Microstructure engineered multiphase tellurides with enhanced thermoelectric efficiency

Abstract

Thermoelectricity is one of the most important and extensively researched ways to recycle waste heat and enable efficient energy conversion without involving any movable parts. Multiphase thermoelectrics (TEs) are found to be better contenders than single-phase TEs, owing to their freedom to tune the interface, phase assemblage, defects, etc. Multiphase TEs provide the advantage of energy filtering, modulation doping, enhanced phonon scattering, higher structural integrity, and magnetic effects over single-phase materials. These engineered microstructural features and atomic arrangements result in a significant effect on the electronic and thermal transport in these materials. Moreover, tellurides among the chalcogenides have gained a lot of attention for their applicability as room-to-medium temperature range TE materials. In the present work, recent advancements in multiphase TEs based on tellurium are reviewed. Thus, this review focuses on some interesting ternary systems (Ag–Bi–Te, Bi–Cu–Te, Bi–Ga–Te, Bi–In–Te, Ga–In–Te, and Sn–Ga–Te) of the Ag–Bi–In–Ga–Sn–Te multicomponent system. Recent advancements in phase diagram engineering, microstructure evolution, processing conditions, and the role of these techniques in optimising the TE performance of multiphase materials belonging to the above-mentioned ternary systems have been discussed.

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1 Introduction

Thermoelectricity, a solid-state mechanism, has shown its impactful influence on waste heat recovery by providing a direct conversion between thermal and electrical energy. The energy conversion efficiency of thermoelectric (TE) devices is directly dependent on the material properties and the thermal gradient across the two ends of the device. A material's ability is defined using a TE figure of merit (zT), which dictates the thermal and electronic transport in the material. Electronic and thermal transport is determined by the material properties like electrical conductivity (σ), Seebeck coefficient (S), and thermal conductivity (κ) of the material. These properties define the zT of a material, as given in eqn (1):¹

As evident in eqn (1), to achieve high zT, high σ and S with low κ are required. Thus, the material must conduct electrons/holes like that in crystalline materials, while heat must be conducted like that in amorphous materials to achieve the Phonon-Glass Electron-Crystal (PGEC) concept.² There are various classes of materials being explored for TE applications, such as clathrates,^3,4 chalcogenides,^5–8 skutterudites,^9,10 Half-Heusler,^11,12 Zintl phases,^13,14 oxide-based,¹⁵ organic,^16,17 nanostructured TEs,^18–21etc. Moreover, these materials can also be classified based on their operational temperature as low temperature (<573 K), room to medium temperature (573–873 K), and high temperature (>873 K) TE materials.² Among the chalcogenides, tellurides are emerging TE materials with a wide application range in the room to medium temperature range.^22–27 These materials offer applications in the fields of automobiles, power plants, medical science, electronics, aerospace, space exploration, solar TE hybrid systems, defence and military, biotechnology, etc.^28,29 There are various telluride phases such as Bi₂Te₃, SnTe, In₂Te₃, In₂Te₅, InTe, GaTe, Ga₂Te₃, GaInTe₂, AgBiTe₂, Ag₂Te, Cu₃Te₂, Cu₂Te, etc.,^30–40 which have shown significant improvement in their TE performance over time. Moreover, depending on the number of phases in a TE material, it can be classified as single-phase or multiphase. Single-phase materials comprise a characteristic crystal structure under specific composition, temperature, and pressure conditions. In the case of single-phase materials, doping, alloying, solid solutioning, changing processing conditions, etc., are usually used to tune charge and phonon transport.⁴¹ In addition, for a single-phase material, electronic band structure and phonon dispersion relationships can be easily generated using computational tools,^42,43 which can aid in understanding the electron and phonon transport. On the other hand, in the case of multiphase materials, consisting of more than one phase (of the same or a different crystal structure), these calculations become computationally expensive and time-consuming due to structural complexity, an increased number of atoms, interfaces, defects, disorder, etc.^44–46 Meanwhile, multiphase systems can aid in achieving selective conduction of high-energy charge carriers along with strong scattering of phonons.⁴⁷ Although incorporating more than one phase in a material can open the freedom to tune the electronic and thermal transport, it also comes with computational and characterisation challenges. The coexistence of two or more phases with different concentrations, types of defects, and interfaces can alter the electronic and thermal transport significantly. Defects in these phases (vacancies, antisites, dislocations, stacking faults, etc.) can enhance or hinder electronic and thermal transport.^48,49 In addition, depending on the type of phase (metallic, semiconductor, or insulator) present in the material, the transport behaviour differs.⁵⁰ Therefore, phase diagrams, defects, and interface engineering are crucial in determining the material's overall TE response while developing multiphase TEs.

This review discusses recent advancements in some of the Telluride-based systems of the Ag–Bi–In–Ga–Sn–Te multicomponent system. In addition, various processing techniques like solid solutioning, directional solidification, heat treatments, etc., which are employed to prepare and tune the TE properties of these multiphase TEs, will be discussed. The review covers various aspects of multiphase TEs in six sections. We will start with a brief introduction to the fundamentals of TEs in Section 2. Section 3 will discuss the importance and the key strategies to develop multiphase TEs. In Section 4, the phase diagram engineering of six interesting ternary systems of the Ag–Bi–In–Ga–Sn–Te multicomponent system will be discussed. Section 5 shows microstructure engineering and its impact on the TE properties of these systems using different phase diagrams, processing conditions, composition control, and types of primary/secondary phases. Section 6 will discuss some of the theoretical models available, which are/can be used to strengthen the understanding of transport phenomena in multiphase TEs. At last, a summary with key achievements and challenges in multiphase TEs will be provided.

2 Fundamentals of thermoelectric properties

As mentioned in eqn (1), zT defines a material's ability to efficiently convert thermal to electrical energy or vice versa. Moreover, it requires a material to have high σ and S with low κ to achieve high zT. Electrical conductivity dictates the ability of a material to conduct electrons/holes under specific conditions (external potential, temperature, photons, etc.). Furthermore, the σ of a material depends on the concentration (n) and the mobility (µ) of charge carriers as described below:⁵¹

σ = n_ee|µ_e| + n_pe|µ_p| (2)

where subscripts ‘e’ and ‘p’ denote electrons and holes, respectively, while ‘e’ is the elementary charge. Thus, both electrons and holes contribute positively to the σ. Moreover, as the number of charge carriers increases, the probability of scattering will also increase, which reduces µ. Generally, strategies like doping (increases the number of free charge carriers), alloying, reducing defects like dislocations, vacancies, and grain boundaries, and approaching single-crystal form (improve µ) are used to enhance the σ of a material.^52–54

Similarly, ‘S’, a material property, defines the ability of a material to generate a voltage (ΔV) under an applied temperature difference (ΔT) across the material, which can be given by eqn (3):²

In addition to this, S can be related to n and the density of states effective mass (m*) through a simplified relationship for metals or degenerate semiconductors using the Mott relationship⁵⁵ provided in eqn (4):

Thus, it is evident from eqn (2) and (4) that n shows direct dependence on σ, and an inverse dependence on S. In addition, the S shows a direct relationship with m*. Thus, optimum n and m* values are desired to have balanced S and σ. It has been observed that the n range lying between 10¹⁹ and 10²¹ cm⁻¹ (semiconductor range) generally leads to optimised S and σ, resulting in the maximum zT. Thus, instead of higher doping levels, it is always an optimum doping level that should provide higher σ without significantly reducing S. In addition to this, S can be related to the density of states and the Fermi distribution function through the Mott relationship:⁵⁵

where k_B, q, E, E_F, σ(E), and ∂f_o(E)/∂(E) correspond to the Boltzmann constant, elementary charge, carrier energy, Fermi energy, a kinetic coefficient (differential electrical conductivity), and energy derivative of Fermi–Dirac distribution, respectively. As evident in eqn (5), S is directly dependent on the normalized energy difference (E − E_F)/k_BT. Thus, it requires the charge carriers of higher energy to conduct to improve the S of a material. Band structure engineering—such as increasing density of states near E_F (increases the asymmetry in energy distribution), blocking low energy charge carriers from conducting, and band convergence—is typically used to enhance the S of the material. These strategies can be applied to the materials experimentally through solid solutions, alloying, doping, nanostructuring/quantum confinements, annealing/heat treatments, interface tuning, and defect engineering.^48,56–60 Moreover, one of the challenges in TE materials is the bipolar conduction, wherein the majority and minority charge carriers both contribute to conduction, which can enhance σ but deteriorates S⁶¹ as S is related to both charge carriers according to eqn (6).where σ_n and σ_p are the electrical conductivity by electrons and holes, respectively, while S_n and S_p represent the Seebeck coefficient due to electrons and holes, respectively. The sign of S will change depending on the charge carrier (− for n and + for p). Thus, in the case of bipolar conduction, S decreases drastically. Moreover, the bipolar effect is usually observed in narrow bandgap materials.

Other than σ and S, κ also plays a significant role in deciding the zT of a material. It usually comprises two contributions: lattice (κ_l) and electronic (κ_e), as shown in eqn (7). In the case of semiconductors and insulators, a large amount of heat is usually carried by the lattice, while for metals and degenerate semiconductors, the electronic contribution is also significant.

κ_tot = κ_e + κ_l (7)

Using the Wiedemann–Franz Law,⁶²κ_e is usually computed using the relationship given in eqn (8), where L is the Lorenz number, which is considered to be 2.44 × 10⁻⁸ WΩK⁻² for metals and degenerate semiconductors.

κ_e = σLT (8)

In addition, Kim et al.⁶³ observed that L and the experimentally determined S can be related using a simple relationship eqn (9)

This relationship holds good for |S| >10 µV K⁻¹.

In the case of narrow bandgap materials, there are enough electron–hole pairs created at high temperatures, which carry heat toward the cold end and recombine to liberate that heat, thus aiding in heat transfer.⁶¹ Therefore, the total electronic contribution to the thermal conductivity can be expressed as in eqn (10):

where the last term represents the bipolar conduction in thermal conductivity (κ_bp).

On the other hand, κ_l dictates how efficiently heat is carried by phonons in a material. Using classical kinetic theory, κ_l can be related to the heat capacity at constant volume (C_v), phonon group velocity (v), and mean free path (l) of the phonon by the following relationship:⁶⁴

Meanwhile, l can be related to relaxation time (τ) by eqn (12)

l = τv (12)

The several scattering mechanisms responsible for limiting the mean-free path can be related to τ by eqn (13)

where i represents the scattering mechanisms such as acoustic phonon scattering, ionized impurity scattering, alloy scattering, grain boundary scattering, impurity scattering, etc.^2,65 In addition, various models and approaches, such as the Callaway model for low temperature κ_l,⁶⁶ first principles Debye–Callaway approach,⁶⁷ Slack's model for high temperature κ_l of non-metallic crystals,⁶⁸ the Cahill–Pohl model for amorphous solids,⁶⁹etc., are used extensively to understand κ_l. All these models are mostly restricted to homogeneous single-phase materials, a single phonon spectrum, and a lot of adjustable parameters. Nevertheless, eqn (11) shows that κ_l is independent of the electronic contribution; thus, reducing κ_l separately can be used as one of the ways to maximize zT. Scattering of phonons is usually enhanced by nanostructuring,⁷⁰ grain size engineering (more than the mean free path of charge carriers, yet less than the mean free path of phonons),⁷¹ nanocomposites,⁵⁹ interfaces,⁷²etc. Crucially, extensive research into the interaction of phonons of various frequencies with defects of all dimensions (from point-like to extended surfaces) reveals a compounding effect that diminishes κ_l.^47–49 As evident from eqn (1)–(13), the TE properties are dependent on each other, and achieving a high-zT material requires an optimum balance between these properties. More elaborate details on the fundamentals of transport properties are provided by Zevalkink et al.⁷³ Thus, a high zT material can be achieved by either maximising the power factor (S²σ) with a minimum increase in κ or reducing κ_l.

