## Introduction

Structural disorder plays a crucial role in understanding the physical properties of a material1. Disorder and defects usually act as scattering centers for quasiparticles, such as electrons and phonons, influencing and governing the transport properties. In particular, the structural disorder is well known to scatter phonons and results in a lower thermal conductivity2,3, an essential property that is fundamentally important in technological applications such as thermal barrier coatings and thermoelectric (TE) energy conversion. For the TE technology directly interconverting heat and electricity, reducing thermal conductivity is an essential strategy to improve the figure of merit zT = α2σT/κ of a TE material, where α, σ, T, and κ represent the Seebeck coefficient, electrical conductivity, absolute temperature, and thermal conductivity, respectively. Structural disorder with great potential in reducing thermal conductivity can be achieved in various ways. In addition to the substitutional disorder commonly used in alloys3,4, the disorder can be realized intrinsically through vacancies, interstitial sites, off-center displacements, rattling atoms, or fast ionic diffusion within the unit cell in many state-of-the-art TE materials5,6,7,8,9.

Understanding, characterizing, and modeling the structural disorder is of significant interest in TE materials with ultralow glass-like thermal conductivity, which poses great challenges to both theorists and experimentalists. Ultralow thermal conductivity is common in large complex crystal structures, whereas it is rare in simple crystal structures. However, in recent years many simple inorganic crystalline solids such as SnSe10, Tl3VSe411, and BaTiS312 have been discovered to show ultralow thermal conductivity approaching or even lower than the glass limit13 for the amorphous and highly disordered solids. Theoretical models11,14 recently developed have been quite successful in modeling ultralow, weak temperature-dependent thermal conductivities of crystalline solids through introducing a wave-like tunneling term for describing the disorder. However, very few detailed experimental characterizations on atomic disorder are available in these simple crystalline solids typically because of great challenges for experimentalists to probe subtle structural disorders. One class of such crystalline solids is the TlSe-type compounds. The TlSe structure type covers a rich variety of binary and ternary compositions, among which TlInTe2, TlGaTe2, TlSe, and InTe have been discovered as promising TE materials with ultralow thermal conductivity15,16,17,18. Ultralow thermal conductivity was typically attributed to strong anharmonic rattling induced by weakly bonded In1+ or Tl1+ atoms with lone pair electrons16,17. On the other hand, the atomic disorder, prone to appear for rattling atoms in oversize cages, should not be overlooked as it may also contribute to low thermal conductivity. In particular, the superionic conductivity19,20 widely observed in this type of materials was suggested to be induced by the structural disorder and ion diffusion. However, no direct experimental evidence of the structural disorder or ion diffusion has been reported so far in any TlSe-type compound.

Here we combine multitemperature single-crystal synchrotron X-ray diffraction (SCSXRD) and the maximum entropy method (MEM) to obtain the electron density distribution for probing structural disorder in an archetypal TlSe-type compound InTe. A detailed structural analysis reveals a significant deficiency on the In1+ site as the dominant intrinsic defect for understanding intrinsic p-type behavior, consistent with the theoretical defect calculations. Moreover, the structure contains two interstitial indium sites between the In1+ sites, forming a one-dimensional (1D) disordered In1+ chain along the c axis. Interestingly, the MEM electron density at elevated temperatures and ab initio molecular dynamics simulations clearly show a static-dynamic transition of interstitial, disordered In1+ ions, suggesting a 1D In1+-ion diffusion/hopping pathway. We attribute such a diffusion pathway to a very low In1+-ion migration energy barrier along the c direction, which is found to be a general feature in many other TlSe-type compounds. The local correlations in the 1D In1+ chains are examined through single-crystal diffuse X-ray scattering, and it is found that the static-dynamic transition is related to the degree of local order in the chains. The direct visualization of a 1D disordered diffusion channel not only accounts for the reported superionic conductivity but also provides a basis for understanding ultralow thermal conductivity and its peculiar, weak temperature dependence in InTe and other TlSe-type compounds.

