Thermoelectric modules are solid-state devices composed of pairs of n- and p-type thermoelectric materials, which enable the conversion of heat into electrical power. This property could therefore be exploited for energy recovery from waste heat. The physical basis of thermoelectric power generation is the Seebeck effect: when a material is subjected to a temperature gradient, an electrical potential is generated. Hence, when n- and p-type thermoelectric materials are paired in a couple (Figure 1.1) and a temperature gradient is applied, electrons and holes conduct heat from the hot to the cold junction, and an electrical current is generated. For power generation, the maximum efficiency of a thermoelectric module, consisting of a series of couples of two materials is given by the expression:¹ 

where T_H and T_C are the temperatures at the hot and the cold junctions, respectively, T̄ is the average temperature and Z_c is the figure of merit of the couple:

This is dependent on the Seebeck coefficient, S, electrical resistivity, ρ, and thermal conductivity, κ, of the materials A and B:

Therefore, the thermoelectric power generation efficiency depends on the Carnot efficiency (ΔT/T_H) and the properties of the materials A and B (i.e. Z_c). The efficiency can hence be improved either by increasing the temperature gradient or by increasing Z_c.

The urgent need to reduce the dependence on fossil fuels has stimulated considerable research efforts for the discovery of materials with better properties, to improve the efficiency of thermoelectric power generation. To facilitate the optimisation of individual thermoelectric materials, the dimensionless figure of merit (ZT) of a material is defined as:

where σ is the electrical conductivity (which is the inverse of the electrical resistivity) and S²σ is known as the power factor. To estimate the efficiency of a thermoelectric device, Z_c may be approximated as the average Z for the n- and p-type materials forming the couples. Measurements of electronic and thermal transport properties are essential to determine Z_c and therefore of the utmost importance for the development of better thermoelectric materials.

Research in thermoelectric materials entails synthesis, structural characterisation over a wide range of length scales and the study of thermal and electronic transport in a broad class of materials, and spans the fields of solid-state chemistry, condensed-matter physics, materials science and engineering. The classes of inorganic materials currently under investigation, which have been reviewed elsewhere,^2,3  include chalcogenides,^4,5  oxides,⁶  oxychalcogenides^7,8  and intermetallic phases such as clathrates,^9,10  half-Heusler¹¹  and Zintl phases.¹²  Recent progress in selected examples of materials is described elsewhere in this volume. This chapter aims to provide an introduction, accessible to those with little experience in the field, to the approaches used for the synthesis of thermoelectric materials and to the measurement techniques employed for the characterisation of their transport properties.

Synthesis is the first essential step in the production of thermoelectric materials. The traditional methods for the preparation of inorganic thermoelectric materials require high temperatures and produce bulk materials with large particle sizes. In recent years, new approaches, such as mechanochemical, solvothermal and colloidal synthetic methods, have developed greatly. These alternative methodologies facilitate the production of nanostructured materials, which are very attractive, given that nanostructuring can be a highly effective approach to increase the thermoelectric figure of merit.¹³  The growth of single crystals is also important as it facilitates the investigation of the anisotropy of the physical properties, which can in turn be exploited to optimise thermoelectric performance. A variety of approaches, ranging from the well-established Czochralski and Bridgman methods to more exploratory methodologies using fluxes or chemical vapour transport, are currently used to grow single crystals of thermoelectric materials. A practical overview of the synthesis techniques identified above, together with examples of their application to relevant thermoelectric materials, is presented here.

High-temperature synthesis, often termed the ceramic method, entails heating of components at elevated temperatures for an extended period of time.^14,15  For instance, the syntheses of Bi₂Te₃, the skutterudite CoSb₃ and Na₂Co₂O₄ can be performed through the following reactions carried out at temperatures in excess of 1000 K for several hours:^16–18 

Reactions between solid-state materials are generally slow and require high temperatures, because they depend on atomic diffusion, which increases with increasing temperature. High temperatures are necessary for sufficient atomic diffusion to take place to enable reaction to occur. If we consider two particles of the elemental solid reagents A and B, in close contact with each other, the reaction to form product AB requires atomic diffusion of A into B and vice versa (Figure 1.2). As the reaction progresses, the product AB is formed at the boundary between A and B, and further reaction requires diffusion of A and B through the product. As the thickness of the product layer increases, the reaction rate will slow, since the product layer acts as a barrier for further diffusion of A into B and B into A. Therefore, in order to reduce reaction times, reagents should be ground to reduce the particle size and ensure intimate mixing, to maximise the contact area. For the same reason, reaction mixtures are often pressed into pellets prior to heating, and samples are removed periodically from the furnace and reground, in order to create new interfaces where reaction can occur. Grinding and reheating two or more times are often required before a phase-pure product is obtained. Although the reactants are usually solids at room temperature, they may melt or even volatilise at the temperatures required for the reaction to occur. For instance, tellurium and antimony melt above 723 and 900 K, respectively, and sulfur becomes a gas above 718 K. As the atomic diffusion rate is higher for liquid or gaseous phases, this will usually aid the reaction. However, phase diagrams should be consulted (when available) to decide on the heating conditions. Compilations of phase diagrams are available,¹⁹  in addition to online databases such as ACerS-NIST PHASE 4.4, the Landolt–Börnstein database (also known as SpringerMaterials, https://materials.springer.com/) and the ASM Alloy Phase Diagram Database. If phases that melt incongruently are formed on heating, phase segregation may occur on cooling. Under those circumstances, quenching of the melt may be required, or ball milling (see Section 1.2.2) may be a more suitable synthetic approach.

Heating of reaction mixtures at temperatures up to 2100 K can be achieved using furnaces that operate by resistance heating. Metallic alloys are used as the heating elements in furnaces operating at temperatures up to 1500 K, and ceramic elements are used for higher temperatures: silicon carbide elements can operate up to 1900 K and molybdenum disilicide elements can reach temperatures of 2100 K. Furnaces that use radiofrequency (RF) induction coils for heating can also be employed. Higher temperatures, up to 3300 K, can be achieved by arc melting, which heats by passing an electric arc through the reagents, causing them to melt. Arc melting is a particularly convenient method for the preparation of Heusler and half-Heusler phases,²⁰  because this class of thermoelectric material contains chemical elements with very high melting points. Detailed descriptions of the design and working principles of different types of furnaces were given by Motzfeldt.²¹ 

Whereas the preparation of oxides is often carried out by heating in air, this is not suitable for the synthesis of chalcogenides or intermetallic phases, which would rapidly oxidise under those conditions. Horizontal tube furnaces can be used to heat reaction mixtures under a flowing gas, with a bubbler attached to the exit to maintain a positive pressure and prevent back-diffusion of air. This enables reactions to be carried out under an inert (e.g. N₂ or Ar), oxidising (O₂) or reducing (e.g. H₂–Ar mixture) atmosphere, in addition to allowing syntheses involving reactive gases to be performed. For instance, sulfides can be prepared in a tube furnace by flowing H₂S or CS₂ mixed with an inert gas, as exemplified by the quaternary sulfide ACuMnS₂ (A = K, Rb, Cs), prepared by reacting A₂CO₃, MnS and CuO under flowing CS₂–N₂ at 1000 K.²²  As both H₂S and CS₂ are highly toxic, caution is required when carrying out such reactions.

A major disadvantage of reactions under a flowing gas is that changes in stoichiometry may occur due to the transport of volatile components. For this reason, the evacuated sealed tube method (Figure 1.3) is commonly used for the synthesis of chalcogenides, oxychalcogenides and intermetallic phases. This entails loading the reaction mixture into a tube made of heat-resistant glass (borosilicate or fused silica, depending on the reaction temperature), which is evacuated using a vacuum line and sealed using a gas–oxygen torch,²³  prior to heating it in a furnace. To reach the temperatures required to seal heat-resistant glass, the gas–oxygen torch will burn a mixture of natural gas (or propane) and oxygen. The reactions described in eqn (1.5) and (1.6) are performed using this method. When using sealed glass ampoules, the melting and boiling points of the reactants must be considered: reagents that volatilise on heating may cause sealed ampoules to explode owing to an increase in internal pressure, and reactants that melt may attack the glass and therefore destroy the ampoule. Prior to raising the temperature to the reaction temperature, it is advisable to heat reaction mixtures for a few hours below the melting or boiling point of any reactant, to enable it to react with the other components, in order to minimise the risk of damage to the sealed ampoule.

As the products of solid-state reactions cannot be easily purified, the purity of the reagents must be assessed thoroughly, to ensure that the initial reaction mixtures have the intended stoichiometry. It should never be assumed that the composition of a reagent is that stated by the supplier. The reagent purity given by the supplier is usually expressed in terms of ‘trace metal basis’. This neglects the possible presence of elements such as oxygen or carbon. Moreover, depending on how reagents are stored, they can with time oxidise or absorb moisture or CO₂ from air. For instance, rare earth oxides stored in air often contain carbonates. Purification of reagents may be required prior to use. For example, heating in air is required to remove carbonates from rare earth oxides, and heating under a reducing atmosphere (e.g. H₂–Ar) may be required to remove oxides from germanium metal prior to use. Traces of oxides from metals such as sodium or barium, which should be stored under an inert atmosphere, can be eliminated by mechanically removing the outer surface inside a glovebox.