In addition to the longitudinal zT of a TE (discussed above), transverse zT (shown in eqn (14)) for transverse thermoelectric materials also gained attention in recent years.^74,75 Transverse thermoelectricity is based on the Nernst/Ettingshausen effect, wherein an electric field develops perpendicular to both the applied temperature gradient and the magnetic field.⁷⁴ This is different from traditional longitudinal thermoelectricity, where the temperature gradient and electrical current flow in the same direction. Recently, a detailed review has been published by Adachi et al.⁷⁶ which summaries the fundamentals, progress and challenges experienced over the years in the field of transverse thermoelectric conversion.

where zT(⊥), σ_yy, S_xy, and κ_xx are the transverse thermoelectric figure of merit, electrical conductivity along the direction of induced voltage, transverse thermopower and thermal conductivity along the direction of heat flow respectively.

3 Why multiphase thermoelectrics

Multiphase materials contain two or more phases simultaneously contributing to thermal and electronic transport. Although the idea of multiphase materials for advancing TEs seems to be intuitive, several challenges have been raised in the community against its implementation. On introducing a second phase, possible scattering of charge carriers at the interface and poor energy band alignment can lead to a reduction in σ and S. Moreover, the phases will have different phase stability, crystal structures, and microstructures, which can significantly deteriorate the TE and mechanical stability. Tuning the volume fraction, grain morphology, atomic defects, and interface to maximise σ and S are some of the key challenges in these materials. Moreover, it was believed that the zT of a multiphase TE can never be beyond the highest value of one of the constituent phases,⁷⁷ till a two-phase composite based on SiGe-Si was reported with enhanced zT compared to that of the individual phases.^78,79 The high zT was attributed to inhomogeneous selective doping of Si via B and P instead of the SiGe phase, resulting in enhanced µ.⁷⁸ Thus suggesting the role of band engineering at the interface, which favoured the easy transfer of charge from one phase to the other, resulting in enhanced zT.⁷⁹ After the success story of achieving enhanced zT in SiGe-Ge multiphase materials, significant interest has emerged among researchers to understand and explore different multiphase TEs.

Although the transport of charge carriers and phonons can be explained by the calculated electronic bands and phonon spectrum in the case of single-phase materials, the presence of secondary phases requires further attention. Understanding individual phases and the interfaces between them is essential to determine the overall behavior of the material. Over the years, it has been observed that multiphase TEs are preferable to single-phase materials as they offer more freedom to tune the transport properties.⁵⁰ The development of multiphase alloys involves phase diagram engineering, band structure engineering and interface engineering, as shown in Fig. 1. Moreover, introducing two or more phases into the systems impacts the band structure of the constituent phases due to possible doping and changes in the solubility of the impurities in the constituent phases. The improvement in the TE behaviour in multiphase TE materials primarily relies on energy filtering, enhanced phonon scattering, modulation doping, and magnetic effects (illustrated in Fig. 1).^47,50,80 As illustrated in Fig. 1, the charge carriers with low energy are filtered out by potential barriers created by the secondary phases, which leads to a high S without much loss of σ.⁵⁰ On the other hand, modulation doping reduces the charge carrier scattering due to the presence of ionized impurities.⁵⁰ These impurities spatially separate the conduction carriers from the host atoms and thus lead to an increase in σ of the material. In addition, for highly anisotropic alloys, crystallite alignments along a preferential orientation can lead to easy charge transport, thus increasing the µ of charge carriers. Moreover, in multiphase TEs, high frequency phonons are scattered at the point defects due to atomic mismatch and the presence of vacancies, and low-frequency phonons are usually scattered by the interface, grain boundaries, etc., which leads to further reduction in κ_l. On the other hand, introducing secondary magnetic phases introduces magnetic ions. Thus, from the hot to cold end of the TE leg, there will be varying magnon density. These magnons can drag the charge carriers along with them, resulting in a significant increase in σ.⁸¹ In addition, more detailed explanations and effects of different magnetic phases on the TE properties are discussed by Fortulan and Yamini.⁵⁰ Fortulan and Yamini⁵⁰ reviewed various success stories and advancements using the above-mentioned effects in multiphase TEs. In addition, the effect of different types of the secondary phase on the electronic band structure is usually understood by heterojunctions between the phases involved. For that, information on the electron affinity, bandgap, and/or work function of the involved phases is usually required.⁸² Thus, it can be inferred that multiphase TEs make it possible to strongly scatter phonons and low charge carriers resulting in enhanced S and σ.

Fig. 1 Schematic illustration of various engineering strategies and effects of secondary phases and interfaces in multiphase TEs.

Moreover, the use of anisotropic or layered multiphase structures can be promising to enhance transverse zT compared to single-phase materials. Ando et al.⁸³ designed a multilayer composite of tilted SmCo₅ (magnetic phase) and Bi_0.2Sb_1.8Te₃ (TE phase) at a particular angle. Thus, an increased transverse voltage, owing to the internal magnetization and strong coupling between Seebeck and anomalous Nernst effects, was observed without the need for a large external magnetic field. Hirai et al.⁸⁴ investigated the tilted multilayers of a magnetic topological phase with non-conductive top layers. They observed a hybrid effect which leads to enhanced S_xy, combining the Berry curvature driven anomalous Nernst voltage with anisotropic transport attributed to the tilt geometry. There are some other interesting studies wherein multiphase materials in transverse TEs have been explored.^85,86 Thus, multiphase TEs provide freedom to change the phase fractions, which can control the anisotropy, carrier concentration and magnetic field response. As summarised from recent studies, TE materials with combined magnetic and semiconductor phases can provide strong magneto–thermoelectric coupling. In addition, other than the properties of the individual phases, the type of interface between these phases can play a vital role in further enhancing the transverse thermopower.

4 Application of thermodynamic databases in thermoelectrics

In recent years, a lot of new and interesting ternary systems have been explored for TE applications. The design and optimisation of new TE materials rely on phase diagrams to determine composition, phase fraction, solidification pathways, and appropriate heat treatment cycles. The CALPHAD (CALculation of PHAse Diagrams) approach⁸⁷ integrates crystallographic data, thermodynamic, and thermochemical information, and physical properties, including the magnetic characteristics of all relevant phases, obtained from both experimental measurements and theoretical calculations, to construct a self-consistent thermodynamic database. The basic principle to determine the phase equilibria involves Gibbs free energy (G) minimization, using a developed database which contains optimised G of all the phases with changing composition, temperature, and pressure of a thermodynamic system. The CALPHAD method depends on experimentally assessed thermodynamic data and parameter optimization, making it highly reliable for multicomponent and industrially relevant systems where accurate phase equilibria are required. In contrast, first-principles or Open Quantum Materials Database (OQMD) derive thermodynamic properties from electronic structure computations, allowing the exploration of new or experimentally inaccessible systems and the prediction of metastable phases. In addition, the formation energies predicted at 0 K determined using OQMD can be used for defining the unknown compounds and the end members of a solution phase and thus, aid in precise modeling of the phases using compound energy formalism (CEF) or other models wherein the Gibbs free energy of the pseudo-compounds or end members is usually not known. Moreover, the 0 K energetics can be extended to the finite temperature using the quasi-harmonic phonon models and entropy corrections, which thus can be directly used as the Gibbs free energy of the phase in CALPHAD. However, OQMD is often computationally intensive and less effective for high-entropy or multi-component systems due to combinatorial complexity. Moreover, these two approaches (CALPHAD and OQMD) are mutually reinforcing within a prediction–validation loop: first-principles calculations provide initial thermodynamic insights and phase stability trends, which can guide CALPHAD assessments; conversely, CALPHAD modeling and experimental data can validate and refine first-principles predictions. This integrated strategy enhances both the predictive accuracy and reliability of computational phase diagram design. Various thermodynamic models are used to fit the available data to optimize the G of all the phases, including pure elements, stoichiometric/non-stoichiometric compounds, solid/liquid solutions, and the gases as a function of temperature, pressure, and composition.^87–89 Moreover, using the CALPHAD approach, solutions in the subsystems are extended to a multicomponent system by appropriate extrapolation techniques.⁹⁰ The developed phase diagrams provide a road map for a variety of material processing operations, including crystal growth, joining, melting, and casting. To improve control over material properties and performance, they will also be essential in understanding phase changes, optimising heat treatments, and directing solid-state reactions. Recently, the addition of the Debye–Einstein model in the C_p (used in determining the κ) estimation widened the applicability of CALPHAD at low temperatures.^91,92 In addition, it also provides information about the crystal structure, homogeneity ranges, site occupancies, first and second order transitions, thermal expansion coefficients (a parameter to decide the compatibility of a TE device), and other physical properties of the different phases in various temperature and pressure ranges.

Moreover, the CALPHAD-based relationship⁹³ with the solubility limit was used to determine the carrier concentration in PbSe, which gives a good fit with the experimentally observed carrier concentration compared to that determined using DFT. In addition, a new CALPHAD method was developed, which uses the heat capacity, thermal expansion, and adiabatic bulk modulus information to determine the κ_l of insulators and semiconductors.⁹⁴ Later, κ of the solid solutions and two-phase materials was modelled using Redlich–Kister (RK) polynomials⁹⁵ by Huang et al.⁹⁶ In addition, they also incorporated the interface scattering parameter with linear temperature dependence for the multiphase materials. Apart from this, various aspects of CALPHAD in the TE field were reviewed by Li et al.⁹⁷ There are various success stories wherein CALPHAD played an important role in discovering new TE materials such as the new filling fraction limit (FFL) of Ce_xCo₄Sb₁₂ from 0.09 to 0.2 at 1123 K, which result in the discovery of a new alloy Ce_0.14Co₄Sb₁₂ with a zT of 1.3 at 850 K,^98,99 similar for FFL of In, Ga, and Yb in CoSb₃.^100–102 Similarly, AgCuX (X = S, Se, Te),^103,104 PbTe–Sb₂Te_3,¹⁰⁵ and Bi_0.5Sb_1.5Te₃–Te¹⁰⁶ are some of the systems, where CALPHAD played a significant role. Thus, phase diagrams dictate the experimental conditions and the composition limits to target a specific phase assemblage and type of reaction by elucidating the presence of miscibility gaps, phase separations, and the presence and stability ranges of intermetallic compounds. Moreover, nanoprecipitate formation in a solution with a miscibility gap can be estimated by selecting the compositions lying in the metastable regions between the binodal and spinodal lines, which are predicted by CALPHAD. These precipitates act as energy filters and phonon scatterers to enhance the S and reduce κ_l.⁶⁰ Even as summarised by Femi et al.,¹⁰⁷ eutectic microstructures can also play a vital role in selectively scattering long-wavelength phonons and maintaining structural integrity. These precipitation and eutectic morphologies can be predicted and understood using phase diagrams.

