Fig. 1 a Relationship between $\sigma$, $S$, $S^{2} \sigma$, $\kappa$, ${\kappa}_{{{\text{lat}}}}$, ${\kappa}_{\text{e}}$, and ZT. b ZTmax and ${\kappa}_{{{\text{lat}}}}$ of advanced thermoelectric materials reported in the past decade. The black line illustrates the approximate inverse relationship between ZTmax and ${\kappa}_{{{\text{lat}}}}$; materials with lower ${\kappa}_{{{\text{lat}}}}$ tend to exhibit relatively higher ZTmax [15,16,17,18,20,21,22,23,24,75]. c The influence of crystal defects and the complexity of the unit cell on ${\kappa}_{\text{lat}}$ is significant. Point defects primarily scatter high-frequency phonons, while dislocations scatter mid-frequency phonons, and precipitates and interfaces scatter low-frequency phonons. As defects accumulate across multiple scales and dimensions, the range of phonon frequencies being scattered increases, leading to a lower lattice thermal conductivity. Simpler unit cells exhibit more symmetrical bonding environments, resulting in better thermal transport capabilities. In contrast, more complex unit cells experience stronger anharmonicity, which causes phonon softening and lower ${\kappa}_{{{\text{lat}}}}$. Compared to PbSe, the complexity of the crystal structures of SnSe and Ag increases, which corresponds to lower ${\kappa}_{{{\text{lat}}}}$. At room temperature, the ${\kappa}_{{{\text{lat}}}}$ of PbSe is 1.5 W·m−1·K−1, while those of SnSe and Ag5-xTe3 are 0.7 W·m−1·K−1 and 0.2 W·m−1·K−1, respectively [28,30,31]

Fig. 2 a Crystal structure along the a-axis: gray, Sn atoms; red, Se atoms. b A Sn atom is coordinated with seven Se atoms, featuring three short and four long Sn-Se bonds. c The lattice thermal conductivity comparison of SnSe along the b-axis (ZTmax = 2.62) and PbTe-4SrTe-2Na (ZTmax = 2.2). d The lattice thermal conductivity and total thermal conductivity of SnSe single crystals along the a, b, and c axes. The out-of-plane thermal conductivity along the a-axis is the lowest, while the in-plane thermal conductivities along the b and c axes are relatively high and show little difference. e Grüneisen dispersion; inset, the average Grüneisen parameters along a, b and c axes. f Phonon dispersion. TA, TAʹ, transverse acoustic phonon scattering branches; LA, longitudinal acoustic phonon scattering branch. The Gruüneisen parameters along all three axes are quite large, with the Gruüneisen parameter along the Γ-X direction (a-axis) significantly higher than those along the Γ-Y (b-axis) and Γ-Z (c-axis) directions, indicating strong anharmonicity. This is also reflected in the phonon dispersion, where the phonon modes along the a-axis are softer [31]

Fig. 3 a Diagram of Seebeck coefficient measurements under pressure. The gray sample was placed in a chamber, which was surrounded by deep blue cubic boron nitride as an insulating layer. Ruby was used to determine the pressure. b Pressure dependence of thermal conductivity and lattice thermal conductivity of PbSe. The inset illustrates the structures of the B1 and Pnma phases before and after applying 5 GPa pressure, with the dashed line indicating the onset of the phase transition to Pnma at 5 GPa. Both total thermal conductivity and lattice thermal conductivity exhibit a similar abrupt change at 5 GPa [42]

Fig. 4 a and b shows HAADF-STEM images of PbSe and Pb1.004Se with additional 0.08 mol% Cu, respectively [51]. The dashed box in Fig. 4b highlights Pb vacancies within the matrix of the PbSe crystal. c and d display STEM-EDS analyses for Pb1.075Se0.8Te0.2 and Cu0.005PbSe0.99Te0.01 [19,50]. In the blue circles, blue Te atoms are indicated as substitutional defects replacing green Se. The red circles denote interstitial Cu atoms shown in red. e Lattice thermal conductivity of intrinsic PbSe compared to PbSe with point defects shown in b, c, and d. The introduction of a significant number of point defects in PbSe enhances phonon scattering, leading to a marked decrease in lattice thermal conductivity. This illustrates the qualitative effect of point defects; other defects present in these three different compositions may also impact thermal transport properties [19,50,51,76]

Fig. 5 a-c TEM images of PbTe with Na doping concentrations of 0.02 mol%, 0.025 mol%, and 0.03 mol%, respectively. In Fig. 5a, dislocations are nearly absent, while a significant number of dislocations are observed in Fig. 5b. In Fig. 5c, the dislocation density is noticeably reduced. As the Na doping concentration increases, the dislocation density gradually rises, peaking at 0.025 mol%, followed by a decline. Figure 5d further illustrates that the introduction of dislocations leads to a substantial reduction in lattice thermal conductivity compared to intrinsic PbTe. PbTe with a doping concentration of 0.025 mol% exhibits the highest dislocation density and the lowest lattice thermal conductivity. e Composition-dependent lattice thermal conductivity for NayEu0.03Pb0.97−yTe at 850 K. The symbols show the experimental results (black), while the curves show the model predictions (with both N- and U-process included) based on a Debye approximation with different types of phonon scattering. Phonon scattering by dislocations contributes the most to the reduction on the lattice thermal conductivity in this material. Dashed lines are used for guiding the eye only [52]

Fig. 6 Compositional analysis on nanostructures in the PbSe0.998Br0.002-2%Cu2Se sample by APT. a Three-dimensional APT reconstruction of the volume showing the spatial distribution of Pb (green), Se (orange), and Cu (red) atoms. Discrete Cu-rich regions are clearly observed in contrast to homogeneous distribution of Pb and Se atoms. b Enlarged view on the Cu-rich region in a (enclosed by the purple circle) revealing the formation of discrete nanoscale single Cu atomic layers. Their interval ranges from 4 to 30 Å. c Magnified view on the single Cu atomic layer marked by the blue rectangle in Fig. 6b, clearly showing Cu atomic layers. d Lattice thermal conductivity of PbSe0.998Br0.002-x%Cu2Se (x = 0-7) with respect to temperature [57]

Fig. 7 a Comparison of lattice thermal conductivity between single crystalline and polycrystalline PbSe [30,61]. b Backscattered electron (BSE) image of the NbCo0.95Pt0.05Sn sample. The gray areas represent the grains, while the white regions indicate the Pt precipitated at the grain boundaries. c The black squares, blue triangles, and red circles represent the lattice thermal conductivity of NbCoSn, NbCo0.95Pt0.05Sn before annealing, and NbCo0.94Pt0.06Sn after annealing, respectively. The inset within the blue and red boxes shows BSE images of NbCo0.95Pt0.05S and NbCo0.94Pt0.06Sn, respectively. Doping with Pt significantly reduces the lattice thermal conductivity, and smaller grain sizes exhibit stronger phonon scattering at the grain boundaries, resulting in even lower lattice thermal conductivity [62]