## Results

### Ideal crystal structure and thermal conductivity

Despite simple crystal structure with relatively light mass, InTe has been reported to show ultralow thermal conductivity values of ~0.7–0.4 W m−1 K−1 at 300–700 K16, much lower than those in many well-known binary tellurides such as PbTe and Bi2Te33. InTe, crystallizing in the TlSe-type structure with the space group I4/mcm21, is generally described by the formula $${{{{{{\rm{In}}}}}}}^{1+}{{{{{{\rm{In}}}}}}}^{3+}{{{{{{\rm{Te}}}}}}}_{2}^{2-}$$. The In3+ ions are tetrahedrally coordinated to Te2− ions forming (InTe2) chains along the c axis, while the In1+ ions with 5s2 lone pair electrons are weakly bound to a cage-like system of eight Te atoms with the square antiprismatic arrangement (Fig. 1a, b). Thermal displacement parameters of In1+ ions were found by Hogg and Sutherland21 to be very large and anisotropic with the maximum vibration along the c axis. In this study, we synthesized large InTe single crystals using the vertical Bridgman method. Energy-dispersive spectroscopy (EDS) mapping was conducted using scanning transmission electron microscopy (STEM) and the result confirmed a uniform distribution of In and Te in the crystal (Supplementary Fig. 1). The chemical composition of the single crystal was determined to be x = 0.98 ± 0.01 in InxTe by inductively coupled plasma optical emission spectrometry (ICP-OES) (Supplementary Table 1). Due to the weak interactions between (InTe2) chains, the crystal tends to cleave along the (hh0) planes (Fig. 1c). The intersection line of two perpendicular cleavage planes (hh0) of A and B was used to determine the crystallographic c direction (Supplementary Fig. 2). With the identified crystal orientation, we then measured the thermal conductivity from 2 to 723 K for the InTe crystals along the c direction (see Fig. 1d). The measured low-temperature thermal conductivity data along the c direction show a very good agreement with those reported in ref. 22. By comparing with the reported data along the [110] direction in ref. 23, it is found that the thermal conductivity of InTe single crystal is clearly anisotropic at room temperature but becomes less anisotropic or even nearly isotropic at high temperatures. A peculiar feature of the thermal conductivity is found with a nearly temperature-independent behavior (~T−0.1) at low temperatures of ~25–80 K, where the extrinsic scattering should be more important. Even with increasing temperature well above the Debye temperature (~120 K, Supplementary Table 2), where the intrinsic phonon-phonon scattering is expected to be dominant, the temperature dependence of thermal conductivity (~T−0.69) still deviates from T−1. Such abnormal behavior, also observed in ref. 22, is typically an indication of the extrinsic scattering induced by structural disorder. In this work, we focus on a proper description of the crystal structure as a basis for a profound understanding of the atomic disorder in thermoelectric InTe.

### Disordered In1+ atoms and intrinsic defects

High-resolution SCSXRD data were used for the detailed crystal structure determination of InTe. We conducted the SCSXRD measurements at the BL02B1 beamline24 from SPring-8 using a high photon energy of 50.00 keV. The data at 25 K were measured with a high resolution of sinθ/λ < 1.67 Å–1 on a small high-quality single-crystal extracted from the as-grown large crystal. A total of 89,683 reflections were collected and reduced to 2723 unique reflections with a high redundancy of 32.9 and Rmerge = 4.13% (Supplementary Table 3). The structure at 25 K was solved with lattice parameters of a = 8.3685(3) and c = 7.1212(2) Å in I4/mcm space group. Accurate structure factors obtained from the SCSXRD data were used for MEM calculations. The MEM can provide a non-biased reconstruction of the most probable electron density from scaled and phased structure factors25,26, making it a powerful tool to study the location of atoms27, structural disorder5,28, and migration pathways of ions29,30.

Combining the MEM electron density and structure refinements, a precise structure model can be derived for understanding the atomic disorder (Fig. 2a, b). A three-dimensional (3D) MEM electron density map of InTe at 25 K (Fig. 2a) clearly reveals two small, nonequivalent additional regions of electron density between the In1+ atoms along the c direction. The two-dimensional (2D) slice and 1D profile of the MEM density further confirm two small electron density maxima between the In1+ atoms (Fig. 2c, d, and Supplementary Figs. 3–5), suggesting two interstitial sites with low occupancy in addition to the three main atomic sites. The localized nature of the extra electron density with distinct peaks along the c direction demonstrates a clear 1D static disordered feature. By analyzing the peak positions with crystal symmetry, two interstitial sites were determined to be 8 f (0.5 0.5 0.5463(4)) and 8 f (0.5 0.5 0.6369(4)). If interstitial indium atoms of Ini(1) and Ini(2) are allocated to the two additional sites, a two interstitial model can be obtained (see Fig. 2b and Supplementary Table 4).

To better elaborate the structure determination, we conducted and compared the refinements of different structure models with intermediate steps towards the two interstitial model (see Table 1 and Supplementary Tables 57). We started with the reported simple full occupancy model21 with three main atomic sites fully occupied, then moved to the vacancy model with the indium vacancy allowed, and finally explored the two interstitial model with two interstitial indium atoms added. Notably, we found that each intermediate step greatly improves the agreement with the experimental SCSXRD data as the RF value drops remarkably from 2.54% to 2.06% to 1.33%, the wRF value from 5.54% to 4.14% to 2.15%, and the goodness-of-fit (GoF) from 3.02 to 2.26 to 1.37. In addition to the significantly improved fit to the experimental diffraction data, only the two interstitial model shows a composition of In0.98Te consistent with the experimental value measured by ICP-OES.

