electron chemical potential, l, in some sources (here, we reserve “Fermi-energy” for 0 K) f ðEÞ ¼
1 e
El kT
þ1
:
(1)
Transport through the benzene ring occurs via an integer number of parallel transport channels, M(E), with welldefined transition probabilities, T(E), through the channels. The current is given by a sum over the transitions, weighted by the difference in occupation statistics between the electrodes. Electrons are always hopping in both directions; there is only a net current, I, if the electron occupation is higher at one electrode (Df ¼ f2 – f1 6¼ 0) so that there are more particles making transitions from that electrode ð 2q 1 I¼ TðEÞMðEÞDfdE: (2) h 1 The transport between the reservoirs can be modeled as ballistic, diffusive, or anywhere in between by tuning the transition probability. Extending the Landauer approach to bulk systems involves considering the current density, J, through an infinitesimal slice of bulk material. The finite difference in particle occupation statistics becomes a gradient in occupation, rf,
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induced by a gradient in chemical potential or temperature. The transport channels and their associated transition probabilities [T(E) and M(E)] can be combined into a positivevalued, energy-dependent transport function G(E) ð1 J ¼ q GðEÞrfdE: (3) 1
Note that the charge associated with a carrier, q, includes its sign (e.g., q for an electron is –e). This equation can be generalized to treat more than just charge transport by electrons. The current density, J, in Eq. (3) is the product of a particle flux and the charge per particle q; we can treat the heat flux from charge carriers by considering the heat carried per particle. In addition, Eq. (3) is applicable to more than just electrons; if the suitable particle distribution function is used, the equation is equally valid for fermions, bosons, and Boltzmann-like particles. Therefore, we can treat more than just conductivity and the Seebeck coefficient with Landauer theory; contributions to the thermal conductivity from electrons and vibrational quanta are accessible. Without specifying any material details—metal or semiconductor, crystalline or amorphous—we can begin to understand transport induced from electric fields and temperature gradients; these gradients will induce gradients in the particle distribution function, which drives transport. Charge transport by electrons will be dealt with first. Then, Eq. (3) will be adapted to consider heat transport. Derivations in
these sections will focus on understanding the underlying transport of measurable quantities such as conductivities and the Seebeck coefficient without specifying any functional form for G(E). In all cases, we will see that transport can be understood from gradients in particle occupation statistics. If you are impatient to see what G(E) looks like, step ahead briefly to Sec. II E, which describes G(E) for an inorganic semiconductor. Those who are interested in measuring the transport coefficients derived in this section should continue to Sec. V. B. Charge transport in an electric field
Characterizing electronic transport induced from electric fields is a typical situation in a thermoelectric laboratory; conductivity measurements are carried out by applying a voltage across a sample at isothermal conditions. An external bias results in a gradient in the electron chemical potential (rl ¼ qrV), which in turn induces a gradient in the electron occupation statistics. The electron distribution functions near the contacts of a sample under applied bias (at isothermal conditions) are depicted in Fig. 1(a). The high energy states near contact two have a greater occupation than those states near contact one. The difference in electron occupation constitutes a driving force to transfer electrons from contact two to contact one. Let us check our intuitive picture against Landauer theory.
FIG. 1. (a) A gradient in chemical potential shifts the Fermi-Dirac distribution. The infinitesimal difference between neighboring occupations is positive for all carrier energies. Transport proceeds down the chemical potential gradient. (b) In contrast, a temperature gradient changes the curvature of the Fermi-Dirac distribution. The selection function is odd around the chemical potential. Carriers above the chemical potential diffuse from hot to cold, while carriers below flow in opposition.
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Landauer theory for bulk materials [Eq. (3)] relates current density to the gradient in the carrier statistics. If we can find a relationship between the spacial gradient of the occupation statistics (rf ¼ @f/@x) and the spacial gradient of the electron chemical potential (rl ¼ @l/@x), we can determine the current density induced by an applied voltage. A chain rule relates the gradient of the occupation statistics and the gradient of the chemical potential (@f =@x ¼ @f =@l @l=@x). However, it is far more convenient to formulate the derivative of the carrier statistics with respect to energy, E, instead of chemical potential l, since the integral for current density is over energy-space. The derivative of the Fermi-Dirac function [Eq. (1)] with respect to chemical potential and energy are opposite in sign (@f =@l ¼ @f =@E). Therefore, a succinct chain-rule relates the spacial gradient of the occupation statistics and the spacial gradient of chemical potential (rf ¼ @f =@E rl); applying this chain-rule to the general current density equation informs us how a material responds to an applied voltage ð1 @f JrV ¼ q2 rV dE: (4) GðEÞ @E 1 The negative derivative of the Fermi-Dirac function (see Fig. 1) and G(E) are positive-valued functions. This means that current flows down a voltage gradient regardless of material type (i.e., metal, n/p-type semiconductor, or insulator). The details of G(E) and the temperature determine the magnitude of the current density. Conductivity is the thermoelectric property of interest when we apply a voltage to a sample. Ohms law (J ¼ –rrV) relates the current density to the magnitude of the applied bias ð1 @f 2 r¼q dE: (5) GðEÞ @E 1 A material’s conductivity, r, is always positive regardless of whether a semiconductor is n- or p-type. The derivative of the electron distribution function (–@f/@E) acts as a selection function that determines which carriers are involved in transport (see Fig. 1). At high temperatures, particles in a wider energy range participate in transport. C. Charge transport in a temperature gradient
The Seebeck effect is a phenomenon where a temperature gradient in a material induces a voltage gradient. Applying a temperature gradient across a sample and measuring the induced voltage in an open-circuit condition define the Seebeck coefficient, a. There are two driving forces for carrier transport during a Seebeck measurement: a temperature gradient and the induced voltage gradient. In the linear regime, we can treat the current density from these two stimuli additively. To understand the Seebeck measurement, we need to first understand transport induced from a temperature gradient in a short-circuit condition (i.e., the electron chemical potential is constant throughout the sample).
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1. Short-circuit condition
Applying a temperature gradient to a sample induces a gradient in the carrier occupation statistics; with increasing temperature, the carrier distribution function broadens around the electron chemical potential. The carrier occupation statistics near the contacts of a sample in a temperature gradient (under short-circuit conditions) are depicted in Fig. 1(b). Regions of the sample at high temperature have more electrons occupying high energy states, and fewer occupying low energy states; there is thus a driving force to transfer high energy electrons from hot to cold and low energy electrons from cold to hot. This intuitive picture will be confirmed below by Landauer theory. Landauer theory for bulk materials [Eq. (3)] relates current density to the gradient in the carrier statistics. Again we use a chain rule to relate the spacial gradient of the occupation statistics with the spacial gradient of the temperature (rf ¼ @f =@T rT ¼ ðE lÞ=T @f =@E rT). Using Eq. (3), the Landauer solution to current density induced from a temperature gradient becomes ð1 E l @f JrT ¼ qrT dE: (6) GðEÞ T @E 1 While the negative derivative of Fermi-Dirac function and G(E) are positive-valued functions, El T is an odd function around the electron chemical potential [see Fig. 1(b)]. An electron above the chemical potential will transport from hot-side to cold-side, while electrons with energies less than the chemical potential will transport from cold-side to hotside. This asymmetry allows us to distinguish n- or p-type semiconductors; the electrons in n-type transport are predominantly above the chemical potential, while the opposite is true for p-type materials. A temperature analog to Ohm’s law (J ¼ –rT) relates the current density to the magnitude of the temperature gradient ð1 E l @f ¼q dE: (7) GðEÞ T @E 1 The intensive, material quantity, , is analogous to an Onsager coefficient that determines the current response from a temperature gradient. Materials may respond differently depending on the nature of G(E). If G(E) is symmetric around the chemical potential, there will be no net current. The selection function for transport from a temperature gradient is (E – l) (–@f/@E). Most of the carriers involved in transport will have energies 1.5 kBT above or below the chemical potential (Fig. 1). It is important (though somewhat unfortunate) that the energy range for temperature driven transport overlaps with the voltage driven transport. 2. Open-circuit condition
Having derived transport from a temperature gradient under short-circuit conditions, we can now revisit the Seebeck effect where both temperature and voltage gradients are present in a material. In the linear regime,11 the current
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densities induced by the temperature and voltage gradients are additive. Historically, these two contributions were termed the diffusion and drift currents, respectively J ¼ JrT þ JrV ¼ rT rrV:
(8)
A Seebeck measurement is defined at open-circuit conditions where the net current density is zero (J ¼ 0). The Seebeck coefficient, a, is given by the ratio of the voltage difference and the temperature difference through the thickness, l, of the sample a¼
DV lrV ¼ ¼ : DT lrT r
(9)
Recall that and r determine how readily a material responds to a temperature and voltage gradient, respectively. From Eq. (9), we see that materials with a large Seebeck coefficient respond more readily to a temperature gradient than an electric field. To have an ideally large Seebeck, a material would not respond to an electric field at all. However, it is impossible for electrons in a real material to respond to a temperature gradient and not a voltage gradient because the carriers involved in the two transport processes are in overlapping energy ranges; this is apparent when one examines the selection functions in Fig. 1. Since the two processes cannot be totally separated, thermoelectric materials must be carefully doped to optimize the position of the electron chemical potential. To read about the role of G(E) in this optimization, continue to Sec. II E.
this model when explaining the entropy associated with fermions and bosons. Imagine a large parking lot with N parking spaces. The parking lot is initially empty, but as the day progresses, cars begin to file in. At noon, there are c identical cars in the parking lot. There are many ways that the cars could be arranged in the parking lot. We can count the number of configurations, X, with the binomial coefficient N! N : (11) X¼ ¼ c ð N cÞ!c! When the parking lot was completely empty in the morning, there was only one way for how the spots could be empty. At noon, if all the spots are taken, then again, there is only one way all the spots can be taken (all of the cars are identical). The number of configurations is maximized if half of the parking places are taken (see Fig. 2). The entropy, S, associated with the cars in the parking lot is given by the logarithm of the number of configurations S ¼ kB log X:
(12)
Now imagine one more car drives into the parking lot in the afternoon. The addition of this one car changes the number of possible configurations in the parking lot! This is analogous to the change in entropy associated with a particle moving through a material. Now that we have considered a toy model for entropy, let us return to discussing fermions and bosons. We will need to consider the number of fermions/ bosons present at each energy level and the density of states,
D. Heat transport
In the preceding sections, we learned that current induced from electric fields and temperature gradients is carried by particles in particular energy ranges set by selection functions. Heat transport is conducted in much the same way: only particles in particular energy windows contribute. The heat carried by each individual particle is intimately tied to the entropy associated with that particle at it moves through a material. The second law of thermodynamics (for a reversible process) relates the entropy, S, associated with a particle to the heat, Q, it carries dQ ¼ T dS:
(10)
We will first explore how entropy associated with a particle determines the amount of heat it carries. Those who are not interested in how entropy and heat are related may wish to skip ahead to Eq. (18), where you can find the satisfactory end-result. The heat carried by a particle will be applied to Eq. (3) to determine the contributions to thermal conductivity from electrons and atomic vibrations. 1. Heat carried by a particle
In this section, we will explore how heat carried by particles is related to their entropy. The statistical mechanics in this section are rather obtuse, so to get a feel for how particle entropy is calculated, we are going to first consider a visual, toy model: cars in a parking lot. We will draw analogies to
FIG. 2. (a) The number of configurations for cars in a parking lot is maximized when the lot is half-full. (b) The entropy is maximized when the number of configurations X is maximized. (c) When cars are added to the lot, the change in entropy per-car depends on the number of cars in the lot.
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g, at each energy level. We will consider each energy level as a separate “parking lot” with its own set of parking spaces. Fermions obey the Pauli-exclusion principle, so only one may occupy a given state (we will account for spindegeneracy in the density of states). In contrast, multiple Bosons can inhabit one state. The occupation statistics for bosons reflects the ability of multiple particles to occupy one state fb ðEÞ ¼
1 E kT
e 1
:
(13)
The number of particles is the product of their density of states and the relevant occupation statistics [c ¼ g(E)f(E)]. Since multiple bosons can occupy one state, the “effective” number of parking spaces increases with the number of bosons (gðEÞ ! gb ðEÞ þ gb ðEÞfb ðEÞ 1) ! gðEÞ ; (14) Xf ðEÞ ¼ gðEÞf ðEÞ ! gb ðEÞ þ gb ðEÞfb ðEÞ 1 : (15) Xb ðEÞ ¼ gb ðEÞfb ðEÞ The number of configurations defines the amount of entropy in the energy level according to Eq. (12). When g(E) 1, Stirling’s approximation significantly simplifies the expression for the entropy in each energy level. Ultimately, the change in energy upon adding another particle is expressed relative to the chemical potential for fermions and on an absolute energy scale for bosons @Sf E l ¼ ; @c T
(16)
@Sb E ¼ : @c T
(17)
We will refrain from expressing the algebraic details that leads to these results, since they do not add to our intuition of transport, which is the purpose of this section. The second law of thermodynamics [Eq. (10)] relates the entropy change from adding particles to the heat carried per particle @Qf ¼ E l; @c
(18)
@Qb ¼ E: @c
(19)
The heat carried by a fermion is referenced to its chemical potential, while the heat carried by a boson is on an absolute energy scale. This is a somewhat satisfying result. The general current density equation [Eq. (3)] can be adopted for heat transport by exchanging the charge per particle with the energy per particle (q ! @Q/@c). Since the heat carried per particle depends on energy, it is pulled into the integral ð1 @Q GðEÞrfdE: (20) JQ ¼ 1 @c
We can now derive the heat transport induced from electric fields and temperature gradients. The derivations will follow the same flow as the sections on charge transport. Chain rules will relate voltage and temperature gradients to gradients in occupation statistics; then, Onsager relations will elucidate measurable quantities. We will encounter two new selection functions for heat transport. 2. Electronic thermal conductivity
In Secs. II B and II C, we derived the current densities induced from voltage and temperature gradients [Eqs. (4) and (6)]. We can apply these results to heat transport by converting the charge carried per-particle to the heat carried perparticle (q ! E – l). Notice that one unit of charge remains in the voltage equation from converting a chemical potential gradient to a voltage gradient (rl ¼ qrV) ð1 @f dE ; (21) GðEÞðE lÞ JQ;rV ¼ qrV @E 1 ð1 ðE lÞ2 @f dE: (22) JQ;rT ¼ rT GðEÞ @E T 1 We have already seen the upper selection function when we examined temperature-driven charge transport. The selection function indicates that electrons above the chemical potential carry heat down a voltage gradient, while electrons below the chemical potential oppose this heat-flow. The lower selection function (E – l)2 (–@f/@E) is an even function around the electron chemical potential, so heat carried by electrons always flows down a temperature gradient (see Fig. 3). Another Onsager coefficient relates the heat current density to an applied temperature gradient (JQ ¼ –jerT) at open-circuit conditions; this coefficient is the electronic contribution to the thermal conductivity, je JQ;rT JQ;rV je ¼ þ (23) : rT J¼0 rT J¼0 There are thus two contributions to je: one driven by a temperature gradient and one driven by a voltage gradient. The temperature term is j0 j0 ¼
ð1 GðEÞ 1
ðE lÞ2 @f dE T @E
(24)
and is independent of an applied voltage. We refer to the voltage driven term as j1 ð1 rV @f dE: (25) j1 ¼ q GðEÞðE lÞ rT J¼0 1 @E We can see that the origin of j1 is heat driven by a voltage gradient. Electronic thermal conductivity is defined at opencircuit conditions where rV/rT is related to the Seebeck coefficient je ¼ j0 a2 rT:
(26)
We can understand Peltier heat currents and heat pumps by relaxing the zero-current condition. Running current through
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FIG. 3. (a) A temperature gradient induces a selection function for electron heat transport and for (b) heatcarrying bosons. The selection function can be interpreted as the per-mode specific heat for phonons. The boson selection function is broader than –@f/ @E (dashed curve).
a material decreases rV in Eq. (25). Decreasing the magnitude of the voltage gradient tends to increase the effective electronic thermal conductivity. The voltage driven term allows thermoelectric devices to be used as heat pumps or Peltier coolers. In one configuration, heat is actively pumped across the device, whereas reversing the applied voltage results in Peltier cooling.