Another frequent source of problems is the container used for high-temperature reactions, as it may lead to contamination of the sample. Ideally, reaction containers must be inert, to avoid reaction with the reactants and products, and must also be able to withstand elevated temperatures. Crucibles and boats can be made of a variety of materials, including alumina, fused silica, graphite and platinum. The maximum operating temperatures of common crucible materials, under different atmospheres, are summarised in Table 1.1. When choosing a crucible, it is essential to ensure that it is chemically compatible with the reaction mixture that it will contain. For example, antimony, bismuth, selenium and tellurium all react readily with a platinum crucible on heating, whereas alkali and alkaline earth metals, titanium and aluminium will attack fused-silica containers at elevated temperatures. To avoid attack on the container, reactions using the evacuated, sealed-tube method may sometimes require the inside of the fused-silica tube to be coated with carbon, which can be achieved by pyrolysis of acetone or toluene. For very reactive reaction mixtures, it may be necessary to place a graphite or boron nitride crucible inside the glass ampoule or to use a welded niobium or tantalum tube, to contain the reaction mixture.

Mechanochemical synthesis, usually implemented in a ball mill, is an increasingly popular method for the preparation of thermoelectric materials, in addition to many other classes of organic or inorganic material.^24–26  The term mechanochemistry, which was first introduced by Ostwald,²⁷  refers to chemical reactions that are induced by mechanical energy,²⁸  instead of heat (thermochemistry) or light (photochemistry). The first mention of a mechanochemical reaction, milling of HgS with copper to produce elemental mercury, appears to have been made by a disciple of Aristotle, Theophrastus of Eresus, around 315 bce.²⁹  The establishment of mechanochemistry as a distinct branch of chemistry at the end of the nineteenth century is sometimes attributed to Matthew Carey Lea, who demonstrated that silver and mercury halides react differently when exposed to heat or to a mechanical action.³⁰ 

Mechanochemical synthesis can promote fast reactions between solids (or between a solid and a gas, for instance for the synthesis of hydrides,³¹ ) resulting in reduced reaction times. In mechanochemical reactions, high-energy impact collisions result in pulverisation, cold welding and re-welding of the grains, and introduce defects, disorder and strain within the crystallites, often leading to the formation of amorphous phases.³²  A number of theories and models for mechanosynthesis have been developed over the years, and these have been reviewed by Baláž.³³  The hot-spot theory proposes that friction or impact between powder particles generates high local temperatures, of over 1000 K, for 10⁻³–10⁻⁴ s, on surfaces of around 1 μm²,³⁴  while the magma–plasma model postulates that a special plasmatic state is formed at the spot where particles collide, which locally reaches temperatures of ca. 10 000 K.³⁵  However, such high temperatures are unlikely to be reached when organic components are involved, as decomposition would occur. In such cases, a mechanism involving the diffusion of reactants through a mobile phase, which may be a gas, eutectic liquid or amorphous solid, has been suggested.²⁶  Some mechanochemical reactions (e.g. synthesis of metal nitrides or sulfides) proceed through an exothermic self-propagating reaction, in which ignition occurs when the powder is activated by milling due to grain size reduction and defect formation.³⁶  Ignition results in an abrupt increase in the temperature of the reaction mixture.

There are different types of mills for mechanosynthesis (Figure 1.4), but all of them adopt the same basic operating principle: the powdered reagents are placed in a sealed jar, together with small balls, which are used to crush and grind the sample. In planetary ball mills, which are the most commonly used type for the synthesis of thermoelectric materials, milling entails planetary-like movement of the jars, which rotate in the opposite direction to the supporting disc. Attrition mills consist of a cylindrical chamber with a central rotating shaft that has impellers. The rotating shaft and impellers create a stirring motion in the powder and balls, causing mixing and grinding. In vibration (shaker) mills, grinding occurs by rapid linear oscillation of the jars, at frequencies of 10–30 Hz. Parameters such as the ball-to-powder ratio, the degree of filling and the milling time and speed are usually varied to optimise the synthesis of materials. To avoid oxidation of air-sensitive reagents or products, milling is usually performed under an inert atmosphere and the jar is loaded and sealed inside a glovebox. Alternatively, gas-tight jars that can be purged with the desired gas are commercially available. The balls and jars are made either of metal (stainless steel or hardened steel) or of a ceramic material (such as agate, zirconia or tungsten carbide) and, when choosing a container, consideration must be given to the possible contamination and reaction of the powdered sample with the jar and grinding balls. Care must also be taken when opening jars after milling, as finely ground metal powders could ignite spontaneously and cause a fire when exposed to air.

As illustrated in Table 1.2, mechanosynthesis has been successfully exploited for the preparation of a wide variety of thermoelectric materials, including Si–Ge alloys,³⁷  tellurides,^38–40  sulfides,^41–43  skutterudites,^44,45  half-Heusler compounds⁴⁶  and oxychalcogenides.⁴⁷  Ball milling can produce either the desired phase in a poorly crystalline form or a finely ground precursor, which will be converted into the desired phase following reactive hot pressing or spark-plasma sintering (SPS), described in Section 1.3. For example, the preparation of Cu₅FeS₄ by ball milling results in the formation of a poorly crystalline phase together with unreacted iron (Figure 1.5), and hot pressing is required to produce highly crystalline Cu₅FeS₄.⁴³  By contrast, filled skutterudites such as Ce_0.8Fe₃CoSb₁₂ and Ce_0.5Yb_0.5Fe_3.25Co_0.75Sb₁₂ form directly on milling.⁴⁴ 

Ball milling can also be exploited as a top-down approach to reduce the particle size in order to produce nanopowders. This is desirable as nanostructuring can lead to a substantial reduction in lattice thermal conductivity due to increased phonon scattering by interfaces.¹³  For example, ball milling of bulk Bi_0.5Sb_1.5Te₃ to produce nanoparticles with an average particle size of 20 nm results in a nanostructured material that exhibits a peak figure of merit, ZT = 1.4 at 373 K, due to the decrease in thermal conductivity.⁴⁸  However, ball milling to produce nanopowders can sometimes lead to contamination or to the formation of oxides on the highly reactive particle surfaces, which degrades the overall thermoelectric performance, as exemplified by a study of the type I clathrate Ba₈Cu_4.5Si₆Ge_35.5.⁴⁹  The consolidation process, which must be fast to avoid grain growth when sintering, is critical for nanopowders, with SPS usually being favoured (see Section 1.3.2). Ball milling is also often exploited as a top-down approach for the preparation of nanocomposites by mechanical mixing. For example, Li et al. achieved ZT = 1.33 at 373 K by preparing a composite of Bi_0.3Sb_1.7Te₃ with 0.4 vol% SiC nanoparticles.⁵⁰  The scalability of mechanochemical synthesis has been investigated by Baláž et al., who successfully produced materials such as tetrahedrite and Cu₂ZnSnS₄ using an industrial eccentric vibratory ball mill.^51,52  This type of mill can be used to produce batches of up to 50 kg of material.

Single crystals are needed to investigate the intrinsic properties of thermoelectric materials and are particularly useful when their properties show marked anisotropy. The Czochralski method⁵³  and the Bridgman technique⁵⁴  have been applied for the growth of large single crystals, whereas smaller crystals may be grown using fluxes^55–57  or by chemical vapour transport.^14,58  In the Czochralski method (Figure 1.6a), a single crystal is grown by dipping a seed crystal into a melt of the feed material. The seed crystal is attached to a moveable rod, which is slowly pulled, while rotating, from the melt. This causes the melt to crystallise at the interface between the seed and the melt, slowly growing a crystal. Crystal growth by the Czochralski method is affected by parameters such as the temperature gradient, pulling rate, the crucible materials and the reaction atmosphere. A detailed discussion of this technique has been provided by Hurle.⁵⁹  Examples of thermoelectric materials that have been successfully grown using this method include Bi₂Te₃ alloys,⁶⁰  Si_1–xGe_x⁶¹  and clathrates such as Ba₈Ga₁₆Ge₃₀.⁶²  Although the Czochralski method permits the growth of large, high-quality crystals, with diameters of the order of 50 mm, it is slow (with a growth rate of a few mm h⁻¹) and the growing equipment is expensive.

The Bridgman method involves the slow movement, typically vertical, of a crucible containing a melt of the feed material between a high- and a low-temperature zone (Figure 1.6b). The low- and high-temperatures zones are set to a temperature below and above the melting point of the material being grown, respectively. The crucible is typically made of fused silica and has a cylindrical shape with a sharp conical tip. When the melt is slowly lowered into the colder region, a crystal starts to grow in the conical tip, although a seed crystal may also be used. The crystal growth rate is in the range 0.1–30 mm h⁻¹. The Bridgman method is widely applicable, and examples of crystals of thermoelectric materials grown by this method include Bi₂Te_3−xSe_x,⁶³  SnSe,⁶⁴  the skutterudite CoSb₃,⁶⁵  Mg₂Si⁶⁶  and the clathrate Ba₈Cu₁₄Ge₆P₂₆.⁶⁷ 

The Czochralski and Bridgman methods work well for materials that melt congruently (i.e. the composition of the melt is identical with that of the solid). For materials that melt incongruently, alternative approaches can be adopted. In the flux method,^55–57  a molten solid, which can be a low-melting point salt or a metal, is used as solvent. The reagents and the flux are heated until the flux melts and reagents dissolve in the flux. On cooling, nucleation of crystals of the product occurs as the solution becomes supersaturated. The crystals can be separated from the flux by washing with a solvent that dissolves the flux, without destroying the crystals, or sometimes by centrifugation of the hot mixture. Crystals with edge lengths up to 10 mm can be grown in this way. Important considerations when choosing a flux are as follows:⁶⁸ 

1. the reagents must be soluble in the flux, with the solubility changing significantly with temperature;

2. the flux should not form stable compounds with the reagents (unless a reactive flux⁵⁶  is sought);

3. the flux should have a reasonably low melting point and low volatility and viscosity at the crystallisation temperature;

4. the flux should not react with the crucible;

5. the flux should be easily separated from the crystals; and

6. the use of toxic fluxes (e.g. mercury) should be avoided.