The new structure determination paves the way to understand the dominant intrinsic defects and persistent p-type behavior of InTe. In contrast to the nearly fully occupied In3+ site, our structure refinements clearly show a significant vacancy on the In1+ site with only ~90% occupancy (Table 1). Moreover, the distances between In1+ and In-interstitial sites are generally too small (<2.2 Å) so that it is unlikely for indium atoms to occupy these sites at the same time, suggesting the formation of Frenkel pairs. The dominant In1+ vacancy along with small proportions of the Frenkel pairs thereby reasonably describes the observed structural disorder. To further elucidate the experimental observation, we conducted formation energy calculations of native defects in InTe using density functional theory. As depicted in Fig. 2e, the vacancy on the In1+ site is the dominant native defect showing a much lower formation energy than other intrinsic defects including the In3+ vacancy, Te vacancy, and antisite defects. The easier formation of the In1+ vacancy may be attributed to very weak adjacent bonds of In1+ atoms, showing large nearest-neighbor distances of 3.56 and 3.54 Å, respectively with In1+ and Te atoms. The dominant native defect of the In1+ vacancy, normally negatively charged, is known to pin the Fermi level close to the valence band maximum, which therefore explains the persistent p-type behavior of InTe reported in previous studies16,23.

In comparison with the In1+ vacancy, the formation of Frenkel pairs including the experimental indium-interstitials alone is energetically unfavorable owing to a bit higher formation energies (Fig. 2e), which is expected due to the resulting small In-In distances. However, on the basis of the In1+ vacancy, the formation of Frenkel pairs becomes more favorable with lower formation energy, indicating that the formation of Frenkel pairs in InTe is vacancy-mediated, consistent with the experimental observation. This may typically be ascribed to the vacancy-induced void space that helps avoid small In-In distances during the formation of Frenkel pairs.

### Temperature-driven 1D diffusive In1+ channel

To explore the temperature dependence of the disordered indium in InTe, we further conducted MEM calculations based on the SCSXRD data at 100–700 K (Supplementary Table 8 and Supplementary Figs. 610). Similarly, the SCSXRD data at 100–700 K were collected with high resolutions of (sinθ/λ)max = 0.97–1.83, as well as high redundancy of 28.8-88.0 (Supplementary Table 3). The 3D and 2D MEM electron density maps at various temperatures of 25–700 K are plotted in Fig. 3a, b for comparison. Strikingly, with increasing temperature, electron densities on the In1+ and interstitial In sites become delocalized and eventually form a smooth, continuous 1D diffusion channel through In1+ atoms along the c direction. In Fig. 3c we compare the normalized 1D MEM electron density profile through In1+ atoms along the [001] direction at 25–700 K (see also Supplementary Figs. 11 and 12). At 25 and 100 K, the electron densities with distinct peaks on the two interstitial In sites are generally well separated, though the electron densities of the In1+ and the nearest interstitial Ini(2) are partially connected. With increasing temperature from 100 to 200 K, there are two notable changes in the 1D electron density profile. One is that the In1+ density expands and merges with the electron density of the Ini(2), resulting in the disappearance of the peak on the interstitial site Ini(2). The other is the onset of the electron density connection between the In1+, Ini(1), and Ini(2) through the c direction (see also Supplementary Fig. 13). The two clear changes in electron density are likely the origin of the kink observed on the resistivity versus temperature curve between 100 and 200 K (Supplementary Fig. 14). The kink in resistivity data has been consistently observed in both single-crystalline22 and polycrystalline31 InTe at low temperatures and is thus not due to a measurement artifact. Finally, as the temperature increases further (≥300 K) the electron density maxima of the Ini(1) are smeared out and flattened, forming a continuous 1D channel connecting In1+ atoms along the c direction.

