Ruddlesden-Popper chalcogenides push the limit of mechanical stiffness and glass-like thermal conductivity in single crystals

Abstract

Insulating materials featuring ultralow thermal conductivity for diverse applications also require robust mechanical properties. Conventional thinking, however, which correlates strong bonding with high atomic-vibration-mediated heat conduction, led to diverse weakly bonded materials that feature ultralow thermal conductivity and low elastic moduli. One must, therefore, search for strongly-bonded single crystals in which heat transport is impeded by other means. Here, we report intrinsic, glass-like, ultralow thermal conductivity and ultrahigh elastic-modulus/thermal-conductivity ratio in single-crystalline Ruddlesden-Popper Ba_n+1Zr_nS_3n+1, n = 2, 3, which are derivatives of BaZrS₃. Their key features are strong anharmonicity and intra-unit-cell rock-salt blocks. The latter produce strongly bonded intrinsic superlattices, impeding heat conduction by broadband reduction of phonon velocities and mean free paths and concomitant strong phonon localization. The present study initiates a paradigm of “mechanically stiff phonon glasses”.

Introduction

Designing materials with ultralow thermal conductivity (κ) without reducing their density and degrading their mechanical properties typically evades century-old theories on microscopic heat conduction. In non-metallic crystals, the primary modes of thermal transport are lattice vibrations, namely phonons. Dating back to theories originally pioneered by Peierls, Leibfreid and Schlomann, Einstein, and Debye, the thermal transport of atomic vibrations is directly related to the interatomic bond strengths^{1,2,3,4,5,6,7,8,9}. Hence, “ultralow" thermal conductivity in solids often comes at the cost of “weak” interatomic-bond strength and “soft” elastic modulus (E), limiting their mechanical performance^{10,11,12,13,14}. In recent years, a family of halide perovskite-structure crystalline solids has been found to feature ultralow thermal conductivities, of order 0.15–0.3 W m⁻¹ K⁻¹, but their weak van der Waals (vdW) bonding leads to very low values of elastic and shear moduli^{15,16,17,18,19,20}. Breaking this Pareto-normality in the design of crystals to create ultrahigh E/κ materials must, therefore, involve different strategies than those traditionally proposed to limit phonon transport. Such decoupling would have major implications for both inorganic and organic materials in many technological applications²¹. As materials such as thermal barrier coatings and thermoelectrics are subjected to large thermal gradients and intensive cycling, low thermal conductivity and high elastic modulus are often desired for better efficiency and crack propagation resistance^{22,23,24,25,26,27,28}. Thermally insulating and mechanically strong materials are also coveted in metal contact adjacents to prevent failure from electromigration^29,30,31.

To overcome the barriers of achieving ultrahigh E/κ materials and create new directions for achieving single crystals with ultralow thermal conductivity, one seeks to inhibit the propagation of phonons in a “hard” lattice. In perfect crystals, phonons travel with intrinsic group velocities and are inhibited by phonon-phonon scattering caused by anharmonic effects. Historically, chalcogenides (S, Se, and Te) have been investigated for ultralow thermal conductivity because of strong anharmonicity (e.g., PbSe, Bi₂Te₃, SnSe)^{32,33,34,35,36}. However, many of these materials tend to crystallize in layered structures with vdW interactions, leading to poor mechanical properties across the layers due to ease of shearing. Thus, covalently-bonded layered chalcogenides present a potential pathway to low thermal conductivity and high elastic modulus in the direction across the layers. We, therefore, present Ruddlesden-Popper (RP) sulfides^37,38 as potential candidates for this scenario. Unlike some RP halide perovskites that are bonded by weak vdW forces,^16,19 the RP sulfides hold both the promise of high anharmonicity and strong bonding in a layered, superlattice-like structure with intrinsic interfaces.

Herein, we report on the ultralow thermal conductivity of RP phases Ba_n+1Zr_nS_3n+1, n = 2 and 3 of barium zirconium sulfide (BaZrS₃)^38,39 single crystals, enabled by strong anharmonicity and a large fraction of localized and low-velocity vibrational modes throughout the entire vibrational spectrum, thus achieving broadband attenuation of thermal transport. We reveal the origin of the ultralow thermal conductivities in these RP derivatives of the chalcogenide perovskite BaZrS₃ using a combination of experiments and first-principles- and machine learning-driven computational approaches. Unlike previously studied vdW layered crystals, including the RP halides,^16,19 the strong Ba-S and Zr-S chalcogenide bonds across the rock-salt layers and the (BaZrS₃)_n layers bring elastic-moduli values of the RP phases nearly an order of magnitude higher than other ultralow-thermal-conductivity inorganic crystals. As a result, the RP phases of these single crystals exhibit record-setting values of E/κ for any single-crystalline material discovered to date, while maintaining glass-like, ultralow thermal conductivities. Such a useful combination of thermal and mechanical properties makes the materials highly desirable in the fields of thermal barrier coatings and thermoelectric materials⁴⁰.

