Introduction

Low thermal conductivity (κ) is crucial for applications in thermal barrier coating1,2, energy gas storage3,4, and thermoelectric energy conversion5,6,7, etc. Typically, reducing κ involves enhancing phonon and carrier scattering8,9,10,11 through approaches like boundary engineering12,13, porosity design14,15, and composition tuning16. Among these, composition tuning is the most widely used17,18, as it can be directly implemented by adjusting material content19. This strategy is generally understood in terms of point defects, assuming a homogeneous composition20.

In addition to homogeneous compositions, inhomogeneous composition distribution is also commonly observed in low κ materials21,22, affecting κ beyond the effects of point defects. Compositional inhomogeneity is typically analyzed by effective κ models23,24,25,26, such as parallel model27, series model27, effective medium theory model28, and Maxwell–Eucken model29. These models are based on the Fourier’s Law27,28,29,30, treating phases with varying volume fractions and the κ as individual components. However, this conventional approach neglects the impact of nanostructural features of the neighboring domains.

In this work, by taking Bi0.4Sb1.6Te3 as a case study, we demonstrated that compositional inhomogeneity with randomly aligned edge dislocations enhances phonon scattering, leading to a lower κ. Compared to a directional composition distribution, a random composition distribution results in lower κ due to a higher density of randomly aligned edge dislocations. These dislocations arise from the lattice misfit due to composition gradient domains between Bi-rich and Sb-rich domains, featuring by a preferential orientation along the composition gradient direction (dCG).

Results and discussion

Composition distribution influences thermal conductivity

To explore the relationship between compositional inhomogeneity and the κ in the as-prepared Bi0.4Sb1.6Te3 pellet, scanning electron microscopy (SEM), energy-dispersive X-ray spectroscopy (EDS) and scanning thermal probe micro-image (STPM) measurements were performed on the same area from a well-polished flat surface, as shown in Fig. 1. The backscattered electron (BSE) image (Fig. 1a) reveals a scattered distribution of bright and dark areas, indicating the inhomogeneous composition31. The bright areas, identified as Bi-rich ones due to Bi higher atomic number, and the dark areas, Sb-rich with a lower atomic number, are corroborated by the EDS maps of Bi, Sb and Te (Fig. 1b, c and Supplementary Fig. 5).

Fig. 1: Influence of composition distribution on thermal conductivity.
Fig. 1: Influence of composition distribution on thermal conductivity.
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a, Backscattered electron (BSE) image showing the composition distribution of Bi0.4Sb1.6Te3 pellet. Energy-dispersive X-ray spectroscopy (EDS) maps of Bi, illustrating areas with random (b) and uniform (c) composition distribution. d Scanning thermal probe micro-image (STPM) depicting the thermal conductivity (κ) distribution corresponding to the area in (a). Schematic illustrations on the influence of edge dislocations with different alignment types in areas exhibiting random (e) and directional composition distribution (f), respectively. g A schematic representation of the preferential orientation of edge dislocations along the composition gradient direction (dCG).