3. Lattice thermal conductivity
In addition to electrons, atomic vibrations also transport heat. Since vibrational quanta have no charge associated with them, they do not respond to an electric field; the only driving force for heat-flux (barring the formation of a polaron) is a temperature gradient. Landauer theory for heat transport [Eq. (20)] relates the heat flux to the gradient of the carrier statistics. A chain rule relates the spacial gradient of Bose-Einstein statistics with the spacial gradient of the temperature (rfb ¼ @fb =@T rT ¼ ET @fb =@E rT). The Landauer solution for bosonic heat-flux induced from a temperature gradient is ð1 E2 @fb dE: (27) GðEÞ JQ;rT ¼ rT T @E 0 Notice that the lower limit of the integral is zero, since vibrational quanta energies are measured on an absolute energy scale. Remarkably, the selection function for heat carrying bosons (E2 @fb =@E) is similar to electrons driven by an electric field (–@f/@E); while the boson selection function is the same shape, it is about twice as broad (see Fig. 3). Therefore, heat carried by bosons always travels down a temperature gradient. Lattice thermal conductivity relates heat flux to an applied temperature gradient (JQ ¼ –jlrT) ð1 E2 @fb dE: (28) GðEÞ jl ¼ T @E 0 Expanding the derivative of Bose-Einstein statistics is instructive, since it reveals the per-mode specific heat
jl ¼
ð1 GðEÞ 0
E2 eE=kT dE: kT 2 ðeE=kT 1Þ2
(29)
Readers will typically find lattice thermal conductivity formulated as the product of the energy-dependent transport function, G(E), and the per-mode specific heat, though you may need to infer G(E). The non-zero energy span for G(E) in the case of phonons or vibrational modes is smaller than that of fermions, which means that most or all vibrational modes will be active in transport above 300 K. We stress that this derivation applies to more than just phonons, which only exist in crystalline materials; this theory treats vibrational quanta in amorphous materials equally well. Phonons are discussed in-depth in Sec. VII.
E. The energy-dependent transport function
In Secs. II A–II D, we found that only carriers in energy ranges set by selection functions are involved in transport. Now, we will explore how the selection functions sample the energy-dependent transport function, G(E). The convolution of the selection functions with G(E) determines all of the transport coefficients we have derived. First, we need to explore the microscopic origins of G(E). Throughout this section, we will examine crystalline materials in the diffusive transport regime; different material classes may have different functional forms for G(E). In crystalline materials, G(E) can be derived from the band structure and scattering theory. To determine G(E), we must consider the collective motion of all the particles at a given energy. Conveniently, the band structure organizes all of the particles (electrons or phonons) by energy in reciprocal space. Each particle travels with a group velocity determined by the slope of the band structure (vg ¼ @E/@k) in the transport direction. If we consider the average squaredvelocity, hv2 i, of the particles at a given energy, then we can account for their collective transport with the density of states, g. On average, each particle scatters after a characteristic relaxation time, s. The diffusivity of each individual particle is described by hv2 is. G(E) is the sum of the
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individual particle diffusivities, since transport occurs in parallel conducting channels GðEÞ ¼ hvðEÞ2 isðEÞgðEÞ:
(30)
This formulation of G(E) describes both electrons and phonons. The key difference between electrons and phonons is the energy dependence of each constituent term. In Secs. II E 1 and II E 2, we will describe simplified models for the electron band structures. We will use these models to explain the properties of high-performing thermoelectric materials. We will focus on aspects of these models that are measurable in the laboratory. For those looking to apply these models, look ahead to Secs. VI and VII on the analysis of transport data and phonon behavior. 1. Role in maximizing power factor
An important quantity for thermoelectric device performance is the power factor, a2r. In a thermoelectric device, the power factor is proportional to the amount of local Joule heating induced by a temperature gradient. We will see in Sec. II F that the optimized output power from a TE device is roughly proportional to the power factor, due to power matching between the TE device and the load it drives. However, we must stress that thermal conductivity is equally important for device performance. Mathematically, we can write the power factor in terms of the Onsager coefficients ( and r) that we derived in Secs. II A–II D. A material’s power factor is a competition between temperature (the numerator) and voltage (the denominator) driven transport "ð a2 r ¼
#2 E l @f dE GðEÞ T @E 1 ð1 : @f dE GðEÞ @E 1 1
(31)
The power factor equation involves several complex integrals, making it difficult to see how we can get a high power factor in real materials. We are going to demystify this
expression by graphically illustrating how G(E) for electrons combines with the selection functions. We will learn where to place the electron chemical potential, l, (with respect to a band edge) to maximize power factor, and then we will examine G(E) as a design metric for achieving high thermoelectric performance. Keep in mind that l can be tuned by doping, while G(E) is material dependent, and can thus be controlled by clever material design. First, we need to derive a functional form for G(E) from the band structure. In crystalline inorganic materials, band edges are modeled as spherical carrier pockets with a parabolic dispersion. The dispersion curvature is parameterized by an effective mass, m (E ¼ h2 jkj2 =2m ). This model is referred to as the single parabolic band or effective mass model12 and determines the energy dependence of both the average velocity hv2 i and density of states, g, in G(E). Qualitatively, the slope of the parabolic dispersion increases into the band, so hv2 i must be an increasing function; the surface-area of the spherical pocket also increases into the band, so g must be an increasing function. Specifically, hv2 i increases linearly into a band, while g increases as the square-root of energy. At temperatures greater than 300 K, the relaxation time, s, is typically limited by phonon scattering, which scales inversely with g. Since the energy dependence of g and s cancel, the electron energy-dependent transport function is linear (Gel ðEÞ ¼ G0 E). Figure 4 depicts the constituent terms in G(E). Now that we have a functional form for G(E), we can see how it interacts with selection functions to produce the variety of transport properties seen in metals, insulators, and semiconductors. Since G(E) is zero inside of a band gap, the most important difference between these three classes of materials is the proximity of a band gap to the electron chemical potential. As we will see, the chemical potential should be located near a band edge to maximize power factor. a. Insulators/semiconductors. The electron chemical potential in insulators and semiconductors is located in proximity to a band gap. The difference between insulators and semiconductors is the size of the bandgap. When either material is degenerately doped, the electron chemical
FIG. 4. (a) For a crystalline semiconductor, the energy dependence of the transport function G(E) is derived from its constituent terms. A spherical, parabolic band edge determines the charge carrier group velocity (v) and density of states (g). The energy dependence of electron-phonon relaxation times (s) is inversely proportional to g. (b) The convolution of G(E) with selection functions determines thermoelectric transport coefficients: including, conductivity r, the Seebeck coefficient a, and the electronic contribution to thermal conductivity je.
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potential resides just inside the band edge (as illustrated in Fig. 4). The voltage induced selection function [denominator of Eq. (31)] partially samples the band edge. Only half of the temperature induced selection function [numerator of Eq. (31)] samples the band edge; since the temperature selection function is odd, this prevents transport cancellation between electrons above and below the chemical potential. In this configuration, we get both high Seebeck coefficient and sufficiently high conductivity. Materials with the electron chemical potential near a band edge tend to make the best thermoelectric materials due to the compromise between temperature and voltage driven transport. Optimizing the location of the chemical potential with respect to the band edge optimizes the power factor. b. Metals. In contrast, the electron chemical potential in metals is deep inside a band. As a result, the voltage selection function samples G(E) where it is largest. However, the temperature selection function induces cancellation between carriers below and above the chemical potential. While the conductivity of metals tends to be high, their low Seebeck coefficients make them poor thermoelectric materials. In addition, a metal’s electronic contribution to the thermal conductivity is high because carriers on both sides of the chemical potential are contributing. We have established that placing the electron chemical potential near a band edge optimizes a material’s power factor; this placement offers a compromise between the temperature/voltage driven transport integrals in Eq. (31). However, experimentally realized optimum power factors are highly variable between material systems. The slope (G0) of G(E) explains the wide range of power factors observed experimentally. Examine Eq. (31), G0 can be pulled from the integrals since it is energy-independent; optimized power factor is roughly proportional to G0! G0 is the intrinsic metric for optimized power factor, and an entire section of this review has been devoted to extracting it from Seebeck and conductivity measurements (in Sec. VI, rE0 ¼ q2 G0 ). rE0 can be used to compare the electronic quality of different materials even if they have not been optimally doped.
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the impact on power factor [Eq. (31)] we find that a delta function collapses the integrals and allows us to write power factor as T j0 [Eq. (24)]. Placing G(E) at the peak of the selection function for j0 optimizes the ratio of the temperature/voltage driven electronic transport. The delta function also minimizes the electronic contribution to thermal conductivity at open circuit conditions [Eq. (26)], since now j1 completely cancels j0. While rather unphysical, this delta function “pass-band” serves as a universal, aspirational goal. The take-away of this thought experiment is quite simple; the optimal thermoelectric material will have an extremely narrow transport pass-band that is an excellent conductor within that energetic window. The chemical potential needs to be near, but not within that pass-band region. As discussed above, current thermoelectric materials are quite different from this picture and many opportunities remain to achieve such narrow pass-band behavior. F. Power from a thermoelectric device
So far, we have considered how gradients in carrier occupation statistics drive all thermoelectric transport; this microscopic perspective illuminated the band structure features that maximize power factor. We will now see why power factor and thermal conductivity are the metrics for thermoelectric performance by examining a thermoelectric device. The following analysis ignores the key role of heat exchangers in thermoelectric performance. The instructive value of this section is understanding the origins of zT rather than system-level design of a thermoelectric device. Thermoelectric materials are used in both Peltier coolers and thermoelectric generators; as a case study, we will investigate power generation from a thermoelectric material with resistance, RTE, connected to a load resistor, RL. The power at the load resistor is generated from a current, ITE, driven by the thermoelectric voltage, VTE ¼ aDT, where DT is the temperature drop across the thermoelectric material. The gener2 ated power, P ¼ ITE RL , becomes P¼
a2 DT 2 ðRTE þ RL Þ2
RL :
(32)
2. The ideal G(E) for electrons
While the simple G(E) shown in Fig. 4 applies to most materials, this is far from the optimal form for G(E). The optimal G(E) is a delta function a few kB away from the chemical potential, as noted by Mahan.13 Often the requirement for a delta function in G(E) is confused for a delta function in density of states g(E); this may tempt one to believe that, for example, isolated f-states offer an avenue for optimal G(E). However, while f-states do have a delta functionlike g(E) (since they have little band dispersion), they also have near-zero average velocity hv2 i, which means the combined G(E) does not actually resemble a delta function. A delta function-like G(E), instead, will require high g and v coupled with an extremely long s at a specific energy. Delta function G(E) is good for thermoelectric performance as it yield both a high power factor and eliminates the electronic contribution to thermal conductivity. Investigating
By being able to freely adjust the RL to RTE ratio (m ¼ RL/ RTE), the power becomes P / a2 rDT 2
m ð1 þ mÞ2
:
(33)
The power is maximized when the power dissipated by joule heating in the thermoelectric material is matched by the power dissipated by the load resistor (m ¼ 1). We see that the maximum power is proportional to the power factor, a2r, and that low thermal conductivity is inherently needed for power generation, because it determines the magnitude of the achievable temperature drop (DT) in the thermoelectric material. For optimizing power in an actual device, considering the heat exchanger and thermal impedance matching introduce interdependencies that make the optimization problem more than just optimizing power factor.14
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In summary, high-performing thermoelectric materials will have low lattice thermal conductivity in addition to high power factor; the material figure-of-merit zT for thermoelectrics reflects both of these electronic and thermal aspects15 zT ¼
a2 r T: je þ jl
(34)
It is somewhat gratifying that all longitudinal thermoelectric devices, regardless of material, conform to the same demands for maximizing electrical power from a given temperature gradient.
III. INORGANIC SYNTHESIS METHODS
In this section, we focus on the synthesis of single-phase bulk polycrystalline materials and large single crystals. Organic thermoelectrics are discussed below in Sec. VIII. The solid state chemist or materials scientist has a wide variety of tools to utilize when approaching a material with no known (or poorly documented) synthesis process. This section provides a guide to some of the most widely used methods in the thermoelectric community, as well as a few powerful, but lesser-known approaches.
