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Research in thermoelectric materials involves the synthesis, structural characterisation and measurement of thermal and electronic transport properties. It spans the fields of solid-state chemistry, condensed-matter physics, materials science and engineering. This chapter provides an introduction, accessible to those with little prior experience in the field, to the synthetic approaches used for the preparation of thermoelectric materials and to the measurement techniques employed for the characterisation of their transport properties. Methods for the preparation of powder and single-crystal samples are presented, together with the processes commonly used for the consolidation of samples into well-densified ingots. The measurement techniques for the properties required for the determination of the thermoelectric figure of merit, ZT, are described, together with their limitations and possible sources of error. Alternative approaches for the direct determination of ZT are also outlined, together with Hall coefficient measurements, reflecting the importance of charge-carrier concentration and mobility to the detailed understanding of thermoelectric materials.

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

Equation 1.1

where TH and TC are the temperatures at the hot and the cold junctions, respectively, is the average temperature and Zc is the figure of merit of the couple:

Equation 1.2

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

Equation 1.3

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

Figure 1.1

Schematic representation of a thermoelectric couple, composed of two semiconducting materials, one p-type and the other n-type, which are joined by a metal strip (black bar), to make a junction. Power generation occurs when one end of the junction is heated.

Figure 1.1

Schematic representation of a thermoelectric couple, composed of two semiconducting materials, one p-type and the other n-type, which are joined by a metal strip (black bar), to make a junction. Power generation occurs when one end of the junction is heated.

Close modal

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:

Equation 1.4

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, Zc 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 Zc 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  oxychalcogenides7,8  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

2Bi + 3Te → Bi2Te3
Equation 1.5
Co + 3Sb → CoSb3
Equation 1.6
Na2CO3 + 2/3Co3O4 → Na2Co2O4 + CO2(g) (in air)
Equation 1.7

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.

Figure 1.2

Schematic representation of the solid-state reaction between particles of the elements A and B, in contact at a common surface, to form a product AB.

Figure 1.2

Schematic representation of the solid-state reaction between particles of the elements A and B, in contact at a common surface, to form a product AB.

Close modal

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.

Figure 1.3

The evacuated sealed-tube method, which involves (a) evacuating a glass ampoule that contains the powdered sample, (b) sealing the ampoule using a gas–oxygen torch and (c) heating the sealed ampoule in a furnace.

Figure 1.3

The evacuated sealed-tube method, which involves (a) evacuating a glass ampoule that contains the powdered sample, (b) sealing the ampoule using a gas–oxygen torch and (c) heating the sealed ampoule in a furnace.

Close modal

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.

Table 1.1

Maximum operating temperatures for commonly used crucible materials.

MaterialTemperature/K
Alumina (Al2O32000
Quartz/silica (SiO21500
Boron nitride (BN) 1173 in air
2273 in Ar–N2
Graphite (C) 673 in air
3000 in Ar–N2
Zirconia (ZrO22300
Platinum 1500
Tantalum 573 in air
1700 in Ar–N2
Zirconium 700 in air
1700 in Ar–N2
MaterialTemperature/K
Alumina (Al2O32000
Quartz/silica (SiO21500
Boron nitride (BN) 1173 in air
2273 in Ar–N2
Graphite (C) 673 in air
3000 in Ar–N2
Zirconia (ZrO22300
Platinum 1500
Tantalum 573 in air
1700 in Ar–N2
Zirconium 700 in air
1700 in Ar–N2

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.

Figure 1.4

Types of ball mills: (a) planetary ball mill; (b) vibration (shaker) mill; (c) attrition mill used, in the mechanosynthesis of thermoelectric materials. Adapted from ref. 25 with permission from the Royal Society of Chemistry.

Figure 1.4

Types of ball mills: (a) planetary ball mill; (b) vibration (shaker) mill; (c) attrition mill used, in the mechanosynthesis of thermoelectric materials. Adapted from ref. 25 with permission from the Royal Society of Chemistry.

Close modal

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 Ce0.8Fe3CoSb12 and Ce0.5Yb0.5Fe3.25Co0.75Sb12 form directly on milling.44

Table 1.2

Selected examples of thermoelectric materials prepared by mechanosynthesis. PM = planetary mill; SM = shaker mill; HP = hot press; SPS = spark-plasma sintering.

MaterialSynthesisMaximum ZTReferences
Si80Ge20P2 PM + HP 1.3 at 1173 K 37
Bi2Se0.15Te2.85, doped with 0.07 wt% SbI3 PM at 400 rpm, 0.7 at 300 K 38
HP at 723 K for 1 h,
annealed at 723 K for 2 h
Tl0.02Pb0.98Te PM for 10–20 h, 1.3 at 673 K 40
HP
La3–xTe4 SM for less than 24 h, 1.13 at 1273 K 39
HP at 1300 K
Ti0.95Nb0.05S2 PM at 600 rpm for 48 h, SPS at 1073 K for 30 min 0.3 at 700 K 41
Cu5FeS4 PM at 500 rpm for 40 h, 0.55 at 543 K 43
HP at 823 K for 30 min
Cu10.4Ni1.6Sb4S13 PM at 600 rpm for 20 h, SPS at 753 K for 30 min 0.75 at 700 K 42
TiNiSn PM at 300 rpm, 0.32 at 785 K 46
SPS at 1123 K for 10 min
CoSb2.75Sn0.05Te0.20 PM at 400 rpm for 10 h, 1.13 at 673 K 45
HP at 873 for 30 min
Ce0.8Fe3CoSb12 PM at 400 rpm for 10 h, 0.68 at 773 K 44
HP at 873 for 30 min
Ce0.5Yb0.5Fe3.25Co0.75Sb12 PM at 400 rpm for 10 h, 0.93 at 823 K 44
HP at 873 for 30 min
BiCuSeO PM at 500 rpm for 13 h, SPS at 973 K for 5 min 0.5 at 773 K 47
MaterialSynthesisMaximum ZTReferences
Si80Ge20P2 PM + HP 1.3 at 1173 K 37
Bi2Se0.15Te2.85, doped with 0.07 wt% SbI3 PM at 400 rpm, 0.7 at 300 K 38
HP at 723 K for 1 h,
annealed at 723 K for 2 h
Tl0.02Pb0.98Te PM for 10–20 h, 1.3 at 673 K 40
HP
La3–xTe4 SM for less than 24 h, 1.13 at 1273 K 39
HP at 1300 K
Ti0.95Nb0.05S2 PM at 600 rpm for 48 h, SPS at 1073 K for 30 min 0.3 at 700 K 41
Cu5FeS4 PM at 500 rpm for 40 h, 0.55 at 543 K 43
HP at 823 K for 30 min
Cu10.4Ni1.6Sb4S13 PM at 600 rpm for 20 h, SPS at 753 K for 30 min 0.75 at 700 K 42
TiNiSn PM at 300 rpm, 0.32 at 785 K 46
SPS at 1123 K for 10 min
CoSb2.75Sn0.05Te0.20 PM at 400 rpm for 10 h, 1.13 at 673 K 45
HP at 873 for 30 min
Ce0.8Fe3CoSb12 PM at 400 rpm for 10 h, 0.68 at 773 K 44
HP at 873 for 30 min
Ce0.5Yb0.5Fe3.25Co0.75Sb12 PM at 400 rpm for 10 h, 0.93 at 823 K 44
HP at 873 for 30 min
BiCuSeO PM at 500 rpm for 13 h, SPS at 973 K for 5 min 0.5 at 773 K 47
Figure 1.5

Powder X-ray diffraction pattern of (a) the initial product from ball milling a mixture of stoichiometry Cu5FeS4 and (b) after hot pressing the initially formed product to produce Cu5FeS4. Unreacted iron in (a) is marked with an asterisk.

Figure 1.5

Powder X-ray diffraction pattern of (a) the initial product from ball milling a mixture of stoichiometry Cu5FeS4 and (b) after hot pressing the initially formed product to produce Cu5FeS4. Unreacted iron in (a) is marked with an asterisk.

Close modal

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 Bi0.5Sb1.5Te3 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 Ba8Cu4.5Si6Ge35.5.49  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 Bi0.3Sb1.7Te3 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 fluxes55–57  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  Si1–xGex61  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.

Figure 1.6

Methods for the growth of large single crystals: (a) Czochralski method; (b) Bridgman method.

Figure 1.6

Methods for the growth of large single crystals: (a) Czochralski method; (b) Bridgman method.

Close modal

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 Bi2Te3−xSex,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 materials69–73  grown in this way are presented in Table 1.3.