Distinguished by the composition distribution pattern, only random composition distribution and directional composition distribution are detected. Bule and orange rectangular areas as shown in Fig. 1a are selected as representatives for random and directional composition gradient areas respectively. Notably, since our sample is prepared by melting method, which can be regarded as a quasi-steady-state process, the compositions of Bi-rich, composition gradient and Sb-rich domains, and the size of composition gradient domains (indicated by average width) can be considered consistent throughout the sample. As a result, magnitude difference between random and directional composition distribution areas are minimal. And the directional feature in composition gradient domains is more dominant than magnitude of variation. The STPM image (Fig. 1d) shows the κ distribution in the same area as in Fig. 1a. To ensure comparability of the result, the κ in random and directional composition distribution areas are measured along the same direction by same method under same condition in different areas. Areas with random composition distribution exhibit lower κ, this is attributed to randomly aligned edge dislocations that enhance phonon scattering (schematically shown in Fig. 1e). Conversely, areas with directional composition distribution display higher κ due to directionally aligned edge dislocations with relatively weaker phonon scattering (schematically shown in Fig. 1f). The correlation between composition distribution and κ difference is observed throughout the bulk sample, which is evidenced by SEM-EDS and STPM characterizations on areas at different depths, obtained by sequentially polishing off surface parts (Supplementary Figs. 6–8). Considering that composition inhomogeneity has minor influence on carrier concentration and corresponding electrical thermal conductivity32, the low κ is mainly contributed by phonon transportation influenced by the random alignment of edge dislocations. This behavior is caused by preferential orientation of edge dislocations along the dCG, as illustrated schematically in Fig. 1g. Notably, these edge dislocations are largely gathered at the composition gradient domains between Bi-rich and Sb-rich domains, being aligned with the macroscale composition gradient distribution. As shown in Supplementary Fig. 5, EDS quantitative analyses on random and directional composition distribution areas show similar compositions, indicating similar density of point defects in these areas. We have excluded the influence of density of point defect on κ from both thermodynamic and kinetic perspectives. From the thermodynamic perspective, composition is a dominant factor influencing the likelihood for point defect formation. According to EDS analysis shown in Supplementary Fig. 5, similar compositions have been observed in random composition distribution area (Bi0.405Sb1.645Te2.95) and directional composition distribution area (Bi0.405Sb1.640Te2.955), implying similar thermodynamic factors. From the kinetic perspective, synthesis conditions (e.g. sintering temperature33, annealing temperature34, cooling rate35) dominate the evolution of point defects. Since both random and directional composition distribution areas are observed in the same sample without special treatment in different areas, the synthesis conditions for them are the same (Supplementary Fig. 3). Therefore, the kinetic factors influencing point defect formation and corresponding density are also the same. Moreover, electron backscatter diffraction (EBSD) analysis is conducted on the same area of STPM measurement. As shown in Supplementary Fig. 9, crystallographic orientations of random and directional composition distribution areas are highly consistent, which can exclude the influence of texture on κ difference. Further details and implications of this alignment will be discussed in subsequent sections.

Composition gradient characterization

To investigate the composition and structure characteristics of the as-prepared Bi0.4Sb1.6Te3 pellet, X-ray diffraction (XRD) and SEM-EDS analyses were performed (Fig. 2). The XRD pattern (Fig. 2a) matches well with the PDF#01-072-1836, with no additional peaks, confirming that sample solely consists of hexagonal Bi0.4Sb1.6Te3 without any secondary phases.

Fig. 2: Structural and compositional characterization of the Bi0.4Sb1.6Te3 pellet.
Fig. 2: Structural and compositional characterization of the Bi0.4Sb1.6Te3 pellet.
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a X-ray diffraction (XRD) pattern matching PDF #01-072-1836, confirming the absence of secondary phases. b Secondary electron (SE) and Backscattered electron (BSE) images highlighting compositional contrast, with Bi-rich domains appearing bright and Sb-rich domains dark. c BSE image of a selected region from (b) with corresponding energy-dispersive X-ray spectroscopy (EDS) maps for Bi, Sb, and Te distribution. d EDS line scan along the selected region in (c), showing compositional variation between Bi-rich and Sb-rich domains. e Representative EDS spectra from Bi-rich and Sb-rich domains in (c), with tables giving the elemental contents shown as insets. f Statistic summary of average atomic percentages of Bi and Sb in Bi-rich and Sb-rich domains.

Figure 2b compares the secondary electron (SE) and BSE images of the sample. While the SE image shows a flat surface without revealing grain boundaries, the BSE image of the same region reveals obvious contrast difference, indicating inhomogeneous composition where the bright areas have higher atomic number than the dark areas. To further analyze the composition distribution, Fig. 2c, d provide an enlarged BSE image of the area in Fig. 2b, along with the corresponding EDS maps and a line scan profile. These results confirm that the bright areas are Bi-rich, while the dark areas are Sb-rich. In contrast, Te is evenly distributed throughout the Bi0.4Sb1.6Te3 pellet. As shown in Fig. 2d, the composition gradually changes from Sb-rich to Bi-rich and back to Bi-rich again. The domain with ~4.5 μm width between Bi-rich and Sb-rich domain is noted as composition gradient domain, which will be further characterized and discussed in subsequent sections. Notably, such composition distribution occurs throughout the sample rather than solely on the top surface as evidenced by additional EDS analysis from multiple perspectives (Supplementary Fig. 10). As shown in Supplementary Fig. 11c, the fracture SEM images indicate the average grain size is >80 μm, which is much larger than the as-selected area. Therefore, grain boundaries are not the reason for the κ difference in random and directional composition distribution areas. Also, according to the XRD and SEM-EDS analyses, the sample is solely composed of Bi0.4Sb1.6Te3 without secondary phases, which can exclude the influence of secondary phase on the κ difference.

