Introduction

The discovery of crystalline materials with ultralow glass-like lattice thermal conductivity (ULG-type κL) is essential for maximizing performance in thermoelectrics1, thermal insulators2, and thermal coating barriers3. Generally, crystalline materials with complex structures, large primitive cells (containing a number of atoms >20), and heavy elements tend to exhibit low \({\kappa }_{{{\rm{L}}}}\)4. However, ultralow \({\kappa }_{{{\rm{L}}}}\)5 can also occur in materials with small primitive cells and light elements, which is due to the presence of rattler atoms6, strong lattice anharmonicity7,8, or bonding heterogeneity9. Recently, significant progress has been made in both experimental and theoretical investigations aimed at uncovering crystalline materials with very low \({\kappa }_{{{\rm{L}}}}\) below 1.0 W/m·K. Experimentally, Mukhopadhyay et al.10 developed a class of thallium selenides and measured an ultralow κL of approximately 0.30 W/m·K at 300 K in Tl3VSe4. This value was attributed to its exceptionally low group velocity and strong anharmonicity. Acharyya et al.11 synthesized a layered halide perovskite Cs3Bi2I6Cl3 single crystal and measured an ultralow thermal conductivity of 0.20 W/m·K along in-/cross-plane directions at 300 K, which was attributed to its low average sound velocity (\(v\)) and weak chemical bonds. Gibson et al.12 found that the 2D layered structure of mixed anion Bi4O4SeCl2 with a specific spatial arrangement of interfaces suppresses acoustic phonon contributions, achieving an ultralow κL of 0.10 W/m·K in the cross-plane direction at 300 K. Interestingly, for the even more complex crystal structure of argyrodite Cu7PS6 with 54 atoms in primitive cell, Shen et al.13 reported an ultralow κL of 0.50 W/m·K at 300 K. Notably, these crystalline materials with ultralow \({\kappa }_{{{\rm{L}}}}\) also generally exhibit a weak/positive temperature-dependent behavior, resembling ULG-type thermal conduction. These experimental observations of ULG-type \({\kappa }_{{{\rm{L}}}}\) raise an open question: how can \({\kappa }_{{{\rm{L}}}}\) be pushed to its lower limit over a broad temperature range based on material properties, such as anharmonicity and structural complexity?

Theoretically, to accurately explain the experimental ULG-type \({\kappa }_{{{\rm{L}}}}\), Simoncelli et al.14 proposed a unified theory of thermal transport that considers both particle-like phonon propagation (phonon contribution) and wave-like tunneling channels (diffuson contribution), successfully reproducing the glass-like temperature dependence of \({\kappa }_{{{\rm{L}}}}\) in crystalline CsPbBr3. Building on the unified theory of thermal transport, Xia et al.15 further incorporated the effects of anharmonic temperature dependence and higher-order phonon scattering, successfully replicating the temperature-independent ultralow κL in crystalline Cu12Sb4S13. Moreover, the well-established unified theory of thermal transport has been successfully applied to explain thermal transport in various materials with complex structures or strong anharmonicity, such as Yb11M4Sb11 (M=Mg and Mn)16, Cu7PS613, NaAg3S217, K2Ag4Se318, YbFe4Sb1219, and MAPbI3 (MA=methylammonium)20. Therefore, exploring materials with ULG-type κL, approaching the glassy limit in crystalline compounds, is not only fundamentally intriguing but also essential for understanding the complex interplay between crystal structure, bonding strength, and anharmonic lattice dynamics. These investigations offer valuable criteria for identifying unknown low-κL crystalline materials and open new avenues for engineering heat transport properties in existing compounds.

Despite significant advancements in experimental observations and theoretical predictions of κL in crystalline materials, the high cost remains a major obstacle to high-throughput calculations and experiments, as well as the rapid screening of crystalline materials with ULG-type κL. To simplify the theoretical prediction process for identifying materials with ultralow κL, Knoop et al.21 introduced the concept of the degree of anharmonicity (DOA), derived from ab initio molecular dynamics (AIMD), to quantitatively evaluate κL in crystalline materials. Using the concept of DOA, Zeng et al.22 theoretically and experimentally reported an ultralow κL of 0.25 W/m·K at 300 K in the simple crystalline material AgTlI2, attributed to its Ag-associated strong anharmonicity. Additionally, considering the inverse relationship between population/diffuson thermal conductivity and number of atoms in a primitive cell (n), they proposed a conceptual diagram to facilitate the search for ultralow κL crystalline materials with a simple crystal structure. Although the AIMD-based DOA concept facilitates the rapid screening of materials with ultralow κL, the lengthy AIMD calculations remain computationally expensive, and the DOA concept is limited to qualitatively evaluating thermal transport properties.

Over the past decades, widely known empirical thermal transport models, such as the Slack23 and Debye-Callaway24 models, have been proposed to estimate the thermal conductivity of crystalline materials. However, these models are limited in accuracy and rely heavily on numerous empirical parameters. To overcome the limitations of the aforementioned empirical models, several have proposed an improved and straightforward minimum thermal conductivity model based on diffuson-mediated thermal transport theory. Furthermore, inspired by the unified theory of thermal transport, Xia et al.25 recently proposed a minimum thermal conductivity model that incorporates contributions from both propagons and diffusons. Meanwhile, Wang et al.26 derived an interpretable formula for κL of crystals by utilizing a large database and building upon the Slack model. Although recently developed thermal transport models have achieved significant success, their complexity25,26 or limited prediction accuracy27 hinders their practical application by experimentalists. A simple unified thermal transport indicator based on experimentally accessible parameters has still not yet been developed, posing a significant challenge to the efficient exploration of crystalline materials with desirable thermal transport properties. In addition, with the rapid advancements in universal machine learning force fields (MLFF), such as MACE28, M3GNET29, CHGNET30, and MatterSim31, integrating MLFF into a universal thermal transport indicator has become essential.

In this work, we introduce a universal macroscopic descriptor (σ) that enables effective screening of ULG-type κL in crystalline materials. This descriptor is expressed in a simple form involving only the sound velocity (\(v\)) and the atomic number of the primitive cell (n). Using a large and diverse set of experimental and theoretical data, we established this universal descriptor by identifying a highly correlated Pearson-type relationship and systematically simplifying the Slack model based on fundamental physical quantities. We further integrated this universal descriptor with high-throughput workflows and universal machine-learning interatomic potentials to screen for materials with ultralow κL across the Materials Project database32. We identified two metal halides in the CsA₂I₃ (A = Ag, Cu) system as promising ULG-type candidates. We subsequently synthesized CsAg₂I₃ and CsCu₂I₃ and experimentally confirmed ULG-type \({\kappa }_{{{\rm{L}}}}\) values of 0.15–0.16 W/m·K between 170 and 400 K and 0.18–0.20 W/m·K between 300 and 523 K, respectively. Moreover, using a combination of state-of-the-art techniques, including electron diffraction, inelastic neutron scattering, and a unified theory of thermal transport, we comprehensively investigated the relationships among crystal structure, lattice dynamics, and microscopic thermal transport in the representative compound CsAg₂I₃. Our work introduces an effective and accessible tool, readily usable by both experimentalists and theoreticians, for screening crystalline materials with targeted ULG-type κL. By establishing a robust correlation between ULG-type thermal transport and laboratory-measurable physical quantities, this study significantly accelerates the discovery of materials for a broad range of thermal-related applications.