Considering the importance of the phase diagrams, a review focusing on the impact of phase diagram engineering in the Ag–Bi–Se, Pb–Te–Ga, Zn–Sb–In, Ga–Sb–Te, Bi–Ga–Te, Bi–Ge–Te, Bi–Cu–Te, and Cu–Ga–Te ternary systems was reported by Deng et al.¹⁰⁸ They reviewed some of the interesting studies, wherein phase diagrams helped to choose compositions, understand the obtained phase assemblage, and correlate them to changes in TE properties. The present work reviews some of the recently explored tellurium-based ternary systems: Ag–Bi–Te, Bi–Cu–Te, Bi–Ga–Te, Bi–In–Te, Ga–In–Te, and Sn–Ga–Te, which have a detailed and almost complete thermodynamic description based on CALPHAD. Moreover, using these developed thermodynamic databases, researchers calculated the specific pseudo-binary phase diagrams of known interesting TE phases to design new multiphase TEs. In the following subsections, we will discuss the latest thermodynamics/phase equilibria advancements in these ternary systems. Fig. 2 shows the liquidus projections of these ternary systems using the developed thermodynamic databases (except in the case of Bi–In–Te, it is experimentally determined).

Fig. 2 Liquidus projections of (a) Ag–Bi–Te,¹¹⁴ Copyright 2024, Elsevier Ltd., (b) Bi–Cu–Te,³⁹ Copyright 2023, Springer Nature (c) Bi–Ga–Te,¹⁸¹ Copyright 2023, Elsevier B.V., (d) Bi–In–Te,¹⁸⁶ Copyright 2017, Elsevier B.V., (e) In–Ga–Te,¹⁹² Copyright 2022, Springer Nature, and (f) Sn–Ga–Te,²⁰⁴ Copyright 2023, ASM International, ternary systems. Symbols marked show the compositions of the investigated alloys and arrows represent the solidification pathways.

4.1 Advancements in phase equilibria of silver–bismuth–tellurium (Ag–Bi–Te)

The Ag–Bi–Te ternary system consists of phases such as Bi₂Te₃, β, Ag₂Te, Ag₅Te₃, AgBiTe₂, etc., which have been reported as potential TE materials.^{35,38,70,109–113} Recently, a complete thermodynamic description of the Ag–Bi–Te ternary system was attempted by Serbesa et al.¹¹⁴ using the CALPHAD approach. They first optimized the binaries (Ag–Te, Bi–Te, and Ag–Bi) and then the corresponding ternary using a Toop-like asymmetric approximation with Te as an asymmetric component.¹¹⁵ They used the G of pure elements from the Scientific Group Thermodata Europe (SGTE) database,¹¹⁶ and the G of pure compounds was fitted using heat capacity, standard enthalpy, and entropy at 298.15 K available in the literature. To explain the diverse nature of the liquid solution (positive change in mixing enthalpy (ΔH_mix) in Bi–Ag and negative ΔH_mix in Bi–Te and Ag–Te) in the binaries and the ternary liquid, the Modified Quasichemical Model (MQM)¹¹⁷ was used. Moreover, the Ag–Te binary system shows a miscibility gap in the liquid phase, which was precisely captured using the MQM. In addition, compound energy formalism (CEF)⁸⁹ was used to model Bi₂Te₃ and BiTe solid solutions. The experimentally observed allotropic forms of the binary compounds were considered while developing the thermodynamic database. They observed a reasonable agreement with the experimental studies on the Ag–Bi–Te ternary by Babanly et al.¹¹⁸ and Stegherr et al.¹¹⁹ Babanly et al.¹¹⁸ developed various alloys in the Ag₅Te₃–Bi₂Te₃, AgBiTe₂–Te, AgBiTe₂–Bi, and Ag–Bi₂Te vertical sections. Moreover, they also constructed an isothermal section based on their experimental results at 400 K. On the other hand, Stegherr et al.¹¹⁹ explored the phase equilibria in the Ag₂Te–Bi₂Te₃ pseudo-binary phase diagram, which showed the formation of a high-temperature phase, AgBiTe₂, at 828 K. The calculated liquidus projections with the primary crystallizing phases from the ternary liquid are shown in Fig. 2(a). Using the developed thermodynamic database, various compositions in the Ag₂Te–Bi₂Te₃ sections were selected, and their TE behavior was explored by Serbesa et al.¹¹⁴ In addition, these compositions were cast, which shows the formation of the AgBiTe₂ phase as predicted by Scheil cooling calculations.

There are various experimental studies on the Ag–Bi, Ag–Te, and Bi–Te binary systems to find stable and reliable phase equilibria. They used a variety of experimental and thermodynamic methods. The Ag–Bi system shows a simple eutectic phase diagram. Early studies by Heycook and Neville,¹²⁰ Petrenko,¹²¹ and Nathans and Leider¹²² showed a eutectic reaction (L → fcc_Ag + rhombo_Bi) at about 535 K and 4.7 at% Ag. They also showed that fcc_Ag has limited Bi solubility.^{121,123–128} Thermochemical data from calorimetry^129,130 and activity measurements^{126,131–140} corroborated these findings, while thermodynamic modelling progressed from the Redlich–Kister approach¹⁴¹ to the MQM.^142,143 The Ag–Te system has been investigated using thermal, X-ray, DTA, and calorimetric methods,^144–153 uncovering various polymorphs (Ag₂Te and Ag₅Te₃) and significant short-range ordering in the liquid phase (minimum at X_Te = 0.33).^150–152 Predel and Piehl's¹⁵⁴ activity measurements and Karakaya and Thompson's¹⁵⁵ and Gierlotka's^156,157 CALPHAD optimisation matched with each other. Mao et al.,¹⁵⁸ Gierlotka et al.,¹⁵⁹ and Kumar et al.¹⁶⁰ have also optimised the Bi–Te system thermodynamically using regular solution, associate, and MQM models, respectively. The parameters based on the MQM model gave the best agreement. Drezewowska and Onderka's¹⁶¹ reassessment suggested more low-temperature phases (Bi₇Te₃ and Bi₂Te), but these were not included by Kumar et al.¹⁶⁰ Together, these studies create a consistent thermodynamic framework for all three binary systems, which is a strong base for modelling the ternary Ag–Bi–Te system.

Similar to the Ag–Bi–Te system, the Ag–Sb–Te system is also an important ternary system for TE applications. There is an interesting phase, AgSbTe₂, which shows a very high TE performance (zT > 1.6). Although there is no complete thermodynamic description available for the system, there are various thermodynamic studies that investigate the Ag₂Te–AgSbTe₂,¹⁶² Ag₂Te–Sb₂Te₃,¹⁶³ and isothermal sections at 523 and 673 K.¹⁶⁴ Thus, a complete thermodynamic description of this system by utilizing the available thermodynamic information can help design multiphase alloys based on Ag–Sb–Te.

4.2 Advancements in phase equilibria of bismuth–copper–tellurium (Bi–Cu–Te)

Another interesting ternary system, Bi–Cu–Te in the field of TEs, has been thermodynamically optimized by Serbesa et al.³⁹ They used the thermodynamic modelling parameters of the Bi–Te binary system from Kumar et al.¹⁶⁵ The Cu–Te binary system was reassessed using the MQM, as earlier studies used associate models, which lag in describing the entropy accurately.^166,167 After a detailed literature survey, they considered four intermetallic compounds: Cu₂Te, Cu₆₃Te₃₇, CuTe, Cu₃Te₂, and Cu₄Te₃ with six, one, two, three, and two polymorphic forms, respectively, in the Cu–Te binary system. They incorporated the Cu₃Te₂ intermetallic for the first time while considering the experimental observations by Yahyaoglu et al.¹⁶⁸ Mao et al.,¹⁵⁸ Gierlotka et al.,¹⁵⁹ and Kumar et al.¹⁶⁰ worked on optimising the Bi–Te system using different models as mentioned in Section 4.1. The developed database by Kumar et al.¹⁶⁰ was the most consistent with experimental data of the Bi–Te system. The Cu–Te system, first modelled by Huang et al.¹⁶⁶ and then improved by Yu et al.,¹⁶⁷ showed that there are many intermetallics (CuTe, Cu₂Te, Cu₄Te₃, and Cu_3−xTe₂) and a lot of polymorphisms, which were observed by both experimental and DFT data.^{127,168–173} Serbesa et al.³⁹ utilised the MQM model to characterise the Cu–Te liquid phase and introduced Cu₃Te₂ for the first time in thermodynamic optimisation based on the experimental observations using microstructural analysis and XRD. On the other hand, Chakrabarti and Laughlin¹⁷⁴ reviewed the Bi–Cu system, while Wang et al.¹⁴³ provide thermodynamic optimisation. The system is a eutectic one with limited solubility between Bi and Cu. Their MQM- and CEF-based model parameters were used without any changes by Serbesa et al.³⁹ In general, these consistent thermodynamic datasets provide a strong basis for accurately predicting and modelling the phase diagram of the Bi–Cu–Te system. In addition, short-range ordering (SRO) in the Cu-rich region of the liquid was precisely captured by the MQM. On the other hand, the Cu–Bi binary system is a simple eutectic with negligible solubility of Cu in Bi or vice versa.¹⁷⁴ Serbesa et al.³⁹ used the thermodynamically optimised parameters developed by Wang et al.¹⁴³ for the Cu–Bi binary system, wherein they used the MQM for the description of the liquid phase. The negative ΔH_mix in Bi–Te and Cu–Te liquids and the positive ΔH_mix in Bi–Cu were captured precisely by the MQM. To explain the experimentally constructed Cu₂Te–Bi₂Te₃ pseudo-binary phase diagram, Serbesa et al.³⁹ used Te as an asymmetric component along with ternary interaction parameters in the liquid phase. Finally, using the developed database, liquidus projections were calculated, and it was observed that the miscibility gap in the Cu–Te binary system was extended to the ternary space (as shown in Fig. 1(b)). The developed database was used to design various alloys for the TE application by Serbesa et al.,³⁹ which will be explained in Section 5.2.

4.3 Advancements in phase equilibria of bismuth–gallium–tellurium (Bi–Ga–Te)

The Bi–Ga–Te ternary system with Bi₂Te₃, Bi₄Te₃, Bi₂Te, Ga₂Te₃, GaTe, Ga₃Te₄, and Ga₂Te₅ attracted a lot of attention in TE applications.^35,175–177 Lin et al.¹⁷⁸ reported an experimentally constructed isothermal section of Bi–Ga–Te at 523 K using the alloys prepared in an ampoule at 1223 K for 12 hours and post-annealed at 523 K for at least 6 months, followed by water quenching. Later, thermodynamic modelling of the Bi–Ga–Te system was performed by Kumar et al.,¹⁶⁵ affording a thermodynamic description throughout the whole composition range. While developing the database, a thorough literature survey of the system was performed. They used the MQM¹¹⁷ to precisely capture the diverse changes in the liquid behaviour (negative change in mixing enthalpy in Ga–Te and Bi–Te and positive mixing enthalpy in Bi–Ga liquid) along with SRO. The Gibbs free energy of the ternary liquid solution in the Bi–Ga–Te system was approximated using the Toop-like asymmetric approximation, where Te was an asymmetric component.¹¹⁵ They did not use any ternary parameter in liquid solution to describe the phase equilibria in the ternary system. They reported a reasonable agreement with the experimentally observed vertical sections such as GaTe–Bi, Ga₂Te₃–Bi₂Te₃, GaTe–Bi₂Te₃, and Ga₂Te₃–Bi sections.^179,180 Moreover, two ternary eutectics were observed in the system: one between L, β, Bi, and Ga₃Te₄, and the other between L, Bi₂Te₃, Ga₂Te₃, and Te phases. They also observed a miscibility gap extended from the Ga–Te binary, as evident in Fig. 2(c). Moreover, the calculated liquidus projection was used by Pal et al.¹⁸¹ with some compositions marked in Fig. 1(c) to design multiphase alloys for TE applications. Using the developed thermodynamic database, the solidification path and the phase fraction evolution of the as-cast microstructures were explained via Scheil cooling calculations.