In Fig. 3d we show the temperature-dependent constrained structure refinements with one interstitial indium manually moving along the 1D channel between the In1+ main sites. At temperatures of 25–200 K, the RF, wRF, and GoF values all show two distinct local minima, the location of which exactly correspond to the two indium interstitial sites. This signifies the generally static feature of the two interstitial indium sites. In contrast, at elevated temperatures of 300–700 K the RF, wRF, and GoF values generally become nearly constant with moving the interstitial indium between the In1+ sites along the c direction, suggesting nearly equal probability for the occupation of the interstitial indium. This result along with the MEM densities indicates a static-dynamic transition of the interstitial indium positions. The dynamic behavior of the interstitial indiums makes the structure refinements with the two interstitial model only feasible up to 200 K (Supplementary Table 7). A sudden increase in the thermal vibration along the c axis (U33) of the interstitial indiums between 100 and 200 K shown in Fig. 3e suggests the onset of the dynamic motion along the c axis, consistent with the MEM density at 200 K (Fig. 3c and Supplementary Fig. 13). It should be noted that we are not able to determine whether this dynamic behavior is induced by the thermal disorder or the dynamically positional disorder since the time-space averaged MEM density is not deconvoluted with the thermal smearing effect. Nevertheless, the continuous 1D MEM density channel above 200 K clearly reveals the probability for In1+ to diffuse or hop between the In1+ sites, indicating the 1D In1+-ion hopping/diffusion pathway along the c direction (Fig. 3a, f). The In1+-ion diffusion/hopping probability increases with increasing temperatures, but is generally not very high as the MEM electron density value of the diffusion channel even at 700 K is about 5.3% of the maximum peak value of the main In1+ site (Fig. 3c, f). The ion diffusion/hopping in InTe is clearly weaker than those in conventional superionic conductors5,9,29, which might pose a challenge to probe the signature of superionic conductivity in InTe. An external driving force such as a higher temperature, a large electric field20, or the increased In1+ vacancy may be required to enhance the diffusion/hopping probability. The In1+ diffusion/hopping may be associated with very large thermal motions of the In1+ and interstitial indiums along the c direction (Fig. 3e and Supplementary Fig. 15). In particular, exceptionally large U33 values of the In1+ atom at elevated temperatures do not extrapolate to zero at 0 K, another clear indication of atomic disorder along the c direction32.

The local structural ordering of vacancies and interstitial sites in the 1D chains was investigated through diffuse X-ray scattering, shown in Fig. 4a (see also Supplementary Fig. 16). At 25 K the diffuse scattering forms 2D planes for even values of L, seen as lines in the figure, indicative of strong correlations along the c direction, and weak correlations in other directions33. At 300 K the planes have disappeared and been replaced by more 3D features with strong maxima at the Bragg peak positions, a typical indication of thermal diffuse scattering from correlated vibrations. This is seen in more detail using the three-dimensional difference pair distribution function (3D-ΔPDF), shown in Fig. 4b (see also Supplementary Fig. 17). The 3D-ΔPDF shows the local correlations in disordered crystals34,35,36,37. It is a map showing which interatomic vectors are more or less present in the real structure compared to the average crystal structure. Positive features show vectors, which separate atoms more frequently in reality and negative features show vectors separating fewer atoms. The 3D-ΔPDF at 25 K shows strong correlations along z, which become weaker at 300 K, showing the vacancies and interstitials to be strongly correlated along the 1D chains at low temperatures, but less so at higher temperatures. This is shown in more detail along the z direction in Fig. 4c. At x, y, z = 0 there is noise due to Fourier ripples. For vectors equal to ½c and 1c (z equal to 3.56 and 7.1 Å), there are positive peaks, indicating that In atoms will most often be separated by these vectors, as they would in the ideal structure. Around these positive peaks, there are negative signals, indicating vectors that do not separate In atoms in the real structure. At low temperature the negative signal around positive peaks is asymmetric, a typical indication of local relaxations from atoms moving slightly towards neighboring vacancies34,37. At 25 K a positive peak is found at 5.8 Å, corresponding to a vector separating an Ini(1) and an Ini(2) interstitial, as illustrated in Fig. 4d, suggesting that the interstitial Frenkel pairs tend to be separated by this distance. At 300 K the asymmetry in the negative signal and the Frenkel pair peak have disappeared, indicating that the positions of vacancies and interstitials are less correlated at high temperatures consistent with the 1D In1+-ion diffusion/hopping behavior.

The experimental observation of the 1D In1+-ion diffusion pathway is further confirmed by the ab initio molecular dynamics simulation. As illustrated in Fig. 5a, b, and Supplementary Fig. 18, the trajectories of the In1+ ions for the simulation at 700 K show a clear hopping/diffusion behavior along the c direction. We note that only a limited number of In1+ ions show 1D diffusion/hopping behavior in the simulated supercell, indicating that the probability of the In1+-ion diffusion is not very significant consistent with the experimental MEM electron density. To understand the 1D diffusion/hopping pathway of the In1+ in InTe, we further conducted nudged elastic band calculations to estimate the energy barriers for the vacancy-mediated In1+-ion migration. Three possible indium ion migration pathways along the [001], [110], and [100] directions were considered (Fig. 5c). Notably, the In1+ migration barrier along the [001] direction is significantly lower than those along the other two directions (Fig. 5d), which reasonably explains the 1D diffusion channel along the c direction in the MEM electron density. The flat energy landscape with a very low energy barrier may be attributed to the weak atomic interaction along the c direction.