Results and discussion

Potential indicator for screening ULG-type κ L materials

Using the oversimplified elementary kinetic transport theory, κL can be computed as \({\kappa }_{{{\rm{L}}}}=\frac{1}{3}{C}_{{{\rm{V}}}}{v}_{{{\rm{g}}}}l\), where Cv is the isochoric heat capacity, \({v}_{{{\rm{g}}}}\) is the phonon group velocity, and l is the phonon mean free path. Therefore, a high \({v}_{{{\rm{g}}}}\) inherently results in a high lattice thermal conductivity33. For instance, Li et al.33 demonstrated a well-linearized relationship between the minimum thermal conductivity and the \({v}_{{{\rm{g}}}}\) for various crystalline materials. Additionally, for crystalline materials with ultralow κL, Agne et al.27 derived a minimum diffuson-mediated thermal transport model: \({\kappa }_{{{\rm{L}}},\min }=0.76{({n}_{{{\rm{d}}}})}^{2/3}{k}_{{{\rm{B}}}}*\frac{1}{3}(2{v}_{{{\rm{t}}}}+{v}_{{{\rm{l}}}})\), where nd is the atomic number density, kB is the Boltzmann constant, and vt and vl are the transverse and longitudinal sound velocities, respectively. Recently, Wang et al.26 also derived an interpretable formula for the κL of crystals: \({\kappa }_{{{\rm{L}}}}=\frac{{Gv}{V}^{1/3}}{n{T}^{\delta }}*{e}^{-r}\), where G, V, r, and δ are shear modulus, volume of the primitive cell, Grüneisen parameters, and scattering parameter (between 1 and 2). This formula achieves high precision and provides fast predictions. Within the framework of Slack model and some predictive models of minimum κL23,25,26,27, various empirical macroscopic quantities are considered as potential correlated parameters influencing ULG-type κL, such as Debye temperature (\({\varTheta }_{{{\rm{D}}}}\)), n, volume per atom (Va), density (\(\rho\)), average mass of atoms (Ma), and isobaric heat capacity (Cp). To evaluate the correlation between these microscopic quantities, we conducted a Pearson correlation matrix analysis.

To propose a model for assessing ULG-type κL in crystalline materials, we analyze the relationship between temperature-dependent κL and n. As illustrated in Fig. 1a, the κL of crystalline materials typically exhibits a universal asymptotic decay, expressed as κL(T) ~ T −m 34. Specifically, the dominant anharmonic three-phonon scattering mechanisms leads to a trend of κL(T ) ∝ T −1 (m = 1)35, as observed in strongly-bonded materials, such as AlN36, Si37, and GaSb38. Strong anharmonic phonon scattering events result in milder decays (m < 1), as seen in κL(T ) behavior of PbTe (T−0.92)39, InTe (T −0.55)40,41, and SrCuSb (T −0.51) compounds42. In contrast, in highly disordered or glassy materials, κL(T ) exhibits a temperature independence or even positive temperature dependence, as seen in Cs3Bi2I943, Cu12Sb4S1344, Ag8GeSe645, KCu5Se346, and Cu7PS613. Macroscopically, compounds with a higher number of n generally show a greater tendency to display weak or positive temperature dependence in κL, which can be considered a critical factor in capturing the glass-like behavior of thermal conductivity.

Fig. 1: Construction of a universal descriptor for screening ULG-type κL.
figure 1

a Experimental κL for various compounds, including Si, AlN, GaSb, PbTe, InTe, SrCuSb, Cs3Bi2I9, Cu12Sb4S13, Ag8GeSe6, KCu5Se3, and Cu7PS6. b The Pearson correlation matrix calculations of over 40 crystalline materials. Note that the crystal anharmonicity was evaluated using the method proposed by Knoop et al.21. c The plot of the experimental κavg vs. κL_300K of over 60 crystalline samples. d The plot of the experimental κL_300K vs. σ of over 60 crystalline samples. Detailed information on the experimental and calculated data shown in Fig. 1b–d are listed in Table S1a, b in the Supplementary Information (SI).

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To qualitatively assess the magnitude of ULG-type κL in crystalline materials, we utilized experimental κavg, obtained as the average of the κL over measured temperatures (Table S1a). We considered the κL values above the Debye temperature for specific crystalline materials, as this temperature range is dominated by phonon-phonon Umklapp scattering. Our Pearson correlation matrix analysis (Fig. 1b) shows that \(\upsilon\) has the strongest positive correlation value with κavg (0.44), suggesting that ultralow \(\upsilon\) can serve as a practical and direct parameter for identifying ULG-type κL. Collectively, \({\varTheta }_{{{\rm{D}}}}\), \(\upsilon\), anharmonicity, crystal symmetry (designated by the space group number), n, Ma, Va, Cp, number of atoms in the chemical formula, and \(\rho\) account for about 98% of the correlation with κavg. In contrast, parameters like formation energy and band gap present negligible correlation (2%).

To simplify the prediction of ULG-type κL using physical quantities, we begin with a classical Slack model of κL47,48, under dominant Umklapp scattering:

$${\kappa }_{{{\rm{L}}}}\approx A\frac{{M}_{a}{{\varTheta }_{{{\rm{D}}}}}^{3}{{V}_{a}}^{1/3}}{{\gamma }^{2}{n}^{2/3}T}$$
(1)

where A is a constant. Given that \(\gamma\) is not constant but typically varies between 1 and 2 in most crystalline materials, we employ a proportional relationship:

$${\kappa }_{{{\rm{L}}}}\propto \frac{{M}_{a}{{\varTheta }_{{{\rm{D}}}}}^{3}{{V}_{a}}^{1/3}}{{n}^{2/3}T}$$
(2)

Considering that optical phonons can contribute significantly to thermal conductivity49,50 and to develop a universal lattice thermal-conductivity descriptor applicable to both diffuson- and phonon-dominated regimes, we adopt the traditional definition of the Debye temperature, rather than the acoustic-mode Debye temperature. The traditional Debye temperature \({\varTheta }_{{{\rm{D}}}}\) can be expressed by the relation51 \({\varTheta }_{{{\rm{D}}}}=\frac{\hslash \upsilon }{{k}_{{{\rm{B}}}}}{\left(\frac{3}{4\pi }\frac{n}{V}\right)}^{1/3}=\frac{\hslash \upsilon }{{k}_{{{\rm{B}}}}}{\left(\frac{3}{4\pi }\frac{1}{{V}_{a}}\right)}^{1/3}\), which we rewrite in the following form:

$${\kappa }_{{{\rm{L}}}}\propto \frac{3{\hslash }^{3}}{4\pi {{k}_{{{\rm{B}}}}}^{3}}\frac{{M}_{{{\rm{a}}}}{\upsilon }^{3}}{{{V}_{a}}^{2/3}{n}^{2/3}T}\propto \frac{{M}_{{{\rm{a}}}}{\upsilon }^{3}}{{{V}_{a}}^{2/3}{n}^{2/3}T}$$
(3)

Figure 1c plots the relationship between \({\kappa }_{{{\rm{L}}}\_300{{\rm{K}}}}\) and \({\kappa }_{{{\rm{ave}}}}\) (Table S1b), which reveals a strong linear positive correlation. This indicates that crystalline materials with low \({\kappa }_{{{\rm{L}}}\_300{{\rm{K}}}}\) also tend to have a low \({\kappa }_{{{\rm{ave}}}}\). Therefore, \({\kappa }_{{{\rm{ave}}}}\) can be approximated by the following expression:

$${\kappa }_{{{\rm{ave}}}}\propto {\kappa }_{{{\rm{L}}}\_300{{\rm{K}}}}\propto \frac{{M}_{{{\rm{a}}}}}{{{V}_{a}}^{2/3}}\frac{{\upsilon }^{3}}{{n}^{2/3}}$$
(4)

After several trials of the possible indicators in formula (4), including Ma*\({(\upsilon /1000)}^{3}\), Ma*\({n}^{-\frac{2}{3}}\), \({n}^{-\frac{2}{3}}{(\upsilon /1000)}^{3}\), Ma*\({(\upsilon /1000)}^{3}\)*\({n}^{-\frac{2}{3}}\), and Ma*\({(\upsilon /1000)}^{3}\)*\({(n*{V}_{a})}^{-\frac{2}{3}}\), we identified an indicator σ, expressed as \({n}^{-\frac{2}{3}}{(\upsilon /1000)}^{3}\), that exhibits a linear relation with κavg, achieving a goodness of fit of 0.9. Other fitting results between \({\kappa }_{{{\rm{L}}}\_300{{\rm{K}}}}\) and indicators (Ma*\({(\upsilon /1000)}^{3}\), Ma*\({n}^{-\frac{2}{3}}\), Ma*\({(\upsilon /1000)}^{3}\)*n-2/3, and Ma*\({(\upsilon /1000)}^{3}\)*\({(n * {V}_{a})}^{-\frac{2}{3}}\)) can be found in Fig. S1. As shown in Fig. 1d, we collected experimental κL_300K, including both ultrahigh and ultralow values, as well as \(\upsilon\) and n values from over 60 crystalline materials with various crystal structures (Table S1b). κL_300K typically follows a linear relationship with the established σ on a log scale. Our modeling indicates that almost all data points fall within the gray region, representing a 95% confidence and prediction band. This behavior strongly supports the correctness of the proposed indicator σ, suggesting its effectiveness in identifying ULG-type κL crystalline materials without the need for time-consuming calculations of κL. However, experimental determination and validation of \(\upsilon\) still require material synthesis and characterization, which can be time- and energy-intensive for cross-checking.

