Hidden structures: a driving factor to achieve low thermal conductivity and high thermoelectric performance

Abstract

The long-range periodic atomic arrangement or the lack thereof in solids typically dictates the magnitude and temperature dependence of their lattice thermal conductivity (κ_lat). Compared to crystalline materials, glasses exhibit a much-suppressed κ_lat across all temperatures as the phonon mean free path reaches parity with the interatomic distances therein. While the occurrence of such glass-like thermal transport in crystalline solids captivates the scientific community with its fundamental inquiry, it also holds the potential for profoundly impacting the field of thermoelectric energy conversion. Therefore, efficient manipulation of thermal transport and comprehension of the microscopic mechanisms dictating phonon scattering in crystalline solids are paramount. As quantized lattice vibrations (i.e., phonons) drive κ_lat, atomistic insights into the chemical bonding characteristics are crucial to have informed knowledge about their origins. Recently, it has been observed that within the highly symmetric ‘averaged’ crystal structures, often there are hidden locally asymmetric atomic motifs (within a few Å), which exert far-reaching influence on phonon transport. Phenomena such as local atomic off-centering, atomic rattling or tunneling, liquid-like atomic motion, site splitting, local ordering, etc., which arise within a few Å scales, are generally found to drastically disrupt the passage of heat carrying phonons. Despite their profound implication(s) for phonon dynamics, they are often overlooked by traditional crystallographic techniques. In this review, we provide a brief overview of the fundamental aspects of heat transport and explore the status quo of innately low thermally conductive crystalline solids, wherein the phonon dynamics is majorly governed by local structural phenomena. We also discuss advanced techniques capable of characterizing the crystal structure at the sub-atomic level. Subsequently, we delve into the emergent new ideas with examples linked to local crystal structure and lattice dynamics. While discussing the implications of the local structure for thermal conductivity, we provide the state-of-the-art examples of high-performance thermoelectric materials. Finally, we offer our viewpoint on the experimental and theoretical challenges, potential new paths, and the integration of novel strategies with material synthesis to achieve low κ_lat and realize high thermoelectric performance in crystalline solids via local structure designing.

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Debattam Sarkar

Debattam Sarkar obtained his BSc degree in chemistry from Visva Bharati University, West Bengal, India, in 2016, and earned his MSc degree in chemistry from the University of Hyderabad, Telangana, India, in 2018. Later he obtained his PhD degree from Jawaharlal Nehru Centre for Advanced Scientific Research (JNCASR), Bangalore, India (2023) under the supervision of Prof. Kanishka Biswas. His research is centred around the modulation of crystal and electronic structures and the reduction of thermal conductivity of crystalline solids to enhance thermoelectric performance, and emerging quantum transport in topological materials.

Animesh Bhui

Animesh Bhui attained his BSc degree in chemistry from Vidyasagar University, West Bengal, India, in 2016, followed by MSc degree in chemistry from the Indian Institute of Technology Guwahati, Assam, India, in 2018. Currently, he is pursuing his PhD under the supervision of Prof. Kanishka Biswas at the New Chemistry Unit, Jawaharlal Nehru Centre for Advanced Scientific Research (JNCASR), Bangalore, India. His research focuses on thermoelectrics, low lattice thermal conductive materials and solid-state pigments.

Ivy Maria

Ivy Maria earned her BSc (Hons) in chemistry from St. Stephens College, University of Delhi, Delhi, India, in 2020, and later completed her MS (part of Int. PhD) in chemical science at Jawaharlal Nehru Centre for Advanced Scientific Research (JNCASR), Bangalore, India, in 2023. Currently, she is actively engaged in her PhD studies under the guidance of Prof. Kanishka Biswas at the New Chemistry Unit, Jawaharlal Nehru Centre for Advanced Scientific Research (JNCASR), Bangalore, India. Her research is centered around charge and phonon transport properties in correlated disorder systems and understanding of chemical bonding and local structure by synchrotron X-ray techniques.

Moinak Dutta

Moinak Dutta received his BSc degree from the University of Calcutta, Kolkata, in 2014, followed by an MSc degree in chemistry from the University of Hyderabad, Hyderabad, in 2016. He obtained his PhD from Jawaharlal Nehru Centre for Advanced Scientific Research (JNCASR), Bangalore, India, in 2021, under the supervision of Prof. Kanishka Biswas. His research primarily focused on unravelling the structure–property relationship of chalcogenides with low thermal conductivity, with potential applications in thermoelectrics and synchrotron X-ray techniques. Currently, he serves as a postdoctoral fellow at the University of Liverpool, United Kingdom.

Kanishka Biswas

Kanishka Biswas obtained his MS and PhD (Int. PhD) (2009) degree (advisor – Prof. C. N. R. Rao) from Indian Institute of Science (IISc), India, and did his postdoctoral research (advisor – Prof. Mercouri G. Kanatzidis, 2009–2012) in Northwestern University, USA. He is a full Professor in Jawaharlal Nehru Centre for Advanced Scientific Research (JNCASR), India. He is pursuing research on solid-state and inorganic chemistry, exploring areas such as thermoelectric energy conversion, 2D materials, topological quantum materials, and perovskite halides. Recognized for his contributions, he is a Fellow of the Indian Academy of Sciences (FASc) in Bangalore and an invited Fellow of the Royal Society of Chemistry (FRSC) in London, UK. He received the prestigious Shanti Swarup Bhatnagar Prize in Chemistry (2021) from the Government of India. He serves as an Executive Editor for ACS Applied Energy Materials, ACS.

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1. Introduction

The endless rise in fossil fuel usage, despite their limited abundance and adverse environmental effects, forces researchers to actively seek out alternative and sustainable energy sources. Most of the energy conversion processes generate heat as a byproduct and finding routes to harness and convert this wasted heat into useful forms of energy requires materials that can impede heat dissipation. Heat transmission occurs through the propagation of free charge carriers and quantized lattice vibrations (i.e., phonons) in crystalline solids, the control of which is extremely crucial for various applications, such as thermal barrier coatings,¹ photovoltaics² or in the field of thermoelectrics.^3–6 Converting waste heat into valuable electrical energy via thermoelectric materials stands out as a highly promising alternative to the global energy crisis. The efficiency of heat-to-electricity conversion in materials is primarily dictated by their thermoelectric figure of merit, zT = S²σT/κ, where S and σ are respectively the Seebeck coefficient and electrical conductivity at the specific operating temperature, T.^3,4 The total thermal conductivity (κ), which consists of electronic (κ_e) and lattice (κ_lat) thermal conductivity, hugely influences this heat to electricity conversion. While σ is controlled by the charge carriers, κ displays a direct dependence on both charge carriers and phonons.^3,7,8 Thus, it is imperative to selectively dampen the propagation of phonons while permitting the unhampered flow of charge carriers. This task has spurred the scientific community to deliberately explore strategies for modulating the lattice dynamics without disrupting the movements of charge carriers in solids. As S, σ and κ_e are complexly entangled and κ_lat remains the only standalone parameter that can be independently tuned, effective suppression of κ_lat to elevate zT has attracted remarkable attention.

The kinetic theory states that κ_lat = (1/3)C_vυ²τ, where C_v, υ and τ are the volumetric heat capacity, phonon group velocity and phonon relaxation time, respectively.⁹ As heavy elements exhibiting weak chemical bonding are ideal for designing low thermally conductive materials.¹⁰ Conversely, reducing τ by increasing phonon scattering has also been recognized as an efficient strategy to suppress κ_lat. These approaches intended at inhibiting phonon propagation primarily fall into two distinct categories: extrinsic and intrinsic pathways. Extrinsic approaches like the introduction of point defects⁴ or all-scale hierarchical nanostructures,^11,12etc., have the unintended impeding effect on electrical mobility. Therefore, it is advantageous to look for solids possessing innately low κ_lat as they simplify the counteracting correlations between the materials’ thermoelectric parameters and permit us to mainly focus on improving their electronic properties. Lattice anharmonicity diminishes the phonon propagation and reduces the phonon mean free path (l_ph). Lattice anharmonicity is quantified by the Grüneisen parameter (γ), which is defined as the variation in phonon frequencies with respect to the changes in the lattice volume γ is strongly correlated with κ_lat as where M_a, a and θ_D are the mean atomic weight, ratio of the volume to number of atoms within a unit cell and Debye temperature, respectively.¹³ The presence of lattice anharmonicity due to chemical bonding hierarchy,^14–19 rattler atoms,^17,20–26 part crystalline-part liquid states,^27–30 ferroelectric lattice instability,^31–34etc., are therefore highly beneficial in disrupting the smooth conduction of phonons and help to achieve low κ. Interestingly, compounds exhibiting such phenomena often also show glass-like temperature dependence of thermal transport (reaching glass limit κ_min³⁵ when l_ph becomes comparable to that of the interatomic distance³⁶), despite possessing high crystallinity or long range periodicity. This observation prompted a rigorous investigation of the crystal structure down to the very atomic level motifs. A host of factors including the presence of stereochemically active lone pair of electrons, the presence of discordant atoms,^{14,18,31,37–65} local disorder in high entropy oxides,^66–68 atomic tunnelling,⁶⁹ atomic rattling,^17,20–26 local atomic ordering,^70–76 creation of local van der Waals gap,^77–82etc., were found to underpin the notion of a locally distorted crystal symmetry whilst retaining the global (average) structural symmetry. These findings subsequently led to the discovery of more complex materials, characterized by structural motifs that manifest partial similarities to both ordered crystals and disordered glasses. Although these phenomena are locally confined (within a few Å), their effects can impact the structure–property relationships to varying degrees. For instance, the local symmetry lowering often introduces high lattice anharmonicity and inhibits the fluent conduit of heat carrying phonons via strong phonon scattering, resulting in glassy thermal transport and high thermoelectric performance (Fig. 1).^{31,47,50,55,70,77,83} Thus, employing hidden structures via local structure engineering can assist in developing an innovative tool for improving thermoelectric performance in crystalline solids. Not only in the field of thermoelectrics, even amongst other functional materials groups, the comprehension and tailoring of materials’ local crystal structure are emerging as new paradigm for illuminating their structure–property relationships.

Fig. 1 Schematic demonstrating several local structural phenomena like local structural distortion (upper left and right), rattling of atoms (lower left), atomic tunnelling (lower right), etc. leading to high thermoelectric performance (centre).^{31,47,50,55,70,77,83} Reproduced with permission from ref. 31, 2019,©, Royal Society of Chemistry, ref. 39 © 2023, ref. 47 © 2018, ref. 50 © 2021, ref. 55© 2021, ref. 83 © 2023, American Chemical Society, ref. 70 © 2021, AAAS, ref. 69 © 2020, ref. 77 © 2022, Nature Publishing Group, and from ref. 20 © 2021, ref. 54 © 2022, Wiley-VCH.

This review seeks to explore the underlying reasons for the low κ_lat and high thermoelectric performance in crystalline solids, primarily from a local structural perspective. Initially, we will delve into the existing knowledge and fundamental principles that dictate how we understand the hidden local structure of a compound. Subsequently, we will examine the commonly utilized techniques for determining these structures, highlighting their advantages and limitations. Our discussion will then shift to the various factors influencing a material's local structure and their consequent effects on thermal transport properties. By showcasing diverse examples, we aim to present an overarching view of the intricate interplay between the local crystal structure, chemical bonding, and thermal transport. Additionally, we will outline forthcoming insights and focus on the obstacles hindering the advancement of high-performance thermoelectrics. These hurdles are tied to our evolving understanding of the intricate relationship between thermal transport and lattice dynamics, as governed by the local crystal structure of solids.

2. Factors governing the local crystal structure

Since the discovery of valence shell electron pair repulsion (VSEPR) theory,⁸⁴ the lone pair of atoms in a molecule has become a well-known feature that affects the electronic structure of molecules. However, the role of lone pairs of electrons is less explored in extended solids, where they have a substantial influence on a variety of structural attributes and functionalities. Lone pairs have arguably played a less evident but no less important role in dictating the physical properties of many crystalline compounds having group IIIA, IVA, and VA elements showing valence state(s) lower than the group valency – termed the ‘inert pair effect’. Today, it is accepted that the off-centering structural distortions in solids can often be traced back to the behaviour and function of their lone pair(s). Orgel explained these structural distortions as originating from the polarizability and mixing of the s and p orbitals of the cations.⁸⁵ He highlighted the stereochemical effects of lone pair of electrons and even connected them to significant dielectric response in thallium chloride and bromide. However, not every relevant compound, say, those having Tl⁺ or Pb²⁺, display off-centric coordination, which indicates that the stereochemical activity of lone pair of electrons also depends on the anion and possibly the coordination geometry.⁸⁶ The stereochemical activity of lone pairs of electrons can indeed be explained by the strength of the interaction and the energy separation between the unoccupied orbitals of the metal-cation (np) and the occupied orbitals of the anion (np) and metal-cation (ns). The p-states of the cation stabilize the anti-bonding state made by the cation s- and the anion p-state.⁸⁷ This leads to lone pair activity (stereochemical expression) in oxides like PbO, SnO, and Sb₂O₃ because of the strong s–p interaction where the anion p-states have a higher energy than the cation s-states. Heavier anions (S, Se, Te) exhibit significantly weaker interaction between the cation s-states and anion p-states, leading to less pronounced hybridization and weaker expression of the underlying lone pair.^87,88 In centrosymmetric systems, for the mixing of s and p orbitals, the inversion centre needs to be lifted. Therefore, it is crucial to have local lattice distortions for breaking the inversion symmetry in centrosymmetric systems to enable the stabilization of the antibonding state (from cation s-anion p) by coupling with the unoccupied cation-np states in the conduction band.

The stereochemical expression of ns² lone pair of electrons gives rise to several fascinating properties, including second harmonic generation, induction of multiferroic properties and even intrinsically low lattice thermal conductivity in crystalline solids beneficial for thermoelectricity.^88–90 Although historically crystallography has been used to characterize global structural distortions caused by lone pairs, recent studies have found the unusual existence of lone pair-induced distortions even in highly symmetrical environments that are not correlated over long length scales. In the crystallographic structure, the average position resulting from the uncorrelated and incoherent distortion of the local environment of cations appears to be at the symmetric position in the coordination polyhedra. Such incoherent distortion in the coordination environment of Pb or Sn in PbS, PbTe,⁴² and SnTe⁴⁴ upon heating has recently been qualified as “emphanisis”. The dynamic switching by the lone pair of Pb or Sn between an inactive and active state with a concomitant change in lattice volume upon warming drives the local distortion of their coordination environments in the above systems. Exactly similar behaviour has also been observed in CsSnBr₃,⁹¹ wherein a stereochemically active 5s² lone pair of Sn²⁺ upon warming produces a locally distorted structure that was crystallographically hidden. Even though there is no indication that the typical (average) cubic structure is altered in any way, the lone-pair induces an asymmetry in the adjacent Sn–Br correlations, which corroborates with the presence of a dynamically off-centered Sn²⁺ in CsSnBr₃. The emergent anharmonicity from the lone pair driven structural distortion reflects in the phonon spectrum and understandably affects phonon lifetimes and related properties. Strong acoustic phonon scattering and subsequent low κ_lat have also been linked to the large fluctuating atomic displacements in such systems at elevated temperatures. Note that this is an aspect of particular importance for developing excellent thermoelectric materials.

Another electronic origin of similar hidden local structural distortion is the variation in the strength of orbital interactions. For example, in AMX₂ (A = Cu and Ag; M = Al, Ga, In, and Tl; X = S, Se, and Te) type diamondoid materials,⁵⁴ to create an ideal tetrahedral coordination with X, Cu and Ag must generate sd³ hybridization. Due to the relativistic effect, Ag has a considerable energy difference between its 4d and 5s electronic orbitals, leading to weaker sd³ hybridization and thus, is prone to distortion. The local distortion and off-centering of Ag from the perfect tetrahedral coordination originate from the above reason, in contrast to Cu where such traits are far less noticed.⁵⁴ Therefore, the chemical nature of orbitals can also induce local structural distortion and emphanisis for strongly influencing the thermal conductivity of chalcopyrites, an aspect that has remained undiscovered over their lengthy history of exploration.

Finally, we mention that changes in the local structure do not always have an electronic origin. The formation of 2D sheets of vacancies at the local scale in oxides and chalcogenides is receiving a growing amount of interest because this type of defect modifies the local atomic arrangement and local chemical bonding simultaneously.^77,78,81,82 Locally ordered structures can be created by cation or anion vacancies – the type of defect commonly described as the van der Waals gaps. The normal strain causes discrete vacancies to coalesce into a van der Waals gap, by means of which the van der Waals gaps further grow, similar to the scrambling of an edge dislocation. For example, in GeTe, the strain developed from the martensitic transformation of cubic-to-rhombohedral structure is the primary cause of the ordering and slip process which results in van der Waals gaps.⁷⁸ While the van der Waals gaps in layered 2D materials are periodic in nature, the vacancy-ordering driven van der Waals gaps are non-periodical local defects. The formation of such local van der Waals gaps can be controlled by tuning the vacancy concentrations and synthesis temperatures. Furthermore, such local planar defects are similar to the quantum gaps characterized by their narrow potential well.⁷⁷ These quantum gaps permit near-perfect transmission of carriers but strongly scatter the phonons, another degree of freedom to optimize the thermoelectric performance. In all, it is crucial to comprehend the impact of both the global average structure and the local structure on phonon dynamics in crystals, especially in situations where disorder or aperiodicity may be present at a local length scale. This necessitates a comprehensive understanding of the thermal transport mechanisms in both crystalline and amorphous solids at different length scales.

3. Thermal transport in crystalline vs. amorphous solids

Crystalline and amorphous solids are generally distinguished by the presence or absence of long-range atomic ordering. A crystal is defined as a periodic arrangement of atoms in a lattice with definitive symmetry. If the position of an atom and its surrounding neighbors are known at a certain lattice point (known as the Wyckoff site), its location can be accurately determined throughout the crystal. A crystal can be single crystalline, indicating that the atoms are arranged periodically across the whole volume or polycrystalline where the atoms are periodic across individual grains. Glasses or amorphous solids on the other hand are devoid of such long-range ordering and exhibit only short to medium range ordering (Fig. 2). The atoms are often connected to their immediate neighbors in the same way as in crystals; however their positions become uncorrelated as the distance increases. A case in point, crystalline silicon exhibits a long-range ordering, wherein a network of tetrahedrally bonded Si atoms exists throughout the crystal. In its amorphous counterpart, however, the same short range order exists, with each Si atom tetrahedrally bonded to four neighboring Si atoms, but their bonding direction changes as we move away from any one central Si atom.⁹² The presence or absence of long-range ordering in solids has been shown to exercise far reaching implications for their physical properties ranging from electrical conductivity, magnetism, thermal conductivity, catalysis, photoelectricity, etc. where the properties are found to be linked to the arrangement and ordering of atoms, thereby illustrating the importance of both synthesis and structural determination of these materials to understand their structure–property correlation.

Fig. 2 Schematic of a single crystal (left), polycrystal (middle) and amorphous (right) solid. A single crystal is characterized by ordered atoms throughout its volume while a polycrystal is ordered in its domains. The amorphous solid has no such ordering.

Determining the structure of a crystal is relatively more straightforward as compared to that of an amorphous solid. Given the periodicity in its structure, constituent atoms occupy specific sites with symmetry-guided multiplicity. Atoms of a crystal being in such an ordered arrangement, their positions can be determined with precision using routine X-ray crystallography. However, the same cannot be said for amorphous solids or glasses. Being devoid of any long-range ordering, X-ray diffraction-based techniques encounter diffuse scattering, rendering their structure determination more complicated. Given that glasses have only short to medium range ordering, they require probes which can accurately investigate their local structure. In this regard, techniques like nuclear magnetic resonance (NMR) spectroscopy, extended X-ray absorption fine structure (EXAFS) spectroscopy, pair distribution function (PDF) analysis, electron microscopy, etc., are being used to characterize the structure of such structurally complicated materials.

However, in recent years it has been found that just developing an ordered crystal or a disordered glass is not enough for technological applications. Solid state energy storage systems require crystalline compounds containing disordered atoms in them (for example Argyrodites).⁹³ Herein, a crystalline motif provides an ordered channel through which the disordered ions flow on application of voltage, thus combining the structural integrity of both a crystal and a glass into one.⁹³ Similarly, thermoelectric materials require high electrical transport like in crystals but low thermal conductivity as that of glasses – an idea known in the community as phonon-glass electron-crystal (PGEC). The notion led to the discovery of more complex materials which contain structural motifs partially resembling those in ordered crystals and disordered glasses. The ordered framework is required for the smooth flow of electrons or holes (to maintain sufficient electrical conductivity) while the disordered part helps to minimize l_ph to achieve minimum possible κ_lat.^35,36 Thus, in order to understand their structure–property correlations, a combination of techniques which probe both their long range and short range structural ordering is necessary. In this review, we focus on crystalline materials which exhibit unusually low thermal conductivity, with potential technological applications mainly in thermoelectric generators. In the subsequent sections, we will discuss how combining structure determining techniques at various length scales can accurately determine the cause of low thermal conductivity and why understanding the structure and the correlated thermal transport properties using only long-range order probes is insufficient for these complex materials.

