1. Introduction

Thermoelectric materials enable direct conversion between heat and electricity, making them promising candidates for waste heat recovery and solid-state refrigeration^[1,2]. Their conversion efficiency is described by the dimensionless figure of merit^[3], ZT = S²σT/(κ_e + κ_L), where S, σ, T, κ_e, and κ_L denote the thermopower, electrical conductivity, absolute temperature, charge-carrier thermal conductivity, and the lattice thermal conductivity, respectively. Notably, S, σ, and κ_e are strongly interdependent and coupled to the carrier concentration, whereas κ_L is a relatively independent parameter that can be minimized to maximize ZT^[4].

Diamondoid compounds have attracted increasing interest in photovoltaics and thermoelectrics due to their eco-friendly constituents and chemically tunable compositions^[5-7]. Through cation-site cross-substitution, a broad family of diamondoid compounds with diverse space groups and stoichiometries can be derived from the sphalerite structure^[8-11], which exhibit rich and intriguing phonon behaviors^[12-14] and yield high ZT values (Figure 1a). Among these, Cu₂SnSe₃ has garnered increased interest for thermoelectric applications, owing to its earth-abundant constituents and the absence of toxic Pb or scarce Te. Theoretical efforts have been made to understand its electrical transport properties. For instance, Xi et al.^[31,32] theoretically revealed that Cu-Se bonds dominate hole transport in Cu₂SnSe₃, forming a three-dimensional electrical conduction network, while Sn atoms primarily provide electrons and stabilize the framework. Ming et al.^[33] further showed that Cu vacancies are the dominant intrinsic point defect, featuring shallow transition energy levels that account for the intrinsic p-type conduction behavior. Concurrently, extensive experimental efforts have focused on enhancing the thermoelectric performance of Cu₂SnSe₃^[34,35]. For example, Shi et al.^[36] found that In doping at the Sn site effectively improves the carrier concentration without disrupting the hole conduction network, attaining a high ZT of 1.14 at 850K. Ming et al.^[37] found that (In, S)-codoping effectively engineers the band structure, promotes multi-band transport, and gives a high ZT of 1.51 at 858 K. By employing an entropy and enthalpy regulation strategy, Zhang et al.^[18] achieved a record-high ZT of 1.76 in Cu_1.73Ag_0.27(In_0.06Sn_0.94)Se₂S at 873 K, which is an extremely high value among diamondoid compounds, as shown in Figure 1a.

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Figure 1. (a) Comparison of maximum ZT (ZT_max) values reported for copper-based diamondoid compounds, including CuInTe₂^[15], CuGaTe₂^[16], Cu₂GeTe₃^[17], Cu₂SnSe₃^[18], Cu₂SnS₃^[19], Cu₃SbSe₄^[20], Cu₃SbS₄^[21], CuFeS₂^[22], CuInSe₂^[23], CuInS₂^[23], Cu₂CoSnSe₄^[24], Cu₂CoGeSe₄^[25], Cu₂CoSnS₄^[24], Cu₂CoGeS₄^[25] and Cu₂ZnSnSe₄^[26]; (b) Temperature-dependent lattice thermal conductivity (κ_L) of copper-based diamondoid compounds, including Cu₂SnSe₃^[27], Cu₂SnS₃^[28], CuFeS₂^[22], CuGaTe₂^[29], CuInTe₂^[15], CuInSe₂^[23], CuInS₂^[23], Cu₂GeTe₃^[17], Cu₃SbSe₄^[30], Cu₂CoSnS₄^[24], Cu₂CoGeS₄^[25], Cu₂CoSnSe₄^[24], and Cu₂CoGeSe₄^[25].