To quantitatively compare the composition between Bi-rich and Sb-rich domains, representative EDS point analyses were conducted, as shown in Fig. 2e (corresponding to points in Fig. 2c). The results reveal that the Bi-rich domain has significantly higher Bi content (10.9 at.%) compared to Sb-rich domains (6.7 at.%). Figure 2f provides a statistical summary of the average elemental content in Bi and Sb in Bi-rich and Sb-rich domains. On average, Bi-rich domains contain 10.4 at.% Bi, 30.1 at.% Sb, and 59.5 at.% Te, while Sb-rich domains contain 7.0 at.% Bi, 33.7 at.% Sb, and 59.3 at.% Te. This demonstrates the clear Bi and Sb composition differences in the neighboring areas, while Te content is similar in both areas.

Overall, Fig. 2 confirms the inhomogeneous Bi and Sb compositional distribution in the as-prepared Bi0.4Sb1.6Te3 pellet without any secondary phases seen. The compositionally distinct areas are visually apparent, with Bi-rich domains appearing bright, and Sb-rich domains dark in the BSE images. Since both random and directional composition distribution areas contain widely distributed composition gradient domains, composition gradient domains are also not the reason for κ difference between random and directional composition distribution areas.

Lattice misfit and strain analysis

To examine the structural characteristics of the composition gradient domain between Bi-rich and Sb-rich domains, Fig. 3 presents the SE image of a focused ion beam (FIB) lamella cut off from the Bi0.4Sb1.6Te3 pellet and corresponding transmission electron microscopy (TEM) analyses. Figure 3a provides an overview of the FIB lamella, spanning the Bi-rich and Sb-rich domains (cut-off location shown in Supplementary Fig. 12). A low-magnification TEM image (Fig. 3b) of the red rectangle area in Fig. 3a reveals a continuous, single grain structure without visible grain boundaries, nanoprecipitates or dislocation networks. Also, the almost even contrast throughout the whole sample reflects closely consistent thickness as shown in Fig. 3a, b. Figure 3c displays a Bi-content contour map of the same area, showing a decrease in Bi content from approximately 18.0 to 11.6 at.% from bottom left to top right. Notably, a composition gradient domain, approximately 4.5 μm wide, was observed between the Bi-rich and Sb-rich domains. These domains have been specified in Fig. 3b, c. Figure 3d–f presents high-resolution TEM (HRTEM) images and corresponding strain maps for the Bi-rich, composition gradient, and Sb-rich domains, respectively. Full strain maps with exx, exy, eyx and eyy for these areas are provided in Supplementary Fig. 13. Indicated by minor strain, the Bi-rich and Sb-rich domains exhibit nearly perfect lattice along the [10 5 \(\bar{1}\)] zone axes (indicated by selected area electron diffraction (SAED) patterns shown in Supplementary Fig. 14a and c). In contrast, the composition gradient domain (Fig. 3e) shows significantly larger lattice strain along the same zone axis (indicated by SAED patten shown in Supplementary Fig. 14b), compared with Bi-rich and Sb-rich domains.

Fig. 3: Analysis of lattice misfit and strain.
Fig. 3: Analysis of lattice misfit and strain.
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a Secondary electron (SE) image of a focused ion beam (FIB) lamella cut off from the Bi0.4Sb1.6Te3 pellet. b Low-magnification transmission electron microscopy (TEM) image of the selected area in (a), specifying Bi-rich, composition gradient and Sb-rich domains. c Contour map showing the distribution of Bi atomic content in the area shown in (b), specifying Bi-rich, composition gradient and Sb-rich domains. High-resolution TEM (HRTEM) images and strain maps of Bi-rich (d), composition gradient (e), and Sb-rich (f) domains, respectively, corresponding to the highlighted areas in (b) and (c). Enlarged HRTEM images and corresponding line profiles of selected areas in Bi-rich (g), composition gradient (h), and Sb-rich (i) domains, respectively. j Statistical comparison of planar distances of (0 1 5) plane (d(0 1 5)) among Bi-rich, composition gradient, and Sb-rich domains.