To overcome this limitation, we conducted high-throughput calculations using a universal machine learning force field method (MACE28) to enhance the efficiency and accuracy of υ estimations. As illustrated in Fig. 2, we utilized the Materials Project database32 containing 153,902 materials and applied various screening criteria. These included selecting only non-magnetic compounds, limiting the number of different element types to less than four, restricting the number of atomic sites to less than 60, requiring a band gap greater than 0 eV, and ensuring the energy above the hull was below 0.1 eV/atom. Applying these criteria, we filtered the dataset down to 25,556 materials. We then integrated the universal machine learning force field package MACE28 with our indicator to predict thermal conductivity. After obtaining the predicted values, we further excluded the low-symmetry systems, i.e., monoclinic and triclinic lattices and considered only crystals identified as experimentally observed in the Materials Project47. As a result, we identified only 336 materials with a thermal conductivity below 1.0 W/m·K. Given that metal halides consistently exhibit low thermal conductivity11,43,50,52,53, we selected their analogs, specifically crystalline CsAg₂I₃ and CsCu₂I₃, from 336 compounds for further experimental investigation.

Fig. 2: Schematic diagram of high-throughput screening process.
figure 2

High-throughput thermal conductivity calculations using MACE28 and our universal indicator. The crystal structures were visualized and exported using the VESTA software package92.

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Structural evolution and thermal transport properties

In combination of high-throughput calculations with machine learning techniques (see the method section), we screened metal halides CsA2I3 (A = Ag, Cu) and synthesized them for characterizations of their structural and physical properties. Prior studies reported that CsAg2I3 adopts an orthorhombic crystal structure of Pnma space group (No 62)54,55, while the structural analog CsCu2I3 crystallizes in the higher-symmetry Cmcm space group (No 63), as plotted in Fig. S2. The corresponding Rietveld refinements of powder X-ray diffraction (PXRD) patterns of the CsAg2I3 and CsCu2I3 samples at 300 K confirm their high-purity (Figs. S3–S4). The refined crystallographic data for the two compounds are provided in Table S2–S3. The microscopic morphology and chemical distributions of all the samples, examined using SEM and SEM-EDX, are presented in Figs. S5–S6.

To analyze the phase stability of CsAg2I3, PXRD patterns were recorded from 250 to 320 K at Cu wavelength with 10 K intervals (Fig. S7). Enlargement of the PXRD patterns reveals that, as the temperature increases, certain diffraction peaks gradually disappear, as exemplified by the peaks observed at 2θ = 22.46°, 27.20°, and 28.07° at T = 250 K. In addition, some peaks shift significantly to lower angles, such as those observed at 2θ = 28.83°, 30.68°, and 31.71° at T = 250 K and at 2θ = 28.55°, 30.41°, and 31.43°, respectively, at T = 320 K. Such peak intensity and position modifications suggest the existence of a structural modification of CsAg2I3 around room temperature. Analyses (Fig. S8, see details in the SI) support the assumption that crystal structure of CsAg2I3 evolves gradually from 250 to 320 K from its distorted variant crystal structure reported in \({Pnma}\) space group (Fig. S9a, b) to that reported for CsCu2I356, a crystal structure of higher symmetry in Bbmm/Cmcm space group (Fig. S9c, d). Besides, a pronounced minor peak was observed at 278 K in the Cp data (Fig. S10), which probably corresponds to a subtle symmetry modification in CsAg2I3, supporting the results of the PXRD data. The second peak over a broad temperature range of 285-310 K originates from the typical transition of addenda for the low-temperature Cp measurement.

The measured κL of all sintered polycrystalline samples is displayed in Fig. 3a. The CsAg2I3 sample exhibits a temperature-independent κL of 0.15–0.16 W/m·K from 170 to 400 K, whereas the CsCu2I3 sample shows values of 0.18–0.20 W/m·K from 300 to 523 K. Notably, compared to other metal halides11,52,57,58,59, the CsA2I3 (A = Ag, Cu) samples exhibit the lowest κL values across a broad temperature range, validating the practical effectiveness of combining a universal descriptor with high-throughput workflows and universal machine-learning interatomic potentials. Our primary experimental objective is to demonstrate the successful identification of crystalline materials with the ULG-type κL; detailed microscopic analysis of the slight differences in κL between CsAg2I3 and CsCu2I3 falls outside the scope of this study. To correlate thermal transport properties to crystal structures, we conducted structural and chemical bonding analyses. Since these two samples share structural similarity in the high-temperature Bbmm/Cmcm phase, we selected a representative compound, CsAg2I3, for further detailed discussions. The Bbmm structure of the 320 K-form of CsAg2I3 (Fig. 3b) consists of [Ag2I3]− double chains of edge-sharing AgI4 tetrahedra running along b (Fig. 3b), interconnected by (100) layers of Cs+ cations. Remarkably, the Ag-I distances, ranging from 2.78 to 2.92 Å, are significantly smaller than that expected from the sum of ionic radii (rAg+= 1 Å, rI-=2.2 Å). The stability of the structure is clearly ensured by the ionic character of the Cs-I bonds between the [Ag2I3]− double chains. Each Cs+ cation exhibits eight nearest I− neighbors with Cs-I distances ranging from 3.93 to 4.18 Å, close to the sum of ionic radii (rCs+=1.7 Å, rI-=2.2 Å), forming almost regular CsI8 bicapped trigonal prisms sharing faces along b and edges along c.

Fig. 3: Thermal conductivities and structural characterizations of CsA2I3 (A = Ag, Cu) compounds.
figure 3

a Comparison κL of CsAg2I3 and CsCu2I3 with other metal halides compounds11,52,57,58,59. Evolution of the crystal structures of CsAg2I3 with temperature, as determined by b PXRD at 320 K, c PXRD at 250 K, and d 3D ED at 100 K. Crystal structure of CsAg2I3 (SG: Pnma) projected along [010], as obtained from e PXRD data recorded at 250 K and f, single-crystal 3D ED data recorded at 100 K with main interatomic distances indicated.

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The Pnma structures of the low temperature form of CsAg2I3 (i.e. 100 K and 250 K, Fig. 3c-d) is very similar to the Bbmm form (Fig. 3b). The [Ag2I3]− double chains of edge-sharing AgI4 tetrahedrons (Fig. 3c) are only slightly distorted with Ag-I distances, ranging from 2.78 to 2.97 Å (Fig. 3e-f). Previously aligned along the c-axis in the Bbmm form (Fig. 3b), the [Ag2I3]− double chains are now tilted around the b-axis and in antiphase along a (Fig. 3c). This tilting is slightly more pronounced at 100 K than at 250 K (Fig. 3c, d), in line with the simultaneous increase in the c lattice parameter and the decrease in the a lattice parameter as the temperature decreases (Fig. S8). It is accompanied by a displacement of Cs+ cations within the (001) plane (Fig. 3c), leading to a significant change in the geometry of the [CsI3] layers (Fig. 3b). Each Cs+ cation now exhibits seven nearest I− neighbors with Cs-I distances ranging from 3.86 to 4.07 Å (Fig. 3e, f), forming quasi-regular CsI7 monocapped trigonal prisms. Two additional I neighbors corresponding to larger Cs-I distances (4.28 to 4.45 Å) are observed which may not be considered as pure ionic bonds. As a consequence, the structure can be described as (100) [CsI3] layers built up of CsI7 monocapped trigonal prisms sharing faces along b and apices along c (Fig. 3c).

Such a soft topotactic transition from the high temperature to the low temperature form is explained by the fact that the I(1a) and I(1b) anions in the low temperature Pnma form at the border of the [Ag2I3]− double chains are only linked to two Ag+ cations and consequently can move easily contrary to the I(2) anions located inside the chains which are blocked by 4 Ag+ cations. This is supported by the atomic displacement parameters (ADPs) obtained from both PXRD (Table S4) and 3D ED (Table S8), which show that the I(1a) and I(1b) atoms have higher ADPs than the I(2) atom (Fig. 3e, f). With 3D ED, anisotropic ADPs could be refined, revealing that while I(2) atoms display nearly isotropic ADPs with only slight anisotropy, I(1a) and I(1b) exhibit pronounced anisotropy (Fig. 3f) along a direction associated with the rotation of the [Ag₂I₃]− double chains.

