Structural Characteristics and Recent Advances in Thermoelectric Binary Indium Chalcogenides

Structural Characteristics and Recent Advances in Thermoelectric Binary Indium Chalcogenides

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Yasong Wu^†, Binjie Zhou^†, Lu Liu, Shengnan Dai^*, Lirong Song^*, Jiong Yang^*

Research. Vol 8 Article ID 0727

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Research. Vol 8 Article ID 0727

• Review Article •

Structural Characteristics and Recent Advances in Thermoelectric Binary Indium Chalcogenides

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Yasong Wu^†, Binjie Zhou^†, Lu Liu, Shengnan Dai^*, Lirong Song^*, Jiong Yang^*

Affiliations

Materials Genome Institute, Shanghai Engineering Research Center for Integrated Circuits and Advanced Display Materials, Shanghai University, Shanghai 200444, China.

Published: 2025-06-10 doi: 10.34133/research.0727

Outline

Abstract

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Thermoelectric (TE) materials have garnered widespread research interest owing to their capability for direct heat-to-electricity conversion. Binary indium-based chalcogenides (In–X, X = Te, Se, S) stand out in inorganic materials by virtue of their relatively low thermal conductivity. For example, In₄Se_2.35 shows a low thermal conductivity of 0.74 W m⁻¹ K⁻¹ and an impressive zT value of 1.48 along the b–c plane at 705 K, as a result of structural anisotropy. Here, we review the structural features and recent research progress in the TE field for In–X materials. It begins by presenting the characteristics of crystal structure, electronic band structure, and phonon dispersion, aiming to microscopically understand the similarity/dissimilarity among these In–X compounds, notably the role of unconventional bonds (such as In–In) in modulating the band structures and lattice vibrations. Furthermore, TE optimization strategies of such materials were classified and discussed, including defect engineering, crystal orientation engineering, nanostructuring, and grain size engineering. The final section provides an overview of recent progress in optimizing TE properties of indium tellurides, indium selenides, and indium sulfides. An outlook is also presented on the major challenges and opportunities associated with these material systems for future TE applications. This Review is expected to provide critical insights into the development of new strategies to design binary indium-based chalcogenides as promising TE materials in the future.

Cite this Article

Yasong Wu, Binjie Zhou, Lu Liu, Shengnan Dai, Lirong Song, Jiong Yang. Structural Characteristics and Recent Advances in Thermoelectric Binary Indium Chalcogenides[J]. Research, 2025 , 8 (6) : 0727 . DOI: 10.34133/research.0727

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Introduction

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Efficient and clean energy is increasingly in demand owing to the growing energy and environmental crisis [1,2]. To tackle the issue of energy supply shortage and prevent further environmental degradation, it is crucial to develop renewable energy devices. One promising solution is the utilization of thermoelectric (TE) materials, which can directly convert thermal energy into electricity [3–5]. TE energy converters have gained substantial interest due to their advantages, for example, no mechanical moving parts, reliability, quiet operation, and environmental friendliness [6,7]. As a result, TE conversion technology has promising applications in aerospace, biomedicine, integrated circuits, etc. [8–11]. However, despite the potential advantages of TE technology, the current conversion efficiency of TE devices remains low, limiting their commercial applications [12]. Therefore, researchers are continuously exploring and developing new high-performance TE materials to attain high conversion efficiency and broaden the application of TE devices.

The efficiency of TEs to convert thermal energy into electricity is crucially indicated by the dimensionless figure of merit, zT, which is expressed by the following formula [13,14]:

zT = S 2 σ T κ e + κ L

(1)

where S, σ, κ _e, κ _L, and T are the Seebeck coefficient, electrical conductivity, electronic thermal conductivity, lattice thermal conductivity, and absolute temperature, respectively. Nevertheless, attaining a high zT value is intricate owing to the interplay between these TE parameters. Boosting the zT value necessitates the strategic decoupling of electrical and thermal transport properties [15], a substantial research hurdle in the field of TEs [16].

Currently, inorganic semiconductor materials such as Bi₂Te₃ [17–19], PbTe [20–22], GeTe [23–25], and SiGe [26–28] are showing great promise in the field of TEs. Among them, binary indium-based chalcogenides (e.g., In₄Se₃ [29] and InTe [30]) have earned substantial attention because of their unique structure and impressive TE performance. Indium exhibits versatility and flexibility in forming binary chalcogenides, enabling the formation of various compounds with different stoichiometric ratios, resulting in a complex material system. Indium tellurides include In₄Te₃, InTe, In₃Te₄, In₂Te₃, In₂Te₅, etc. Indium selenides contain In₄Se₃, InSe, In₆Se₇, In₃Se₄, In₂Se₃, etc. Indium sulfides comprise InS, In₆S₇, In₃S₄, and In₂S₃. In binary chalcogenides, indium can bond to other atoms via ionic or covalent bonds, and generally tend to exhibit mixed valence states. For example, InTe, i.e., In⁺In³⁺Te₂, contains both In¹⁺ and In³⁺ cations. Owing to exceptional nonlinear effects, high damage threshold, remarkable photoresponsivity, and optimal band gap, InX compounds (X = Te, Se, S) are extensively applied in fields such as optoelectronic devices, nonlinear optics, and ultrafast lasers [31–35]. Nonetheless, binary indium-based chalcogenides, especially In–Te and In–Se systems, have attracted incremental consideration in the TE research field (Fig. 1A), owing to their intrinsically low thermal conductivity of <2 W m⁻¹ K⁻¹ at 300 K. Moreover, several pure In–X materials (e.g., InTe and In₄Se₃) exhibit the maximum zT (zT _max) value higher than 0.5 (see Fig. 1B).

This review offers a thorough overview of recent advancements in TE research on binary indium-based chalcogenides. We first focus on presenting and comparing the crystal structures for those binary indium-based chalcogenides that have the same In-to-chalcogen ratio at ambient conditions, e.g., InTe versus InSe versus InS. Afterward, the band structures and phonon dispersions are analyzed to better understand the role of unconventional bonds present in certain In–X compounds. Moreover, this review also analyzes and summarizes the current experimental TE optimization strategies of these compounds, including defect engineering, crystal orientation engineering, nanostructuring, and grain size engineering. Additionally, the review discusses the existing challenges and future development of binary indium-based chalcogenides, with the expectation of providing some new insights on how to further enhance their TE properties and promote their large-scale application. Overall, this Review serves as a valuable reference for those working in the TE field and offers a thorough overview of the progress and potential of binary indium-based chalcogenides as TEs.

Crystal Structure

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In₄Te₃ and In₄Se₃

The In₄Se₃ compound crystallizes in an orthorhombic system with the space group of Pnnm (no. 58), as determined by Hogg et al. [36]. The lattice parameters are a = 15.297 Å, b = 12.308 Å, and c = 4.081 Å. In₄Se₃ exhibits layered structure with distorted layers of nonplanar surfaces connected by weak interactions. The structure is a result of the bonding between atoms with mixed valence states, specifically In₄Se₃ = [In]⁺[In₃]⁵⁺[Se²⁻]₃ [37]. Referring to Fig. 2A, each quasi-2-dimensional In/Se layer consists of one-dimensional In/Se chains, and adjacent layers are stacked along the a axis through van der Waals interactions. The one-dimensional In/Se chain comprises trinuclear [(In₃)⁵⁺(Se₃)⁶⁻] clusters [38], where In1, In2, and In3 atoms form a quasi-one-dimensional chain with metallic bonding interactions. These In atoms are covalently bonded respectively to 3 differently positioned Se atoms (Se1 to Se3) in the b–c plane [38]. In1 occupies an edge position in the (In₃)⁵⁺ chain, tetrahedrally surrounded by 3 selenium atoms (2 Se1 and 1 Se2) as well as 1 In2. In2 is positioned at the center of the (In₃)⁵⁺ chain, encircled by 2 Se2 atoms, 1 In1, and 1 In3. Similar to In1, In3 is in the position of the chain termini and exhibits a tetrahedral coordination with 3 selenium atoms (2 Se3 and 1 Se1) and 1 In2. The In4 atom is located in the interlayer region, where there is a notable density of valence electrons [39,40]. The nearest indium counterpart, another In4, is situated at a notably substantial distance, exceeding 3 Å. Its distance to selenium atom is also relatively elongated (the shortest with Se3, approximately 3 Å). A Peierls distortion of the one-dimensional In/Se chain, which is caused by the charge density wave (CDW) along the b–c plane [40,41], can reduce the thermal conductivity in the b–c plane [42]. The relatively low band gap of In₄Se₃ compared to other In–Se compounds is mainly attributed to its unique crystal structure with the In–In bonding feature [41]. In₄Te₃ shows a similar crystal structure (orthorhombic Pnnm space group), with lattice parameters a = 15.730 Å, b = 12.784 Å, and c = 4.434 Å [43].