Results and discussion

To investigate the impact of sub-unit cell structures on thermal conductivity, we consider perovskite BaZrS₃ and two RP phases, Ba₃Zr₂S₇ and Ba₄Zr₃S₁₀, whose crystal structures are investigated by X-ray diffraction in Supporting Information section S1. The perovskite shown in Fig. 1a consists of tilted ZrS₆ octahedra (Fig. 1c) and BaS₁₂ polyhedra (Fig. 1d). The RP-phase Ba₃Zr₂S₇ shown in Fig. 1b contains two perovskite sections (red brackets) that are separated by rock-salt packed layers (Fig. 1e). Ba₄Zr₃S₁₀ differs from Ba₃Zr₂S₇ by adding one more BaS and ZrS₂ atomic layer to each perovskite section. Fig. 1f shows an iDPC image of BaZrS₃ and Fig. 1g, h show the layered periodic stacking of perovskite layers in the two RP phases with the enlarged regions of interest, emphasizing one of two rock-salt layers that are present in a single unit cell. The elemental maps generated by electron-energy-loss spectroscopy (EELS) and the corresponding atomic-number-contrast (Z-contrast) image shown in Fig. 1(i-l) show the high degree of chemical ordering in each sublattice and the change in local symmetry at the rock-salt layers.

Fig. 1: Structure of perovskite BaZrS₃ and Ruddlesden-Popper Ba_n+1Zr_nS_3n+1.

Ball-and-stick model of (a) Pnma perovskite BaZrS₃ and (b) I4mmm Ruddlesden-Popper Ba₃Zr₂S₇ showing gray ZrS₆ octahedra and green BaS bonds. Red markers in (b) indicate the perovskite blocks of the RP phase separated by the rock-salt blocks. Blue dotted lines between BaS atomic planes indicate the midplane of the rock-salt blocks. Ball-and-stick model of a (c) Zr octahedra, (d) undistorted Ba polyhedral, and (e) rock-salt building block resulting from layering in Ba₃Zr₂S₇. Integrated differential phase contrast images of the (f) BaZrS₃, (g) Ba₃Zr₂S₇, (h) Ba₄Zr₃S₁₀ crystals. Enlarged regions from the cyan annotations are shown below each image. In the enlargements, two perovskite unit cells are annotated with Ba (green), Zr (gray) and S (yellow) circles. (i) Z-contrast image and (j) BaZr composite image from a STEM-EELS spectrum image. Intensity maps of (k) Ba-M₄₅, (l) Zr-M₂₃ background-subtracted edges. m, n Section views along (200) and (110) of the valence electron density of the RP-phase Ba₃Zr₂S₇ calculated by DFT, respectively. Red marks and blue dashed lines help to correlate atomic structure to that of panel (b). o A line profile of the valence electron density along the white dashed line in (m). Blue bars illustrate the rock-salt-block regions.

To understand the bonding in the RP phases, density functional theory (DFT) calculations were performed on Ba₃Zr₂S₇ to calculate the valence electron density, as shown in Fig. 1m, n. A line profile along the dashed white line in Fig. 1m is shown in Fig. 1o for a more quantitative evaluation. The valence electron density within the rock-salt regions is non-zero and comparable to those inside the perovskite blocks, which suggests similar intra- and inter-perovskite-block bonding strength. In other words, the bonding in the rock-salt regions, namely across the gaps highlighted in Fig. 1(b), is not of the weak, vdW type. The overall strong bonding is also reflected in the calculated elastic moduli, which have comparable values along the cross-plane and in-plane directions. The elastic-moduli values are three to four times higher than those observed in other ultralow-thermal-conductivity halide perovskites, such as, Cs₃Bi₂I₆Cl₃,¹⁷ Cs₃Bi₂I₉,¹⁸ the RP-phase Cs₂PbI₂Cl₂¹⁶ and several metal halide perovskites (see Supplemental Table S3). The presence of strong intra- and inter-perovskite-block bonding strength in the sulfide RP phases described here mitigate the role of bond strength in the observed ultralow thermal conductivity.