2. Preparation of reagents
Proper preparation of reagents is key to performing repeatable, reliable syntheses. Reagents will often come with a quoted purity in either a “metals basis” or “REM basis.” A metal’s basis purity indicates that the quoted number (e.g., 99.999%) does not take into account non-metal components, such as O, C, and N, which might be present in the form of oxides, carbides, nitrides, and organics. Quoted REM purities are only referenced to the total fraction of rare-earth metals, and will not reflect impurities of other elemental species. Consider cleaning reagents in-house, even when quoted purities are high. In cases when the oxidation is primarily on the surface of the material (e.g., K, Na, Ba, Sr, and transition metal powders such as Fe, Co, and Ni), the oxide layer can sometimes be removed in an inert atmosphere. For example, the surface of Ca or Sr pieces can be polished with a metal wire brush, and the surfaces of Na cubes can be cut, exposing a fresh, shiny surface. Transition metal oxides can be reduced by annealing in a hydrogen gas flow. This process has its limitations because main group metals cannot be reduced in this way (e.g., Ca, Al, and Mg). There are many clever ways of purifying or cleaning sub-par reagents in the literature, and the experimentalist is encouraged to seek additional methods by contacting the authors. B. Basic powder synthesis routes
A. General guidelines 1. Reporting and literature
In many modern journals, the fine details of the synthesis are downplayed and may not be amenable to repetition by a researcher not “skilled in the art.” Consequently, the researcher is encouraged to effectively communicate their own synthesis methods in journal communications via “Electronic Supplementary Information,” which is a prime location to give a detailed breakdown of the synthetic procedures. Remember to always include • • • • • •
The sources and purities of the starting materials, along with any treatments used to clean them. The chemical nature and volume of reaction vessel and/or crucible. Reaction temperature, heating and cooling rates, annealing temperatures and times. Any post-synthetic treatments (e.g., washing, drying, polishing, storage, etc.). Any special techniques you may have used that were key in the process. Air and moisture sensitivity of the sample.
Certain compounds may not only degrade or decompose under exposure to ambient atmosphere but also release hazardous volatile products, such as H2S. In addition, we encourage reaching out to authors and asking for more detailed information as they are usually more than willing to discuss. For the work reported decades ago, doctoral and master theses may often contain additional unpublished information.
Synthesizing dense, polycrystalline samples for electronic and thermal transport measurements is typically, at minimum, a two-step process involving: (1) preparation of feedstock (typically single phase) in the form of a powder and (2) consolidation of the powder into a dense sample suitable for transport measurements. The first synthesis can be classified by the type of chemical reaction used: addition, decomposition, or substitution. Within thermoelectrics, synthesis by addition is most common, while synthesis by decomposition or substitution reactions currently play a role only in specialized instances. 1. Ball milling
Ball milling is one of the most powerful tools for producing bulk thermoelectric powders in a time-efficient and industrially scalable process. A vast number of published thermoelectric compounds utilize ball milling as the key synthesis step.16 Commercial ball mills are available from a host of companies (SPEX Sample Prep, RETSCH, etc.). While the type of mill may vary by rotation type (high energy, planetary, etc.), the operating principle remains the same—using a series of small balls contained in a vial to crush, react, and evenly disperse the sample. The ball and vial type varies widely, and both metals (stainless-steel, hardened steel) and ceramics (tungsten carbide and zirconia) are available. Steel and stainless-steel vials are the economical option, though they can introduce contaminants (e.g., Fe, Ni, and Cr) into the samples.16 Ceramic vial sets are generally more efficient at grinding and introduce less contaminants, but are often more expensive and more prone to breakage.
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In many materials, ball milling alone can result in near phase-pure powder of the target compound (a process known as mechanical alloying). However, ball milling introduces an enormous degree of disorder and strain, leading to an amorphous component in most samples.17 To overcome this, ball milling is followed by one of a variety of thermal treatments to obtain a dense sample for transport measurements which leads to full crystallization and grain coarsening. For synthesis of more complex materials (e.g., Zintl compounds), thermal treatment is often required to obtain the desired phase, as ball-milling alone yields only a homogeneous amorphous mixture, as shown in Fig. 5 for Ca5Ga2Sb6. It is therefore typical to combine ball milling and reactive hot pressing (to be discussed later) to both transform and consolidate samples simultaneously. Although the principle behind ball milling is quite simple, considering the following factors can dramatically improve milling success rates: •
Mechanical properties of reagents. Most non-metals and some metals (e.g., Mn) are brittle and easily pulverized, but large chunks of malleable metals (e.g., Sn, Zn, and Ag) may take an inordinate amount of time to react fully in a ball mill. If unreacted metals remain after milling
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•
•
•
•
•
(i.e., metallic particles in an otherwise matte powder), try reducing the initial size of the malleable reagents via cutting, filing, or purchasing powdered reagents. Weigh reactants. This should be completed before and after every milling step to determine the fraction lost to the vial. It is possible for a large fraction of the initial load to be lost to the vial/balls (up to 25%). Scraping the vial walls midway through the milling process to dislodge and re-mix powder that is cold-welded to the vial may alleviate this issue. Consider reagent order. Unreactive (Cu, Zn, Sn, Ag, and Fe) or very soft (K, Na, Ba, and Yb) metals may accumulate inside the vial or fail to mix, especially when the metal/anion ratio is large. One solution is to mill the anion alone (e.g., Sb shot) for a few minutes and then add the metallic reagent (e.g., Cu granules). The anion will already have coated the vial surfaces, reducing metal adhesion with the vial walls. Further, it is advisable to add any dopants as early in the process as possible to encourage homogeneous mixing. Consider reactions with the vial. Relatively corrosive anions (Te, Se, Sb, and As) may slowly react with the vial/balls and contaminate the sample. Use vials that are chemically resistant to attack from the sample, mill for the minimum time necessary, and monitor contamination with a spectroscopic technique such as electronic dispersive spectroscopy. Leaking of air. Air present in the vial can lead to sample oxidation. Careful vial maintenance (e.g., threads, o-rings), and cap tightness can reduce most oxidation concerns. Pre-reaction of reagents. Elemental reagents that are difficult to handle due to air sensitivity, hardness, or stickiness, can be replaced with pre-reacted compounds (e.g., use of GaSb instead of Ga þ Sb). Another possibility is to use compounds which will generate the desired element upon decomposition, such as hydrides.
2. Case study: Incremental ball-milling synthesis of KAlSb4 and KGaSb4
FIG. 5. (a) Ball milling Ca5Ga2Sb6 directly from the elements for up to four hours leads to an amorphous powder. Only 10 min of applied pressure and heat during spark plasma sintering was necessary to obtain a dense, phase pure sample. (b) In contrast, just one hour of ball milling Bi2Se3 yields phase pure powder.
Synthesis of the n-type Zintl compounds KAlSb4 and KGaSb4 highlight the precautions described above. For ballmill synthesis of KAlSb4 or KGaSb4, we must first consider the reactivity and mechanical properties of the raw elemental precursors. Potassium is soft, sticky, and will easily smear, but is also incredibly reactive. It has a tendency to become impacted at the corners and edges of the vial, where it fails to mix homogenously. The same can be said for aluminum, but aluminum is even more problematic as its reactivity is relatively low. One solution is to add the Al and Sb to the vial alone. With a 1:4 ratio of Al:Sb, the reaction proceeds fairly quickly ( 75 lV=K: mS
2=3 h2 3nH jaj pffiffiffi exp 2 0:17 ; (50) 2kB T 16 p ðkB =eÞ
and, when jaj < 75 lV=K mS
2=3 3h2 jaj 3nH 2 : 8p kB T ðkB =eÞ p
(51)
Here, nH is the Hall carrier concentration. A heavier mS gives higher jaj for a given nH (alternatively, a higher nH for a given jaj). A plot of jaj with respect to nH is called a “Pisarenko plot,” by which one can determine mS from a set of data points. The inertial mass mI is not easily separable from the relaxation time, and we thus keep it in the form l0 ¼ es0 =mI . We can nevertheless understand how band structure impacts l0 using the deformation potential model120 to expand the relaxation time peh4 Cl : l0 ¼ pffiffiffi 3=2 2mI mb ðkB TÞ3=2 N2
(52)
Here, N is the deformation potential and Cl is the longitudinal elastic constant. The combination of Eqs. (50)–52 helps one understand what type of band is good for thermoelectrics lw / ðmS =mb Þ3=2 =mI :
(53)
Suppose that symmetry provides NV multiple bands with the same dispersion (multi-valley degeneracy). Then, l0 of a single band is identical to the l0 of all the multiple bands together. On the other hand, mS is larger than that of a single 2=3 band by NV because the density-of-states [and thus nH in Eqs. (50) and (51)] is larger by a factor of NV. Overall, lw (and, thus rE0 and B) scales with NV =mI lw / NV =mI :
(54)
Therefore, multi-valleys and lighter bands (small mI ) are advantageous for thermoelectrics. In general, when multiple bands contribute to transport, they are not necessarily identical or aligned; nevertheless, the trend of mS and lw both increasing simultaneously with advantageous band complexity remains similar,121 allowing one to relate transport measurements to understandings of the electronic structure. Therefore, it is best to keep track of both mS and lw when analyzing transport data. An advanced example would be a case when two conduction bands have their band edges offset by a small amount, on the order of a few kBT. When EF is below the lower band, the upper band would not contribute significantly to transport. With doping, once EF moves within a few kBT to the edge of the upper band, both bands would start contributing. In experimental characterization, one would observe mS and lw both increasing at a threshold of g, where the threshold indicates the extent the bands are offset. Such a signature would be a strong motivation to further investigate the band structure using more specific methods such as optical absorption. The usefulness of mS , or any m , in general, comes from the fact that it is a convenient metric to characterize an electronic structure and so used to characterize diverse measurements such as the electronic specific heat, plasma frequency, as well as Seebeck coefficient. Mathematically, the procedure could be understood as a change of variables. While E, k, rE, s, or density-of-states change dramatically with experimental variables such as doping or temperature, the various m ’s as defined though different measurements (Seebeck, specific heat, plasma frequency, etc.) remain relatively constant and thus m ’s are typically reported as results of such measurements. Just as m ’s are reported rather than specific values of electronic specific heat (e.g., heavy fermion metals) or optical absorption (plasma frequency measurements), it would be more useful to report mS than specific values of a in many insulators and semiconductors. All of these effective masses are expected to change somewhat with doping, temperature and even alloying and structural modification. In fact, observing and quantitatively characterizing how m changes might be the best way to identify changes in parabolicity122 or multiple band effects.123,124 In this way, the
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effective m approach does not simply assume, or impose an approximation of, a single parabolic band, but rather provides a helpful means to characterize data and identify deviations from single or parabolic electronic structures. VII. MEASURING PHONON BEHAVIOR IN CRYSTALLINE MATERIALS
In any type of engine, energy that is expended without performing useful work reduces its efficiency. In the context of a thermoelectric generator, heat that is conducted through the material from the hot to the cold side is wasted energy. The thermal conductivity, j, of thermoelectric materials should thus be as low as possible to maximize efficiency. Although any type of mobile species can contribute to j, in thermoelectric materials, electrons and phonons are the primary carriers. Since the electronic thermal conductivity, je, is tied to the electrical conductivity [Eq. (45)], emphasis is primarily placed on minimizing the lattice contribution, jL. The lattice thermal conductivity is a function of the phonon scattering rates, group velocities, and heat capacity, summed over all phonon modes in the crystal.125,126 To control jL, it is necessary to understand the underlying lattice dynamics (i.e., the forces experienced by atoms in a crystal and the resulting atomic motion). Experimental characterization of phonons are invaluable to better understand thermal transport and for benchmarking first-principles simulations. Currently, computations of phonon frequencies and group velocities using density functional theory (DFT) can often yield results within 15% of experimental values, for systems that are not too strongly anharmonic or strongly correlated. This level of agreement, reaching phonon energies of a few meV, is a towering achievement for DFT. Yet, the phonon group velocities, which scale with the frequencies, enter in jL at the second power, meaning that small deviations can lead to substantial errors in thermal transport predictions. Thus, direct evaluation of predictions against measured phonon dispersions and scattering rates remains a priority to establish the validity of first-principles thermal transport models. This section begins by introducing the fundamentals of lattice dynamics and then discusses two important techniques for characterizing the phonon properties that control jL: (1) ultrasonic determination of sound velocity and (2) inelastic neutron scattering measurements to measure the phonon dispersion. The first method is accessible and can be used for almost any sample type and geometry. It can be carried out in minutes, and requires little preparation, yet provides critical information about the velocity of long wavelength phonons and the elastic constants of the material. Readers who are interested primarily in ultrasonic measurements can safely skip ahead to Secs. VII B and VII C of this chapter. Inelastic neutron scattering (INS), in contrast, requires careful planning, a rigorous understanding of lattice dynamics, and necessitates instrument time at a national neutron source. However, successful measurements and analysis can yield the complete phonon dispersion of a crystal, from which one can extract the group velocity and even the anharmonicity of the entire phonon spectrum. We discuss these topics in the context of crystalline solids; however, many of the principles
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herein can be extended to amorphous materials such as polymers and glasses. A. Basics of lattice dynamics
The term “lattice dynamics” refers to the forces and resulting motion experienced by atoms in a solid. Introductory treatments found in solid-state physics textbooks127,128 approach the subject using a 1D chain of atoms connected by harmonic springs. The equations used in the following sections are generalized for 3D. Figure 18 provides a guide to the indexing conventions used to specify atoms and phonon modes in 3-dimensional systems, as keeping track of these can quickly become onerous for newcomers. For detailed accounts on the theory of lattice dynamics, the reader is encouraged to consult the seminal books of Born and Huang129 or Maradudin,130 as well as the more recent books by Br€uesch131 and Dove.132 1. Interatomic force-constants and dynamical matrix
Lattice dynamics modeling generally starts with the use of the Born-Oppenheimer approximation, decoupling fastadapting electrons from the ionic motions, to define an effective external potential for the nuclei Vðfrl;j ðtÞgÞ, where rl,j(t) is the instantaneous position of atomic nucleus of index j in unit cell l of the crystal. It is assumed that the atoms oscillate around their equilibrium positions with amplitudes u that are small compared to the interatomic separations 0 þ ul;j ðtÞ: rl;j ðtÞ ¼ rl;j
(55)
This assumption of small ionic displacements is generally safe at low temperatures, but tends to break down close to the melting point, near superionic transitions, or in the vicinity of some structural phase transitions with large lattice reorganizations. The potential V can then be Taylor expanded about the equilibrium configuration 1 X ð2Þ U ðlja; l0 j0 a0 Þ ulja ul0 j0 a0 V ¼ V0 þ 2! lja;l0 j0 a0 þ
X 1 Uð3Þ ðlja; l0 j0 a0 Þ ulja ul0 j0 a0 ul00 j00 a00 3! lja;l0 j0 a0 ;l00 j00 a00
þ ;
(56)
where a denotes the Cartesian components of ul,j. The linear first-order term vanishes as a result of the condition of mechanical stability of the lattice (zero forces) in the reference equilibrium configuration. This expression defines the harmonic (2nd order) and higher-order anharmonic interatomic force constant tensors, U(2), U(3), etc. We begin by considering the harmonic approximation which truncates the potential energy at the second order129 (anharmonic effects will be discussed later). The harmonic force-constant tensor @2V ð2Þ 0 0 0 ; (57) U ðlja; l j a Þ ¼ @ulja @ul0 j0 a0 0 can be thought of as a collection of spring-like stiffnesses of interatomic bonds between pairs of atoms (lj and l0 j0 ).