InX (X = Te, Se, S)

InTe exhibits a TlSe-type structure under ambient conditions, belonging to the tetragonal system (space group I4/mcm). Its lattice parameters are a = b = 8.454(2) Å and c = 7.152(6) Å [44]. The mixed-valence formula of InTe, In¹⁺[In³⁺Te²⁻ _4/2]⁻, suggested that In atoms occupy 2 different positions within the structure [45] (Fig. 2B), and the distances between indium atoms are exceedingly large, i.e., the nearest distances of In⁺-In⁺, In³⁺-In³⁺, and In⁺-In³⁺ are approximately 3.576, 3.576, and 4.227 Å, respectively. The In³⁺ cation is tetrahedrally coordinated to the Te²⁻ anions and forms an (InTe₂)⁻ chain along the c axis [Fig. 2B(ii)]. The In¹⁺ cation with 5s² lone pair electrons is located in the center of a cage-like framework formed by the 8 surrounding Te atoms arranged in a square antiprismatic geometry [46], as shown in Fig. 2B(iii).

InSe, unlike InTe, can crystallize into both hexagonal and rhombohedral layered crystal structures, i.e., β-InSe and γ-InSe phases, respectively [41], as shown in Fig. 2C and D. β-InSe has lattice parameters of a = 4.005 Å and c = 16.640 Å, with the space group P6₃/mmc [47,48]. γ-InSe crystallizes in the space group R3m, with lattice parameters a = 4.0046 Å and c = 24.960 Å [41]. As shown in Fig. 2C(i) and Fig. 2D(i), both structures are triangular biconical, which results in similar band gaps (will be further shown in the third section). The layers in β-InSe exhibit an ABAB stacking pattern, while in γ-InSe, the stacking pattern is ABCABC. Every layer of InSe is constituted by a network of covalently bonded Se–In–In–Se, with weaker van der Waals bonding between the layers [33]. Within γ-InSe, the In–In bonds are slightly shorter. Each indium atom has a tetrahedral configuration that is similar to the In1 and In3 atoms in In₄Se₃.

In 2000, Hollingsworth et al. [49] identified 2 different crystalline phases of InS compounds: a more-stable network structure and a higher-energy (metastable) layered structure. These phases consist of the S–In–In–S basic structural units [35]. Later in 2014, Kushwaha et al. [50] demonstrated that the most stable crystalline phase was recognized to show a network structure belonging to an orthorhombic crystal system (space group Pnnm) (Fig. 2E), by growing InS single crystals using the In flux method. The lattice parameters were a = 4.4506(2) Å, b = 10.6503(4) Å, and c = 3.9455(2) Å. Notably, the minimum distances between In–In and S–S atoms (interlayer) were determined to be 2.806 Å and 3.088 Å, respectively.

In₆Se₇ and In₆S₇

In₆Se₇ has a monoclinic structure with a space group of P2₁/m [51]. The unit cell parameters are a = 9.433 Å, b = 4.064 Å, c = 17.663 Å, and β = 100.92°. The compound In₆Se₇ consists of In ions in various valence states (+1, +2, and +3) and can be represented as In⁺[In₂]⁴⁺(In³⁺)₃(Se²⁻)₇ [52]. It seems to contain the characteristics of the In atom found in the InSe (triangular biconical), In₄Se₃ (one-dimensional In⁺), and In₃Se₄ (octahedral In) structures. The [In₂]⁴⁺ and In³⁺ ions are distributed over 2 and 3 distinct lattice sites [53], respectively, as shown in Fig. 2F. In₆S₇ also shows the monoclinic structure (space group P2₁/m). Two formula units are contained within the unit cell, and all In and S atoms are located at the special position 2e [54]. The cell parameters are a = 9.088 Å, b = 3.887 Å, c = 17.166 Å, and β =101.92° [55].

In₃X₄ (X = Te, Se, S)

In₃Se₄ has a layered rhombohedral crystal structure with a space group of R

3 ¯

m. Its lattice parameters are a = 3.964 ± 0.002 Å and c = 39.59 ± 0.02 Å [56]. The crystal structure of In₃Se₄ is characterized by the stacking of Se–In–Se–In–Se–In–Se layers along the c axis [Fig. 3A(i)]. The In atoms are coordinated in an octahedral geometry, surrounded by 6 neighboring Se atoms that form the vertices of the octahedron [57] [Fig. 3A(ii)]. This structure of In₃Se₄ resembles the one described for In₃Te₄ by Geller et al. [58] [space group R

3 ¯

m, lattice parameters a = b = 4.27(1) Å and c = 40.90(10) Å]. But according to Karakostas et al. [59], In₃Te₄ crystallizes in a tetragonal system with lattice parameters a = 6.173 Å and c = 12.438 Å. There is limited information available on the structural studies of both In₃Te₄ and In₃S₄. Zavrazhnov et al. [60] reported that the crystal structure of In₃S₄ adopts a cubic system (point group of m3m) with an isotropic lattice parameter of 10.736₁ Å.

In₂X₃ (X = Te, Se, S)

At atmospheric pressure, In₂Te₃ exists in 2 polymorphic forms: α-In₂Te₃ (stable at lower temperatures) and β-In₂Te₃ (the high-temperature phase) [10,61]. α-In₂Te₃ has a defective fluorite structure, while β-In₂Te₃ has a defective zincblende structure (Fig. 3B and C). The transition from α-In₂Te₃ to β-In₂Te₃ occurs at temperatures above 733 K. Both α-In₂Te₃ and β-In₂Te₃ possess face-centered cubic (FCC) lattice (space group F

4 ¯

3m), with lattice parameters of a = b = c = 18.50 Å and a = b = c = 6.16 Å [62], respectively. Due to the presence of vacancies in one-third of the cationic sites, the defect concentration in In₂Te₃ reaches 10²¹ cm⁻³. In α-In₂Te₃, these vacancies are orderly distributed, whereas in β-In₂Te₃, they are randomly distributed [63]. At room temperature, each indium atom in the α-In₂Te₃ structure is exclusively bonded to 4 Te atoms, arranged in a tetrahedral configuration.

In₂Se₃ compounds also exhibit different crystalline phases at different temperatures. Eight distinct crystalline phases have been reported, comprising 3 predominant phases (α, β, and γ) and 5 less frequently observed phases (δ, κ, α′, β′, and γ′) [64–66]. According to the stacking order, the α phase has 2 stacking modes, hexagonal (2H) and rhombohedral (3R), while the β phase exists in 3 stacking modes, i.e., triangular (1T), 2H, and 3R. At room temperature, the stable structures of In₂Se₃ are α(2H) and α(3R). The α(2H) phase of In₂Se₃ crystallizes in the P6₃/mmc space group, characterized by lattice parameters a = 4.025 Å and c = 19.235 Å [47,67]. In contrast, the α(3R) polymorph adopts the R3m space group with the lattice parameters of a = 4.026 Å and c = 28.750 Å [68]. Both α(3R)-In₂Se₃ and α(2H)-In₂Se₃ exhibit a layered structure composed of quintuple Se–In–Se–In–Se layer blocks arranged in an ABBCA sequence, with van der Waals forces bonding the interlayers [69]. Each layer comprises both tetrahedrally and octahedrally coordinated indium atoms, as illustrated in Fig. 3D. The 2H and 3R structures show different stacking arrangements. Unlike the “zigzag” pattern characteristic of the 2H structure, the layers in the 3R structure are distinguished solely by a translation in the ab plane [68]. The exploration of 3R structure began with Osamura et al. in 1966 [70], who first proposed a model featuring InSe₄ tetrahedra and InSe₆ octahedra, but with questionable details of interatomic distances. Later, another structure with In atoms coordinated solely in tetrahedral arrangements was suggested by Ye et al. [71], but with highly questionable singly coordinated Se atoms. In 2015, Debbichi et al. [72] introduced a new model [Fig. 3D(i)] through quantum-chemical calculations, which was finally confirmed by Zhou et al. [73] in 2017 through electron microscopy. Compared to purely octahedral coordination, the mixed coordination in 3R is more conducive to the stability for both covalent and ionic contributions [68].

In₂S₃ exhibits 3 different crystalline phases: α-type (cubic structure between 717 and 1,049 K), β-type (tetragonal structure below 717 K), and γ-type (trigonal structure above 1,049 K) [74] (Fig. 3E to G). The low-temperature phase β-In₂S₃ can be described with a defective spinel-like structure with orderly arranged indium vacancies. It consists of octahedral and tetrahedral In atoms, bonded exclusively to S atoms. The space group is I4₁/amd (no. 141), and the lattice parameters are a = 7.6231(4) Å and c = 32.358(3) Å. The space group of α-In₂S₃ is Fd

3 ¯

m (no. 227), with a lattice parameter of 10.8315(2) Å. α-In₂S₃ shows a random distribution of indium vacancies over all tetrahedral sites. The high-temperature phase γ-In₂S₃ adopts a layered structure with the space group of P

3 ¯

m1 (no. 164) and lattice parameters of a = 3.8656(2) Å and c = 9.1569(5) Å. In γ-In₂S₃, the S–In–S–In–S slabs consist of close-packed sulfur layers, with indium atoms having octahedral coordination.

In₂Te₅

In₂Te₅ has a layered structure with zigzag layers (as shown in Fig. 3J), which are formed by edge-sharing In–Te tetrahedra and connected Te–Te bonds [75]. There are 2 distinct crystalline forms of In₂Te₅, characterized by space groups Cc and C2/c [76], respectively. These 2 forms differ in their stacking sequence (Fig. 3H and I). The structure with space group Cc has lattice parameters of a = 4.39 Å, b = 16.39 Å, and c = 13.52 Å, and the structure belonging to the space group C2/c exhibits lattice parameters of a = 16.66 Å, b = 4.36 Å, and c = 41.34 Å [77].