This structural analysis emphasizes the key role of iodine vibration in the thermal conductivity of the low temperature form of CsAg2I3 compound. It suggests that I− governs the appearance of ultralow thermal conductivity in connection with the one-dimensional character of the structure. The 1D character of the double [Ag2I3]− anionic chains facilitates the phonon propagation along the b axis, so that higher \({\kappa }_{{{\rm{L}}}}^{{{\rm{P}}}}\) can be expected along this direction. These anisotropic transport properties are moreover reinforced by the fact that the central “Ag-I(2)-Ag” chains form very strong bonds as the I(2) atom is blocked between four Ag+ cations. In contrast, the I(1a) and I(1b) atoms located at the junction between the CsI7 monocapped prisms and the [Ag2I3]− chains are freer due to the fact that they are linked to only two Ag+ cations. Consequently, larger ADPs likely contribute to pronounced anharmonicity at these atomic positions. Such a feature is favorable to the suppression of phonon propagation, leading to lower \({\kappa }_{{{\rm{L}}}}^{{{\rm{P}}}}\) values in the directions transversal to the b-axis. In the high temperature Bbmm form the distribution of the I− anions is more symmetric, leading to more regular AgI4 tetrahedrons and a more symmetric distribution of the I(1) anionic species (instead of I(1a) and I(1b)). These features are expected to facilitate the phonon transport, increasing \({\kappa }_{{{\rm{L}}}}^{{{\rm{P}}}}\) along the a- and c-axis. As shown in Table S9a, the experimental average sound velocity (\({\nu }_{{{\rm{a}}},\exp }\)) of 1188 m/s is considerably lower than that of most thermoelectric materials28. The measured \({\nu }_{{{\rm{a}}},\exp }\) aligns well with the calculated value of 1125 m/s, as listed in Table S9b, indicating the weak bonding and soft atomic lattices of CsAg2I3 compound, which might contribute to the low \({\nu }_{{{\rm{a}}},\exp }\), reduced elastic properties, significant anharmonicity, and ultimately ultralow κL.

Lattice dynamics and strong anharmonicity

For better understanding the ultralow κL of CsAg2I3, we conducted temperature-dependent inelastic neutron scattering (INS) to study the lattice dynamics properties. Details on the INS technique and data processing are provided in the SI. Measurements were performed with two incident energies, Ei = 12.5 and 30.0 meV, across a wide temperature range of 1.5-300 K, enabling us to analyze the complete generalized density of states (GDOS) as well as the temperature evolution of low-energy phonons. As illustrated in Fig. S11, contour maps of neutron scattering intensity, the dynamic structure factor S( | Q | , E) measured with a higher incident energy Ei = 30.0 meV, reveals prominent low-energy phonon modes below 5 meV at 1.5 K. Furthermore, we processed the calculated phonon density of states (PDOS) at 0 K by convoluting it with the experimental resolution and applying elemental neutron weighted cross-sections. Note that the calculated neutron weighted PDOS is in close agreement with the experimental GDOS obtained with Ei = 30.0 meV at 1.5 K (see Fig. 4d).

Fig. 4: Temperature evolution of phonons in CsAg2I3.
figure 4

a Experimental S( | Q | , E) with Ei = 12.50 meV at T = 1.5 K. The blue and white solid lines denoted as the acoustic and optical phonons, respectively. b Cp/T vs. T2 and c Cp/T3 vs. T using the Debye-Einstein model. d Experimental GDOS determined with Ei = 12.5 meV (marked as the solid blue scatter points) and Ei = 30.0 meV (marked as the hollow blue scatter points) at T = 1.5 K. The calculated PDOS convoluted with a 0.6 meV resolution resulted in a solid black line. The calculated neutron-weighted partial phonon density of states for Cs (gray solid line), Ag (purple solid line), and I (pink solid line) atoms from 1.5–300 K. The experimental data for e temperature-dependent phonon modes of DHO1 and DHO2. The lines here are guides to the eye. Error bars represent standard deviations.

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Figure 4a depicts S( | Q | , E) measured with Ei = 12.5 meV at 1.5 K. The measured signal discloses two low-energy phonon bands of high intensity located at about 2 and 3.3 meV and two less intense modes centered around 5 and 7 meV. They are visible as clearly separated peaks in the corresponding GDOS presented in Fig. 4d. The computed element-projected PDOS shown with the experimental data suggests that these low-energy flat bands are dominated by Ag- and I-weighed phonon modes. Interestingly, as indicated by the computed phonon dispersion (blue lines denote acoustic phonon branches; white lines denote optical phonon branches) in Fig. 4a, b strongly dispersive feature can be observed around |Q | = 2.5–3.0 Å−1. It is characteristic of acoustic phonons. These acoustic phonons exhibit thus an exceptionally low cut-off frequency by crossing the low-energy optical phonons at 1 ~ 2 meV. We expect this phonon dispersion to be a favorable configuration for efficient phonon-phonon scattering.

Additionally, we conducted a fitting analysis of the low-temperature Cp using the Debye-Einstein model. Detailed discussions of the fitting process and the Debye-Einstein model can be found in the referenced study13. The Cp/T versus T2 plot in Fig. 4b illustrates a well-fitted profile, indicating three localized low-energy Einstein modes, with values E1 = 19.0 K (1.6 meV), E2 = 34.0 K (2.9 meV), and E3 = 60.0 K (5.2 meV). Importantly, these Einstein modes derived from Cp are also consistent with the INS findings obtained using high incident energy Ei = 30.0 meV and high-resolution Ei = 12.5 meV, further supporting the presence of low-energy optical modes in CsAg2I3. Figure 4c displays the excellent fitting profile of Cp/T3 vs. T plot with Debye and three Einstein modes from 2 to 23 K. Notably, a distinct Boson peak is observed in the Cp/T3 vs. T plot, revealing the presence of low-energy excitations13.

To investigate the temperature dependence of low-energy phonons, we analyzed the constant-momentum spectra at |Q | = (3.2 ± 0.1) Å−1 (as displayed in Fig. 4a), where the low-energy phonon signal is prominent. These constant |Q| spectra (Fig. S12) were fitted using a combination of functions: four damped harmonic oscillator (DHO) functions33, a delta function for the elastic line (convoluted with a Gaussian-shaped resolution of 0.6 meV full-width at half-maximum, FWHM), and a background term. The four DHO components, labeled DHO1 to DHO4 in ascending energy, were used to characterize the phonon modes. The analysis revealed that the characteristic energies of DHO1 and DHO2 undergo continuous softening of the low-energy phonon peaks (Fig. 4e). Specifically, DHO1 and DHO2, with initial energies of 2 and 3.3 meV at 1.5 K, respectively, develop into overdamped phonons as temperature approaches 300 K. These low-energy modes can be viewed as soft phonons, which are closely associated with the displacive phase transition, such as superionic transitions60. The appearance of soft phonons in CsAg2I3 is likely linked to the structural transition from Pnma to Cmcm around 300 K. Besides, the characteristic energies of the high-energy optical phonons, referred as DHO3 and DHO4, exhibit a phonon softening behavior but without fully softening (Fig. S13) as observed in DHO1 and DHO2. Therefore, the presence of soft phonons and phonon softening in CsAg2I3 underscores its intrinsically strong anharmonic feature, contributing to its exceptionally low κL.