Table 1.3

Selected examples of single-crystal thermoelectric materials prepared using a molten flux.

MaterialFluxFlux removalReferences
BiCuSeO NaCl/KCl Water 69,70
AxBa8−xAl14Si31 (A = Sr, Eu) Al Decanted, followed by 5 M NaOH 71
Ca3Co2O6 K2CO3 Water 72
Yb14MnSb11 Sn Centrifuge 73
MaterialFluxFlux removalReferences
BiCuSeO NaCl/KCl Water 69,70
AxBa8−xAl14Si31 (A = Sr, Eu) Al Decanted, followed by 5 M NaOH 71
Ca3Co2O6 K2CO3 Water 72
Yb14MnSb11 Sn Centrifuge 73

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:

Equation 1.8

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 Bi0.5Sb1.5Te3, 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 MnSi1.73 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

Figure 1.7

Schematic diagram of a Teflon-lined stainless-steel autoclave, composed of (a) stainless-steel lid; (b) pressure plate; (c) corrosion and rupture discs; (d) stainless-steel body; (e) Teflon liner; (f) spring; (g) solvent; (h) solid reagents.

Figure 1.7

Schematic diagram of a Teflon-lined stainless-steel autoclave, composed of (a) stainless-steel lid; (b) pressure plate; (c) corrosion and rupture discs; (d) stainless-steel body; (e) Teflon liner; (f) spring; (g) solvent; (h) solid reagents.

Close modal

Solvothermal methods have been used to prepare thermoelectric materials such as Bi2Te3,85,86  BixSb2−xTe3,87  PbTe,85,88  Cu2Se,89  Sn1−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:

Te + 2e → Te2−
Equation 1.9
Se4+ + 6e → Se2−
Equation 1.10

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

Table 1.4

Selected examples of nanostructured thermoelectric materials prepared by solvothermal synthesis, together with the maximum figure of merit, ZT (when reported). EG = ethylene glycol; en = ethylenediamine; PVP = polyvinylpyrrolidone; PEG = poly(ethylene glycol).

MaterialReagentsSolventT/KMorphologyMaximum ZTReferences
Bi2Te3 Bi2(C2O4)3 + Te en 413 Flakes — 85
Bi2Te3 Bi2O3 + TeO2 + NaOH + PVP EG 473 Nanoplates — 86
BixSb2–xTe3 Bi(NO3)3·5H2O + SbCl3 + Na2TeO3 + NaOH EG 503 Nanoplates 1.2 at 320 K 87
Bi2Se3 Bi2(C2O4)3 + Se en 413 Flakes  85
Bi2S3 Bi2(C2O4)3 + S en 413 Flakes — 85
Bi2S3 BiCl3 + thiourea Ethanol 413 Needles — 96
PbTe PbC2O4 + Te en 413 Nanocubes — 85
PbTe Pb(C2H3O2)2 + Te + hydrazine + PEG + NaOH Water/ethanol 433 Nanoboxes — 88
Cu2Se CuO + SeO2 + NaOH + PVP EG 503 Nanoplates 1.82 at 850 K 89
Sn1–xSe SnCl2·2H2O + Se + NaOH Water 403 — 2.1 at 873 K 90
CoSb3 CoCl2 + SbCl3 + NaBH4 Ethanol 523 Granules 0.61 at 725 K 91
FeSb2 Fe(CH3COO)2 + Sb(CH3COO)3 + NaBH4 Ethanol 493 Polygonal — 92
MaterialReagentsSolventT/KMorphologyMaximum ZTReferences
Bi2Te3 Bi2(C2O4)3 + Te en 413 Flakes — 85
Bi2Te3 Bi2O3 + TeO2 + NaOH + PVP EG 473 Nanoplates — 86
BixSb2–xTe3 Bi(NO3)3·5H2O + SbCl3 + Na2TeO3 + NaOH EG 503 Nanoplates 1.2 at 320 K 87
Bi2Se3 Bi2(C2O4)3 + Se en 413 Flakes  85
Bi2S3 Bi2(C2O4)3 + S en 413 Flakes — 85
Bi2S3 BiCl3 + thiourea Ethanol 413 Needles — 96
PbTe PbC2O4 + Te en 413 Nanocubes — 85
PbTe Pb(C2H3O2)2 + Te + hydrazine + PEG + NaOH Water/ethanol 433 Nanoboxes — 88
Cu2Se CuO + SeO2 + NaOH + PVP EG 503 Nanoplates 1.82 at 850 K 89
Sn1–xSe SnCl2·2H2O + Se + NaOH Water 403 — 2.1 at 873 K 90
CoSb3 CoCl2 + SbCl3 + NaBH4 Ethanol 523 Granules 0.61 at 725 K 91
FeSb2 Fe(CH3COO)2 + Sb(CH3COO)3 + NaBH4 Ethanol 493 Polygonal — 92

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.

Figure 1.8

Schematic diagram illustrating colloidal synthesis. In the ‘heat-up’ method, heating occurs after adding all reagents. In the ‘hot-injection’ method, the final reagent is added once the reaction mixture in the round-bottomed flask is hot.

Figure 1.8

Schematic diagram illustrating colloidal synthesis. In the ‘heat-up’ method, heating occurs after adding all reagents. In the ‘hot-injection’ method, the final reagent is added once the reaction mixture in the round-bottomed flask is hot.

Close modal

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 Bi2Te3−xSex,110  PbTe–PbS and PbS–Ag nanocomposites,111,112  Cu2SnSe3,113  Cu2CdSnSe4114  and SnSe115,116  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 BixSb1–x.117

Table 1.5

Selected examples of thermoelectric materials prepared using a colloidal method, together with the maximum figure of merit, ZT (when reported). R =reduction; T = thermal decomposition; PVP = polyvinylpyrrolidone; TEG = tetraethylene glycol; OAm = oleylamine; TOP = trioctylphosphine; ODE = 1-octadecene; HDA = hexadecylamine; ODPA = octadecylphosphonic acid; EG = ethylene glycol; OA = oleic acid.

MaterialReagentsSurfactant/ligandsSolventReaction typeMaximum ZTReferences
CoSb3 CoCl2·6H2O  +  SbCl3 + NaBH4 PVP TEG — 107
SnSe Bis[bis(trimethylsilyl)amino]tin(ii) + TOP:selenium TOP OAm — 115
SnSe SnCl2·2H2O + NaOH + Se + NaBH4 — Water — 116
Cu2SnSe3 CuCl + SnCl4·5H2O + SeO2 HDA, ODPA ODE 0.3 at 730 K 113
Cu2CdSnSe4 CuCl + CdO + SnCl4·5H2O + SeO2 HDA, ODPA ODE 0.71 at 685 K 114
Bi2Te3–xSex Bi(NO3)3·5H2O + Na2SeO3 + Na2TeO3 + KOH PVP EG 1.31 at 438 K 110
PbTe/PbS 1. PbO  +  TOP:Te OA ODE 1.07 at 700 K 111
2. Thioacetamide
PbS/Ag 1. PbO  +  S OA ODE 1.7 at 850 K 112
2. AgNO3  +  Fe(NO3)3·9H2OLA
MaterialReagentsSurfactant/ligandsSolventReaction typeMaximum ZTReferences
CoSb3 CoCl2·6H2O  +  SbCl3 + NaBH4 PVP TEG — 107
SnSe Bis[bis(trimethylsilyl)amino]tin(ii) + TOP:selenium TOP OAm — 115
SnSe SnCl2·2H2O + NaOH + Se + NaBH4 — Water — 116
Cu2SnSe3 CuCl + SnCl4·5H2O + SeO2 HDA, ODPA ODE 0.3 at 730 K 113
Cu2CdSnSe4 CuCl + CdO + SnCl4·5H2O + SeO2 HDA, ODPA ODE 0.71 at 685 K 114
Bi2Te3–xSex Bi(NO3)3·5H2O + Na2SeO3 + Na2TeO3 + KOH PVP EG 1.31 at 438 K 110
PbTe/PbS 1. PbO  +  TOP:Te OA ODE 1.07 at 700 K 111
2. Thioacetamide
PbS/Ag 1. PbO  +  S OA ODE 1.7 at 850 K 112
2. AgNO3  +  Fe(NO3)3·9H2OLA

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 Ba0.22Co4Sb12.125

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

Figure 1.9

Schematic representation of the three stages of sintering, together with the corresponding densification curve.

Figure 1.9

Schematic representation of the three stages of sintering, together with the corresponding densification curve.

Close modal

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

Figure 1.10

Schematic diagrams of (a) a hot-press apparatus and (b) a spark-plasma sintering (SPS) apparatus, which differ in the method used to heat the sample.