For binary indium-based chalcogenides, a comprehensive summary of the various materials typically involves the conventional In–X bonding, which serves as the cornerstone of these structures. Nevertheless, we have noted that most indium-based chalcogenides feature unique structural motifs, such as In–In bonding, Se–Se bonding, planar-coordinated Te–Te bonding, and In3 chain. Additionally, the mixed valence states of indium, for example, In⁺, In²⁺, and In³⁺, importantly contribute to the structural stability, electrical transport, and thermal transport properties of these materials. The interplay between different types of bonding and valence states may give rise to the diverse and complex behaviors observed in these materials, making them a subject of great interest for further research and application.

Electronic and Phonon Band Structures

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To investigate the role of the aforementioned unconventional bonds, a consistent computational approach was adopted for the calculation of band structures, phonon dispersions, and related orbital and vibrational properties of these binary InX compounds. The Vienna ab initio simulation package [78] was utilized for all ab initio calculations based on density functional theory, employing the projector-augmented wave method [79]. The revised Perdew–Burke–Ernzerhof for solids (PBEsol) functional [80] was employed. Considering the layered crystal structure in most systems, a van der Waals correction was implemented using the DFT-D3 method [81]. The cutoff of plane-wave energy was 520 eV. The energy convergence criterion used for electronic band structure was 10⁻⁴ eV. Structural optimization was conducted until residual forces on all atoms fell below 0.01 eV/Å, ensuring equilibrium geometries for both lattice constants and atomic coordinates. The electronic structures were analyzed employing the modified Becke–Johnson method (mBJ) [82,83] to more accurately evaluate the band gaps. Bonding analysis was performed via the band-resolved projected crystal orbital Hamilton population (pCOHP) method [84], using the Local Orbital Basis Suite Towards Electronic-Structure Reconstruction package [85]. For lattice dynamics, phonon dispersions were computed via the PBEsol functional within the PHONOPY code [86], employing stringent convergence thresholds of 5 × 10⁻⁸ eV for energies and 10⁻⁵ eV/Å for atomic forces. The VibCrystal tool was utilized to visualize the lattice vibrations [87].

Electronic band structure

The space groups and lattice constants (with calculated results presented in unit cell form for direct comparison with experimental data) of these calculated room-temperature phases are listed, and the computed band gaps are compared with experimental data in Table. Computational results exhibit good consistency with experimental band gap values across most systems. Additionally, Heyd–Scuseria–Ernzerhof (HSE06) [88] hybrid functionals were implemented for band structure analysis for some systems (as shown in Figs. S1 to S3). For certain systems, such as α(3R)-In₂Se₃, the band gap obtained using the mBJ method demonstrates better alignment with experimental values than that using the HSE06 functional (Fig. S2). Considering the computational cost and accuracy for systems with more than 20 atoms, the results derived from the mBJ approach were ultimately adopted. The specific band structures will be elucidated in Indium tellurides, Indium selenides, and Indium sulfides. Concurrently, we focus on the comparative bonding analysis of indium selenides as a representative example.

Indium tellurides

Figure 4 displays the calculated band structures of In₄Te₃, InTe, α-In₂Te₃, Cc, and C2/c phases of In₂Te₅, all computed using the mBJ potential based on primitive cells. Selected wavefunctions associated with the special bonding systems under investigation are also included. Among the 5 indium telluride compounds, 4 (excluding InTe) exhibit indirect band gaps ranging from 0.2 to 0.3 eV.

For In₄Te₃, our calculation yielded an indirect band gap of 0.285 eV, close to the experimental data (0.29 eV [89]). The valence band maximum (VBM) occurs along the Γ–X direction, while the conduction band minimum (CBM) is along the Y–Γ direction. Observations reveal the formation of a camel's back-like structure near the Γ point, in close proximity to the VBM. This feature forms a pocket-like configuration. Each pocket-like structure exhibit pronounced contributions to the power factor (PF) [90]. More pockets in proximity to the Fermi surface are advantageous for enhancing the Seebeck coefficient, as the enhanced density of states (DOS) near the Fermi level [91].

For InTe, it does not exhibit a band gap, which differs from the small band gaps (0.06 to 0.3 eV) reported in other studies [30,32,92]. This discrepancy may be due to the incorporation of van der Waals correction that leads to a reduced lattice constant, and InTe does not exhibit the pronounced layered properties. After excluding the effects of van der Waals interaction, we employed the advanced r²SCAN functional [93] to relax the structure and calculate the electronic structure of InTe, revealing a bandgap of approximately 0.16 eV (see Fig. S4). The VBM resides at the M point (−0.5, 0.5, 0.5), while the CBM is located along the X–P path, in agreement with previously reported results [30,45].

For α-In₂Te₃, the VBM is along the K–Γ direction, while the CBM is located at X (0.5, 0, 0.5) point, with an indirect band gap of 0.318 eV. The valence band edge exhibits little dispersion, leading to a large effective mass that impedes electrical transport. Reports on this phase are limited.

For Cc-In₂Te₅, the VBM is along the Γ–X direction (as indicated by the red marking in Fig. 4D), while the CBM is located at Γ, with an indirect band gap of 0.288 eV (dark green circle). For C2/c-In₂Te₅, both the VBM and CBM are not located at the high symmetry point, with an indirect band gap of 0.399 eV. It is worth noting that both phases exhibit notable band edge dispersion, particularly the M-shaped VBM in the C2/c phase, which is conducive to the electrical transport properties. Analysis of the wavefunctions at the band edge (Fig. 4E, F, H, and I) reveals highly similar compositions in these 2 phases. Notably, both show significant contributions from planar-coordinated Te (p-orbital) at VBM, while the CBM is mainly composed of conventional In (s-like orbital) and Te–p orbital states.

Indium selenides

Indium selenides are a class of materials with varying compositions and crystal structures, significantly impacting their physical properties. Elucidating the electronic structural nuances of indium selenides may facilitate the exploration of potential methodologies for the enhancement of their TE performance. The calculated band structures and partial wavefunctions of In–Se compounds are shown in Fig. 5.

Similar to In₄Te₃, In₄Se₃ also has shaper camel' s back-like VBM, and CBM is along the Y–Γ direction, exhibiting an indirect band gap of 0.221 eV. As shown in Fig. 5D, wavefunction analysis reveals that the VBM of In₄Se₃ primarily consists of In–s orbitals (particularly from the isolated In4 atoms, represented by purple spheres in the figure) and Se–p orbitals, consistent with the findings reported by Losovyj et al. [94]. For InSe, β and γ phases show comparable indirect band gaps. Despite belonging to distinct space groups and having different high-symmetry points, the valence and conduction band edges of these compounds exhibit analogous shapes. The wavefunction distributions (Fig. 5E and F) further demonstrate similar band-edge compositions between these 2 phases. The VBM in both cases predominantly arises from Se–p orbitals with potential interlayer Se interactions, while the CBM likely originates from In–In bonding interactions combined with Se–p orbital contributions. These features will be further analyzed through subsequent pCOHP calculations. This similarity arises from the identical structural motifs within their individual layers, differentiated solely by the stacking sequences, as previously described in the structural context.

In₆Se₇ has no band gap according to our calculation in both mBJ and HSE06 potential (see Fig. S2D). For α(3R)-In₂Se₃, its VBM is located along Γ–X, with the CBM situated at Γ point, featuring an indirect gap of 1.450 eV. As illustrated in Fig. 5I, the VBM of α(3R)-In₂Se₃ comprises Se–p orbital characteristics. It has the same space group as γ-InSe, which is equivalent to adding an additional Se layer on the basis of the latter. Compared with γ-InSe, the band structure has undergone changes, particularly along the segments L–B ₁|B–Z and Γ–X|Q–F, resulting in the emergence of a multi-valley valence band edge. This may be attributed to the variations in the unusual Se–Se and In–In bonding within the material, which will be specifically analyzed later.

To further elucidate the influence of bonding beyond the conventional indium–chalcogenide interactions within binary indium chalcogenides, a case study of indium selenide compounds was carried out. Specifically, γ-InSe and α(3R)-In₂Se₃ were selected due to their appropriate system size for chemical bonding analysis. The results of band-resolved pCOHP calculations are shown in Fig. 6, where red and blue represent anti-bonding and bonding, respectively. According to Fig. 6A and B, the VBM of both γ-InSe and α(3R)-In₂Se₃ exhibits anti-bonding interactions stemming from interlayer Se–Se interactions (consistent with the results of the VBM orbital in Fig. 5F and I), with the latter exhibiting a more pronounced effect. The multi-valley feature of the VBM in In₂Se₃ is likely attributed to the shortened interlayer Se–Se bond length (3.176 Å in the In₂Se₃ phase, as opposed to 3.262 Å in the InSe phase). The anti-bonding interaction of Se–Se increases the band energy along Γ–L, B–Z, and Γ–X, which leads to a more effective convergence of the band at the VBM [95]. In addition to the Se–Se interactions, our investigation reveals that the In–In bonding within InSe also exerts certain effects in valance band, characterized by the bonding interactions that arise from the proximity of In–In atoms (Fig. 6C). In contrast, within the In₂Se₃ phase, the absence of neighboring In–In layers excludes the presence of such interaction (see almost green area in Fig. 6D). Consistent with the previous wavefunction result, the cyan lines at the L point near 1 eV in Fig. 6C also validate the minimal In–In bonding interactions at the CBM of InSe.