## Discussion

Interstitial, disordered, diffusive atoms with large thermal motions are known to be a highly effective mechanism for reducing thermal conductivity. One remarkable example is Zn4Sb3, which shows extremely low thermal conductivity due to multiple disordered Zn interstitials that substantially reduce the phonon mean-free path5. Interstitial, disordered atomic positions are commonly observed in large complex crystal structures such as Zn4Sb35 and oxide-ion conductors30, whereas they are rarely found in small simple crystal structures. The 1D interstitial, disordered, diffusive In1+ ions discovered in InTe are quite noteworthy, given the simple structure with merely 8 atoms in primitive cell. The lattice thermal conductivity in both single-crystalline and polycrystalline InTe is unexpectedly low at high temperatures16,23, which has been shown to reach the theoretical glass limit (~0.3 W m−1 K−1) proposed by Cahill13 for the amorphous and disordered solids.

The interstitial, disordered, diffusive In1+ ions could be considered as an important origin of the ultralow, weak temperature-dependent lattice thermal conductivity in InTe. The correlated disorder is known to lead to the broadening of phonon linewidth that is inversely proportional to lattice thermal conductivity38. Correlated disorder of indium vacancies and interstitials in 1D In1+ chains could potentially broaden the phonon linewidth for the low-energy In1+ vibration modes, resulting in the suppression of phonon lifetime, phonon mean-free path, and thereby lattice thermal conductivity. In particular, the structural disorder of In1+ ions was suggested by Misra et al.22 to be a key origin of the broadening of In1+-weighted low-energy optical modes and the limited energy range of heat-carrying acoustic phonons, lowering the lattice thermal conductivity. Moreover, the plateau with nearly temperature-independent behavior (~T−0.1) in thermal conductivity at ~25–80 K may be attributed to the correlated static disorder in 1D In1+ chains revealed by the MEM density and 3D-ΔPDF, similar to those observed in strongly disordered materials39,40,41. With increasing temperature above the Debye temperature of ~120 K, as the intrinsic phonon-phonon scattering begins to be dominant thermal conductivity shows a clear decreasing trend but its temperature dependence is still a bit weaker than T−1, which is likely induced by the wavelike tunneling11,14 contribution from interstitial, disordered, diffusive In1+ ions. However, the exact impact of the structural disorder with interstitial, disordered, diffusive In1+ ions comparing with the anharmonic phonon-phonon scattering on lattice thermal conductivity in InTe still requires further systematic theoretical and experimental investigations.

We expect the 1D disordered diffusion channel to be transferable to many other ABX2 compounds with the TlSe-type structure. In Fig. 5e we show the calculated A1+-ion migration energy barrier along the [001] direction in ten other ABX2 compounds with the TlSe-type structure in comparison with that in InTe. Notably, nearly all these compounds show comparable or lower migration energy barriers than that of InTe. This suggests that the 1D disordered diffusion channel of the A1+-ions along the c direction is very likely to appear in many other TlSe-type compounds, given commonly observed large thermal displacements and low migration energy barriers of the A1+-ions along the c direction. The intrinsic p-type transport behavior in nearly all other TlSe-type compounds may be understood by the dominant A1+ cationic vacancy as described in this work. Considering the A1+-ion diffusion has been widely mentioned in previous studies19,20 without any experimental evidence to date, the direct observation of the 1D hopping/diffusion cationic channel in our work thereby provides long-awaited experimental evidence for understanding the electric-field enhanced superionic conductivity19,20 observed in many TlSe-type compounds. Moreover, the proper description and fine details of the atomic disorder and hopping/diffusion in this work could be helpful for theory in developing models to understand ultralow thermal conductivity with weak temperature-dependent behavior in InTe and other TlSe-type chain-like materials.