4. Structure determining techniques

4.1 Limitations of average structure determination

A comprehensive understanding of the structure of compounds is acknowledged as a powerful tool for engineering their properties. Yet today the bigger question is, what does it truly entail to comprehend the structure of a compound? One of the earliest indications that we might be missing out on the bigger picture came while attempting to explain the properties of ferroelectric materials. After decades of research, it had come to light that the intriguing properties in ferroelectrics were born out of atomic ordering that persisted only at the local scale, i.e., of the order of a few unit cells, while it averaged out on the global structural scale.⁹⁴ What followed was the revelation of the implications of the local structure via disorder, randomness, short range order, etc., for the various material properties like ferroelectricity,^94–97 dielectric behavior,⁹⁸ catalytic activity,^99–102 upconversion emission,^103–105 electrical and thermal transport,^106,107 superionic conduction,^108–110 superconductivity,^111–114 optoelectronics,^115–117etc.

Why is it imperative to study the local structure of materials? To answer this, it is worthwhile to remember that the atomic level crystal chemistry is closely related to the local order, playing a pivotal role in rationalizing the origin of macroscopic properties at the most elementary level. As alluded to before, ferroelectrics provide an exemplary system to highlight the importance of investigating local structural aspects separate from the average structure to comprehend the origin of intriguing properties in materials. Thus, we take them up as an examplar to illustrate the idea. For applicability as a ferroelectric material, we gauge on a material's ability to generate large polarization and piezoelectric strain but suffer low dielectric loss. In this regard, a central idea behind the origin of ferroelectricity is that there is some form of an ion displacement which acts as the inception of the local separation of opposite charges to generate polarization. In the related class of relaxor ferroelectrics, the reason for the diffuse and frequency dependent ferroelectric transition is often attributed to the disorder persisting in the system.⁹⁶ What is fascinating here is the fact that the very presence of ferroelectricity means that the displacing ions are interacting, but the observation of disorder highlights the lack of any long-range order induced by these displacive movements. Note that the above observation still does not rule out the existence of any local ordering that may be present in these systems. The aforementioned puzzle can be addressed if we acknowledge the knowledge gap in our structure-determination techniques.

The visionary insights and the elegant interpretation of the X-ray diffraction patterns from crystalline solids by Laue and Bragg are the cornerstone of every solid-state diffraction-based structure determination technique. However, despite all their merits, in cases where the assumption of a perfect periodic lattice is compromised, these techniques are unfortunately less than stellar. To elaborate, conventional crystallographic structure solving paradigms are founded on the assumption that a crystal can be considered as a three-dimensional (3D) array of identical units (unit cells). Hence, we consider an average over a multitude of local (ordered) domains to construct what is called the average crystal structure. In this exercise, we lose the intricacies of local structure and the associated structural chemistry because the average may be over many reasonable local configurations. For instance, consider a disordered cubic system where a constituent atom displaces along the 8 possible 〈111〉 directions. If the proportion of atoms displacing along each of these directions is nearly equal, the average structure would converge at an ‘ordered’ structure but with a higher atomic displacement parameter (ADP) for that particular atom type. Thus, information about the very existence of disorder is lost due to the configurational averaging. Note that this is not a limitation of the experiment but of the mathematical modelling. In fact, deviations from such an ideal arrangement of atoms (as encompassed by the average structure) appear as weak diffused scattering intensities along with the sharp Bragg peaks in diffraction patterns – making them the go-to tool for studying structural disorder (discussed later).¹¹⁸

Having emphasized the importance of studying structures at the local length scale, we also highlight the difficulty in probing the same. In general, local structural/diffused scattering analysis demands advanced characterization techniques and notoriously rigorous mathematical modelling for gathering useful insights.¹¹⁹ Further, mind that pursuing local structure characterization is about entering a realm where the very crux of solid-state chemistry, i.e., the idea of the existence of a unit cell that can generate the overall crystal structure upon 3D repetition, is only partly valid.

4.2 Advanced local structure determination techniques

In the current era of high-performance functional material synthesis, understanding the structure–property relationships in materials is a prime research goal. Continuing from the discussions in previous sections, accomplishing this task demands studying the crystal structure at different length scales. Moreover, the complexity from intricate crystal structures, inherent atomic disorder, or novel bonding preferences has rendered the applicability of conventional structure determination techniques like single crystal X-ray diffraction (SCXRD) largely limited. Due to the incompetence of classical crystallographic techniques to resolve the local structure and its deviations from the average structure (long-range atomic arrangement), researchers today are employing advanced structure-characterization techniques,^120–124 which form the focus of our discussion in this section.

4.2.1 X-ray absorption spectroscopy. X-ray absorption spectroscopy (XAS) (Fig. 3a)¹²⁰ covers two very powerful techniques to study the atom-level structure of compounds, viz., X-ray absorption near edge spectroscopy (XANES) and EXAFS. While XANES specializes at the investigation of the valence state of ions making use of the sharp absorption of characteristic X-rays (close to the ionization energy of the probed ion), EXAFS is sensitive to and focusses on the local structural aspects like coordination number, bond length, degree of disorder, etc., in materials by deciphering X-rays from the scattering centers after they interact with their neighbors. Since local structure determination is our goal, we pursue only EXAFS further from here. Typically, the EXAFS spectra reflect the scattering-induced oscillations of the generated photoelectron.¹²⁵ These oscillations encompass information about the scatterers residing in the immediate neighborhood of the absorber atom (<5 Å).

Fig. 3 Local structure determination techniques: (a) X-ray absorption spectroscopy,¹²⁰ (b) pair distribution function (PDF) analysis using X-ray/neutron/electron total scattering (the inset shows a typical X-ray total scattering set-up (left) and the 3D-ΔPDF pattern (right)), (c) solid state nuclear magnetic resonance (ssNMR) spectroscopy,¹²¹ (d) Raman spectroscopy,¹²² (e) angstrom beam electron diffraction (ABED),¹²³ and (f) electron ptychography.¹²⁴ Fig. (a), (c) and (d) are reproduced with permission from ref. 120 © 2011, ref. 121 © 2015 and ref. 22 © 2022 respectively, American Chemical Society, Fig. (e) from ref. 123 © 2011, Nature Publishing Group, and Fig. (f) from ref. 124 © 2021, AAAS.

EXAFS is a very powerful probe for distinguishing anharmonic effects that manifest vibrational disorders in particular atomic correlations. Understandably, a tremendous amount of literature exists upon the application of EXAFS to determine the local structure of functional materials, some of which we will glance upon in this section. To begin with, by employing XAS and PDF analysis, Rodrigues et al.¹²⁶ showed that the local structure involving the shape of [Sb₄] rings plays a crucial role in determining the thermoelectric properties of CoSb₃-based Skutterudite materials. In another work, the presence of rhombohedral distortions (Sn off-centering) in a typical thermoelectric material SnTe even above the ferroelectric transition temperature was revealed by Fornasini et al.,¹²⁷ supporting the partial order–disorder contribution to the transition. Another exemplary case study of structural modifications imparting relaxor ferroelectric properties to 30 mol% Sn doped BaTiO₃ was undertaken by Surampalli et al.¹²⁸ Through complementary local structure probes, the authors unraveled the dipole dynamics underlying the dielectric behavior and evolution of the polar nanoregions: the combination of off-centering displacement of Ti and local structure modifications around Sn that is indicative of the influential role of phonon dynamics of the Sn sublattice, herein.

Although EXAFS is an impressive technique for local structure determination, it is incomplete and restrained in being capable of providing only short-range information, i.e., only about the immediate neighbors around an atom and not about the correlations that spread over a few angstroms or nanometers beyond. It is in this regard that the next technique based on analyzing diffused scattering of X-rays/electrons/neutrons from solids comes handy.

4.2.2 Pair distribution function (PDF) analysis. In total scattering-based techniques, essentially, we look not only at the intensities of Bragg reflections (following the Q⃑ = G⃑ condition, where G⃑ is the reciprocal lattice vector) as in conventional diffraction-based techniques, but also at the intensities at other scattering vectors (diffused scattering). Hence, analysis of X-ray/neutron/electron total scattering intensities can avail information from both shorter (∼Å) and longer length scales (∼nm) unlike that achieved from X-ray absorption studies. Presently, analysis of single crystal diffused scattering is the gold-standard amongst the available techniques for studying structural disorders.¹¹⁹ So how exactly does diffused scattering become a diagnostic tool for probing disorder and provide information about the local structure? To answer that, let us first understand how general crystallographic analysis based on diffraction works.

In the classical approach, the crystal structure is reconstructed from a small subset of possible Fourier components that lie on the reciprocal lattice by experimentally measuring the scattering intensity at Bragg points. Here, the reciprocal lattice in the first place is calculated by considering a long-range ordered structure. Thus, any information that we gain is about the average structure which is a series of single-body averages. This is a phenomenological technique which can be used to accurately determine the position of a particular site, its average occupancy, mean deviation from the Wyckoff position in terms of the atomic displacement parameter (ADP), etc. Along with phase identification and quantification, information on defects and dislocations, crystallite sizes, texturation and micro-/macro strains can be unsheathed by modelling the peaks observed in the experimental data.^129,130 However, the model unfortunately cannot comment about conditional variables, for example, how the occupancy of a site is affected/influenced by the neighboring sites. On the other hand, while analyzing diffused scattering intensities, since we are not sampling only at the Bragg points, insights into two-body correlations can also be extracted.⁹⁴ However, although diffused scattering is aware of average two-body correlations, it makes no attempt at describing many-body correlations unless in special cases.¹³¹

One of the most widely used structure characterization techniques exploiting diffused scattering is the total scattering experiments through PDF analysis (Fig. 3b). PDF analysis as the name suggests is a statistical approach that inspects the weighted distribution of various inter-atomic (particle) correlations present in a system to attain information about both the local and global/average structure.¹³² It is a very powerful technique since the analysis methodology presumes no long-range periodicity to begin with and therefore, there is no information loss due to configurational averaging as in regular crystallography. Thus, systems with varying degree of periodicity, like amorphous solids, nanocrystals, quasi crystals, and crystalline solids, can be analyzed. Conceptually, PDF uses the Fourier transform of the total scattering intensity from a solid to analyze its structure. Since the total scattering intensity has contributions from both the Bragg scattering, which holds information about the average structure, and the diffused scattering, which holds information about the deviations at local length scales, PDF analysis is a one-pot recipe to a complete and comprehensive structural analysis.^132–134 Further, the lack of correlation between particle size and peak width makes it extremely useful for comparative studies between nano and bulk samples also.

Amongst the multitude of structural information that PDF analysis gives, perhaps the one most important for interpreting the local structure is the information on bond lengths, bond angles, lattice parameters, correlation lengths, localization of constituent atoms, correlated motion of atomic pairs, etc.¹³⁵ Since powder PDF analysis uses rotational projections of 3D interatomic vectors onto a one-dimensional Patterson space, the technique is limited and offers no spatial information.¹³⁶ Thus, interatomic vectors of similar magnitude but oriented in different directions can superimpose and get masked under a single unresolved peak.¹³⁷ To overcome this, 3D-PDF and its variant 3D-ΔPDF were developed and are increasingly attracting attention.^136–143 Here, since the primary data collected are inherently 3D, PDF analysis yields both angular information and the magnitude of interatomic vectors. It is a relatively new technique, a brief overview of which is provided in the following section.

Disorder may be understood as a collection of non-periodic deviations from an ideal periodic lattice (average structure).¹³⁷ Thus, the electron density of a disordered system, ρ(r), can be expressed as the sum of contributions from a periodic structure, [small rho, Greek, macron](r), and a non-periodic term to account for the deviations from ideality, Δρ(r). Likewise, the total scattered intensity, I(u), can be expressed as the sum of Bragg scattering, |F̄(u)|², and diffused scattering, |ΔF(u)|².

By Fourier transforming the scattering intensities, we obtain the PDF as follows:
Here P_tot(r) is the generalized Patterson function while P_hkl(r) and ΔP(r) are the periodic Patterson function and the auto-correlation function of the non-periodic component respectively. Note that P_tot(r) involves the complete reciprocal space while P_hkl(r) considers only the integral reciprocal lattice vectors. Since the diffused scattering intensity is several orders of magnitude less than Bragg intensities, the effect of diffused scattering is often masked under the Bragg peaks. Therefore, 3D-ΔPDF focuses specifically on ΔP(r) for analyzing the exclusive contribution from local structure features. 3D-ΔPDF describes the frequency distribution of spatially resolved interatomic vectors in the true crystal structure relative to the average structure. For instance, positive values for any position vector indicate that the particular bond length appears more in the actual crystal structure as compared to the (ideal) average structure.¹³⁶ Further, the appearance of an alternating pattern of positive and negative correlation values in 3D-PDF suggests preferential ordering/arrangement of atoms different from that suggested by the conventional crystal structure. With its advantages, 3D-PDF also comes with some demanding requirements like the need for high-quality single crystal diffraction data, low background noise, high signal to noise ratio, highest possible maximum diffraction angle for enhanced pixel resolution, coverage of at least one asymmetric reciprocal unit cell by the recorded oscillation range, etc.¹³⁷

The realm of perovskites is very special and deserves to be mentioned here for its role in bringing PDF analysis into the characterization of crystalline materials and in the process, being revolutionized with the attained insights. Some of the earliest applications of PDF analysis are found amongst the literature on the local structure of perovskite oxides acknowledged for colossal magnetoresistance¹⁴⁴ and dielectric/relaxor ferroelectric properties.^145–149 Presently, (organic/hybrid/all-inorganic) halide perovskites are also immensely being studied for their photovoltaic applications. It is here again that PDF analysis has been handy for deciphering the complex structure–property relationships. Further, owing to the local structure revealed by PDF analysis, today there is increasing consensus that owing to the atom off-centering displacements, octahedral tilting, rotation, etc., the structure of halide perovskites must be viewed as a polymorphous network of low-symmetry motifs with complex lattice dynamics instead of the classical monomorphous interpretation.^150,151 Additionally, the underappreciated role of stereochemically active lone-pair of electrons in affecting intriguing properties like broadband emission, defect tolerance, dielectric response, carrier mobility, thermal conductivity, etc., in halide perovskites is being uncovered via rigorous PDF analysis.^17,18,89 In later sections we will look at examples where PDF analysis has been used from the perspective of a characterization tool to investigate the versatile origin of intrinsically ultralow thermal conductivity in crystalline thermoelectric materials.

4.2.3 Solid state NMR spectroscopy (ssNMR). ssNMR spectroscopy exploits the transition between quantized spin states of nuclei that can generate measurable magnetization upon application of an external magnetic field (Fig. 3c).¹²¹ Since the immediate local environment and interactions influence the spin of nuclei, ssNMR is an excellent technique to probe atomic-level dynamics. Limited by the spectral resolution, sensitivity, and complexity of the solid state, ssNMR matured slowly as compared to solution-state NMR spectroscopy. However, technological developments are pushing it to greater heights. Now, more than ever, ssNMR is inching closer to achieving the intrinsic resolution limits.¹⁵² The technique has even enabled an emerging field termed as NMR crystallography.¹⁵³

Although ssNMR is limited by the anisotropic nature of spin interactions, since it is a spectroscopic technique that is blind to any periodicity, the technique is equally applicable to characterize disordered solids. Thus, ssNMR is a valuable complement to structure determination techniques based on Bragg diffraction. Moreover, ssNMR has the ability to probe local structure dynamics at even ps timescales and distinguish static and dynamic disorder in solids. In ssNMR, the number of peaks, the magnitude of chemical shift and linewidth of the spectra, etc., are analyzed to gather information about the local structure around the atom under study. There are several inhomogeneous factors that broaden the ssNMR spectra like deviation from crystal symmetry, magnetic anisotropy due to different effective local magnetic field, dipolar coupling, quadrupolar interactions, etc. Often, the anisotropic nature of interactions broadens spectral lines and hinders site-specific information. On the bright side, quantifying these directional interactions can open new avenues of structural information.

ssNMR can also delve into probing disorders in systems. As alluded to before, ssNMR can directly distinguish between static and dynamic disorder. Here, it is worth noting that a perfectly symmetric crystal would imply that identical atoms at equivalent positions in the crystal will have the same resonant frequency and therefore give rise to a well-resolved NMR spectrum. However, spectra from disordered materials can exhibit large linewidth broadening and if it exceeds the intrinsic line width, the very shape of the spectral peaks/peak profile can give local structural information. Further, since J couplings are dependent on the spatial arrangement of magnetic spins, measurement of the coupling strength can give valuable structural information like the bond angles, bond lengths, etc., for characterizing challenging compositions. Quite contrary to static disorder, depending on the time scales, dynamic disorder gives rise to spectral narrowing due to the averaging effect on many NMR parameters. The scope of ssNMR is so diverse that a comprehensive review of its application space is impractical and far from the scope of this review. However, we glance over a few recent works that establish the strength of the technique in probing the local structure of crystalline solids to understand the structural origin of their functionality.

Numerous examples can be found in the literature where ssNMR has been pivotal in revealing insights into the local structure and sometimes even the crystal structure of controversial systems. Amongst the thermoelectric community, ssNMR has been utilized for measuring carrier concentration using spin-relaxation time and Knight shifts amongst others. The very low spin-relaxation time (5.3 ms) found using ¹²⁵Te NMR in GeTe pointed at its very high carrier concentration (∼10²⁰ cm⁻³).¹⁵⁴ A unique feature of ssNMR is its extreme sensitivity to local electronic inhomogeneity around the probed atom. This aspect was exploited by Levin et al.¹⁵⁵ to demonstrate that unlike GeTe which was electronically homogeneous, PbTe was inhomogeneous and possessed high (∼10¹⁸ cm⁻³) as well as low (∼10¹⁷ cm⁻³) carrier concentration components, the latter of which remained unfound by transport measurements. The authors suggested that the reason behind the observed inhomogeneity in PbTe is due to the spatial variation of Pb:Te concentration in PbTe leading to different spin relaxation times for the ¹²⁵Te nuclei depending on their unique local environment. In a similar study using ¹²⁵Te and ²⁰⁷Pb NMR, the electronic inhomogeneity and Ag:Sb imbalance in Ag_1−yPb₁₈Sb_1+zTe₂₀ (LAST-18) were presented.¹⁵⁶ Further, the local probe nature of ssNMR was employed to reveal the ‘phase inhomogeneous thermoelectric alloy’ in the PbTe_1−xS_x material class using ¹²⁵Te NMR.¹⁵⁷ The study suggested that adherence to Vegard's law doesn’t necessarily rule out local phase separation and nanostructuring in solid-solutions.

4.2.4 Raman spectroscopy. Owing to its ability to provide accurate structural fingerprints, Raman spectroscopy is increasingly being developed and finding application in local structure determination of materials. Raman spectroscopy relies on extracting information about the local structure by analyzing the scattered photons after they interact with the optical phonons in the system. By careful analysis using Group theory, the detected modes can be related to the local structure dynamics of the system under study. It is noteworthy that as compared to synchrotron X-ray/neutron/electron-based techniques which rely on expensive area detectors to allow spatially resolved analyses, Raman spectroscopy offers hyperspectral imaging after coupling with a simple microscope (Fig. 3d).¹²² Moreover, due to the lower coherence length of Raman spectroscopy (∼nm) as compared to XRD-based techniques, the former is highly sensitive to the local structure of materials. Today, a wide variety of sub-techniques like resonant Raman, micro-Raman, tip-enhanced Raman, etc., are available under the umbrella of Raman spectroscopy. In this section we shall see how Raman studies can probe the local structure by studying the phonon modes begot by lattice disorder and address the multi-length scale problems like those in electroceramics.

The scattered light in Raman spectroscopy shows higher/lower wavelengths w.r.t. those present in the incident light depending on whether the photons lost/gained energy quanta from their interaction with phonons. Ideally, owing to the momentum selection rule, only the q⃑ = 0 phonons contribute to the scattering at the zone center. However, under special circumstances like electron–phonon coupling or when the phonon coherence length is compromised due to particle size or other disorder-induced effects, q⃑ ≠ 0 phonons also might start contributing and cause asymmetric spectral shapes. Mention must be made at this juncture that the presence of new Raman modes or changes in the peak profile are indicative of internal or external perturbations in the system. Of particular interest is the presence of symmetry-forbidden modes that may be explored to gain insights into the dynamics of a perturbed system.