Examples of fluxes include eutectic LiCl–KCl and NaCl–KCl mixtures, with melting points of 625 and 931 K, respectively,⁵⁵  metals such as Sn, Ga and In (melting points 505, 303 and 430 K, respectively)⁵⁶  and polychalcogenide fluxes A₂Q–Q (where A is an alkali metal and Q a chalcogen).⁵⁷  Halide and polychalcogenide fluxes can be easily removed by washing with water, whereas a tin flux for instance can be removed by washing with dilute hydrochloric acid. An introductory account of the practical aspects of crystal growth using fluxes, focused on oxides and intermetallics, was given by Tachibana.⁶⁸  Selected examples of single crystals of thermoelectric materials^69–73  grown in this way are presented in Table 1.3.

Chemical vapour transport reactions, which were developed by Schäfer in the 1950s and 1960s,⁷⁴  involve a condensed phase that is volatilised in the presence of a gaseous transporting agent and subsequently deposited in another region of the reaction vessel, often as single crystals. For example, single crystals of ZnS can be grown using I₂ as the transporting agent.¹⁴  The transport reaction is carried out in an evacuated sealed ampoule, under a temperature gradient, which can be generated using a tube furnace with two independently heated zones. For the transport reaction to occur, a mixture of solid ZnS and the transport agent are placed at the hot end of the tube, at 1173 K, while the other end is maintained at 1073 K. The vapour transport reaction can be described by the following equilibrium:

where I₂, ZnI₂ and S₂ are all gases at the reaction temperature. This reaction is endothermic and, therefore, according to Le Chatelier's principle, the partial pressures of ZnI₂ and sulfur will be lower at the colder end of the tube, where ZnS will recrystallise. Endothermic reactions will always transport from the hot to the cold region, whereas for exothermic reactions transport will occur from the cold to the hot region. Common transporting agents, which have been discussed in detail by Binnewies et al.,⁵⁸  include Cl₂ and O₂ (auto transport following decomposition) for oxides, I₂ or HCl for chalcogenides and I₂ or SiI₄ for silicides. Although chemical vapour transport has not been widely used for thermoelectric materials, crystals of a very large number of binary, ternary and even quaternary chalcogenides have been grown using this method.⁵⁸  For instance, chemical vapour transport, using BiBr₃ as transport agent,⁷⁵  produces single crystals of Bi_0.5Sb_1.5Te₃, and 20 mm long needles of the sulfide Cu₂CdGeS₄ can be produced using CdI₂.⁷⁶  Controlling the reaction conditions permits the growth of nanowires instead of larger single crystals, as exemplified by those of lead chalcogenides.⁷⁷  Thermoelectric silicides can also be grown using this approach and, for example, single crystals of MnSi_1.73 and nanowires of CrSi₂ have also been reported.^78,79  Software to predict optimum reaction conditions for chemical vapour transport is available,⁸⁰  and extensive tables of previously-reported conditions for the growth of a large number of oxides, chalcogenides and intermetallics have been published.⁵⁸ 

Although solvothermal synthesis has traditionally been exploited for the growth of single crystals of a wide range of materials [e.g. quartz, zeolites, metal–organic frameworks (MOFs), metal oxides, metal chalcogenides], its main application in thermoelectrics is for the preparation of nanostructured powders. A solvothermal process is a heterogeneous reaction occurring in a sealed vessel and in the presence of a solvent heated above its boiling point and under pressure.^81,82  Under solvothermal conditions, it becomes possible to dissolve and crystallise materials that are insoluble at ambient temperature and pressure, due to changes in the physicochemical properties of the solvent (viscosity, polarity, dielectric constant, etc.). For example, the viscosity of water decreases markedly with increasing temperature and pressure. This accelerates ion diffusion and hence the solubilisation of the reactants and the growth of the final product. When the solvent is water, the process is termed hydrothermal synthesis. A wide range of minerals grow under hydrothermal conditions in Nature. The term solvothermal refers to the use of a non-aqueous solvent. Commonly used solvents include liquid ammonia, methanol, ethanol, ethylene glycol and ethylenediamine.^83,84  Given that solvothermal reactions are performed in sealed vessels, the pressure–temperature behaviour of the solvent at constant volume becomes important,⁸²  with the solvent reaching supercritical conditions in some reactions. For example, for water, the critical temperature and pressure are 647 K and 221.2 bar, respectively, and NH₃ reaches the critical point at 405 K and 111 bar. Below those conditions, a liquid and a gas phase coexist, but above the critical point only one phase, the supercritical phase, exists.

In order to carry out solvothermal reactions, vessels capable of containing a corrosive solvent at high temperatures and pressures are required. Small-scale reactions can be performed using sealed thick-walled Pyrex ampoules (or borosilicate vials with Teflon caps for reactions under mild conditions). Owing to its ease of use, the vessel most commonly employed for solvothermal reactions is the general-purpose autoclave (Figure 1.7), which consists of an inner Teflon cup and lid, enclosed in an outer stainless-steel body. The capacities of these autoclaves range between 23 and 125 mL, with a recommended maximum filling of two-thirds of the total volume. Autoclaves suitable for heating in a microwave oven are also available. Although the use of a Teflon liner provides inertness and avoids reactions with the metal container, it also limits the maximum operating temperature to ca. 523 K, as Teflon starts to deform above 473 K. Autoclaves capable of operating at higher temperatures and pressures have been described in detail by Byrappa and Adschiri.⁸¹ 

Solvothermal methods have been used to prepare thermoelectric materials such as Bi₂Te₃,^85,86  Bi_xSb_2−xTe₃,⁸⁷  PbTe,^85,88  Cu₂Se,⁸⁹  Sn_1−xSe,⁹⁰  CoSb₃⁹¹  and FeSb₂⁹²  (Table 1.4), with different sizes and morphologies. Surfactants and polymers are sometimes used as additives to control the morphology. Most of these solvothermal reactions involve redox processes. Whereas metal cations (e.g. Bi³⁺, Pb²⁺) may remain in their initial oxidation state, the chalcogen sources usually undergo reduction. For instance, when Te and SeO₂ are used as chalcogen sources, the following reductions occur:

Reducing agents, such as hydrazine and sodium borohydride, are added when reactions with large reduction potentials are involved. In other cases, the solvent plays a role in the redox processes. Amines and alcohols such as oleylamine and ethylene glycol often act as weak reducing agents. It is also known that sulfur readily dissolves in ethylenediamine and liquid ammonia, producing highly reactive species through a disproportionation process.^93,94  Alternatively, the decomposition of compounds such as thiourea and thioacetamide has been used to generate sulfide anions in situ.^95,96 

Although solvothermal synthesis provides a good degree of control over the particle size and morphology, the control over the stoichiometry of the final product may be more limited. Controlling the charge-carrier concentration through the introduction of dopants can also be challenging, as the composition of the final product may differ from that of the initial reaction mixture. The few studies describing the synthesis of doped thermoelectric materials using a solvothermal method include lead doping in Cu₂Se,⁹⁷  Cu₃SbSe₄⁹⁸  and SnSe⁹⁹  and copper doping in SnSe.¹⁰⁰  Moreover, to ensure that well-densified ingots are produced, complete removal of solvent, surfactants and any soluble products from the surface of the nanoparticles is required. This may require extensive washing, followed by heating under vacuum or an inert atmosphere,⁹⁸  which may lead to nanoparticle growth. A potential application of solvothermal synthesis is in the preparation of nanocomposites, either by coating a bulk material with nanoparticles¹⁰¹  or by preparing nanoparticles which are then mechanically added to a bulk material.¹⁰² 

Solution-based approaches permit the simultaneous growth of large numbers of nanoparticles with narrow size and shape distributions and are therefore of interest for the preparation of nanostructured thermoelectric materials.¹⁰³  The colloidal synthesis method, which entails the reaction of precursors in a high-boiling point organic solvent, in order to produce a colloidal suspension of nanoparticles, has been extensively developed for the preparation of metal and chalcogenide nanoparticles.^104–106  In this method, which can be carried out by a ‘heat-up’ or a ‘hot-injection’ technique (Figure 1.8), nanoparticles are produced through rapid nucleation followed by a growth process, with subsequent isolation from the reaction mixture and extensive washing. Reactions are typically carried out in an oxygen-free environment, using a three-necked round-bottomed flask connected to a condenser and Schlenk line, allowing purging with an inert gas.