To understand phonon transport through the structures, we measured the cross-plane (c-axis) thermal conductivity using time-domain thermoreflectance (TDTR) from 100 to 400 K, as shown in Fig. 2a. The thermal conductivities exhibit several unusual features for single-crystalline materials. First, the thermal conductivity of crystalline BaZrS₃ increases from 100 to 250 K, then remains relatively temperature independent. Such trend is observed in amorphous materials and disordered crystals, but it is rare in single crystals⁷. The weak or negligible temperature dependence cannot be explained by prior first-principles three-phonon scattering calculations⁴¹ (the purple dashed curve in Fig. 2a). In contrast, our DFT-based machine-learning-interatomic-potential (MLIP)-driven molecular-dynamics (MD) simulation results (orange open dots in Fig. 2a) show good overall agreement with experimentally measured data, with only small differences above 200 K. This agreement demonstrates the accuracy of MLIP-MD (or MLMD) simulations, which implicitly capture all the atomic-vibration contributions to thermal conductivity, including the effects of three-phonon scattering, four-phonon scattering, finite-temperature phonon renormalization, temperature correction to force constant, and diffuson contributions^42,43. This agreement also confirms that the glass-like thermal conductivity trend of single-crystalline BaZrS₃ is not from extrinsic defects, but rather mechanisms intrinsic to the crystal. However, the exact mechanism behind the temperature trend in the thermal conductivity of single-crystalline BaZrS₃ remains an open question. Supporting Figure S11 shows that perovskites have been reported to exhibit both glass- and crystalline-like thermal conductivity trends. Isolating the mechanisms that underlie the thermal conductivities observed in this class of materials is beyond the scope of the present study, as we focus on the RP phases.

Fig. 2: Thermal conductivities - experimental and simulated.

a TDTR-measured cross-plane thermal conductivity of BaZrS₃ and its RP derivatives. For comparison, we also include the three-phonon and MLMD predicted thermal conductivity of BaZrS₃ and RP phases. The three-phonon prediction is adopted from Osei-Agyemang et al.⁴¹ The minimum limit and diffuson limit refer to Cahill et al.⁴ and Agne et al.'s⁶ theoretical models. b Thermal conductivity distribution of BaZrS₃ and RP phases as a function of heavy ion-irradiation doses. The thermal conductivity of the crystalline BaZrS₃ is lower compared to panel (a) due to the presence of nano-domains (see Supporting Information for details). c Anisotropic thermal conductivity of Ba₃Zr₂S₇ measured by TDTR at room temperature. The uncertainty/error bars of the measurements are discussed in the Supporting Information.

The RP phases Ba₃Zr₂S₇ and Ba₄Zr₃S₁₀ possess ultralow thermal conductivities, i.e., 0.45 ± 0.07 and 0.42 ± 0.05 W m⁻¹ K⁻¹, respectively, over a relatively large temperature range. These values are ~3.5 times lower than those of single-crystalline BaZrS₃, in agreement with the Agne et al.’s diffuson limit,⁶ and lower than Cahill et al.’s glass limit⁴. These two limits are two of the most commonly used theoretical models to predict the lowest possible thermal conductivity of a crystalline material. Additionally, the thermal conductivities of Ba₃Zr₂S₇ (green diamonds) and Ba₄Zr₃S₁₀ (blue triangles) show glass-like temperature trends, comparable to that of amorphous BaZrS₃ (black circles). The MLMD thermal conductivity simulations of Ba₃Zr₂S₇ show quantitative agreement with the experimental data. The cross-plane thermal conductivities of the sulfide RP phases are even lower than those of vdW layered materials, except when thin film samples and interlayer rotations are involved^10,44. On the other hand, the sulfide RP phases feature far superior mechanical properties of all vdW layered materials.