Microscopic mechanisms of ultralow glass-like κ L

To better understand thermal transport in crystalline CsAg2I3 with space groups Pnma and Cmcm, we calculated \({\kappa }_{{{\rm{L}}}}\) using the unified theory of thermal transport14, incorporating both population \({\kappa }_{{{\rm{L}}}}^{{{\rm{P}}}}\) and diffuson contributions \({\kappa }_{{{\rm{L}}}}^{{{\rm{C}}}}\), as shown in Fig. 5a. It is worth noting that the calculated \({\kappa }_{{{\rm{L}}}}\) of the CsCu2I3 compound was reported61 while this paper was in preparation, therefore, we focus here only on the theoretical study of the CsAg2I3 compound. When both 3-phonon (3 ph) and 4-phonon (4 ph) scattering processes are considered, the \({\kappa }_{{{\rm{L}}}}^{{{\rm{C}}}}\) significantly dominates the total \({\kappa }_{{{\rm{L}}}}\) along all crystallographic axes. We predict a room-temperature \({\kappa }_{{{\rm{L}}}}^{{{\rm{C}}}}\) of 0.09 W/m·K along the x-axis (0.11 and 0.30 W/m·K along the y- and z-axes, respectively), while the corresponding \({\kappa }_{{{\rm{L}}}}^{{{\rm{P}}}}\) values are 0.02 W/m·K along the x-axis (0.03 and 0.02 W/m·K along the y- and z-axes, respectively) (see Fig. 5a).

Fig. 5: Phonon transport properties of CsAg2I3.
figure 5

a The calculated axis-dependent \({\kappa }_{{{\rm{L}}}}\), including both population \({\kappa }_{{{\rm{L}}}}^{{{\rm{P}}}}\) and coherence \({\kappa }_{{{\rm{L}}}}^{C}\) contributions, for crystalline CsAg2I3 with space group of \({Pnma}\) and \({Cmcm}\), respectively. Since our experimental \({\kappa }_{{{\rm{L}}}}\) was measured in polycrystalline CsAg2I3, the averaged total thermal conductivity \({\kappa }_{{{\rm{L}}},{{\rm{ave}}}}^{P+C}\) was computed for a fair comparison and shows good agreement with the experimental data. b The calculated phonon lifetime as a function of phonon frequency at T = 300 K, accounting for 3-phonon processes and both 3- and 4-phonon processes, respectively. The solid green line represents the Wigner limit in time14, while the black dashed line indicates the Ioffe-Regel limit in time34. c,d The calculated spectral and cumulative population and diffuson thermal conductivity \({\kappa }_{{{\rm{L}}}}^{P/C}\left(\varOmega \right)/{\kappa }_{{{\rm{L}}}}^{P/C}\) along the x- axis at T = 300 K, considering 3 ph and both 3- and 4 ph scattering processes, respectively. e, f The same as c-d, but for spectral and cumulative population and diffuson thermal conductivity \({\kappa }_{{{\rm{L}}}}^{P/C}\left(\varOmega \right)/{\kappa }_{{{\rm{L}}}}^{P/C}\), along the y-axis.

Full size image

Interestingly, even when considering both phonon and diffuson channels, we observe an exceptionally low total \({\kappa }_{{{\rm{L}}}}\) of 0.11 and 0.13 W/m·K along the x-axis and y-axis, respectively, which is comparable to the experimentally recorded ultralow \({\kappa }_{{{\rm{L}}}}\) of 0.1 W/m·K in crystalline Bi4O4SeCl212. To validate our predicted \({\kappa }_{{{\rm{L}}}}\), we averaged the computed total \({\kappa }_{{{\rm{L}}}}\) across all crystallographic orientations and compared it with our experimentally measured \({\kappa }_{{{\rm{L}}}}\) for the polycrystalline CsAg2I3 sample. From Fig. 5a, we observe that the comparison of \({\kappa }_{{{\rm{L}}}}\) between experiment and theory shows good agreement, although the theoretical values are slightly higher than the experimental measurements.

We next examine the glass-like behavior observed in both the experimentally measured and theoretically predicted \({\kappa }_{{{\rm{L}}}}\), as illustrated in Fig. 5a. Notably, both the experimentally measured and theoretically predicted \({\kappa }_{{{\rm{L}}}}\) exhibit temperature-independent behavior over the range of 170–400 K. This temperature-independent behavior of \({\kappa }_{{{\rm{L}}}}\) has also been observed in cubic tetrahedrite Cu12Sb4S1315, although crystalline CsAg2I3 exhibits even lower thermal conductivity. Structural complexity can enhance thermal transport in materials with ultralow thermal conductivity14 and may cause \({\kappa }_{{{\rm{L}}}}\) to exhibit positive temperature dependence13 due to the dominant role of the diffuson channel14. The structural complexity of CsAg2I3, with 12(24) atoms per primitive cell (moderate complexity), may represent a critical threshold to suppress lattice \({\kappa }_{{{\rm{L}}}}\) to an ultralow limit.

To gain deeper insight into the microscopic mechanisms of thermal transport in crystalline CsAg2I3, we calculated both the phonon lifetimes and axis-dependent group velocities, as illustrated in Fig. 5b and Fig. S14, respectively. In Fig. 5b, most phonons have lifetimes shorter than the Wigner limit in time34, highlighting the dominant role of diffuson contribution in the thermal transport of CsAg2I3, as shown in Fig. 5a. Notably, another eye-catching characteristic is that most phonon lifetimes approach the Ioffe-Regel limit62, particularly for phonons with frequencies between 0.2–2.2 THz (approximated 1-9 meV, see Fig. 5b), indicating the strong lattice anharmonicity and main contributor to ultralow \({\kappa }_{{{\rm{L}}}}\) in CsAg2I3.

Combined with the structural complexity in crystalline CsAg₂I₃, the observation of strong anharmonicity further underscores the minor contribution of the particle-like phonon channel to the total \({\kappa }_{{{\rm{L}}}}\). Additionally, from Fig. S14, we observe low \({v}_{{{\rm{g}}}}\) in crystalline CsAg2I3. Specifically, the highest \({v}_{{{\rm{g}}}}\) are 2200 m/s along the x-axis (1800 and 1200 m/s along the y- and z-axes, respectively). These low \({v}_{{{\rm{g}}}}\) further suppress phonon thermal transport in crystalline CsAg2I3, consistent with observations reported in the perovskites50,63. This unique characteristics of \({v}_{{{\rm{g}}}}\) leads to distinct features in the spectral \({\kappa }_{{{\rm{L}}}}^{{{\rm{P}}}}\) along different crystallographic orientations, as shown in Fig. 5c, e and Fig. S15a.

For instance, the spectral \({\kappa }_{{{\rm{L}}}}^{{{\rm{P}}}}\) along the x-axis is predominantly contributed by phonons with frequencies ranging from 0 to 4.6 THz (approximated 0-19 meV), whereas the \({\kappa }_{{{\rm{L}}}}^{{{\rm{P}}}}\) along y- and z-axes is mainly contributed by phonons with frequencies below 1 THz (approximated 4 meV). In contrast, the spectral \({\kappa }_{{{\rm{L}}}}^{{{\rm{C}}}}\) along all crystallographic orientations is contributed by all phonons, unaffected by the material’s structure, due to its diffuson characteristics, as shown in Fig. 5d, f and Fig. S15b. By carefully comparing the atomic participation ratio (APR) (see Fig. S16), phonon lifetimes (see Fig. 5b), and spectral \({\kappa }_{{{\rm{L}}}}^{{{\rm{P}}}}\) (see Fig. 5c, e and S15a), we find that the strong anharmonicity primarily originates from the I atoms. Notably, the atom-resolved PDOS (see Fig. S17) confirms that the low-energy phonon modes are primarily governed by the I-related atoms, further corroborating the large atomic vibration observed in the structural analysis. Additionally, the flattening modes associated with Cs atoms contribute significantly to phonon scattering rates. A similar phenomenon has also been observed in perovskites50 and tetrahedrites15. Overall, the combination of weak bonding (low \({v}_{{{\rm{g}}}}\)) and strong anharmonicity results in the record-low thermal conductivity observed in crystalline CsAg2I3. Finally, from Fig. 5a, we observe a distinct difference in the phonon and coherence contributions to the total thermal conductivity between the Pnma and Cmcm phases. In the Cmcm phase, phonons contribute significantly to the total thermal conductivity, whereas in the Pnma phase, coherence effects dominate. This disparity can be attributed to structural complexity: the Pnma phase contains 24 atoms per unit cell, while the Cmcm phase contains only 12. The increased structural complexity in Pnma suppresses particle-like phonon transport and enhances diffuson contributions to thermal transport14,34.