Disorder has a significant impact on Raman scattering. At the very outset, disorder affects the local environment and thus alters the intrinsic lattice vibrations and consequent phonon dispersion of the system, which in turn reflect as changes in mode frequency, amplitude, lifetime, etc. Often, new modes are observed when there are defects (vacancies, guest atoms, etc.) in the system. Additionally, two-mode behavior in Raman spectra may be indicative of atom clustering/short range ordering as expected due to the vibrational changes caused by the changes in the effective mass of the vibrating cluster.¹⁵⁸ Likewise, there can be additional phonon–electron scattering processes in addition to phonon–phonon scattering processes, especially in strongly correlated systems, that can amplify the peak broadening/asymmetry.¹⁵⁹

Further, symmetry breaking can activate new phonon modes by relaxing the selection rules. By exploiting this idea, investigating new or symmetry forbidden modes is an appealing way to study the local structure of disordered systems. Since displacive transitions as those found in ferroelectrics have an optical mode that softens to zero near the transition, Raman spectroscopy has been extensively used to study ferroelectric phase diagrams.^160–164 Despite the challenges in analysis posed by peak broadening and mixing (quasi modes), Raman spectroscopy is an established technique to study complex disordered solids. A novel work from Yang et al.¹⁶⁵ realized nanoscale spatial resolution to reveal local structural heterogeneity and visualize short-range atomic ordering in amorphous Si using tip-enhanced Raman spectroscopy. In another study, two extra modes were observed in the polarized Raman spectra of NaLnF₄ (Ln: La, Ce, Pr, Sm, Eu, Gd)¹⁰³ which were attributed to the cation disordering within the P6̄ space group that otherwise generated 18 vibrational modes. Atom ordering and local structure studies in high T_C superconductors have also been undertaken with Raman spectroscopy.^166–168 Further, Raman spectra had shed light on the medium range order and the increase in the extent of disorder with Ge concentration in chalcogenide glasses (Ge₂S₃)_x(As₂S₃)_1−x and (GeS₂)_x(As₂S₃)_1−x¹⁶⁹ while it revealed the ring-chain formation in As–Se–S glasses.¹⁷⁰ The local phenomena in complex oxide perovskites are also increasingly being probed via Raman studies revealing local chemical ordering, atom off-centering, ferroelectric distortions, etc.^{106,171–175}

4.2.5 Electron diffraction-based techniques. In this section we will be concentrating on three techniques, viz., (i) fluctuation electron microscopy (FEM), (ii) convergent-beam electron diffraction (CBED) and (iii) electron ptychography.
Fluctuation electron microscopy (FEM). FEM is a transmission electron microscopy-based diffraction technique that is used to study ordering regimes by measuring the statistical fluctuations in electron scattering owing to structural variance at the nanometer length scale of disordered materials. A major advantage of FEM over PDF is that it analyzes many-body correlation functions unlike only two-body correlation functions used by PDF. Thus, FEM overcomes the problem of statistical sampling which renders PDF insensitive to nanoscale ordering.¹⁷⁶ The differences in structural arrangement and orientation of sub-volumes within a thin sample can be efficiently studied using FEM. Herein, the width of the investigated volume is determined by the point spread function/resolution of the microscope.

FEM is performed in two modes: transmission electron microscopy (TEM) mode and scanning transmission electron microscopy (STEM) mode. In the TEM mode, tilted dark field images from the sample are collected using a small aperture. In disordered sample specimens, the images appear to be speckly, owing to variation in the scattering intensity from regions having width comparable to the point spread function. The variation originates from the fluctuations in the coherence of the scattering between atoms lying in the probed sub-volumes in the sample. These fluctuations could be due to either local structure ordering or statistical/randomized arrangements of atoms. The variation of image intensity is a measure of the speckiness of an image. Note that upon changing the angle of illumination, total scattering intensity also varies due to the underlying changes in the atomic scattering factors with angle. Here random and ordered arrangements are differentiated based on the behavior of the normalized image intensity variance as a function of angle.¹⁷⁶ On the other hand, in the STEM mode, a microdiffraction pattern is obtained at each probed point. A collection of such microdiffraction patterns at the end of scan is analyzed and the normalized variance of these patterns is used to extract analogues information as in the TEM mode. This technique has often been used to study highly disordered materials, often non-crystalline.^176–183 Recently, using spherical aberration-corrected STEM, Ge et al. have shown that the off-centering behaviour of Pb in Cu_0.0029Pb(Se_0.8Te_0.2)_0.95 plays a significant role in dictating its thermal transport and thermoelectric performance.⁶⁴ A classic example of the strength of FEM in characterizing complex structures is from Treacy and Borisenko who used it to reveal the controversial amorphous network of Si.¹⁷⁸ Further, an FEM study on a metallic glass, Al_88−xY₇Fe₅Ti_x, revealed that microalloying of 0.5% Ti for Al in this alloy enhanced its short- and medium-range order.¹⁸⁰ Zhu et al. revealed the structural origin of spatial inhomogeneity in a hyperquenched metallic glass, Zr₅₃Cu₃₆Al₁₁, using HAADF-STEM with angstrom beam electron diffraction.¹⁷⁹

Convergent beam electron diffraction (CBED). Characterizing the structure of materials with compromised periodicity, i.e., those lacking translational/rotational symmetry, even at the sub-nanometer scale owing to disorder is an extremely difficult feat. Founded on the seminal work of Kossel and Möllenstedt¹⁸⁴ on mica flakes with an aluminum cone sealed inside a wine bottle as the electron source, CBED has revolutionized sub-angstrom characterization of materials with advancements in electron sources, imaging techniques, aberration corrected lenses, etc. The intensity profile within the discs can then be mapped to identify the point group, space group, etc., of the crystal. Further, since CBED uses dynamic diffraction approximation unlike the kinematic diffraction approximation in X-ray diffraction, it can distinguish between polar and non-polar specimens in addition to providing information about atomic positions, lattice parameters, stacking faults, Debye–Waller factors, etc.^185,186

An impressive application of CBED was the direct observation of static nanoscale local structures with rhombohedral symmetry in an archetypal ferroelectric, BaTiO₃, that settled the debate over the nature of its transition (displacive or order–disorder) from a paraelectric cubic phase to the various ferroelectric phases.¹⁸⁷ In a similar work from the same group, existence of local polarization clusters originating from symmetry breaking in the cubic phase of KNbO₃ was revealed.¹⁸⁸ Depending on the electron beam size used, there are multiple variants of CBED based techniques, viz., nano beam electron diffraction (NBED), angstrom beam electron diffraction (ABED), etc. NBED uses a coherent electron beam <1 nm to capture 2D diffraction patterns that enables probing of nanoscale regions to investigate the actual local structure of disordered solids. The well-defined points in an NBED image are the signature of local atomic order in the investigated material. A statistical analysis of the scattering intensity may also be used to derive quantitative information about the underlying local order. Further progress in the field saw the development of the ABED method (electron beam size: <4 Å) which uses a sub-nanometer electron beam to study local atomic configurations (Fig. 3e).^123,189 In an exemplary work, Hirata et al.¹⁹⁰ demonstrated the geometric frustration-driven preference for distorted icosahedra in a candidate metallic glass, Zr₈₀Pt₂₀, using ABED with an electron beam of ∼0.36 nm.

Electron ptychography. Conceptualized by the pioneering works of Hoppe in the 1960s,¹⁹¹ ‘ptychography’ refers to a computational microscopic imaging technique where numerous overlapping diffraction patterns are collected by systematically shifting either the sample or the beam and subsequently reconstructing the phase using phase retrieval algorithms.^192–195 In electron ptychography (Fig. 3f),¹²⁴ a coherent convergent beam of electrons is used in STEM mode to scan the sample and collect 2D diffraction patterns. Next, direct or iterative inverse analysis is used to computationally reassemble the phase (or more correctly, the complex amplitude of the diffracted electron beam) enabling high resolution spatial (structural) analysis. Remarkably, the probing beam size can be made smaller than the separation between the scatterers itself, implying that the subatomic diffraction imaging can potentially be used to engineer functionalities at the atomic-scale itself.¹⁹⁶

For a very long time, super-resolution imaging crucial for investigating local symmetry breaking was beyond electron ptychography due to the intrinsic/extrinsic misorientation between the crystal axis and the electron beam limiting both accuracy and resolution. However, Sha et al.¹⁹⁷ overcame this hurdle by developing adaptive propagator ptychography (APP), wherein the Fresnel propagator which is generally considered fixed during reconstruction was modified to include sample tilt as well in the multi-slice simulation of dynamic diffraction. Furthermore, in a cutting-edge work from Chen et al., electron ptychography achieved atomic-resolution limits set by the inherent vibrations of atoms itself in PrScO₃.¹²⁴

We acknowledge that the above discussion is not a comprehensive overview of the characterization techniques available for local structure determination. With the current technologies advancing at an unprecedented speed, new methods are being invented and old ones being further improved even as we write, rendering the above discussion obsolete. Nonetheless, our aim while working on this review was to provide the reader with a crisp and concise summary of the existing methods/approaches with hand-picked examples from recent literature. How these local structure determining techniques are valuable in investigating the origin of low thermal conductivity in high performance crystalline thermoelectric materials will be further elaborated in the later sections of this review.

4.3 Local structure induced structure–property relationships

Understanding and tailoring the local structure is becoming a new paradigm in realizing and rationalizing the structure–property relationships in functional materials. The understanding of local structure engineering is an effective strategy for tuning the upconversion emission and it is becoming a game-changer in lanthanide-doped nanocrystals.¹⁰³ It is worthwhile noting that lanthanides, having large quantum numbers, 4f electronic configuration and spin–orbit coupling, are rich in spectroscopic terms and have numerous consequent electronic transitions. However, owing to the Laporte rule, the intra-configurational transitions within the 4f levels are parity forbidden. In this regard, by breaking down the local site symmetry, we can alleviate the forbidden nature of these transitions. The importance of the local structure around the Ln³⁺ ions in determining the emission properties of nanocrystals doped with them is therefore apparent. Consequently, local structure engineering is potentially a powerful handle for tuning the properties of lanthanide doped emission upconversion nanocrystals. Corroborating with the above discussion, in this class of materials, modulating the local structure has indeed been shown to improve the upconversion emission intensity and selectivity and tune the wavelength shift and lifetimes.¹⁰³ Local structure modulation is understood to be acting by varying the degree of parity hybridization and tuning the energy level splitting of the lanthanide ions facilitating greater interionic energy transfer efficiency. Another example to illustrate the importance of the local structure in determining material properties comes from Pb-based ferroelectrics.^94,96 Herein, the origin of ferroelectric properties was traced to short range order present in these systems despite the disordered average structure.^94,96 Thus, studies focused on Pb²⁺ local structure proved that tuning of the ferroelectric properties in Pb-based ferroelectrics is possible by engineering the size of the Pb²⁺ environment, B-site cation order (stoichiometric ratio) and the relative size of the B-site cation with respect to Pb²⁺.¹⁹⁸ Apart from these, altering the local structure from its global average structure has a strong influence on catalytic activity,¹⁰² sodium storage in hard carbon anodes in sodium-ion batteries,¹⁹⁹ in the field of metallic glasses,^179,180 superionic conduction,^108–110 superconductivity,^111–114 optoelectronics,^115–117etc. The tuning of the local structure plays a profound role in enabling numerous applications which are beyond the scope of this review. However, we conclude this section by pointing to the prospect of exploiting the local structure as a fundamental handle to tune material functionality and a paradigm shift to engineering disorder in crystalline materials is thus called upon.²⁰⁰ In the subsequent sections we will discuss about how the local structure plays a pivotal role in governing the thermoelectric performance of a material.

5. Local structure relation to thermal transport in crystalline solids

5.1 Local off-centering of atoms

In simple terms, deviation of an atom from its parent site in the lattice can be termed as off-centering. This deviation brings forth a reduction in crystal symmetry and induces lattice strain which in turn hinders the phonon flow.^201,202 However, in some cases it has been observed that the deviation is so slight that the overall symmetry does not change; however, the local co-ordination becomes asymmetric. This local off-centering of atoms can be due to a host of reasons, including the presence of stereochemically active lone pair of electrons, discordant atoms in the lattice, presence of smaller atoms in a large void, weak hybridization, etc.

Clathrates form an important member of the thermoelectric family due to their glass-like thermal conductivity, despite being crystalline. Sales et al., while investigating the thermal conductivity of Eu₈Ga₁₆Ge₃₀, Sr₈Ga₁₆Ge₃₀, and Ba₈Ga₁₆Ge₃₀, found that Eu₈Ga₁₆Ge₃₀ and Sr₈Ga₁₆Ge₃₀ possess glass-like thermal conductivity while the temperature dependence of thermal conductivity in Ba₈Ga₁₆Ge₃₀ resembles that of a crystal (Fig. 4a).^203,204 From single crystal diffraction it has been found that Eu and Sr remain off-centered in their large cages to four nearby positions while Ba remains mostly in the center of the cage. This off-centering of guest Eu and Sr atoms creates tunneling states which are probably behind the glass-like thermal conductivity in an otherwise crystalline compound.²⁰³ Several other factors do contribute to lowering of thermal conductivity in compounds which show such frameworks containing guest atoms in oversized cages, including rattling of the guest atoms,²⁰⁵ avoided crossing of the guest optical mode with the heat carrying acoustic modes,²³ phonon charge carrier scattering,²⁰⁶etc. Nevertheless, it has been observed that off-centering of atoms within a host cage plays a pivotal role in inducing glass-like plateau in thermal conductivity. To determine the role of off-centering in inducing glass-like thermal conductivity, Christensen et al. synthesized Sr₈Ga₁₆Ge₃₀ using the flux growth method and the zone-melting process. Sr₈Ga₁₆Ge₃₀ belongs to clathrate type I class and crystallizes in the Pm3̄n space group. It consists of two types of cages, (i) two dodecahedral cages, comprising of a 20-atom framework made up of Ga–Ge bonding, and (ii) six tetrakaidecahedral cages, comprising of a 24-atom framework made up of Ga–Ge bonding (Fig. 4b). The flux-grown sample exhibits glass-like thermal conductivity while the zone-melted sample induces a crystalline peak. A combination of the aforementioned two processes exhibits an intermediate thermal conductivity.³⁷ Single crystal neutron diffraction studies revealed that the transformation of crystalline to glass-like thermal conductivity is correlated with the spatial distribution of the Sr atoms in the dodecahedral cages, shifted from the central site (Fig. 4c and d). The off-centering of Sr atoms is found to be 0.24 Å for the zone-melted sample, 0.36 Å for the sample realized by a combination of zone-melting and flux growth, and 0.43 Å for the flux-grown sample. This increase in off-centering of the Sr atom strongly suggests that the glass-like plateau as observed for the flux-grown Sr₈Ga₁₆Ge₃₀ is directly correlated with the magnitude of the off-centering of the Sr atom, thereby validating its role in lowering κ_lat.³⁷

Fig. 4 (a) Temperature dependent lattice thermal conductivity (κ_lat) of the five clathrates given in the legend. The dotted and dashed lines correspond to a-Ge and a-SiO₂, respectively.²⁰⁴ (b) Structure type belonging to clathrate type I class, M₈III₁₆IV₃₀, and crystallizing in cubic space group Pm3̄n. Here M is a divalent cation, and III and IV correspond to group 13 and group 14 atoms respectively. The structure consists of two dodecahedral cages comprising of 20 atoms, hosting the M1 guest atom on the 2a site, and six large tetrakaidecahedral cages, comprising of 24 atoms each hosting an M2 guest atom.³⁷ Panels (c) and (d) are representative figures of the dodecahedral cage framework where the large void formed by III–IV host atoms, while the guest atom either resides in the center (6d site) or is disordered over the surrounding 24k sites.³⁷ NCSI-TEM images²⁰⁷ of (e) 20 at% Yb filled Co₄Sb₁₂ n-type and (f) 100 at% Ce filled (CoFe₃)Sb₁₂ p-type along the 〈001〉 crystal axis. (g) Magnified images of individual cages marked by d1–d4 in figures (e) and (f) representing the 4 representative configurations of the filler atoms. From d4, we can clearly observe the off-centering of Ce atoms from their mean position. Fig. (a) is reproduced with permission from ref. 204 © 1999, American Physical Society. Fig. (b)–(d) are reproduced with permission from ref. 37 © 2016, American Physical Society. Fig. (e)–(g) are reproduced with permission from ref. 207 © 2022, Wiley VCH.

Skutterudites which also have a similar host–guest framework exhibit off-centered rattlers leading to ultralow κ_lat. Ge et al., in their recent work on Yb partially filled and Ce fully filled skutterudites, have directly measured the off-centering of the guest atom using atomic-resolution negative spherical aberration imaging TEM (NCSI-TEM) (Fig. 4e and f).²⁰⁷ NCSI-TEM, having picometer precision in resolving the structure of the compound, directly evidences the off-center shifts of the Yb and Ce filler atoms by 0.37 Å for 20 at% Yb filled Yb_0.2Co₄Sb₁₂ and 0.60 Å for 100 at% Ce filled Ce(CoFe₃)Co₁₂ skutterudites (Fig. 4g). Using the modified Debye–Callaway model, they have quantified the contribution from off-centering and found out that off-center rattling of atoms (Yb/Ce) strongly scatters a broader phonon frequency range compared to other phonon scattering processes like on-center rattling, resonant scattering, point defect scattering, etc., thereby quantifying the role of off-centering in inducing ultralow κ_lat.²⁰⁷ LaFe₃CoSb₁₂,²⁰⁸ PrOs₄Sb₁₂,²⁰⁹ NdOs₄Sb₁₂,²¹⁰ LaOs₄Sb₁₂,²¹¹ Ce_1−xYb_xFe₄P₁₂,²¹² and SnFe₄Sb₁₂²¹³ are some of the notable examples where the off-centered filler atoms in skutterudites play an important role in lowering κ_lat to an ultralow value.

Similar observations of local structural distortions and cation off-centering in larger co-ordination spheres are also made in perovskite halides.⁸⁹ Cs in CsSnBr_3−xI_x resides in a cubo-octahedral void formed by SnX₆ octahedra (X = Br or I). Cs is found to be residing in an off-centered position having enhanced thermal vibration, thus acting as an off-centered rattler.^21,214 This off-centering tendency of the Cs atom combined with the distorted octahedra of SnX₆⁹¹ induces low Debye temperature and phonon velocity, thereby rendering ultralow κ_lat in CsSnBr_3−xI_x.²¹ Similar off-centering phenomena lowering κ_lat in perovskites are also observed in Cs₃Bi₂I₆Cl₃,¹⁸ BaTiS₃,⁶⁹ CsSnI₃,²¹⁵etc., which also possess locally distorted frameworks.

The role of off-centered atoms in lowering thermal conductivity is not only limited to small hosts in large, oversized cages, but is also seen in materials containing stereochemically active lone pair of electrons. AgSbSe₂ crystallizes in a rock-salt cubic lattice where Ag and Sb are positionally disordered in the cation position (0, 0, 0) and Se takes up the anion position (½, ½, ½).^216–218 Despite being a crystalline compound, the thermal conductivity of AgSbSe₂ exhibits a plateau in its temperature dependence resembling that of a glass.^13,40 The total scattering technique using X-ray PDF (X-PDF) revealed the presence of locally distorted coordination around the cations (Ag/Sb) in a globally symmetric rock-salt cubic lattice. X-PDF revealed asymmetry in the first peak (Fig. 5a). In an ideal rock-salt crystal, the cations are surrounded equidistantly by six Se atoms forming an octahedra, thereby exhibiting a symmetric first peak in X-PDF. However, the asymmetry observed in the first X-PDF peak indicates the presence of distortion in the first coordination sphere of AgSbSe₂. Refinement of the X-PDF data using the rock-salt model refines the average structure appreciably conforming to the overall global structure being Fm3̄m. However, the poorer fit of the first peak indicates local distortion in AgSbSe₂ within an average cubic lattice (Fig. 5b). The fit improves substantially when the cations were distorted by ∼0.20 Å along the crystallographic 〈100〉 direction (Fig. 5c). The result of this distortion is the creation of multiple short and long bonds (Fig. 5d), which creates an aperiodicity in the lattice, thereby enhancing the phonon scattering. This local distortion is caused by the presence of stereochemically active 5s² lone pair of electrons on Sb which is observed from the electron localization function (ELF) (Fig. 5e). The ELF shows the asymmetric localization of charge, indicating the preference of the lone pairs to occupy a certain space in the coordination sphere. As a result, these lone pairs exert electrostatic repulsion to their neighboring bond pairs resulting in a locally distorted octahedron, creating aperiodicity in the lattice vibration, and thereby glass-like thermal conductivity in AgSbSe₂.⁴⁰ Lone pair induced local symmetry breaking and its subsequent influence in lowering κ_lat are also seen in PbQ (Q = S, Se, Te),^42,43 SnQ (Q = Se, Te),^44,45 AgBiS₂,^41,219 AgPbBiSe₃,¹⁴ Sn_1−xGe_xTe,³¹ Ge₂Sb₂Te₅,⁴⁶etc.²⁰²

Fig. 5 (a) X-ray PDF of AgSbSe₂ corresponding to the local structure (left) and the average structure (right) at different temperatures. The asymmetry in the first peak indicates a local distortion in the globally symmetric rock-salt AgSbSe₂.⁴⁰ (b) Fitting of the first two peaks (local structure) considering an undistorted rock-salt model.⁴⁰ (c) Fitting of the first two peaks (local structure) considering a cation distortion along the crystallographic 〈100〉 direction.⁴⁰ (d) Coordination of the locally distorted cations (Ag/Sb) with corresponding bond lengths.⁴⁰ (e) Electron localization function (ELF) of AgSbSe₂ on a (121) plane showing the selective preference of the stereochemically active 5s² lone pairs of Sb, resulting in a distorted octahedron. Grey, orange and green spheres represent Ag, Sb and Se atoms, respectively.⁴⁰ The figure is reproduced with permission from ref. 40 © 2022, Wiley VCH.