Defects in materials have also been known to lead to ultralow thermal conductivity and glass-like temperature trends⁷. X-ray diffraction data mentioned earlier (shown in Supporting Information section S1), combined with the agreement of the simulations of the thermal conductivity with the experimental data, prove the high quality of the single crystals used in this study, which precludes the attribution of the low thermal conductivity to disorder and defects. To study how the introduction of defects would change thermal conductivity, we irradiate the BaZrS₃ and Ba₄Zr₃S₁₀ crystals with high energy gold ions. The measured cross-plane thermal conductivities of the heavily ion-irradiated crystals as a function of ion dose are shown in Fig. 2b. The thermal conductivity of BaZrS₃ exhibits a sigmoidal reduction, typically characteristic of irradiated crystalline materials^45,46. At low doses, irradiation introduces low concentrations of clustered point defects and vacancies. The overall crystal structure remains relatively unchanged, whereby the thermal conductivity is nearly constant. At high doses, point-defect concentrations increase and damaged regions overlap, which gradually decreases the thermal conductivity to that of an amorphous solid.

Compared to BaZrS₃, a completely different thermal conductivity trend is observed vs. ion dose in the RP phases; the thermal conductivity of the Ba₄Zr₃S₁₀ crystals remains nearly constant regardless of gold ion dose. TEM micrographs show that the layering of Ba₄Zr₃S₁₀ crystals remains uninterrupted throughout the range of doses, although high doses can introduce amorphous pockets (see Supporting Information). In vdW layered materials with interlayer rotation, ion irradiation can lead to increased thermal conductivity due to increases in interatomic bonding¹⁰. However, due to the already strong bonding of the RP phases, no such trend is observed in the present study. The resistance of the RP crystals to irradiation damage makes them a highly suitable thermal barrier coating in deep space applications in radiation environments.

To gain insight into the role of anisotropy in ultralow thermal conductivity of the RP phases, we measure the thermal conductivity of Ba₃Zr₂S₇ along the in-plane direction (perpendicular to c-axis) at room temperature as shown in Fig. 2c. The in-plane thermal conductivity is 1.06 ± 0.14 W m⁻¹ K⁻¹ which is ~2.5 times higher than the cross-plane thermal conductivity (0.45 ± 0.07 W m⁻¹ K⁻¹). It is noteworthy that the MLMD-simulated in-plane and cross-plane thermal conductivities are 1.02 ± 0.17 and 0.53 ± 0.02 W m⁻¹ K⁻¹, respectively, showing excellent quantitative agreement with the experimental anisotropy. The agreement combined with the X-ray diffraction data rule out the impact of defects on measured thermal conductivities. The major structural difference between the in-plane and cross-plane directions is the periodic rock-salt and perovskite layers, relative to continuous layers in-plane. This result suggests that, despite the strong bonding, the rock-salt layers are causing the ultralow and anisotropic thermal conductivity in the RP phases.

To understand how the introduction of rock-salt layering leads to the ultralow thermal conductivities in the RP phases compared to that of crystalline BaZrS₃, we consider three factors that may contribute to ultralow thermal conductivity: 1) anharmonic scattering, 2) decreased phonon group velocities (v_g), and 3) phonon localization within unit cells⁴⁷.

To assess the role of anharmonic scattering, we calculated the spectral energy density (SED) of BaZrS₃ and Ba₃Zr₂S₇ based on MLMD, as shown in Fig. 3a, b. Compared to a typical semiconductor (e.g., silicon, gallium nitride), the SEDs of both BaZrS₃ and Ba₃Zr₂S₇ show much more blurred and broadened linewidths, indicating strong anharmonicity and large phonon scattering rates^47,48. These features could be a critical reason for the ultralow thermal conductivity. The large overlapping of branches due to the anharmonic broadening also indicates that the interband tunneling (i.e., diffusons) should be significant, based on the Wigner formalism^49,50. This feature could be a reason for the glass-like thermal conductivity exhibited by the experimental data. Recent calculations using the Wigner formalism show that diffuson contributes 30% to thermal conductivity of BaZrS₃. We expect the diffuson contribution to be even larger in Ba₃Zr₂S₇ since the broadening of SED is more significant.

Fig. 3: Evaluation of factors that influence thermal conductivity.

Spectral energy density calculated from MLMD for (a) BaZrS₃ and (b) Ba₃Zr₂S₇. The overlays are the harmonic dispersions of phonons in BaZrS₃ and Ba₃Zr₂S₇, which contrast with the blurred SED (SEDs for materials with negligible anharmonic effects look very much like the corresponding harmonic dispersions, as we show in the case of silicon in Supplemental Figure S19). c v_g of the BaZrS₃ and RP structure along the cross-plane direction as a function of frequency with the dashed black line serving as a reference for air. d Cumulative number of phonon modes having up to a given group velocity as a function of frequency for the crystals along cross- (z) and in- (x) plane directions. e Participation ratio (PR) as a function of frequency for BaZrS₃ and (b) Ba₃Zr₂S₇. f Cumulative number of phonon modes having up to a given PR as a function of frequency.