Despite these notable advances, the intrinsic phonon transport properties of Cu₂SnSe₃ have remained largely unexplored. Using the Debye-Callaway model, Qin et al. demonstrated that introducing grain boundaries^[38], high-density stacking faults^[39], or nanoinclusions^[40] can effectively scatter the heat-carrying phonons in Cu₂SnSe₃. However, such analyses provide only semi-quantitative insights into the effect of additional defects on phonon scattering. Figure 1b compares the κ_L of copper-based diamondoid compounds. Cu₂SnSe₃ exhibits an exceptionally low κ_L^[27] compared with other ternary copper-based diamondoid compounds, including CuFeS₂^[22], CuGaTe₂^[29], CuInTe₂^[15], CuInSe₂^[23], CuInS₂^[23], Cu₂GeTe₃^[17], and Cu₃SbSe₄^[30]. At 300 K, Cu₂SnSe₃ shows a κ_L of 2.6 W m^-1 K^-1[27], substantially lower than the values of 8.4 and 6.12 W m^-1 K^-1 reported for CuFeS₂ and CuGaTe₂^[22,29], respectively. More intriguingly, the κ_L of these ternary diamondoids deviates from the classical T^-1 temperature dependence. For example, κ_L of CuFeS₂ follows a strongly temperature-dependent T^-1.8 relationship, while that of Cu₂SnSe₃ scales as T^-1.2. Therefore, elucidating the origin of both the strong temperature dependence and the exceptionally low κ_L in Cu₂SnSe₃ is of fundamental importance.

The temperature dependence of κ_L in several typical diamondoid compounds has been previously investigated. For instance, Liu et al.^[41] reproduced the strong temperature dependence of κ_L for diamondoid CuInSe₂ (~ T^-1.8) and CuInTe₂ (~ T^-2) by considering temperature-dependent force constants within the framework of three-phonon scattering alone. They attributed this behavior to the dominant role of Cu atomic motion in generating large anharmonicity at elevated temperatures, which makes anharmonic renormalization indispensable for accurately predicting lattice dynamics and thermal conductivity^[41]. Similarly, Yu et al.^[42] examined the nonclassical temperature dependence of CuInTe₂ using first-principles calculations combined with experimental Raman spectroscopy. Their results show that incorporating both three-phonon (3ph) and four-phonon (4ph) scattering with temperature-independent force constants successfully reproduces the experimentally observed strong temperature dependence. Moreover, their calculated scattering rates indicate that 4ph scattering reaches the same order of magnitude as 3ph at 300 K and even exceeds it at higher temperatures. More importantly, their work is further supported by temperature‑dependent Raman analyses, which demonstrate that the temperature-induced Raman shift and linewidth broadening can be well reproduced only when both 3ph and 4ph scattering processes are taken into account.

In this work, we theoretically investigate the κ_L of Cu₂SnSe₃ using a two-channel thermal transport model. Specifically, the phonon contribution to thermal transport is decomposed into a particle-like contribution (κ_p) and the coherence contribution (κ_c)^[43,44], such that κ_L = κ_p + κ_c. A complete theoretical description should incorporate both aspects. Moreover, previous studies have shown that κ_c can even dominate heat transport in certain materials, such as Cu₁₂Sb₄S₁₃^[45] and diamondoid AgInSnSe₄^[46]. Furthermore, given that high-order phonon scattering strongly influences the temperature dependence of κ_L, both 3ph and 4ph scattering are incorporated into our calculations. Our results reveal that κ_p dominates κ_L and decreases with increasing T, whereas κ_c exhibits the opposite trend. When both 3ph and 4ph scattering are considered, κ_p displays a strong temperature dependence of T^-1.1, closely matching the experimental ~ T^-1.2 behavior. This confirms that 4ph scattering is the primary origin of the anomalous temperature dependence. The strong 4ph scattering is attributed to the flat optical phonon modes in the frequency range of 1.2-2.2 THz, which provide a large 4ph scattering phase space. The markedly low κ_L of Cu₂SnSe₃ relative to other ternary copper-based diamondoids arises predominantly from the asymmetric potential energy profiles of Cu atoms, leading to large atomic displacement parameters and strong anharmonicity of the Cu-related low-frequency optical phonon modes. Furthermore, qualitative analyses of cation disorder and anion substitution show that disorder leads to significant phonon broadening, while replacing Se with S hardens the acoustic branches and shifts the optical modes markedly upward. This study elucidates the intrinsic phonon transport properties of Cu₂SnSe₃ and establishes a foundation for the rational design of diamondoid compounds with intrinsically low κ_L.

2. Experimental Section

Detailed experimental procedures, computational details, convergence tests, and additional characterization results are provided in the Supplementary materials.