## Methods

### Single crystal growth and characterization

Large InTe single crystal was synthesized with the vertical Bridgman method. Indium shots (99.999%, ChemPUR) and tellurium pieces (99.999%, ChemPUR) were combined in stoichiometric ratio and placed into a carbon-coated quartz ampoule. The ampoule was flame-sealed after being evacuated to a pressure below 10−4 mbar. The sealed ampoule was then put into a box furnace, heated to 1023 K in 10 h with a dwell time of 24 h, and then slowly cooled to room temperature in 10 h. The obtained polycrystalline ingot was crushed into small pieces and loaded into a carbon-coated quartz ampoule with a conical tip, which was subsequently evacuated and sealed. For the crystal growth, the sample was kept at 1030 K for 24 h for homogenizing the melt and then slowly cooled directionally from the melt with a temperature gradient of ~6 K cm-1 and a sample moving rate of 2 mm h-1. To determine the c axis of the single crystal, X-ray diffraction data of cleavage surfaces were measured on a Rigaku Smartlab in the Bragg-Brentano geometry using a Cu Kα1 source. The intersection line of the two perpendicular cleavage planes (hh0) defines the c axis. The chemical composition of the InTe single crystal was determined using the inductively coupled plasma optical emission spectrometry (ICP-OES). The sample was dissolved with PlasmaPure aqua regia and diluted to 1% acid concentration with MilliQ-water. The ICP-OES measurement was carried out on a Spectro ARCOS ICP-OES equipped with a Burgener Nebulizer and Cyclonic Spray Chamber with an ASX-520 Auto sampler. For quantification of the elements, the standard curves series was measured. The standard series consists of concentrations for each element with 0, 0.01, 0.05, 0.1, 0.5, 1, 5, 10, 20, and 50 ppm. The homogeneous elemental distribution of the InTe single crystal was confirmed with the Scanning Transmission Electron Microscopy Energy Dispersive Spectroscopy (STEM-EDS) by elemental mapping analysis using an FEI Talos F200X microscope operated at 200 kV. The microscope is equipped with an X-FEG electron source and Super-X EDS detector system. Small single crystals were extracted from the as-grown big crystal and screened with a Bruker Kappa Apex II diffractometer equipped with a Mo source (λ = 0.71073 Å). A small high-quality needle-shaped single crystal (~20 × 20 × 200 μm3) was selected for the single-crystal synchrotron X-ray diffraction.

### Single-crystal synchrotron X-ray diffraction

High-resolution single-crystal synchrotron X-ray diffraction data of InTe were collected at the BL02B1 beamline24 from SPring-8 using a photon energy of 50.00 keV with a Pilatus3 X 1 M CdTe (P3) detector, which recently has been found capable of achieving extremely high quality for the electron density data42. The low-temperature data at 25, 100, 200, 300, and 400 K and the high-temperature data at 500, 600, and 700 K were collected with two different wavelengths of 0.2480 Å and 0.2509 Å on two different beam times, respectively. The collected frames were converted to the Bruker.sfrm format42, which were integrated using SAINT-Plus43. After the integration, the data were scaled and corrected for absorption and other random errors using SADABS44. Subsequently, the data averaging and the uncertainty estimation were conducted using SORTAV45. The structure was solved with SHELXT46, and the detailed structure refinement was conducted using JANA200647 (Supplementary Note 1 and Supplementary Tables 5–7). Details of the single-crystal diffraction data collection and reduction are summarized in Supplementary Table 3. The high-quality single-crystal synchrotron data here ensure accurate structure factors for obtaining high-quality, smooth MEM electron density.

### MEM density analysis

Structure factors extracted from the structure refinement of high-resolution single-crystal synchrotron X-ray diffraction data were used for the calculations of MEM electron densities. The Sakato-Sato algorithm implemented in the BayMEM software48 was used to conduct the MEM calculations. The unit cell of InTe was divided into $${N}_{pix}=168\times 168\times 144$$ pixels along the a, b, and c directions with a fine grid size of ~0.05 Å. Using scaled, phased, “error-free” structure factors as input, MEM calculations reconstruct and determine the electron density of the pixel i ($${\rho }_{i}$$) that maximizes the information entropy, defined as

$$S=-{\sum }_{i=1}^{{N}_{pix}}{\rho }_{i}\,{{{{\mathrm{ln}}}}}\left(\frac{{\rho }_{i}}{{\rho }_{i}^{prior}}\right),$$
(1)

under the constraints based on the observed structure factors

$${\chi }^{2}=\frac{1}{{N}_{F}}{\sum }_{i=1}^{{N}_{F}}{\left(\frac{|{F}_{obs}-{F}_{MEM}|}{\sigma ({F}_{obs})}\right)}^{2}.$$
(2)

Here $${N}_{F}$$ is the number of the observed structure factors, $${F}_{obs}$$ is the observed experimental structure factors with standard uncertainties $$\sigma ({F}_{obs})$$, $${F}_{MEM}$$ is the structure factors corresponding to the final calculated electron density, and $${\rho }_{i}^{prior}$$ is the prior density that introduces prior information available for the system. Either a uniform prior density or a non-uniform prior density based on the independent spherical atom model may be used for the MEM calculations. In this study, the tests of MEM calculations with both a uniform (flat) prior and a nonuniform prior were conducted and the corresponding results were confirmed to be virtually identical (see Supplementary Fig. 4). For simplicity, the results of MEM calculations with uniform prior density were adopted for discussion in the main text. To ensure the optimal MEM density, the optimal stopping criterion χ2 of the MEM calculation was determined with the fractal dimension analysis of the residual density49 (Supplementary Note 2 and Supplementary Figs. 9 and 10). The MEM electron density is independent of different structural models, which is elucidated by virtually the same results using different structure models as input (Supplementary Fig. 3). For consistency, here we use the MEM electron density calculated with the structure factors from the full occupancy model for discussion in the main text. The 3D MEM electron density maps and crystal structures were visualized by VESTA50.