Figure 1.10

Schematic diagrams of (a) a hot-press apparatus and (b) a spark-plasma sintering (SPS) apparatus, which differ in the method used to heat the sample.

Close modal

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 Li1.3Fe0.3Ti1.7(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 Cu2−xS or Cu2−xSe by SPS. However, this does not appear to have been investigated.

Many thermoelectric materials have highly anisotropic properties64,151–153  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

Equation 1.11

where

Equation 1.12

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.

Table 1.6

Anisotropy ratio of the electrical (ρ, S) and thermal (κ) transport properties for selected thermoelectric materials at room temperature, determined from data for single crystals or highly textured samples. The subscripts ‖ and ⊥ refer to in-plane and out-of-plane properties, respectively.

Materialρ/ρS/Sκ/κReferences
Bi0.4Sb1.6Te3 0.34 0.96 2.44 16
SnSe 10 0.9 1.4 64
Bi0.875Ba0.125CuSeO 0.7 151
MnSi1.73 0.13 0.5 1.7 153
TiS2  ∼ 0.002 a 1.6 152
Materialρ/ρS/Sκ/κReferences
Bi0.4Sb1.6Te3 0.34 0.96 2.44 16
SnSe 10 0.9 1.4 64
Bi0.875Ba0.125CuSeO 0.7 151
MnSi1.73 0.13 0.5 1.7 153
TiS2  ∼ 0.002 a 1.6 152
a

Owing to the size of the single crystals, only in-plane measurements were reported for S.

Figure 1.11

Schematic representation of the hot forging process, which entails the use of a die larger than the ingot: (a) a consolidated ingot (pale blue) is placed inside a die of larger diameter; (b) pressure and temperature are applied, resulting in (c) the formation of a consolidated ingot with a larger diameter and with a higher degree of preferred orientation.

Figure 1.11

Schematic representation of the hot forging process, which entails the use of a die larger than the ingot: (a) a consolidated ingot (pale blue) is placed inside a die of larger diameter; (b) pressure and temperature are applied, resulting in (c) the formation of a consolidated ingot with a larger diameter and with a higher degree of preferred orientation.

Close modal

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 Bi2Te2.7Se0.3, 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 Bi1.95Sb0.05Te2.3Se0.7 with ZT ≈ 1.3 at 450 K, due to the presence of multiscale microstructures.156  For Bi1.95Sb0.05Te2.3Se0.7, 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 κL = 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 Bi0.875Ba0.125CuSeO, hot forging increases the ratio σ/κtotal 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 Bi0.6Sb1.4Te3 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:

σ = neμ
Equation 1.13

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

Equation 1.14

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

Equation 1.15

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

Equation 1.16

where σ0 is a proportionality constant, Eg is the bandgap energy, kB 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, EF, 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

Equation 1.17

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

Equation 1.18

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

Equation 1.19

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).

Figure 1.12

The three common techniques for the measurement of the electrical resistivity of a material: (a) two-probe method; (b) four-probe method; (c) van der Pauw method.

Figure 1.12

The three common techniques for the measurement of the electrical resistivity of a material: (a) two-probe method; (b) four-probe method; (c) van der Pauw method.

Close modal

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:

Vtotal = IR + SΔT
Equation 1.20

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

Equation 1.21

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 IV 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 RAB,CD and RBC,DA are then defined as

Equation 1.22

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

Equation 1.23

where d is the thickness of the sample and f is a function dependent on RAB,CD and RBC,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:

dV = SABdT
Equation 1.24

where SAB is the relative Seebeck coefficient (SAB = SASB) between materials A and B.

Figure 1.13

Schematic diagram of a thermocouple formed by two dissimilar materials, A and B, joined at the junctions 1 and 2.

Figure 1.13

Schematic diagram of a thermocouple formed by two dissimilar materials, A and B, joined at the junctions 1 and 2.

Close modal

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

Table 1.7

Seebeck coefficients, S, at room temperature for selected examples of metals and semiconductors.

MaterialS/μV K−1References
Pt  − 5 179
Cu  + 1.8 180
Ag  + 1.5 180
Bi2Te3  + 230 181
PbTe  + 300 181
MaterialS/μV K−1References
Pt  − 5 179
Cu  + 1.8 180
Ag  + 1.5 180
Bi2Te3  + 230 181
PbTe  + 300 181

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

Equation 1.25

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

Equation 1.26

where σn and σp are the electrical conductivities of electrons and holes, respectively. In intrinsic semiconductors, both types of charge carriers contribute to the electrical conduction and, as Sn and Sp 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 EFkBT, the Seebeck coefficient will be given by:182

Equation 1.27

where EF is the Fermi energy, kB the Boltzmann constant and kB/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 EFkBT) 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

Equation 1.28

and therefore semiconductors exhibit Seebeck coefficients larger than 87 μV K−1 (since EgkBT 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

Equation 1.29

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, |Smax|, can be related to the bandgap by the following expression:185

Eg ≈ 2e|Smax|Tmax
Equation 1.30

where Tmax is the temperature at which |Smax| 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 |Smax| < 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 = T2T1) is applied at the average temperature of interest, Tave = (T1 + T2)/2 (where Tave ≫ ΔT), and the Seebeck coefficient at Tave is obtained as VT. 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 Tave 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.

Figure 1.14

Probe arrangements for Seebeck coefficient measurements: (a) two-probe and (b) four-probe. The heaters to create a temperature gradient are shown in red and blue and the thermocouples to measure V and ΔT are shown as black lines.

Figure 1.14

Probe arrangements for Seebeck coefficient measurements: (a) two-probe and (b) four-probe. The heaters to create a temperature gradient are shown in red and blue and the thermocouples to measure V and ΔT are shown as black lines.

Close modal

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 Ix. When a magnetic field, Bz, is applied along the z-direction, charge carriers are deflected along y, producing a Hall voltage, VH. 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 Ix, Bz and the thickness of the sample, d, is given by the expression

Equation 1.31

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

Equation 1.32
Figure 1.15

Schematic diagram of the Hall effect. Electrons and/or holes in the current, Ixx, are deflected by the magnetic field Bz. This generates the Hall voltage, VH.

Figure 1.15

Schematic diagram of the Hall effect. Electrons and/or holes in the current, Ixx, are deflected by the magnetic field Bz. This generates the Hall voltage, VH.

Close modal

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

Equation 1.33

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:

Equation 1.34

where the heat flux, Jx(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.

Table 1.8

Thermal conductivities, κ, at room temperature for selected materials.

Materialκ/W m−1 K−1References
Silicon 124 197
Al2O3 36 196
Cu 398 196
SiO2 (amorphous) 1.38 196
Bi2Te3 2.0 197
PbTe 2.3 197
Materialκ/W m−1 K−1References
Silicon 124 197
Al2O3 36 196
Cu 398 196
SiO2 (amorphous) 1.38 196
Bi2Te3 2.0 197
PbTe 2.3 197

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:

κ = κL + κe
Equation 1.35

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

κe = L0σT
Equation 1.36

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 κe are directly related, a commonly used approach to improve the figure of merit is to reduce κL. The value of κL 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 κL. A straightforward approach to estimate the value of L0, from the Seebeck coefficient, is through the expression198

Equation 1.37

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.

Table 1.9

Commonly used methods for the measurement of thermal conductivity in bulk materials.

ApproachMethod
Comparative
Parallel conductance
Transient (frequency domain) Pulse power (Maldonado technique)
Transient (time domain) Hot wire
Laser flash
Transient plane source (hot disk)
ApproachMethod
Comparative
Parallel conductance
Transient (frequency domain) Pulse power (Maldonado technique)
Transient (time domain) Hot wire
Laser flash
Transient plane source (hot disk)

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

κ = DdCp
Equation 1.38

where d is the sample density and Cp 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, t½, according to202

Equation 1.39

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 Co0.97Ni0.03Sb3,190  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

Figure 1.16

Schematic diagram of a laser-flash instrument for the measurement of thermal diffusivity.

Figure 1.16

Schematic diagram of a laser-flash instrument for the measurement of thermal diffusivity.

Close modal

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, Cv, 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

Equation 1.40

The Dulong–Petit heat capacity is related to Cp by205

Equation 1.41

where α is the coefficient of thermal expansion, βT the isothermal compressibility and d the density. Whereas C$vDP$ is constant with temperature, Cp 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

Equation 1.42

where Va is the adiabatic voltage and Vi 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 Va and Vi).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.