To further analyze lattice structural features, planar distances of (0 1 5) planes (d(0 1 5)) were examined in the Bi-rich, composition gradient, and Sb-rich domains (Fig. 3g–j). Enlarged HRTEM images (Fig. 3g–i) and typical line profiles (averaged over 10 consecutive planes) reveal d(0 1 5) values of 3.27 Å, 3.14 Å and 3.04 Å for the Bi-rich, composition gradient and Sb-rich domains, respectively. Figure 3j provides a statistical summary of d(0 1 5) measurements averaged over 10 selected areas in the Bi-rich, composition gradient, and Sb-rich domains (locations shown in Supplementary Fig. 15). There is a clear trend of decreasing d(0 1 5) with lowering Bi content from the Bi-rich domain through the composition gradient domain to the Sb-rich domain, driven by the difference in atomic radius between Bi (1.56 Å) and Sb (1.40 Å). Additionally, the composition gradient domain exhibits stronger fluctuations of d(0 1 5) (± 0.07 Å), compared to the Bi-rich (± 0.03 Å) and Sb-rich domains (± 0.03 Å), indicating denser lattice distortion. This is consistent with the denser lattice strain observed in the composition gradient domain.

Dislocation characterization and formation mechanism

To further elucidate the nanostructure characteristics of Bi-rich, composition gradient, and Sb-rich domains, additional HRTEM images and inverse fast Fourier transform (IFFT) images were analyzed, as shown in Fig. 4a–c. Edge dislocations are highlighted in the IFFT images. Both Bi-rich and Sb-rich domains exhibit nearly perfect lattice with a negligible amount of edge dislocations, whereas the composition gradient domain shows a significant accumulation of edge dislocations, indicative of a large lattice misfit. As shown in Fig. 4b, dislocations in composition gradient domain are highlighted. Based on the direction of extra half-planes (EHPs), dislocations with EHPs parallel to dCG (dEHPs ∥ dCG) are highlighted in red and dislocations with EHPs perpendicular to dCG (dEHPs ⊥ dCG) are highlighted in blue. As can be calculated, the overall dislocation density of the composition gradient domain is ~1.6 × 1012 cm-2. Among them, ~77% are dEHPs ∥ dCG, the other ~ 23% are dEHPs ⊥ dCG. Such preferential orientation of edge dislocation along the dCG can be attributed to the directional lattice distortion caused by the directional composition gradient, as illustrated in Fig. 4d. Considering dislocation density is concentrated in composition gradient domains, which widely exist in both random and directional composition distribution areas, dislocation density is also not the reason for the κ difference. Such preferential orientation of edge dislocation along the dCG can be attributed to the directional lattice misfit caused by the directional composition distribution, as illustrated in Fig. 4d. To support this observation, density functional theory (DFT) calculations were conducted to evaluate the formation energy of edge dislocations (EDF) under different conditions. As shown in Fig. 4e, the EDF,∥ (dEHPs ∥ dCG) is much lower than EDF,⊥ (dEHPs ⊥ dCG), which accounts for the preferential orientation of edge dislocations along the dCG. Notably, when applying electron sectioning method on composition gradient domain36, high-resolution scanning transmission electron microscopy (HRSTEM) images with focal planes at different depths indicate that no screw dislocation is observed in the composition gradient domain (Supplementary Fig. 16). Therefore, screw dislocations or mixed-type dislocations are not the reason for the κ difference in random and directional composition distribution areas.

Fig. 4: Edge dislocation characterization and analysis.
Fig. 4: Edge dislocation characterization and analysis.
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High-resolution TEM (HRTEM) images and corresponding inverse fast Fourier transform (IFFT) images of Bi-rich (a), composition gradient (b), and Sb-rich (c) domains. d Schematic diagram illustrating the preferential orientation of edge dislocations along with composition gradient direction. e Density functional theory (DFT) calculated edge dislocation formation energy (EDF) for extra half-planes (EHPs) grown along and perpendicular to composition gradient direction (dCG), respectively.