3. Results and Discussion

Figure 2a shows the monoclinic crystal structure of Cu₂SnSe₃, which contains two types of metal-centered tetrahedra, [CuSe₄]^7- and [SnSe₄]^4-, connected by corner-sharing Se atoms. Figure 2b presents the negative integrated crystal orbital Hamiltonian populations (-ICOHP) for Cu-Se and Sn-Se bonds in monoclinic Cu₂SnSe₃, with the insert showing the [CuSe₄]^7- and [SnSe₄]^4- tetrahedra. The Cu-Se bond lengths within a [CuSe₄]^7- tetrahedron range from 2.431 to 2.468 Å, while those of Sn-Se within a [SnSe₄]^4- tetrahedron vary from 2.596 to 2.693 Å. These varying bond lengths give rise to differences in bonding strength and are typically associated with anisotropic potential energy. Furthermore, the smaller -ICOHP value for Cu-Se compared to Sn-Se bonds indicates that the Cu-Se bond is considerably weaker, which is expected to contribute to the low κ_L of Cu₂SnSe₃. This weak bonding originates from the strong antibonding interactions between the Cu 3d and Se 4p states, as confirmed by Figure S4.

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Figure 2. (a) Crystal structure of monoclinic Cu₂SnSe₃; (b) -ICOHP of Cu-Se and Sn-Se bonds in monoclinic Cu₂SnSe₃. The insert shows the [CuSe₄]^7- and [SnSe₄]^4- tetrahedra; (c) Temperature-dependent κ_p (3ph), κ_c (3ph), and κ_L (3ph) by considering the three-phonon process (3ph), and κ_p (3 + 4ph), κ_c (3 + 4ph), and κ_L (3 + 4ph) by considering three- and four-phonon process (3 + 4ph) for Cu₂SnSe₃. Here, κ_p and κ_c denote particle-like propagation and wave-like coherent contribution to the lattice thermal transport. The dots denote the experimental κ_L^[27,37]; (d) The spectral ${\kappa }_p\left({\kappa }_p^s\right)$ and ${\kappa }_c\left({\kappa }_c^s\right)$ as a function of phonon frequency at 300 K for Cu₂SnSe₃. -ICOHP: negative integrated crystal orbital Hamiltonian populations; 3ph: three-phonon.

To elucidate the origins of both the strong temperature dependence and the exceptionally low κ_L in Cu₂SnSe₃ (Figure 1b), we theoretically investigate its lattice thermal transport using a two-channel thermal transport model. Within the Wigner formalism^[43,44], κ_L is decomposed into two contributions: κ_L = κ_p + κ_c, where κ_p corresponds to the particle-like propagation of phonon wavepackets discussed by Peierls’s semiclassical picture, and κ_c is the wave-like coherent component arising from tunneling and loss of coherence between different phonon branches. We further incorporate both 3ph and 4ph scattering processes into our calculations. As shown in Figure 2c, κ_p is the dominant contributor to κ_L and decreases as T increases, while κ_c follows the opposite trend and can be neglected at 300 K. When both 3ph and 4ph scattering are included, κ_p(3 + 4ph) displays a strong temperature dependence of T^-1.1, closely matching the experimentally observed ~ T^-1.2 behavior. This confirms that 4ph scattering is the primary origin of the anomalous temperature dependence.

Figure 2d displays the spectral ${\kappa }_p\left({\kappa }_p^s\right)$ and ${\kappa }_c\left({\kappa }_c^s\right)$ as a function of phonon frequency at 300 K for Cu₂SnSe₃. κ_p is predominantly contributed by the phonons in the 0-3 THz range, whereas κ_c arises mainly from the phonons between 1 and 5.5 THz. However, the latter contributes negligibly to the total κ_L. Further discussion is provided in Section 6 of the Supplementary materials. Accordingly, our subsequent analysis focuses on the phonon dispersions in the 0-3 THz range, as shown in Figure 3a, which includes phonon frequency shifts and broadening, with the black solid line denoting the harmonic phonon dispersion at 300 K. In Figure 3a, the low-lying optical phonon modes in the 1.2-2.2 THz region show substantial phonon broadening, corresponding to strong scattering rates and short phonon lifetimes. The projected phonon density of states (phDOS) further shows a pronounced contribution from Cu atoms, confirming their dominant role in these low-frequency modes. Figure 3b displays the harmonic phonon dispersion at 300 K with projected group velocities (v_g). The highly dispersive acoustic branches, characterized by large v_g, are redirected toward lower frequencies via avoided crossings (highlighted by purple circles). These strong avoided crossings, occurring between both acoustic-optical and optical-optical branches, significantly suppress the corresponding v_g, as marked by the purple arrows in Figure 3b. Figure S6 presents the phonon dispersions weighted by the contributions of Cu, Sn, and Se atoms. As clearly shown in Figure S6a, the avoided crossings are associated with Cu-dominated optical phonon modes (marked by purple circles).