1.
D. M.
Rowe
,
Thermoelectrics Handbook: Macro to Nano
, ed. D. M. Rowe,
CRC Press, Taylor and Francis
,
Boca Raton
,
2006
, ch. 1
2.
Sootsman

J. R.
Chung

D. Y.
Kanatzidis

M. G.
Angew. Chem., Int. Ed.
2009
, vol.
48
pg.
8616

3.
Tan

G.
Zhao

L.-D.
Kanatzidis

M. G.
Chem. Rev.
2016
, vol.
116
pg.
12123

4.
Powell

A. V.
J. Appl. Phys.
2019
, vol.
126
pg.
100901

5.
A. V.
Powell
and
P.
Vaqueiro
,
Thermoelectric Materials and Devices
,
2017
, ed. I. Nandhakumar, N. M. White and S. Beeby, RSC Energy Environment Series,
vol. 17
, p. 27
6.
Koumoto

K.
Wang

Y.
Zhang

R.
Kosuga

A.
Funahashi

R.
Annu. Rev. Mater. Res.
2010
, vol.
40
pg.
363

7.
Luu

S. D. N.
Vaqueiro

P.
J. Materiomics
2016
, vol.
2
pg.
131

8.
Zhao

L.-D.
He

J.
Berardan

D.
Lin

Y.
Li

J.-F.
Nan

C.-W.
Dragoe

N.
Energy Environ. Sci.
2014
, vol.
7
pg.
2900

9.
Kleinke

H.
Chem. Mater.
2010
, vol.
22
pg.
604

10.
Christensen

M.
Johnsen

S.
Iversen

B. B.
Dalton Trans.
2010
, vol.
39
pg.
978

11.
Zeier

W. G.
Schmitt

J.
Hautier

G.
Aydemir

U.
Gibbs

Z. M.
Felser

C.
Snyder

G. J.
Nat. Rev. Mater.
2016
, vol.
1
pg.
16032

12.
Toberer

E. S.
May

A. F.
Snyder

G. J.
Chem. Mater.
2010
, vol.
22
pg.
624

13.
Vaqueiro

P.
Powell

A. V.
J. Mater. Chem.
2010
, vol.
20
pg.
9577

14.
J. D.
Corbett
,
Solid State Chemistry Techniques
, ed. A. K. Cheetham and P. Day,
Clarendon Press
,
Oxford
,
1995
, ch. 1
15.
U.
Schubert
and
N.
Hüsing
,
Synthesis of Inorganic Materials
,
Wiley
,
Weinheim
,
2000
, ch. 1, pp. 5–15
16.
H.
Scherrer
and
S.
Scherrer
,
Thermoelectrics Handbook: Macro to Nano
, ed. D. M. Rowe,
CRC Press, Taylor and Francis
,
Boca Raton
,
2006
, ch. 27
17.
Ballikaya

S.
Uzar

N.
Yildirim

S.

J. R.
Uher

C.
J. Solid State Chem.
2012
, vol.
193
pg.
31

18.
Kurosaki

K.
Muta

H.
Uno

M.
Yamanaka

S.
J. Alloys Compd.
2001
, vol.
315
pg.
234

19.
ASM Handbook Volume 3: Alloy Phase Diagrams
, ed. H. Okamoto, M. E. Schlesinger and E. M. Mueller,
ASM International
,
2016
20.
Chen

S.
Ren

Z.
Mater. Today
2016
, vol.
16
pg.
387

21.
K.
Motzfeldt
,
High Temperature Experiments in Chemistry and Materials Science
,
John Wiley & Sons
, 3rd edn,
2013
, ch. 2
22.
Oledzka

M.
Lee

C. L.
Ramanujachary

K. V.
Greenblatt

M.
Mater. Res. Bull.
1997
, vol.
32
pg.
889

23.
G. S.
Coyne
,
The Laboratory Companion. A practical Guide to Materials Equipment and Technique
,
John Wiley & Sons
,
New Jersey, US
,
2006
, ch. 8
24.
B. A.
Cook
and
J. H.
Harringa
,
Thermoelectrics Handbook: Macro to Nano
, ed. D. M. Rowe,
CRC Press, Taylor and Francis
,
Boca Raton
,
2006
, ch. 19
25.
Baláž

P.
Achimoviĉová

M.
Baláž

M.
Billik

P.
Cherkezova-Zheleva

Z.

J. M.
Delogu

F.
Dutková

E.
Gaffet

E.
Gotor

F.
Kumar

R.
Mitov

I.
Rojac

T.
Senna

M.
Streletskii

A.
Wieczorek-Ciurowam

K.
Chem. Soc. Rev.
2013
, vol.
42
pg.
7571

26.
James

S. L.

C. J.
Bolm

C.
Braga

D.
Collier

P.
Friščići

T.
Grepioni

F.
Harris

K. D. M.
Hyett

G.
Jones

W.
Krebs

A.
Mack

J.
Maini

L.
Orpen

A. G.
Parkin

I. P.
Shearouse

W. C.
Steed

J. W.

D. C.
Chem. Soc. Rev.
2012
, vol.
41
pg.
413

27.
W.
Ostwald
,
Handbuch der Allgemeinen Chemie, Band 1
,
,
Leipzig
,
1919
,
vol. 70
28.
IUPAC
,
Compendium of Chemical Terminology
, (the ‘Gold Book’), A. D. McNaught and A. Wilkinson,
Blackwell Scientific Publications
,
Oxford
, 2nd edn,
1997
, Online version (2019-) created by S. J. Chalk. ISBN 0-9678550-9-8. https://doi.org/10.1351/goldbook
29.
Takacs

L.
J. Met.
2000
, vol.
52
pg.
12

30.
Lea

M. C.
Am. J. Sci.
1893
, vol.
46
pg.
413

31.
Chen

Y.
Williams

J. S.
J. Alloys Compd.
1995
, vol.
217
pg.
181

32.
Suryanarayana

C.
Ivanov

E.
Boldyrev

V. V.
Mater. Sci. Eng., A
2001
, vol.
304–306
pg.
151

33.
P.
Baláž
,
Mechanochemistry in Nanoscience and Minerals Engineering
,
Springer-Verlag
,
Berlin Heidelberg
,
2008
, ch. 1, pp. 8–14
34.
F. P.
Bowden
and
A.
Yoffe
,
Fast Reactions in Solids
,
Butterworths Scientific Publications
,
London
,
1958
35.
P. A.
Thiessen
,
K.
Meyer
and
G.
Heinicke
,
Grundlagen der Tribochemie
,
,
Berlin
,
1967
36.
Takacs

L.
Prog. Mater. Sci.
2002
, vol.
47
pg.
355

37.
Wang

X. W.
Lee

H.
Lan

Y. C.
Zhu

G. H.
Joshi

G.
Wang

D. Z.
Yang

J.
Muto

A. J.
Tang

M. Y.
Klatsky

J.
Song

S.
Dresselhaus

M. S.
Chen

G.
Ren

Z. F.
Appl. Phys. Lett.
2008
, vol.
93
pg.
193121

38.
Yang

J. Y.
Aizawa

T.
Yamamoto

A.
Ohta

T.
J. Alloys Compd.
2000
, vol.
312
pg.
326

39.
May

A. F.
Fleurial

J.-P.
Snyder

G. J.
Phys. Rev. B: Condens. Matter Mater. Phys.
2008
, vol.
78
pg.
125205

40.
Yu

B.
Zhang

Q.
Wang

H.
Wang

X.
Wang

H.
Wang

D.
Wang

H.
Snyder

G. J.
Chen

G.
Ren

Z. F.
J. Appl. Phys.
2010
, vol.
108
pg.
016104

41.
Bourgès

C.
Barbier

T.
Guélou

G.
Vaqueiro

P.
Powell

A. V.
Lebedev

O. I.
Barrier

N.
Kinemuchi

Y.
Guilmeau

E.
J. Eur. Ceram. Soc.
2016
, vol.
36
pg.
1183

42.
Barbier

T.
Rollin-Martinet

S.
Lemoine

P.
Gascoin

F.
Kaltzoglou

A.
Vaqueiro

P.
Powell

A. V.
Guilmeau

E.
J. Am. Ceram. Soc.
2016
, vol.
99
pg.
51

43.
Guélou

G.
Powell

A. V.
Vaqueiro

P.
J. Mater. Chem. C
2015
, vol.
3
pg.
10624

44.

J.
Vaqueiro

P.
Nuttall

C.
Potter

R.
Powell

A. V.
J. Alloys Compd.
2017
, vol.
695
pg.
3598

45.