In conclusion, as exemplified by a Bi0.4Sb1.6Te3 compound, we identified the mechanism of low κ in materials with an inhomogeneous composition distribution. This can be explained by the randomly aligned edge dislocations at the composition gradient domains, between domains with different compositions. At the microscale, Bi-rich and Sb-rich domains exhibit nearly defect-free structures. However, in the composition gradient domain between Bi-rich and Sb-rich domains, a significant atomic radius difference between Bi and Sb results in a large lattice misfit. This misfit leads to the formation of a dense array of edge dislocations, which preferentially align along the composition gradient direction (dCG) due to a lower edge dislocation formation energy parallel to this orientation (EDF,∥) compared to that perpendicular to it (EDF,⊥). At the macroscale, random composition distribution can lead to randomly aligned edge dislocations, contributing to strengthened phonon scattering, and lower κ compared to areas with directional composition distribution. In our discussion, influences of nanostructures, point defects, screw dislocation, dislocation density, composition gradient domains, grain boundaries or phase boundaries have all been well excluded. Therefore, low κ in random composition distribution area should be dominated by the random alignment of edge dislocations induced by random alignment of composition gradients. This insight reveals the origin of low κ in compositional inhomogeneous materials from the microstructure viewpoint, which can provide valuable guidance for the design of future materials with optimized thermal properties.

Methods

Synthesis and characterization

Polycrystalline Bi0.4Sb1.6Te3 samples were synthesized by melting stoichiometric ratios of Bi, Sb, Te (99.99%, Chengdu optoelectronic materials Co., Ltd.) in evacuated and carbon-coated silica tubes. The mixtures were heated up to 773 K at 2 K min−1, held for 2 h, then further heated to 1223 K at 1 K min−1, maintained for 5 h, and quenched in ice water. The acquired ingots were ground into powder using an agate mortar and sieved through a 50-mesh screen. The powders were consolidated by spark plasma sintering (SPS; LABOX-110H, Sinter Land, Japan) at 723 K under 50 MPa in a Ø12.7 mm graphite mold for 5 min. (Supplementary Fig. 3)

The phase composition of samples was inspected by X-ray diffraction (XRD) (SmartLab, Cu Kα line, Rigaku, Japan). The κ distribution was measured by calibrated Scanning Thermal Probe Micro-image (STPM-1000 Advance Riko, Japan). After calibration, thermal conductivities of standard materials are verified by Laser Flash Analysis (LFA 467 Hyperflash, Netzsch, German). Secondary electrons (SE) and backscatter electrons (BSE) images were taken by field emission scanning electron microscopy (FESEM) (SU7000, Hitachi, Japan) with energy-dispersive X-ray spectroscopy (EDS) (Ultim Max 100, Oxford, British) and field-free Ultra-High Resolution (UHR) FESEM (S8000X, TESCAN, Czech Republic). The characterization of the sample’s atomic structure was carried out using high-resolution transmission electron microscopy (HRTEM) (JEM-2100 and JEM-ARM200F, JEOL, Japan). The samples for TEM observations were prepared by focused ion beam (FIB) milling (S8000X, TESCAN, Czech Republic).

DFT calculations

DFT calculations were performed using a projector-augmented wave (PAW) method implemented in the Vienna Ab initio Simulation Package (VASP). The fully relativistic Perdew–Burke–Ernzerh generalized gradient approximation functional (GGA-PBE) was employed to model exchange correlation interactions. A fundamental supercell comprising of 90 atoms without edge dislocation was constructed by cleaving six atomic layers in (0 1 5) planes from a 3 × 3 × 3 Bi0.4Sb1.6Te3 supercell, followed by the addition of a 15 Å vacuum slab37. Supercells with edge dislocation parallel and perpendicular to the composition gradient direction were constructed by slipping the (0 1 5) and (1 \(\bar{1}\) 5) planes by 50% of planar distance along respective directions38. The supercell energy was calculated with a plane wave cut-off energy of 400 eV, based on which the dislocation formation energy of the two configurations were further evaluated.

STPM analysis

The STPM measures the Seebeck coefficient and thermal properties distribution at the sample surface by using a thermal probe and measuring the voltage and temperature of the probe. As can be seen in Supplementary Fig. 1, STPM measures κ along the out-of-plane direction as the temperature gradient is applied perpendicular to the sample surface. The spatial resolution of the equipment is 20 μm. In order to improve the accuracy of STPM, 7 standard materials (Si, Ta, Ge, SUS304, SrTiO3, SiO2 glass and Vespel) are used to calibrate the equipment before sample measurement. After calibration, the κ of each standard materials are verified by Laser Flash Analysis (LFA). As a result, the accuracy of STPM can reach ~13%. More detailed discussions are available in Supplementary Note 1.