Indium sulfides

The calculated band structures of InS, In₆S₇, and β-In₂S₃ in the mBJ potential are shown in Fig. 7. For InS, the VBM lies along the Γ–X direction, while the CBM is located at the R point (0.5, 0.5, 0.5), with an indirect band gap of 0.421 eV. As shown in Fig. 7B, the orbital compositions at the band edge of InS show clear features: The VBM is dominated by bonding interactions between In atoms and dumbbell-shaped orbitals localized on S atoms, while the CBM encompasses spherical In orbitals combined with contributions from S–p orbitals. For In₆S₇, both the VBM and CBM are located away from high-symmetry points. It exhibits a narrow indirect band gap of 0.031 eV, and its band structure is similar to that of In₆Se₇. In contrast, β-In₂S₃ reveals an indirect gap of 2.346 eV, with dispersionless VBM and a Γ-point CBM. Owing to its suitable wide band gap, β-In₂S₃ thin films have been utilized as an efficient electron transport layer for perovskite solar cells [96].

Phonon band structure

The computational phonon dispersions with total and partial phonon DOS for several systems, chosen for their fewer atoms in the primitive cells, are shown in Fig. 8. Note that InTe exhibits imaginary frequencies (~−0.2 THz) near the Γ point (see Fig. 8A), which is consistent with the computational results of other studies [97], unless under a pressure of 3 GPa [32,98]. The In³⁺ and Te atoms are the primary contributing factors of the low-frequency phonons. It is noteworthy that the special In⁺ in InTe (red line in Fig. 8A) is active in the mid-frequency optical branch (with frequencies of 1.78 and 2.04 THz, and the specific vibrational modes at Γ point can be seen in Fig. S5). Little interaction of In⁺ with other atoms was observed, which indicates the weakly bound “rattling” In⁺ atoms in the structure. This characteristic is expected to increase the anharmonicity of the system, thereby enhancing the Grüneisen parameter, which effectively reduces the lattice thermal conductivity [32,97].

Regarding In₄Se₃, its small phonon group velocity can be estimated from the flat dispersion of the phonon dispersion (see Fig. 8B). Typically, the lattice thermal conductivity is predominantly attributed to the acoustic branch. The relatively small slope of the acoustic phonon branch in this system suggests a low phonon group velocity, which serves as an indicator of its low lattice thermal conductivity. Similar to InTe, isolated In⁺ atom is found in In₄Se₃ (red line in Fig. 8B) and is active in the low- to mid-frequency region of the phonon dispersion, e.g., 2.08 THz at Γ point (Fig. S6). Furthermore, our analysis reveals the presence of In–In interactions within the high-frequency optical branch. Notably, the highest optical mode at Γ point, with a frequency of 7.04 THz, demonstrates a relative motion confined to the (In₃)⁵⁺ chain (Fig. S6). The high-frequency stretching and low-frequency collective motion of the (In₃)⁵⁺ chain confirm the strong chemical bonding between the In–In atoms in the (In₃)⁵⁺ chain [99,100].

Due to the difference in their layer stacking, the phonon dispersion of β-InSe and γ-InSe exhibits distinct features (see Fig. 8C and D); however, certain characteristics, such as the gap between phonon branches, remain similar. In contrast to the bulk InTe, both InSe materials display distinct layered vibrational modes at Γ point, encompassing intra- and interlayered motion. In β-InSe, the low-frequency optical phonons correspond to motions within the ab plane (Fig. S7). For example, in the phonon mode with a frequency of 0.96 THz, the Se–In–In–Se layers undergo relative shear motion against each another, resembling 2 interlayer slips. Such mode, with energies in close proximity to acoustic phonons, can interact strongly with them. This interaction leads to strong phonon–phonon scattering and an enhancement of anharmonicity, thereby resulting in a low thermal conductivity [101]. In the slightly higher mode at 1.23 THz, every 2 layers of In–Se move in opposite directions within the ab plane, corresponding to intralayer shear. At 3.54 THz, every 2 layers of In–Se move in opposite directions along the c axis. In the highest optical mode (7.52 THz), adjacent atomic layers (whether In or Se layers) vibrate toward each other along the c axis. Similarly, in the γ phase, the low-frequency region (1.34 THz) exhibits intralayer sliding at Γ point, while the mid-frequency region (4.3 THz) and the highest mode (7.38 THz) involve vibrations along the c axis (Fig. S8).

In α(3R)-In₂Se₃, a tiny imaginary mode appears around the zone center (see Fig. 8E). This mode cannot be related to any structural transition, as its frequency is found to be negligible. All phonons at Γ point below 3.6 THz correspond to motions within the ab plane (Fig. S9). Specifically, the phonon at 3.52 THz, which exhibits relatively flat dispersion, represents the vibration of the octahedral In–Se atoms (also blue line in DOS of Fig. 8E). The branch at 3.78 THz encompasses the relative motion between tetrahedral Se and octahedral Se. The concerted motion of these 2 types of Se–Se interactions occurs at 5.60 THz. The highest optical mode at Γ point (7.68 THz) is the reverse motion between 2 neighboring atomic layers along the c axis, except octahedral In. This is also reflected in the phonon DOS, where octahedral In shows no contribution at high frequencies.

As for InS (Fig. 8F and Fig. S10), phonons at Γ point below 2.25 THz are associated with motions confined to the ab plane, while those between 2.37 and 3.90 THz correspond to breathing-mode vibrations along the c axis, such as the 3.13-THz mode shown in Fig. S10. The pendular motion of the S atoms at Γ point predominantly governs the phonon modes between 4.50 and 7.50 THz, including the 7.34-THz mode shown in Fig. S10.

In summary, the presence of unconventional bonds (e.g., In–In, Te–Te, and Se–Se) in In–X compounds may influence the electronic structure near the band edges. Phonon dispersions, DOS, and their corresponding vibrational modes further highlight the effects of unconventional bonds on the lattice vibrations. Meanwhile, the isolated In⁺ atom vibrating at the low-to-mid frequency enhances the anharmonicity of the system, and the softened low-frequency optical branch, with energies similar to those of acoustic phonons, increases phonon–phonon scattering. These features both serve as indications of low thermal conductivity. However, the underlying mechanisms contributing to the low thermal conductivity of these materials still require further investigation. Our computational results provide microscopic insights into the role of unconventional bonds in shaping the electronic and phononic transport characteristics of In–X compounds.

Experimental Strategies for Optimizing TE Properties

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At the experimental level, we have summarized the TE optimization strategies employed in recent years for binary indium-based chalcogenides. These approaches can be classified into 4 categories: defect engineering, crystal orientation engineering, nanostructuring, and grain size engineering, as shown in Fig. 9A. These strategies have effectively improved thermal transport, electrical transport, or both simultaneously, leading to a significant enhancement in the zT value. For thermal transport, point defects (e.g., vacancies [32,42] and impurity atoms [102]), line defects (dislocations [30]), and nanoprecipitates [103,104] can effectively scatter phonons, thereby reducing the lattice thermal conductivity (Fig. 9B). Doping, a common approach to generate point defects, enhances phonon scattering through lattice distortions caused by solute atoms with mismatched ionic radii (Fig. 9C and D). For electrical transport, the carrier concentration can be controlled by adjusting the doping level. For example, Zhou et al. [105] reduced the carrier concentration by introducing the donor impurity Pb at the In⁺ site in p-type InTe materials. In contrast, Zhai et al. [106] increased the carrier concentration by doping Sn in β-InSe, which also resulted in a narrowed band gap. Additionally, doping can induce energy filtering effects (Fig. 9E). For instance, Luo et al. [107] found that the carrier concentration in Pb/Cu co-doped samples was lower than that in Pb single-doped samples, while the carrier mobility exhibited an opposite trend. This behavior was primarily attributed to the higher work function of Cu (4.65 eV) compared to In₄Se₃ (4.30 eV), forming an energy barrier that filters out low-energy electrons and resulting in a higher S at elevated temperatures in the co-doped samples. In addition, the temperature dependence of electrical conductivity can be tuned by increasing the grain size [30,92], and leveraging the intrinsic anisotropy of the material may optimize thermal or electrical properties [42,108], thereby enhancing TE performance. These methods, either individually or in combination, have been successfully implemented in indium-based chalcogenide compounds. Accordingly, this section reviews recent experimental advances in these modulation strategies and examines the underlying physical mechanisms.

Defect engineering

Self-vacancy manipulation

In order to optimize the TE performance, self-vacancy is widely adopted as an effective strategy in InTe and In₄Se₃ systems. For p- and n-type semiconductors, self-vacancy has the unique advantage of tailoring the carrier concentration and phonon scattering while minimizing the impact on crystal structure.