J.
Phillips

M.
Vaqueiro

P.
Min

G.
Powell

A. V.
ACS Appl. Energy Mater.
2018
, vol.
1
pg.
6609

46.
Zou

M.
Li

J.-F.
Du

B.
Liu

D.
Kita

T.
J. Solid State Chem.
2009
, vol.
182
pg.
3138

47.
Wu

J.
Li

F.
Wei

T.-R.
Ge

Z.
Kang

F.
He

J.
Li

J.-F.
J. Am. Ceram. Soc.
2016
, vol.
99
pg.
507

48.
Poudel

B.
Hao

Q.
Ma

Y.
Lan

Y.
Minnich

A.
Yu

B.
Yan

X.
Wang

D.
Muto

A.
Vashaee

D.
Chen

X.
Liu

J.
Dresselhaus

M. S.
Chen

G.
Ren

Z.
Science
2008
, vol.
320
pg.
634

49.
Yan

X.
Prokofiev

A.
Bauer

E.
Rogl

P.
Bernardi

J.
Paschen

S.
J. Alloys Compd.
2017
, vol.
725
pg.
783

50.
Li

J.
Tan

Q.
Li

J.-F.
Liu

D.-W.
Li

F.
Li

Z.-Y.
Zou

M.
Wang

K.
2013
, vol.
23
pg.
4317

51.
Baláž

P.
Guilmeau

E.
Daneu

N.
Dobrozhan

O.
Baláž

M.
Hegedus

M.
Barbier

T.
Achimovičová

M.
Kaňuchováf

M.
Briančin

J.
J. Eur. Ceram. Soc.
2020
, vol.
40
pg.
1922

52.
Baláž

P.
Hegedus

M.
Baláž

M.
Daneu

N.
Siffalovic

P.
Bujňáková

Z.
Tóthová

E.
Tešínsky

M.
Achimovičová

M.
Briančin

J.
Dutková

E.
Kaňuchová

M.
Fabián

M.
Kitazono

S.
Dobrozhan

O.
Prog. Photovolt. Res. Appl.
2019
, vol.
27
pg.
798

53.
Müller

G.
Cryst. Res. Technol.
2007
, vol.
42
pg.
1150

54.
P.
Capper
,
Springer Handbook of Electronic and Photonic Materials
, ed. S. Kasap, and P.Capper,
Springer
,
2017
, ch. 12
55.
Liu

X.
Fechler

N.
Antonietti

M.
Chem. Soc. Rev.
2013
, vol.
42
pg.
8237

56.
Kanatzidis

M. G.
Pöttgen

R.
Jeitschko

W.
Angew. Chem., Int. Ed.
2005
, vol.
44
pg.
6996

57.
Kanatzidis

M. G.
Chem. Mater.
1990
, vol.
2
pg.
353

58.
M.
Binnewies
,
R.
Glaum
,
M.
Schmidt
and
P.
Schmidt
,
Chemical Vapor Transport Reactions
,
De Gruyter
,
Berlin, Boston
,
2012
59.
D. T. J.
Hurle
,
Crystal Pulling from the Melt
,
Springer
,
Berlin
,
1993
60.
Svechnikova

T. E.
Shelimova

L. E.
Konstantinov

P. P.
Kretova

M. A.
Avilov

E. S.
Zemskov

V. S.
Stiewe

C.
Zuber

A.
Muller

E.
Inorg. Mater.
2005
, vol.
41
pg.
1043

61.
Abrosimov

N. V.
Rossolenko

S. N.
Alex

V.
Gerhardt

A.
Schröder

W.
J. Cryst. Growth
1996
, vol.
166
pg.
657

62.
Toberer

E. S.
Christensen

M.
Iversen

B. B.
Snyder

G. J.
Phys. Rev. B: Condens. Matter Mater. Phys.
2008
, vol.
77
pg.
075203

63.
Keawprak

N.
Lao-ubol

S.
Eamchotchawalit

C.
Sun

Z. M.
J. Alloys Compd.
2011
, vol.
509
pg.
9296

64.
Zhao

L.-D.
Lo

S.-H.
Zhang

Y.
Sun

H.
Tan

G.
Uher

C.
Wolverton

C.
Dravid

V. P.
Kanatzidis

M.G.
Nature
2014
, vol.
508
pg.
373

65.
Caillat

T.
Fleurial

J.-P.
Borshchevsky

A.
J. Cryst. Growth
1996
, vol.
166
pg.
722

66.
Yoshinaga

M.
Iida

T.
Noda

M.
Endo

T.
Takanashi

Y.
Thin Solid Films
2004
, vol.
461
pg.
86

67.
Wang

J.
Lebedev

O. I.
Lee

K.
Dolyniuk

J.-A.
Klavins

P.
Bux

S.
Kovnir

K.
Chem. Sci.
2017
, vol.
8
pg.
8030

68.
M.
Tachibana
,
Beginner's Guide to Flux Crystal Growth
,
NIMS Monographs
,
Springer
,
Japan
,
2017
69.
Richard

A. P.
Russell

J. A.
Zakutayev

A.
Zakharov

L. N.
Keszler

D. A.
Tate

J.
J. Solid State Chem.
2012
, vol.
187
pg.
15

70.
Dong

S.-T.
Lv

Y.-Y.
Zhang

B.-B.
Zhang

F.
Yao

S.
Chen

Y. B.
Zhou

J.
Zhang

S.-T.
Gu

Z.-B.
Chen

Y.-F.
CrysEngComm
2015
, vol.
17
pg.
6136

71.
Condron

C. L.
Kauzlarich

S. M.
Inorg. Chem.
2007
, vol.
46
pg.
2556

72.
Takahashi

J.
Yamane

H.

M.
Jpn. J. Appl. Phys.
2004
, vol.
43
pg.
L331

73.
Rauscher

J. F.
Cox

C. A.
Yi

T.
Beavers

C. M.
Klavins

P.
Toberer

E. S.
Snyder

G. J.
Kauzlarich

S. M.
Dalton Trans.
2010
, vol.
39
pg.
1055

74.
Schäfer

H.
Angew. Chem., Int. Ed.
1971
, vol.
10
pg.
43

75.
Koyano

M.
Tanaka

J.
Suekuni

K.
Ariga

T.
J. Electron. Mater.
2012
, vol.
41
pg.
1317

76.
Davydyuk

G. Y.
Parasyuk

O. V.
Romanyuk

Y. E.
Semenyuk

S. A.
Zaremba

V. I.
Piskach

L. V.
Koziol

J. J.
Halka

V. O.
J. Alloys Compd.
2002
, vol.
339
pg.
40

77.
Fardy

M.
Hochbaum

A. I.
Goldberger

J.
Zhang

M. M.
Yang

P.
2007
, vol.
19
pg.
3047

78.
Girard

S. N.
Chen

X.
Meng

F.
Pokhrel

A.
Zhou

J.
Shi

L.
Jin

S.
Chem. Mater.
2014
, vol.
26
pg.
5097

79.
Szczech

J. R.
Schmitt

A. L.
Bierman

M. J.
Jin

S.
Chem. Mater.
2007
, vol.
19
pg.
3238

80.
G.
Krabbes
,
W.
Bieger
,
K.-H.
Sommer
,
T.
Söhnel
and
U.
Steiner
,
Computer program TRAGMIN
, version 5.0, IFW Dresden, University of Dresden, HTW Dresden,
2008
, https://www.htw-dresden.de/en/luc/forschung/chemieingenieurwesen/festkoerperchemie/translate-to-english-tragmin (accessed 23 September 2021)
81.
Byrappa

K.