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FIG. 18. The indexing conventions used to describe the lattice dynamics in crystals can easily discourage newcomers to this field from a deep understanding of the theory. To clarify the indices used in Eqs. (55)–(63), we show (a) real-space structure with atomic position, unit cell, displacement, and coordinate indices labeled; (b) a hypothetical phonon dispersion for a material with two atoms per unit cell (r ¼ 2); and (c) the simplified Debye model dispersion for the same system.
These “springs” define the dynamics of nuclei of masses mlj, which in turn determine the intrinsic lattice thermal conductivity of a material. The magnitude of bond stiffnesses tends to decrease with increasing distance between atoms. With modeling force constants, however, it is still generally necessary to include a large number of bonds to faithfully capture the lattice dynamics, and convergence studies are used to determine an acceptable cutoff distance. The symmetries of the crystal lattice impose constraints on U(n), limiting the number of its independent and non-zero elements.129,130 The atomic motions are obtained by solving Newton’s equations of motion in a classical framework, although U(n) encapsulates the chemical information about interatomic bonding from the quantum mechanical electron states. To solve, X Uð2Þ ðlj; l0 j0 Þ ul0 j0 (58) mj u€lj ¼ l0 j 0
in matrix notation, one assumes periodic boundary conditions for the crystal and seeks traveling wave solutions to describe the linear oscillatory motion of atom j in the unit cell 1=2 u6 ej ðj; qÞeiðqrl 6xq;j tÞ ; lj ¼ AðNmj Þ
(59)
where xq,j is the vibration frequency for the jth phonon mode of wavevector q, and ej(j, q) are the phonon polarization vectors. Inserting in Eq. (58) leads to the introduction of the dynamical matrix D(q), and the reformulation as an eigenvalue problem for polarization vectors and associated frequencies X Dðjj0 ; qÞ ¼ ðmj m0j Þ1=2 Uð2Þ ð0j; l0 j0 Þ eiqðrl0 r0 Þ ; (60) l0
DðqÞ eðq; jÞ ¼ x2q;j eðq; jÞ ;
(61)
where e(q, j) collects all three vectors ej(q, j) for all atoms in the unit cell, and similarly D is in block-form collecting all Dðjj0 Þ. Thus, the dimension of D(q) is 3r 3r if there are r
atoms in the unit cell (1 j r). The knowledge of U(2) from first-principles or phenomenological modeling thus allows calculation of phonon frequencies at arbitrary wavevectors q by diagonalizing D(q), which also yields the patterns of atomic motions in real space (polarizations). 2. Phonon density of states and dispersions
Phonon dispersion plots are used to illustrate the relationship between the frequency, xq,j, and wavenumber, q, for the normal modes of the crystal. The number of individual dispersion branches depends on the number of atoms in the unit cell. For a unit cell with r atoms, there will be 3r solutions (phonon modes) for a given q, so that x ¼ xj(q), j ¼ 1…3r. Figure 18(b) shows the calculated dispersion for a material with r ¼ 2, which has 3r ¼ 6 branches in the phonon dispersion. The three modes beginning at x ¼ 0 are always referred to as acoustic modes, and all higher modes are called optical modes. Knowledge of dispersions is particularly important to evaluate the phonon group velocities, which largely control the thermal conductivity. The group velocity is given by the slope of the dispersion, (or gradient, in 3D): vg(q, j) ¼ rqxj(q). The phonon dispersions (and polarizations) are identical in every Brillouin zone of the reciprocal lattice. If we write Q, an arbitrary wave vector in reciprocal space, and s the nearest reciprocal lattice vector (defining which BZ contains Q), then the reduced wave vector q is defined as Q ¼ q þ s. Phonons are identical around any lattice vector s. However, we stress that neutron (or X-ray) scattering measurements do not obey this periodicity of the crystal’s reciprocal space (see Sec. VII E below). From the phonon energies Eq;j ¼ hxq;j evaluated on a dense set of q points, we can obtain the frequency distribution (the phonon density of states), which is useful to calculate the thermodynamic properties. The phonon density of states, g(E), is defined so that g(E)DE is the fraction of normal modes with vibrational frequencies in the interval between E and E þ DE
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gðEÞ ¼
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1 X dðE Eq;j Þ ; 3N q;j
(62)
where d is the Dirac delta function. This expression is normalized to a unit integral, but other normalizations are sometimes used. One can also define a partial phonon density of states, gj, which is defined as the vibrational contribution of atom j to the total phonon density of states, g(E), as X jej ðq; jÞj2 dðE Eq;j Þ ; (63) gj ðEÞ / q;j
where Eq,j and eq,j are the phonon energies and eigenvectors.131 B. Debye model
If no detailed lattice dynamics are available (whether from experiment or computation), which is typically the case when studying new materials, the Debye model provides a useful starting point. The Debye model assumes a phonon dispersion consisting of three linear acoustic modes: one with longitudinal and two with transverse polarization [see Fig. 18(c)]. The longitudinal and transverse sound velocities, long and trans, respectively, are used to approximate the slopes of these modes. This model can be further simplified by employing a single effective sound velocity and by assuming that the sound velocity is isotropic. Thus, we obtain the simplified phonon dispersion illustrated in Fig. 18(c), which bears only a faint resemblance to the real phonon dispersion. The maximum phonon frequency in the Debye model is referred to as the Debye frequency, given by xD ¼
6p2 V
1=3 eff ;
(64)
where V is the average volume per atom, and the effective velocity in this case is given by 3 1 2 ¼ 3 þ 3 : 3 eff long trans
(65)
The corresponding Debye temperature, TD, and Debye energy, ED, are given by ED ¼ kB TD ¼ hxD . The Debye phonon density of states is given by: gD ðEÞ ¼ 3 E2 =E3D for E ED and gD (E) ¼ 0 for E > ED. Often, the speed of sound reasonably approximates the group velocity of the acoustic phonons, but drastically overestimates the velocity of optical phonons. The relative contribution of the acoustic branch diminishes with increasing atoms per unit cell, r, since the number of acoustic modes remains constant (there are always three of them) and the number of optical modes increases as 3(r–1). In materials with large unit cells, the Debye model is thus sometimes combined with the Einstein model (phonon branch of constant energy, independent of q, corresponding to independent oscillators), to represent the optical modes. At the limit of r ¼ 1, one recovers Cahill’s approximation for the minimum lattice thermal conductivity.133 This approach, as well as other strategies for modeling jL within a Debye-Callaway
framework are described in the following review by Toberer et al.134 C. Characterization of the speed of sound
To model lattice thermal conductivity within a Debye model, knowledge of the transverse and longitudinal sound velocity, at a minimum, is needed. Indeed, the sound velocities play the same role in the Debye model that the effective mass plays in the single parabolic band model (i.e., effective mass model). The measured sound velocity is used as a proxy for all phonons in the solid, just as the measured effective mass is used to approximate the behavior of all electrons or holes. Measurements of the sound velocity and measurements of the elastic constants are synonymous, as long as the density, d, of the material is known. In an isotropic material— i.e., a polycrystalline sample with randomly oriented grains—two independent elastic constants such as the shear modulus, G, and either the bulk modulus, B, or Young’s modulus, K, are sufficient to fully describe the elastic tensor of a material. These can be calculated from the transverse and longitudinal sound velocities using the following equations: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u uK þ 4 G t 3 ; (66) long ¼ d rffiffiffiffi G trans ¼ : (67) d It is important to remember that the elastic tensor is a rank 4 tensor.135 Even in a cubic material, it is not fully isotropic. In single crystals or textured polycrystalline samples, the sound velocity depends on both the orientation of the sample and the polarization of the wave (longitudinal or transverse), and several independent elastic constants are needed (up to 21 in a triclinic crystal structures) to completely describe the elastic tensor. Reference 136 provides an excellent summary of the relationships between speed of sound and the elastic tensor elements in lower symmetry materials. 1. Ultrasonic methods
Ultrasonic measurements provide an accurate and nondestructive approach to measure the elastic moduli and speed of sound, particularly for inorganic materials, which are often small and brittle. Ultrasonic measurements fall into one of two categories: the first involves direct measurement of the speed of sound by observing the transit time of an acoustic wave—referred to as a time-of-flight measurement. The second involves measurement of the vibrational frequencies of a free body. The following is a discussion of the experimental setup and the advantages and disadvantages of these two approaches. a. Pulse-echo ultrasound. This is a conceptually simple method that falls into the time-of-flight category. It is used across many industries for non-destructive materials
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testing137–139 and can also be found in undergraduate teaching labs. In pulse-echo ultrasound measurements, a piezoelectric transducer is coupled to a sample prepared with two parallel faces. During the measurement, a short wave packet, or pulse, is emitted by the transducer and the transit time for the pulse to reflect from the opposite face and arrive at the original transducer is recorded. Alternatively, two transducers (a drive and a pickup) can be placed on opposite sides of the sample [Fig. 19(a)], in which case the transit time from the drive to the pickup transducer is measured. The wave velocity is readily obtained by dividing the transit time by the thickness of the sample. The wave frequency generated by the transducers is typically on the order of 107 Hz. This is extremely small compared to the Debye frequency of most materials (on the order of 1012 Hz), meaning that only the phonon velocity for the acoustic mode very near to the center of the Brillouin zone (q ¼ 0) is measured. To obtain the velocity of both the longitudinal and transverse acoustic modes, two separate measurements must be made; one with a transducer capable of producing a longitudinal (compression) wave and another producing a transverse (shear) wave. Pulse-echo ultrasound measurements are flexible regarding sample geometry, the analysis of the data is straightforward, and the necessary equipment is widely available. However, a degree of measurement uncertainty stems from the inherent width of the pulse, which can make the beginning and end of the pulse difficult to distinguish.140 As the sample size decreases, uncertainty caused by the spreading of the pulse increases. Further, a coupling medium is needed to facilitate the transmission of vibrations between the transducer and the sample—typically a moderately viscous, nonreactive gel or paste. Honey is an effective and commonly used couplant, but as it contains water, it is not appropriate
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for acutely moisture sensitive samples,141 for measurements in an inert glove box, or at extreme temperatures.142 b. Laser ultrasound. In laser ultrasound measurements, a surface acoustic wave is generated by a short laser pulse illuminating and rapidly heating a local area on the surface of the sample.143,144 The laser-induced wave is then detected either by a transducer with a piezoelectric element or by another laser. Laser ultrasound has several advantages, the most important being that no contact to the sample is needed. The technique can be applied remotely via a window or optical fiber to a sample in an oven or vacuum chamber.145 Further, laser ultrasound is insensitive to sample geometry, making it valuable for irregularly shaped samples and thin films.144 The primary challenge in laser ultrasonic measurements is obtaining a sufficiently high signal to noise ratio. The theory behind this technique and a discussion of further advantages and disadvantages can be found in a review by Scruby et al.146 c. Resonant ultrasound spectroscopy (RUS). This method makes use of the fact that the resonant frequencies (i.e., standing waves) of an object are a function of the object’s geometry, density and elastic constants.140,147,148 RUS measurements are carried out using a carefully cut regularly shaped sample, often a parallelepiped, which is mounted with two opposing corners lightly held between two piezoelectric transducers. An alternative experimental configuration, preferable for cylindrical samples, involves balancing the sample on top of three transducers, one of which is the drive transducer and two of which are pickup transducers [Fig. 19(b)]. The driving transducer sweeps across a range of frequencies (typically from 0 to 1000 kHz) over a time
FIG. 19. (a) In pulse-echo ultrasound measurements, the time required for an acoustic wave to travel across the sample is measured. Different transducers are required to generate the longitudinal or transverse waves. A single transducer can be used to emit (drive) and read (pickup) the signal, or separate transducers can be employed. (b) In resonant ultrasound, the drive transducer emits acoustic waves over a frequency spectrum. Peaks correspond to the resonant frequencies of the sample, with predicted peak locations shown as the black circles. The inset shows the shift of the peaks towards lower frequencies with increasing temperature, corresponding to the softening of the sample.
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period of several minutes while the pickup transducer records and amplifies the sample response. When the drive frequency is equal to one of the sample’s resonant frequencies (the eigenfrequencies), the pickup transducer records a sharp peak, leading to a spectrum like that shown in Fig. 19(b). Each eigenfrequency can be a function of either the longitudinal or shear modulus, or both. The analysis of RUS data is therefore more involved than that of pulse-echo ultrasound measurements, yet when carefully carried out, yields very high accuracies. Experimentally, one measures the geometry, density and the eigenfrequencies of the sample. We begin our analysis by guessing the elastic constants and solving the “forward problem”—in which the eigenfrequencies belonging to a given geometry and set of elastic constants are calculated. The predicted frequencies are compared to the observed values, and the elastic constants are iteratively adjusted until a final solution is obtained.140,149 Assuming that the peaks are correctly identified, measurement uncertainty stems from differences between the predicted and observed peak positions, and any error in measurement of the geometry. Although in principle the forward problem can be solved for arbitrary sample geometries, in practice, spheres, cylinders, and parallelepipeds are used. Compared with the above “transit time” measurement methods, resonant ultrasound measurements offer a critical advantage. The complete elastic tensor of the sample can be obtained from a single measurement, because the resonant frequencies are a property of the entire sample. For isotropic samples, this means that the shear and longitudinal velocities can be determined simultaneously. For single crystal samples, RUS is even more powerful, since the complete elastic tensor (up to 21 independent components) can, in principle, be obtained from a single measurement.150 The main disadvantage of RUS is the need for samples with well-defined geometry and a reasonable initial guess for the elastic constants, which can be inconvenient when studying new materials.