For p-type InTe, Jana et al. [32] successfully enhanced the PF (PF = σS²) and reduced the lattice thermal conductivity by creating In deficiency (Fig. 10A to C). As a result, the zT value of the In_0.997Te sample was significantly enhanced, peaking at approximately 0.9 at 600 K. This represents a notable improvement compared to that of undoped InTe. The increase in the PF was mainly driven by the improved carrier mobility and carrier concentration. The large reduction in lattice thermal conductivity was expected to arise from phonon scattering caused by vacancy-type point defects. Similarly, the researchers also experimented with Te deficiency [109]. The zT values of InTe_1−δ were substantially enhanced within the mid-temperature range (T ≤ 500 K), due to a decrease in lattice thermal conductivity and an increase in the PF.

The introduction of Se deficiency into n-type In₄Se₃ has been a well-established strategy. In 2009, Rhyee et al. [42] successfully synthesized 2 Se-deficient crystals (In₄Se_2.78 and In₄Se_2.35). The In₄Se_2.35 crystal exhibited an unusually high zT value of 1.48 along the b–c plane at 705 K, surpassing the zT value of 1.1 in the In₄Se_2.78 crystal along the same plane at the same temperature. This zT enhancement primarily resulted from greater Se deficiency in the In₄Se_2.35 crystal, which amplified phonon scattering, thereby leading to a significant reduction in thermal conductivity (see Fig. 10D to F). Later, Rhyee et al. [110] reported that increasing the Se deficiency (x) of In₄Se_3−x from 0.02 to 0.05 resulted in an enhancement of zT from ~0.40 to 0.63 at 710 K, as there is a reduction in the band gap and an increase in the PF. Moreover, the thermal conductivity was lowered due to the disordered phonon scattering caused by the Se-deficient sites. In addition, Alsharafi et al. [111] found that increasing the Se deficiency in the Pb-doped In₄Se₃ polycrystalline samples (In₄Pb_0.01Se_3−x, x = 0 to 0.1) increased the Hall carrier concentration, which consequently lowered the electrical resistivity. Furthermore, the lattice thermal conductivity was also reduced. Ultimately, the sample with x = 0.07 exhibited a peak zT of 0.95 at 690 K, outperforming the sample without Se vacancies (x = 0) by approximately 15% (zT = 0.83).

Doping engineering

Doping engineering has become a key approach for optimizing the TE properties. By adopting the doping strategy, the electrical conductivity and Seebeck coefficient of a material can be improved due to the optimization of carrier concentration, and the thermal conductivity can also be reduced because of increasing the number of scattering centers for phonons.

Misra et al. [102] reported that the atomic disorder was induced in the structure by doping Pb into InTe, which increased phonon scattering and decreased the lattice thermal conductivity to 0.22 W m⁻¹ K⁻¹. Consequently, In_0.999Pb_0.001Te achieves a peak zT of 1.05 at 790 K. Nevertheless, the use of Pb element raises concerns about environmental and health hazards. Therefore, researchers have been exploring for less toxic alternatives like Cu and Na [112], but generally achieving slightly poorer performance than Pb doping. Until 2022, Li et al. [30] significantly improved the TE properties of InTe through the synergistic optimization of electrical and thermal transport properties. They found that through Ga doping in InTe, the grain boundary scattering (GBS) was purified and weak phonon–electron coupling was induced, thereby improving carrier concentration and mobility. Thus, the PF value was increased to 8.9 μW cm⁻¹ K⁻² at 500 K. Transmission electron microscopy (TEM) characterization results revealed dense in-grain dislocation arrays along with multiple lattice domains with pronounced structural distortions [30] (Fig. 11A and B), which effectively enhanced the scattering of intermediate-frequency phonons, which reduced the lattice thermal conductivity (Fig. 11C). As a result, the In_0.99Ga_0.01Te sample achieved a peak zT value of 1.2 at 648 K and an average zT of 0.8 over the temperature range 300 to 650 K [30], outperforming all other known InTe-based materials (Fig. 11D). Other doping elements such as Bi, Ag, Mn, Sn, and Sb also have been investigated by Song et al. [45] for InTe. Among these doped InTe samples, In_0.99Sn_0.01Te exhibited the highest zT value of 0.64 at 725 K, corresponding to a >50% improvement over undoped InTe. The TE performance enhancement of these doped InTe samples is primarily due to the reduction in thermal conductivity.

Luo et al. [107] introduced multiple doping of Pb, I, and Cu in In₄Se_2.5, which improved the electrical conductivity through simultaneous improvement in carrier concentration and Hall mobility (Fig. 11E). As a result, the highest values of PF and zT were achieved in the (Pb + Cu + I)-doped In₄Se_2.5 polycrystalline sample from 300 to 723 K (zT = 1.4 at 723 K) (Fig. 11F and G). Co-doping of Pb and Sn in In₄Se₃ also leads to a larger peak zT of 1.4 at 733 K, mainly because Pb and Sn act as effective electron donors, which significantly enhance the carrier concentration by an order of magnitude compared to the undoped sample [103]. The PF value of In₄Pb_0.01Sn _ySe₃ (y = 0.03 or y = 0.04) is ∼10 μW cm⁻¹ K⁻² at 733 K, representing an enhancement of nearly 80% compared to undoped In₄Se₃ [103]. He et al. [113] found that Yb doping at the In site of In₄Se₃ leads to a lower thermal conductivity, resulting in a zT value of 0.81 at 703 K for the In_3.97Yb_0.03Se₃ and In_3.95Yb_0.05Se₃ samples, which is an improvement of approximately 30% compared to pure In₄Se₃ [113]. In addition, other dopants, e.g., Ag [37], Sn [40], CuI [114], and Cl [115] in In₄Se_3−x, and Sn [116] and Cu [117], have also been demonstrated to improve the TE performance of In₄Se₃.

InSe has a low intrinsic carrier concentration (~10¹⁴ cm⁻³) due to its relatively large band gap [118], and thus, the electrical conductivity needs to be increased. In 2013, Zhai et al. [106] prepared polycrystalline n-type In_1.3−xSn _xSe samples (x = 0, 0.05, 0.1, and 0.2). By doping Sn into β-InSe, a combination of the increased carrier concentration due to a narrowed band gap and the reduced lattice thermal conductivity led to the highest zT value of 0.66 at 700 K for the In_1.25Sn_0.05Se sample (Fig. 12). This value is 57% higher than that of the undoped sample (zT = 0.42). Although doping studies on InSe have continued in recent years, e.g., doping with elements such as Si [118,119] and Cu [120] at the cationic sites, and Cl [121] and Te [122] at the anionic sites, the zT value has not been significantly improved.

In₆Se₇ can undergo a transition from p-type to n-type conduction after Sn or Pb doping [52,53]. In pure In₆Se₇, In exhibits multiple valence states (+1, +2, and +3). But Pb²⁺ and Sn⁴⁺ are inclined to occupy the In⁺ sites, thus resulting in the p–n transition. This p–n transition leads to a significant enhancement in the Hall carrier concentration. Consequently, the Sn-doped sample In_5.9Sn_0.1Se₇ achieved a zT value of 0.28 at 833 K, representing an approximately 19-fold enhancement compared to the undoped In₆Se₇, which exhibited a zT of only 0.015 at 640 K [52]. The In_5.5Pb_0.5Se₇ sample exhibited the highest zT value of 0.4 at ~850 K [53] (Fig. 13).

Cui et al. [123] studied Cu doping in n-type α-In₂Se₃, which led to a band gap reduction. Although the transition from an amorphous-like structure to a distinct polycrystalline morphology increased the lattice thermal conductivity (Fig. 14A, B, and E), the PF values increased by 3 to 4 times due to the improved electrical conductivity as a result of the band gap reduction, and eventually achieving a peak zT value of approximately 0.55 in the In_1.8Cu_0.2Se₃ sample (x = 0.2) at 846 K (Fig. 14). In another study by Cui et al. [124] in 2015, a non-equilibrium fabrication technology (NEFT) was used to achieve Zn/S doping in In₂Se₃. This doping improved the electrical conductivity of the material. Zn doping created an anti-site defect Zn_In as a donor, resulting in a peak zT value of 1.23 (± 0.02) at 916 K in the In_1.99Zn_0.01Se₃ sample. Doping isoelectronic S atoms at the Se site helped avoid the annihilation of donor defects (i.e., V_In and In_i), resulting in a zT peak value of 0.67 in the α-In₂S_0.05Se_2.95 sample at 923 K. This represents an enhancement of approximately 2.8 times compared to the undoped α-In₂Se₃, which exhibited a zT of only 0.24 [125]. Additionally, doping Si at the In site effectively reduces the thermal conductivity, reaching ~0.35 W m⁻¹ K⁻¹ for the β-phase In_2−xSi _xSe₃ (x = 0.005) at 500 K [126].

For β-In₂S₃, substituting In atoms with Mg atoms (In_2−xMg _xS₃, 0 ≤ x ≤ 0.20) can effectively reduce the lattice thermal conductivity [127]. The In₂S₃ samples with x ≥ 0.30 underwent a phase transition to the cubic α phase, which has too high electrical resistivity to obtain meaningful electrical transport properties. For the In_1.95Mg_0.05S₃ sample, a high zT value of 0.53 was achieved at 700 K, which represents a 1.4-fold increase compared to the undoped sample.

Crystal orientation engineering

Misra et al. [108] grew InTe single crystals using the Bridgman–Stockbarger method and demonstrated a noticeable anisotropy in electrical resistivity and Seebeck coefficient when comparing the c axis and [110] directions of the crystal structure (Fig. 15A to C). Furthermore, the lattice thermal conductivity along the [110] direction is exceptionally low, only 0.32 W m⁻¹ K⁻¹ at 780 K. A combination of the extremely low thermal conductivity and relatively high PF yields a high zT value of 0.61 at 780 K.