T.
Prog. Cryst. Growth Charact.
2007
, vol.
53
pg.
117

82.
U.
Schubert
and
N.
Hüsing
,
Synthesis of Inorganic Materials
,
Wiley
,
Weinheim
,
2000
, ch. 4, pp. 180–191
83.
Li

J.
Chen

Z.
Wang

R.-J.
Proserpio

D. M.
Coord. Chem. Rev.
1999
, vol.
190–192
pg.
707

84.
Sheldrick

W. S.
Wachhold

M.
Angew. Chem., Int. Ed.
1997
, vol.
36
pg.
206

85.
Yu

S.-H.
Yang

J.
Wu

Y.-S.
Han

Z.-H.
Lu

J.
Xie

Y.
Qian

Y.-T.
J. Mater. Chem.
1998
, vol.
8
pg.
1949

86.
Zhang

G.
Wang

W.
Lu

X.
Li

X.
Cryst. Growth Des.
2009
, vol.
9
pg.
151

87.
Hong

M.
Chen

Z. G.
Yang

L.
Zou

J.
Nano Energy
2016
, vol.
20
pg.
144

88.
Wang

W.
Poudel

B.
Wang

D.
Ren

Z. F.
2005
, vol.
17
pg.
2110

89.
Yang

L.
Chen

Z.-G.
Han

G.
Hong

M.
Zou

Y.
Zou

J.
Nano Energy
2015
, vol.
16
pg.
367

90.
Wei

W.
Chang

C.
Yang

T.
Liu

J.
Tang

H.
Zhang

J.
Li

Y.
Xu

F.
Zhang

Z.
Li

J.-F.
J. Am. Chem. Soc.
2018
, vol.
140
pg.
499

91.
Mi

J. L.
Zhu

T. J.
Zhao

X. B.
Ma

J.
J. Appl. Phys.
2007
, vol.
101
pg.
054314

92.
Datta

A.
Nolas

G. S.
Eur. J. Inorg. Chem.
2012
pg.
55

93.
Thomson

J. W.
Nagashima

K.
Macdonald

P. M.
Ozin

G. A.
J. Am. Chem. Soc.
2011
, vol.
133
pg.
5036

94.
MacColl

R.
Windwer

S.
J. Phys. Chem.
1970
, vol.
74
pg.
1261

95.
Wu

C.
Yu

S.-H.
Chen

S.
Liu

G.
Liu

B.
J. Mater. Chem.
2006
, vol.
16
pg.
3326

96.
Yu

S. H.
Qian

Y. T.
Shu

L.
Xie

Y.
Yang

L.
Wang

C. S.
Mater. Lett.
1998
, vol.
35
pg.
116

97.
Li

H.
Du

X.
Cao

L.
Guo

X.
Yuan

Z.
J. Mater. Sci.
2020
, vol.
55
pg.
2905

98.
Wang

B.
Wang

Y.
Zheng

S.
Liu

S.
Li

J.
Chang

S.
An

T.
Sun

W.
Chen

Y.
J. Alloys Compd.
2019
, vol.
806
pg.
676

99.
Tang

G.
Liu

J.
Zhang

J.
Li

D.
Rara

K. H.
Xu

R.
Lu

W.
Liu

J.
Zhang

Y.
Feng

Z.
ACS Appl. Mater. Interfaces
2018
, vol.
10
pg.
30558

100.
Shi

X. L.
Zheng

K.
Hong

M.
Liu

W. D.
Moshwan

R.
Wang

Y.
Qu

X.-L.
Chen

Z. G.
Zou

J.
Chem. Sci.
2018
, vol.
9
pg.
7376

101.
Ji

X.
Alboni

P.
Su

Z.
Gothard

N.
Zhang

B.
Tritt

T. M.
Kolis

J. W.
Phys. Status Solidi RRL
2007
, vol.
1
pg.
229

102.
Xin

J.
Yang

J.
Li

S.
Basit

A.
Sun

B.
Li

S.
Long

Q.
Li

X.
Chen

Y.
Jiang

Q.
Chem. Mater.
2019
, vol.
31
pg.
2421

103.
Ortega

S.
Ibáñez

M.
Liu

Y.
Zhang

Y.
Kovalenko

M. V.

D.
Cabot

A.
Chem. Soc. Rev.
2017
, vol.
46
pg.
3510

104.
Park

J.
Joo

J.
Kwon

S. G.
Jang

Y.
Hyeon

T.
Angew. Chem., Int. Ed.
2007
, vol.
46
pg.
4630

105.
Cushing

B. L.
Kolesnichenko

V. L.
O'Connor

C. J.
Chem. Rev.
2004
, vol.
104
pg.
3893

106.
Coughlan

C.
Ibáñez

M.
Dobrozhan

O.
Singh

A.
Cabot

A.
Ryan

K. M.
Chem. Rev.
2017
, vol.
117
pg.
5865

107.
Yang

L.
Hng

H. H.
Cheng

H.
Sun

T.
Ma

J.
Mater. Lett.
2008
, vol.
62
pg.
2483

108.
Murray

C. B.
Norris

D. J.
Bawendi

M. G.
J. Am. Chem. Soc.
1993
, vol.
115
pg.
8706

109.
Lee

J. M.
Miller

R. C.
Moloney

L. J.
Prieto

A. L.
J. Solid State Chem.
2019
, vol.
273
pg.
243

110.
Liu

Y.
Zhang

Y.
Lim

K. H.
Ibáñez

M.
Ortega

S.
Li

M.
David

J.
Martí-Sánchez

S.
Ng

K. M.
Arbiol

J.
Kovalenko

M. V.

D.
Cabot

A.
ACS Nano
2018
, vol.
12
pg.
7174

111.
Ibáñez

M.
Zamani

R.

S.
Fan

J.
Ortega

S.

D.
Morante

J. R.
Arbiol

J.
Cabot

A.
ACS Nano
2013
, vol.
7
pg.
2573

112.
Ibáñez

M.
Luo

Z.
Genc

A.
Piveteau

L.
Ortega

S.

D.
Dobrozhan

O.
Liu

Y.
Nachtegaal

M.

M.
Arbiol

J.
Kovalenko

M. V.
Cabot

A.
Nat. Commun.
2016
, vol.
7
pg.
10766

113.
Ibáñez

M.

D.
Anselmi-Tamburini

U.
Zamani

R.

S.
Li

W.
Lopez

A. M.
Morante

J. R.
Arbiol

J.
Cabot

A.
J. Mater. Chem. A
2013
, vol.
1
pg.
1421

114.
Ibáñez

M.

D.
Zamani

R.
García-Castelló

N.
Izquierdo-Roca

V.
Li

W.
Fairbrother

A.

J. D.
Shavel

A.
Arbiol

J.
Pérez-Rodríguez

A.
Morante

J. R.
Cabot

A.
Chem. Mater.
2012
, vol.
24
pg.
562

115.
Baumgardner

W. J.
Choi

J. J.
Lim

Y.-F.
Hanrath

T.
J. Am. Chem. Soc.
2010
, vol.
132
pg.
9519

116.
Han

G.
Popuri

S. R.
Greer

H. F.
Bos

J.-W. G.
Zhou

W.
Knox

A. R.
Montecucco

A.
Siviter

J.
Man

E. A.
Macauley

M.
Paul

D. J.
Li

W.-G.
Paul

M. C.
Gao

M.
Sweet

T.
Freer

R.
Azough

F.
Baig

H.
Sellami

N.
Mallick

T. K.
Gregory

D. H.
Angew. Chem., Int. Ed.
2016
, vol.
55
pg.
6433

117.
He

M.
Protesescu

L.
Caputo

R.
Krumeich

F.
Kovalenko

M. V.
Chem. Mater.
2015
, vol.
27
pg.
635

118.
Scheele

M.
Oeschler

N.
Meier

K.
Koronowski

A.

C.
Weller

H.
2009
, vol.
19
pg.
3476

119.
Liu

Y.
Garcia

G.
Ortega

S.

D.
Palacios

P.
Lu

J.
Ibáñez

M.
Xi

L.
De Roo

J.
Lopez

A. M.
Martı-Sanchez

S.
Cabezas

I.
de la Mata

M.
Luo

Z.
Dun

C.
Dobrozhan

O.
Caroll

D. L.
Zhang

W.
Martins

J.
Kovalenko

M. V.
Arbiol

J.
Noriega

G.
Song

J.
Wahnon

P.
Cabot

A.
J. Mater. Chem. A
2017
, vol.
5
pg.
2592

120.
De Trizio

L.
Manna

L.
Chem. Rev.
2016
, vol.
116
pg.
10852

121.
Sotelo

A.
Rasekh

Sh
Torres

M. A.
Bosque

P.