Group velocity (v_g) is another important parameter to understand the diffusivity of vibrational modes in a material¹³. The group velocities along the cross-plane direction (v_g,z) of BaZrS₃ and Ba₃Zr₂S₇ are compared in Fig. 3c. Despite the presence of strong intra- and inter-perovskite-block bonding strength in Ba₃Zr₂S₇, its cross-plane phonon modes have smaller group velocities at nearly all frequencies with relatively flat dispersions. Though weak cross-plane bonding (e.g., in van der Waals layered materials) results in relatively flat bands, the reverse is not true as strongly bonded materials often have flat bands. For example, optical phonon bands are often relatively flat in strongly bonded materials, e.g., GaAs. Another strongly bonded material, β-Ga₂O₃, has a relatively large number of flat bands and thus low thermal conductivity. The origin of relatively flat bands in strongly bonded materials is not always clear.

In Fig. 3d, we plot the cumulative number of phonon modes as a function of group velocity to quantify the number of modes having ultralow group velocities. We find that 80% of phonon modes of Ba₃Zr₂S₇ have a v_g,z lower than the sound speed of air. In comparison, the x component shows a larger group velocity, which is similar to that of BaZrS₃. This feature would indicate that the in-plane thermal conductivity of Ba₃Zr₂S₇ is similar to that of BaZrS₃ but the cross-plane thermal conductivity would be much lower, which is consistent with both the experimental observations and the simulations. Thus, the ultralow phonon velocities, induced by the presence of periodic building blocks in unit cells, is another key contributing factor to the ultralow thermal conductivities.

The third possible factor contributing to ultralow thermal conductivity is phonon localization. We note that the ultralow group velocities of Fig. 3c, d do not inherently imply phonon localization. For example, phonons in relatively flat optical bands in GaAs have small group velocities, but are not localized anywhere because the primitive unit cell has only two atoms. To study the spatial localization of phonon waves within a unit cell of BaZrS₃ and Ba₃Zr₂S₇, we calculate the participation ratio (PR) of the vibrational modes. Localized vibrational modes are usually defined as having a participation ratio lower than 0.1⁵¹. As shown in Fig. 3(e,f), the participation ratio of Ba₃Zr₂S₇ is significantly lower than that of BaZrS₃ across all frequencies. It is noteworthy that some low-frequency modes (<2 THz) in the RP phase have a participation ratio that is smaller than 0.1, which is comparable to the localization expected for locons in amorphous materials^51,52. This result provides evidence that the presence of rock-salt building block layers in the RP phases causes a significant number of vibrational modes to become highly localized¹⁸.

To further show the localization of the vibrational modes, we estimate the average mean free path of phonons in Ba₃Zr₂S₇ in Supporting Figure S12. Assuming the diffuson thermal conductivity is zero, the average mean free path of phonons is estimated as 1 nm, to match with experimental thermal conductivity. Since diffuson contribution is nonzero, the actual mean free path of phonons should be smaller than 1 nm, the inter-gap thickness. These considerations indicate that the phonons are localized inside the rock-salt layers of Ba₃Zr₂S₇ by the gaps, being consistent with the participation ratio results.

In summary, the presence of intra-unit-cell rock-salt blocks in the sulfide RP phases derived from BaZrS₃ and the corresponding selenide and telluride RP phases effectively produces strongly bonded, intrinsic superlattices with different periodicities and interfacial regions that largely reduce phonon velocities and mean free paths, inducing strong localization. Combined with the strong anharmonicity that is known to be intrinsic to chalcogenides,^{32,33,34,35,36} the chalcogenide RP phases are a class of single-crystalline materials that can achieve broadband restriction of thermal transport, leading to ultralow thermal conductivities, while sustaining high elastic moduli and hence high E/κ ratios.