4.4 Advancements in phase equilibria of bismuth–indium–tellurium (Bi–In–Te)

The Bi–In–Te system constitutes InBi, In₅Bi₃, In₂Bi, Bi₂Te₃, In₂Te₃, InTe, In₄Te₃, In₂Te₅, (Bi₂)_m(Bi₂Te₃)_n, etc. phases which are important from the TE perspective.^182–185 No complete thermodynamic modelling is available for this ternary. However, experimentally, there is a detailed investigation by Chen et al.¹⁸⁶ to construct the phase diagram of the system. They reported a ternary compound BiIn₂Te₄ for the first time in some of the alloys annealed at 423 K. To construct the liquidus projections, they prepared 88 alloys and tried to understand the liquid to solid transformations (the constructed liquidus projections are shown in Fig. 2(d)). In addition, Bi₂Te₃ also shows a solubility of about 8.7 at% In in some of the investigated alloys, which can be incorporated using CEF. As evident, there is a miscibility gap in the In–Te binary system, which was extended to the ternary system. Thus, using the experimentally observed phase transformation and the liquidus projections, a self-consistent thermodynamic database can be developed. There are studies in the Bi₂Te₃–In₂Te₃ pseudo-binary phase diagram that show a eutectic at 844 K and 66 ± 0.5 at% Bi₂Te₃.^187,188 A maximum solubility of 20 at% In₂Te₃ in Bi₂Te₃ was observed at the eutectic temperature. Moreover, they performed thermal analysis, microstructural examination, and XRD to construct pseudo-binary phase diagrams. Apart from these studies, there are some experimental investigations on InTe–Bi and InTe–InBi isopleths, which can be considered while developing a thermodynamic model for the system.^189,190

4.5 Advancements in phase equilibria of indium–gallium–tellurium (In–Ga–Te)

The In–Ga–Te ternary system has interesting phases, which have been explored for TE applications, such as In₂Te₃, Ga₂Te₃, InTe, In₃Te₄, In₂Te₅, etc.^{7,31,40,182,191} A complete thermodynamic description of the system was provided by Pal et al.,¹⁹² in which they have used the CALPHAD approach to develop a thermodynamic database. In–Te and In–Ga binaries were first modelled by Pal et al.¹⁹² with the liquid phase being modelled by the MQM¹¹⁷ while CEF⁸⁹ used for the solid solutions. The optimized parameters for the Ga–Te binary developed by Kumar et al.¹⁶⁵ were used to develop the complete thermodynamic description of the ternary. The experimentally investigated vertical sections of GaTe–InTe by Kuliev et al.¹⁹³ show the limited solubility due to the difference in crystal structures. On the other hand, Ga₂Te₃–In₂Te₃ by Wooley et al.¹⁹⁴ and by Yamanaka et al.¹⁹⁵ shows a complete solid solubility of both the phases throughout the temperature range owing to their same crystal structure (defect zinc blende cubic). In addition, R. Blachnik and E. Klose¹⁹⁶ reported ΔH_mix of the different alloy compositions at 973, 1073, and 1173 K, which were also considered by Pal et al.¹⁹² In addition, a ternary line compound GaInTe₂ was also incorporated in the database after its observation in an alloy. They incorporated the (Ga,In)₂(Te)₃ and (In,Ga)(Te) solid solutions to explain the solubilities in Ga₂Te₃ and In₂Te₃, and in InTe and GaTe. Ternary liquid phase was extrapolated using ‘Toop-like’ asymmetric approximation¹¹⁵ considering Te as an asymmetric component. No ternary interaction parameter was required in the liquid phase to fit the available experimental data. As seen in Fig. 2(e), there is a miscibility gap in the liquidus projection extending from both In–Te and Ga–Te binaries. In addition, the symbols marked are the compositions investigated using the developed thermodynamic database, while the arrows show the solidification path.

4.6 Advancements in phase equilibria of Tin–gallium–tellurium (Sn–Ga–Te)

Another interesting ternary system, comprising SnTe, is a promising candidate for replacing PbTe in the thermoelectric field.¹⁹⁷ In addition to SnTe, the other phases, such as Ga₂Te₃ and GaTe, with narrow to wide bandgaps.^185,198 Kumar et al.¹⁹⁹ provided a complete thermodynamic database for the ternary system by considering almost all the available experimental and theoretical examinations. They also used the MQM to describe Sn–Te, Ga–Sn, and Ga–Te liquid solutions, while SnTe and BCT solid solutions were modelled using CEF. The description of Ga–Te was included from their previous work.¹⁶⁵ Moreover, a similar Toop-like interpolation method with Te as an asymmetric component was able to describe the ternary liquid phase. In addition, there are various vertical sections such as Ga₂Te₃–SnTe^200,201 and GaTe–SnTe,^200,202 which have been experimentally explored in the system. A ternary compound, Ga₆SnTe₁₀, which has been reported experimentally by different researchers,^201,203 has been incorporated by Kumar et al.¹⁹⁹ in the thermodynamically optimised database. There are ternary eutectics, one at the Te-rich side, between SnTe, Te, and Ga₂SnTe₁₀ phases, and another at the Sn-rich side (between SnTe, BCT, and GaTe phases), which were further observed experimentally by Pal et al.,²⁰⁴ as evident in Fig. 2(e). In addition, the arrows marked show the direction of the solidification pathways or the change in liquidus with decreasing temperature.

As observed in all the reviewed developed thermodynamic databases based on tellurides, a Toop-like symmetric approximation with Te as an asymmetric component can explain the behaviour of the ternary liquid. Thus, the ternary system Bi–In–Te can be optimised using a similar approach. At last, all these developed thermodynamic databases, along with other ternaries and the quaternary systems, can be combined and incorporated into the Ag–Bi–In–Ga–Sn–Te multicomponent system.

5 Microstructure-processing conditions and TE properties

Most of the above-mentioned phase diagrams are explored around a limited pseudo-binary phase diagram that comprises phases with already known TE properties. In the subsequent sections, we will review some of the interesting studies that investigate the electronic and thermal transport in these ternary systems over recent years.

5.1 Silver–bismuth–tellurium (Ag–Bi–Te) thermoelectrics

Ag–Bi–Te, as seen in the previous section, consists of some of the interesting phases such as AgBiTe₂, Ag₂Te, and Bi₂Te₃. Takigawa et al.¹¹² reported (AgBiTe₂)_1−x(Ag₂Te)_x based composites for TE applications. They developed these alloys using a vertical furnace wherein they held the liquid at a higher temperature to allow Ag₂Te crystallisation with the high-temperature AgBiTe₂ phase also. Fig. 3(a) shows the microstructure with Ag₂Te along with the AgBiTe₂ phase in the (AgBiTe₂)_0.25(Ag₂Te)_0.75 alloy. They observed a significant change in the grain size from 200 microns to 10 microns by changing the Ag₂Te proportion from 0.75 to 0.5. They observed the highest σ and S of single-phase Ag₂Te at room temperature compared to that of the composite, while the presence of 0.75 Ag₂Te along with Bi₂Te₃ leads to the lowest κ_l due to scattering of long-wavelength phonons at the grain boundaries. The highest zT of 8 × 10⁻⁴ was observed for the Ag₂Te alloy, but the alloy with 0.5 Ag₂Te was found to have about three times higher zT than that of AgBiTe₂. Later, they tried to understand µ_H in these composites, wherein they observed the highest µ_H at 243 K for the alloy with 0.875 Ag₂Te, which was higher compared to that of the constituent phases, suggesting electron transfer from AgBiTe₂ to the Ag₂Te phase in the alloy.²⁰⁵

Fig. 3 SEM-BSE micrograph of (a) (AgBiTe₂)_0.25(Ag₂Te)_0.75,¹¹² Copyright 2011, The Materials Research Society, (b) BSE micrograph of Ag-37.5Bi-55Te (at%), (c) TEM analysis of Bi₂Ag_0.03Te_2.97, (d) lattice thermal conductivity(κ_l) mapping at 300 K superimposed on the isothermal section,³⁸ Copyright 2021, Elsevier Ltd., SEM micrograph of (e) Ag_19.3Bi_28.6Te₅₂ and (f) Ag₃₉Bi_16.5Te_44.5,¹¹⁴ Copyright 2024, Elsevier Ltd., (g) differential charge density of the (GeSe)_0.09(AgBiTe₂)/SnSe composite,²⁰⁸ and (h) TE properties of different multiphase TEs as a function of temperature.^{36,38,114,207,211}

To understand the effect of Ag₂Te in the Bi₂Te₃ matrix, Drzewowska et al.¹⁶¹ prepared alloys with varying Ag₂Te compositions from 0 to 0.648 using a pseudo-binary section of Bi₂Te₃–Ag₂Te.²⁰⁶ They used spark plasma sintering (SPS) to develop these alloys from the powders of Bi₂Te₃ and Ag₂Te. They observed an interesting jump in the σ vs. T curve for the alloys with higher Ag₂Te content between 413 and 423 K due to the monoclinic to cubic phase transformation of Ag₂Te. Moreover, they observed an n-type conduction for all the alloys, with the highest zT of 0.43 at 373 K for Ag₅Bi₃₇Te₅₈ at room temperature in the presence of Bi₂Te₃ and Ag₂Te phases (shown in Fig. 3(h)).

Yen et al.³⁸ reported an extensive study of the Ag–Bi–Te ternary system using phase diagram engineering and carrier optimisation. They used 35 alloys, annealed at 523 K for six months, to construct an isothermal section. As shown in Fig. 3(b), the microstructure shows the formation of elongated Bi₂Te₃, Ag₂Te (dark), and Bi₄Te₅ phases in the Ag-37.5 at.%Bi-55 at.%Te alloy. Instead of the secondary phases, they reported that the presence of Ag in the Bi₂Te₃ can produce Ag_i defects in the van der Waals gaps, which generate an extra electron, while the Ag_Bi defects introduce 2 holes. To further understand the defect formations, higher resolution transmission electron microscopy (TEM) showed the formation of Ag₂Te nanoprecipitates for the Bi₂Ag_0.03Te_2.97 alloy (shown in Fig. 3(c)). Moreover, they observed the presence of misfit dislocations between Bi₂Te₃ and Ag₂Te in region I (shown in Fig. 3(c)1) and piled-up stacking faults at the Bi₂Te₃/Ag₂Te interfaces in region II (shown in Fig. 3(c)2). Thus, the presence of nanoprecipitates of Ag₂Te and defects between these phases resulted in a significant decrease in κ_l. Fig. 3(d) shows the change in κ_l with the change in the phase assemblage at 300 K; the lowest κ_l was obtained for alloys with Bi₂Te₃–Ag₂Te and Bi₂Te₃–Bi₄Te₃–Ag₂Te phase combinations. As evident, the regions at the centre of the phase diagram with varying phase fractions of Ag₂Te, Bi₂Te₃, and Bi₄Te₅ show a κ_l lower than 1. In addition, the variation in the TE properties with temperature is shown in Fig. 3(h), and Bi₂Ag_0.03Te_2.97 shows the highest zT of 1.4 at 363 K, with an average zT of 1.