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temperature-independent. This assumption, however, is contradicted by experimental evidence in nearly every class of solids. High-temperature elastic moduli are available for only a small percentage of materials, but the results tell a consistent story of decreasing elastic moduli and sound velocities with increasing temperatures. This trend is illustrated by Fig. 20, which shows the results of RUS measurements carried out at high temperature. High-temperature measurements are possible using any of the above methods,158–161 however, RUS has been the most widely utilized method to study thermoelectric materials. Here, we show the change in Young’s modulus as a function of temperature for Si0.8Ge0.2,153 Mg2Si0.4Sn0.6,152 MnSi1.85,152 PbTe,151 and SnTe.154 There is a strong correlation between the rate of softening and the degree of anharmonicity in the chemical bonding, as quantified by the Gr€ueneisen parameter, c (c ¼ 1.06, 1.17, 2.1, 1.45 in Si,162 Mg2Si,163 PbTe162 and SnTe,134 respectively). This observation was established empirically by Wachtman in 1959.164 We discuss the connection between bond softening and anharmonicity further below. D. Anharmonicity and the quasiharmonic approximation
In the theory of lattice dynamics presented in Sec. VII A of this chapter, the interatomic potential was truncated at the quadratic order. This allows us to identify decoupled vibration modes of the crystal, or phonons, which depend only on the (harmonic) force-constants, U(2). The phonon frequencies and velocities so-derived do neither depend explicitly on the amplitudes of the vibrations nor on the temperature. The harmonic approximation stems from the proposition that the displacements of the nuclei away from their equilibrium positions are small and that higher order terms in the expansion of the potential are negligible. Although these
2. Considerations for porous samples
It is important to consider that the bulk elastic properties, and thus the speed of sound, depend strongly on the porosity and microstructure of the sample.138,155,156 Pores and grain boundaries both have the effect of decreasing the overall elastic stiffness of the sample. There have been numerous proposed relations between porosity and elastic moduli.156 In general, sound velocity will decrease approximately linearly with increasing porosity when the overall porosity is low. For very porous samples however (> 10%), the speed of sound will depend on the details of the pore structure.157 If corrections are used, however, it is good practice to also report the uncorrected values along with the porosity of the samples. When comparing to literature values, you should assume that no correction has been made unless otherwise stated. 3. Temperature-dependent speed of sound measurements
In the thermoelectric literature, the speed of sound and elastic constants are almost always treated as if they were
FIG. 20. Elastic moduli of several thermoelectric materials have been measured using resonant ultrasound (PbTe,151 Mg2Si0.4Sn0.6,152 MnSi1.85,152 Si0.8Ge0.2,153 SnTe154) The Young’s modulus, Y, is shown normalized by the room temperature value, Yo. This data highlights the significant softening of bonds with increasing temperature, an effect that appears to correlate strongly with Gr€ uneisen parameter.
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conditions are verified in many solids at low temperature, they tend to become less valid as the amplitude of the displacements increases at high temperatures. If the harmonic model were valid, there would be (1) no thermal expansion, (2) no phonon-phonon scattering processes, (3) force-constants and elastic constants would not vary with temperature or pressure and (4) the heat capacities at constant pressure and constant volume would be equal. Clearly, these consequences of the harmonic approximation are not physically satisfying and one needs to extend the harmonic phonon picture to better describe the thermal behavior of materials. The physical underpinnings of anharmonic effects are the shifts in phonon frequencies with volume, V, or temperature, T. To capture the dependence of the phonon frequencies on volume for each phonon mode, (q, j), the mode Gr€uneisen parameters, cq,j, are introduced cq;j ðV; TÞ ¼
@ ln xq;j ðV; TÞ : @ ln V
(68)
Often, the Gr€uneisen parameter is quite weakly dependent on temperature, except in cases of strong intrinsic anharmonicity (T dependence of xq,j at fixed V). This assumption is commonly referred as the quasi-harmonic approximation (QHA), which amounts to considering that phonons are harmonic (infinitely long-lived), albeit with frequencies depending on volume. The volume dependence of cq,j is often also weak at ambient pressure, but would have to be included if one considers large volume changes in highpressure studies. Thus, cq,j at atmospheric pressure is commonly approximated as a constant for each mode. Experimentally, it is challenging to access cq,j for every mode. For this reason, an average quantity, or thermodynamic Gr€ uneisen parameter is often used instead cG ðV; TÞ ¼
3 aVBT 3 aVBS ¼ ; CV CP
(69)
where a is the coefficient of linear thermal expansion, V is the specific volume, BT and BS are the isothermal and isentropic bulk moduli, respectively, and CV and CP are the heat capacity at constant volume or constant pressure, respectively. The difference between heat capacity at constant volume and at constant pressure is often a source of confusion. In general, heat capacity is measured at constant (ambient) pressure. Measurements of CP are made either at low temperature using a physical property measurement system (PPMS), or at high temperature using differential scanning calorimetry (DSC). As illustrated in Fig. 21, the experimental heat capacity is mainly accounted for by three components, the harmonic phonon contribution, Cph,har, the dilational (quasiharmonic) phonon contribution, CD, and the electronic contribution, Cel.165,166 Additional contributions from intrinsic anharmonicity (temperature induced shifts in phonon frequencies at constant volume) or electron-phonon coupling are typically much smaller. The harmonic phonon contribution is the primary contribution and can be expressed directly from the phonon DOS as
Cph;har ðTÞ ¼ ¼
ð Emax
char ðE;TÞgðEÞdE 0 ð Emax 2 0
E kb T
expðE=kb TÞ
½expðE=kb TÞ 12
gðEÞdE; (70)
where char (E, T) is the harmonic heat capacity of a single mode of energy E. One must pay attention to the normalization of g(E) in this expression: if g is normalized to a unit integral, then the result for Cph, har (T) is in units of kB/atom per degree of freedom, which tends to 1 at high T. To recover the Dulong-Petit value of 3kB/atom, one must take into account the three degrees of freedom per atom. Because phonons do not depend on P or V in the harmonic approximation, Cph,har is the same at either constant P or V. Of course, the real heat capacities at constant P or V are not equal (CP 6¼ CV). This fact actually underlies the dependence of phonon frequencies on volume, which also causes thermal expansion of the lattice. The difference between CP and CV can be expressed both in terms of the thermodynamic Gr€uneisen parameter introduced earlier, and standard thermodynamic quantities, as165,166 CP CV ¼ 3 acG CV T ¼ 3 a2 VBT T:
(71)
A convenient estimate for the experimental heat capacity at constant pressure is given by the sum of Eqs. (70) and (71). At high temperatures well above the Debye temperature, Eq. (70) can be substituted with the Dulong-Petit value. We must also stress that the concept of volumetric heat capacity (heat capacity per unit volume) is not the same as heat capacity at constant volume, CV, a common misconception. In particular, one should not attempt to relate CV and CP through only the volume density, d, as CV 6¼ dCP! This fact is important to obtain the thermal conductivity from thermal diffusivity measurements performed at constant P, but often expressed in volumetric units (instead of normalizing by unit mass or number of atoms).
FIG. 21. Example of analysis of heat capacity for La2.86Te4 (from Ref. 167) (a) La3Te4 CP curves, measured at low (“PPMS”) and high temperatures (“DSC”), and calculated CP components from harmonic phonons (Cph, H), electrons (Cel), and dilation of the lattice (CD). (b) Measured thermal expansion dL/L0 of La2.86Te4 (markers) and linear thermal expansion coefficient, a.
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Following the quasi-harmonic approximation, one can also estimate the expected temperature dependence of phonon frequencies (and thus phonon velocities) using the experimental volume thermal expansion.168 Assuming that cG is constant over the range of V and T measured, a crude approximation for the T-dependence of the average phonon frequency and speed of sound can be obtained as Dhxi= hxi ’ cG DV=V. This is relatively straightforward in the case of cubic systems, but much less so for an anisotropic material, in which thermal expansion affects different crystal directions unequally. In the latter case, one needs to consider anisotropic strain dependences (instead of overall volume) for the Gr€ uneisen parameter. It is ultimately more rigorous to numerically build and minimize a free energy function.169 Also, it is important to keep in mind that a single Gr€uneisen parameter approximation (i.e., employing cG) is a severe one, as it is known from experiments and DFT simulations that in all materials, the Gr€uneisen parameters show large variations between various phonon modes (q, j). E. Neutron scattering measurements of phonons
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scattering channel (even in a single crystal). Similarly, the inelastic component of the scattering cross-section is a smoothly varying function of Q, which does not depend on the direction of the momentum transfer vector Q, but only on its magnitude Q ¼ jQj. In a polycrystalline (powder) sample of a material whose cross-section does contain a coherent component, the information about Q is also limited to the magnitude Q ¼ jQj (we do not consider texture effects in powders). While in this case of a coherently scattering powder, Bragg diffraction rings are observed (similar to conventional powder x-ray diffraction), if the experiment is performed such that many Brillouin zones are averaged over a given Q value, the scattering cross-section can be approximated by the expression for an incoherent sample. This approximation is commonly performed to extract the phonon density of states DOS from powder measurements [as shown in Fig. 22 for SnSe], since it is convenient to relate the incoherent scattering function to the phonon DOS, as follows:177 SðQ; EÞjincoh /
X i
Presently, the most powerful technique for the study of phonons is the inelastic scattering of thermal neutrons. The technique directly determines the dispersion relation, that is, the relationship between the frequency x and propagation vector q of the phonons. The triple-axis neutron spectrometry (TAS) developed by Brockhouse has historically been the work-horse technique for dispersion measurements. More recently, intense pulsed neutron sources have enabled timeof-flight spectrometry (TOF) to map single-crystal excitations across large volumes in reciprocal space, offering great complementarity with the more focused TAS measurements.171–174 Additionally, inelastic X-ray scattering, based on principles similar to the triple-axis neutron technique, now enables the study of considerably smaller crystals.169 Before pursuing neutron scattering measurements, it is always recommended to collect as much information as possible using accessible and inexpensive macroscopic measurements. For example, in addition to the speed of sound measurements described above, the phonon contribution to the heat capacity can be isolated from other contributions, providing important information on average phonon energy, and direct measurement of the frequencies of long-wavelength optical phonons can be obtained by Raman scattering and infrared spectroscopy.175,176 These experiments provide essential information for solids and should be performed, if possible, before one undertakes a time-consuming and expensive detailed neutron scattering study. 1. Powder or incoherently scattering samples
For single-phonon scattering, the inelastic neutron scattering cross-section is proportional to the scattering function, S(Q, E). In the case of an incoherently scattering sample (such as elemental vanadium), the phase information of the neutron waves at each scattering center (atomic nucleus) is lost, and there is no constructive/destructive interference possible over the entire crystal lattice or grains in the sample. As a result, no sharp diffraction peaks occur in the elastic
ri
h2 Q2 2Wi gi ðEÞ e ½nT ðEÞ þ 1 ; 2mi E
(72)
where Q and E are the momentum and energy transfer to the sample, ri, mi, and exp ð2Wi Þ are the neutron scattering cross-section, mass, and Debye-Waller factor for atom i, respectively, and nT(E) is the Bose occupation factor.177 It is
FIG. 22. Phonon DOS of SnSe as a function of temperature (different temperatures are offset for clarity), measured with INS (Reprinted with permission from D. Bansal et al., Phys. Rev. B 94, 054307 (2016). Copyright 2016 American Physical Society). Notice the pronounced softening and broadening of optical modes around 17 meV with increasing temperature, as well as the fact that not all features behave the same way, which reflects differences in anharmonicity between modes. Also note the difference in agreement between two different flavors of DFT simulations and the experimental result (LDA: local density approximation and GGA: generalized gradient approximation). The DFT simulations were corrected for neutron-weighting and experimental resolution for direct comparison with INS data.170
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important to note that in this expression, we only keep track of the magnitude of the momentum transfer, Q ¼ jQj, and thus the orientation of the vector direction is lost, which amounts to performing an orientation average. Also, while in all rigor, one should use the incoherent component of the neutron cross-section of each species present in this expression (ri,inc); when utilizing the incoherent approximation for a sample with both coherent and incoherent scattering, one should replace ri,inc with the total scattering cross-section ri;tot ¼ ri;inc þ ri;coh :
inelastic scattering by neutrons provides a direct measure of S(Q, x), providing detailed insight to the microscopic dynamics of real systems. We emphasize that, both in the case of coherent scattering or incoherent scattering, the cross-section for the inelastic scattering processes (phonon creation or annihilation) is much weaker than the elastic scattering cross-section. In the case of coherent scattering, the Bragg peaks are generally several orders of magnitude more intense than the phonon signal.
(73)
The neutron scattering lengths and cross-sections of elements (or specific isotopes) are tabulated and available electronically via NIST. It is also important to note that, from a practical standpoint, measurements are easier as T increases, since the intensity scales with [nT(E) þ 1], which is roughly linear in T at high T (conversely, the measured phonon intensities are weak at low temperatures). The energy transfer measured is related to the phonon frequency as E ¼ hx.