Colloidal synthesis can involve redox reactions, although it often entails thermal decomposition of organometallic precursors.¹⁰⁴  For example, CoSb₃ can be prepared by a modified polyol ‘heat-up’ method, using sodium borohydride as a reducing agent for SbCl₃,¹⁰⁷  while the synthesis of CdSe¹⁰⁸  entails the injection of the organometallic precursor dimethylcadmium and trioctylphosphine selenide directly into a hot solvent, with thermal decomposition resulting in rapid nucleation and subsequent growth to generate CdSe nanoparticles. Commonly used precursors for the synthesis of chalcogenide nanoparticles include tertiary phosphine chalcogenides (e.g. trioctylphosphine selenide) and alkylamine chalcogenides (e.g. oleylamine sulfur), which are produced in situ by dissolving the chalcogen in a phosphine or an alkylamine.¹⁰⁹  Phosphines and amines also act as surface ligands, coordinating to the surfaces of the nanoparticles once formed, to stabilise them and prevent Ostwald ripening.¹⁰⁶  Hard–soft acid–base theory provides guidance on the selection of suitable ligands for cations and chalcogens.^106,109  Thermoelectric chalcogenides such as Bi₂Te_3−xSe_x,¹¹⁰  PbTe–PbS and PbS–Ag nanocomposites,^111,112  Cu₂SnSe_3,¹¹³  Cu₂CdSnSe₄¹¹⁴  and SnSe^115,116  have been prepared by colloidal synthesis (Table 1.5). The synthesis of nanoparticles of intermetallic compounds is significantly less developed. Metal alkyamides M(NR₂)_n and silylamides M[N(SiMe₃)₂]_n, although offering potential as precursors for metal and metalloid nanoparticles, are both expensive and extremely air and moisture sensitive, owing to their high reactivities. Amides, which can be formed in situ by reaction of metal chlorides with long-chain primary or secondary amines in the presence of a strong Brønsted base, have been proposed as precursors for the synthesis of monodisperse nanoparticles of metals and metalloids such as In, Sn, Bi, Sb, Ga, Cu and Zn and binary compounds such as Cu₆Sn₅, Cu₂Sb and Bi_xSb_1–x.¹¹⁷ 

As is the case with nanoparticles produced by solvothermal synthesis, removal of solvent, ligands and any soluble products from the surface of the nanoparticles is required prior to consolidation.¹⁰¹  This may be achieved, for instance, by treatment with hydrazine,¹¹⁸  although a widely used approach for the removal of organic impurities is annealing, which may result in particle growth, under an inert gas or vacuum.¹⁰¹  Doping of nanoparticles produced by colloidal synthesis, to control the charge-carrier concentration, is also challenging.¹⁰³  Strategies that have been adopted include the incorporation of a dopant precursor in the initial reaction solution, as exemplified by the synthesis of Sn- and Bi-doped Cu₃SbSe₄ nanoparticles.¹¹⁹  It is also possible to replace a fraction of the cations from the starting nanomaterial with a new type of cation, through a partial ion-exchange process.¹²⁰  Alternatively, a dopant can be added, in the form of a salt, to the nanopowder to be consolidated.¹⁰⁶  An advantageous application of colloidal synthesis may be for the production of inks for printing flexible thermoelectric devices.¹⁰³ 

Precursor methods are well developed for oxides, but there are very few examples of their use for other classes of thermoelectric materials. The use of precursors allows intimate mixing of the reagents, to facilitate atomic diffusion and reduce the temperature and time required for product formation, in comparison with the ceramic method (see Section 1.2.1). Precursor methods can simply entail coprecipitation, as exemplified by the synthesis of Ca₃Co₄O₉, through precipitation of a precursor from an aqueous solution of calcium carbonate and cobalt nitrate, following the addition of ammonium carbonate.¹²¹  Heating of this precursor in air results in decomposition, giving the final product, Ca₃Co₄O₉. However, when using a coprecipitation method, it may be difficult to ensure that the precursor and therefore the final product have the desired stoichiometry, as not all cations will precipitate at the same pH value. This disadvantage may be avoided by using a sol–gel method,¹²²  in which a sol, consisting of a concentrated solution or a stable suspension of colloidal particles of the reagents, is prepared. Gelation of the sol by concentrating it or maturing it produces a gel, which can be described as a porous and continuous solid network surrounding a continuous liquid phase. The solid network in the gel is either an agglomeration of colloidal particles or, in polymeric gels, an aggregation and entanglement of polymer chains. Decomposition of the gel by heating gives the final product.

Traditionally, the term ‘sol–gel’ has been applied to syntheses employing metal alkoxides. In this case, a sol is formed through hydrolysis and condensation of alkoxide precursors, with cross-linking resulting in the formation of a metal–oxane gel. However, more recently, the term ‘sol–gel’ has been extended to cover a wide range of chemistries, which have been discussed in detail by Danks et al.¹²³  Five key types of gels have been identified in sol–gel chemistry:

1. Colloidal gels, which consist of colloidal particles connected by van der Waals interactions or by hydrogen bonding. A colloidal gel has been used, for instance, to produce Al-doped ZnO.¹²⁴ 

2. Metal–oxane polymer gels, which are inorganic polymers. An alkoxide sol–gel method has been used to prepare TiO₂ nanoparticles dispersed in Ba_0.22Co₄Sb₁₂.¹²⁵ 

3. Metal complex gels, which consist of weakly-bonded metal complexes. This involves the use of small molecules (often chelating), such as citric acid. For example, the citrate sol–gel method has been exploited for the synthesis of thermoelectric Ca₃Co₄O₉.¹²⁶ 

4. ‘Pechini’ method gels, formed by in-situ polymerisable metal complexes. The Pechini method entails a transesterification between metal citrates and ethylene glycol, to form an extended covalent network.

5. Coordinating and cross-linking polymer gels. These contain polymers, such as alginate or polyvinylpyrrolidone, that can coordinate to metal cations.

Although sol–gel methods have been primarily developed for oxides, there are some approaches that are applicable to the synthesis of nitrides¹²⁷  and chalcogenides.^128,129  For instance, nitrides can be produced by ammonolysis of dialkylaminosilanes with ammonia, to form the sol, followed by condensation to make imide (–NH–)linkages and form a gel.¹²⁷  Sulfides can be made by an alkoxide sol–gel route, by replacing the hydrolysis by water which is used to produce oxides, with thiolysis with H₂S.^128,129  In the colloidal gel approach, metal chalcogenide nanoparticles are produced and then condensed to form a gel.^128,129 

Precursor or sol–gel methods have only rarely been used for non-oxide thermoelectric materials. Examples include the synthesis of Bi₂Te₃ and the skutterudite CoSb₃ by a coprecipitation route, followed by calcination and reduction under a hydrogen atmosphere.^130,131  Oxide precursors can also be reduced to intermetallic skutterudites by heating them on a magnesium bed, under an inert atmosphere.¹³² 

Following the synthesis of a thermoelectric material (see Section 1.2), consolidation is generally required to produce dense ingots for property measurements. The consolidation of powders is achieved by sintering, which can be defined as ‘the extension of the contact area between powder particles in the solid state, by the transport of material across or around pores, under appropriate conditions of time, temperature, pressure and atmosphere’.¹³³  Sintering of crystalline materials occurs in three stages (Figure 1.9).¹³⁴  In the initial stage, adjacent particles bond through the formation and growth of contact areas between particles, but densification is limited. In the intermediate stage, significant densification occurs and the contact areas between particles grow, but there is still a continuous network of pores. In the final stage, pore shrinkage and closure occur, the remaining pores become isolated and there is significant grain growth. The contact areas, also known as necks, grow mainly by atomic diffusion mechanisms, which are caused by gradients in vacancy concentration. In some cases, evaporation–condensation mechanisms or a liquid phase may also be involved in the growth of contact areas. The driving force for sintering is the decrease in surface area that occurs with increased densification. During the sintering process, the powder decreases its total surface free energy by reducing the total surface area. The densification rate increases with increasing temperature and time, and also with a decrease in the particle size of the ‘green’ (unsintered) powder. Sintering is usually carried out at temperatures in the range of 67–80% of the melting point of the solid.¹³⁵ 

In principle, cold pressing using a stainless-steel die, followed by annealing at high temperature in a furnace, is the simplest approach to consolidate samples. In practice, this method often results in samples with low relative densities (with respect to the crystallographic density) and poor mechanical strength. Thermoelectric materials are usually consolidated by pressure-assisted sintering, using either hot pressing (HP) or spark-plasma sintering (SPS). Both methods are discussed below. The use of hot forging to increase the texture of thermoelectric materials will also be described.

Hot pressing (HP) entails the application of a uniaxial external pressure while simultaneously heating the sample, although isostatic HP, in which the pressure is applied uniformly with a gas, can also be used.¹³⁶  HP increases the densification kinetics and, given that grain growth is not directly linked to the applied pressure, it can be used to limit grain growth and to consolidate materials using shorter sintering times and lower temperatures. Whereas diffusion is the main densification mechanism in pressureless sintering, plastic deformation and creep can become major contributors under an applied pressure. A schematic representation of a uniaxial hot press is shown in Figure 1.10. A die, which is usually made of graphite, is loaded with the powder to be consolidated, between two plungers. To facilitate the extraction of the sample after pressing, disposable graphite foil liners are often inserted. These can be placed on the top and bottom of the sample and also around the die walls. Pressure is applied while the die is heated, under vacuum or in a controlled atmosphere, using resistance heaters. Usually, the heating elements are located in the walls of the chamber where the die is contained and, as a consequence, heating rates are low. Alternatively, the heat can be supplied by RF induction, to allow rapid consolidation and heating rates, comparable to those achieved by SPS.¹³⁷ 

The use of steel dies is not advisable, as steel deforms owing to creep at high temperatures, and also the powdered sample may react with the steel at high temperatures. Under an inert atmosphere or vacuum, graphite will be stable at the temperatures required for the HP of most thermoelectric materials. However, care should be taken to avoid the presence of air/oxygen, as graphite oxidises above 400 °C, which can result in the release of carbon monoxide. As graphite can withstand only relatively low pressure loads before failure occurs, its use imposes a limit on the pressures that can be applied to the sample, which are typically between 0 and 100 MPa. Designs of dies capable of operating at higher pressures (up to 400 MPa), which would be suitable for HP and SPS, have inserts made of silicon carbide or tungsten carbide.^138,139  Alternatively, the use of dies made of carbon fibre-reinforced graphite has been suggested.¹³⁸ 

HP can successfully produce ingots with densities above 97% of the theoretical (crystallographic) density, once the optimum conditions of temperature, pressure and time have been identified. Temperature is often the critical parameter that needs to be adjusted, because atomic diffusion increases exponentially with increasing temperature. If the temperature is too high, sample decomposition or changes in composition due to evaporation of more volatile elements may occur. If the temperature is too low, the density of the ingot will be too low for transport property measurements (see Section 1.4) and the mechanical strength of the ingot will also be poor. The effect of the atmosphere used during pressure-assisted sintering should also be considered. For instance, oxides consolidated under vacuum or a reducing atmosphere may have an increased concentration of oxygen vacancies.