In a recent study by Serbesa et al.,¹¹⁴ different alloy compositions were selected using the thermodynamically optimised phase diagram of Ag–Bi–Te. They were successful in attaining the AgBiTe₂ phase in the as-cast alloys with low Ag content. In addition, they attained the high-temperature cubic phase of Ag₂Te in these alloys. As evident in Fig. 4(e) and (f), the long needle-like morphology of Bi₂Te₃ along with AgBiTe₂ and AgBiTe₂/Ag₂Te phases, was obtained, respectively. These microstructural features resulted in a significant decrease in κ (<1 W m⁻¹ K⁻¹), resulting in a zT of about 0.42 at 550 K (shown in Fig. 3(h)). On the other hand, the alloy with Ag₂Te and AgBiTe₂ shows comparatively higher σ but very low S, resulting in the highest zT of 0.15 at 375 K, but still higher than reported by Takigawa et al.¹¹² for composites based on these phases. Other than the Ag–Bi–Te ternary, there are various reports with AgBiTe₂ being used along with other phases, such as SnTe, to enhance the TE performance. A similar attempt was made by Zhao et al.,²⁰⁷ wherein they successfully attained the high-temperature cubic AgBiTe₂ phase at room temperature by SnTe alloying. By adding SnTe, they observed an increased configurational entropy in the system, which increased the thermodynamic stability of the AgBiTe₂ phase. Moreover, they also added Cd and Br to further enhance the TE performance of the alloys. Variations in the TE properties with temperature for the same are shown in Fig. 3(h). The highest zT of 0.21 at 423 K and 0.33 at 381 K was achieved for (AgBiTe₂)_0.7(SnTe)_0.3-Br 6% and (AgBi_0.99Cd_0.01Te₂)_0.6(SnTe)_0.4 alloys.

Fig. 4 SEM-BSE micrograph of the as-grown (a) (Bi₂Te₃)_0.91(Cu₂Te)_0.09 and (b) EDS mapping at the grain boundary in the (Bi₂Te₃)_0.95(Cu₂Te)_0.05 alloy,²¹² Copyright 2018, Elsevier Ltd., and BSE micrograph of (c) Cu_10.7Bi_32.4Te_56.9 and (d) Cu₃₇Bi₁₅Te_48,³⁹ Copyright 2023, Springer Nature, BSE micrograph of Bi₂Te₃-5at%Cu-(e) brine quenched and (f) oil quenched, along with EDS mapping, (g) bright field image showing high defect density under the zone-axis conditions of brine-quenched Bi₂Te₃-5at%Cu,²¹⁵ Copyright 2024, Springer Nature, (h) SEM-BSE micrograph of Bi₂Te₃-5at%Cu directionally solidified at 20 µm s⁻¹ parallel to growth direction,²¹⁶ Copyright 2024, Elsevier Ltd., (i) SEM micrograph of Cu_0.06Bi₂Te_3.17 along with EDS mapping,²¹⁷ Copyright 2019, American Chemical Society, (j) SEM image of a Bi₂Te₃ nanowire with Cu_2−xTe cubic particles, (k) energy band diagram at a heterojunction between Cu_2−xTe and Bi₂Te₃, charge distribution, and energy band diagram of a Bi₂Te₃ nanowire in contact with a Cu_2−xTe nanoparticle,⁵⁹ Copyright 2021, Elsevier B.V., and (l) variation of TE properties with temperature.^{39,59,212,215–217}

Moreover, the addition of AgBiTe₂ in SnTe has been reported to be a significant improvement when compared to AgSbTe₂ due to relatively effective hole neutralisation by Bi.³⁶ Although there was no secondary phase observable using XRD and SEM techniques, TEM analysis revealed nano-precipitate in the AgSn₁₅BiTe₁₇ alloy. Thus, a solid solution and the secondary Ag-rich nanoprecipitate in the alloy lead to a significant reduction in κ_l. Moreover, AgBiTe₂ addition increases the number of charge carriers in the alloy with deteriorating mobility, leading to a significant decrease in the overall σ of these alloys. Thus, a potential lead-free AgSn₁₅BiTe₁₇ alloy with the highest zT of 1.1 at 775 K was developed. A recent study also shows the impact of interfaces between AgBiTe₂ and GeSe to achieve high TE performance.²⁰⁸Fig. 3(g) shows differential charge density analyses that show 2 different types of interfaces, one with covalent bonding (interface 1) and the other with ionic bonding (interface 2), with red regions showing charge accumulation, while blue regions show the depletion of charge.

Thus, thermoelectric performance in multiphase Ag–Bi–Te alloys arises from a combination of microstructural, defect, and interfacial mechanisms that collectively tune electrical and thermal transport. Takigawa et al.¹¹² demonstrated that varying Ag₂Te content in (AgBiTe₂)_1−x(Ag₂Te)_x composites controls grain size and interfacial density, where fine-grained microstructures (10 µm) enhance phonon scattering and reduce lattice thermal conductivity (κ_l), while charge transfer from AgBiTe₂ to Ag₂Te owing to the difference in the chemical potentials. Similarly, Drzewowska et al.¹⁶¹ showed that in the Bi₂Te₃–Ag₂Te pseudo-binary system, the monoclinic–cubic transition of Ag₂Te near 413–423 K modulates σ(T), and the two-phase nanostructure suppresses κ_l through boundary scattering. Yen et al.³⁸ further revealed that Ag-induced point defects (Ag_i, Ag_Bi) in Bi₂Te₃ tailor carrier polarity, while nanoscale Ag₂Te precipitates and misfit dislocations at Bi₂Te₃/Ag₂Te interfaces effectively scatter phonons, achieving κ_l < 1 W m⁻¹ K⁻¹ and zT = 1. Serbesa et al.¹¹⁴ obtained similar κ_l suppression (<1 W m⁻¹ K⁻¹) in AgBiTe₂–Ag₂Te multiphase alloys due to needle-like Bi₂Te₃ morphologies and coherent interfaces. Alloying-based phase stabilization and defect control were emphasized by Zhao et al.,²⁰⁷ where SnTe addition increased configurational entropy and stabilized cubic AgBiTe₂ at room temperature, while Cd and Br doping tuned carrier concentration and reduced κ_l, yielding zT ∼0.3–0.4. In the AgSn₁₅BiTe₁₇ alloy, Bi effectively neutralized holes and introduced Ag-rich nano-precipitates that reduced κ_l, enabling zT = 1.1 at 775 K.³⁶ Furthermore, first-principles analysis of AgBiTe₂/GeSe interfaces²⁰⁸ revealed that covalent versus ionic bonding at phase boundaries governs charge accumulation and phonon scattering. Thus, it can be inferred that the understanding of interfaces and phase engineering plays a vital role in understanding the transport behaviour in the multiphase TEs.

Similar to AgBiTe₂, AgSbTe₂ is also an interesting ternary compound that has been explored extensively. In a recent review by Li et al.,²⁰⁹ it has been summarised that pristine AgSbTe₂ can provide a zT of around 1.7 owing to the PGEC transport mechanism. Moreover, the work done by Wang et al.²¹⁰ explores the role of the presence of different phases along with AgSbTe₂, which results in a significant change in the TE performance of the multiphase alloys. They reported the presence of the secondary phases along with AgSbTe₂ such as Sb₇Te and Ag₅Te₃, which play a significant role in determining the TE properties. Further increments in the alloys can be made by understanding and optimising the secondary phases.

5.2 Bismuth–copper–tellurium (Bi–Cu–Te) thermoelectrics

In the Bi–Cu–Te ternary systems, there are various phases such as Bi₂Te₃, CuTe, Cu₇Te₅, Cu₃Te₂, Cu₂Te, etc., which are of interest for TE applications. Among the various sections, the pseudo-binary phase diagram of Bi₂Te₃–Cu₂Te has been extensively explored. Wu and Yen²¹² prepared two series of alloys (Bi₂Te₃)_1−x(Cu₂Te)_x (where x = 0.01 to 0.09) and Cu_yBi_2−yTe₃ using the Bridgman method. They maintained the growth rate of 4.2 K h⁻¹ from 903 K to a temperature of 803 K, where alloys were solidified. In addition, they also constructed an isothermal section at 523 K using various alloys annealed at 523 K for six months. Fig. 4(a) and (b) show the microstructure of the as-grown Cu_0.09Bi_1.91Te₃ alloy, which revealed a eutectic lamellar of Bi₂Te₃/Cu₇Te₅ along with primary Bi₂Te₃. They observed the maximum solubility of Cu in Bi₂Te₃ to be 3 at%, which can be accommodated in the van der Waals gaps of Bi₂Te₃. In the Bi₂Te₃ phases, it has been observed that Cu, when intercalated at the van der Waals gap between two Bi₂Te₃ layers, acts as a donor while its presence at the Bi site makes it an acceptor.²¹³ In addition, the presence of a small fraction of Cu_2−xTe phase (having a low work function) is found to inject charge carriers into the Bi₂Te₃ matrix.²¹⁴ Wu and Yen²¹² observed with increasing Cu₂Te content (from 0 to 0.09), there is precipitation of Cu₇Te₅ phase along with Bi₂Te₃, resulting into a complete p to n transition. Moreover, due to lamellar arrangements of the phases, the interface-enhanced phonon scattering leads to a significant reduction in κ, resulting in a high zT of 1.07 at 360 K. In addition, due to the increase in the fraction of Cu₇Te₅, the overall PF was observed to be constant with temperature, resulting in an interesting average zT of around 1 for the whole investigated temperature range (shown in TE properties in Fig. 4(l)).

Serbesa et al.³⁹ reported a complete thermodynamic description of the Bi–Cu–Te ternary system, which helped them to choose specific compositions. They prepared the as-cast alloys with varying Cu fractions from 0 to 37 at% in Bi₂Te₃. As seen in the microstructure in Fig. 4(c), there is a eutectic growth between the Cu₃Te₂ (dark phase) and Bi₂Te₃ (grey colored) phases along with primary Bi₂Te₃ grains in Cu_10.7Bi_32.4Te_56.9 and the formation of dendritic Cu₃Te₂, Bi₂Te₃, and the eutectic mixture of Bi₂Te₃ and Cu₃Te₂ in Cu₃₇Bi₁₅Te₄₈ (shown in Fig. 4(d)). The variation of the volume fraction of phases and the change in the interlamellar spacing resulted in decreased κ for the alloys. Recently, Legese et al.²¹⁵ further investigated the effect of the cooling rate on the microstructure and TE properties of Bi₂Te₃-5at%Cu, wherein they used brine-ice and food-grade oil as quenching media. They observed a change in stoichiometry from Cu_1.25Te (brine, shown in Fig. 4(e) and (e).1) to Cu_1.5Te in the oil-quenched alloy, as shown in Fig. 4(f). In addition, there was precipitation of Bi and BiTe phases along with Bi₂Te₃ and Cu_1.5Te in the oil-quenched sample due to a slower cooling rate (as shown in Fig. 4(f)1). Fig. 4(g) shows the presence of a large number of dislocations in the quenched alloy, which reduced κ_l for the alloy. In addition, they also observed a p–n transition for the alloy, while the oil-quenched alloy showed n-type conduction throughout the investigated temperature range (shown in Fig. 4(l)).