The TE properties of polycrystalline InTe can be improved through texture modulation [128]. The main step was through the oriented crystal hot-deformation method, which is a process of taking the top part of the InTe single crystal and applying a uniaxial pressure in the direction perpendicular to the (110) plane. The texture degrees were evaluated by analyzing the relative peak intensities from the powder x-ray diffraction (PXRD) patterns of the samples. The results indicated that the ZM-HP sample obtained from the single crystal treated with hot pressing (i.e., hot-deformation method) exhibited the highest texture of (110), as shown in Fig. 15D. It has been previously reported that the lattice thermal conductivity along the [110] direction is lower than that along the [001] direction due to the bonding asymmetry and lattice anharmonicity in InTe [32,46,102]. Consequently, the lattice thermal conductivity decreases with the increase of the [110] texture degree, as shown in Fig. 15E. Finally, the ZM-HP sample exhibited a peak zT value of 0.95 at 623 K, as shown in Fig. 15F.

Rhyee et al. [42] discovered that the CDW instability induced a significant anisotropy in the electrical and thermal transport properties of In₄Se_3−x crystals (Fig. 16). Over the temperature range of 300 to 700 K, In₄Se_2.35 crystals exhibited considerably lower thermal conductivity along the b–c plane than along the a-b plane. The thermal conductivity along the b–c plane is ≤1.2 W m⁻¹ K⁻¹ at 300 K and 0.74 W m⁻¹ K⁻¹ at 705 K. This decrease in the thermal conductivity is caused by Peierls lattice distortion in the b–c plane due to the CDW effect. Furthermore, the Peierls distortion effect causes a decline in carrier concentration along the b–c plane. Ultimately, an exceptionally high zT value of 1.48 at 705 K was obtained along the b–c plane for In₄Se_2.35 crystals.

Nanostructuring

As shown in Fig. 17A to D, Zhu et al. [129] introduced trace amounts of Sb nanoprecipitates into InTe, successfully converting the dominant scattering mechanism from intervalley scattering to acoustic phonon scattering (APS) at elevated temperatures. As a result, carrier mobility and PF show notable improvement, and ultimately, the InTe-Sb_0.01 sample achieved a peak zT value of 0.8 at 623 K.

Nanostructuring has also been shown to be an effective method for enhancing the TE performance of the In₄Se₃ system. When In₄Se_3−x samples were synthesized through mechanical alloying and hot-pressing techniques, nanoscale In precipitates were easily formed, which increased phonon scattering and resulted in a reduction in thermal conductivity. Specifically, the thermal conductivity of the In₄Se_2.2 sample reaches a minimum value of 0.41 W m⁻¹ K⁻¹ and a maximum zT value of 1.13 at 723 K [104]. In a separate study, Lin et al. [103] observed the In nanoparticles (i.e., 5- to 70-nm particles in the Pb/Sn co-doped In₄Se₃ sample and 3- to 100-nm particles in the Pb-doped sample) using TEM (Fig. 17E to H). These nanoparticles significantly enhance phonon scattering, resulting in a decrease in lattice thermal conductivity. The thermal conductivity can reach approximately 0.56 W m⁻¹ K⁻¹ at 733 K in the Pb/Sn co-doped sample, which is notably lower than those of undoped In₄Se₃ (0.65 W m⁻¹ K⁻¹) and Pb-doped In_4−xPb _xSe₃ (0.75 W m⁻¹ K⁻¹). Other approaches, such as the Cu intercalation and In nanoprecipitation produced by Cu embedding and Br substitution in In₄Se_2.5 polycrystals [130], Cu nanoincorporation in In₄Se₃ bulk materials through thermal decomposition of Cu(OAc)₂ [117], and the formation of biphasic structure by combining In₄Se₃ and In nanostructures [131], have also been found to effectively promote the reduction of lattice thermal conductivity in In₄Se₃.

Grain size engineering

The grain size of undoped and Cd-doped InTe samples was notably enlarged by extending the annealing time from 2 to 7 d (Fig. 18A and B) [92]. The InTe-7d sample exhibits higher carrier mobility at room temperature (11 cm⁻² v⁻¹ s⁻¹), ~110% enhancement compared to the InTe-2d sample. This increase in carrier mobility was attributed to the larger grain size and reduced scattering effect from the ionizing impurities [92], resulting in a substantial improvement in electrical conductivity (Fig. 18C) and PF. Specifically, after annealing for 7 d, the peak zT values at 773 K of the undoped InTe sample and the In_0.98Cd_0.02Te sample are 0.70 and 0.87, respectively, obviously higher than those of the samples annealed for 2 d (zT ~ 0.5 at 773 K) (Fig. 18D). Li et al. [30] also demonstrated that grain coarsening in the InTe material can achieve a transition from dominant GBS to APS, thus improving the Hall mobility and PF. As a consequence, the coarse-grain InTe sample exhibited a peak zT of 0.85 at 650 K, outperforming the fine-grain InTe sample (zT = ~0.6 at 650 K). Similarly, Feng et al. [128] also found that grain growth not only suppressed GBS, thereby enhancing electrical conductivity, but also exerted negligible influence on the density-of-states effective mass (m*_DOS). This implies that enhancing electrical conductivity through grain growth has little impact on S, which is advantageous for optimizing electrical performance. Furthermore, attenuating GBS through grain enlargement significantly enhances the TE properties of InTe materials, particularly at low temperatures. Moreover, Zhou et al. [105] concluded that both grain size and carrier concentration notably influence the GBS behavior in InTe sample. They tuned the carrier concentration by substituting Pb for In⁺ and found that GBS is predominantly governed by grain size when the carrier concentration exceeds 0.7 × 10¹⁹ cm⁻³, whereas at lower carrier concentrations, it is primarily influenced by the carrier concentration itself. Therefore, by modulating both grain size and carrier concentration, the TE performance of InTe materials can be optimized across a broad temperature range, leading to enhanced energy conversion efficiency in practical applications.

Summary and Outlook

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Indium tellurides

The emergence of InTe in the TE field has been attributed to its relatively high electrical conductivity and very low thermal conductivity [108]. The low thermal conductivity is mainly due to the strong anharmonicity resulting from weakly bound In¹⁺ ions exhibiting dynamic lone pair expression [46]. Experimental studies have shown that doping InTe with elements such as Pb, Ga, Cd, and Sb significantly improves the TE performance. Figure 19A shows the temperature dependence of the zT values for representative InTe-based samples reported in literature. Manipulating the grain size of InTe for the purification of GBS has also been demonstrated to enhance the TE performance. All experimental studies on InTe thus far have exclusively reported p-type behavior, which is primarily attributed to the prevailing In¹⁺ vacancy defect that stabilizes the Fermi level near the valence band edge [32,45,46,108]. It is predicted that if InTe can be n-doped, an enhanced PF and zT could be achieved in n-type InTe, owing to its high valley degeneracy of the CBM [45]. For future research, exploring new dopants, grain size engineering, or realizing the n-type transition may be considered to manipulate the TE performance of InTe.

There are limited reports on other In–Te systems (i.e., In₂Te₃, In₄Te₃, In₃Te₄, and In₂Te₅) in the TE field. The electrical properties of In₂Te₃ are generally poor, making it challenging to enhance its TE performance [132,133]. Leveraging its inherent structural vacancy, In₂Te₃ has been utilized as an alloying component in various material systems, including Cu₂SnSe₄ [134], CuGaTe [135], SnTe [136], GeTe [137], and InSb [138], to form tunable solid solutions. The vacancy-mediated solid solution not only facilitates carrier concentration optimization but also introduces additional scattering centers. This strategic incorporation enables band structure engineering while simultaneously suppressing lattice thermal conductivity via enhanced phonon scattering mechanisms. Similarly, In₂Te₃ nanowires, as a relatively underexplored one-dimensional nanomaterial, offer a potential for future scientific exploration [139]. Additionally, while the thermal conductivity of In₄Te₃ [89] and In₂Te₅ [140] can be reduced to very low levels, their high electrical resistivity limits the increase of the zT value. Thus, reducing the electrical resistivity could be the starting point for research with the aim of significantly enhancing the TE performance.

Indium selenides

In₄Se₃ exhibits excellent TE performance in both single crystals and polycrystals. The reported zT values as a function of temperature for the representative samples are plotted in Fig. 19B. In₄Se₃ shows anisotropy as a result of its distinctive crystal structure. The Peierls distortion of the crystal lattice induced by the CDW along the b–c plane results in strong electron–phonon coupling. Accordingly, the In₄Se_2.35 single crystal demonstrates exceptionally low thermal conductivity along the b–c plane, achieving a high zT value of 1.48 at 705 K [42]. In addition to monocrystalline In₄Se_2.35, polycrystalline In₄Pb_0.01Sn _ySe₃ (y = 0.03, 0.04) has demonstrated a high zT value of 1.4 at 733 K [103], mainly because of the successful doping of Pb/Sn as an electron donor in In₄Se₃. The Pb/Sn doping increases the Hall carrier concentration and decreases the thermal conductivity. Previous experimental researches on In₄Se₃ have demonstrated that its TE properties can be greatly enhanced via Se deficiency manipulation, doping engineering, nanostructuring, or employing a combination of these strategies. Se deficiency has been extensively investigated as a prominent approach. Nanostructuring, such as generating nanoscale precipitates, has been designed to scatter phonons effectively, resulting in reduced thermal conductivity. Future research could focus on the exploration of Pb-free single-element doping or multi-element co-doping, as well as nanostructuring, with the aim of discovering innovative methods for boosting the TE performance of In₄Se₃.