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Figure 3. (a) Phonon dispersion and density of states of Cu₂SnSe₃ at 300 K, including phonon frequency shifts and broadening. The black solid line denotes the harmonic phonon dispersion; (b) Harmonic phonon dispersion at 300 K with projected group velocities (v_g) (left panel), and the corresponding v_g variation with phonon frequencies (right panel). The color bar, ranging from white (0) to red (4), denotes the phonon group velocity in km/s; (c) Eigenvector visualizations of representative phonon modes in Cu₂SnSe₃.

Representative eigenvector visualizations for the low-lying optical modes (# 1-3) and the avoided-crossing modes (# 4-6) are provided in Figure 3c. Modes # 1 and # 2 feature antiphase vibrations between [CuSe₄]^7- and [SnSe₄]^4- tetrahedra. Mode # 3 specifically involves twisting motions of Cu and Se atoms within [CuSe₄]^7- tetrahedra. Furthermore, modes # 4-6 clearly reveal the participation of Cu atoms, which perturb the highly dispersive acoustic branches.

Figure 4a,b,c present the 3ph and 4ph scattering rates (SRs) of Cu₂SnSe₃ at 300, 500, and 800 K. In the 1.2-2.2 THz range, corresponding to the flat optical phonon region, both 3ph and 4ph scattering rates are substantially high, with 3ph SRs being roughly one order of magnitude larger than their 4ph counterparts at 300 K. As temperature increases from 300 to 800 K, however, the 4ph scattering rates rise sharply and eventually become comparable in magnitude to the 3ph rates at 800 K (Figure 4c). Such pronounced scattering rates can be attributed to the combination of a large scattering phase-space volume and a strong Grüneisen parameter (γ).

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Figure 4. (a-c) Three- (3ph) and four-phonon (4ph) SRs for Cu₂SnSe₃ at (a) 300 K, (b) 500 K, and (c) 800 K; (d) Dimensionless phonon scattering phase space volumes (P₃, P₄) at 300 K; (e) Phase space volume for three-phonon scattering (P₃); (f) Scattering phase space volume for four-phonon scattering (P₄); (g) Grüneisen parameter (γ) as a function of phonon frequency; (h) Potential energies of Cu, Sn, and Se along x, y, and z directions in Cu₂SnSe₃; (i) Atomic displacement parameters for Cu, Sn, and Se in Cu₂SnSe₃. 3ph: three-phonon; SRs: scattering rates.

The scattering phase-space volume, which quantifies the number of scattering processes available to each phonon while obeying momentum and energy conservation, depends solely on the phonon dispersion, unlike the scattering rates. We calculated their 3ph and 4ph scattering phase-space volumes (P₃ and P₄), as shown in Figure 4d. In the flat frequency region, P₃ and P₄ exhibit comparable magnitudes. For 3ph scattering, the large P₃ predominantly arises from the absorption process (Figure 4e), i.e., the coalescence of two phonons into a single phonon. For 4ph scattering, the large P₄ in the same frequency range is attributed to the redistribution process (Figure 4f), where two incoming phonons annihilate and generate two outgoing phonons, thereby redistributing energy across different phonon branches. This redistribution process not only enlarges the total 4ph scattering phase space but also introduces additional scattering channels that are particularly effective at high temperatures, thereby contributing to the deviation from the classical T^-1 behavior observed in the κ_L.