Spark-plasma sintering (SPS), also known as field-assisted sintering (FAST) or electric current-activated sintering (ECAS), differs from HP in the method used to heat the sample. In SPS, the mechanical loading system also acts as a high-power electrical circuit, and heating is achieved by passing a pulsed DC electrical current through the sample (if it is conductive) or through the die. The applied voltage, which is generally below 10 V, produces large currents, typically between 1 and 10 kA, resulting in efficient Joule heating and therefore extremely fast heating rates, of up to 1000 K min⁻¹. SPS permits significant reductions in the total duration of the sintering process to be achieved. This offers the benefit of reducing grain growth, a particularly desirable feature when consolidating nanopowders. A schematic diagram of an SPS system, highlighting the similarities and differences between HP and SPS, is shown in Figure 1.10b. Despite its name, there is no evidence of the presence of a plasma during the SPS process.¹⁴⁰  There are a number of comprehensive reviews on SPS.^141–144 

Most materials can be consolidated equally well by HP or by SPS. For instance, both approaches are suitable for obtaining well-densified Ca₃Co₄O₉ ceramics¹⁴⁵  or Na-doped PbTe,¹⁴⁶  although differences in the microstructure may result in slightly different thermoelectric performances. For colusite (Cu₂₆V₂Sn₆S₃₂), Bourgès et al. have shown that HP at 1023 K produces an improvement of around 300% in the figure of merit at 675 K, in comparison with SPS at 873 K.¹⁴⁷  This has been attributed to the formation of disordered domains and sulfur volatilisation due to the higher temperature used for HP processing.

There are some specific issues related to the nature of the heating in SPS that need to be considered. For electrically conductive materials, the flow of electrical current through the powder will not occur homogeneously; rather, it will depend on the packing of particles in the unsintered body. This will result in the formation of local hot spots, and this local variation in temperature may influence the final microstructure of the consolidated material, which can sometimes show a current percolation pattern.¹⁴¹  Moreover, for thermoelectric materials, the flow of a DC current through the powder during the SPS process may result in uneven heating, due to the Peltier effect, hence large differences in grain size between the top and bottom faces of the consolidated sample can occur. Peltier heating or cooling occurs if there is a discontinuity in the Peltier coefficients of two materials in contact. As semiconductors have much higher Peltier coefficients than the electrodes and the die, Peltier heating or cooling will occur at those interfaces. It has been estimated that for semiconductors, Peltier heating can reach up to 10% of the total heating delivered by SPS.¹⁴¹ 

The large DC currents used in SPS may also drive the transport of mobile ions, resulting in inhomogeneous samples, with compositional gradients. For example, transport of Li⁺ ions occurs when sintering Li_1.3Fe_0.3Ti_1.7(PO₄)₃ by SPS. This results in a colour gradient between the opposite faces of the consolidated ingot due to compositional changes and partial decomposition of the material.¹⁴⁸  The diffusion of Na⁺ ions under SPS conditions has been used to produce the intermetallic clathrate Na₂₄Si₁₃₆ from a Na₄Si₄ precursor.¹⁴⁹  Given the known issues of degradation of phonon–liquid electron–crystal (PLEC) thermoelectric materials under a DC electric current,¹⁵⁰  diffusion of Cu⁺ ions may also occur during the consolidation of materials such as Cu_2−xS or Cu_2−xSe by SPS. However, this does not appear to have been investigated.

Many thermoelectric materials have highly anisotropic properties^64,151–153  and therefore, for polycrystalline samples, preferential alignment of the crystallites along a favourable transport direction can be exploited to enhance the figure of merit. As shown in Table 1.6, there can be fairly large variations between the in-plane and out-of-plane electrical resistivities, whereas the anisotropy in the Seebeck coefficient is often less marked. Hence alignment along the direction with the lowest electrical resistivity should be advantageous. One of the most straightforward approaches to induce preferential orientation is that of hot forging, which entails repeated consolidation of the ingot by HP or SPS, using dies with progressively larger diameters (Figure 1.11). Following hot forging, comparison of the X-ray diffraction pattern of the consolidated ingots with that of a randomly oriented powder sample, through the calculation of the Lotgering factor, F, can be used to assess the degree of texture.¹⁵⁴  In the Lotgering method, the orientation factor is defined as

where

I(00l) and I(hkl) are the measured intensities of reflections with (00l) and (hkl) Miller indices, respectively, and p₀ is the value of p for a randomly oriented powder. The orientation factor takes values of F = 0 for a randomly oriented sample and F = 1 for a fully oriented or textured sample.

Hot forging has been investigated for a number of thermoelectric materials. For example, a 22% improvement in the maximum figure of merit of n-type Bi₂Te_2.7Se_0.3, which increases from ZT = 0.85 to 1.04 at 400 K in the direction perpendicular to pressing, can be achieved by hot forging samples prepared by ball milling.¹⁵⁵  Hu et al. reported hot-forged n-type Bi_1.95Sb_0.05Te_2.3Se_0.7 with ZT ≈ 1.3 at 450 K, due to the presence of multiscale microstructures.¹⁵⁶  For Bi_1.95Sb_0.05Te_2.3Se_0.7, five successive hot-forging steps were used to increase the Lotgering factor F to 0.79, and also to introduce multiscale structural defects, which resulted in a lattice thermal conductivity as low as κ_L = 0.31 W m⁻¹ K⁻¹ at 405 K.¹⁵⁶  Hot forging of Ca₃Co₄O₉ enhances the electrical conductivity without lowering the Seebeck coefficient, and therefore improves the power factor, which reaches a value of S²σ = 0.8 mW mK⁻² at 1073 K.¹⁵⁷  In Bi_0.875Ba_0.125CuSeO, hot forging increases the ratio σ/κ_total in the out-of-plane direction and leads to an increase in the figure of merit from ZT ≈ 1.1 to ZT ≈ 1.4 at 923 K after three hot-forging steps.¹⁵¹ 

Most commercial thermoelectric modules are based on Bi₂Te₃ alloys, for which aligned polycrystalline ingots and single crystals are grown from a melt, for instance by zone melting.¹⁵⁸  The n- and p-type legs for the modules are produced by cutting from large ingots, without any need for powder consolidation. Owing to the limitations of HP and SPS for large-scale production, hot extrusion, which entails HP of the thermoelectric material through a draw die, is another method used commercially to produce Bi₂Te₃ thermoelements. Hot extrusion introduces texture and, more importantly, improves the mechanical properties of Bi₂Te₃.¹⁵⁸  A study by Ivanova et al.¹⁵⁹  has shown that hot extrusion of p-type Bi_0.6Sb_1.4Te₃ produces materials with figures of merit comparable to those of Czochralski-grown single crystals. Hot extrusion has also been applied to produce n-type PbTe rods of up to 2 kg, which are less brittle than conventionally processed materials and exhibit a thermoelectric performance comparable to those of Bridgman-grown materials.¹⁶⁰ 

High-pressure torsion, in which the sample undergoes severe plastic deformation when torsional strain is applied under pressure between two anvils,¹⁶¹  has also been exploited in an effort to improve the figure of merit. High-pressure torsion reduces the grain size and produces a high concentration of deformation-induced defects, lowering the lattice thermal conductivity. For example, skutterudites with figures of merit approaching ZT = 2 have been produced using this method.¹⁶² 

Prior to measurements of physical properties, it is good practice to determine the density of the consolidated materials and to compare this with the theoretical (crystallographic) density, determined by the crystal structure of the material. Property measurements should be carried out on samples with densities as close as possible to the theoretical density. Figures of merit for porous samples (non-fully densified) often differ markedly from those of well-consolidated samples, as porosity increases the electrical resistivity and decreases the thermal conductivity.^163,164  After consolidation, it is also advisable to assess the sample purity and homogeneity using X-ray diffraction, scanning electron microscopy (SEM) and energy-dispersive X-ray analysis (EDAX), as the presence of impurities introduced during the consolidation process (see Section 1.3) may affect the charge-carrier concentration of the main phase. Similarly, sample inhomogeneities could result in errors in the figure of merit if the contributing properties were to be measured on different samples;¹⁶⁵  it is preferable to measure all properties on the same ingot wherever possible. When property measurements are carried out along different directions of the consolidated ingots, consideration must be given to the texture of the samples. For materials with anisotropic properties, measurement of the electrical and thermal conductivities along different directions could result in an overestimation of the figure of merit by up to 60%.¹⁶⁶ 

Determination of the figure of merit usually involves measuring the electrical conductivity, σ, Seebeck coefficient, S, and thermal conductivity, κ, although direct measurement of ZT is also possible (see Section 1.4.5).¹⁶⁷  For the complete characterisation of thermoelectric materials, Hall coefficient measurements (see Section 1.4.3), which provide information on charge-carrier concentration and mobility, are also desirable. In the following sections, the principles on which these measurements are based will be discussed, together with their limitations, sources of error and important considerations for the reliable determination of thermoelectric properties. A discussion of available standards for these measurements can be found in ref. 168.