### Diffuse scattering and 3D-ΔPDF

In addition to the normal Bragg diffraction data for structural analysis and MEM calculations, X-ray diffuse scattering data of InTe single crystal at 25, 100, and 300 K were collected on the same setup at the BL02B1 beamline. Here 4 runs were measured, each a 180° Ω rotation with 900 frames, for χ = 0° and χ = 40°, with the detector at 2θ = 0° and 2θ = 20°. An exposure time of 2 s per frame was used. Background and air-scattering were measured using the same exposure time and detector positions as for the crystal. For each combination of these, 200 frames of air scattering were measured and averaged. The data were converted to reciprocal space using a custom Matlab script. During this process the data were corrected for Lorentz and polarization factors, the background scattering from air was subtracted, and a solid angle correction was applied as the detector is flat. The resulting scattering data were reconstructed on a 901 × 901 × 901 point grid with each axis spanning ± 28.3 Å−1. The resulting data were symmetrized using the 4/mmm point symmetry of the Laue group, and outlier rejection was used in the symmetrization, such that symmetry equivalent voxels were compared and rejected if they deviated by more than two standard deviations from the median of the equivalents45. The Bragg peaks were punched and filled. Because a large degree of diffuse scattering is observed at the positions of the Bragg peaks, care has to be taken to remove all Bragg scattering while leaving as much of the diffuse scattering as possible. To accomplish this, the Bragg peaks were masked in the raw data frames before conversion using a Python script which masks sharp and strong peaks in the data. This is possible as the diffuse scattering here is slowly-varying compared to Bragg peaks. The script compares the measured pixels to all neighboring pixels, and if a pixel is relatively much stronger (a factor of 2.5 was used here) than its neighbors, it is marked as part of a Bragg peak. To avoid weak noise from being marked as peaks, a lower bound for the absolute intensity needed for a peak is also used. A high-intensity cutoff is also used, for which pixels larger than this value are automatically marked as peaks. Once peak pixels have been marked this way, the pixels neighboring these are also marked as being part of peaks. This approach effectively masks most Bragg peaks, but leaves some of the almost-zero-intensity Bragg peaks at large scattering vectors. To fill in the masked regions and remove remaining weak peaks, a small punch was applied to the allowed Bragg positions after conversion to reciprocal space and linear interpolation was used to fill. A constant value was also filled into the regions where no data has been measured to minimize Fourier ripples. The 3D-ΔPDF is obtained as the inverse Fourier transform of the diffuse scattering intensity, Id, and is given by the autocorrelation of the deviations in electron density from the average crystal structure34:

$${{{{{\rm{3D}}}}}}\mbox{-}\Delta {{{{{\rm{PDF}}}}}}={ {\mathcal F} }^{-1}[{I}_{d}]=\langle\delta \rho \otimes \delta \rho \,\rangle.$$
(3)

Here $$\delta \rho ({{{\bf{r}}}},t)=\,\rho ({{{\bf{r}}}},t)-{\rho}_{periodic}({{\bf{r}}})$$ is the difference between the total electron density of the crystal and the periodic average electron density. The brackets $$\langle \ldots \rangle$$ indicate time averaging, $$\otimes$$ denotes cross correlation, and $${\mathcal F}$$ is the Fourier transform.

### Thermal conductivity measurements

A small bar-shaped sample obtained by cutting the large single crystal along the c axis was used for low-temperature thermal conductivity measurement. Low-temperature thermal conductivities from 2 to 300 K were measured with a Physical Property Measurement System (PPMS, Quantum Design, US) using the Thermal Transport Option (TTO) under high vacuum. A flat sample (~ 6 × 6 × 1.9 mm3) with the surface normal along the c axis was extracted from the large single crystal and used for high-temperature thermal diffusivity measurement. The thermal diffusivity (D) measurement from 300 to 723 K was carried out on a Netzsch LFA457 setup using the laser flash method. High-temperature thermal conductivity was then determined using $$\kappa \,=\,{{{\it{dD}}}}{C}_{{{\rm{P}}}}$$, where the heat capacity $${C}_{{{\rm{P}}}}$$ was estimated using the Dulong-Petit law $${C}_{{{\rm{P}}}}=3{k}_{{{\rm{B}}}}$$ per atom and the sample density d was measured with the Archimedes method.