M. A.
Diez

J. C.
J. Solid State Chem.
2015
, vol.
221
pg.
247

122.
U.
Schubert
and
N.
Hüsing
,
Synthesis of Inorganic Materials
,
Wiley
,
Weinheim
,
2000
, ch. 4, pp. 192–222
123.
Danks

A. E.
Hall

S. R.
Schnepp

Z.
Mater. Horiz.
2016
, vol.
3
pg.
91

124.
Cai

K. F.
Müller

E.
Drašar

C.
Mrotzek

A.
Mater. Sci. Eng., B
2003
, vol.
104
pg.
45

125.
Xiong

Z.
Chen

X.
Zhao

X.
Bai

S.
Huang

X.
Chen

L.
Solid State Sci.
2009
, vol.
11
pg.
1612

126.
Liu

Y.
Lin

Y.
Shi

Z.
Nan

C.-W.
J. Am. Ceram. Soc.
2005
, vol.
88
pg.
1337

127.
Hector

A. L.
Chem. Soc. Rev.
2007
, vol.
36
pg.
1745

128.
A. C.
Pierre
,
Introduction to Sol-Gel Processing
,
Springer Nature
,
Switzerland
,
2020
, ch. 4
129.
Brock

S. L
Arachchige

I. U.
Kalebaila

K. K.
2006
, vol.
27
pg.
1

130.
Saleemi

M.
Toprak

M. S.
Li

S.
Johnsson

M.
Muhammed

M.
J. Mater. Chem.
2012
, vol.
22
pg.
725

131.
Khan

A.
Saleemi

M.
Johnsson

M.
Han

L.
Nong

N. V.
Muhammed

M.
Toprak

M. S.
J. Alloys Compd.
2014
, vol.
612
pg.
293

132.
Le Tonquesse

S.
Alleno

E.
Demange

V.
Dorcet

V.
Joanny

L.
Prestipino

C.
Rouleau

O.
Pasturel

M.
J. Alloys Comp.
2019
, vol.
796
pg.
176

133.
Walker

R. F.
J. Am. Ceram. Soc.
1955
, vol.
38
pg.
187

134.
S. L.
Kang
,
Sintering: Densification, Grain Growth and Microstructure
,
Elsevier
,
Oxford
,
2004
, ch. 5
135.
A. C.
Reardon
,
Metallurgy for the Non-Metallurgist
,
ASM International
,
Ohio
, 2nd edn,
2011
, ch. 6, p. 135
136.
Atkinson

H. V.
Davies

S.
Metall. Mater. Trans. A
2000
, vol.
31
pg.
2981

137.
LaLonde

A. D.
Ikeda

T.
Snyder

G. J.
Rev. Sci. Instrum.
2011
, vol.
82
pg.
025104

138.
Grasso

S.
Yoshida

H.
Porwal

H.
Sakka

Y.
Reece

M.
Ceram. Int.
2013
, vol.
39
pg.
3243

139.
Sokol

M.
Kalabukhov

S.
Dariel

M. P.
Frage

N.
J. Eur. Ceram. Soc.
2014
, vol.
34
pg.
4305

140.
Hulbert

D. M.
Anders

A.
Dudina

D. V.

J.
Jiang

D.
Unuvar

C.
Anselmi-Tamburini

U.
Mukherjee

A. K.
J. Appl. Phys.
2008
, vol.
104
pg.
033305

141.
Guillon

O.
Gonzalez-Julian

J.
Dargatz

B.
Kessel

T.
Schierning

G.
Räthel

J.
Herrmann

M.
2014
, vol.
16
pg.
830

142.
Orrù

R.
Licheri

R.
Locci

A. M.
Cincotti

A.
Cao

G.
Mater. Sci. Eng. R Rep.
2009
, vol.
63
pg.
127

143.
Garay

J. E.
Annu. Rev. Mater. Res.
2010
, vol.
40
pg.
445

144.
Grasso

S.
Sakka

Y.
Maizza

G.
2009
, vol.
10
pg.
053001

145.
Kenfaui

D.
Chateigner

D.
Gomina

M.
Noudem

J. G.
J. Alloys Compd.
2010
, vol.
490
pg.
472

146.
Lee

M. H.
Park

J. H.
Park

S.-D.
Rhyee

J.-S.
Oh

M.-W.
J. Alloys Compd.
2019
, vol.
786
pg.
515

147.
Bourgès

C.
Bouyrie

Y.
Supka

A. R.
Al Orabi

R. A. R.
Lemoine

P.
Lebedev

O. I.
Ohta

M.
Suekuni

K.
Nassif

V.
Hardy

V.
Daou

R.
Miyazaki

Y.
Fornari

M.
Guilmeau

E.
J. Am. Chem. Soc.
2018
, vol.
140
pg.
2186

148.
Perez-Estébanez

M.

M.
Caballero

A. C.
Palomares

F. J.
Nygren

M.
Isasi-Marín

J.
Bol. Soc. Esp. Ceram. V
2016
, vol.
55
pg.
38

149.
Beekman

M.
Baitinger

M.
Borrmann

H.
Schnelle

W.
Meier

K.
Nolas

G. S.
Grin

Y.
J. Am. Chem. Soc.
2009
, vol.
131
pg.
9642

150.
Dennler

G.
Chmielowski

R.
Jacob

S.
Capet

F.
Roussel

P.
Zastrow

S.
Nielsch

K.
Opahle

I.

G. K. H.
2014
, vol.
4
pg.
1301581

151.
Sui

J. H.
Li

J.
He

J. Q.
Pei

Y. L.
Berardan

D.
Wu

H. J.
Dragoe

N.
Cai

W.
Zhao

L. D.
Energy Environ. Sci.
2013
, vol.
6
pg.
2916

152.
Imai

H.
Shimakawa

Y.
Kubo

Y.
Phys. Rev. B: Condens. Matter Mater. Phys.
2001
, vol.
64
pg.
241104R

153.
M. I.
Fedorov
and
V. K.
Zaitsev
,
Thermoelectrics Handbook: Macro to Nano
, ed. D. M. Rowe,
CRC Press, Taylor and Francis
,
Boca Raton
,
2006
, ch. 31
154.
Lotgering

F. K.
J. Inorg. Nucl. Chem.
1959
, vol.
9
pg.
113

155.
Yan

X.
Poudel

B.
Ma

Y.
Liu

W. S.
Joshi

G.
Wang

H.
Lan

Y.
Wang

D.
Chen

G.
Ren

Z. F.
Nano Lett.
2010
, vol.
10
pg.
3373

156.
Hu

L.
Zhang

Y.
Wu

H.
Liu

Y.
Li

J.
He

J.
Ao

W.
Liu

F.
Pennycook

S. J.
Zeng

X.
2018
, vol.
28
pg.
1803617

157.
Mikami

M.
Guilmeau

E.
Funahashi

R.
J. Mater. Res.
2005
, vol.
20
pg.
2491

158.
Y. M.
Belov
,
S. M.
Maniakin
and
I. V.
Morgunov
,
Thermoelectrics Handbook: Macro to Nano
, ed. D. M. Rowe,
Taylor & Francis, CRC Press
,
Boca Raton
,
2006
, ch. 20
159.
Ivanova

L. D.
Petrova

L. I.
Granatkina

Y. U. I.
Zemskov

V. S.
Sokoov

O. B.
Skipidarov

S. Y. A.
Duvankov

N. I.
Inorg. Mater.
2008
, vol.
44
pg.
687

160.
Vasilevskiy

D.
Masut

R. A.
Turenne

S.
J. Electron. Mater.
2012
, vol.
41
pg.
1057

161.
Zehetbauer

M.
Stüwe

H. P.
Vorhauer

A.
Schafler

E.
Kohout

J.
2003
, vol.
5
pg.
33

162.
Rogl

G.
Grytsiv

A.
Rogl

P.
Peranio

N.
Bauer

E.
Zehetbauer

M.
Eibl

O.
Acta Mater.
2014
, vol.
63
pg.
30

163.
Euler

F.
J. Appl. Phys.
1957
, vol.
28
pg.
1342

164.
Zhang

C. C.
Fan

X. A.
Hu

J.
Jiang

C. P.
Feng

B.
Xiang

Q. S.
Li

G. Q.
Li

Y. W.
2016
, vol.
18
pg.
1777

165.
Borup

K. A.
de Boor

J.
Wang

H.
Drymiotis

F.
Gascoin

F.
Shi

X.
Chen

L.
Fedorov

M. I.
Müller

E.
Iversen

B. B.
Snyder

G. J.
Energy Environ. Sci.
2015
, vol.
8
pg.
423

166.
Shen

J. J.
Hu

L. P.
Zhu

T. J.
Zhao

X. B.
Appl. Phys. Lett.
2011
, vol.
99
pg.
124102

167.
Harman

T. C.
J. Appl. Phys.
1958
, vol.
29
pg.
1373

168.
E.
Müller
,
C.
Stiewe
,
D. M.
Rowe
and
S. G. K.
Williams
,
Thermoelectrics Handbook: Macro to Nano
, ed. D. M. Rowe,
Taylor & Francis, CRC Press
,
Boca Raton
,
2006
, ch. 26
169.
Austin

I. G.
Mott

N. F.
1969
, vol.
18
pg.
41

170.
N. F.
Mott
and
E. A.
Davis
,
Electronic Processes in Non-Crystalline Materials
,
Clarendon Press
,
Oxford
,
1979
171.
G. S.
Nolas
,
J.
Sharp
and
H. S.
Goldsmid
,
Thermoelectrics: Basic Principles and New Materials Developments
,
Springer
,
2001
, ch. 4
172.
T. M.
Tritt
,
Thermoelectrics Handbook: Macro to Nano
, ed. D. M. Rowe,
Taylor & Francis, CRC Press
,
Boca Raton
,
2006
, ch. 23
173.
Wang