On the other hand, a study investigated the role of solidification velocity (V) in the Bi₂Te₃-5at%Cu alloy using the Bridgman technique.²¹⁶ They used three V: 50, 20, and 10 µm s⁻¹ to change the grain morphology and the elemental distribution. In all the alloys, they observed the formation of Bi₂Te₃ and Cu₃Te₂ phases as shown in Fig. 4(h). They reported that the increase in V led to finer grains, which led to reduced κ due to pronounced scattering of high-wavelength phonons. In addition, the highest zT of 0.93 at 442 K was observed for the alloy solidified at 20 µm s⁻¹; changes in the TE properties with temperature are provided in Fig. 4(l).

The work reported by Cha et al.²¹⁷ shows the influence of the excess Te condition in Bi₂Te₃ along with the presence of Cu. In addition, they observed the formation of elongated rod-shaped regions (Cu rich) in Cu_0.06Bi₂Te_3.17 alloys prepared using SPS, as shown in Fig. 4(i). The presence of the secondary phase led to an enhancement of S and σ due to the energy filtering effect. In addition, the κ at room temperature for this alloy was also low compared to other Te-excess alloys with low Cu concentration. The highest zT of 0.85 was observed for the alloy at RT. Other than multiphase approaches, an interesting study to create nanodomains of Cu₂Te in n-type Bi₂Te₃ was carried out by Zhang et al.,⁵⁹ wherein they used nanoparticles of Cu₂Te on Bi₂Te₃ as shown in Fig. 4(j). They observed very low κ < 0.9 W m⁻¹ K⁻¹ for all the investigated samples. The energy filtering effect due to the nanodomains of Cu_2−xTe resulted in selective charge carrier filtering, thus enhancing the overall zT. Moreover, the heterojunction between Cu_2−xTe and Bi₂Te₃ shows the formation of an accumulation layer, as shown in Fig. 4(k), due to band bending at the interface. The presence of Cu_2−xTe cubic particles over the Bi₂Te₃ nanowire shows electron movement from Cu_2−xTe to the Bi₂Te₃ nanowire, which can be viewed as the Fermi level (E_F) lying in the conduction band at the interface. In addition, using ultraviolet photoelectron spectroscopy (UPS), Cu_2−xTe is found to possess a lower work function compared to Bi₂Te₃, and thus the movement of electrons from Cu_2−xTe to Bi₂Te₃ will be favoured. Similar effects of the Cu_2−xTe phases in the above-discussed Bi–Cu–Te-based multiphase TE reports can be linked. Thus, from bulk to nano-dimensions, multiphase TEs in the Bi–Cu–Te ternary have shown significant improvement in terms of heat and charge carrier transport.

Wu and Yen²¹² observed that Cu incorporation in Bi₂Te₃ up to 3 at% occurs via intercalation in the van der Waals gaps, where Cu acts as a donor, whereas substitutional Cu on Bi sites behaves as an acceptor. The coexistence of Bi₂Te₃ and Cu₇Te₅ lamellae led to p–n conduction switching and strong interface-induced phonon scattering, yielding a high zT of 1.07 at 360 K. Serbesa et al.³⁹ showed eutectic coupling between Bi₂Te₃ and Cu₃Te₂ and variation in interlamellar spacing, effectively suppressed κ_l. Legese et al.²¹⁵ revealed that faster quenching (brine) produced Cu₁.₂₅Te with high dislocation density and lower κ_l, while slower cooling (oil) resulted in Cu₁.₅Te with additional Bi and Bi_Te precipitates, causing a conduction-type reversal and enhanced carrier scattering. In a related study, varying solidification velocity during Bridgman growth²¹⁶ refined grain size and improved phonon scattering, where an optimal velocity (20 µm s⁻¹) balanced electrical and thermal transport, achieving zT = 0.93 at 442 K. Cha et al.²¹⁷ demonstrated that Cu addition under Te-rich conditions produced Cu-rich rod-like secondary phases, which promoted energy filtering of charge carriers and lowered κ, enhancing zT to 0.85 at room temperature. At the nanoscale, Zhang et al.⁵⁹ embedded Cu_2−xTe nanodomains into Bi₂Te₃, forming heterojunctions with band bending and electron accumulation at the interface, which induced energy-dependent carrier filtering and drastically reduced κ (<0.9 W m⁻¹ K⁻¹). Overall, these studies show that in Bi–Cu–Te multiphase systems, thermoelectric enhancement comes from a combination of Cu-induced carrier modulation, interfacial phonon scattering, and energy-filtering mechanisms that work from bulk to nanoscale architectures.

5.3 Bismuth–indium–tellurium (Bi–In–Te) thermoelectrics

In this particular system, Bi₂Te₃, In₂Te₃, In₂Te₅, and other phases are available, which in combination, have shown significant enhancement in the TE properties. Among the various possible combinations, Bi₂Te₃–In₂Te₃ has been explored extensively due to its interesting crystal structure. Nagao and Ferhat²¹⁸ reported the TE properties of Bi_2(1−x)In_2xTe₃ alloys. They observed the highest zT of 0.6 at 300 K with x = 0.2, attributed to high S and reduced κ due to the presence of In₂Te₃ along with Bi₂Te₃.

Liu et al.¹⁸³ investigated the effect of growth and kinetics in Bi-7.5In-60Te (at%) of the Bi₂Te₃-In₂Te₃ pseudo-binary phase diagram (shown in Fig. 5(a)). Alloys were prepared using the seeding zone melting technique.¹⁸⁸ This alloy was annealed at different temperatures (773 K and 573 K) for 72 h, and step annealing was first performed at 573 K and then at 773 K. The step annealing was to modify the nucleus density of In₂Te₃ and the lamellar structure and chemical composition uniformity. The microstructure shows longer lamellar structures of In₂Te₃ and Bi₂Te₃ by step annealing, as shown in Fig. 5(b). Although there is not much change in κ_l for the two-step annealed alloy and the alloy with annealing at 573 K, as κ was limited by bipolar conduction, the two-step annealing shows a significant decrease in S (shown in Fig. 5(m)). They observed a zT of 0.045 at 773 K for the two-step annealed alloy. On the other hand, the alloy annealed at 573 K shows a zT of 0.065 at the same temperature. Later, Liu et al.²¹⁹ investigated the effect of varying annealing temperature (773 and 723 K for 3 days, while at 673 K for 7 days) in the Bi_1.625In_0.375Te₃ alloy to understand the change in the microstructure and corresponding change in TE properties. The initial alloy was prepared using seeding zone melting. The microstructures revealed the change in the phase fractions due to the change in In solubility from 6.2 to 3 at% as the temperature of annealing was lowered from 773 to 673 K (Fig. 5(c) to (e)). In addition, they observed a coherent interface between these phases with crystallographic orientation of <−211>In₂Te₃//<1–100>Bi₂Te₃, {111}In₂Te₃//{0001}Bi₂Te₃ as evident in Fig. 5(f) and (g). Thus, using proper heat treatment, a higher interface density was achieved, which increases zT from 0.065 to 0.26.¹⁸³

Fig. 5 (a) Pseudo-binary phase diagram of Bi₂Te₃–In₂Te₃, (b) BSE micrograph of Bi_1.625In_0.375Te₃ annealed at 673 K,¹⁸³ Copyright 2015, The Minerals, Metals and Materials Society, BSE micrograph of Bi_1.625In_0.375Te₃ annealed at (c) 773 K, (d) 723 K, and (e) 673 K, (f) TEM image and (g) selected area diffraction pattern (SADP) in Bi_1.625In_0.375Te₃ annealed at 673 K,²¹⁹ Copyright 2016, American Chemical Society, BSE micrographs of (Bi,In)₂Te₃(∼3 at% In); (h) 4 at% In, and (i) 6 at% In, (j) HRTEM image of the Bi₂Te₃/In₂Te₃ interface suggesting a sharp interface, and (k) schematic illustrating the band bending at the Bi₂Te₃/In₂Te₃ interface. E_g: band gap; E_C: conduction band; E_V: valence band; E_F: Fermi level; ΔE_C: offset of the conduction band; ΔE_V: offset of the valence band,²²⁰ (l) schematic illustrating the crystallographic orientation relationship (001)_Bi2Te3//(031)_In2Te3, [010]_Bi2Te3//[100]_In2Te3,²²¹ Copyright 2018, Elsevier B.V., and (m) variation of TE properties with temperature.^183,219,221

In all the studies reported by Liu and co-researchers (discussed above), the initial alloys do not exhibit grains oriented in a single preferred direction after solidification.^183,219 In a later report,²²⁰ they annealed (Bi_1−x,In_x)₂Te₃ solid solution (grown parallel to {001} plane) at 673 K for 6 days. To understand the effect of interface density, samples with 3, 4, 6, and 7.5 at% In were prepared using the seeding zone melting technique (as shown in Fig. 5(h) and (i)). They observed a slightly higher Te content (>60 at%) in Bi₂Te₃ and a lower Te content in In₂Te₃ (<60 at%). In addition, the TEM image shown in Fig. 5(j) shows a sharp and coherent interface between Bi₂Te₃ and In₂Te₃, which leads to an electron accumulation layer. On increasing the In fraction, the alloys show a significant decrease in S with increasing σ. The alloy with the highest interface density, i.e., eutectic composition, resulted in a significant decrease in κ_l, which leads to the maximum zT of 0.29 at 348 K along the growth direction. Compared to the single-phase solid solution, the Bi₂Te₃–In₂Te₃ multiphase shows enhanced zT. There was a drastic decrease in the zT along the direction perpendicular to growth due to a drastic decrease in σ. In addition, the variation of the TE properties with temperature for the same alloy is shown in Fig. 5(m). The heterojunction of Bi₂Te₃–In₂Te₃ was supplemented to understand the possible band bending (Fig. 5(k)). The flow of electrons from In₂Te₃ to (Bi,In)₂Te₃ leads to balancing E_F, which in turn yields band bending. In addition, the metal-like behaviour surface state of Bi₂Te₃ is one of the reasons for the increase in σ with increasing interface density.

In their later work,²²¹ they prepared Bi-13.5In-60Te and Bi-16.5In-60Te using induction melting followed by a Bridgman-type furnace. The growth was performed at 20 K min⁻¹ and 10 µm s⁻¹ velocities. To understand the interface, an alloy with maximum In solubility, i.e., 8.5 at%, was prepared using seeding zone melting. They observed a superstructure In₂Te₃ phase with an orthorhombic crystal structure, which was not observed in other solid-state transformations. The corresponding change in the crystallographic relationship between the two phases as observed by TEM analysis ((003)_Bi2Te3//(031)_In2Te3 and [010]_Bi2Te3//[100]_In2Te3) is illustrated using a schematic in Fig. 5(l). This change in the crystallographic orientation and the interface density observed a significant improvement in TE performance with the highest S, lowest σ and κ (<0.6 W m⁻¹ K⁻¹) for the eutectic composition, resulting in the highest zT of 0.12. The change in the TE properties of the various alloys investigated with temperature is shown in Fig. 5(m).