In Fig. 4, we compare the E/κ of BaZrS₃ and its RP derivatives with a wide range of single crystals. The materials shown here range from soft, insulating crystals (e.g., Co₆S₈) to stiff, conductive crystals (e.g., diamond). The elastic moduli of BaZrS₃ and RP phases are significantly higher than those of other ultralow-thermal-conductivity single crystals (e.g., superatoms, metal halide perovskites, and layered perovskites) and are surpassed only by oxides and some semiconductors. Despite such strong bonding, the RP phases possess an ultralow thermal conductivity. As a result, the E/κ ratio of the Ba₃Zr₂S₇ single crystal is the highest reported to date. Though some polycrystalline materials have larger E/κ ratios,^24,53 the record E/κ ratio shown in Fig. 4 among single-crystalline solids is very significant in its own right as not only it eliminates any extrinsic spurious influences in E/κ values, but also allows us to study the fundamental mechanisms behind the mechanically stiff phonon glass paradigm. Additionally, the E/κ ratio of the RP phase is ~3 times higher than that of BaZrS₃.

Fig. 4: Elastic modulus/thermal conductivity (E/κ) ratio for a wide range of single-crystalline materials at room temperature.

The crystals are grouped into superatoms, metal halide perovskites, semiconductors, oxides, layered materials, and BaZrS₃ and its RP derivatives. The thermal conductivity and elastic modulus data of the single crystals are provided in the Supporting Information.

As shown in Fig. 4, oxides generally possess a high elastic modulus. By replacing oxygen with sulfur, we reduce thermal conductivity significantly with moderate reduction in the elastic modulus. Introduction of strongly bonded periodic interfaces can further reduce the thermal conductivity without sacrificing stiffness proportionately. The sulfides studied here sit in an ideal regime of relatively high elastic modulus and low thermal conductivity thereby opening a paradigm for finding high E/κ ratio materials. For example, RP phases of BaHfS₃ are also likely to exhibit a high E/κ ratio. Moreover, the study of chalcogenide perovskites has gained momentum only recently. We anticipate the discovery and synthesis of a broad range of layered phases such as Aurivillius phases and Dion Jacobson phases in the near future^54,55. These layered materials will present additional opportunities to test and further the paradigm of “mechanically stiff phonon glass”.

The RP phases of BaZrS₃ is found to possess ultralow thermal conductivity and ultrahigh modulus-to-thermal conductivity ratio. We find that the rock-salt layers separating the perovskite sections of the RP structure lead to highly anisotropic thermal conductivity, with the cross-plane reaching values comparable to the amorphous solid despite similar and highly strong bonding across the full unit cell. Together with simulations, our results provide evidence that the rock-salt layers in the single crystal RP phases lead to ultralow phonon velocities, ultrashort phonon mean free paths, and strong localization within rock-salt layers, leading to ultralow, glass-like thermal conductivity. Our study provides a detailed overview of the mechanisms needed to achieve ultralow thermal conductivity in a non-vdW, strongly bonded, layered material.

Methods

Time-domain thermoreflectance (TDTR)

We use a two-tint time-domain thermoreflectance (TDTR) setup to measure the thermal conductivity of the crystalline and amorphous BaZrS₃, Ba₃Zr₂S₇, and Ba₄Zr₃S₁₀ specimens^56,57. In our TDTR setup, a Ti:sapphire oscillator (80 MHz, ~808 nm central wavelength, and ~14 nm full width at half maximum) emanates subpicosecond laser pulses that are split into a high-power pump and a low-power probe beam. The pump beam is modulated at a frequency of 8.4 MHz by an electro-optic modulator (EOM) to create oscillatory heating events at the sample surface. The probe beam is then directed through a mechanical delay stage to detect the temporal change in thermoreflectivity which is related to the surface temperature change. Using a lock-in amplifier and a balanced photodetector, the probe beam measures the temperature decay up to 5.5 ns. The TDTR data are analyzed by fitting a cylindrically symmetric, multilayer thermal model to the ratio of in-phase to out-of-phase signal (–V_in/V_out) from the RF lock-in amplifier^58,59,60,61.

Ion irradiation

The BaZrS₃ and Ba₄Zr₃S₁₀ single crystals are irradiated with gold (Au) ions at an energy of 2.8 MeV using a 6 MV tandem Van de Graaff accelerator. The ion implantation depths are calculated via SRIM simulations for an ion energy of 2.8 MeV. Details of the SRIM simulations for determining the stopping range of ions can be found in previous publications^46,62. The implantation depths of the Au ions are greater than 450 nm. This length scale is much larger than the thermal penetration depth of TDTR measurements⁶³. Therefore, the thermally probed region and the measured thermal conductivity are of the defected region pre-end-of-range^45,64.