3. Practical considerations
Neutron experiments require access to large-scale experimental neutron facilities, which are considerably more scarce than synchrotrons for X-ray experiments, and as a result, access is generally quite competitive. In addition to the requirement of access to neutron facilities, several aspects of neutron scattering for phonon measurements are worth highlighting: •
2. Single-crystals
For the treatment of phonon dispersion measurements in single-crystal samples (of coherently scattering elements), one must consider the coherent scattering process. Coherent neutron scattering from a crystal is localized in the fourdimensional Q – E space, as it is constrained to the phonon dispersion surfaces and thus presents sharp features in Scoh (Q, E) (peaks) as one crosses these surfaces. This is analogous to coherent elastic scattering (diffraction) from a single crystal, which produces intensity localized in sharp, intense Bragg spots. It is possible that the single crystal also exhibits an incoherent component, if it contains specific elements with a sizeable incoherent cross-section, or if some chemical disorder is present at specific crystalline sites, as in an alloy. This incoherent signal will contribute a smooth “background,” but the sharp features of coherent scattering are generally clearly identifiable on top of this signal. In the case of creation (Stokes process) of a single phonon quasiparticle through the interaction with the neutron, the dynamical structure factor S(Q, E) is given by177 d2 r k0 ð2pÞ3 1 / Rs Rs dXdE coh k 2v0 xs 2 bd p ffiffiffiffiffiffi ffi
R d expðWd Þ expðiQ dÞðQ eds Þ M d
hns þ 1idðE Es ÞdðQ q sÞ rcoh k0 NSðQ; xÞjcoh ; ¼ 4p k
(74)
where k0 and k are the final and incident wave vector of the scattered particle, q is the phonon wave vector (reduced to the first BZ), Es is the eigenvalue of the phonon corresponding to the branch index s, s is the reciprocal lattice vector, d is the atom index in the unit cell (d is its internal coordinate vector), bd is the coherent neutron scattering length of the corresponding element, Wd is the corresponding DebyeWaller factor, and Q is the wave vector transfer. Thus,
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Neutrons interact weakly with matter (except for some elements with strong absorption, see below). In addition, even the brightest neutron beams currently available are many orders of magnitude weaker than synchrotron X-ray beams. As a result, large samples are necessary, and even with appropriate samples, data collection times are long (days). Sample sizes of at least 1 cm3 are necessary, and larger sizes are advantageous. One needs to evaluate the neutron scattering properties of elements in a material of interest, by consulting tables of neutron cross-sections. Some elements or isotopes are so strongly absorbing as to make neutron measurements impossible, or extremely challenging. Common culprits include B, Cd, Li, In, and Co, and many lanthanides or actinides. Absorption cross-sections higher than 30 to 100 barns (1 b ¼ 1024 cm2), in concentrations larger than 20%, can impede quality data collection. The analysis of neutron scattering data is involved, and an appropriate commitment (or collaboration) is necessary to extract meaningful results from the experiments. Complementary measurements (such as Raman and ultrasound) and phonon simulations are often helpful, as shown for FeSi in Figs. 23(a) and 23(b). Measurements of phonon dispersions in single-crystals are considerably more complex and involved than measurements of the phonon DOS in powder samples. This is somewhat analogous to the difference between routine powder X-ray diffraction and single-crystal structure determination. The growth of appropriate single crystals is often a major impediment to detailed phonon dispersion studies, and for this reason, it is common to limit investigations to phonon DOS measurements. Single-crystals for phonon dispersion measurements need not only be sufficiently large, they must also be of high crystalline quality (crystal mosaic of order 18 or better), with a single crystalline domain. For this reason, thorough prior screening of samples with both X-ray diffraction or dedicated neutron stations is an important first step. Once the crystal quality is firmly established, the sample needs to be oriented and
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•
are of order 5–10%, and the E and Q resolution are coupled. A good understanding of the neutron scattering technique in general, and of the specifics of the instrument, are necessary to interpret the data. Measurements of phonon linewidths, which require the best possible crystals, remain arduous as resolution effects on dispersion measurements need to be carefully accounted for. To date, linewidth measurements, which are needed to estimate mode Gr€uneisen parameters, have only been successfully performed in a handful of materials. Artifacts are common in neutron scattering measurements, and are often referred to as “spurions.” These can arise because of multiple scattering effects owing to the large size of the samples and intrusive sample environments, or because of the incident neutron beam itself being contaminated by multiple wavelengths, or yet because of divergence effects. Thus, surprising initial results should always be considered circumspect, and corresponding measurements should be tested with different instrument conditions, or on several instruments.
VIII. INTRODUCTION TO ORGANIC THERMOELECTRICS
FIG. 23. Example of dynamical structure factor, S(Q, E), for FeSi singlecrystal, comparing INS measurement and DFT simulation (reprinted from178 with permission). (a) Data measured with INS (T ¼ 300 K) for Q along a path ½3; 3; L; 1 6 L 6 1 in reciprocal space. Along this particular path, acoustic and optical branches (above 20 meV) of transverse character have strong intensity, while longitudinal modes are weaker. (b) Our DFT simulation, with phonon thermal occupation factor set to T ¼ 300 K. Intensity color bar is on a logarithmic scale. Note that the strong horizontal bar of intensity for E < 1.5 meV in (a) corresponds to experimental background signal.
•
•
•
mounted with a pre-determined crystallographic orientation, as the actual measurement occurs preferentially in the horizontal plane. Dedicated planning for the experiment is another important prerequisite. The interaction of the neutron with the sample does not follow the translational periodicity of the reciprocal space. Thus, not every Brillouin zone will provide the same information about phonon dispersions, and for single-crystal experiments, a realistic model of S(Q, E) is necessary. An intimate knowledge of crystallography and understanding of the crystal structure (and possible defects) in a sample are key. Instrumental resolution effects are often considerable. Typical relative energy resolution of neutron spectrometers
Organic semiconductors have recently received increasing attention as thermoelectric materials,59,179–181 due to characteristics (inexpensive, earth-abundant constituents with scalable manufacturing) that lend themselves to applications inaccessible to many traditional inorganic materials.180,182 The thermoelectric behavior of several classes of organic materials including crystalline small molecules183,184 polymers,59,180 fullerene derivatives,185–187 and carbon nanotubes188–190 have been explored. We focus here on noncrystalline materials including semiconducting polymers and single-walled carbon nanotubes (SWCNTs), which have promising carrier transport properties and are amenable to chemical doping. In both cases the low dimensional building blocks of the material, a polymer chain or a nanotube, are formed into a van der Waals solid. Here, the ordering is imperfect due to the distribution in length of the building blocks and limitations in processing methods. Because of the low dimensionality of the constituents, these materials systems have pronounced anisotropies in their electrical and thermal properties.180,191,192 A. Chemical and electronic structure of carbon-based materials
The electronic properties of semiconducting polymers are defined by their molecular structure, comprising electronically conjugated cyclic, or heterocyclic, rings in the backbone with substituents that help provide processability and structural ordering (Fig. 24).193 Because of this molecular structure, the electronic bands of a hypothetical perfect crystal have substantially different dispersions along the different crystallographic axes [Fig. 24(b)].194,195 The gap between the valence and conduction levels of most semiconducting polymers ranges from 1–3 eV and the effective masses of holes and electrons can be as small as 0.1me, where me is the
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FIG. 24. (a) Chemical structures of some representative p- and n-type semiconducting polymers. (b) Schematic illustrating the chemical structure motif (top), an idealized ordered solid-state structure (middle), and the resulting electronic band dispersion and transport anisotropy (bottom) for a representative semiconducting polymer.
mass of an electron, in the chain direction. In real polymer samples, however, there is a distribution of the lengths of the molecules (i.e., the molecular weight) and the structural order is imperfect due to bends or folds in the main chain. Even if extended-state conduction occurs along a chain, the carrier must hop between chains, cross over domain boundaries, or be scattered by defects, leading to electronic states that tail into the gap. SWCNTs can essentially be thought of as long, cylindrical “molecules” where the electronic structure is dictated by the symmetry of the carbon network. The structure of a SWCNT can be conceptualized as a graphene sheet (a single-layer, hexagonal lattice of sp2-hybridized carbon atoms) “rolled up” to form a seamless cylinder, with a large number of SWCNT chemical structures possible due to the periodic nature of the graphene lattice.196,197 Many of the basic physical properties of SWCNTs vary strongly with the chiral indices (n, m) or diameter (e.g., the electronic structure can vary from metallic to semiconducting with diameter-dependent direct bandgaps ranging from 0.1 to 1.5 eV).197 The single-particle electronic bands of SWCNTs are dominated by sharp van Hove singularities that arise from 2D quantum confinement [Fig. 25(a)]. The carrier effective mass is relatively low (in the range of 0.3–0.5me) and nearly equivalent for both electrons and holes. As a
result of the extended conjugated p electron network, carriers can be delocalized over many bonds, leading to high carrier mobility and small bond reorganization energies.198 Thus, transport along a single SWCNT is characterized by delocalized charge carriers (l 104 105 cm2 V 1 s1 ), while the macroscopic transport within a thin film is determined in large part by the barriers encountered by charge carriers when hopping or tunneling from nanotube to nanotube (l 100 102 cm2 V 1 s1 ; r 1 6 105 Sm1 ). Coupled with their extraordinary electrical transport properties,199 the prediction that reducing the dimensionality of a semiconductor can lead to large enhancements to the thermoelectric properties200,201 suggests that SWCNTs could potentially enable thermoelectric materials with large power factors. Theoretical calculations have confirmed that semiconducting (s-SWCNTs) species have the potential to exhibit impressive thermopowers (jSj),188,202 which were recently confirmed experimentally.188 B. The problem of polydispersity
Both semiconducting polymers and SWCNTs suffer from their as-prepared materials consisting of a variety of structurally dissimilar components (i.e., polydispersity and purity). In the case of semiconducting polymers, the
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to expect that the presence of m-SWCNTs is likely to limit the thermoelectric performance [Fig. 25(a) illustrates the electronic density of states and calculated thermopower for a m-SWCNT and a s-SWCNT of similar diameter], these advances open up the possibility of determining the limits of the performance of SWCNT-based thermoelectric materials. Several experimental studies have confirmed that the elimination of m-SWCNTs from the sample results in a significant enhancement in the thermopower (jSj).188,209,210 Figure 25(b) illustrates the thermopower of a SWCNT network containing both m-SWCNTs and s-SWCNTs, assuming a model composed of both parallel and series conducting pathways with nanotube-nanotube junctions that contribute to the thermopower.209,210 Multi-walled carbon nanotubes (MWCNTs) are currently less expensive than their single-walled counterparts suggesting that they may represent a commercially viable alternative for low-cost thermoelectric applications. Their electronic structure, however, suggests that their performance may be severely limited. For instance, statistical arguments suggest that 56% of double-walled carbon nanotubes (DWCNTs) should have metallic character, but interwall interactions can further complicate their electronic structure properties.211 2. Electronic bandgap of SWCNTs
FIG. 25. (a) Electronic density of states and associated theoretical thermopower for (left) metallic and (right) semiconducting SWCNTs as a function of the position of the Fermi level. (b) Thermopower, normalized to the value expected for a purely semiconducting SWCNT network, as a function of the fraction of semiconducting SWCNTs in the mixed network.
polydispersity originates from a distribution in molecular weights (i.e., chain lengths), which is dependent on the type of synthetic process used for polymerization. A typical molecular weight is 20 kDa, which corresponds to a molecular length of 20 nm. The synthetic approach can also lead to variations in the chemical structures within a single sample, such as the regioregularity of the coupling between asymmetric monomer units or the formation of random copolymers of two (or more) sub-units.203 The synthetic conditions employed for SWCNTs also result in variations in the lengths of the nanotubes, although the situation is further complicated by the ability for the graphene lattice to form nanotubes with different electronic character and/or diameter (vide infra). 1. Electronic character of SWCNTs
Over the last decade significant effort has been devoted to developing a variety of non-destructive strategies to break apart the SWCNT bundles and separating the metallic (mSWCNT) and semiconducting (s-SWCNT) species,196,204–206 and even to selectively extract small subsets of isolated s-SWCNTs from the raw material.207,208 Since it is intuitive
Recently, it was also demonstrated that the thermoelectric performance, as defined by the power factor, in polymerenriched s-SWCNT networks, is strongly dependent on the s-SWCNT diameter (electronic bandgap), with the optimum diameter appearing to be approximately 1.1 nm.188 Smaller diameter s-SWCNTs suffer from lower electrical mobility, despite the potential for very large thermopowers, whereas larger diameter s-SWCNTs may be limited by the smaller thermopowers that result from their reduced electronic bandgap. C. Processing organic thermoelectric materials
The properties of semiconducting polymers depend on the deposition conditions due to their strong influence on the orientation of polymer chains, the extent of chain folding and other defects, and the size of ordered domains in semicrystalline polymers.212 Films are frequently deposited by processes such as spin-casting or blade-casting into thin films (50–1000 nm in thickness) that tend to lead to alignment of the polymer chains parallel to the substrate. In contrast, thicker samples (> 1 lm) formed by processes such as dropcasting may lead to other morphologies. Measurements of the thermoelectric properties of thick and thin samples can therefore result in different values depending on the method, making it difficult to disentangle the influence of processing and molecular structure. SWCNTs are insoluble in almost all solvents, necessitating surfactants or polymers to disperse them for subsequent solution-phase processing. The identity of the dispersing agent and the ability to fully remove it from the final film may play a significant role in the performance. The processing conditions can be equally important for SWCNT networks, since they can influence the SWCNT packing density
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and porosity, the SWCNT bundle size, and the degree of inplane alignment of the SWCNTs. While all of these properties may influence the performance, the effect of thin-film morphology has not been examined in detail for SWCNT networks. However, several recent studies demonstrated that removing the polymer used to extract the s-SWCNTs from the raw material can significantly improve the power factor [Fig. 26(a)],189 due to increased charge carrier mobility, provided that the extent of nanotube bundling can be controlled during the network formation process.190 Recently, floating catalyst chemical vapor deposition (FC-CVD) has emerged as an alternative to solution-phase processing routes, since it allows for the direct production of thermoelectric SWCNT buckypapers/mats213,214 or yarns215 absent of insulating “fillers” that inhibit carrier transport and limit thermoelectric performance. D. Charge carrier doping
Analogous to inorganic semiconductors, one of the main strategies to afford fundamental insight into the transport properties and optimize the thermoelectric performance of organic semiconductors is the ability to effectively tune the carrier concentration. Both semiconducting polymers and s-SWCNTs are intrinsically undoped, although the former
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often have a non-negligible dark carrier density (1016 carriers cm3) that has been attributed to chemical defects216 and the latter are prone to adventitious doping due to adsorption of oxygen onto the nanotube surface.217 Doping of both polymer and SWCNTs is typically achieved by chargetransfer from a dopant, where the dopant is chosen such that its redox properties are appropriate to inject electrons into the lowest unoccupied molecular orbital (i.e., conduction band) or extract electrons from the highest occupied molecular orbital (i.e., valence band) to facilitate n- or p-type conductivity, respectively. Figure 26(b) illustrates the charge-transfer doping process for a s-SWCNT, including the impact that removal of electron density from the highest occupied molecular orbital has on the position of the Fermi level and the thermopower predicted based on the s-SWCNT electronic density of states. In all cases, the doping process results in the formation of a ground-state charge-transfer complex, with either partial or quantitative transfer of charge between the material and dopant. In some cases, the doping mechanism proceeds via (partial) decomposition of the dopant,188,218 although the charge density in the SWCNT must still be stabilized by the presence of a counterion. The doping process forms polaronic carriers in polymers due to the geometric distortion of the backbone on oxidation or reduction.219 These polarons move through a disordered electronic DOS resulting from the imperfect structural order that can dominate the polaronic behavior.220 The shape of the resulting DOS is idealized to be Gaussian or to have an exponential tail in most studies, but one expects the shape to depend on the specifics of a material and also details in processing. Despite these issues, the electrical conductivity after doping can reach 102 to 103 S cm1 in many materials with appropriate processing methods.180,221 For SWCNTs, the charge-transfer doping strategy relies on the fact that nanotubes typically form highly porous random networks, whereby the dopant can easily penetrate the network and has easy access to the SWCNT surface. However, two recent studies demonstrated that the presence of a wrapping polymer, or a significant amount of SWCNT bundling, significantly reduces the nanotube surface area available to the dopant, resulting in a lower doping efficiency and limited electrical conductivity.189,190 1. Measuring charge carrier concentrations
FIG. 26. (a) The dependence of the (left) thermopower and (right) thermoelectric power factor on the electrical conductivity, determined by increasing the carrier concentration by charge-transfer doping, in polymer-wrapped and polymer-free s-SWCNT networks. (b) Schematic cartoon showing p-type charge-transfer doping of a s-SWCNT, including the removal of electron density from the highest occupied molecular orbital and the subsequent impact on the thermopower.