In summary, informed by the classical Slack model and a unified perspective on minimum κL, we derived a simple yet powerful universal descriptor, \({n}^{-\frac{2}{3}}{(\upsilon /1000)}^{3}\), for rapidly identifying crystalline materials exhibiting ULG-type κL. To avoid the time- and resource-intensive requirement of experimentally or theoretically determining \(\upsilon\), we built a global database of υ for inorganic crystalline materials using high-throughput calculations and machine-learning methods, thereby accelerating the discovery of materials with targeted thermal transport properties. Guided by the universal descriptor and established global \(\upsilon\) database, we identified the CsA₂I₃ (A = Ag, Cu) halide family as a promising materials system, including the compounds CsAg₂I₃ and CsCu₂I₃. We synthesized polycrystals and characterized their thermal conductivities, which displayed record ULG-type κL of 0.15–0.20 W/m·K in the temperature of 170–523 K. To investigate the microscopic relationships between crystal structure, vibrational properties, and phonon transport, we further conducted in-depth PXRD, electron diffraction, INS, and unified theory to a representative compound CsAg2I3.

The temperature-dependent PXRD confirmed the orthorhombic crystal structure of CsAg2I3 characterized by the Pnma space group below 300 K and revealed a structural transition to Bbmm/Cmcm space group above 300 K. This compound, with a moderate complexity of n = 24 for Pnma and n = 12 for Cmcm, can be viewed as consisting of a building unit [Ag2I3]− anionic chains (1D character), along with Cs+ playing the role of counter-cations. The structural analysis emphasizes the key role of iodine vibration and weak Ag-I bonds in the thermal conductivity of the low-temperature form, resulting in extremely low \(\upsilon\) and large anharmonicity. Moreover, our temperature-dependent INS experiments revealed the presence of soft optical phonons located at ~2 meV as well as a phonon softening feature at elevated temperatures, further supporting a strong anharmonic feature. Our unified theory of thermal transport also highlights the dominant diffuson thermal transport properties, and the coexistence of extremely low \({\kappa }_{{{\rm{L}}}}^{{{\rm{C}}}}\) and \({\kappa }_{{{\rm{L}}}}^{{{\rm{p}}}}\) in CsAg2I3. These features, enabled by moderate complexity and strong anharmonicity, are responsible for ULG-type κL of this material. Thus, the discovery of record-low ULG-type κL in the CsA₂I₃ (A = Ag, Cu) halide system provides compelling evidence for the utility of our universal descriptor, high-throughput methodology, and universal machine-learning potentials in identifying crystalline materials exhibiting ULG-type \({\kappa }_{{{\rm{L}}}}\). The applicability and versatility of our descriptor are anticipated to bridge the gap in evaluating ultralow \({\kappa }_{{{\rm{L}}}}\) in crystalline inorganic materials across a broad temperature range. By offering a low-cost and easily deployable approach for both experimentalists and theorists, this work enables accelerated discovery of thermally functional materials for diverse applications.

Methods

Synthesis

The metal halide compounds CsAg2I3 and CsCu2I3 were prepared using stoichiometric amounts of precursors CsI (powder, 99.9%), AgI (powder, 99.9%), and CuI (powder, 99.9%). The mixtures of precursors were ground into fine powders in an Ar-filled glove box. The resulting powders were then cold-pressed into pellets and placed in carbon-coated silica tubes. These tubes were subsequently evacuated and sealed under a vacuum of ~10−3 Pa. The sealed tubes were heated to 623 K over a period of 7 h and held at this temperature for an additional 24 h, followed by cooling down within 7 h. The obtained ingots of CsAg2I3 and CsCu2I3 were ground into fine powders and loaded into a 10 mm graphite die, followed by densification through spark plasma sintering (SPS) at 473 and 523 K, respectively, for 5 min under an applied pressure of 64 MPa. The resultant SPS-ed samples were confirmed to be a highly densified pellet with a 98% of the theoretical density.

X-ray diffraction

Temperature-dependent high-resolution PXRD of the synthesized SPS-ed CsAg2I3 powder were conducted from 250 to 320 K utilizing a Smartlab 9 kW Phenix chamber system equipped with Cu Kα1 radiation (λ = 1.5406 Å) and Johansson Ge (111) monochromator. High-resolution PXRD data of the synthesized SPS-ed CsCu2I3 powders were recorded at 300 K using a Bruker D8 Advance Vario 1 two-circle diffractometer (θ − 2θ, Bragg−Brentano mode) equipped with Cu Kα1 radiation (λ = 1.5406 Å) and a Ge (111) monochromator (Johansson type). Rietveld refinements were carried out using the FullProf64 and WinPlotr65 software packages, incorporating the refinement of parameters, such as zero-point shift, unit cell, peak shape, asymmetry, atomic coordinates, and Biso values of each atoms.

3D electron diffraction and scanning electron microscopy

3D Electron Diffraction (3D ED)66 is a technique that uses electron diffraction patterns collected from a single crystal to determine its crystal structure. For this study, precession-assisted 3D ED data were collected using a JEOL F200 transmission electron microscope at 200 kV, equipped with an ASI Cheetah M3 detector and a Nanomegas Digistar precession unit. A CsAg₂I₃ pellet was ground in ethanol with an agate mortar, and a drop of the resulting suspension was deposited onto a holey carbon membrane on a Cu mesh grid. 3D ED data were recorded at 100 K using a GATAN ELSA tomography holder and the Instamatic program67, with a precession angle of 1.25° and a tilt step of approximately 2° between patterns. Data processing was performed with PETS268, and structure refinements were carried out in Jana202069, accounting for both dynamical diffraction effects and electron beam precession. The refined structural parameters of CsAg₂I₃ are summarized in Table S8. Scanning electron micrographs and electron energy dispersive spectroscopy (EDS) analyses of samples were performed using a JEOL JSM-7200F-SEM equipped with an EDX X-Flash Bruker detector.

Heat capacity and thermal conductivity measurements

The thermal conductivity (\(\kappa\)) was calculated employing the formula \(\kappa=\rho {C}_{p}d\). The wide-range thermal diffusivity (d) of the CsAg2I3 sample was measured using a Netzsch LFA 467 Hyper flash system under a nitrogen atmosphere from 170 to 400 K, while high-temperature d of the CsCu2I3 samples were conducted utilizing a Netzsch LFA 457 laser flash system under a nitrogen atmosphere from 300 to 523 K. The density (ρ) was determined via the Archimedes method. Cp measurement of the CsAg2I3 sample were conducted in the temperature range of 2 to 350 K using a conventional relaxation method with the dedicated 4He option of the PPMS.

Sound velocity measurements

The pulse-echo method was employed to measure the longitudinal and transverse sound velocities of the CsAg2I3 sample at 300 K. A small quantity of grease was applied to ensure effective contact between the sample and the piezoelectric transducers.

Inelastic neutron scattering

Inelastic neutron scattering experiments were conducted on a powder sample weighing ~10 g. The sample was measured at temperatures between 1.5 and 300 K using the thermal neutron time-of-flight spectrometer PANTHER at the Institut Laue Langevin in Grenoble, France. We employed nominal incident energies Ei of 30.0 meV to examine the full phonon spectrum and 12.5 meV to monitor low-energy phonons. We utilized a standard cryostat, with the powder sample contained in an aluminum sample holder. The acquisition periods varied from 1 to 2 hours, depending on Ei and temperature, which influenced the inelastic intensity.

The evolution of the low-energy phonons was recorded during cooling from 300 K to 1.5 using an Ei = 12.5 meV. An acquisition period of 10 min led to a temperature difference of 5 K or less between spectra. To ensure accurate data analysis, we performed additional measurements on empty sample holders, the sample environments, and vanadium. These measurements were essential for standardizing our data correction procedure and converting the corrected signal to the dynamic structure factor S( | Q | , E) and the GDOS g(n)(E). We utilized the Mantid software package for the correction and conversion computing70.

For a given polyatomic material, GDOS can be approximated from the atomic partial phonon density of states (gi(n)(E)) of element i as Eq. (5)60:

$${g}^{\left(n\right)}(E)={\sum }_{i} \; {f}_{i}\frac{{\sigma }_{i}}{{m}_{i}}{g}_{i}(E)\exp (-2{W}_{i})$$
(5)

Where i, \({f}_{i}\), \({\sigma }_{i}\), mi, and Wi denote the different elements, atomic concentration, neutron total cross-section, mass, and Debye-Waller factor of element i, respectively. The values of \(\frac{{\sigma }_{i}}{{m}_{i}}\) for Cs, Ag, and I are 0.03, 0.046, and 0.03 barn/amu.