SnSe-based binary semiconductor materials are among the best thermoelectric materials in the mid-temperature range (zT > 2.5; Fig. 6a), primarily due to their inherently low κ_lat.^220–226 SnSe adopts an orthorhombic crystal structure (Pnma space group) at low temperatures, and it undergoes a structural phase change at a temperature T_s ∼ 800 K to a higher symmetry structure but still in an orthorhombic phase (Cmcm space group).²²⁷ In SnSe, a complex interaction exists between lattice and electronic degrees of freedom, involving van der Waals-like bonding, multiple band edges, and lone pair of electrons of Sn 5s².²²⁸ There has been evidence of a significant lattice anharmonicity caused by soft optical phonon modes evidenced by inelastic neutron scattering.²²⁹ Recent first-principles simulation showed that Jahn–Teller-like instability in the electronic band structure induces a lattice distortion near the phase transition which makes the crystal lattice significantly anharmonic with ultralow κ_lat.²³⁰ Bozin et al. have characterized the local structure of SnSe throughout the Pnma to Cmcm phase transition using X-PDF analysis to unveil the basic origin of low κ_lat in the vicinity of this transition.⁴⁵ The atomic structure of SnSe is well characterized on a global scale both above and below the transition temperature. The unit cell of the Pnma structure comprises two SnSe bilayers with ferro-ordered Sn displacements within the intra-bilayer and anti-ferro-coupling among the inter-bilayers.⁴⁵ The main reason for the distortion is a displacement of Sn from the square's centre caused by the nearly coplanar neighbouring Se atoms (Fig. 6f), whereas the distortion vanishes in the Cmcm global structure, and Sn remains bonded to its neighbouring coplanar Se atoms with four equal bond lengths, placing it in the centre of the square (Fig. 6f).⁴⁵ Although κ_lat increases with increasing lattice symmetry, this is not the case for SnSe. To investigate the local to long-range evolution of the structure, Bozin et al. have fitted X-PDF data of SnSe at 300 K (Fig. 6b) and 1070 K (Fig. 6c).⁴⁵ The Pnma model fits the X-PDF well throughout its entire range at 300 K, as observed from the good agreement factor (7.1%) and absence of structure in the residual plot (green line, DG). Conversely, fitting of the measured X-PDF using the Cmcm model at high temperature yields a satisfactory fit in the high-r (>10 Å) region but a poor fit in the low-r (<10 Å) region. Therefore, in the Cmcm phase the local structure is poorly described by the average Cmcm structure. Fig. 6e highlights the failure of the fitting and exhibits an absence of intensity at 3.07 Å, which is the average Sn–Se distance in the square-planar region. So, even in the high-temperature Cmcm phase, the Sn atom in the local structure is still off-centered in its square of Se ions.⁴⁵

Fig. 6 (a) Temperature dependent thermoelectric figure of merit (zT) for single crystalline SnSe along different crystallographic directions.²²⁰ In (b) and (c), the fits of average structure models across the entire r range at 300 K and 1070 K are displayed for SnSe. The difference traces (ΔG) in green illustrate the variance between calculated and measured PDF. In the low-r region as shown in (d) and (e), the same fits reveal a noticeable inconsistency between the Cmcm model and the 1070 K data on a sub-nm scale, leading to an increase in R_w. The Sn environments anticipated from the average structure through Bragg scattering are portrayed in (f). In the Pnma structure, Sn is laterally displaced from directly beneath Se, while in the Cmcm average structure, Sn is positioned directly below it. Labels A and P denote apical and in-plane Se, respectively. The arrow in (e) highlights a feature expected in the Cmcm symmetry model, associated with Sn being centered in the Se plaquette, which is evidently absent in the data. (g) and (h) Comparisons of PDFs at different temperature for SnSe in the range 1 < r < 10 Å using the split-site model. (i) Reduced temperature dependent Sn displacements (magnitude) away from the center.⁴⁵ Fig. (a) is reproduced with permission from ref. 220 © 2014, Nature Publishing Group. Fig. (b)–(i) are reproduced with permission from ref. 45 © 2023, American Physical Society.

Furthermore, unlike the other ADPs, the in-plane component of the Sn ADP is larger and rises as the material enters the Cmcm phase, reinforcing the notion that Sn is primarily found in locations distant from the Se plaquette's centre. The basic rationale for such observed behaviour is that the structural phase transition possesses an order–disorder characteristic and the Pnma distortions persist at higher temperature but remain disordered within the various structural variants. More intricate modelling was introduced by considering a split-site model (inset of Fig. 6i) where every crystallographic Sn and Se site was divided into four equally populated sites within the Cmcm crystal symmetry.⁴⁵ With the split-site model, the fit had a noticeably better residual (5.8%) (Fig. 6g) and much lower ADPs. The model produced displacements (δ) of Sn of ∼0.25 Å and considerably lesser Se displacements of ∼0.02 Å (Fig. 6i). This implies that in the Cmcm phase, dynamic movement of displaced Sn atoms occurs in a manner akin to a Mexican hat potential. Modelling of the high temperature (above T_s) data shows that Sn displacements were almost temperature independent, but below T_s, the fit deteriorates, inferring that the local symmetry begins to change near T_s. Further short-range modelling suggests that contrary to the 3D-like intra-bilayer interaction seen in the Pnma phase, at high temperatures on a one nanometer length scale the 2D dipole ordering within the bilayers persists; however, the 3D interaction no longer exists.⁴⁵ Eventually, SnSe is nanostructured with local symmetry lowering electronically induced inherently fluctuating Sn dipolar-character displacements near and above the phase transition, resulting in extremely low κ⁴⁵ and exceptional thermoelectric figure-of-merit (Fig. 6a).^220,225,226

5.2 Effect of discordant atoms in the lattice

Sometimes the incorporation of a dopant atom into a compound can be hindered by its ability to fit into the lattice structure, especially if the dopant cation coordinates differently with the anions as compared to the norm. This in turn causes discordant atomic off-centering and acts as a strong phonon scattering center.^47–52 For instance, HgSe always tends to adapt a zinc blende-type structure, whereas the coordination environment of Pb in PbSe is found to be octahedral. In either solids or molecules, it is extremely rare to observe Hg exhibiting octahedral coordination, and thus within the Se sublattice, Hg²⁺ cations sit at an off-centered position. This local off-centering breaks the crystal symmetry and allows the hybridization of orbitals which is otherwise forbidden, driving the system to be more strongly bonded.^87,231

The help of ssNMR has been taken to locally probe the coordination environment of both Hg and Se for which Hodges et al. employed ¹⁹⁹Hg and ⁷⁷Se NMR.⁴⁷ In HgSe, the ¹⁹⁹Hg NMR spectrum has 2 components, where the main one resonates at −1642 ppm and has a shift anisotropy of 27 ppm (Fig. 7a). This is ascribed to Hg atoms present in tetrahedral coordination. This degree of shift anisotropy is extremely reliant on the local structural symmetry, and it stays small at 27 ppm for highly symmetric tetrahedral coordination.²³² On the other hand, the secondary component in the ¹⁹⁹Hg NMR spectrum is seen at −1720 ppm resonance position exhibiting a substantial shift anisotropy of 200 ppm. This significant anisotropy indicates a linear coordination environment for Hg which can be present in the trace amounts of cinnabar-type HgSe.²³³ During the examination of the ¹⁹⁹Hg NMR spectrum of PbSe–6%HgSe, a sole component with a shift of −1860 ppm (shift anisotropy of 65 ppm) is observed. Such difference in these shifts between HgSe and PbSe–6%HgSe can be assigned to different coordination environments present in these two systems. Particularly, the enhanced shift anisotropy of Hg in the case of PbSe–6%HgSe as compared to HgSe suggests a less symmetric circumstance with some local structural distortion.²³⁴ A similar situation occurs in the ⁷⁷Se NMR spectra, where like in the case of HgSe, a single but sharp resonance can be noticed at −245 ppm with a lesser shift anisotropy of 2.4 ppm, while the resonance takes place at −710 ppm for PbSe–6%HgSe, with a larger shift anisotropy of 46 ppm (Fig. 7b). This phenomenon again indicates the presence of a less symmetric structural environment for Se in PbSe–6%HgSe than HgSe. Thus, the ssNMR data imply that within the PbSe matrix, Hg does not occupy the center of an octahedron, supporting its familiar chemical inclination to avoid such a coordination environment.²³⁵ Such asymmetric coordination geometry of Hg in PbSe is further supported by using density functional theory (DFT). The calculated total energy of the Pb₂₆Se₂₇Hg₁ supercell where Hg is placed at different locations of the octahedral and tetrahedral sites demonstrates that the system undergoes an energetic minimum when Hg is placed ∼0.2 Å away from the octahedral center, corroborating the off-centered position. Such a presence of a discordant atom can suppress phonon velocities and initiate additional phonon scattering mechanism(s) as well as modulate the electronic band structure benefiting the electronic properties of semiconductors. As a result, a highly suppressed κ_lat of 0.68 W m⁻¹ K⁻¹ at ∼950 K (Fig. 7c) is observed in 2 mol% HgSe doped Pb_0.98Na_0.02Se with an excellent zT of ∼1.7 (Fig. 7d).⁴⁷ Cd, the same group element of Hg, also acts in a similar way in PbSe. The large shift anisotropy in both the cases of ¹¹¹Cd and ⁷⁷Se NMR spectra in wurtzite CdSe, PbSe, PbSe–3%CdSe and PbSe–10%CdSe, respectively, strongly suggests the different coordination environment present for Cd in PbSe–CdSe compared to that of pristine PbSe. This shows a similar discussed discordant behaviour of Cd in PbSe which acts as an excellent phonon scattering center.⁴⁹

Fig. 7 (a) ¹⁹⁹Hg and (b) ⁷⁷Se Carr–Purcell–Meiboom–Gill (CPMG) static NMR spectra together with the corresponding spectral simulations for HgSe and PbSe–6%HgSe.⁴⁷ Temperature variation of (c) lattice thermal conductivity (κ_lat) and (d) thermoelectric figure of merit (zT) for PbSe–2%Na + x%HgSe.⁴⁷ The figure is reproduced with permission from ref. 47 © 2018, American Chemical Society.

Following the above discussed studies, several research works were undertaken to support the claim of a discordant atom. Luo et al. recently have demonstrated that the variation in the energy profile of PbS and PbSe alloyed with Ge, as the Ge atom shifts away from the octahedral centre, undergoes a minimum, indicating an off-centering of the Ge atom with the magnitude of 0.14 Å and 0.3 Å (Fig. 8a and b) respectively along the 〈111〉 direction.^48,236,237 The 4s² lone pair of electrons in Ge has a strong tendency to stereochemically express itself which causes GeS/Se/Te to adopt distortion and avoid the rock salt structure. Such discordant behaviours have also been noticed for Sn in Cu₂SnSe₃–Cu₃SbSe₄,²³⁸ Gd in PbTe,⁵⁰ Ag in PbSe,⁵¹ Zn in PbTe,⁵²etc. Similarly, the PDF analysis of neutron total scattering data unveils that GeMnTe₂ is locally distorted wherein the Ge²⁺ cations are discordant and shifted ∼0.33 Å from the center of the octahedron along the [100] direction, leading to remarkable phonon scattering.⁸³ As a result of this off-centered dopant, phonon scattering enhances, resulting in a suppressed κ_lat of ∼0.36 W m⁻¹ K⁻¹ for Pb_0.9955Sb_0.0045Se–12%GeSe at 573 K⁴⁸ which is very close to the theoretically minimum achievable value of ∼0.38 W m⁻¹ K⁻¹ for PbSe (Fig. 8c).³⁵ This low κ_lat drives its thermoelectric figure of merit to ∼1.54 at 773 K in n-type Pb_0.9955Sb_0.0045Se–12%GeSe (Fig. 8d).⁴⁸ This strategy is found to be extremely efficient to reduce phonon group velocity and κ_lat in chalcopyrite like CuFeS₂ upon introducing In on the Cu site (Fig. 9a). In⁺ has stereochemically active 5s² lone pair which is incapable of adopting tetrahedral coordination geometry necessary in chalcopyrite sites.²³⁹ The energetically favoured bonding environment (Fig. 9b) for In⁺ – called a seesaw geometry with 4 connected ligands – creates a heavily distorted local structure. This weakens the In–S and Cu–S bonds inducing low energy phonons which interact with the acoustic phonons of the host lattice leading to flattening of the phonon dispersion. Moreover, κ_lat for Cu_1−xIn_xFeS₂ is much lower as compared to CuFe_1−xIn_xS₂. This is due to the dissimilar chemical state of Cu⁺ and Fe³⁺ in CuFeS₂ (Fig. 9c–f). When In is doped at the Cu site, it stays monovalent preserving its stereochemically active lone pair of electrons and causing strong local distortion (inset of Fig. 9b). Therefore, 3 mol% In doping results in 42% drop in κ_lat from pristine CuFeS₂ (Fig. 9a).²³⁹

Fig. 8 (a) The change in the energy profile of PbSe–GeSe as a function of coordinates, from the Pb site substituted with Ge along the 〈111〉 direction to the center of the tetrahedral site. (b) Demonstration of the Ge off-centered crystal structure. The bond length between off-center Ge and three closest Se is 2.77 Å and for other three pairs is 3.13 Å. The usual bond length of Pb–Se and Ge–Se (within the same layer in orthorhombic Pnma structure) is 3.05 and 2.60 Å respectively. Temperature variation of (c) lattice thermal conductivity (κ_lat) and (d) thermoelectric figure of merit (zT) for Pb_0.9955Sb_0.0045Se–x%GeSe.⁴⁸ The figure is reproduced with permission from ref. 48 © 2018, Royal Society of Chemistry.

Fig. 9 (a) Variation of room temperature lattice thermal conductivity (κ_lat) with the doping concentration of In in Cu_1−xIn_xFeS₂ and CuFe_1−xIn_xS₂. (b) Energy profile diagram of In⁺ substitute of Cu in CuFeS₂ at different off-centered positions. The bonding nature of In⁺ in (c) Cu_1−xIn_xFeS₂ and (e) CuFe_1−xIn_xS₂. (d) and (f) Schematic of the distorted weak chemical bond induced localized resonant state.²³⁹ The figure is reproduced with permission from ref. 239 © 2019, American Chemical Society.

5.3 Local disorder in high entropy systems

Ioffe–Regel crossover²⁴⁰ delineates the frequency at which the phonon mean free path in a material becomes equal to its typical atomic spacing.²⁴¹ In proximity to this critical frequency, also lies the crossover frequency for ballistic to diffusive phonon transport. This crossover has remarkable implications, especially in the context of the Boson peak observed from low temperature heat capacity measurements in glasses. Here, the localized disorder results in significant scattering at small wave-vectors which is also relevant in the context of defective or high entropy crystals, wherein vacancies, interstitials, strain fluctuation, local atomic off-centering, discordant atoms, etc., act as local scattering centers.^242–247 It is known that the robust local disorder caused by degenerate doping in the polysilicon layers along with small grain size enforces a severe limitation to the smooth passage of phonons.²⁴⁸ On the other hand, ternary oxide compounds like Y₄Al₂O₉ exhibiting a complex crystal structure (with space group P2₁/c) illustrate a high level of local disorder and large diversity in the inhomogeneity of chemical bonding which yields very low thermal conductivity values.⁶⁸ Similar observations have also been made for Yb₂₁Mn₄Sb₁₈ and Eu₂ZnSb_2−xBi_x as well.^249,250 Recently, Hori et al. have demonstrated that the existence of the local disorders in PbTe reduces the lattice thermal conductivity in the phonon frequency regime >8 Hz without sacrificing the low frequency phonon scattering that is triggered by nanostructuring. Thus, in the entire frequency regime, the thermal conductivity gets diminished, signifying the dual applicability of phonon scattering from nanostructures and local disorder.²⁵¹ Such local disorder not only has profound impact over the phonon dynamics but often affects the magnetic properties also drastically. For instance, systems like RE(Fe, Al)₂ (RE = Gd, Dy) hold properties akin to amorphous magnets. This disorder is primarily attributed to local variations in the ionic radii of the substituting elements causing local lattice distortions which determine the local crystal field. Subsequently, the inherent anisotropy and axis of magnetisation vary over a small space of just a few lattice constants.²⁵²

Recent advancements in the techniques of synthesizing materials have made it possible to generate entropy-stabilized ceramics, which bids a prospect to delve into the influences of disorder on heat propagation. By investigating the structural, mechanical, and thermal properties of single crystalline entropy stabilized oxides (ESOs), it becomes apparent that the irregularities in localized ionic charge can remarkably diminish thermal transport while preserving mechanical rigidity. These materials sometimes exhibit the characteristics of thermal transport properties that resemble those of amorphous materials, aligning with the theoretically minimum attainable limit. While there has been extensive research into the microstructure and mechanical properties of high-entropy alloys (HEAs), their thermal conductivity has gained relatively less attention. The complex atomic arrangements in HEAs, due to their extreme configurational disorder, offer an intriguing opportunity for examining the impact of disorder on heat transport.⁵ However, the metallic behaviour of most HEAs provides significant electronic contribution to the thermal conductivity that can obscure the role of lattice thermal conductivity. When disordered solid solutions involve non-metallic components, the dynamics of heat transport are primarily dictated by phonons.⁶⁶ Lately, the concept of entropy stabilized ceramics has opened new avenues for investigating the influence of mass and interatomic force disorder, surpassing previous boundaries. Recently, Braun et al. have investigated the thermal transport properties of ESOs such as J14: Mg_xNi_xCu_xCo_xZn_xO (x = 0.2), J30: Mg_xNi_xCu_xCo_xZn_xSc_xO, J31: Mg_xNi_xCu_xCo_xZn_xSb_xO, J34: Mg_xNi_xCu_xCo_xZn_xSn_xO, J35: Mg_xNi_xCu_xCo_xZn_xCr_xO and J36: Mg_xNi_xCu_xCo_xZn_xGe_xO (x = 0.167) and observed glass-like temperature dependence of thermal conductivity.⁶⁶ Thermal conductivity undergoes a substantial decline when comparing J14 with all six-cation oxides. Among the latter, the deviations in thermal conductivity are consistent, with each oxide exhibiting values within twenty percent of the rest. These values line up with the expected pattern of declining thermal conductivity as the mean cation mass becomes heavier.