The electrical conductivity, σ, is an intrinsic property of matter that quantifies how well a substance conducts an electrical current. The electrical conductivity depends on the number of carriers, n, and their mobility, μ:

where e is the charge of the carrier. The electrical conductivity is inversely proportional to the electrical resistivity, ρ:

The electrical resistivity depends on the resistance to flow of an electrical current, R, and the dimensions of the material:

where R is given in ohms (Ω)and L and A are the length and the cross-sectional area of the sample, respectively. The SI unit of electrical resistivity is Ω m and the electrical conductivity is given in S m⁻¹ (or Ω⁻¹ m⁻¹). For intrinsic (non-doped) broadband semiconductors, electrical conduction requires thermal excitation of charge carriers across a bandgap, and therefore the electrical conductivity may be given by

where σ₀ is a proportionality constant, E_g is the bandgap energy, k_B is the Boltzmann constant and T is the absolute temperature. The electrical conductivity of intrinsic semiconductors increases with increasing temperature. However, thermoelectric materials are heavily doped and often behave as degenerate semiconductors. In a degenerate semiconductor, the level of doping is sufficiently high that the Fermi level, E_F, is located in either the valence or the conduction band. This causes the material to behave like a metal, with the electrical conductivity exhibiting a metal-like decrease with increasing temperature.

In materials with narrow bands, where the charge carriers behave as localised particles (small polarons), electronic conduction may occur by hopping. In this case, the electrical conductivity may be given by¹⁶⁹ 

where W is the activation energy for hopping. A variable-range hopping (VRH) behaviour may be found in materials that are disordered and where the charge carriers are hopping between localised states. This will result in the following temperature dependence for the electrical conductivity:¹⁷⁰ 

where T₀ denotes the hopping energy and j refers to the dimensionality of VRH conduction.

The most straightforward approach to measure electrical resistivity is a two-probe method (Figure 1.12a), in which two measuring probes are connected to the sample, and the voltage drop across the sample is measured when a current passes. According to Ohm's law, the resistance of the sample will be given by

where V and I are the voltage and electrical current, respectively, and therefore the resistivity can be determined using eqn (1.15). However, two-probe measurements include contact resistances, in addition to contributions from the resistance of the wires and from the measuring equipment, which increase the measured resistance above its true (intrinsic) value. Although the two-probe technique could be used for samples with high resistivities, it is unsuitable for thermoelectric materials, which by their very nature have low electrical resistivities. For accurate measurement of such materials, a four-probe method should be used (Figure 1.12b).

In the four-probe method, the electrical current flows between two contacts at the ends of the sample under test, and the voltage drop is measured between two inner contacts. This arrangement eliminates the contributions described above. However, to ensure that there is a uniform current distribution between the two inner voltage contacts, these must be placed at a distance at least equivalent to one width of the sample from the ends.¹⁷¹  Poor thermal contact between the sample and the thermocouple can lead to inaccurate temperature readings, which can introduce errors into the measurements.¹⁶⁵  Another potentially important source of error in the case of thermoelectric materials is the presence of a temperature difference across the sample, which can arise due to existing thermal gradients, Joule heating or the Peltier effect. If the temperature at each of the two inner contact points is different, an additional Seebeck voltage will be generated and, given that thermoelectric materials necessarily exhibit reasonably large Seebeck coefficients, this could be appreciable. The total voltage measured across the sample is the sum of the resistive voltage, IR, and the Seebeck voltage, SΔT:

This problem can be minimised by determining the average voltage when the same current is passed in opposite directions (I⁺ and I⁻), as this will remove the Seebeck contribution. The average voltage will be given by¹⁷² 

where I⁺ is the positive current and I⁻ the negative current. The quality of the electrical contacts, which should be ohmic, can be tested with an I–V sweep. This requires measurements of current and voltages over a range of currents; a plot of V versus I will be linear if the contacts are ohmic. The values of resistivity determined by the four-probe method are sensitive to errors in the measurement of the dimensions of the sample. This can introduce an error of up to 10% in the measured resistivity.¹⁷³ 

An alternative approach to the four-probe method, which is less sensitive to errors in geometric factors, is the van der Pauw method (Figure 1.12c).¹⁷⁴  This allows measurements to be made on thin samples of arbitrary shapes, provided that they have a uniform thickness. Four contacts (A, B, C and D) are placed on the edges of the sample and the resistances R_AB,CD and R_BC,DA are then defined as

where V_D − V_C is the voltage difference between D and C, I_A−B is the current between A and B, etc. The electrical resistivity of the material is given by

where d is the thickness of the sample and f is a function dependent on R_AB,CD and R_BC,DA, details of which can be found in ref. 175.

For measurements at low temperatures, electrical contacts can be made using electrically conductive epoxies or solders, whereas at high temperatures, contacts are made by pressing probes directly onto the surface of the sample. Tritt provided a useful summary of contact techniques.¹⁷²  Most measurements reported for thermoelectric materials are carried out using commercial instruments [Quantum Design Physical Properties Measurement System (PPMS) for low temperatures and Linseis LSR series or Ulvac ZEM series for measurements above room temperature], all of which use a four-probe method. There are also reports of custom-made apparatus that can accommodate a variety of sample sizes and geometries and often combine resistivity measurements with those of other properties.^176,177 

The Seebeck effect,¹⁷⁸  which lies at the heart of thermoelectric power generation, can be described by considering a thermocouple, formed by two dissimilar materials, A and B, joined at two junctions, labelled 1 and 2 (Figure 1.13). If junctions 1 and 2 are at different temperatures, T₁ and T₂, an electrical potential will develop between points 3 and 4. The voltage difference (dV) between points 3 and 4 will be directly proportional to the temperature difference (dT), with the coefficient of proportionality being the Seebeck coefficient, S:

where S_AB is the relative Seebeck coefficient (S_AB = S_A − S_B) between materials A and B.

Although the SI units of the Seebeck coefficient are V K⁻¹, values are usually presented in the more convenient units of μV K⁻¹. In order to determine the absolute Seebeck coefficient for a given material, A, the Seebeck coefficient for the reference material, B, must be known over the temperature range of the measurements. Usually the reference materials (i.e. the measurement wires) used for Seebeck coefficient measurements are metals with small values of the Seebeck coefficient. For example, platinum exhibits an absolute Seebeck coefficient at room temperature of only −5 μV K⁻¹.¹⁷⁹  Representative values of the Seebeck coefficient for some typical metals and semiconductors are given in Table 1.7.^180,181 

From a physical point of view, the Seebeck coefficient is the heat per carrier over temperature or the entropy per carrier, and can be expressed as

where c is the heat capacity (i.e. heat/temperature) and q is the charge of the carrier.¹⁸²  Therefore, the Seebeck coefficient provides information about the sign of the charge carriers. Materials in which electrical conduction is dominated by electrons have a negative Seebeck coefficient and when conduction is dominated by holes, the Seebeck coefficient is positive. If both types of charge carrier are present, the Seebeck coefficient is given by their average, weighted by their electrical conductivities:¹⁷⁸ 

where σ_n and σ_p are the electrical conductivities of electrons and holes, respectively. In intrinsic semiconductors, both types of charge carriers contribute to the electrical conduction and, as S_n and S_p have opposite signs, the total Seebeck coefficient will be reduced. Therefore, in order to maximise the Seebeck coefficient, it is necessary to minimise the contribution of the minority charge carriers.

Since the Seebeck coefficient is related to the heat transported per carrier, if we consider a metal as a degenerate electron gas, with E_F ≫ k_BT, the Seebeck coefficient will be given by:¹⁸² 

where E_F is the Fermi energy, k_B the Boltzmann constant and k_B/e ≈ 87 μV K⁻¹ is the Seebeck coefficient of a classical electron gas. Therefore, the Seebeck coefficient of a metal will be much less than 87 μV K⁻¹ (as E_F ≫ k_BT) and will increase with increasing temperature. Typically, the absolute value of the Seebeck coefficient of a metal, |S|, is between 1 and 10 μV K⁻¹.

For an extrinsic semiconductor with one type of charge carrier, the following expression has been derived:¹⁸² 

and therefore semiconductors exhibit Seebeck coefficients larger than 87 μV K⁻¹ (since E_g ≫ k_BT to avoid intrinsic conduction) that increase with decreasing temperature. It should be noted, however, that the temperature dependence of the Seebeck coefficient can show greater complexity; models for semiconductors containing more than one type of carrier or for conduction through hopping have been discussed by Chaikin.¹⁸²  For instance, when hopping conduction occurs, the Seebeck coefficient is given by the Heikes formula:¹⁸³ 

where x and 1 − x represent the content of the lower (Mⁿ⁺) and higher (M⁽ⁿ⁺¹⁾⁺) oxidation states present in the material. This expression is widely used to interpret Seebeck data for transition metal oxides.¹⁸⁴ 

In semiconductors, the onset of bipolar conduction occurs when the temperature is high enough for thermal excitation of electrons across the bandgap to take place, and is characterised by a change in the temperature dependence of the Seebeck coefficient. The maximum absolute value of the Seebeck coefficient, |S_max|, can be related to the bandgap by the following expression:¹⁸⁵ 

where T_max is the temperature at which |S_max| is reached. Although eqn (1.30) provides a convenient way to estimate the bandgap, significant deviations from the true value of the bandgap can occur, particularly when |S_max| < 150 μV K⁻¹.¹⁸⁶ 

There are two approaches to measurement of the Seebeck coefficient: the integral and the differential methods.¹⁸⁷  In the integral method (or large ΔT), one end of the sample under test is maintained at a constant temperature, T₁, while the temperature at the other end, T₂, is varied through a temperature range, T₂ = T₁ + ΔT. In the differential method, which is the more commonly used approach in both commercial and custom-made equipment, a small temperature gradient (ΔT = T₂ − T₁) is applied at the average temperature of interest, T_ave = (T₁ + T₂)/2 (where T_ave ≫ ΔT), and the Seebeck coefficient at T_ave is obtained as V/ΔT. This procedure is repeated at different average temperatures, in order to determine the Seebeck coefficient as a function of temperature.