Density functional theory calculations

In this work, all the ab initio calculations for crystalline CsAg2I3 were conducted using density functional theory (DFT)71, as implemented in the Vienna Ab initio Simulation Package (VASP)72. The projector-augmented wave (PAW)73 pseudopotentials were utilized to explicitly treat the 5s25p66s1, 4d105s1 and 5s25p5 electrons as valence states for Cs, Ag, and I atoms, respectively. For the exchange-correlation functional in all DFT calculations, we employed the revised Perdew-Burke-Ernzerhof (PBE) version for solids, i.e., PBEsol functional74, within the generalized gradient approximation (GGA) framework75. In both structural relaxation and self-consistent DFT calculations, tight convergence criteria were applied, with force convergence set to 10−5 eV·Å−1 and energy convergence to 10−8 eV. For crystalline CsAg2I3 with a space group of \({Pnma}\), a kinetic energy cutoff of 600 eV and a Γ-centered 8 × 10 × 8 Monkhorst-Pack k-mesh were used to sample the Brillouin zone in the primitive cell, which contains 24 atoms. The resulting fully relaxed lattice constants are a = 13.9250 Å, b = 5.8845 Å and c = 11.3099 Å. Similarly, for crystalline CsAg2I3 with a space group of \({Cmcm}\), the same kinetic energy cutoff and a Γ-centered 10 × 10 × 10 Monkhorst-Pack k-mesh were employed to sample the Brillouin zone in the primitive cell containing 12 atoms. The fully optimized primitive-cell lattice constants are a = b = 8.9727 Å and c = 5.8862 Å. For the corresponding conventional unit cell, the lattice constants are a = 11.2991 Å, b = 13.9416 Å, and c = 5.8862 Å.

Effective harmonic force constants extraction

To obtain the effective harmonic phonon frequency at finite temperatures, the temperature-dependent effective potential (TDEP) method76 was used to fit first-principles forces to an effective Hamiltonian (\(H\)),

$$H={U}_{0}+{\sum }_{i}\frac{{{{\boldsymbol{P}}}}_{i}^{2}}{2{m}_{i}}+\frac{1}{2}{\sum }_{{ij}\alpha \beta }{\Phi }_{{ij}}^{\alpha \beta }{u}_{i}^{\alpha }{u}_{j}^{\beta },$$
(6)

where \({U}_{0}\) denotes the potential energy, \(i\) and \(j\) denote the atomic indeces, \({p}_{i}\) denotes the momentum associated with atom \(i\), \({m}_{i}\) denotes atomic mass, \({u}_{i}\) denotes atomic displacment. \({\Phi }_{{ij}}^{\alpha \beta }\) denotes the effective harmonic interatomic force constants (IFCs), associated with the Cartesian indices \(\alpha\) and \(\beta\).

To prepare the finite-temperature displacement-force dataset for IFCs extraction, stochastic sampling of the canonical ensemble was employed to generate perturbed supercells77,78, followed by precise self-consistent DFT calculations to determine the forces on all atoms. The Cartesian displacement (\({u}_{i}^{\alpha }\)) is normally distributed around the mean thermal dispacement and is expressed as

$${u}_{i}^{\alpha }={\sum }_{q}{e}_{q}^{i\alpha }\left\langle {A}_{{iq}}\right\rangle \sqrt{-2{\mathrm{ln}}{\zeta }_{1}}\sin (2\pi {\zeta }_{2})$$
(7)

with the thermal amplitude \(\langle {A}_{{iq}}\rangle\) given as77,78,79

$$\left\langle {A}_{{iq}}\right\rangle=\sqrt{\frac{\hslash (2{n}_{q}^{0}+1)}{2{{m}_{i}\omega }_{q}}},$$
(8)

where q represents the phonon mode, acting as a composite index that combines the wavevector q and phonon branch s, \({e}_{q}\) represents the eigenvector, \({\zeta }_{1}\) and \({\zeta }_{2}\) represent stochastically sampled numbers between 0 and 1, \(\hslash\), \({n}_{q}\) and \({\omega }_{q}\) represent the Planck constants, the occupation number following the Bose-Einstein distribution and the phonon frequency, respectively.

In this work, a 1 × 2 × 2 supercell containing 96 atoms of crystalline CsAg2I3 with space group of \({Pnma}\) was used to perform calculations iteratively, starting with 500 thermally perturbed snapshots. The resulting 500 atomic configurations were then used to generate the displacement-force dataset through precise DFT calculations with a Γ-centered 4 × 4 × 4 Monkhorst-Pack k-point density grid. Each iteration involves key procedures, including computing phonon normal modes, generating perturbed snapshots, calculating precise DFT forces, and fitting effective IFCs. To ensure the convergence of finite-temperature IFCs, the final iteration at each temperature (50, 100, 200, 300, and 400 K) was performed using 1500 snapshots.

For the crystalline CsAg2I3 with space group of \({Cmcm}\), a 2 × 2 × 2 supercell containing 96 atoms was used for the iterative calculations of effective IFCs. Similarly, the iteration was initiated with 500 thermally perturbed snapshots generated from zero-K phonon dispersions. Precise forces were obtained through self-consistent DFT calculation using a Γ-centered 4 × 4 × 4 Monkhorst-Pack k-point density grid. The final iteration at each temperature (300 and 400 K) was conducted with 1500 snapshots to ensure the convergence of finite-temperature IFCs. In this study, the temperature-dependent effective potential calculations were performed using the ALAMODE package80.

Effective anharmonic force constants extraction

Instead of the least-squares approach for extracting the effective harmonic interatomic force constants (IFCs), the Compressive Sensing Lattice Dynamics (CSLD) method81 was employed to efficiently and accurately extract the anharmonic IFCs. The CSLD method efficiently identifies the physically significant terms from a large set of irreducible anharmonic IFCs using a limited displacement-force dataset81. To extract the effective anharmonic IFCs, we sampled a dataset consisting of 300 configurations, with each configuration containing up to 96 displacement-force pairs, from the previous harmonic canonical ensemble at finite temperatures76,82. Subsequently, the effective harmonic IFCs and the displacement-force dataset were used as an input to extract anharmonic IFCs up to the sixth order. The anharmonic IFCs of crystalline CsAg2I3 with both space groups \({Pnma}\) and \({Cmcm}\) were extracted using the least absolute shrinkage and selection operator (LASSO) technique83, with real-space cutoff radii of 7.41 Å, 6.35 Å, 4.23 Å, and 3.17 Å for the cubic, quartic, quintic, and septic IFCs, respectively. In this work, the IFCs fitting process was performed using the ALAMODE package80,81,84.

Anharmonic phonon scattering rates

With the effective interatomic force constants (IFCs) available and using the Fermi’s golden rule85, the phonon scattering rates for three-phonon (3 ph) \({\varGamma }_{q}^{3{ph}}\) and four-phonon (4 ph) \({\varGamma }_{q}^{4{ph}}\) processes under the single-mode relaxation time approximation (SMRTA) are expressed as15,85

$${\Gamma }_{q}^{3{ph}}={\sum }_{q{\prime} q{\prime} {\prime} }\left\{\frac{1}{2}\left(1+{n}_{q{\prime} }^{0}+{n}_{q{\prime} {\prime} }^{0}\right){{{\mathscr{L}}}}_{-}+\left({n}_{q{\prime} }^{0}-{n}_{q{\prime} {\prime} }^{0}\right){{{\mathscr{L}}}}_{+}\right\},$$
(9)
$${\Gamma }_{q}^{4{ph}}= {\sum }_{q{\prime} q{\prime} {\prime} q{\prime} {\prime} {\prime} }\left\{\frac{1}{6}\left(\frac{{n}_{q{\prime} }^{0}{n}_{q{\prime} {\prime} }^{0}{n}_{q{\prime} {\prime} {\prime} }^{0}}{{n}_{q}^{0}}\right){{{\mathscr{L}}}}_{--}+\frac{1}{2}\left(\frac{\left(1+{n}_{q{\prime} }^{0}\right){n}_{q{\prime} {\prime} }^{0}{n}_{q{\prime} {\prime} {\prime} }^{0}}{{n}_{q}^{0}}\right){{{\mathscr{L}}}}_{+-}\right. \\ \left.+{\frac{1}{2}\left(\frac{{\left(1+{n}_{q{\prime} }^{0}\right)\left(1+{n}_{q{\prime} {\prime} }^{0}\right)n}_{q{\prime} {\prime} {\prime} }^{0}}{{n}_{q}^{0}}\right){{\mathscr{L}}}}_{++ }\right\},$$
(10)