To comprehend the reduction in thermal conductivity when transitioning from 5 to 6 cations, the thermal conductivity measurements on J14 and J35 (for the temperature range 78–450 K) have been performed as shown in Fig. 10a. J14 and J35 have closely identical average mass, thickness, and sound velocity, and thus are ideal for comparison. The reduction in thermal conductivity in J35 compared to J14 has been comparable throughout the measured temperature range. Both the compounds show amorphous behaviour in terms of thermal transport despite exhibiting crystallinity, unlike the typical crystalline thermal conductivity (distinct Umklapp scattering trend αT⁻¹) observed for 2-cation oxides, like Cu_0.2Ni_0.8O, Zn_0.4Mg_0.6O, and Co_0.25Ni_0.75O. Further, the magnitude of thermal conductivity of J35 aligns with the amorphous J14 (a-J14), which displays a characteristic amorphous thermal conductivity over the temperature. The employed model of phonon scattering and estimated temperature variation of thermal conductivity using virtual crystal approximation (VCA) does not go in accordance with the observed experimental trend. The incapability of VCA to capture the trend of thermal conductivity of J14 and J35 could be described by the recent comprehension which states that when disorder reaches significant levels, non-propagating vibrational modes, i.e., diffusons, hold an extensive share of the modes contributing to thermal transport.^253,254 This lends proof to the conception that either diffusons or phonons with a short mean free path/propagons are the governing factors for thermal transport in these compounds. Otherwise, it could be concluded that, if VCA is valid, the impact of temperature independent Rayleigh scattering may be significant enough to suppress Umklapp scattering. Certainly, if the Umklapp scattering from the VCA model can be eliminated, the thermal conductivity as a variation of temperature can be closely approximated, demonstrating that Rayleigh scattering is the major phonon scattering mechanism. Moreover, albeit anharmonic phonon scattering can occur due to significant mass disorder,²⁵⁵ the determined thermal expansion coefficients from temperature variation of XRD for these ESOs are found to be comparable to the values obtained for the constituent oxides.²⁵⁶ This indicates that anharmonic scattering is not abnormally evident in ESOs. Therefore, the mechanism accountable for ultralow glass-like thermal conductivity in ESOs, particularly reducing thermal conductivity from 5- to 6-cation configurations, seems to be mainly attributed to interatomic force constant (IFC) disorder induced Rayleigh scattering. The VCA fitted models show that IFC disorder induced scattering is ∼2.8 times more dominant in J35 than J14. It is crucial to emphasize that J14 and J35 exhibit nearly alike average mass. Furthermore, the systematic enhancement in mass disorder among 6-component oxides by elevating the sixth cation's mass accelerates the reduction in thermal conductivity; however, this suppression is less than 20%, which is unable to explain the disparity detected with J14. Thus, the local ionic charge disorder induced pronounced IFC disorder is the primary factor behind the increased suppression in thermal conductivity which can only be identified upon investigating the local crystal structure.

Fig. 10 (a) Temperature dependent lattice thermal conductivity (κ_lat) of six oxides given in the legend. EXAFS for (b) J14 and (c) J35. The data and the fitting model are shown for both magnitude (top) and imaginary component of the real-space function χ(R), with respect to the radius from the Co absorber (R). (d) and (f) and (e) and (g) represent the local structural changes around the Co octahedra for J14 and J35 respectively.⁶⁶ The figure is reproduced with permission from ref. 66 © 2018, Wiley-VCH.

The employment of EXAFS to examine the local coordination surrounding Co absorbers reveals the alterations in J14 and J35. Fits to the phase-uncorrected, self-absorption-corrected magnitude and imaginary part of the real space function χ(R) are illustrated in Fig. 10b and c for J14 and J35. Each χ(R) pattern shows features consistent with a metal oxide system, where the initial peak corresponds to the scattering interactions between the absorber and atoms (1st coordination shell) and the second peak corresponds to the 2nd coordination shell. For J14, the coordination environment surrounding Co correlates with the experimentally observed lattice parameters a = 4.21 Å and c = 4.29 Å of the tetragonal unit cell (Fig. 10d). However, upon the addition of the 6th cation in J35, the absorber octahedron experienced a noteworthy transformation which no longer aligns with the lattice parameters as it exhibits a tetragonally compressed unit cell (a = 4.21 Å and c = 4.08 Å) (Fig. 10e). The half-scattering path lengths between the Co absorber and the 6th nearest oxygens reveal a significantly compressed octahedron having 4 planar and 2 axial oxygens with bond lengths 1.93 and 1.96 Å respectively. These EXAFS findings indicate the presence of a substantial strain in the 6-cation ESOs, resulting in the distortion of oxygen atoms from their ideal positions. This pronounced distortion assists the strong IFC disorder in J35 (Fig. 10g) when compared to J14 (Fig. 10f). Such atomic distortion disrupts the smooth propagation of phonons and scatters the heat carrying phonons heavily, explaining the observed lower thermal conductivity in J35 compared to that of J14. Further, these compounds exhibit a relatively high elastic modulus. This exceptional combination of properties is feasible due to the presence of highly disordered interatomic forces originating from charge disorder among bonds. These exceptional thermophysical properties of ESOs contribute to their potential applications in thermoelectrics and thermal barrier coatings.

Zhang et al., on the other hand, recently have successfully synthesized and investigated the thermoelectric properties of novel perovskite-type high-entropy ceramics: (Ca_0.25Sr_0.25Ba_0.25La_0.25)TiO₃ and (Ca_0.25Sr_0.25Ba_0.25Ce_0.25)TiO₃.⁶⁷ The Raman spectrum was employed to detect structural disorder induced variations in local symmetry interactions. According to the selection rule, the first order Raman-active vibrations should not get detected in an ideal cubic perovskite structure as for SrTiO₃-based compounds.²⁵⁷ However, the presence of first-order peaks at 112, 543, and 795 cm⁻¹ can be ascribed to TO₁, TO₄ (transverse optical) and LO₄ (longitudinal optical) phonon modes, respectively.²⁵⁸ These modes indicate the broken centre of symmetry caused by the localized disorder. The reduced symmetry most probably originates from the coexistence of several elements at the A-site, forming a chemically disordered lattice, which enhances lattice anharmonicity. The wavenumbers <200 cm⁻¹ are indicative of vibrations associated with the A–O bonds, encompassing those that include Ca/Sr/Ba/La/Ce–O bonds (where A = Ca/Sr/Ba/La/Ce). The band at 543 cm⁻¹ is highly sensitive to crystal structure and mainly governed by the O atoms of the TiO₆ octahedra.²⁵⁹ This analysis underlines the enhanced disorder resulting from lattice distortion and strain effects, which arise from the existence of multiple elements at the same cationic site and the local surroundings of Ti⁴⁺ in high entropy ceramics. Further, the introduction of multiple elements with different mass and size into the same lattice sites certainly creates local stress fields, advancing the scattering broad spectrum of phonons. As a result, (Ca_0.25Sr_0.25Ba_0.25La_0.25)TiO₃ exhibits a minimum κ_lat of ∼2.26 W m⁻¹ K⁻¹ which is substantially lower in comparison to the perovskite oxides with a zT of ∼0.18 at 1073 K.

5.4 Emphanisis

Structural transition generally involves evolution from a lower symmetric ordered phase to a higher symmetric disordered phase on warming. However, recently it has been observed that a counter-intuitive phenomenon, i.e., a symmetry lowering transition upon warming, termed as emphanisis, can also occur.⁴² Emphanisis was first observed in PbTe and PbS, both of which possess a global rocksalt cubic lattice. However, when the local structure was probed using synchrotron X-PDF, it was revealed that the system undergoes a symmetry lowering transition at 100 K, with Pb being off-centered from its parent site at high temperatures.⁴² Using off-centered models, it was found that Pb tends to off-center along the crystallographic 〈100〉 direction (Fig. 11a) and the magnitude of off-centering increases with temperature, indicating a transition from a more symmetric to a less symmetric structure, thereby exhibiting emphanitic behavior (Fig. 11b).⁴² The origin of this emphanisis stems from the 6s² lone pair of electrons on Pb, which stereochemically expresses itself, and drives Pb to remain in an off-centered position to reduce the electrostatic repulsion between the lone pairs and bond pairs. This observation has come under heavy introspection over the past decade. Inelastic neutron scattering (INS) on PbTe shows formation of a new TO mode above 100 K, indicative of a broken symmetry in PbTe²⁶⁰ along with the other highly anharmonic TO phonon modes that can be ascribed to the locally undistorted structure.²⁶¹ However, investigation into the local structure using EXAFS revealed no such Pb off-centering and the intrinsically low κ_lat in PbTe is mainly due to thermally induced disorder.²⁶² Recently, through single crystal X-ray diffuse scattering in PbTe, Sangiorgio et al. have observed correlated dipole formation which is associated with the reduction in the local symmetry due to Pb off-centering which supports emphanisis.²⁶³ Holm et al., while studying the temperature dependent (30–622 K) 3D-ΔPDF, observed an increase in the antiphase displacement with the increase in temperature, thereby further confirming the occurrence of emphanisis in PbTe.^264,265 This local off-centering is associated with the softening of the TO modes, which in turn explains the unusual low κ_lat in an otherwise globally high symmetric structure. Similar observations of emphanisis in simple binary compounds are also made in PbS,⁴² PbSe,⁴³ SnTe,⁴⁴ GeTe,²⁶⁶etc., all of which show intrinsically low κ_lat.

Fig. 11 (a) Rock-salt structure PbTe where Pb is undistorted (top) and distorted along 〈100〉 by a distance Δr (bottom).⁴² (b) Pb displacement as a function of temperature. Pb displacement increases with increasing temperature, exhibiting emphanisis.⁴² (c) Evolution of the first peak in (SnSe)_0.5(AgSbSe₂)_0.5 with temperature. The peak becomes more asymmetric as the temperature increases.³⁸ (d) Magnitude of Se off-centering along the 〈100〉 direction with temperature.³⁸ (e) Local environment of Ag in AgGaTe₂ (left) and direction of local distortion of Ag due to weak sd³ hybridization and correlated distortion of Te (marked by arrows) viewed from the (00l) direction.⁵⁴ (f) Temperature dependent displacement of Ag and Te. The dashed lines indicate the linear extrapolation at low temperatures.⁵⁴ (g) Ni₂Se₂ tetrahedra of KNi₂Se₂ at T = 300 K. Splitting of the Ni ions from the high symmetry 4d site to a lower symmetry 8g site is necessary to account for the electron density from synchrotron powder X-ray diffraction (SXRD). On the right side, Fourier difference maps between experimental data at T = 300 K and split site (top right) and single site (bottom right) models are shown. For a single site, a clear electron density is observed away from Ni, indicating its preference to split and remain in 8g sites.²⁶⁸ Fig. (a) and (b) are reproduced with permission from ref. 42 © 2010, AAAS. Fig. (c) and (d) are reproduced with permission from ref. 38 © 2021, American Chemical Society. Fig. (e)–(g) are reproduced with permission from ref. 54 © 2022 and ref. 268 © 2022, Wiley VCH.

Halide perovskites of the formula ABX₃ (A⁺ = Cs, CH₃NH₃ (MA) or CH(NH₂)₂ (FA); B²⁺ = Sn or Pb; X⁻ = Br or I) in their cubic symmetry also display the emphanitic phenomenon.^91,267 Inorganic halide perovskite CsSnBr₃, crystallized in the cubic Pm3̄m space group, consists of an A site cation, i.e., Cs⁺, residing in the void formed by the edge sharing SnBr₆ octahedra. X-PDF indicates peak asymmetry in the 300–420 K range, which can be accounted using the locally displaced Sn²⁺ cation along the 〈111〉 direction. This displacement is found to increase with temperature resembling emphanisis. Ab initio studies revealed the presence of stereochemically active lone pair of electrons in Sn²⁺ which drives the local distortion in an otherwise globally periodic CsSnBr₃.⁹¹ Hybrid perovskites, namely MABX₃ and FABX₃, also exhibit similar dynamic off-centering wherein the preferential arrangement of the large central cation (i.e., MA and FA) coupled with the stereochemically active lone pair of electrons on Sn²⁺ and Pb²⁺ drives the local displacement in the Sn/Pb–Br₆/I₆ octahedra.²⁶⁷

Recently, globally rocksalt cubic (SnSe)_0.5(AgSbSe₂)_0.5 was found to exhibit intrinsically ultralow κ_lat in the temperature range of 295–725 K. Total structure investigation using synchrotron X-PDF revealed asymmetry in the first peak signifying a locally distorted octahedron. This asymmetry is found to increase with increasing temperature (Fig. 11c).³⁸ The reduction in the local symmetry is found to be because of off-centering of Se along the 〈111〉 direction within an average rock-salt structure. This off-centering of Se is found to be temperature dependent, and increases with increasing temperature, thereby resembling emphanisis (Fig. 11d). Se off-centering creates a local aperiodicity in the bonding with 3 short and 3 long bonds in the octahedra, thereby impeding the phonon flow. Theoretical phonon dispersion further revealed that this local off-centering is associated with the damping of the lowest energy phonon mode at S and Z points of the Brillouin zone, predominantly involving vibrations from Se atoms. The phonon damping caused by the locally off-centered Se atoms is responsible for the intrinsically ultralow κ_lat of 0.49–0.39 W m⁻¹ K⁻¹ in the temperature range of 295–725 K, which aids in improving the thermoelectric performance, with a high zT of ∼1.05 in 6 mol% Ge doped (SnSe)_0.5(AgSb_1−xGe_xSe₂)_0.5 at 706 K.³⁸

Generally, emphanisis is mainly mediated by the stereochemical activity of lone pair of electrons. However, recently, locally broken symmetry in AgGaTe₂ was observed where the Ag atoms tend to remain in an off-centered position (Fig. 11e).⁵⁴ This is unusual given that Ag does not possess any stereochemically active lone pair or reside in large voids. The origin of the local off-centering of Ag is ascribed to the large energy difference between the valence s and d electronic orbitals leading to weak sd³ hybridization. Low energy optical phonons manifested from this local off-centering couple with the heat carrying acoustic phonons resulting in ultralow κ_lat in AgGaTe₂.⁵⁴ Similar to PbTe and (SnSe)_0.5(AgSbSe₂)_0.5, the magnitude of Ag off-centering increases with the increase in temperature (Fig. 11f) and this off-centering results in a strong acoustic optical phonon scattering, thereby reducing κ_lat to 0.26 W m⁻¹ K⁻¹ at 850 K.⁵⁴ Similar local off-centering of Ag was observed in AgMnSbTe₃,⁵⁵ and here also it is possibly due to similar weak sd³ hybridization as observed in AgGaTe₂. KNi₂Se₂ is another example where emphanisis is observed not due to stereochemical expression of lone pairs but due to the spontaneous formation of Ni–Ni bonds. At low temperatures (<10 K) KNi₂Se₂ resides in a highly symmetric structure (of ThCr₂Si₂-type) with Ni occupying a distinct crystallographic 4d Wyckoff site (0, 0.5, 0.25). However, upon warming, Ni off-centers from the 4d site to split into two 8g sites with 50% occupancy, suggestive of lowering of symmetry (Fig. 11g). Further analysis using EXAFS, Raman spectroscopy and X-PDF all pointed to the symmetry lowering transformation in KNi₂Se₂ with increasing temperature thereby exhibiting emphanitic behavior.²⁶⁸ Although the phenomenon is relatively new, given the plethora of compounds currently showing emphanisis, it is expected to be more ubiquitous than currently realized, highlighting the importance of local structural analysis in functional materials for a more accurate structure–property description.

5.5 Splitting of atomic sites

Recently, there have been very few studies conducted to explore atomic tunnelling as a plausible explanation for glassy thermal transport in solids at low temperature.^69,204 Through neutron scattering techniques, however, it is not feasible to directly investigate compounds consisting of heavy atoms undergoing tunnelling. This is due to their extremely low tunnelling splitting energies (sub-GHz).²⁶⁹ These energies are very low to contribute to thermal conduction, unless the temperature drops to ≤1 K. A particle in a double or multi-well potential, resulting from a two-level system created by an atom located at different crystallographic positions (referred to as site-splitting), has the potential to exhibit tunnelling characteristics.⁶⁹ For example, such atomic tunnelling can be observed in ammonia's ‘umbrella inversion’.²⁷⁰ In solids, with atoms heavier than hydrogen, it is an extremely rare phenomenon and has not been evidently identified using neutron scattering techniques.²⁶⁹ These kinds of systems exhibiting tunnelling can scatter heat carrying phonons and have been employed to interpret thermal transport mechanisms of solids at very low temperatures.^{204,271–273}

Recently Sun et al. have demonstrated glass-like thermal conductivity in single crystalline hexagonal perovskite BaTiS₃ (space group: P6₃mc) despite exhibiting a highly symmetric and simple primitive cell.⁶⁹ Ti in BaTiS₃ is surrounded by face-sharing octahedra of S atoms, leading to a chain like structure along the c-axis (Fig. 12a). The thermal conductivity of BaTiS₃ is found to be 0.39 W m⁻¹ K⁻¹ at room temperature. Further, the temperature variation of thermal conductivity follows the typical glassy behaviour despite having single crystalline nature. To investigate the local crystal structure of BaTiS₃, neutron powder diffraction and neutron PDF have been performed where a substantial peak anisotropy can be observed in the negative peaks of Ti–S (Fig. 12b). Here mention must be made that, Ti exhibits negative neutron scattering length; thus Ti–S and Ti–Ba peaks appear in the negative region in PDF. The intensity in the pair–pair correlation peaks decreases with temperature for both Ba–S and S–S which is the typical nature of most of the compounds due to thermal motion. However, surprisingly this is not the case for Ti–S and Ti–Ba peaks in BaTiS₃. Lack of temperature dependency in the intensity of these peaks indicates the complex crystallographic nature of the Ti atom in BaTiS₃. The PDFs were refined using both P6₃/mmc and P6₃mc structures; yet the choice of structure did not alter the resulting local structure pair distribution. The anisotropic displacements of Ti are extended along the c-direction exhibiting a large atomic displacement parameter Ti–U₃₃ = 0.2(0.07) Ų at 100 K, as compared to that of Ti–U₂₂ = 0.03(0.005) Ų (Fig. 12c). Similar elongation was also observed for S atoms with S–U₃₃ = 0.07(0.01) Ų as compared to S–U₂₂ = 0.014(0.002) Ų. However, upon heating, these large values of U₃₃ reduce to 0.15(0.05) Ų (Ti) and 0.055(0.01) (S) Ų respectively. This nature is quite contradictory to what is anticipated on the basis of thermal vibrations. Neither in terms of the magnitude of thermal displacement (of Ba atoms) nor its temperature dependency, such a behaviour has been noticed. Along the c-direction, splitting the position of the Ti atom into two sites on the other hand, separated by 0.15 Å, improves the fitting significantly at 100 K (inset of Fig. 12c). This tendency of Ti sites to develop bimodal distributions upon cooling suggests that Ti atoms exist in shallow double-well potentials.

Fig. 12 (a) Crystal structure of BaTiS₃. Blue, orange and green spheres denote Ba, S and Ti atoms, respectively. (b) PDF analysis for BaTiS₃ at several temperatures (upper panel) and a breakdown of the PDF by atomic pairs (all pairs, Ti–S, Ti–Ba, Ti–Ti, Ba–S, Ba–Ba and S–S are denoted by red, blue, violet, cyan, green, chartreuse and black colour respectively) from the refinement at 100 K (lower panel). (c) Temperature dependent ADP of different elements. (d) High energy resolution INS spectra measured at 2.4 K for BaTiS₃.⁶⁹ The figure is reproduced with permission from ref. 69 © 2020, Nature Publishing Group.

The site splitting is further evident from the inelastic neutron scattering measurements performed on BaTiS₃ powders. At 2.4 K the appearance of sharp low-energy excitation at 0.46 meV (Fig. 12d) corresponds to a separation distance which is close to 0.5 times the maximum extent of the thermal ellipsoid for Ti as obtained from PDF analysis. The tunnelling offers a clarification for the temperature variation of U₃₃ of Ti. At elevated temperatures, Ti gets the freedom to stay at all allowed positions within a harmonic potential; thus, a population distribution is formed centred around the equilibrium position. On cooling, the occupation of the ground and first excited tunnelling states becomes dominant. Therefore, the probability of finding the atom gets divided between the two off-centered positions, which causes a significant ADP than at the average position (center of the well). Such a phenomenon may arise from a dynamic instability resulting from the formation of a double-well potential.^274,275

A similar observation has been made for CsAg₅Te_3−xS_x as demonstrated by Hodges et al.²⁷⁶ CsAg₅Te₃ exhibiting a complex 3D structure shows high thermoelectric performance with zT ∼ 1.5 at 730 K.²⁷⁷ On the other hand, when Te is substituted by S, the material formed, i.e., CsAg₅TeS₂ (space group: P4/mmm), crystallizes in a 2D structure having lattice parameters a = 4.3160(3) Å and c = 11.2486(8) Å. Layers of Cs⁺ and [Ag₅TeS₂]⁻ are alternatively stacked in CsAg₅TeS₂. The two different chalcogen atoms, Te and S, in CsAg₅TeS₂ occupy dissimilar sites, whereas Te cuboctahedrally coordinates with Ag as it binds to 4 and 8 atoms of Ag1 and Ag2 atom types respectively. The S atoms are attached to 4 Ag1 type of atoms. Mention must be made that the [Ag₅TeS₂]⁻ layer exhibits two distinct Ag sites: Ag1 atoms are coordinated to Te in a square-planar fashion, while on either side of this net there exists a layer of Ag2 atoms, followed by another layer of S atoms that ends the slab. X-PDF analysis of CsAg₅TeS₂ shows that the medium-range ordering (2.0–10 Å) is in accordance with the structure obtained from the single crystal structural determination (space group: P4/mmm) with goodness of fit (R_w) ∼ 20%. However, the asymmetry can be observed at shorter distance ∼2.5 Å of the main peak at ∼3 Å, for which the previous model does not work. This was resolved by changing the space group to I4/mcm where the heteroleptic Ag2 atom distorts from its ideal tetrahedral position towards a trigonal geometry. This leads to the change in the Ag1–S bond distance to ∼2.54 Å which is in good agreement with the shoulder at ∼2.5 Å. More fascinatingly, at both sides of the perfect tetrahedral site, Ag2 is disordered between two equivalent positions. For the Ag2 atom, the rate of ADP elevation is noticeably higher compared to Cs, Ag1, Te, and S, delivering indication for dynamic disorder.^278,279 The atom-projected phonon density of states further supports the dynamic disorder or rattling like vibration of Ag which induces low frequency optical phonon modes. As a result, the value of κ_lat stays as ultralow as ∼0.36–0.40 and 0.25–0.32 W m⁻¹ K⁻¹ along the perpendicular and parallel directions to the spark plasma sintering (SPS) pressing direction respectively in the temperature range 300–800 K. Further, this kind of atomic site splitting induced dynamic disorder or rattling has been also investigated in the past for clathrates like (Sr/Ba)–Ga–Ge^280,281 and sulphides like Cu₄SnS₄,²⁸² in which the thermal conductivity was found to be extremely low. The role of such rattler atoms in impeding the passage of phonons has been further elaborated in the next section.