5.4 Bismuth–gallium–tellurium (Bi–Ga–Te) thermoelectrics

The Bi–Ga–Te ternary system, as discussed in the phase diagram section, comprises interesting semiconductor phases such as Bi₂Te₃, Ga₂Te₃, GaTe, Bi₂Te, etc. Hazama et al.²²² tried to understand the possible energy filtering/tunneling effect by introducing the nanosized Ga₂Te₃ phase in the Bi₂Te₃/Ga₂Te₃ composite. They prepared Bi_2−xGa_xTe₃ (0 ≤ x ≤ 0.25) alloys using SPS. Fig. 6(a) shows the Bi₂Te₃ grains along with the Ga₂Te₃ phase for the Bi_1.85Ga_0.15Te alloy prepared using SPS. On annealing this alloy at 773 K, the precipitation of Te was observed, as shown in Fig. 6(b). In addition, they performed the DFT simulation using PBE-GGA to determine the energy vs. momentum diagram for the Ga₂Te₃ and Bi₂Te₃ phases. Using the band structure information of both the phases, a possible Bi₂Te₃/Ga₂Te₃ heterojunction was designed (shown in Fig. 6(c)), wherein the (0001) direction of Bi₂Te₃ aligned with the (111) direction of the Ga₂Te₃. They considered the experimentally observed band gaps of 0.2 and 1.2 eV for the Bi₂Te₃ and Ga₂Te₃ phases, respectively. In addition, the Ga₂Te₃ phase produces halo rings indicating non-crystallinity; the effect of this non-crystallinity on electronic transport was not discussed by the authors (shown in Fig. 6(d)). As per Hazama et al.²²² taking into account the asymmetry of the density of states concerning E_F, a reduction in the Ga₂Te₃ content can result in a further increase in the PF because the number of electron carriers passing through the Ga₂Te₃ potential height increases when the potential width is decreased due to the tunnelling effect. Variation in TE properties for the alloy with the highest zT is shown in TE properties Fig. 6(k). The highest zT of 0.38 was obtained for the Bi_1.88Ga_0.12Te₃ alloy annealed at 673 K.

Fig. 6 BSE micrograph of Bi_1.85Ga_0.15Te₃ prepared using (a) SPS and (b) SPS and annealing at 773 K, (c) band offset between Ga₂Te₃ and Bi₂Te₃, (d) TEM image and the selected area electron diffraction (SAED) patterns of the Bi_1.9Ga_0.1Te₃ alloy,²²² Copyright 2017, Elsevier B.V., (e) BSE micrograph of the directionally grown Bi_1.91Ga_0.09Te₃ alloy, (f) BSE micrographs of thermally equilibrated Bi-20Ga-60Te, Bi-15Ga-57.5Te, Bi-10Ga-45Te, Bi-20Ga-45Te, Bi-30Ga-50Te, and Bi-20Ga-30Te (at%) alloys, (g) magnified isothermal section with Seebeck coefficient mapping at 300 K,¹⁷⁸ Copyright 2020, American Chemical Society, BSE micrograph of the Bi₂Te₃–25Ga₂Te₃ alloy directionally solidified at 10 µm s⁻¹ (h) parallel to the growth direction with EDS mapping, (i) perpendicular to the growth direction,²²³ Copyright 2024, IOP Publishing Ltd., and (j) SAED pattern of the ternary Bi₂Te₃–In₂Te₃–Ga₂Te₃ material,²²⁴ Copyright 2017, Elsevier Ltd., and (k) variation of TE properties with temperature.^178,223,224

In a recent study by Lin et al.,¹⁷⁸ 34 alloy compositions of Bi–Ga–Te were investigated to develop an isothermal section at 573 K. The microstructure of Bi_1.91Ga_0.09Te₃ reveals well-defined elongated Bi₂Te₃ grown parallel to the growth direction with the formation of Bi₂Te₃/Ga₂Te₃ eutectic at grain boundaries (shown in Fig. 6(e)). They performed TE measurements of these alloys and reported a significant change in TE transport with changing the phase assemblage. As reported, they observed σ and S decoupling mainly due to the excess Te and the additional impurity levels due to Ga in the Bi₂Te₃ phase. The alloys were prepared using the Bridgeman technique. In addition, the presence of secondary phase Ga₂Te₃, Te, and Bi₄Te₃, along with the Bi₂Te₃, affects the performance of the multiphase TE. They observed the highest S for the (Bi₂Te₃)_0.93(Ga₂Te₅)_0.07 alloy with Ga₂Te₃ and Te as the secondary phases along with Bi₂Te₃ (shown in Fig. 6(k)). In addition, it also shows high σ and low κ, leading to a zT of 1.5 at 300 K. The Ga⁺ or Ga³⁺ substitution at Bi³⁺ can lead to an expanded ab plane of Bi₂Te₃, which expands the basal plane. As the Ga addition increases up to 0.03 in Bi_2−bGa_bTe₃, a p–n transition was observed. Similar transitions were observed for the other three alloy series (Bi₂Te₃)_1−a(Ga₂Te₅)_a, (Bi₂Te₃)_1−c(GaTe)_c, and Bi₂Ga_dTe_3−d. They developed various alloys with three phases: Bi₂Te₃ + Bi₄Te₅ + Ga₂Te₃, Bi₂Te + Bi₄Te₃ + Ga₂Te₃, and GaTe + Ga₂Te₃ + Bi₄Te₃ and two phases: Bi₇Te₃ + Ga₂Te₃ and Bi + Bi₇Te₃ + GaTe (as shown in Fig. 6(f)). A color map in Fig. 6(g) shows the variation of S on the changing phase assemblage at 300 K. Recently, using the thermodynamically optimised Bi–Ga–Te thermodynamic database, the eutectic composition in the Bi₂Te₃–Ga₂Te₃ pseudo-binary phase diagram was obtained.²²³ To understand the effect of the change in the interface between Bi₂Te₃ and Ga₂Te₃, the alloy was directionally solidified using the Bridgeman technique to alter the microstructure. As shown in Fig. 6(h), the alloy solidified at 10 µm s⁻¹ revealed a Ga₂Te₃ needle-like structure aligned parallel to the growth direction. These changes in the lamellar spacing and crystal orientations affected the mobility of the charge carriers. In addition, there is a lateral growth of Ga₂Te₃ in the eutectic colonies observed along the transverse direction (shown in Fig. 6(i)). The TE properties measured along the growth direction show a significant change in σ and S, resulting in a maximum zT of 0.38 for the alloy solidified at 20 µm s⁻¹ solidification velocity. In addition, a significant drop in κ_l was observed for the alloy with finely distributed eutectic lamellae. Moreover, Kim et al.²²⁴ reported an interesting study on Bi₂Te₃–In₂Te₃–Ga₂Te₃ thin films on a PET substrate. As shown in Fig. 6(j), they observed (−11 0), (2 2 1), and (1 3 1) planes for the Bi₂Te₃ phase suggesting a rhombohedral crystal structure; (5 1 1), (8 2 2), and (9 3 3) planes for In₂Te₃; and the (2 0 0), (2 2 0), and (3 1 1) planes in Ga₂Te₃. The variation of the TE properties of some alloys is shown in Fig. 6(k). As evident, the highest zT for n-type conduction was observed for (Bi₂Te₃)_0.93(Ga₂Te₅)_0.07 at 300 K and for p-type Bi_1.99Ga_0.01Te₃.

5.5 Tin–gallium–tellurium (Sn–Ga–Te) and indium–gallium–tellurium (In–Ga–Te) thermoelectrics

In the Sn–Ga–Te ternary system, among all the phases, SnTe is one of the most explored phases as a potential replacement for PbTe.¹⁹⁷ However, SnTe exhibits challenges such as a high concentration of Sn vacancies, which reduces S and increases κ.²²⁵ Additionally, SnTe's narrow band gap (∼0.18 eV) and the substantial energy disparity (0.3–0.4 eV) between its light and heavy hole bands diminish the contribution of heavy holes to electronic transport, thereby compromising its TE performance relative to PbTe.²²⁶ Hong et al.¹⁷⁷ investigated Ga_1.8SnTe_5.2, Ga₂SnTe₅, and Ga_2.2SnTe_4.8 alloy compositions. The alloys were prepared by heating the elements in an ampoule at 1273 K, followed by SPS. They observed the formation of Ga₂Te₅, SnTe, and Te phases using XRD. SEM analysis shows the formation of Ga_0.74SnTe_2.01Ga and Ga_10.91SnTe_6.62 regions in the Ga₂SnTe₅ alloy, as shown in Fig. 7(a). In addition, Ga₂SnTe₅ showed the highest zT of 0.16 at 549 K, among the other investigated alloys. The other TE properties of the Ga₂SnTe₅ alloy are shown in Fig. 7(f). Orabi et al.²²⁷ investigated the Sn_1.03−xGa_xTe (x = 0, 0.02, 0.03, 0.05, 0.07, and 0.1) alloy series. They observed an interesting ultra-low κ_l below the amorphous limit at elevated temperature. They observed a complete solubility of Ga up to 0.05, beyond which GaTe crystallizes as nanoprecipitates (shown in Fig. 7(b)). To further understand the effect of Ga substitution at the Sn site, they performed a density functional theory analysis (DFT). As indicated by first-principles calculation, Ga substitution decreases the number of electrons, leading to a E_F close to the valence band, which leads to the activation of several hole pockets in the valence band. Remarkably, S improved significantly by Ga-doping and GaTe precipitation. The presence of GaTe can be seen as an energy filter, which leads to an increase in S at higher temperatures. Moreover, the variation of the TE properties of alloy with a Ga content of 0.07 in Sn_1.03−xGa_xTe is shown in Fig. 7(f). Other than the crystalline phases, various amorphous TE alloys based on Sn–Ga–Te systems are explored. Zhang et al.²²⁸ prepared bulk glass of [(Ga₂Te₃)₃₄(SnTe)₆₆]_100−x-Sn_x (x = 0 to 10 mol%) using melt spinning followed by SPS, wherein they reported an ultra-low κ of 0.16 W m⁻¹ K⁻¹. Moreover, they observed the formation of Ga₆SnTe₁₀ along with SnTe on annealing the as-spun [(Ga₂Te₃)₃₄(SnTe)₆₆]₉₂-Sn₈, as shown in Fig. 7(c). In another report,²²⁹ Ag_x[(Ga₂Te₃)₃₄(SnTe)₆₆]_100−x(x = 0 to 18) based amorphous alloys were fabricated using melt-spinning and the formation of Ga₆SnTe₁₀ was reported (shown in Fig. 7(d)). Moreover, the addition of Ag affected the SRO, which resulted in improved S and κ. The highest zT of 0.09 was observed for the addition of Ag at 465 K. In addition to these studies, there are various reports wherein multiphases and doping in SnTe have shown tremendous improvement in TE performance by bandgap and interface engineering, which has been reviewed extensively by Moshwa et al.¹⁹⁷