There has been some research on InSe in recent years, but its TE performance has not been significantly improved. Currently, the highest zT value of 0.66 was reported in In_1.25Sn_0.05Se at 700 K [106]. InSe possesses a relatively wide energy gap of ~1.2 eV, leading to a low intrinsic carrier concentration of around 10¹⁴ cm⁻³ [118]. Consequently, the lower carrier concentration limits the electrical conductivity and Seebeck coefficient. In contrast with In₄Se₃, In₂Se₃ has been less studied, but it has shown promise with a maximum zT of 1.23 [124]. This opens up more possibilities for studying this material system. Additionally, In₆Se₇ has successfully undergone a transition from p-type to n-type conduction, offering further opportunities for regulating its TE properties. Overall, for these materials, methods such as doping, nanostructuring, or achieving p–n transition could be applied to further enhance their TE properties. In addition to their TE performance, the unique plastic deformation capability of InSe warrants comprehensive investigation. Recent studies [101,141] have shown that the interlayer slip mechanism and unusually high ductility offer a robust foundation for the mechanical performance of flexible electronic devices.

Indium sulfides

Up to now, there has been limited research on TE properties of In₂S₃. The lattice thermal conductivity of In₂S₃ can be effectively reduced by substituting Mg for In. The optimized composition In_1.95Mg_0.05S₃ achieves a peak zT of 0.53 at 700 K, which is primarily attributed to its exceptional PF at this temperature [127]. Additionally, theoretical calculations have shown that the lattice thermal conductivity of 2-dimensional InS is relatively low at room temperature, ~0.6 W m⁻¹ K⁻¹ [142]. Since both InS and In₂S₃ show low thermal conductivity, one can start with optimizing the electrical transport properties through various possible methods (e.g., doping) so as to enhance their TE performance. The exploration of In₆S₇ and In₃S₄ for TE applications remains largely underexplored, highlighting the need for a comprehensive research strategy that combines cutting-edge experimental techniques with first-principles theoretical investigations.

Despite the substantial progress in optimizing the TE properties of the binary indium-based chalcogenides described above, their practical application still faces several key challenges: (a) regarding material stability, some indium-based chalcogenides exhibit pronounced temperature-dependent phase transitions (e.g., In₂Te₃ [10,61], In₂Se₃ [64–66], and In₂S₃ [74]), which may restrict their operating temperature range in TE applications; (b) regarding scalability, it remains challenging to balance batch preparation and microstructure control using current synthesis methods (e.g., melting combined with spark plasma sintering [30,111] and NEFTs [124]); (c) in terms of cost-effectiveness, the scarcity of high-purity indium and the complexity of the doping process substantially increase manufacturing costs. To address these challenges, future research could focus on several key directions: developing synergistic phase transition modulation strategies [143–146], such as suppressing harmful transitions to expand the operational temperature range of materials; exploring new scalable synthesis routes, such as batch production processes used in Mg–Si–Sn [147], Mg–Si [148], and Bi–Sb–Te [149,150] systems; and reducing material costs by developing recycling techniques for waste materials and investigating low-cost element substitutes, such as Cu and Ag. In addition, it is recommended to integrate advanced computational and experimental approaches within these systems [151–153]. These include first-principles calculations, high-throughput screening, automated synthesis, and characterization techniques. Such integration would enable the systematic optimization of material compositions and processing parameters, thereby accelerating the pace of materials research and development.

Overall, this Review elaborates on the crystal structure, electronic structure, and phonon dispersion for binary indium-based chalcogenides, subsequently summarizing TE optimization strategies, i.e., defect engineering, crystal orientation engineering, nanostructuring, and grain size engineering. The perspectives on the challenges and opportunities in the TE field for binary indium-based chalcogenides are also discussed. Crucially, the comprehensive computational study presented in this Review, integrating electronic structure calculations, orbital and bonding analysis, and phonon dispersion evaluations, provides a robust framework for comprehending the intricate structure–property relationships inherent in these materials and establishes a guide for innovative strategies aimed at boosting the TE performance.

Funding

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National Key Research and Development Program of China(No. 2024YFF0505900)
National Natural Science Foundation of China(Grant No. 52172216 and 92163212)
Shanghai Magnolia Talent Program
Shanghai Technical Service Center of Science and Engineering Computing in Shanghai University
Hefei Advanced Computing Center
Shanghai Engineering Research Center for Integrated Circuits and Advanced Display Materials

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doi: 10.34133/research.0727

Receive Date：2025-03-28
Online Date：2025-07-21
Published：2025-06-10

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Received：2025-03-28
Revised：2025-04-25
Accepted：2025-05-12

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National Key Research and Development Program of China(No. 2024YFF0505900)

National Natural Science Foundation of China(Grant No. 52172216 and 92163212)

Shanghai Magnolia Talent Program

Shanghai Technical Service Center of Science and Engineering Computing in Shanghai University

Hefei Advanced Computing Center

Shanghai Engineering Research Center for Integrated Circuits and Advanced Display Materials

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Materials Genome Institute, Shanghai Engineering Research Center for Integrated Circuits and Advanced Display Materials, Shanghai University, Shanghai 200444, China.

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^* Address correspondence to: musenc@shu.edu.cn (S.D.); lirongsong@shu.edu.cn (L.S.); jiongy@t.shu.edu.cn (J.Y.)

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表12种不同金属材料的力学参数

科 Family	属数 Number of genus	种数 Number of species	占总种数比例 Percentage of total species (%)	属 Genus	种数 Number of species	占总种数比例 Percentage of total species (%)
鹅膏菌科Amanitaceae	2	11	5.26	鹅膏菌属 Amanita	10	4.78
小菇科 Mycenaceae	2	12	5.74	丝盖伞属 Inocybe	5	2.39
多孔菌科 Polyporaceae	8	14	6.70	蜡蘑属 Laccaria	5	2.39
红菇科 Russulaceae	3	23	11.00	小皮伞属 Marasmius	6	2.87
				小菇属 Mycena	11	5.26
				光柄菇属 Pluteus	5	2.39
				红菇属 Russula	17	8.13
				栓菌属 Trametes	5	2.39

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Table. Calculated lattice parameters and band gaps of binary indium-based chalcogenides

Type	Formula	N atoms in primitive cell	Crystal structure	Calculated lattice parameters (Å)			Band gap (eV)
Type	Formula	N atoms in primitive cell	Crystal structure	a	b	c	This work	Experiment
Indium telluride	In₄Te₃	28	Pnnm (no. 58)	12.050	14.161	4.411	0.285	0.29 [89]
	InTe	8	I4/mcm (no. 140)	7.593	7.593	6.931	0	0.22 [92]
	α-In₂Te₃	45	F $4 ¯$ 3m (no. 216)	17.521	17.521	17.521	0.318	-
	In₂Te₅	14	Cc (no. 9)	4.094	15.087	12.590	0.288	-
	In₂Te₅	42	C2/c (no. 15)	15.151	4.138	37.607	0.399	0.21–0.27 [77,160]
Indium selenide	In₄Se₃	28	Pnnm (no. 58)	11.688	13.985	4.022	0.221	0.42 [89]
	β-InSe	8	P6₃ /mmc (no. 194)	3.816	3.816	15.865	1.269	1.2 [106]
	γ-InSe	4	R3m (no. 160)	3.871	3.871	23.111	1.222	-
	In₆Se₇	26	P2₁/m (no. 11)	9.031	3.876	16.642	0	0.54 [161]
	α(3R)-In₂Se₃	5	R3m (no. 160)	3.937	3.937	26.447	1.450	1.42 [124]
Indium sulfide	InS	8	Pnnm (no. 58)	3.834	10.730	3.829	0.421	1.9 [31]
	In₆S₇	26	P2₁/m (no. 11)	8.746	3.730	16.087	0.031	0.64–0.75 [162]
	β-In₂S₃	40	I4₁/amd (no. 141)	7.498	7.498	30.172	2.346	2.0–2.2 [163]

Fig. 1. (A) Timeline of publications and citations for In–X (X = Te, Se, S) material systems in the TE field (data retrieved from Web of Science). (B) Reported total thermal conductivity at 300 K (with ±1% error bars) and zT _max values (with ±5% error bars) for pure binary indium-based chalcogenides [53,77,89,102,103,106,124,127].

Fig. 2. (A) Crystal structure of In₄(Se/Te)₃. (B) Crystal structure of InTe. (i) Single-unit cell, (ii) diagram of the InTe structure with a covalent chain of [In³⁺Te²⁻ _4/2]⁻ along the c axis, and (iii) the In¹⁺ centered distorted square antiprismatic configuration of Te atoms. (C) (i) Side view and (ii) top view of β-InSe. (D) (i) Side view and (ii) top view of γ-InSe. (E) Crystal structure of InS. (F) Crystal structure of In₆(Se/S)₇.