To understand the impact of charge carrier doping on thermoelectric behavior it is important to be able to effectively quantify the carrier concentration. In disordered materials it can be difficult to accurately determine the carrier concentration due to the presence of free and trapped carriers, particularly when there is a broad energetic distribution of electronic states. In non-crystalline materials, the Hall coefficient has not been a generally reliable methodology to extract the carrier density.222 Recently, there have been studies to suggest that ac-Hall measurements can provide reasonable carrier concentrations, but there are not enough examples yet to come to a consensus on its applicability to semiconducting polymers.223–225 In many cases,
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de la Concentración de Portadores de Carga
measurements of the spin density can provide an estimate, but many doped polymers show smaller concentrations due to pairing of the carriers into bipolaronic species.225 Optical spectroscopy has the potential to act as a sensitive tool to qualitatively demonstrate the extent of charge carrier doping, since the positions of the optical absorption bands are often well-defined for both the doped and undoped species. However, the quantitative accuracy is limited by the need to have well-defined optical attenuation (extinction coefficients) for the doped and undoped species.226 Because of these difficulties, the thermopower of semiconducting polymers and SWCNTs is compared to the electrical conductivity in many cases rather than a value for the carrier concentration.188,189,227,228 2. Dopants and doping stability
Common p-type dopants for both semiconducting polymers and SWCNTs are oxidizing agents, such as organometallic salts,229,230 or molecular charge transfer dopants, such as fluorinated TCNQ.231,232 A widely studied polymer, PEDOT, is synthesized in a p-doped state and requires a polyelectrolyte or solubilizing strong acid (e.g., toluene sulfonic acid) to allow it to be processed.233 There are fewer stable n-type dopants than p-type, but there have been recent advances in reactive dopants and air-stable organometallic dopants,234–237 as well as organophosphorus and aminebearing compounds.232 One aspect of the charge-transfer doping strategy that must be considered, particularly with long-term applications in mind, is the relative stability of the chosen dopant or the charge-transfer complex formed between the organic semiconductors and the counterion. Stable n-type doping is a particular concern for SWCNTs, since the redox properties of the SWCNTs make them prone to doping compensation due to adsorption of oxygen into the doped network.238 Encapsulation of the dopant/counterion into the endohedral volume of the SWCNTs239–241 or coating the SWCNT network with a thin encapsulation layer (e.g., Al2O3)190 can effectively prevent, or at least retard, the loss of dopant molecules from the SWCNT network. A second approach is to employ a salt (e.g., NaCl, NaOH, KOH, etc.) to n-type dope the SWCNTs,242 where the electron density injected into the nanotubes is subsequently stabilized either by a crown ethercomplexed alkali metal or Onium ions.
Carriers Concentration
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conductivity.228 More recently, there have been efforts to further develop transport models for structurally and energetically disordered material systems, so that further insight can be gained regarding their thermoelectic behavior.114,244,245 1. Thermoelectric power factor of organic semiconductors
There are currently no clear predictive models that encompass the effects of morphology and disorder that allow easy prediction of the thermopower and power factor of organic semiconductors. Figure 27 and Table II summarize selected thermoelectric properties of p-type and n-type semiconducting polymers and SWCNTs. At electrical conductivities above 100 S cm1, the magnitude of the thermopower of many organic semiconductors is usually < 100 lV K1 with power factors (PF) surpassing 100 lW m1 K2.59,180,188 In contrast, polyaniline has a thermopower that is 10 times smaller at a comparable electrical conductivity to other semiconducting polymers,246 which may be due to the different doping mechanism (protonation of backbone nitrogen atoms). Similar to inorganic semiconductors, the magnitude of the thermopower of organic semiconductors tends to decrease with charge carrier concentration, while the electrical conductivity increases.188,189,228 However, the role of morphology in both semiconducting polymer and SWCNT networks means such relationships are not always apparent. For example, the electrical conductivity of PEDOT:PSS can range from 0.1 to 103 S cm1 despite similar charge carrier concentrations, whereas the thermopower appears less sensitive to processing methods.247–249 The difference occurs due to changes in the morphology of films, where carriers must move through the conductive regions, PEDOT, in the presence of the solubilizing and counter charged polymer PSS. In heterogeneous blended
E. Thermoelectric properties
Due to the interest in both semiconducting polymers and SWCNTs as active components in electronic (i.e., transistors) and optoelectronic (i.e., light-emitting diodes and photovoltaics) applications there has been significant effort in understanding charge carrier transport within these materials.220,243 For semiconducting polymers, foundational work focused on heavily doped materials, whereas more recent work has been aimed at intrinsic materials. In both cases, models from non-crystalline materials such as hopping or mobility-edge formalisms have been used to interpret transport measurements. Unfortunately, these models have not yet provided consistent fits to thermopower and electrical
FIG. 27. A compilation of literature data showing the empirical relationship between the power factor, a2r, and the electrical conductivity, r, for a variety of semiconducting polymers and organic molecular materials compiled from literature.227,228,247,248,250–261 The solid line indicates the empirical relationship PF / r1=2.
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TABLE II. Summary of selected published thermoelectric results in studies of organic semiconductors.
p-type SPs
n-type SPs
p-type SWCNTs
n-type SWCNTs
a (lV K1)
r (S m1)
ra2 (lW m1 K2)
PEDOT-Tosylate
70
66 000
324
PEDOT-Tosylate
117
92 300
1270
PEDOT-Tosylate
55
1 50 000
454
PEDOT:PSS
72.6
89 000
469
P3HT
38
100
0.14
PBTTT-C14
13.5
5400
0.98
P3HT PBTTT-C12
39.5 27
200 190
31 14
PDPP3T PBTTT-C14
70 14.4
50 1 30 000
25 27.6
PBTTT-C14
32.7
1 00 000
109
P(NDIOD-T2)
–850
0.8
0.6
BDPPV
–141
1400
28
BBL
–60
100
0.43
s-SWCNTs
76.2
18 500
108
s-SWCNTs
88
27 000
209
Mixed SWCNTs
65
51 400
218
s-SWCNTs
91
41 500
345
s-SWCNTs
56
1 27 000
398
Mixed SWCNTs
88
3 20 000
2478
Mixed SWCNTs
100
2 20 000
2200
Mixed SWCNTs
78
3 02 000
1840
s-SWCNTs
69
1 47 500
706
Mixed SWCNTs
57
7 47 600
2390
Mixed SWCNTs
–52
9500
25
Notes Polymerized from a solution of 3,4-ethylene dioxythiophene (EDOT), iron(III) tosylate, and pyridine; spin coated film; annealed at 110 C; reduced in tetrakis(dimethylamino)ethylene (TDAE) vapor247 Polymerized from a solution of EDOT, iron(III) tosylate, pyridine, and poly(ethylene glycol-propylene glycol-ethylene glycol) (PEPG) triblock copolymer; spin coated film; annealed at 70 C; electrochemically optimized262 Vapor polymerized by exposure of a film of iron(III) tosylate and PEPG to a EDOT in a vacuum oven; washed with ethanol249 Spin-coated from a PEDOT:PSS suspension containing either ethylene glycol (EG) or dimethylsulfoxide; annealed at 130 C; de-doped by immersion in EG248 Drop-cast film; doped by immersion in solution of nitrosonium hexafluorophosphate (NOPF6)257 Drop-cast film; annealed at 150 C; doped by immersion in solution of NOPF6258 Drop-cast films; doped by immersion in a solution of iron(III) triflimide58 Spin-coated film; annealed at 180 C; doped by immersion in a solution of 4-ethyl-benzenesulfonic acid (EBSA)250 Spin-coated film; annealed at 180 C; doped by exposure to (tridecafluoro-1,1,2,2-tetrahydrooctyl) trichlorosilane (FTS) vapor250 Film spin-coated from solution of P(NDIOD-T2) and (4-(1,3dimethyl-2,3-dihydro-1H-benzoimidazol-2-yl)phenyl)dimethylamine (N-DMBI); annealed at 150 C263 Film spin-coated from solution of BDPPV and N-DMBI; annealed at 120 C264 Spin-coated film; dried at 70 C and annealed at 110 C; doped by exposure to TDAE vapor265 s-SWCNTs extracted using density gradient ultracentrifugation (DGU) of deoxycholate sodium salt solutions; annealed under vacuum at 500 C; doped by immersion in solutions of HCl, HNO3, or H2SO4209 s-SWCNT suspensions purchased directly from NanoIntegris, Inc.; films prepared by vacuum filtration; dried at 80 C210 Suspensions prepared by probe sonication of SWCNTs and conjugated polyelectrolyte; films prepared by vacuum filtration; dried at 80 C266 s-SWCNTs extracted using fluorene-based conjugated polymers; films ultrasonically sprayed onto heated substrate (130 C); excess polymer removed by immersion in toluene at 78 C; doped by immersion in solutions of triethyloxonium hexachloroantimonate (OA) at 78 C188 s-SWCNTs extracted using hydrogen-bonding fluorene-based polymer; films ultrasonically sprayed onto heated substrate (130 C); excess polymer removed by immersion in solution of trifluoroacetic acid at 78 C; doped by immersion in solutions of OA at 78 C189 SWCNT buckypaper prepared directly using FC-CVD; likely doped by O2-adsorption213 Fiber twisted from SWCNT buckypapers prepared directly using FCCVD; likely doped by O2-adsorption213 SWCNT buckypaper prepared directly using FC-CVD; likely doped by O2-adsorption214 s-SWCNTs extracted using cleavable fluorene-based polymer; films ultrasonically sprayed onto heated substrate (130 C); excess polymer removed by immersion in solution of trifluoroacetic acid at 78 C; doped by immersion in solutions of OA at 78 C190 Strongly aligned SWCNT yarn prepared directly using FCCVD; doped by immersion in FeCl3 in ethanol; dried in air215 Suspensions prepared by homogenization of SWCNTs and 1,3-bis (diphenylphosphino)propane; films prepared by vacuum filtration; dried at 808 C 232.