Fig. 7 (a) SEM-BSE micrograph of Ga₂SnTe₅,¹⁷⁷ Copyright 2012, Northwest Institute for Nonferrous Metal Research (China) Elsevier Ltd., (b) SEM micrograph of the Sn_0.93Ga_0.1Te alloy with EDS mapping,²²⁷ Copyright 2017, American Chemical Society, TEM image and corresponding SAED pattern of [(Ga₂Te₃)₃₄(SnTe)₆₆]₉₂-Sn₈ annealed at (c)513 K and (d) 533 K,²²⁸ (e) TEM images and electron diffraction patterns of Ga_1−xIn_x−2Te₃ (x = 0, 0.25, 0.5, 0.75, and 1),¹⁹⁵ Copyright 2009, TMS, and (f) TE properties of different alloys with temperature.^{177,195,227,229}

In the In–Ga–Te ternary system, In₂Te₃, Ga₂Te₃, InTe, etc., are some of the interesting phases explored for TE applications. Yamanaka et al.¹⁹⁵ investigated the Ga₂Te₃–In₂Te₃ binary system. The alloys were prepared by melting the elements in stoichiometry, followed by annealing at 973 K for 2 weeks, followed by hot pressing. They observed interesting two-dimensional vacancy planes with 3.5 nm intervals at the (111) plane in Ga₂Te₃. On the other hand, they observed the triple periodicity along the [111] direction of In₂Te₃ as shown in Fig. 7(e). The alloys in between the end members showed a random periodicity of these 2D vacancy planes. Moreover, they observed a complete solubility of phases in the investigated alloys due to the similar crystal structures of both phases. They found the minimum κ of 0.25 W m⁻¹ K⁻¹ to 0.5 W m⁻¹ K⁻¹ for x = 0.25 and 0.5 in (Ga_1−xIn_x)₂Te₃. Ga₂Te₃ showed the highest zT of 0.16 when compared to In₂Te₃ and other solid solutions. Variations of TE properties for solid solutions are shown in the TE Fig. 7(f). In addition, Gojayev et al.¹⁹¹ investigated the InGaTe₂ phase using experimentation and DFT calculations to understand the electronic band structure. They prepared the alloy by heating the elements in stoichiometry at 1273 K for 24 h, followed by homogenization at 973 K for 8 h in an ampoule. From the obtained XRD spectrum, there is a clear formation of secondary phases such as GaTe and Te, which can be explained by the developed thermodynamic database.¹⁹² In addition, the calculated band structure shows a direct bandgap of 0.56 eV for InGaTe₂. Jacobsen et al.¹⁸⁵ investigated the effect of high pressure on the transport and structural properties of InTe, GaTe, and GaInTe₂ phases. They observed that tetragonal GaInTe₂ undergoes a structural transformation at 9.25 GPa, which changes the conduction type. For some of these materials (InTe and InGaTe₂), this yields a lower efficiency than ambient conditions. However, for GaTe, an increase in zT of 14 times at 8 GPa was observed when compared to that without applied external pressure.

6 Theoretical models for multiphase TEs

The electronic and thermal transport behaviour of new phases is usually explored theoretically by first-principles calculations. These calculations are used to develop electronic band structure and phonon dispersion relationships to understand the new phases with/without defects. These calculations are usually computationally expensive, while performing these simulations for multiple phases is comparatively more challenging and expensive. Instead of using density functional theory, empirical relationships to understand the TE properties are usually applied. The simplest approach to understanding transport phenomena at high temperature is performed using Single Parabolic Band (SPB) approximation.⁶¹ In SPB, a single parabolic band with acoustic phonon scattering as the dominant scattering mechanism is assumed. The model uses experimentally observed temperature-dependent transport properties like S, σ, and R_H to determine the band structure information. In addition, for materials with roles of multiple bands in the electronic transport, a Multiband Parabolic Model (MBP) was proposed by Agrawal et al.,²³⁰ which provides valuable insights into the band structure. In the case of multiphase TEs, the effective transport property of the multiphase is usually averaged between the constituent phases using their volume fraction and transport property. This is one of the easiest approaches, but it always results in a lower value for the composite compared to individual phases.

For understanding the electronic properties in the multiphase TEs, effective medium theory (EMT) was proposed by Bruggeman, wherein the effective electrical and thermal conductivities of the composite comprising isotropic components are related to the phase fraction and the individual component property.²³¹Eqn (15) relating the volume fraction (v_i) and individual phase properties (ζ_i) to that of the effective multiphase TE property (ζ) is one of the most widely used relationships:

Landauer²³² used this equation by replacing ζ with σ to evaluate the effective σ for the composite with metallic phases. In a similar study by Odelevskii,²³³ the effective κ of the multiphase was evaluated by replacing ζ with κ. Moreover, it was observed that the effective S can be determined by replacing ζ by σ/S.²³⁴ Sonntag used the EMT formalism for the heat flow and chemical potential to obtain a relationship for metals or degenerate semiconductors.^235,236 EMT works well for phases with similar properties but fails when the fraction approaches the percolation threshold. Thus, McLachlan et al.²³⁷provided a generalized effective medium theory (GEMT), which incorporated percolation theory as shown in eqn (16):

where ‘t’ corresponds to the microstructure asymmetry, while A depends on the percolation threshold ‘p_c’ (A=(1 − p_c)/p_c), which is limited to the lattice type and the dimensions of the network. These parameters are determined by fitting the experimental data. In addition, to make the model applicable for non-degenerate multiphase materials, the heat flux used by Sonntag^235,236 was replaced by entropy flux.²³⁸ GEMT works well for the phases with significant differences in the transport properties. In addition to GEMT, the Webman–Jortner–Cohen (WJC) model used the EMT to Onsager equations to develop a relation for thermopower as provided by eqn (17):where Δ₀=(2σ + σ_i)(2κ + κ_i). Vaney et al.²³⁹ prepared multiphase materials based on glassy (Si₁₀As₁₅Te₇₅) and crystalline (Bi_0.4Sb_1.6Te₃) phases and observed that the EMT and GEMT are not able to explain the transport behaviour, while WJC provided a better fit to the experimental data. In addition to these models, a recent study by Rösch et al.²⁴⁰ proposed a more accurate way to relate the individual phase properties to those of multiphase TEs by using an expanded nodal analysis of random resistor networks. This model handles the effects of cluster formation, anisotropic phases, and the phases with significant changes in their transport properties. More details about the developed models are provided elsewhere.²⁴⁰

7 Current application scenario and challenges of multiphase TEs

In the current usage, multiphase TEs are used as TE devices for cooling and power generation applications. Moreover, these can be used in the field of human body energy harvesting using wearable and flexible electronics. The radioisotope TE generators (RTGs) in aerospace, high-temperature energy harvesting, power plants, turbines, and utilizing geothermal energy are some of the applications of multiphase TEs. Other than generators, TE coolers for thermal management in satellites, microelectronics, etc., are some of the other application fields. When compared to the well-known single phases like SnTe, In₂Te₃, and Bi₂Te₃, the multiphase materials provide a pathway to attain zT higher than 1.5 with engineered phase fraction, interfaces, and the involved individual phases. Achieving the high conversion efficiency in multiphase TEs comes with some challenges, such as phase segregation. Repetitive use of the devices based on multiphase TEs can result in phase separations and interfacial reactions, which can deteriorate the TE performance of the material. Moreover, the self-doping, impurity levels, and defects in these materials require precise control over the manufacturing process and the post-processing conditions, which is one of the key challenges in these materials. In addition, the two phases might consist of different mechanical strengths, which will challenge the mechanical stability of the material. At last, some of the elements mentioned in the current review are mostly costly due to their limited abundance. In addition, some of these materials are toxic, thus challenging environmental sustainability. Although the review discusses more about Te based TE materials, the research focus should also be centred on TE devices wherein efforts are being made to optimise the TE device for enhanced efficiency.^241–243 As mentioned, there are four layers starting with a metallization layer on the top of the TE leg, followed by a solder layer, a conducting strip and a ceramic plate at top.²⁴⁴ One of the challenges is to decide on the conductive strip material for a particular TE material, which should offer higher electrical and thermal conductivity, a minimum coefficient of thermal expansion difference (between the contact material and TE material), and retain higher mechanical stability at the operational temperature.²⁴⁴ Thus, the type of interface between the TE material and the contact also affects the electronic and thermal transport.²⁴⁵ In the case of multiphase TE materials, this can be more challenging owing to the presence of more than one phase. Thus, the selection of the contact material requires greater attention to overcome several engineering challenges including interfacial stability, diffusion control, and long-term reliability under thermal gradients. On the other hand, the phase transitions with temperature are also an important aspect of the TE behaviour as observed for single phase TEs.²⁴⁶ As there will be more than one phase in the multiphase TE material, the effect of phase transitions on each phase will contribute to the overall electronic and thermal transport. Such transitions can be leveraged to modulate carrier concentration and phonon scattering, providing new pathways for improving the efficiency and adaptability of TE materials.

8 Summary and future outlook

In the current paper, we have reviewed the phase diagram, microstructure evolution, and interface engineering in six interesting Te-based ternary systems. As observed, these engineering techniques play a crucial role in deciding the transport behaviour of multiphase TEs. Some of the key observations from the present review are summarised below:

• The developed thermodynamic databases of the ternary systems can be combined to build a complete thermodynamic database for the Te-based multicomponent system using appropriate extrapolation techniques. This can aid in designing new multicomponent TE materials with the desired phase assemblage.

• The CALPHAD-based model has shown its applicability from phase equilibria to the determination of the carrier concentration and the thermal conductivity. Thus, this method can be further used to design TE materials.

• A more generalised Gibbs Free energy expression with incorporated electrical resistivity and thermal conductivity can help in predicting the phases with desired properties at specific temperatures.

• The microstructure consisting of different phase assemblages, phase fractions, interface densities, defect concentrations, and types of defects is one of the key microstructural features that play an important role in deciding the transport properties.

• In the case of multiphase TEs, the interface between the two phases can be viewed as a heterojunction, which shows the possible alignment of E_F.

• The interface tuning thus helped in selectively filtering high-energy charge carriers with limited conduction of heat, leading to a simultaneous increase in S and decrease in κ.

• Better contact materials and robust interfaces are necessary for advancements in TE device engineering, particularly for multiphase Te-based systems where temperature-driven transitions along with various phases are present. There are encouraging opportunities to enhance thermoelectric efficiency, stability, and long-term device reliability by controlling the interfacial and phase-related effects.

• Multiphase TEs, comprising magnetic and semiconducting phases with engineered microstructures, can be used to exploit the electron–phonon–spin interactions to maximise the transverse zT.

Author contributions

Varinder Pal: writing – original draft, review and editing, and conceptualization. Duraisamy Sivaprahasam: writing – review and editing. Manas Paliwal and Chandra Sekhar Tiwary: supervision and writing – review and editing.

Conflicts of interest

There are no conflicts to declare.

Data availability

No primary research results, software or code have been included and no new data were generated or analysed as part of this review.

Acknowledgements

The authors are thankful to the Metallurgical and Materials Engineering Department of Indian Institute of Technology Kharagpur for the resources.

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