5.6 Rattling of atoms

Crystalline solids supporting extremely low and glass-like κ_lat, along with the unhindered flow of electrons, are ideal for thermoelectrics. Such crystals are referred to as PGEC.²⁸³ Following the work on PGEC, Slack et al. have demonstrated that clathrate compounds are potential candidates as efficient thermoelectric materials.^283,284 The origin of high thermoelectric performance in these materials has been attributed to the encapsulated ‘rattlers’ in them. When a weakly constrained atom inside an enormous covalently connected cage vibrates with a substantially bigger amplitude than the cage-forming atoms with thermal perturbation, the term ‘rattler’ is employed while characterizing that atom. These guest rattler atoms in filled skutterudites and clathrates remain weakly bonded, ensuring their large amplitude of vibration.^23,285–287 The vibration of the rattler atoms itself does not act as a scattering center. Instead, it is the avoided crossing between the acoustic phonon and nearly flat soft optical phonon modes, induced by the vibration of the guest atom, that suppresses the acoustic modes further. These low-energy optical phonon modes confine the phonon group velocity, leading to low κ_lat.²³ Additionally, these rattling modes are found to be strongly anharmonic and suppress the phonon relaxation time significantly.²⁸⁸ Consequently, lowering of κ_lat has been observed for Ba₈Cu₁₄Ge₆P₂₆ (∼0.77 W m⁻¹ K⁻¹ at 300 K),²⁸⁹ Ba₈Ga₁₆Ge₃₀ (∼1.3 W m⁻¹ K⁻¹ at 300 K),²⁹⁰ Ba₂Zn₅Sb₆ (∼0.85 W m⁻¹ K⁻¹ at 300 K)²⁹¹ and Ce(CoFe₃)Co₁₂ (∼1.2 W m⁻¹ K⁻¹ at 300 K).²⁰⁷ Recent studies further show that certain materials exhibit the rattling mode feature even in the absence of an obvious cage, especially those that have at least one univalent component, such as TlInTe₂,²⁰ TlSe,²⁶ InTe,²⁹² Na_1−xCoO₃,²⁹³ Cu_1.6Bi_4.8S₈,²⁴ CsAg₅Te₃,²⁷⁷ Cs₂SnI₆,²² Cu₄Bi₄Se₉,²⁹⁴ AgBi₃S₅,²⁹⁵etc. A fundamental comprehension of the interaction between low energy phonon modes and the local structural evolution of these compounds with temperature is indispensable for rationalizing the role of rattler atoms in lowering κ_lat.²⁹⁶

In this context, Zintl compounds offer a fascinating example where the rattling of a weakly bonded atom that is connected to a covalently bonded cage-like polyanionic substructure strongly suppresses κ_lat (Fig. 13a).^20,25 TlInTe₂, a Zintl type compound, has weak (ionic) and strong (covalent) substructures within it.^20,25 Under ambient conditions, TlInTe₂ has a body-centered tetragonal structure with I4/mcm space group, which consists of tetrahedral anionic [InTe₂]_nⁿ⁻ 1D chains that extend along the c-direction (inset of Fig. 13a). At the same time, Tl⁺ ions remain surrounded by eight Te atoms in a slightly deformed square-antiprismatic configuration, often termed as a Thompson cage. Interestingly the In–Te bond distance (2.82 Å) in TlInTe₂ indicates the presence of a strong covalent anionic framework whereas the Tl–Te bond (3.59 Å) is ionic in nature. Furthermore, the volume of the cage is ∼3.75 times larger than that of a Tl⁺ ion, suggesting that there is more space for the Tl⁺ ion to vibrate from its mean equilibrium position. Such a bonding heterogeneity and rattling of Tl⁺ ions have been observed experimentally through the local structure analysis using synchrotron X-PDF analysis. The peaks in the synchrotron X-PDF data of TlInTe₂ at 300 K (Fig. 13b), fitted using I4/mcm space group below 5 Å, provide information about the local structure, whereas the peaks at higher r define the average structure of TlInTe₂. The first three peaks correspond to In–Te bonding in anionic [InTe₂]_nⁿ⁻ chains, Tl–Te bonding and nearest neighbor In–Tl interactions, respectively (Fig. 13c). The change in the temperature dependent peak intensity demonstrates that among these first few peaks, the intensities of II and III peaks decrease much faster than that of I. This indicates a weaker interaction of Tl with the rest of the lattice, while the In–Te bond is strongly covalent in nature. As a result, the lattice exhibits considerable bonding heterogeneity, which enhances the phonon scattering and lowers κ_lat. Also, it is evident from the ADP analysis that Tl atoms vibrate to a greater extent than In and Te atoms (Fig. 13d). This finding provides more evidence for the rattling behaviour of Tl⁺ ions in TlInTe₂. Rattling of these Tl⁺ ions induces several low energy optical modes observed through inelastic neutron scattering (INS) measurements. These Einstein rattlers disrupt the smooth flow of the acoustic phonon modes, resulting in a low κ_lat of ∼0.46–0.31 W m⁻¹ K⁻¹ in the 300–673 K temperature range.²⁰

Fig. 13 (a) Temperature dependent lattice thermal conductivity (κ_lat) of TlInTe₂.²⁵ (Inset) The structure of TlInTe₂ viewed along the crystallographic c axis. (b) Fitting of synchrotron X-ray PDF data (at 300 K) using the I4/mcm space group of TlInTe₂.²⁰ (c) Temperature variation of G(r) for the first few peaks of TlInTe₂.²⁰ (d) Temperature dependent ADP for Tl, In and Te in TlInTe₂ along the crystallographic c-direction (U₃₃). (e) X-ray PDF of Cs₂Sn_1−xTe_xI₆ fitted using the vacancy-ordered double perovskite cubic structure with isotropic, harmonic ADP. The data, fitting and difference are represented by black circles, coloured lines, and grey colour respectively. The left panel demonstrates the low-r pair correlations, specifically the asymmetry at r ∼ 4.1 Å. The goodness of fit (R_wp) for each composition is shown as well.²⁹⁹ (f) Crystal Structure of Cs₂SnI₆.²⁹⁹ Fig. (a) is reproduced with permission from ref. 25 © 2017, American Chemical Society. Fig. (a)–(d) are reproduced with permission from ref. 20 © 2021, Wiley VCH. Fig. (e) and (f) are reproduced with permission from ref. 299 © 2018, Royal Society of Chemistry.

Cs₂SnI₆, a vacancy-ordered double perovskite, has recently been observed to exhibit an ultralow κ_lat of ∼0.29–0.22 W m⁻¹ K⁻¹ in the 296–423 K temperature limit.²² The isolated octahedral units [SnI₆]²⁻ in Cs₂SnI₆ remain ionically bonded to the Cs⁺ cations in the face-centered cubic framework.^297,298 The lack of polyhedral connectivity in this vacancy-ordered double perovskite provides extra degrees of dynamic freedom, making it a perfect structural framework for studying structure–property correlation. The anharmonic rattling nature of Cs⁺ cations inside the cuboctahedral void formed by [SnI₆]²⁻, together with SnI₆ octahedral rotation, generates several low-frequency localized optical phonon modes which causes significant acoustic phonon dampening to lower the phonon group velocity and leads to low κ_lat.²² The local bonding environment study of Cs₂Sn_1−xTe_xI₆ (x = 0–1) by Maughan et al. using synchrotron X-PDF analysis reveals asymmetry in the Cs–I/I–I pair interactions (r ∼ 4.1 Å), which gradually diminishes as the tellurium content increases (Fig. 13e), in line with the almost constant decline in R_wp.²⁹⁹ The temperature-dependent total neutron scattering implies that the asymmetry in X-PDF is caused by the anharmonic displacement of Cs⁺ and octahedral tilting (Fig. 13f). Furthermore, bond valence sum analysis indicates that anharmonicity is reduced in the case of Cs₂TeI₆ as the size of the cuboctahedral void satisfies the bonding preferences of Cs⁺, whereas it is dissatisfied in the case of Cs₂SnI₆.²⁹⁹ Such low κ_lat is critical for thermoelectric energy conversion, as demonstrated in CsBi₄Te₆, with a peak zT of ∼0.8 at 225 K achieved after SbI₃ doping.³⁰⁰ Similarly, intrinsic rattling induced ultralow κ_lat resulted in remarkable zT in other notable compounds, e.g., KCu₅Se₃ (∼1.3 at 950),³⁰¹ CsAg₅Te₃ (∼1.5 at 727 K),²⁷⁷ Yb₁₄MnSb₁₁ (∼1 at 1223 K),³⁰² AgBi₃S₅ (∼1 at 800 K),²⁹⁵ Na_2.19Ga_2.19Sn_3.81 (∼1.20 at 387 K),³⁰³ AgGaTe₂ (∼1 at 873 K),³⁰⁴etc.

5.7 Liquid-like ion diffusion

The notion of ‘phonon-liquid electron-crystal (PLEC)’, which emerged from PGEC, states that κ_lat can be further reduced even below the glass limit in superionic materials.^27,28,30 The uniqueness of the ‘part-solid part-liquid’ hybrid feature of such systems is that the liquid-like ions are loosely bound and disordered within the rigid sublattice, which can strongly scatter lattice phonons and shorten the phonon mean free path. The liquid-like ions, on the other hand, can hop sequentially from one equilibrium position to another, exhibiting ‘fluid’ behaviour, which may restrict the propagation of transverse waves, as a fluid cannot withstand shear stress. This leads to a diminution of specific heat, C_v, from 3N_Ak_B to 2N_Ak_B (Dulong-Petit limit for solid, N_A and k_B denote Avogadro number and Boltzmann constant respectively) and eventually κ_lat decreases. So far, a number of cutting-edge superionic TE materials have been identified and developed with exceptionally low κ_lat and superior TE performance, with the maximum zT_s reaching 2.0 or above, e.g., AgCrSe₂,²⁸ CuCrSe₂,^305,306 InTe,^140,292 Ag₂(S, Se, Te),^307–312 Cu_2−δ(S, Se, Te),^{27,29,313–321} Cu or Ag-based argyrodites,^322–326 AgCuTe,³²⁷etc. Experimental characterization through X-ray or neutron PDF analysis to probe the subtle structural disorders and atomic dynamics provides a basis for understanding ultralow thermal conductivity in such PLECs.

Liquid-like properties are also exhibited by a crystalline solid like AgCrSe₂,^328–331 which is made up of alternating layers of Ag and CrSe₆.²⁸ At low temperatures, all Ag ions occupy the tetrahedral α sites, which are coordinated by Se atoms, and the structure adopts the space group R3m. The Ag ions become redistributed and disordered across tetrahedral α sites and the symmetry equivalent β sites above an order–disorder transition temperature of 475 K (superionic transition), causing the space group to change to R3̄m (Fig. 14a).^328,330 To track the temperature-dependent local structural evolution, Li et al. performed temperature-dependent X-PDF analysis, followed by G^X(r) analysis (Fig. 14a).²⁸ Within the ordered crystal model, the closest neighbouring Cr–Se, Ag–Se, and Ag–Cr correlations are superposed at the first peak (∼2.5 Å). The nearest neighbouring Cr–Cr and Se–Se correlations mainly contribute to the second peak. The occupational disorder causes the uniform nearest neighbouring Ag–Ag distance to split into three sets, and the subsequent nearest neighbouring Ag-related correlations also successively become diverse. The partial X-PDF of Ag-related correlations, which is displayed in the lower portion of Fig. 14b, indicates that Ag-related peaks at 4.5, 13, and 19 Å have significantly higher susceptibility to temperature changes. The integrated intensity comparison (Fig. 14c) of the peak at 4.5 Å (Ag-correlation-rich) with the 3.5 Å peak (Ag-correlation-poor) displays a clearly defined critical-like behaviour. The decrease in intensity implies that Ag-related pairs slowly lose their coordination as they approach the transition. Fitting of the experimental G^X(r) at 623 K with the R3m crystal model shows that Ag-related pairs are found to be associated with discrepancies (Fig. 14d). Since the real disorder effect reduces coordination, the model generates a higher intensity. The temperature-dependent goodness of fit, R_w, between experimental data and this model in Fig. 14e shows a lower value of ∼0.14 near room temperature but tends to reach saturation following a sharp increase near the transition temperature (T_c). All of these observations indicate that Ag has a full occupational disorder above T_c. According to density–functional–perturbation-theory (DFPT) calculations, Ag motions alone are dominant in low-frequency intense transverse acoustic (TA) phonons. Phonons are considerably attenuated with increasing temperature and over T_c. The longitudinal acoustic (LA) mode persists over the transition temperature, while the ultrafast atomic fluctuations suppress the transverse vibrations of the Ag atoms. Furthermore, a much shorter relaxation time of disordered Ag⁺ ions (∼0.4 ps) compared to the TA phonon timescale (∼1.2 ps) implies that these transverse vibrations are unable to last.²⁸ The heat flow is significantly suppressed in the absence of TA modes, resulting in a low κ_lat of ∼0.5 W m⁻¹ K⁻¹ at 523 K.^28,328 The (AgCrSe₂)_0.5(CuCrSe₂)_0.5 composites showed a superior zT ∼ 1.4 at 773 K because of the further lowering of the κ_lat (∼0.2 W m⁻¹ K⁻¹) value, caused by the multiscale hierarchical design.²⁸

Fig. 14 (a) Crystal structures of α-phase (ordered phase) and β-phase (superionic phase) of AgCrSe₂. (b) The X-ray PDF data of AgCrSe₂ obtained through X-ray scattering at various temperatures, extending up to 20 Å. Below this, there is a composite representation of partial PDF for pairs involving Ag (Ag–Ag, Ag–Cr, and Ag–Se), calculated based on the R3m crystal model. (c) The integrated intensity of PDF for the Ag-correlation-poor peak at 3.5 Å and the Ag-correlation-rich peak at 4.5 Å, identified by a circle and a square, respectively. (d). Real-space refinement of the PDF at 623 K based on the R3m crystal model. (e) The quality of this real-space refinement plotted against temperature. The vertical shaded bars in panels (b) and (d) emphasize positions where Ag-related correlations dominate. In panels (c) and (e), T_c is indicated by a dashed line.²⁸ Fig. (b)–(e) are reproduced with permission from ref. 28 © 2018, Nature Publishing Group.

Reduced thermal conductivity can be achieved through a highly effective mechanism involving diffusive, disordered, interstitial atoms with large thermal motions. Large complex crystal structures like Zn₄Sb₃³³² and oxide-ion conductors³³³ frequently exhibit interstitial, disordered atomic positions, but in small, simple crystal structures, they are rarely observed. Zhang et al. recently demonstrated ultralow κ_lat in single crystalline InTe with direct observation of one-dimensional interstitial, disordered, and diffusive In¹⁺ ions, noteworthy upon considering the simple and small lattice framework of InTe.¹⁴⁰ InTe is a mixed valent compound with the formula In¹⁺In³⁺Te₂²⁻, and it crystallizes in tetragonal symmetry having I4/mcm space group,^140,292 analogous to TlSe (Fig. 15a). The In³⁺ ions are tetrahedrally bonded to Te²⁻ constructing (InTe₂)⁻ chains along the c-axis, whereas the In¹⁺ ions with 5s² lone pair of electrons are weakly connected to eight Te atoms in a cage-like structure arranged in a square antiprismatic manner. A distinct characteristic of κ_lat is observed at low temperatures (Fig. 15b) between 25 and 80 K. Herein, extrinsic scattering plays a larger role causing nearly temperature-independent behaviour (∼T^−0.1). The temperature dependence of κ_lat (∼T^−0.69) deviates from T⁻¹ even at temperatures well above the Debye temperature (∼120 K), where the intrinsic phonon–phonon scattering was predicted to be dominant. Usually, such aberrant behaviour is a sign of extrinsic scattering brought on by the structural disorder. Electron density distribution analysis using the maximum entropy method (MEM) demonstrates the presence of two interstitial In sites (In_i(1) and In_i(2)) along the c-direction between the In¹⁺ atoms, indicating the presence of the 1D static disorder feature in InTe. Further structural analysis proves that the structural disorder is due to the presence of In¹⁺ vacancies and the formation of vacancy-mediated Frenkel pairs in InTe.¹⁴⁰ The local structural ordering of In¹⁺ vacancies and the presence of interstitial sites have been observed experimentally by diffuse X-ray scattering. Diffuse scattering measured at 25 K shows 2D planes for even values of l in the [0 0 l] plane (Fig. 15c), which indicates that there are strong correlations along the c-direction and poor correlations in all other directions. The planes vanish at 300 K and are replaced by more three-dimensional features that have strong maxima at the locations of the Bragg peaks. The local correlations in disordered crystals are displayed by the 3D-ΔPDF. Positive features in 3D-ΔPDF denote interatomic vectors (atom separations) that appear more often in the actual crystal structure than in the average structure. Similarly, negative values correspond to radial vectors that occur less frequently in the actual crystal structure as compared to the average crystal structure. At 25 K, 3D-ΔPDF demonstrates strong correlations along z, which weaken at 300 K (Fig. 15d), indicating that at low temperatures, vacancies and interstitials are strongly correlated along the 1D chains but not as much at higher temperatures. Temperature-dependent 3D-ΔPDF (25, 100 and 300 K) along z demonstrates (Fig. 15e) negative signals surrounding the positive peaks, representing the vectors in the actual structure that do not separate In atoms. The asymmetric negative signal surrounding positive peaks at low temperatures is typically an indication of local relaxations caused by atoms moving slightly closer to nearby vacancies. As shown in Fig. 15f, a positive peak is detected at 5.8 Å at 25 K, which corresponds to a vector separating an In_i(1) and In_i(2) interstitial, implying that the interstitial Frenkel pairs are typically separated by this gap. The asymmetry in the negative signal and the Frenkel pair peak vanished at 300 K, suggesting a lower correlation between the positions of vacancies and interstitials at high temperatures, indicating 1D In¹⁺-ion diffusion/hopping behaviour.¹⁴⁰ In¹⁺ ion trajectories calculated from molecular dynamics simulation at 700 K exhibit a clear hopping behaviour, similar to the experimental evidence. Nudged elastic band calculations further exhibit a much lower migration barrier for In¹⁺ along the [001] direction compared to the [110] and [100] directions, indicating the presence of a 1D diffusion channel along the c-direction. In addition to explaining the observed superionic conductivity, the direct visualization of a 1D disordered diffusion channel offers a framework for comprehending ultralow κ_lat and its unusual and weak temperature dependence in InTe.¹⁴⁰ However, at room and high temperature, In⁺ behaves as an intrinsic rattler probably between the different vacancy sites which enables superionic type conduction and ultralow κ_lat.²⁹² Owing to such ultralow κ_lat, a peak zT value of ∼1.05 at 790 K was achieved through Pb doping in InTe.³³⁴

Fig. 15 (a) Crystal structure of InTe with possible migration pathways of In¹⁺ along [001], [110] and [100] directions. (b) Temperature dependent thermal conductivity of single crystalline InTe.^140,447,448 (c) Diffuse X-ray scattering measured in the (HHL) plane at 25 and 300 K for InTe. 3D-ΔPDF in the x0z plane is presented in (d) at 25 and 300 K. A closer examination of features in the 3D-ΔPDF along the z-axis is shown in (e) at 25, 100, and 300 K. (f) An illustration of the 1D chain with the characteristic distance between interstitial sites, corresponding to the positive peak identified in (e) by a black arrow.¹⁴⁰ The figure is reproduced with permission from ref. 140 © 2021, Nature Publishing Group.