Fig. 3. (A) (i) Crystal structure of In₃Se₄ and (ii) polyhedral view of the octahedral arrangement of In atoms with their surrounding Se atoms. Crystal structures of (B) α-In₂Te₃ and (C) β-In₂Te₃. (D) (i) Crystal structure of α(3R)-In₂Se₃ and (ii) polyhedral view of the tetrahedral/octahedral arrangement of In atoms with their surrounding Se atoms. Crystal structures of (E) α-In₂S₃, (F) β-In₂S₃, and (G) γ-In₂S₃. Crystal structures of (H) Cc and (I) C2/c phases of In₂Te₅, and (J) (i) a close-up view of the layered structure, with the planar-coordinated Te structural unit emphasized, and (ii) view with only Te atom rows. Reproduced with permission [76]. Copyright 2020 American Chemical Society.

Fig. 4. Calculated band structures of (A) In₄Te₃, (B) InTe, (C) α-In₂Te₃, (D) Cc, and (G) C2/c phases of In₂Te₅, and the Fermi levels are set to zero. The corresponding wavefunctions of (E) Cc-In₂Te₅ at VBM, (F) Cc-In₂Te₅ at CBM, (H) C2/c-In₂Te₅ at VBM, and (I) C2/c-In₂Te₅ at CBM. Colored circles mark the positions of wavefunctions.

Fig. 5. Calculated band structures of (A) In₄Se₃, (B) β-InSe, (C) γ-InSe, (G) In₆Se₇, and (H) α(3R)-In₂Se₃, and the Fermi levels are set to zero. The corresponding wavefunctions of (D) In₄Se₃ at VBM, (E) β-InSe at VBM and CBM, (F) γ-InSe at VBM and CBM, and (I) α(3R)-In₂Se₃ at VBM. Colored circles mark the positions of wavefunctions.

Fig. 6. Band-resolved pCOHP of (A) Se–Se in γ-InSe, (B) Se–Se in α(3R)-In₂Se₃, (C) In–In in γ-InSe, and (D) In–In in α(3R)-In₂Se₃. The Fermi level is set to zero. Red represents anti-bonding, and blue represents bonding.

Fig. 7. Calculated band structures of (A) InS, (C) In₆S₇, and (D) β-In₂S₃, and the Fermi levels are set to zero. The corresponding wavefunctions of (B) InS at VBM and CBM. Colored circles mark the positions of wavefunctions.

Fig. 8. Calculated phonon band structures with total and partial phonon DOS of (A) InTe, (B) In₄Se₃, (C) β-InSe, (D) γ-InSe, (E) α(3R)-In₂Se₃, and (F) InS.

Fig. 9. (A) Experimental strategies for optimizing TE properties through defect engineering, crystal orientation engineering, and nanostructure and grain size engineering. Upper-left figure: Reproduced with permission [154]. Copyright 2020 American Chemical Society. Reproduced with permission [105]. Copyright 2023 by Zhou et al. Lower-left figure: Reproduced with permission [105]. Copyright 2023 by Zhou et al. Reproduced with permission [92]. Copyright 2020 Elsevier B.V. Upper-right figure: Reproduced with permission [117]. Copyright 2013 The American Ceramic Society. Reproduced with permission [103]. Copyright 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim. Reproduced with permission [129]. Copyright 2019 the Royal Society of Chemistry. Lower-right figure: Reproduced with permission [108]. Copyright 2021 the Royal Society of Chemistry. Reproduced with permission [128]. Copyright 2023 the Royal Society of Chemistry. (B) Schematic representation of the behavior of carriers and phonons. Reproduced with permission [155]. Copyright Science China Press 2024. (C and D) Schematic representation of local distortions caused by solid solution of solute atoms with different radii. Reproduced with permission [156]. Copyright 2023 Wiley-VCH GmbH. (E) Schematic representation of the energy filtering effect. Reproduced with permission [157]. Copyright 2021 Chinese Chemical Society.

Fig. 10. Temperature-dependent (A) PF, (B) κ _L, and (C) zT (with 10% error bars) of In_1−xTe samples. Reproduced with permission [32]. Copyright 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim. TE properties of Se-deficient In₄Se_3- _δ crystals. (D) Temperature-dependent thermal conductivity and (E) temperature-dependent electrical resistivity and PF (inset). (F) Temperature-dependent zT and Hall carrier concentration n _Hall (inset). Reproduced with permission [42]. Copyright 2009 Macmillan Publishers Limited.

Fig. 11. (A and B) Inverse fast Fourier transform patterns of high-resolution TEM lattice images of the ( $2 ¯$ 0 $1 ¯$ ) and (110) planes for the In_0.99Ga_0.01Te sample. Temperature-dependent (C) κ _L and (D) zT of In_1−xGa _xTe samples. Reproduced with permission [30]. Copyright 2022 Wiley-VCH GmbH. (E) Carrier concentration and Hall mobility at room temperature, and an inset illustrating the schematic diagram of the energy filtering mechanism. Temperature-dependent (F) PF and (G) zT for the undoped/doped In₄Se_2.5. Reproduced with permission [107]. Copyright 2022 the Royal Society of Chemistry.

Fig. 13. Temperature-dependent (A) S, (B) σ, where the thermal activation energy ΔE _a < 0 as T ≥ 835 to 840 K, (C) κ _L, where the top right inset is the total thermal conductivity, and (D) zT, where the inset shows zT versus n _H (Hall carrier concentration) for the In_6−xPb _xSe₇ samples. Reproduced with permission [53]. Copyright 2012 American Chemical Society.

Fig. 14. (A) The amorphous-like structure of α-In₂Se₃ is revealed by the high-resolution TEM image. (B) High-resolution TEM image of In_1.8Cu_0.2Se₃ shows its polycrystalline nature with grain sizes of 5 to 15 nm (gray-lined areas). The upper left inset in (B) shows the energy-dispersive x-ray spectroscopy results with a molar ratio of In:Cu:Se = 2.02:0.28:2.78. The upper right inset in (B) displays the polycrystalline form from electron diffraction. Temperature-dependent (C) σ, (D) PF, (E) κ _L, and (F) zT values for In_2−xCu _xSe₃ bulk materials. Reproduced with permission [123]. Copyright 2011 American Institute of Physics.

Fig. 15. Temperature dependence of (A) σ, (B) S, and (C) PF along the c and [110] direction for single crystals of InTe. Reproduced with permission [108]. Copyright 2021 the Royal Society of Chemistry. (D) PXRD patterns of the <50-HP, 100-HP, Ingot-HP, and ZM-HP InTe samples. Comparison of the temperature-dependent (E) total thermal conductivity and (F) zT of these InTe samples as well as the previously reported InTe samples. Reproduced with permission [128]. Copyright 2023 the Royal Society of Chemistry.

Fig. 16. (A and B) High-resolution TEM images and corresponding electron diffraction patterns of In₄Se_3−δ (δ = 0.22). (A) The a-b plane. The diffraction patterns exhibited quasi-one-dimensional chains aligned along the b axis. The presence of Bragg spots with superstructure peaks along the b axis is attributed to commensurate lattice distortions. (B) The cross-sectional plane of the crystal. The red rhomboid highlights the diffraction pattern corresponding to the unit cell. (C to F) Anisotropic TE properties as functions of temperature, and angle-averaged results from Boltzmann transport calculations for In₄Se_3−δ (δ = 0.65) crystal. Open and filled symbols represent measurements taken before and after the heat treatment (450 °C, 24 h), respectively. Reproduced with permission [42]. Copyright 2009 Macmillan Publishers Limited.

Fig. 17. (A) Diagram illustrating nonequivalent intervalley scattering, identified as the primary scattering mechanism for the carriers at elevated temperatures and (B) different forms of intervalley scattering in InTe. Based on the TEM observations, (C) a portion of Sb appears as nanoprecipitates in the matrix at 300 K, and (D) gradually integrates into the lattice at temperatures exceeding 585 K. Reproduced with permission [129]. Copyright 2019 the Royal Society of Chemistry. Nanostructures embedded in In₄Pb_0.01Sn_0.03Se₃ and In_3.96Pb_0.04Se₃ samples: (E and F) TEM images revealed spherical or ellipsoidal nanoparticles. (G and H) High-resolution TEM images revealed embedded In nanoprecipitates within the host matrix, exhibiting coherent or semi-coherent interfaces. Reproduced with permission [103]. Copyright 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim.

Fig. 18. Field emission scanning electron microscopy (FESEM) images of InTe with different annealing times of (A) 2 d and (B) 7 d. Temperature-dependent (C) σ and (D) zT compared to the previously reported values [32,109]. Reproduced with permission [92]. Copyright 2020 Elsevier B.V. (E) Schematic of Pb substitution at the In⁺ site in InTe. Temperature-dependence of (F) σ, (G) n _H, and (H) S. [102] Reproduced with permission [105]. Copyright 2023 by Zhou et al.

Fig. 19. (A) Temperature-dependent zT values of the reported InTe-based samples [30,32,45,92,102,108,112,128,129]. (B) Temperature-dependent zT values of the reported In₄Se₃-based single crystals [29,42,158] and polycrystals [103,107,111,113,159].

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