with

$${{{\mathscr{L}}}}_{\pm }=\frac{\pi \hslash }{4N}{\left|{V}^{\left(3\right)}\left(q,\pm {q}^{{\prime} },-q{\prime \prime} \right)\right|}^{2}{\Delta }_{\pm }\frac{\delta \left({\Omega }_{{{\boldsymbol{q}}}}\pm {\Omega }_{{{\boldsymbol{q}}}{{{\prime} }}}{-\Omega }_{{{\boldsymbol{q}}}{{{\prime \prime} }}}\right)}{{\Omega }_{{{\boldsymbol{q}}}}{\Omega }_{{{\boldsymbol{q}}}{{{\prime} }}}{\Omega }_{{{\boldsymbol{q}}}{{{\prime \prime} }}}},$$
(11)

and

$${{{\mathscr{L}}}}_\pm \pm=\frac{\pi {\hslash }^{2}}{8{N}^{2}}{\left|{V}^{\left(4\right)}\left(q,\pm {q}^{{\prime}},\pm {q}^{{\prime \prime}},-q^{\prime \prime \prime} \right)\right|}^{2}{\Delta }_{\pm \pm }\frac{\delta \left({\Omega }_{{{\boldsymbol{q}}}}\pm {\Omega }_{{{\boldsymbol{q}}}{{{\prime} }}}{\pm {\Omega }_{{{\boldsymbol{q}}}{{{\prime \prime} }}}-\Omega }_{{{\boldsymbol{q}}}^{{{\prime \prime \prime } }}}\right)}{{\Omega }_{{{\boldsymbol{q}}}}{\Omega }_{{{\boldsymbol{q}}}{{{\prime} }}}{\Omega }_{{{\boldsymbol{q}}}{{{\prime \prime} }}}{\Omega }_{{{\boldsymbol{q}}}^{{{\prime \prime \prime} }}}}$$
(12)

where \({\varOmega }_{q}\) denotes the effective phonon frequency, \({V}^{\left(3\right)}\left(q,\pm {q}^{\prime },-q^{\prime\prime} \right)\) and \({V}^{\left(4\right)}\left(q,\pm {q}^{{\prime} },\pm {q}^{{\prime\prime} },-{q}^{\prime\prime\prime} \right)\) denote the reciprocal forms of 3rd- and 4th-order effective IFCs86, i.e., the strength of the 3 ph and 4 ph scattering matrices, respectively. In 3 and 4 ph scattering processes, energy conservation is enforced using the delta function \(\delta\), while momentum conservation is imposed via the Kronecker delta \(\Delta\).

The extrinsic phonon-isotope scattering rate \({\varGamma }_{q}^{{iso}}\) also plays a critical role in thermal transport and can be expressed as87

$${\Gamma }_{q}^{{iso}}=\frac{\pi {\Omega }_{q}^{2}}{2N}{\sum }_{i\in u.c.}g\left(i\right){\left|{e}_{q}^{*}\left(i\right)\cdot {e}_{q{\prime} }\left(i\right)\right|}^{2}\delta \left(\Omega -\Omega^ {\prime} \right)$$
(13)

Where \(g\left(i\right)\) denotes the Pearson deviation coefficient. Under the assumption of Matthiessen’s rule, the total phonon scattering rate \({\varGamma }_{q}\) for a specific mode \(q\) is expressed as

$${\varGamma }_{q}={\varGamma }_{q}^{3{ph}}+{\varGamma }_{q}^{4{ph}}+{\varGamma }_{q}^{{iso}}$$
(14)

The unified theory of thermal transport

Given the strong anharmonicity and structural complexity of crystalline CsAg₂I₃, the unified theory of thermal transport proposed by Michelle et al.14,34, which accounts for both \({\kappa }_{{{\rm{L}}}}^{{{\rm{C}}}}\) and \({\kappa }_{{{\rm{L}}}}^{{{\rm{P}}}}\), was employed to calculate the total lattice thermal conductivity \({\kappa }_{{{\rm{L}}}}\). Within the SMRTA framework, the unified theory of thermal transport is expressed as14,34

$${\kappa }_{L}^{P/C}= \frac{{\hslash }^{2}}{{k}_{B}{T}^{2}{VN}}{\sum }_{{{\boldsymbol{q}}}}{\sum }_{s,s{\prime} }\frac{{\Omega }_{{{\boldsymbol{q}}}s}+{\Omega }_{{{\boldsymbol{q}}}s{\prime} }}{2}{\cdot \nu }_{{{\boldsymbol{q}}}{ss}{\prime} }\bigotimes {\nu }_{{{\boldsymbol{q}}}{s}^{{\prime} }s} \\ \cdot \frac{{\Omega }_{{{\boldsymbol{q}}}s}{{{\rm{n}}}}_{{{\boldsymbol{q}}}s}\left({{{\rm{n}}}}_{{{\boldsymbol{q}}}s}+1\right)+{\Omega }_{{{\boldsymbol{q}}}{s}^{{\prime} }}{{{\rm{n}}}}_{{{\boldsymbol{q}}}{s}^{{\prime} }}\left({{{\rm{n}}}}_{{{\boldsymbol{q}}}{s}^{{\prime} }}+1\right)}{4{\left({\Omega }_{{{\boldsymbol{q}}}s}+{\Omega }_{{{\boldsymbol{q}}}{s}^{{\prime} }}\right)}^{2}+{\left({\Gamma }_{{{\boldsymbol{q}}}s}+{\Gamma }_{{{\boldsymbol{q}}}{s}^{{\prime} }}\right)}^{2}}\cdot \left({\Gamma }_{{{\boldsymbol{q}}}s}+{\Gamma }_{{{\boldsymbol{q}}}{s}^{{\prime} }}\right),$$
(15)

where \({k}_{{{\rm{B}}}}\), \(T\), \(V\), and \(N\) are the Boltzmann constant, the cell volume, the absolute temperature and the number of sampled phonon wave vectors, \(\nu\) is the inter- and intra-band group velocity matrix and can be expressed as88

$${\nu }_{{qss}^{\prime} }=\frac{{{\langle }}{e}_{{qs}}\left|\frac{\partial D\left(q\right)}{\partial q}\right|{e}_{{qs}^{\prime} }{{\rangle }}}{2\sqrt{{\varOmega }_{{qs}}{\varOmega }_{{qs}^{\prime} }}}$$
(16)

Where \(D\left(q\right)\) denotes the dynamical matrix. The interband term (\(s=s{\prime}\)) results in diagonal terms of heat flux operators and contributing to \({\kappa }_{{{\rm{L}}}}^{{{\rm{P}}}}\). In contrast, the intraband term (\(s\ne s^{\prime}\)) gives rise to off-diagonal terms of heat flux operators and contributing to \({\kappa }_{{{\rm{L}}}}^{{{\rm{c}}}}\). To calculate the total lattice thermla conductivity \({\kappa }_{{{\rm{L}}}}\) of crystalline CsAg₂I₃ with space group of \({Pnma}\)(\({Cmcm}\)), which contains 24(12) atoms in the primitive cell, a \(q\) mesh of 8 × 8 × 8 (12 × 12 × 12) was used for both 3 ph and 4 ph scattering processes, with a scalebroadening parameter of 0.1(0.2). The parameters chosen for Eq.(15) were thoroughly tested for the convergence and yield well-converged results. In this work, lattice thermal conductivity calculations, including population and coherence contributions, were performed using the ShengBTE89 and FourPhonon86 packages, along with our in-house code49,50.

The high-throughput calculation and universal machine-learning force field: MACE

MACE is an equivariant message passing neural network potential, using higher order messages combined with Atomic Cluster Expansion (ACE), a method for deriving an efficient basis to represent functions of atomic neighborhoods. Note that the version of MACE used in this work is the MACE-MP-0 (medium) (https://github.com/ACEsuit/mace-foundations), as implemented in the atomate2 package90.