5.8 Local atomic ordering

In compounds with positionally disordered atoms in their lattice, it has been observed that some of these compounds contain nanoscale regions wherein the disordered atoms occupy preferential sites, thereby forming an ordered sub-region.³³⁵ This local or short-range ordering has recently been classified as another degree of freedom for achieving better thermoelectric performance. In bulk polycrystalline AgSbTe₂, Ag and Sb inherently occupy the same Wyckoff site in the rocksalt framework, resulting in a disordered lattice.^216,336 Ma et al. showed that formation of a nanoscale superstructure leads to glass-like thermal conductivity in single crystalline off-stoichiometric AgSbTe₂.³³⁷ Although initially it was thought that the intrinsically low κ_lat originates only from the lattice anharmonicity, detailed high-resolution TEM study with INS data analysis confirmed that the scattering of phonons was mainly caused by the spontaneously formed nano-scale superstructure,³³⁷ which was formed by local cation ordering evidenced later.^70,338 This ultralow like κ_lat with a room temperature value of ∼0.6 W m⁻¹ K⁻¹ is the key to achieving an excellent zT of ∼1 at 450 K in the AgSbTe₂ system.³³⁹ Although nanoscale superstructure formation has been observed in single crystalline samples,³³⁹ it remains elusive in the polycrystalline counterpart. Roychowdhury et al. recently showed enhanced local cation ordering in Cd-doped polycrystalline AgSbTe₂.⁷⁰ The intermediate-sized Cd²⁺ reduces the potential fluctuation caused by significant charge and size differences between Sb³⁺ and Ag⁺ ions located in the octahedral sites of AgSbTe₂ within the rock salt lattice and thus promotes cation ordering. Theoretical calculations indicated that Cd prefers to occupy disordered Sb sites, which reduces the formation energy of ordered cationic configurations and favours their occurrence.³³⁸ High-resolution scanning TEM with high-angle annular dark field (HRSTEM-HAADF) images of Cd-doped AgSbTe₂ (Fig. 16a) demonstrated the enhanced cation ordering induced nanoscale superstructure domain formation with ∼2–4 nm size.⁷⁰ Selected area electron diffraction (SAED) patterns display the presence of low-intensity superlattice spots exactly halfway between the fundamental diffraction spots (Fig. 16b). Following Cd doping-induced cation ordering, these spots developed as a result of cell doubling. Such cation ordering has a consequence in enhancing electrical transport by delocalizing the electronic states with a concomitant shift of the Fermi level, which is far away from the mobility edge. Furthermore, the presence of the different ordered structure induces significant lattice strain, and such strain ripples have been observed experimentally using inverse fast Fourier transformation (IFFT). κ_lat simultaneously decreased to an ultralow value of ∼0.16 W m⁻¹ K⁻¹ (diffuson limit of thermal conductivity, κ_diff, ∼0.19 W m⁻¹ K⁻¹) for 6 mol% Cd-doped polycrystalline AgSbTe₂ by the formation of the nanoscale superstructure domains and the resulting lattice strain.⁷⁰ As a result of significantly reduced κ_lat and simultaneously enhanced power factor owing to the local cation ordering and concomitant formation of the nanoscale superstructure, an ultra-high zT of ∼2.6 (Fig. 16c) is achieved at 573 K in 6 mol% Cd-doped AgSbTe₂. Additionally, a high conversion efficiency of ∼9.8% at a temperature gradient of 300 K illustrates the potential of local cation ordering to produce thermoelectric materials with superior performance.⁷⁰ Hg³⁴⁰ or Yb³⁴¹ doping in AgSbTe₂ in a similar way curtails the structural disorder partially, and results in a maximum zT of ∼2.4 at ∼570 K. Interestingly, isovalent Yb³⁺ doping in the place of Sb³⁺ in AgSbTe₂ provides enhanced atomic ordering that induces PGEC-like thermoelectric transport by enlarging the gap between phonon's and charge carrier's mean free path. Enhanced carrier mobility through the delocalization of electronic states via Yb³⁺ in place of Sb³⁺ keeps the crystal-like electrical transport in Yb-doped AgSbTe₂ and facilitates significant enhancement of the power factor and zT.³⁴¹ Recently, Pathak et al. demonstrated that removal of disordered Ag-atoms from the AgSbTe₂ lattice resulted in a similar local cationic ordering.⁷² The order–disorder optimization aided to achieve a zT of ∼2.3 at 573 K in Ag_0.98SbTe₂.⁷²

Fig. 16 Enhanced cation ordering in AgSb_0.94Cd_0.06Te₂ illustrated through (a) a high-resolution scanning transmission electron microscopy (HR-STEM-HAADF) image and (b) the corresponding selected-area electron diffraction (SAED) pattern. The nanoscale regions enclosed by dashed lines in (a) depict typical cation-ordered areas. Weak spots marked by yellow arrows in (b) result from cell doubling. (c) Temperature variation of zT is presented for AgSb_1−xCd_xTe₂ (x = 0–0.06).⁷⁰ ED patterns along (d) [1 0 0] and (e) [1 1 1] for Nb_0.8CoSb.⁷¹ (f) Temperature dependent lattice thermal conductivity (κ_lat) of defective half-Heusler (HH) and usual stoichiometric 18-electron HH compounds.^{71,343,345–350} Fig. (a)–(c) are reproduced with permission from ref. 70 © 2021, AAAS. Fig. (d)–(f) are reproduced with permission from ref. 71 © 2019, Royal Society of Chemistry.

X_1−xYZ defective half-Heusler materials, like Nb_1−xCoSb, exhibit remarkable thermoelectric characteristics due to short-range localized atomic order distinct from their mean periodic order.^71,342,343 The high defective nature (X-site vacancies) of a nominal 19-electron half-Heusler system having ordered face-centered cubic (fcc) sublattices facilitates achieving a stable ground state with 18 valence electrons in these compounds.³⁴⁴ Owing to such defective nature, these 19-electron-based half-Heusler compounds (e.g., Nb_0.8CoSb, Ti_0.9NiSb, V_0.9CoSb) show much lower κ_lat in comparison to the normal 18-electron based half-Heusler systems (Fig. 16f).^{71,343,345–350} Still, such a low κ_lat value cannot be explained by considering the presence of only nano-precipitates and lattice dislocation induced by a large number of vacancies. Electron diffraction patterns of Nb_0.8CoSb along [100] and [110] directions (Fig. 16d and e) exhibit the presence of a circular-like diffuse band at the centre of the marked red square together with the fundamental diffraction spots.⁷¹ The diffraction spots/peaks become diffusive in nature in the reciprocal space as the long-range ordering is destroyed in real space. Thus, the appearance of these diffuse bands indicates that the vacancy distribution in Nb_0.8CoSb exhibits order–disorder coexistence and possibly indicates a local or short-range order pattern. Diffuse bands of this type are discernible in the other defective nominal 19-electron half-Heusler compounds, e.g., Ti_0.9NiSb and V_0.9CoSb.⁷¹ This suggests that these compounds share a common characteristic, not been documented in other normal stoichiometric half-Heusler compounds. Furthermore, the developed short-range order observed in Nb_0.8CoSb is stable throughout the temperature range of 97–1073 K and no long-range ordering was created. Depending on the sample stoichiometry, short- or long-range vacancy ordering can be obtained, e.g., the intensity of diffuse bands reduces for Nb_0.82CoSb and entirely vanishes in the Nb_0.84CoSb and NbCoSb system. A superstructure starts to develop from short-range ordering when the number of Nb vacancies decreases, which has been experimentally verified through the development of multiple spots in the electron diffraction pattern for Nb_0.84CoSb.⁷¹ A recent study by Roth et al. also showed that depending on the thermal treatment of the sample, different local-structure states are reached, and they appear to have a substantial impact on the transport properties.^73,351 While the long-range ordering in defective half-Heusler compounds offers a periodic crystalline structure advantageous for charge carrier transport, vacancy-induced formation of short-range order scatters electrons and phonons simultaneously. Therefore, due to local vacancy ordering, Nb_0.8CoSb exhibits a slightly lower κ_lat than those with higher Nb contents (Fig. 16f).^{71,343,345–350} Benefiting from its low κ_lat, a zT of ∼0.9 at 1123 K was attained in carrier concentration optimized Ni-doped Nb_0.8CoSb.⁷¹

The local ordering was also identified in a disordered Zintl phase, Eu₂ZnSb₂, by Yao et al.^76,352 Eu₂ZnSb₂ has a globally disordered hexagonal crystal structure (space group: P6₃/mmc) with 50% Zn vacancies.⁷⁵ Electron microscopy imaging of bulk Eu₂ZnSb₂ by Yao et al. found Zn vacancy ordering with a consequent topological electronic transition.⁷⁶ Intrinsic nanostructuring formed by the coexistence of vacancy order on a local scale and Zn-site disorder on a global scale causes an ultralow κ_lat (∼0.4 W m⁻¹ K⁻¹ at 300 K) while retaining a decent carrier mobility (∼50 cm² V⁻¹ s⁻¹ at 300 K).⁷⁵ Due to a decrease in thermal conductivity without a proportional decrease in electrical conductivity, a maximum zT of ∼1 at 823 K was obtained in hole-doped Eu₂ZnSb₂.^74,75

5.9 Local van der Waals gap

Defect engineering is widely used in the field of thermoelectrics because defects have a significant impact on the electronic and thermal transport properties of solids. Stacking faults, twin boundaries, and grain boundaries are the most prevalent types of two-dimensional (2D) planar defects, well-known in thermoelectrics.^4,353 Beyond these well-researched 2D planar defects, 2D layers of vacancies are garnering more interest because of their ability to alter the local chemical bonding as well as the local atomic arrangement. Small amounts of Sb or Bi alloying can form such local 2D defects in SnTe and GeTe^78–82 and Sn/Ge vacancies can form locally ordered structures in SnTe/GeTe-based thermoelectric materials. These defects are commonly referred to as van der Waals (vdW) gaps (Fig. 17a).^78,81 2D vdW gaps are reported in intergrowth nanostructures (Fig. 17b and c) of (SnTe)_m(Bi/Sb₂Te₃)_n embedded in SnTe, which tremendously suppressed the lattice thermal conductivity and boosted zT to 1 at 800 K in (SnTe)_m(Sb₂Te₃)_n.^354,355

Fig. 17 (a) Predicted crystal structure of SnTe and Sn–Sb–Te. (b) High-resolution transmission electron micrograph (HRTEM) of Sn_0.85Sb_0.15Te shows the presence of layered intergrowth nanostructure domains. (c) Fast Fourier transform (FFT) image of (b) showing superstructure ordering spots (white circles) at ½ (h, k, l).³⁵⁴ The false-colour STEM HAADF image of (d) Sb₂Te₃(SnTe)₈ and (e) Sb₂Te₃(Sn_0.996Re_0.004Te)₈. The white-boxed area in (e) shows the extended gap-like structure. (f) Temperature dependent zT of SnTe, Sb₂Te₃(SnTe)₈ and Sb₂Te₃(Sn_0.996Re_0.004Te)₈.⁸² Fig. (a), (d)–(f) and Fig. (b) and (c) are reproduced with permission from ref. 82 © 2020 and ref. 354 © 2016 respectively, Royal Society of Chemistry.

The presence of local vdW gap structures was demonstrated in SnTe when alloyed with Sb₂Te₃ by Xu et al. via aberration-corrected STEM-HAADF experiments.⁸²Fig. 17d shows a false-colour STEM-HAADF image of the cubic Sb₂Te₃(SnTe)₈ sample viewed along the [110] direction. The occurrence of line-like (blue arrow) as well as gap-like defects (green ellipse) along the [110] direction implies the prevalence of cation vacancies along the same direction. In Sb₂Te₃(SnTe)₈, the gap-like structure prefers to be generated on local scales with an estimated density of the gaps close to 0.003 nm². The formation of intrinsic Sn-vacancies by Sb₂Te₃ alloying in SnTe evolves into a van der Waals gap-like structure. Such kinds of local defects are sensitive to the transport of heat-carrying intermediate and high-frequency range acoustic phonons, which helps to reduce κ_lat. In comparison to the κ_lat of ∼2.8 W m⁻¹ K⁻¹ for pure SnTe,³⁴ Sb₂Te₃(SnTe)₈ exhibits a much-lowered κ_lat of ∼0.9 W m⁻¹ K⁻¹ at 300 K. Moreover, rhenium (Re)-doping in Sb₂Te₃(SnTe)₈ for carrier concentration optimization further modulates the microstructure of the van der Waals gap in Sb₂Te₃(Sn_0.994Re_0.006Te)₈.⁸² STEM-HAADF images (Fig. 17e) of the Re-doped sample display that the van der Waals gap-like structure is much more evident than that of Sb₂Te₃(SnTe)₈ which may be due to a more significant number of vacancy generation after Re-doping. The white-boxed region (Fig. 17e) exhibits an extended gap-like structure which typically indicates a multi-layer characteristic, and the length of every individual gap increases with Re-doping. Re-doping induced the formation of a more profound and larger van der Waals gap-like defect causing a significant local lattice strain and suppressing κ_lat further. Eventually, a high zT of ∼1.4 (Fig. 17f) at 773 K and a high zT_avg of ∼0.83 in the 323–773 K temperature range were achieved in a 0.4 mol% Re-doped sample.⁸²

There have been a few reports of an atomic-scale van der Waals defect in GeTe-based thermoelectric materials.^78–81 Yu et al. represented the local van der Waals gaps as quantum gaps characterized as a nano-sized potential well.⁸¹Fig. 18a illustrates the structure of a quantum gap in a Ge–Bi–Te alloy along the [110]_PC zone axis. An atomically narrow Gaussian-shaped quantum well is produced from the quantum gaps. The carriers can transmit perfectly along the direction perpendicular to these quantum gaps, but the phonons are scattered strongly (Fig. 18b). The presence of a quantum gap from the missing layer of Ge has been evidenced from an atomic-resolution HAADF image viewed along the [110]_PC zone axis as shown in Fig. 18c. An atomic model with red arrows shows that Ge atoms close to the quantum gap move towards the direction of gaps.

Fig. 18 (a) The structural representation of a quantum gap (QG) in the Ge–Bi–Te alloy as observed along the [110] zone axis. (b) Diagram illustrating the transport characteristics of the QG, allowing carriers to move freely while significantly scattering phonons. The backdrop features a false-colour high-angle annular dark-field scanning transmission electron microscopy (HAADF-STEM) image captured from [110]. (c) An atomic-resolution HAADF image showcasing a quantum gap in Ge_0.927Bi_0.049Te, viewed along the [110] zone axis. (d) The reconstructed two-dimensional map depicting the potential distribution of a quantum gap in Ge_0.867Re_0.003Bi_0.087Te at 673 K, obtained through electron holography. Temperature dependent (e) lattice thermal conductivity (κ_lat) and (f) zT for Ge_0.913Bi_0.087Te (without QG), Ge_0.913Bi_0.087Te (with QG) and Ge_0.867Re_0.003Bi_0.087Te (more disordered QG).⁷⁷ The figure is reproduced with permission from ref. 77 © 2022, Nature Publishing Group.

Differential phase-contrast (DPC) imaging using STEM shows the presence of a non-centrosymmetric electric field near the quantum gap. Electric field mapping results show the formation of electric dipole near the quantum gap due to negative and positive charge centers. Furthermore, temperature-dependent in situ electron holography (Fig. 18d) under TEM confirms the formation of an atomically thin potential well at the quantum gap area. The transmission coefficient of charge carriers through the quantum well is ∼1 (100%), calculated from the measured potential using electron holography. For Ge_0.870Bi_0.087Te material at 10 K, the mean free path of the holes is approximately 10 nm, which is larger than the spacings of most quantum gaps supporting scattering less carrier flow.⁸¹ A higher anharmonic scattering rate of phonons results from the enhanced bond anharmonicity at quantum gaps caused by the van der Waals-like bonding and broken local translational symmetry. Furthermore, the dipoles in the vicinity of quantum gaps may also be involved in phonon scattering. Eventually, the samples containing quantum gaps have a higher Grüneisen parameter and lower average sound velocity. This resulted in suppressed κ_lat of ∼0.8 W m⁻¹ K⁻¹ (Fig. 18e) in Ge_0.870Bi_0.087Te with Ge-vacancy induced quantum gaps in comparison to the κ_lat of ∼1.1 W m⁻¹ K⁻¹ in Ge_0.913Bi_0.087Te without a quantum gap, whereas electrical conductivity and the power factor show a reverse trend.⁸¹ Further disordering of these quantum gaps’ distribution via Re doping in Ge_0.870Bi_0.087Te suppresses κ_lat to a larger extent and as a result of the ultralow κ_lat, the thermoelectric performance enhances (Fig. 18e), resulting in a maximum zT of ∼2.6 (Fig. 18f) at 723 K and zT_avg of ∼1.6 in the 300–723 K temperature range.⁸¹ Therefore, such aperiodic local van der Waals gaps, much different from the typical van der Waals gaps in layered materials, can be considered a novel tool to tune the thermoelectric properties.

5.10 Charge density wave

The recent discovery of the charge density wave (CDW) in some thermoelectric materials^356–359 has raised the question of whether there exists a correlation between a material's ability to harbor a charge density wave state and its thermoelectric properties. The CDW is an exemplary manifestation of the low dimensional transport phenomenon arising from strong electron–phonon coupling that breaks the translational symmetry of a lattice.³⁶⁰ The renewed interest in investigating the CDW w.r.t. potential thermoelectric applications is due to the fact that the consequent lattice distortion may be a route to realize extremely low thermal conductivity in materials whilst retaining high power factor.^359,361,362 Building on this idea, Rhyee et al. showed that Peierls’ distortion can be a route to high thermoelectric performance after they realized an extremely low thermal conductivity (0.74 W m⁻¹ K⁻¹ at 705 K) and a maximum zT of 1.48 at 705 K in an n-type CDW material, In₄Se_3−δ.³⁵⁶ Although the authors of this work had proposed the existence of a CDW based on the superstructure spots in HRTEM images and the singularities in the calculated Linhard function (electron susceptibility), its reality was later questioned. Despite rigorous phase analytical investigations on several In₄Se_3−δ samples, Osters et al. did not only observe any unusual displacement parameters that indicate a CDW but also went on to propose that perhaps it is the elemental indium present in the form of nano-precipitates in these nominally Se deficient samples that imparts them the attractive thermoelectric properties.³⁶³ Further, Cho et al.³⁵⁸ investigated the possibility of coexistence of CDW and topological surface states in Bi₂Te_2.1Se_0.9via CuI doping. The enhancement of zT_avg(1.0) was attributed to the simultaneous increase in the power factor and the decrease in κ_lat due to the formation of CDW. One reason why there are very few reports on thermoelectric performance of CDW materials is probably due to the fact that such a directed exploration is relatively new and ongoing. Another contributing factor might be the symmetric electron–hole band dispersion which can lower the Seebeck coefficient. Additionally, the high thermal conductivity owing to the high carrier concentration in several well-known CDW materials also suppresses their overall thermoelectric performance.³⁶¹ However, with new mechanisms/origin for atomic-scale thermopower³⁶⁴ and lowering of thermal conductivity due to electron–phonon coupling,^365–368 phonon puddles,³⁶⁴ local structure distortions,^368–372etc., being discovered even in conventional CDW materials, the field is still ripe for future exploration.

Amongst the phonon scattering mechanisms, perhaps the one that has not received much attention in describing the κ_lat of semiconductors is the scattering due to electron–phonon interactions (EPI). EPI acts to lower thermal conductivity by lowering the phonon frequency and consequently the group velocity (commonly known as the Kohn anomaly) and/or by increasing the phonon scattering rate through electron–phonon coupling. The lack of attention to the effects of EPI is partly due to their lesser significance in semiconductors owing to the low carrier concentration along with the dominance of Umklapp scattering at technologically relevant temperatures (often ≫θ_D) therein. The lack of experimental support is the other reason.³⁷³ It is worth mentioning that in systems where CDW is driven by Fermi surface nesting (i.e. when there exist large parallel sections on the Fermi surface that can be nested with certain fixed wavevectors, q⃑), strong scattering of phonon modes with momentum ℏq⃑ due to the carriers is expected. However, when phonon dispersion shows modes with anomalously large line widths (inversely related to phonon relaxation time) around the nesting vectors, it can be considered as an indication of significant electron–phonon coupling in the system.^{365–367,374} However, considering the coexisting phonon-scattering mechanisms and the purported phonon–electron soup³⁷⁴ that exists near the CDW transition temperature, deconvoluting and quantifying the exclusive contribution from EPI to κ_lat remains a formidable task.