H.
Porter

W. D.
Böttner

H.
König

J.
Chen

L.
Bai

S.
Tritt

T. M.
Mayolet

A.
Senawiratne

J.
Smith

C.
Harris

F.
Gilbert

P.
Sharp

J. W.
Lo

J.
Kleinke

H.
Kiss

L.
J. Electron. Mater.
2013
, vol.
42
pg.
654

174.
van der Pauw

L. J.
Philips Res. Repts.
1958
, vol.
13
pg.
1

175.
ASTM Standard F76-86
,
Standard Method for Measuring Hall Mobility and Hall Coefficient in Extrinsic Semiconductor Single Crystals
,
Annual Book of ASTM Standards American Society for Testing Materials
,
,
1991
176.
Burkov

A. T.
Heinrich

A.
Konstantinov

P. P.
Nakama

T.
Yagasaki

K.
Meas. Sci. Technol.
2001
, vol.
12
pg.
264

177.
Martin

J.
Nolas

G. S.
Rev. Sci. Instrum.
2016
, vol.
87
pg.
015105

178.
Wood

C.
Rep. Prog. Phys.
1988
, vol.
51
pg.
459

179.
Moore

J. P.
Graves

R. S.
J. Appl. Phys.
1973
, vol.
44
pg.
1174

180.
Cusack

N.
Kendall

P.
Proc. Phys. Soc.
1958
, vol.
72
pg.
898

181.
G. S.
Nolas
,
J.
Sharp
and
H. S.
Goldsmid
,
Thermoelectrics: Basic Principles and New Materials Developments
,
Springer
,
2001
, ch. 5
182.
P. M. Chaikin,
Organic Superconductivity
, ed. V. Z. Kresin and W. A. Little,
Plenum
,
New York
,
1990
, p. 101–115
183.
R. H.
Heikes
and
R. W.
Ure
,
Thermoelectricity
,
Interscience
,
New York
,
1961
184.
Hébert

S.
Berthebaud

D.
Daou

R.
Bréard

Y.
Pelloquin

D.
Guilmeau

E.
Gascoin

F.
Lebedev

O.
Maignan

A.
J. Phys.: Condens. Matter
2016
, vol.
28
pg.
013001

185.
Goldsmid

H. J.
Sharp

J. W.
J. Electron. Mater.
1999
, vol.
28
pg.
869

186.
Gibbs

Z. M.
Kim

H.-S.
Wang

H.
Snyder

G. J.
Appl. Phys. Lett.
2015
, vol.
106
pg.
022112

187.
Martin

J.
Tritt

T.
Uher

C.
J. Appl. Phys.
2010
, vol.
108
pg.
121101

188.
Martin

J.
Meas. Sci. Technol.
2013
, vol.
24
pg.
085601

189.
Martin

J.
Wong-Ng

W.
Green

M. L.
J. Electron. Mater.
2015
, vol.
44
pg.
1998

190.
Alleno

E.
Bérardan

D.
Byl

C.
Candolfi

C.
Daou

R.
Decourt

R.
Guilmeau

E.
Hébert

S.
Hejtmanek

J.
Lenoir

B.
Masschelein

P.
Ohorodnichuk

V.
Pollet

M.
Populoh

S.
Ravot

D.
Rouleau

O.
Soulier

M.
Rev. Sci. Instrum.
2015
, vol.
86
pg.
011301

191.
W. D.
Callister
and
D. G.
Rethwisch
,
Materials Science and Engineering: An Introduction
,
Wiley
,
US
, 10th edn,
2018
, ch. 18
192.
E. H.
Putley
,
The Hall Effect and Related Phenomena
,
Butterworths
,
London
,
1960
193.
Borup

K. A.
Toberer

E. S.
Zoltan

L. D.
Nakatsukasa

G.
Errico

M.
Fleurial

J.-P.
Iversen

B. B.
Snyder

G. J.
Rev. Sci. Instrum.
2012
, vol.
83
pg.
123902

194.
Koon

D. W.
Bahl

A. A.
Duncan

E. O.
Rev. Sci. Instrum
1989
, vol.
60
pg.
275

195.
Lupu

N. Z.
Tallan

N. M.
Tannhauser

D. S.
Rev. Sci. Instrum.
1967
, vol.
38
pg.
1658

196.
M. A.
White
,
Properties of Materials
,
Oxford University Press
,
New York
,
1999
,
Chapter 8
197.
G. S.
Nolas
and
H. J.
Goldsmid
,
Thermal conductivity: theory, properties and applications
, ed. T. M. Tritt,
Springer
,
2004
, ch. 1.4
198.
Kim

H.-S.
Gibbs

Z. M.
Tang

Y.
Wang

H.
Snyder

G. J.
APL Mater.
2015
, vol.
3
pg.
041506

199.
J.
Goldsmid
,
Introduction to thermoelectricity, Springer Series in Material Science
,
Springer
,
Springer-Verlag Berlin Heidelberg
,
2010
, ch. 8
200.
Zhao

D.
Qian

X.
Gu

X.
Jajja

S. A.
Yang

R.
J. Electron. Packag.
2016
, vol.
138
pg.
040802

201.
T. M.
Tritt
and
D.
Weston
,
Thermal Conductivity: Theory, Properties and Applications
, ed. T. M. Tritt,
Springer
,
2004
, ch. 2.1, pp. 187–202
202.
Parker

W. J.
Jenkins

R. J.
Butler

P.
Abbott

G. L.
J. Appl. Phys.
1961
, vol.
32
pg.
1679

203.
Cowan

R. D.
J. Appl. Phys.
1963
, vol.
34
pg.
926

204.
Clark III

L. M.
Taylor

R. E.
J. Appl. Phys.
1975
, vol.
46
pg.
714

205.
Wang

H.
Porter

W. D.
Böttner

H.
König

J.
Chen

L.
Bai

S.
Tritt

T. M.
Mayolet

A.
Senawiratne

J.
Smith

C.
Harris

F.
Gilbert

P.
Sharp

J.
Lo

J.
Kleinke

H.
Kiss

L.
J. Electron. Mater.
2013
, vol.
42
pg.
1073

206.
Wang

H.
Bai

S.
Chen

L.
Cuenat

A.
Joshi

G.
Kleinke

H.
König

J.
Lee

H. W.
Martin

J.
Oh

M.-W.
Porter

W. D.
Ren

Z.

J.
Sharp

J.
Taylor

P.
Thompson

A. J.
Tseng

Y. C.
J. Electron. Mater.
2015
, vol.
44
pg.
4482

207.
ASTM E1269-11(2018)
,
Standard Test Method for Determining Specific Heat Capacity by Differential Scanning Calorimetry
,
ASTM International
,
West Conshohocken, PA
,
2018
208.
Thompson

F.
Phys. Educ.
2008
, vol.
43
pg.
396

209.
Wei

T. R.
Guan

M.
Yu

J.
Zhu

T.
Chen

L.
Shi

X.
Joule
2018
, vol.
2
pg.
2183

210.
Vasilevskiy

D.
Simard

J.-M.
Masut

R.
Turenne

S.
J. Electron. Mater.
2014
, vol.
44
pg.
1733

211.
Champness

C. H.
Parekh

P. C.
Can. J. Phys.
1965
, vol.
43
pg.
1589

212.
Kwon

B.
Baek

S. H.
Keun Kim

S.
Kim

J.-S.
Rev. Sci. Instrum.
2014
, vol.
85
pg.
045108

213.
Vasilevskiy

D.
Simard

J.-M.
Caillat

T.
Masut

R. A
Turenne

S.
J. Electron. Mater.
2016
, vol.
45
pg.
1540

214.
Downey

A. D.
Hogan

T. P.
Cook

B.
Rev. Sci. Instrum.
2007
, vol.
78
pg.
093904

215.
M. E.
Orazem
and
B.
Tribollet
,
Electrochemical impedance spectroscopy
,
John Wiley & Sons, Inc
.,
Hoboken, New Jersey
,
2008
216.

J.
Min

G.
J. Appl. Phys.
2014
, vol.
116
pg.
174510

217.
Beltrán-Pitarch

B.

J.
Powell

A. V.
Ziolkowski

P.

J.
J. Appl. Phys.
2018
, vol.
124
pg.
025105

218.
Beltrán-Pitarch

B.

J.
Powell

A. V.
Martínez-Julián

F.