### Theoretical calculations

All density functional theory calculations were conducted with the Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation51 based on the projector-augmented wave method52 (PAW) as implemented in the Vienna ab initio simulation package53 (VASP). Defect calculations were conducted in a supercell of 2 × 2 × 2 conventional unit cells. A Γ-centered 4 × 4 × 4 k-point mesh was used for the structure optimization as well as the total energy calculation. A plane wave cutoff energy of 400 eV and an energy convergence criterion of 10−4 eV were applied. For the structural optimization of defect supercells, all atomic positions were relaxed into their equilibrium positions until the Hellmann-Feynman force was converged to be smaller than 0.01 eV Å−1 while the lattice parameters were fixed at the optimized values from the perfect supercell. For the defect supercells with interstitial indiums (Ini(1) and Ini(2)), the positions of interstitial indiums were fixed at the experimental values while the atomic positions of other atoms were fully relaxed. The formation energy of a native point defect was simply calculated using54:

$$\varDelta {E}_{f}^{d}={E}_{tot}^{d}-{E}_{tot}^{bulk}-{\sum }_{i}{n}_{i}{\mu }_{i},$$
(4)

where $$\varDelta {E}_{f}^{d}$$ is the total energy of the defect supercell, $${E}_{tot}^{bulk}$$ is the total energy of the bulk supercell without any defects, $${n}_{i}$$ denotes the number of atoms of type i that is added to (nі > 0) or removed from ($${n}_{i}$$ < 0) the bulk supercell, and $${\mu }_{i}$$ represents the atomic chemical potential. The chemical potentials of In and Te in InTe should fulfill the thermodynamic condition:$$\,\varDelta {\mu }_{{{{{{\rm{In}}}}}}}+\varDelta {\mu }_{{{{{{\rm{Te}}}}}}}=2\varDelta {H}_{f}({{{{{\rm{InTe}}}}}})$$. Under the In-rich condition, the chemical potentials are limited by the formation energy of the secondary phase In4Te3 ($$4\varDelta {\mu }_{{{{{{\rm{In}}}}}}}+3\varDelta {\mu }_{{{{{{\rm{Te}}}}}}}\le 7\varDelta {H}_{f}({{{{{{\rm{In}}}}}}}_{4}{{{{{{\rm{Te}}}}}}}_{3})$$). Under the In-poor condition, the chemical potentials are limited by the formation energy of the competing phase In3Te4 ($$3\varDelta {\mu }_{{{{{{\rm{In}}}}}}}+4\varDelta {\mu }_{{{{{{\rm{Te}}}}}}}\le 7\varDelta {H}_{f}({{{{{{\rm{In}}}}}}}_{3}{{{{{{\rm{Te}}}}}}}_{4})$$).

The calculations of minimum-energy diffusion pathways and migration barriers of In1+ ions in InTe were performed with the climbing image nudged elastic band (CI-NEB) method55. The calculations were conducted in a 2 × 2 × 2 supercell of the conventional unit cell, where one vacancy was introduced on the In1+ sites. We calculated the diffusion energy barriers for a single nearest indium ion diffusing to the created vacancy. The atomic positions were relaxed until the residual forces were within 10 meV Å−1. A total energy convergence criterion of 10−6 eV and a Γ-centered 2 × 2 × 2 k-point mesh were used for the CI-NEB calculations. The same calculation procedure was applied to calculate the migration barriers of the A1+ ion along the [001] direction in several other ABX2 compounds (including InGaTe2, TlGaTe2, TlInTe2, InGaSe2, TlAlSe2, TlGaSe2, TlInSe2, TlInS2, TlSe, and TlS) with the TlSe-type structure.

Ab initio molecular dynamics (MD) calculations were performed using the PBE functional as implemented in the VASP code. The MD simulations at 700 K were conducted with the NVT ensemble using a Nosé thermostat with the default Nosé mass. A supercell with 127 atoms (2 × 2 × 2 conventional cells) containing one In1+ vacancy was used so that the simulated composition In63Te64 (In0.984Te) is close to the experimental observation. A 300 eV cutoff energy and an energy convergence criterion of 10−5 eV along with the Γ-point sampling were adopted for the simulations. The simulations were run for 130 ps for statistical analysis with a time step of 2 fs, and the first 10 ps was disregarded for equilibrium. The trajectories of the MD simulations were visualized by VMD56.