Measurements using the differential method can be carried out under steady-, quasi-steady- and transient-state conditions, all of which have been discussed in detail by Martin et al.¹⁸⁷  A brief description of the steady-state differential method follows, in order to highlight potential sources of error in Seebeck coefficient measurements. In the steady-state differential method, several (V, ΔT) data points are collected, while ensuring that the sample temperature is stabilised before each data point collection. The Seebeck coefficient at T_ave is calculated as the slope of a V versus ΔT plot. This eliminates offset errors [as in practice the line of best fit will not pass through the origin, (V = 0, ΔT = 0), due to thermocouple inhomogeneities and non-equilibrium contact interfaces]. The disadvantage of the steady-state method is that equilibration of the sample before each (V, ΔT) data point measurement is time consuming, hence quasi-steady-state conditions are often used. A fairly recent study has shown that there is little difference between the steady-state and quasi-steady-state methods if thermal and electrical contacts are good.^188,189  If possible, it is advisable to measure (V, ΔT) data points for both an increasing and a decreasing ΔT. Any hysteresis in the data may be an indicator of poor thermal contact.¹⁶⁵ 

Seebeck coefficient measurements can be carried out using two- or four-probe arrangements (Figure 1.14). These two arrangements have arisen in part from the desire to combine Seebeck coefficient measurements with the determination of either thermal or electrical conductivity. In the two-probe arrangement, V and ΔT are measured on the two probes located at either end of the sample (Figure 1.14a). In this arrangement, the thermocouple and voltage wires are not in direct contact with the sample and are instead embedded in metal blocks at either end. Good thermal and electrical contact with the sample is therefore needed. In the four-probe arrangement, which is required for simultaneous measurements of electrical conductivity and Seebeck coefficient, the two end probes are used to create a temperature gradient, while V and ΔT are measured at two points on the sample separated from the two end probes (Figure 1.14b). In this arrangement, the two probes used for the V and ΔT measurements are in direct contact with the sample.

There has been very little comparison of the results obtained using these two arrangements. According to recent work,^188,189  the difference between Seebeck coefficient values measured with a two-probe arrangement and those measured with a four-probe arrangement increases with increasing temperature. Owing to cold finger effects, the four-probe contact geometry overestimates the Seebeck coefficient, whereas a two-probe geometry underestimates it. Round-robin studies, in which the same sample is measured in several laboratories world-wide, indicate that a variation of ±5% in the measured Seebeck coefficient will occur at room temperature and that this variation increases with increasing temperature.^173,190  For high-temperature Seebeck measurements, commercial instruments (Linseis LSR or Ulvac ZEM series), which perform simultaneous Seebeck coefficient and electrical conductivity measurements using a four-probe arrangement, are widely used. An overview of custom-made apparatus for Seebeck measurements can be found in ref. 187.

Hall effect measurements permit the determination of the charge-carrier type, together with the carrier density and mobility. Although this information is not required to calculate the figure of merit, ZT, it is often important in order to gain a good understanding of a thermoelectric material. The power factor usually reaches an optimal value at charge-carrier concentrations of around 10¹⁹ cm⁻³, therefore the optimisation of the figure of merit requires doping of thermoelectric materials. The physical basis of the Hall effect is as follows:^191,192  when an electrical current flows through a material and a magnetic field is applied perpendicular to the current, a voltage, known as the Hall voltage, is produced in the direction perpendicular to both the electric and magnetic fields. This is illustrated in Figure 1.15, where an electrical field, applied along the x-direction, results in a current I_x. When a magnetic field, B_z, is applied along the z-direction, charge carriers are deflected along y, producing a Hall voltage, V_H. As shown in Figure 1.15, electrons and holes are deflected in opposite directions, hence the sign of the Hall voltage provides information on the identity of the charge carrier. The magnitude of the Hall voltage, which depends on I_x, B_z and the thickness of the sample, d, is given by the expression

where the Hall coefficient, R_H, is related to the concentration of charge carriers, n, by

The Hall coefficient is negative for conduction by electrons and positive when the charge carriers are holes. The charge-carrier mobility, μ, may be determined using

It should be noted, however, that the equations above are based on the free electron model, hence for materials with complex band structures, the mobility and charge-carrier concentration determined from Hall effect measurements, could differ significantly from the true values of charge-carrier concentration and mobility.¹⁶⁵  Nonetheless, Hall measurements provide a useful comparison of compositionally induced changes in charge-carrier densities on chemical substitution within a given material system.

The most common arrangement for Hall coefficient measurements is a van der Pauw geometry (Figure 1.12c), where the same four contacts are used for resistivity and for Hall coefficient measurements.^174,193  The problems associated with measurements of the Hall effect are similar to those found for the measurement of electrical resistivity (see Section 1.4.1). Errors will arise if the contacts are non-ohmic or if the contacts are displaced from their correct positions.¹⁹⁴  As the Hall voltage is often very small, other contributions, such as the Seebeck voltage arising from a temperature gradient in the sample, can be significant. Hall measurements can be performed in DC mode and, by repeating the measurement with the magnetic field reversed, most offset voltages are removed (with the exception of a small contribution from the Ettinghausen effect).¹⁷²  Many thermoelectric materials have large charge-carrier mobilities, which can be measured using a DC Hall effect technique, but for measurements on materials with low mobilities an AC method is more suitable.¹⁹⁵ 

Thermal conductivity is a transport property that describes the movement of heat from high- to low-temperature regions. The thermal conductivity of a material, κ, is defined as the coefficient of proportionality between the heat flux and the temperature gradient across the material. For a material with a temperature gradient along the x-direction, the following expression can be written:

where the heat flux, J_x(Q), is the amount of heat that flows per unit time per unit area (measured in W m⁻²), κ is the thermal conductivity and dT/dx is the temperature gradient along the x-direction. The thermal conductivity of a material determines whether it is a thermal conductor (large κ) or a thermal insulator (low κ). Typical values of thermal conductivities are given in Table 1.8.^196,197  The units generally used for thermal conductivity are W m⁻¹ K⁻¹.

Heat in solids is transported principally by quantised vibrations, termed phonons, and by electrons. The total thermal conductivity can be considered to be the sum of two contributions:

where κ_L is the lattice thermal conductivity, due to phonon transport, and κ_e is the electronic thermal conductivity. The electronic thermal conductivity, κ_e, and the electrical conductivity, σ, are related through the Wiedemann–Franz law:

where L₀ is the Lorenz number, which for most metals takes a value of L₀ = 2.44 × 10⁻⁸ W Ω K⁻².

A high thermoelectric figure of merit requires a material to possess a low total thermal conductivity and a high electrical conductivity. Since σ and κ_e are directly related, a commonly used approach to improve the figure of merit is to reduce κ_L. The value of κ_L is normally estimated using eqn (1.35) and (1.36). Although some thermoelectric materials have a Lorenz number close to the degenerate limit of L₀ = 2.44 × 10⁻⁸ W Ω K⁻², this is not always the case, and an incorrect value of the Lorenz number can result in erroneous values of κ_L. A straightforward approach to estimate the value of L₀, from the Seebeck coefficient, is through the expression¹⁹⁸ 

where the units of L₀ are 10⁻⁸ W Ω K⁻² and those of S are μV K⁻¹. The estimated error in the Lorenz number determined using eqn (1.37) is below 20% for the most common band structures (parabolic or non-parabolic) and scattering mechanisms (acoustic phonon, ionised impurities, polar and alloy scattering) found in thermoelectric materials.¹⁹⁸ 

Compared with the measurement of electrical conductivity or Seebeck coefficient, reliable measurement of thermal conductivity is more challenging, as it is difficult to avoid heat losses completely. As discussed by Goldsmid,¹⁹⁹  radiative heat transfer will still occur even when a material is placed under the high vacuum necessary to prevent heat transfer by conduction and convection.

The most commonly used technique for the determination of the thermal conductivity of thermoelectric materials is the laser-flash method, which will be discussed here. However, it should be noted that other methods are available (Table 1.9), which can be classified into the two broad categories of steady-state and transient techniques. In the steady-state method, the thermal conductivity is determined using a temperature difference that remains unchanged with time, whereas transient techniques measure a time-dependent energy dissipation process.²⁰⁰  These methods, which are beyond the scope of this chapter, have been reviewed in detail in ref. 200 and 201.