Another emerging outlook on the thermal transport in CDW materials concerns the impact of local structural distortions after their occurrence was revealed in some of the classical CDW systems.^368–372 In a recent study on an archetypal CDW system, GdTe₃,³⁶⁸ it was demonstrated that structural reminiscences of the CDW state persist even above the T_CDW (∼380 K) in the form of local lattice distortions. To elaborate, while attempting to study the lattice dynamics of GdTe₃ across its CDW transition using temperature-dependent X-PDF, the authors observed that the Te square nets of GdTe₃ (Fig. 19a) remained distorted even at ∼80 K above the T_CDW, as evident from the evolution of the low-r shoulder in Fig. 19b. In addition, a rather unexpected high-r feature near 3.4 Å which gradually vanished as the system entered deeper into the CDW state upon lowering the temperature below T_CDW was observed. Interestingly, the longer bond length matches with the Te–Te distances in the oligomeric telluride chains of various polytelluride compounds reported with distorted Te nets.³⁷⁵ Thus, it was proposed that the observed local lattice distortions which persist above the T_CDW are most likely from the ensemble of precursor motifs that undergo long-range ordering at T_CDW as the system transitions into the macroscopic symmetry broken state of the CDW phase of GdTe₃. Extensive studies are invited to confirm if these local lattice distortions are also playing a role in engendering the large lattice anharmonicity (calculated cross-plane γ of GdTe₃ = 2.42) and consequently low cross-plane lattice thermal conductivity (k_lat = 0.7 W m⁻¹ K⁻¹ at 673 K) and unusually large thermal transport anisotropy [(k_lat)_cross-plane/(k_lat)_in-plane ≈ 5.18 at 300 K] in GdTe₃ (and in similar CDW systems) (Fig. 19c). While investigating other plausible driving forces for CDW like electron–phonon coupling in addition to Fermi surface nesting in GdTe₃ (Fig. 19d), the individual contributions of different scattering mechanisms to the phonon relaxation time in GdTe₃ were estimated using the Debye–Callaway formalism³⁷⁶ (Fig. 19e). The mode-separated phonon relaxation time revealed that EPI indeed played a tantamount role with Umklapp scattering in reducing κ_lat, even at temperatures ≫θ_D, calling for a revisit to our understanding of the origin and nature of CDW especially in systems with high carrier concentration.³⁶⁸

Fig. 19 (a) The structure of GdTe₃ demonstrating a natural heterostructure of charged and van der Waals layers. The double corrugated slabs of GdTe and Te square net sheets stack along the c-axis and are held together by strong electrostatic and weak van der Waals interaction. (b) Temporal evolution (100–460 K) of the first peak in X-ray PDF of GdTe₃. The low-r shoulder arising from the shorter Te–Te bonds on the Te square net is indicated by the arrow. (c) Lattice thermal conductivity of GdTe₃ measured along the parallel (red spheres) and perpendicular (blue spheres) direction to the SPS pressing direction. (d) The diamond-shaped Fermi surface of GdTe₃ exhibiting nesting by wavevectors q₁ and q₂ along the Γ–S direction. (e) Relaxation time as a function of the normalized phonon frequency of GdTe₃.³⁶⁸ Note that the phonon frequency (ω) is normalized with respect to the Debye frequency (ω_D). Significant electron–phonon scattering in addition to the Umklapp process is evident. The figure is reproduced with permission from ref. 368 © 2023, Wiley VCH.

5.11 Metavalent bonding

The discovery of novel bonding types always excites chemists. However, ever since it was first proposed in 2018 by Wuttig et al.,³⁷⁷ metavalent bonding (MVB) has been an apple of discord amongst material chemists^378–380 especially concerning its distinction from hyperconjugation³⁸¹ and resonant bonding.³⁸² While the uniqueness of MVB remains a vexata quaestio, motivated by the numerous recent reports on its efficacy towards engineering functionality in extended structured advanced materials,^383–388 we pursue here a discussion on its applicability towards enhancing thermoelectric properties of materials.

Founded on the pillars of five characteristic descriptors (Fig. 20a),³⁷⁷viz., violation of the 8-N rule, moderate electrical conductivity (10²–10⁴ S cm⁻¹) and high mode-specific Grüneisen parameter (>2), optical dielectric constant (>15) and Born effective charge (4–6), MVB is still mostly understood in the property space. The bonding is characterized by a distinct bond breaking mechanism^377,389 that corroborates with a multicentric bonding³⁸⁷ achieved via a fine-tuning between electron localization (covalency) and electron delocalization (metallicity) as inferred from its distinct locus in the electron-sharing vs. electron donation map.^390,391 Thus, metavalent bonded materials exhibit properties of both metallic- and covalently bonded systems in addition to other unique attributes. It must be emphasized that the transition from conventional bonding mechanisms (ionic, covalent, metallic) to MVB is largely unclear. However, from a survey of known metavalent bonded systems, it appears that half-filled p-bands and weak Peierls’ distortions are prerequisites for MVB. Arora et al.^387,391 proposed that MVB emerges due to the weak symmetry breaking in the parent simple-cubic structure of group V metalloids. The highly degenerate nested Fermi surface of the parent metal drives the spontaneous breaking of its translational symmetry in the presence of optimal structural/chemical fields to open an energy gap that facilitates strong coupling between the electrons in valence and conduction bands. Consequently, metavalent bonding harbors sensitive bond lengths (lattice anharmonicity) alongside large polarizability and electrical conductivity. Additionally, metavalent bonded systems can host long-range electron transfer in response to polar fields owing to the underlying bonding involving a typical alternating bonding–antibonding pairwise interaction along linear chains of at least 5 atoms (Fig. 20b–k). To understand the multicentric nature of MVB and relevant orbital interactions, consider an archetypal metavalent system PbTe. The Wannier functions (WF) and their projections onto the atomic orbitals of PbTe reveal the multicentric nature of MVB as directed by the mixed bonding and antibonding σ and π interactions (Fig. 20b–e). The WF with symmetry of the p_x orbital centered on the Te atom reveals p–p σ-bonding interaction with axial Pb neighbors and weak π-bonding interaction with transverse Pb neighbors along with the antibonding σ* and π* interactions between cations and anions present in the next shell (Fig. 20b). Similarly, the WF with symmetry of the s-orbital centered on the Pb atom exposes its anti-bonding σ* interaction with the s and p-orbitals of neighboring Te atoms and σ bonding interaction between ions of the next shell (Fig. 20d). Note that the WFs are considerably delocalized and extend to subsequent neighbors revealing π* pp (Fig. 20b) and σ pp interactions (Fig. 20d). The isosurfaces of WFs with symmetry of the p_x (p_y) orbital centered on the Te atom in PbTe and distorted PbTe structure with Pb displacements show the electronic transfer from the Pb atom on the right to that on the left through p–p σ–(p–p π–) bonding interaction (Fig. 20f–i). Readers are encouraged to appreciate the hopping interaction in PbTe (Fig. 20j) and similar metavalent systems. Further, it is worthwhile to note the qualitative differences between a symmetric atomic arrangement (in the absence of charge transfer) and an asymmetric atomic arrangement (expected when there is an electronic redistribution due to charge transfer or Peierls distortion) while rationalizing the degree of lattice distortion expected in metavalent systems (Fig. 20k).

Fig. 20 (a) Descriptors for metavalent bonding (MVB). Panels (b) and (d) show the isosurfaces of various Wannier functions (WF) in PbTe and panels (c) and (e) show their corresponding schematic. Isosurfaces of WFs with symmetry of the p_x orbital centered on the Te atom in (f) PbTe and (g) distorted PbTe. Isosurfaces of WFs with symmetry of the p_y orbital centered on the Te atom in (h) PbTe and (i) distorted PbTe. (Wavefunctions with negative and positive signs are shown in blue and red, respectively.) (j) Schematic picture shows electron transfer across one unit cell in the direction opposite to displacements of Pb atoms. Red (solid) and black (dashed) arrows show charge transfer through σ and π channels respectively for the hopping interaction.³⁸⁷ (k) Schematic diagram of the (001) plane of PbTe, displaying the σ-bonds formed from the p-orbitals of Pb and Te, which are responsible for the octahedral-like atomic arrangement in PbTe. The middle sketch panel shows the symmetric atomic arrangement without charge transfer, while the distribution of electrons changes either by a Peierls distortion (lower panel) or electron transfer (top panel). Fig. (b)–(j) are reproduced with permission from ref. 387 © 2023, Wiley VCH.

Moving on, it is well understood that thermoelectric materials seek a very unusual property portfolio^70,392 which is perhaps why the search for novel thermoelectric materials is often paralleled to the proverbial search for a needle in a haystack. From the previous discussion, it is evident that MVB showcases a unique mélange of material properties, motivating the exploration of their (and of systems likely to show MVB) thermoelectric properties. Indeed, recent reports have evidenced that metavalent systems feature electronic band structures that engender large band degeneracy and high band anisotropy for enhancing the power factor alongside low κ_lat consequent of the soft and anharmonic nature of the underlying chemical bonds. Thus, MVB is predicted (and increasingly being proven) to be a one-pot recipe for realizing high thermoelectric performance.^{15,33,39,165,388,393–399} Since how MVB impacts thermoelectric performance is not our focus, we are not covering the band picture and the associated mathematical formalism which describes the exact mechanism by which MVB can entangle the contradictory dependencies and simultaneously tune both the electrical and thermal transport properties essential for zT (thermoelectric figure of merit) enhancement. Rather, we look at some of the recent reports that validate the claims of MVB towards novel thermoelectric materials. We emphasize here that readers interested in a more rigorous treatment are encouraged to look at the seminal works from Wuttig et al.^377,383,388

There are only a few reports on the experimental evidence for the efficacy of MVB in tailoring the thermoelectric properties in novel thermoelectric materials.^15,33,39 In a rare example of accomplishing simultaneous structural (orthorhombic to cubic) and bond-transformation (covalent to metavalent), a maximum zT of ∼1.35 at 627 K³³ was realized in GeSe (pristine zT ∼ 0.2) via AgBiTe₂ alloying. Metavalent bonding was also shown to be an effective handle to tune the lattice anharmonicity without adversely affecting the electrical transport in a single crystal of cubic AgSnSbTe₃ which exhibited a high zT of ∼1.2 at 661 K.¹⁵ Additionally, the recent reports on Hg- or Yb-doped AgSbTe₂ (zT ∼ 2.4 at 570 K)^340,341 are testimonial to the competence of engineering disorder in metavalent systems for high performance thermoelectric materials.

Striding a different direction into topological materials, the work on the topological insulator, TlBiSe₂,³⁹ was aimed at forging a connection between topology, metavalent bonding and thermoelectric properties. According to the phase diagram constructed by Raagya et al.,³⁸⁷ there are possible non-zero Berry phases along the closed loops around the metallic point, which suggests that metavalent states can host non-trivial electronic topology. An exemplary revelation of emergent properties in solids was the observation of an intrinsic lattice shearing in TlBiSe₂ (Fig. 21a and b) cracked through synchrotron X-PDF analysis and DFT-based energetic studies. The origin of the unexpectedly low κ_lat (Fig. 21c) was traced back to the underlying long-range anharmonic pathway in TlBiSe₂ manifested via the MVB-mediated propagation of dual 6s² lone pair-induced structural distortions over longer distances herein. Thus, corroborating with chemical intuition based on the shared design principles of topological and thermoelectric materials, augmented by the predicted metavalent bonding in TlBiSe₂,⁴⁰⁰ the highest n-type zT (∼0.8 at 715 K, Fig. 21d) was achieved in a Tl-based thermoelectric material. Additionally, the work predicted the occurrence of a unique phonon scattering mechanism in systems where the structure can shuttle between numerous energetically nearly degenerate states.

Fig. 21 X-ray PDF plot of TlBiSe₂ at 300 K fitted against (a) thermal parameters only refined R3̄m model and (b) Tl, Bi, Se positions refined R3̄m model for total structure (2.3–30.0 Å). The insets show the corresponding fittings for the local structure (2.3–5.0 Å) and the right panel shows the corresponding visualizations of the undistorted and distorted structures of TlBiSe₂. Temperature dependent (c) lattice thermal conductivity (κ_lat) and (d) thermoelectric figure of merit (zT) of TlBiSe_2−xS_x (x = 0, 1) samples.³⁹ The figure is reproduced with permission from ref. 39 © 2023, American Chemical Society.

Finally, we draw the reader's attention to emerging evidence on some other unusual implications of MVB for crystalline compounds. Numerous theoretical studies have predicted that MVB is the origin of phonon anomalies in many of the classical (and some potential) thermoelectric materials from the chalcogenide family.^400–402 MVB was also shown to improve the solubility of Ag (from 0.5% to >7%) in SnTe by AgSbTe₂-alloying.³⁹⁴ Finally, in a notable work by Wu et al.⁴⁰³ the strong carrier scattering at the grain boundaries of the champion thermoelectric material PbTe was traced down to the breakdown of MVB owing to the extreme Peierls’ distortion present therein. Herein, the grain boundaries were carefully mapped using atom probe tomography to reveal the subtle link between local chemical bonding and carrier transport in PbTe.

6. Future outlook

200 years since the discovery of the Seebeck effect,⁴⁰⁴ the field of thermoelectrics has come a long way and accomplished much. What began as a pursuit of curiosity in the 1800s has borne the fruit of a sustainable green energy generation technique today.⁴⁰⁵ As a result of the collaborative efforts between chemists, material scientists and physicists, currently, we have materials with zT > 3,²²⁶ outperforming the unitary benchmark which was once believed to be the maximum achievable zT. Acknowledging all the efforts and brilliant strategies devised around the globe in terms of novel synthetic strategies, conceptual advancements, or device engineering, we believe that we still have room for innovation and improvement. The earliest efforts at enhancing thermoelectric performance via the incorporation of dopants and/or all-scale hierarchical heterostructures were essentially aimed at damping the lattice thermal conductivity by increasing phonon scattering due to structural disorder. However, disorder is no longer any new concept in the thermoelectric context. Intrinsically reduced κ_lat by employing dimensionality reduction, mass-mismatch, bonding hierarchy, soft anharmonic lattices, ferroelectric instability, etc., were explored only relatively recently. Note that a common feature of all the above intrinsic strategies is the macroscopic nature of the imparted disorder due to their large coherence lengths. Logically, the maximum thermoelectric performance in materials will be realized only when we work with their intrinsic limits for minimum thermal conductivity and maximum electrical transport. This idea practically translates to engineering the structure at the atomistic level, i.e., by designing structural disorder(s) that specifically scatters phonons without affecting the charge carriers.

This review has provided how hidden structures availed by local structure engineering can achieve the aforementioned task and become a novel tool to enhance the thermoelectric performance in crystalline solids. Thermoelectric properties from a chemist's perspective to decipher the structure design principles by combining the classic chemical bonding and insights from the crystal to glass transition emphasize the importance of controlled structural disorder at the local length scale for accomplishing PGEC. In this review, we have summarized the advancements, current pursuits, and the prospects in the field by exploiting the above strategies by tweaking the local structure of materials for synergistically tailoring their thermoelectric properties; however, we believe there is a long way to go. Upholding the central theme of local structure engineering, we have discussed numerous strategies to lower κ_lat and thereby improve thermoelectric performance by exploiting versatile chemical handles like local disorder, liquid like sublattices, atom off-centering, discordant atoms, site splitting, rattler ions, charge density waves, metavalent bonding, atom ordering, local van der Waals gap, etc., but mind that these strategies have been investigated in only a few materials to date. For example, all-inorganic halide perovskites also exhibit ultralow κ due to their elastically soft lattice, bonding hierarchy, local structural distortion, rattling of atoms, etc.^18,19,21 The possibility of attaining high thermoelectric performance in them has arose in recent investigations but remains highly uncharted.²¹ On the other hand, the design and development of misfit compounds can also be an attractive route to obtain low κ_lat. The major benefit of this class of compounds is their widespread compositional variety, wherein layers of distinct chemical arrangements can be stacked along a particular crystallographic direction. This can have vast implications for thermoelectrics, as the electronic as well as thermal transport can be simultaneously controlled by modulating the stacking. Recent few studies show that misfit compounds exhibit local structural distortion, yet further investigation in this area remains limited.^406–411 Misfit materials such as (BiS)_1.2(TiS₂)₂, (SnS)_1.2(TiS₂)₂, etc., have shown potential in the field of thermoelectrics due to their ‘phonon-blocking, electron-transmitting’ ability, which suppresses κ but does not influence the carrier mobility.^94,412

The provided glance over the plethora of state-of-the-art local structure determination techniques that are being employed and developed to characterize and quantify disordered solids in this review is clearly inadequate. Merging diverse techniques utilized for detecting local phenomena into a unified structure resolving approach can prove highly advantageous for future generations. While the recently explored local structural phenomena have occasionally been employed to illuminate the glass-like temperature dependence of κ in some intrinsically low thermally conductive compounds, it is vital to note that the appearance of such phenomena in a compound does not always imply a glass-like temperature variation in thermal transport. This prompts investigations into the presence of other local phenomena which are yet to be discovered. Further, investigating non-traditional chemical bonds, structural attributes, and physical properties could prove beneficial in enhancing phonon scattering. The promising idea of metavalent bonding which builds a bridge between order and disorder or distortion^377,383,413 could offer constructive insights for developing low thermally conductive and high performance thermoelectric materials.^33,39,340 Lately, machine learning has demonstrated its potential for screening materials which possess the aforementioned structural descriptors, thus accelerating the discovery of new materials with potential high thermoelectric performance.^414–419 The incorporation of information pertaining to the previously mentioned local phenomena as a descriptor into machine learning can play a crucial role in the advancement of thermal insulators or in the field of thermoelectrics.

A survey of the literature on thermoelectric materials reveals that often, there is at least an order of magnitude disparity between the theoretically predicted and the experimentally realized zT values. Consider for instance the case of Bi₂Te₃, the front-runner in room temperature thermoelectrics. In Bi₂Te₃, although theory has predicted a max zT of 14,⁴²⁰ experimental efforts have reached only 2.4 so far.⁴²¹ While this is the story with the conventional thermoelectric materials, the field of organic/hybrid thermoelectric materials seems nearly stagnated at max zT < 1 although theoretically zT as high as 4.8⁴²² has been predicted. Nonetheless, with examples of systems like SnSe, the material with the highest experimentally reported zT (∼3.2) yet,²²⁶ the maximum theoretical prediction lies relatively close at 4.6⁴²³ indicating that experiments can indeed realize theoretical predictions. The question remains whether the disparity is due to the limitations on the experimental front or due to the inaccuracy of the approximations used in theoretical calculations. Our aim here is not to address this aspect but rather highlight the abundant scope left in the thermoelectric field which is sometimes perceived to be reaching its saturation by novice researchers. In the following section we outline some focus areas which we believe deserve maximum attention and will script the future of thermoelectrics in the upcoming days.

A recurring theme in thermoelectrics is the notion that the best-known thermoelectric materials are all restricted to heavy metal chalcogenides, and thus, most of the chemical space remains unexploited/unexplored. This is understandably due to the requirement for low bandgap, heavy atoms, octahedral coordination, etc. which are met by the binary/ternary/multinary phases of the elements from the p-block of the periodic table.⁴²⁴ To move the existing frontiers in thermoelectrics, we call for a conceptual advancement by deviating from the conventional approaches and look for new theories or approaches that can decouple the electrical and thermal transport so that they can be independently optimized. This would also lead to novel thermoelectric compounds. Some promising directions in the chemical library are along chiral phonon manipulation,⁴²⁵ phonon confinement via dimensionality reduction⁴²⁶ (nanowires/nanocrystals),⁴²⁷ metal halide perovskites,^428–430 topological quantum materials,^431–433 ionic and organic thermoelectrics,^434–436 entropy engineering,^437–440 spin-thermoelectrics using skyrmions,^441–443etc. Furthermore, it is high time we explore the emerging novel theories for enhancing thermoelectric performance like topological phonon, spin-Seebeck, topological electron, nanophononic metamaterials, phonon coherence, etc.^444–446

Conflicts of interest

The authors declare no conflict of interest.

Acknowledgements

K. B. acknowledges support from the Swarna-Jayanti fellowship grant, the Science and Engineering Research Board (SERB) (SB/SJF/2019-20/06), I-HUB Quantum Technology Foundation (IHUBQTF) (I-HUB/SPIKE/2023-24/008), and Sheikh Saqr laboratory, JNCASR. D. S. and A. B. thank CSIR for the fellowship.

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