Figure 4g shows γ as a function of phonon frequency, with high γ values in the flat region enhancing scattering rates. These large γ values stem from weak Cu-Se bonding, as supported by the phDOS (Figure 3a) and eigenvector visualizations (Figure 3c). Figure 4h presents the potential energy profiles of Cu, Sn, and Se along the x, y, and z directions. The Cu profile is considerably flatter than that of Sn, consistent with the COHP analysis (see ICOHP values in Figure 2b), confirming weaker Cu-Se bonds. Quantitative fitting of these profiles using a third-degree polynomial (y = a₁x + a₂x² + a₃x³) yields parameters listed in Table S1, revealing a much shallower potential well for Cu (Figure S7) and significantly smaller a₂ coefficients (1.6-2.1 for Cu vs. ~ 3.8-5 for Sn and ~ 3-3.3 for Se), indicative of lower bond stiffness. Moreover, the Cu potential is strongly anharmonic, especially along x and z, in contrast to the nearly harmonic potentials of Sn and Se; this anharmonicity arises from the variation in bond lengths and strengths among the four Cu-Se bonds within each tetrahedron (Figure 2b). The asymmetric potential around Cu also leads to large atomic displacement parameters (ADPs) and pronounced anharmonicity in the Cu-related low-frequency optical branches, as confirmed by Figure 4i, where Cu atoms exhibit substantially larger ADPs than Sn and Se.

Beyond the particle-like propagation contribution κ_p, we also analyzed the coherent (wave-like) contribution κ_c. Phonons can be classified according to their lifetimes (τ) relative to the Wigner (τ_Wigner) and Ioffe-Regel (τ_Ioffe-Regel) limits: modes with τ > τ_Wigner predominantly contribute to particle-like transport κ_p, whereas those satisfying τ_Ioffe-Regel < τ < τ_Wigner give rise to wave-like coherent transport κ_c. With increasing temperature from 300 to 800 K, an increasing number of phonons enter the coherent transport regime (Figure 5a), accounting for the observed enhancement in κ_c (Figure 2c). The inclusion of 4ph scattering further promotes this trend by enlarging the phonon population that contributes to κ_c (Figure 5b).

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Figure 5. Phonon lifetimes (τ) in Cu₂SnSe₃ considering (a) only three-phonon (3ph) interactions and (b) both three- and four-phonon (3 + 4ph) interactions, at 300 K, 500 K, and 800 K. The horizontal lines indicate the Wigner limit, τ_Wigner = N/ω_max, where N is the number of phonon branches, and ω_max is the maximum phonon frequency. The solid diagonal lines mark the Ioffe-Regel limit (τ_Ioffe-Regel = 1/ω). Contour plots of the coherent thermal conductivity contribution (κ_c (3 + 4ph)) as a function of phonon-frequency pairs (ω, ω_s) for Cu₂SnSe₃ at (c) 300K and (d) 800K. 3ph: three-phonon.

To gain deeper insight into the frequency-selective nature of coherent transport, we examined the contribution of phonon branch pairs (ω, ω') to κ_c. At both 300 and 800 K, κ_c is dominated by quasi-degenerate phonon pairs with similar frequencies (ω ~ ω'), underscoring the critical role of small interband frequency spacing in facilitating coherence (Figure 5c). As the temperature rises, off-diagonal contributions (from pairs with different frequencies) gradually increase. This broadening of the contributing frequency space is a direct consequence of enhanced phonon broadening, which becomes more pronounced at higher temperatures and more effectively bridges the frequency gaps between distinct phonon modes in Cu₂SnSe₃ (Figure 5d).

Raman spectroscopy provides direct verification of this phonon broadening. We measured the Raman spectra of Cu₂SnSe₃ across the temperature range from 93 K to 773 K (Figure S3). The selected spectra at representative temperatures are shown in Figure S3. With increasing temperature, the peaks around 180 cm^-1 exhibit pronounced frequency softening (marked by the red arrow) and broadening, ultimately becoming indiscernible above 603 K. A detailed discussion of the Raman spectra is provided in Section 4 of the Supplementary materials.

Cu₂SnSe₃ exhibits polymorphism, with cation-ordered monoclinic and cation-disordered cubic phases, and is also closely related to its sister compound Cu₂SnSe₃. This motivates a qualitative analysis of how cation disorder and anion substitution affect its phonon dispersions. To specifically evaluate the impact of cation disorder on phonon dispersions and thermal transport, we computed the phonon spectra of both the ordered monoclinic and disordered cubic polymorphs. The primitive cell of the monoclinic phase (Cu₄Sn₂Se₆) contains 12 atoms (Figure S8), giving rise to 36 phonon branches, whereas that of the cubic phase (Cu_0.667Sn_0.333Se) comprises only 2 atoms (Figure S8) and thus 6 branches. Figure 6a shows that the monoclinic phase has complex dispersion, with low-lying optical modes (1.3-2.3 THz, yellow shaded) that scatter heat-carrying acoustic phonons, as discussed earlier. Furthermore, although the longitudinal acoustic (LA) branch folds back to lower frequencies owing to acoustic-optical coupling, an optical mode retains the highly dispersive character of the LA branch, extending it up to ~ 3.1 THz (black dashed line).