The laser-flash method measures the thermal diffusivity, D, which is related to the thermal conductivity by

where d is the sample density and C_p is the heat capacity at constant pressure. Therefore, in order to determine the thermal conductivity from diffusivity data, additional measurements of both density and heat capacity are required. In the laser-flash method, one face of the sample is irradiated with a short pulse of heat (generated with a xenon flash lamp or a laser), while an infrared (IR) detector monitors the temperature on the reverse face (Figure 1.16). As this method assumes only an axial flow of heat, with no lateral heat losses, the sample is required to be in the form of a thin disk, with a large cross-sectional area relative to its thickness. To maximise the absorption of radiation and the emissivity, and hence the signal received by the IR detector, the top and bottom sample faces are often coated with a thin layer of graphite. Careful sample preparation is essential, as thick graphite coatings or poor adhesion of the graphite to the sample can introduce significant errors.¹⁶⁵  The top and bottom faces of the sample must be parallel, in order to minimise the error in sample thickness. The thermal diffusivity of the sample is related to the time required to reach half of the maximum temperature rise, t_½, according to²⁰² 

where l is the sample thickness. Commercial laser-flash instruments, which normally perform a least-squares fit to the transient curve to determine the diffusivity, can also apply corrections (due to Cowan,²⁰³ and to Clark and Taylor²⁰⁴ ) that take into account heat losses through the top and bottom faces and from the sides of the sample. The scatter in thermal diffusivity measurements between laboratories worldwide has been found to be of the order of ±8% for Bi₂Te₃²⁰⁵  and ±9.5% for Co_0.97Ni_0.03Sb₃,¹⁹⁰  although smaller variations of ±1.9% at 323 K and ±3.7% at 773 K were reported from a round-robin study on an n-type half-Heusler material, in which greater care was taken with the determination of sample thickness.²⁰⁶ 

In order to extract thermal conductivity values from measurements made by the laser-flash method, heat capacity measurements are required. Although commercial laser-flash instruments can measure heat capacities by a comparative method, using a reference material of known heat capacity, the accuracy of these measurements can be low. A Quantum Design Physical Properties Measurement System (PPMS) with the heat capacity option can be used for heat capacity measurements at low temperatures and differential scanning calorimetry (DSC) is frequently used to determine heat capacities above room temperature. DSC heat capacity measurements should follow the ASTM standard E1269,²⁰⁷  which requires three separate scans: a baseline (empty pan), a sapphire standard and the sample. It is also advisable to remeasure the baseline after completing measurements on the sample under test, as changes in the baseline could affect the determination of heat capacity. Experienced DSC operators can achieve variations as low as ca. ±4% in heat capacity measurements, although often the scatter is significantly higher.²⁰⁵  For measurements above the Debye temperature of the material, a quality check of the heat capacity data can be readily carried out by comparing the measurements with the calculated Dulong–Petit value. According to the Dulong–Petit law, the specific heat under constant volume, C_v, is 3R per mole of atoms. For a material with a molar mass M containing n atoms per formula unit, the Dulong–Petit heat capacity, C$vDP$, in J g⁻¹ K⁻¹ can be expressed as

The Dulong–Petit heat capacity is related to C_p by²⁰⁵ 

where α is the coefficient of thermal expansion, β_T the isothermal compressibility and d the density. Whereas C$vDP$ is constant with temperature, C_p should be slightly higher and increase with increasing temperature. A round-robin study suggested that the experimental heat capacity should be expected to be within ±5% of the Dulong–Petit value, for high-quality measurements.²⁰⁵  For materials with Debye temperatures significantly lower than the measurement temperatures, it may be a reasonable approximation to calculate the thermal conductivity using the Dulong–Petit value when DSC measurements are not available or when heat capacity measurements appear to be unreliable.²⁰⁶ 

Density measurements are also needed to calculate the thermal conductivity. For samples with a regular shape, the geometric density, calculated using the sample volume and mass, can be used. Alternatively, the Archimedes method, in which the sample is immersed in a liquid, can be employed.²⁰⁸  For the calculation of the thermal conductivity, the density is usually treated as being independent of temperature. Toberer et al. considered the effect of correcting the density, heat capacity and diffusivity measurements for thermal expansion, and showed that, in the case of Ba₈Ga₁₆Ge₃₀, the variation in thermal conductivity would be 1.2% at high temperature.⁶²  This value is significantly smaller than the uncertainty in thermal conductivity values. Taking into account error propagation from the individual measurements of diffusivity, density and heat capacity, the uncertainty in the derived thermal conductivity values has been estimated to range between ±6.3 and ±10.4%, with the uncertainty increasing with increasing temperature.²⁰⁶ 

The measurement uncertainties of commercial instruments for the individual physical properties that contribute to the figure of merit are, according to Wei et al.,²⁰⁹  ±3% for electrical conductivity, ±4% for Seebeck coefficient, ±1% for density, ±3% for thermal diffusivity and ±5% for heat capacity. Round-robin studies have revealed significant variations in measured values for individual properties.^{171,190,205,206}  The combination of these individual uncertainties will result in a significant error in the figure of merit, which has been estimated as ±11.5 to ±16.4% for an n-type half-Heusler material,²⁰⁶  or ±11.7 to ±20.9% for Bi₂Te₃,²⁰⁵  depending on the temperature. Given that the uncertainties in the individual measurements contribute to a large uncertainty in the figure of merit, methods for the direct measurement of ZT are clearly attractive.

In the Harman method, ZT is determined by measuring the voltage across the sample under isothermal conditions (i.e. in the absence of a temperature gradient) and under adiabatic conditions.¹⁶⁷  To reach adiabatic conditions, an electrical current is passed through the sample until a steady state, where the heat transported due to the Peltier effect along one direction of the sample is equal to the heat transported in the opposite direction by thermal conduction, is reached. Neglecting Joule heating and heat losses, it can be shown that ZT is given by¹⁷¹ 

where V_a is the adiabatic voltage and V_i is the isothermal voltage. A theoretical treatment to account for heat losses and Joule heating has also been discussed by Nolas et al.¹⁷¹  In order to apply the Harman technique, contact effects, sample resistance and sample heating from the contacts must be negligible, and the sample typically needs to exhibit a figure of merit ZT ≥ 0.1 (to produce the required large difference between V_a and V_i).²⁰¹  Implementations of the technique that allow the determination of the electrical conductivity, thermal conductivity and Seebeck coefficient, together with Z, have been described.^171,201,210  Parasitic heat-exchange effects between the sample and its environment are critical for measurement accuracy: if parasitic thermal effects are not taken into account, measured values of Z determined by the Harman method could vary by up to 50% when the sample dimensions²¹¹  or the electrical contacts are changed.²¹²  A two-sample system calibration, which permits the evaluation of thermal losses during measurements, has been developed.²¹⁰  The accuracy of measurements using the two-sample system calibration has been estimated to be better than 5% for figures of merit at temperatures up to 700 K.²¹³  More extensive testing of this approach, using a wide range of materials, is needed.

Impedance spectroscopy, a technique widely used in fields such as photovoltaics, fuel cells and supercapacitors, can also be exploited for the characterisation of both thermoelectric materials and complete devices.²¹⁴  This technique is based on the application of a small sinusoidally varying voltage to the material or device under investigation and the analysis of the response current, which may be in or out of phase with respect to the voltage, as a function of frequency.²¹⁵  The impedance is given by the AC voltage divided by the AC current. In complex number notation, the impedance, Z*, can be expressed as: Z′ = |Z*|cosθ and Z″ = |Z*|sinθ, where θ is the phase between voltage and current and Z′ and Z″ are the real and imaginary parts of the impedance, respectively. Data are conveniently presented in a Nyquist plot of Z″ as a function of Z′. Equivalent circuits that allow the analysis of the Nyquist plots for thermoelectric materials are available;²¹⁶  the Seebeck coefficient, electrical resistivity, thermal diffusivity, specific heat, thermal conductivity and figure of merit at a given temperature can all be extracted from these data. An experimental setup that permits measurements between 323 and 523 K has recently been described.^217,218  This technique is also useful as a rapid test to evaluate the quality of thermoelectric module fabrication.⁴⁵ 

The methods that are most commonly used for the synthesis, consolidation and measurements of thermoelectric materials have been described. Common problems encountered with each technique, together with their advantages and disadvantages, have been discussed. Much of the research on thermoelectric materials is driven by efforts to increase the maximum figure of merit, ZT, which is related to the efficiency of thermoelectric energy recovery. However, the efficiency of a thermoelectric device [eqn (1.1)] depends on the average figure of merit, (ZT)_ave, across the temperature range of operation. For this reason, increasing (ZT)_ave (rather than the maximum figure of merit) has a greater impact on the efficiency of energy recovery. This is particularly important for materials that show a sudden increase in ZT, for instance, due to a phase transition. Researchers in this field should therefore consider the (ZT)_ave of their materials, in addition to the maximum value.

To ensure that meaningful results in materials development are obtained, careful characterisation of the materials is essential. This extends from the assessment of sample purity and homogeneity following synthesis and consolidation, given that the presence of secondary phases could affect the charge-carrier concentration, to the measurements, which should be of the highest quality, with errors as low as possible. The reproducibility of results should be evaluated and instruments should be regularly checked with standards, to ensure that, for example, thermocouple degradation has not occurred. The thermal stability of thermoelectric materials across the temperature range of operation should also be considered. Oxidation or volatilisation of certain elements can occur at elevated temperatures. For instance, the composition of skutterudites and sulfides may change on heating due to volatilisation of antimony and sulfur, respectively. Thermogravimetric analysis (TGA) of materials under different atmospheres (e.g. nitrogen, air) can provide valuable information about stability. Compositional changes when a thermoelectric device is exposed to elevated temperatures can result in degradation of materials and hence a fall-off in performance. Repeated measurements of the properties, to ensure that the thermoelectric performance does not degrade following the initial measurement, should be performed.

The discovery and thermoelectric characterisation of a promising new thermoelectric material is only the first step. Incorporation of a new material into a thermoelectric device is technologically challenging. In a device, the n- and p-type legs are interconnected by soldering to a metal strip, and contact layers, with low contact resistances, are required. Contact layers suitable for existing materials such as Bi₂Te₃, may not be transferable to a newly discovered material. Reaction of the thermoelectric materials with the contact layers and/or solders can lead to degradation or failure of the device. Matching the coefficients of thermal expansion of the different device components is also essential, to avoid stress and possible fracture of the device during operation. To achieve progress in the practical implementation of thermoelectric technology, successful research efforts resulting in the discovery of a new thermoelectric material must be followed by work that addresses the device-level challenges.