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TABLE II. (Continued.) a (lV K1)
r (S m1)
ra2 (lW m1 K2)
Mixed SWCNTs
–41.8
43 200
75.4
Mixed SWCNTs
–41
10 500
17.8
Mixed SWCNTs
–33
2 05 000
220
Mixed SWCNTs
–38
71 600
103
Mixed SWCNTs
–64
3 63 000
1490
s-SWCNTs
–78.5
1 19 000
730
Mixed SWCNTs
–56
7 85 000
2460
materials like PEDOT:PSS, power factors as high as 500 lW m1 K2 have been reported,248 whereas for single component polymers, the power factor can reach values as high as 100 lW m1 K2 for heavily doped PBTTT.250 As discussed earlier, the morphology of s-SWCNT networks also has an impact on the thermoelectric properties. Removal of the solubilizing polymer, while maintaining relatively small bundles results in a significant increase in the power factor to >700 lW m1 K2 for both p-type and n-type networks.190 Currently the limits of the power factor for semiconducting polymers and SWCNTs, as well as the physics determining those limits, are not known. 2. Ionic transport in semiconducting polymers
Because polymers are “soft” materials, ions can move by mass diffusion at relatively low temperatures leading to both an electronic and ionic contribution to the Seebeck coefficient. If the ionic and electrical conductivity are independent, one expects the thermopower to be a weighted average of the ionic and electronic components (i.e., S ¼ ðSionic rionic þSelec relec Þ=ðrionic þ relec ÞÞ. The ionic thermopower can be large which can make the ionic contribution to the power factor significant, even if the ionic conductivity is small (103 S cm1). The timescales of ion motion are commensurate with those of thermopower measurements, leading to time-dependent behavior in some systems.205,267,268 Thermopower decays of minutes to hours have been observed in some samples, making it difficult to define the power factor of a material because it could depend on the history of operation. Additionally, hygroscopic materials such as PEDOT:PSS have varying ionic conductivity in humid atmospheres leading to significant variation in the apparent thermopower depending on the measurement conditions.269
Notes SWCNTs filled with cobaltocene by heating mixed powders under vacuum at 100 C for 3 days; suspensions prepared by bath sonication; films prepared by vacuum filtration; dried at 80 C240 Suspensions prepared by probe sonication of SWCNTs and conjugated polyelectrolyte; films prepared by vacuum filtration; dried at 80 C266 Suspensions prepared by probe sonication of SWCNTs; films prepared by vacuum filtration; doped by immersion in a solution of KOH and 18-crown-6-ether241 SWCNTs dispersed in a solution of 1,1’-bis(diphenylphosphino)-ferrocene; films prepared by vacuum filtration; dried at room temperature followed by annealing under vacuum at 200 C242 SWCNT buckypaper prepared directly using FC-CVD; doped by polyethyleneimine (PEI) in ethanol; dried at 50 C for 5 min214 s-SWCNTs extracted using cleavable fluorene-based polymer; films ultrasonically sprayed onto heated substrate (130 C); excess polymer removed by immersion in solution of trifluoroacetic acid at 78 C; doped by immersion in solutions of benzyl viologen in toluene in a N2 glovebox190 Strongly aligned SWCNT yarn prepared directly using FCCVD; doped by immersion in PEI in ethanol; dried in air215
3. Thermal transport
Thermal transport in polymers is less well understood than in inorganic materials due to structural disorder. The thermal conductivity of amorphous polymers is usually low (0.2–0.3 W m1 K1) due to the weak coupling between polymer chains,270 but can be much higher in oriented semicrystalline polymers along the chain direction (1–5 W m1 K1).269,271 One therefore expects that locally the thermal conductivity of ordered regions of polymer films should be anisotropic and depend strongly on the chain orientation. Similarly, an individual SWCNT exhibits very long phonon mean free paths and near-ballistic thermal transport (>2000 W m1 K1) along the nanotube axis.272 However, the bulk thermal transport orthogonal to the SWCNT axis, as well as in the plane of a random SWCNT network, is limited by weak inter-nanotube coupling. For instance, the in-plane thermal conductivity of undoped s-SWCNT thin films is reduced from the individual nanotube value but can still be fairly high (16 W m1 K1).188 Interestingly, the thermal conductivity drops to 2–4 W m1 K1 when identically prepared thin films are doped partially or heavily p-type.188–190 The thermal conductivity anisotropy in SWCNT networks is a relatively unexplored topic.242 Time-domain thermoreflectance (TDTR) is a sensitive technique that has been used to measure the out-of-plane thermal conductivity of thin films.185,269,273 TDTR can be used to measure the thermal conductivity in the in-plane direction if samples can be appropriately sectioned. In general, however, methods to measure the in-plane thermal conductivity of thin films are limited. Laser-based methods sensitive to transient thermal gratings have significant potential.274 Platform-based techniques are also useful methods for in-plane measurements of j, where a collection of resistors, which act as heaters or thermometers, are patterned onto a micromachined, thin, suspended platform of a
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material with low thermal conductivity (e.g., Si-N).275 The semiconductor is then either suspended between two islands on the platform or deposited onto a suspended Si-N beam between them. This technique has been utilized successfully for both PEDOT:PSS,276 where the in-plane thermal conductivity of the doped polymer was found to be significantly higher than estimated from 3x measurements, and s-SWCNT samples.188–190 4. Consequences for anisotropic transport
For both semiconducting polymers and SWCNTs, typical thin-film fabrication techniques result in the formation of solid-state structures where the polymer chains or SWCNTs are oriented parallel to the substrate surface. This structural anisotropy, which manifests severely anisotropic transport properties,191 must be considered, both when measuring material properties and designing functioning thermoelectric generators. Early efforts to construct thermoelectric devices based on both semiconducting polymers and CNTs have been fabricated using alternating strips of p- and n-type thin films, where the thermal gradient is established within the plane of the films. For instance, simple devices have been constructed from SWCNT films, either as a free-standing architecture [Fig. 28(a)]240 or deposited onto flexible substrates [e.g., polyimide; Fig. 28(b)].232,266 Novel devices have also been demonstrated where the individual legs of the thermoelectric device are printed in thin stripes and then assembled into the final structure by rolling or corrugation [Fig. 28(c)].180,192,277 IX. GRAND CHALLENGES AND OPPORTUNITIES
The diverse topics discussed in this review are a testament to the breadth of the thermoelectric research field. Each subdiscipline naturally faces its own challenges with regard to synthesis, characterization, and modeling of thermoelectric materials and their behavior. Below, we have highlighted some of the major challenges that we face as a scientific community and research areas that remain virtually uncharted today. A. Mysteries of quasiparticles
Understanding the transport behavior of electrons and phonons lies at the heart of the study of thermoelectric materials. Accurately predicting the ground state electronic and phonon band structures of perfect crystals at absolute zero is rapidly improving. However, despite a wealth of data, there
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are still major gaps in our understanding and predictive abilities; we need an improved understanding of how chemistry (composition, structure, and bonding) controls critical properties such as the band effective mass, electronic band degeneracy, and band gap. Further, the relationship between anisotropic bonding and the carrier effective mass and phonon velocities is not straightforward, and intuitive chemical guidelines are still lacking. Our models and our current understanding break down further as conditions deviate from an idealized, static system, for example, at temperatures above absolute zero, or when appreciable disorder is present. For example, transport in organics materials, which, in principle, can be described by the same Landauer framework as inorganic materials, is challenging to model due to the lack of periodicity. In inorganic solids, finding ways to move beyond our current reliance on the ground state electronic and phonon band structures and on approximations such as the rigid band model will be key to future progress in this area. The temperature- and composition-dependence of the electron and phonon energies are just one part of the transport equation. Accurately modeling energy-dependent electron and phonon scattering is an even greater challenge. In this area, we have more questions than answers: How is scattering controlled by native chemistry, interfaces, and defects? Can we design materials with targeted interfacial scattering rates to optimize the transport function, G(E), described in Chapter II? How can we design materials with weak inter-valley scattering, so that we can take full advantage of high valley degeneracy? Can we control and/or decouple anisotropic scattering of electrons and phonons in materials with layered or quasi-1D structures? Often, a material system is ideal in many regards, and yet we find ourselves at an impasse due to difficulties with doping. This is certainly not a unique challenge to thermoelectrics. Detailed defect calculations are becoming more common to help find potential dopants and to identify intrinsic “killer” defects that pin the Fermi level. However, these calculations are sensitive to errors in the band edge energies and can be qualitatively incorrect if not done carefully.278 With a suite of accurate defect/dopant calculations, it may be possible to extract general design principles for such imperfections. Spin and other quasiparticles. This review has largely focused on weakly interacting quasiparticles that respond to - Electrons & temperature or voltage gradients. However, the realm of con- - Phonons densed matter physics has a rich variety of phenomena
FIG. 28. Prototype SWCNT thermoelectric generators employing (a) standalone240 and (b) substrate-supported232 p-type and n-type legs. (c) Rolled semiconducting polymer thermoelectric modules using PEDOT:PSS as p-type and CPE/CNT nanocomposite as n-type legs respectively.192
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involving charge and entropy transport. For example, materials with unpaired spins enable the formation of collective spin waves (i.e., magnons) and a diverse array of magnonelectron-phonon interactions. Likewise, the interaction between spin and charge across interfaces enables the creation of spin caloritronic devices.279 Moving to even more exotic quasiparticles, we have recently seen the emergence of superconducting heterostructures as a potential route to thermoelectric phenomena.280,281 While G(E) is a symmetric function in a superconductor, breaking this symmetry (e.g., via superconductor-ferromagnet tunnel junctions) allows for the emergence of a thermoelectric current.282 There are continued opportunities for highly entangled quantum systems where a quasiparticle description no longer holds; such materials will push our abilities to predict, synthesize, and characterize their thermoelectric response. B. Large-scale calculations
Calculations offer the opportunity to consider the components of G(E) as well as adjusting EF through native defects and dopants; likewise calculations can shed light on lattice dynamics. However, significant challenges remain. For materials discovery, calculations can be implemented in a high throughput methodology to screen large chemical spaces. Such calculations can identify materials with intriguing g and v, but are not presently able to directly incorporate scattering. Likewise, defect and dopant calculations are not presently scalable to large suites of materials. Perhaps most critically, examples of new scientific insight from large databases of high throughput calculations remain limited to date. Rather, the focus has been on ranking candidate materials. Within this effort, there remain opportunities to bring insight from thermochemical calculations into the search space to identify phase competition and defects. C. Synthetic chemistry
Many of the challenges that we face in synthetic chemistry today are issues that have been facing us for decades. These include, for example, the obstacles inherent to the growth of high quality single crystals that are necessary for characterization of intrinsic materials properties (e.g., measurement of phonon dispersions, anisotropic transport). Efforts to exploit anisotropic materials to decouple transport properties (i.e., SnSe, Bi2Te3) will stall unless we can either scale up single crystal growth or develop novel routes to synthesize highly textured polycrystalline materials. On the other side of spectrum, synthesis of materials with interfaces that are effective as filters for phonons or charge carriers remains frustratingly imprecise. The theoretical advantages of nano-particles, interfaces, etc., are well-established, but in practice it is still an enormous challenge to experimentally disentangle interfacial and bulk effects. If we wish to better understand transport in composite or nano-structured materials, there is a need for fabrication of extremely wellcontrolled interfaces. In a similar vein, bulk materials with well-defined defects are critical for probing the impact of defects on the components of G(E); phase boundary mapping is one emerging technique in this area.283,284 Another area of
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opportunity is that of high-throughput synthesis, which would complement high-throughput calculations. Although there are a number of approaches for high-throughput thinfilm samples, these techniques cannot readily be extended to bulk materials. D. Measurement techniques
Improving the accuracy of characterization of thermoelectric transport coefficients (a, r, and j) is critical, requiring new and improved methods of measuring temperature and in measuring thermal properties at high temperature. Further, there are a number of state of the art techniques that remain underutilized in the thermoelectric community (e.g., resonant X-ray scattering, ARPES, Nernst, XANES, EXAFS, inelastic neutron scattering, Raman, etc.) and thus present immediate opportunities for fundamental insight into underlying phenomena. Considering measurement in the aggregate, many researchers are already investing in machine learning and in the collection and curation of high quality data sets to aid in the search for higher performing thermoelectric materials. This requires researchers to publish reliable supplemental data and synthesis/processing descriptions. It may also prove useful to create a shared online repository for unpublished failed attempts and data obtained from poorly performing materials. E. Thermoelectric devices
Development of new, high zT materials has far outpaced the integration of new materials into thermoelectric devices. If we are to translate improved zT into improved device efficiency, obstacles arising from the brittle nature of the materials and the high operating temperatures must be addressed. High fracture toughness, for example, is needed to withstand stresses caused by thermal expansion. This is a tedious property to measure experimentally,285 but luckily it appears that these values can be computed,286 which may greatly accelerate progress in both understanding and engineering. The long term reliability of materials in use for decades at high temperature is typically limited by sublimation and creep strength. Mechanical creep relieves some of the high temperature stresses but also leads to microstructure evolution and crack growth at the hot side interface. A “mechanical property” figure of merit that balances the tradeoffs between the most important mechanical performance metrics may help focus the field’s efforts. From a manufacturing perspective, new techniques are needed to realize further cost reductions and to enable new device form factors (e.g., transverse thermoelectrics). The extension of additive manufacturing techniques to thermoelectrics, as one such example, could further enable functionally graded materials.287 Atomic diffusion poses a major concern for high-temperature devices. Diffusion in thermoelectric materials and complex semiconductors in general is a largely unexplored area that is relevant not only to the fabrication and lifetime of modules, but may lead to new insight into the complex coupling of thermal, electrical and mass transport and long-term evolution of microstructure in a electrochemical potential gradient. An advanced understanding
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of mixed electronic and ionic conductors exists in the solid state ionics community but rarely is extended to thermoelectrics.288,289 Along these lines, the development of diffusion barriers (e.g., at hot-side metal contact) is a potentially fruitful area of study.290 Finally, improved methods to assess the performance of thermoelectric modules are needed, as direct comparison of the performance of different modules remains challenging. F. Uncharted terrain in organic materials
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Solar-Thermal Energy Conversion Center (S3TEC), an Energy Frontier Research Center funded by the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES), under Award # DE-SC0001299/DEFG02-09ER46577. E.S.T. and G.J.S. acknowledge support from NSF Award #1729487, and EST acknowledges support from NSF Award # 1334713. T.D.S. acknowledges support from NSF CAREER Award # 1651668. 1
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The limits of the thermoelectric performance of organic materials are currently unknown, in part because materials have not been explicitly designed to have high thermopower. The engagement of multidisciplinary efforts to design new organic semiconductors that have molecular level control of doping is likely to lead to new advances. The anisotropic physical properties and dimensionality of organic semiconductors, suggest that chemical/material science strategies aimed at exploiting these characteristics to manipulate the electronic and/or phonon density of states could enable significant improvements in thermoelectric performance. If such advances are accomplished, the flexibility and low density of organic semiconductors open up opportunities to utilize thermoelectric energy harvesting or Peltier temperature control in portable and/or remote applications, where conformal and lightweight devices are desired. If organic semiconductors can realize zT values approaching those of traditional inorganic semiconductors, the specific power (W/kg) delivered by the thermoelectric generator could be dramatically higher. ACKNOWLEDGMENTS
A.Z. acknowledges funding from the National Science Foundation (NSF) under Award # DMR SSMC-1709158 and the editorial contributions of Mack Marshall and Jason Mueller. J.L.B. and A.J.F. gratefully acknowledge financial support from NREL’s Laboratory Directed Research and Development (LDRD) program. The NREL is supported by the U.S. Department of Energy under Contract No. DEAC36-08GO28308 with Alliance for Sustainable Energy, LLC, the Manager and Operator of the NREL. The U. S. Government retains (and the publisher, by accepting the article for publication, acknowledges that the U. S. Government retains) a non-exclusive, paid up, irrevocable, worldwide license to publish or reproduce the published form of this work or allow others to do so, for U. S. Government purposes. M.L.C. acknowledges support from DOE BES award #SC0016390. J.W. and K.K. are grateful to current and former Kovnir Group members for the all the syntheses, crystal growth, in-situ studies, and some of the photos used in this review. Kovnir group research in thermoelectrics is supported by the DOE-BES, Division of Materials Sciences and Engineering, under Award DESC0008931. Beamline 17-BM scientists W. Xu and A. Yakovenko and Advanced Light Source at Argonne National Laboratory are acknowledged for the development and support of the in-situ XRD capabilities. O.D., S.D.K., and G.J.S. acknowledge funding as part of the Solid-State
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