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Figure 6. Phonon dispersions and density of states of (a) Monoclinic Cu₂SnSe₃; (b) Cubic Cu₂SnSe₃; (c) Monoclinic Cu₂SnS₃; (d) Cubic Cu₂SnS₃; (e) Orthorhombic Cu₂SnS₃ at 0 K; (f) Comparison of phonon group velocities.

In stark contrast, the cubic phase (Figure 6b) shows only three discernible acoustic branches in the low-frequency region; its LA branch reaches ~ 3.0 THz, slightly lower than that of the monoclinic phase, implying a reduced v_g (as confirmed by Figure 6f). More importantly, significant broadening is observed in both the high-frequency acoustic and optical modes in the cubic phase, which arises directly from cation disorder. Although no low-frequency optical modes below 3 THz exist to scatter acoustic phonons, the pronounced broadening in the 1-2.3 THz region (yellow shaded area) indicates strong inter-acoustic scattering, and even the transverse acoustic (TA) modes exhibit notable broadening (dashed circle). Collectively, these features suggest that the cubic phase should possess a lower κ_L than the monoclinic phase, due to both reduced v_g and enhanced phonon scattering rates.

To systematically investigate the anion effect, we extended our phonon calculations to Cu₂SnS₃, which exhibits even richer polymorphism, including cation-ordered orthorhombic and monoclinic, as well as cation-disordered cubic and orthorhombic structures (Figure S8). The general trend observed for Cu₂SnSe₃ is preserved in the sulfide. The cation-ordered monoclinic Cu₂SnS₃ displays complex dispersions with low-lying optical modes in the 1.7-4.2 THz range (Figure 6c), while the disordered cubic phase shows only three acoustic branches (Figure 6d); the orthorhombic dispersion (Figure 6e) resembles that of the monoclinic counterpart but has only 18 branches.

Comparing the two compounds (Figure 6), three key differences emerge: (i) the maximum phonon frequency, (ii) the phonon band gap, and (iii) the cut-off frequency of the dispersive branches. For the monoclinic phase, the highest frequency increases from 6.4 THz in Cu₂SnSe₃ (Figure 6a) to 10 THz in Cu₂SnS₃ (Figure 6c), and the same trend holds for the cubic phases (Figure 6b,d). Since these high-frequency modes are dominated by anion vibrations, this shift is attributed to the lighter mass of S. Consequently, the upward shift of the optical modes in Cu₂SnS₃ opens a much wider phonon gap of 2.3 THz (between 4.3 and 6.5 THz, Figure 6c), compared with the narrow gap (4.1-4.3 THz) in Cu₂SnSe₃, implying weaker scattering across the gap in the sulfide. Moreover, the cut-off frequency of the dispersive branches, which reflects the upper limit of the LA-like modes, is ~ 3 THz for Cu₂SnSe₃ in both polymorphs, but rises to 3.4-4 THz for Cu₂SnS₃. This increase originates from the shorter and stronger Sn-S and Cu-S bonds relative to their Se counterparts, as verified by the harmonic interatomic force constants (Figure S9), and is also consistent with higher v_g (Figure 6f). Therefore, for the same crystal structure, Cu₂SnSe₃ is expected to exhibit lower κ_L than Cu₂SnS₃, owing to its lower phonon frequencies, narrower gap, and reduced cut-off frequency.

4. Conclusion

In summary, we have theoretically unraveled the microscopic origin of the anomalously low and strongly temperature-dependent κ_L in Cu₂SnSe₃ by employing a two-channel thermal transport model that incorporates both three- and four-phonon scattering processes. Our calculations show that κ_c is negligible at 300 K but becomes progressively more significant at elevated temperatures, while κ_p remains the dominant contributor to κ_L over the entire temperature range. Furthermore, four-phonon scattering, rather than three-phonon scattering, is the primary factor responsible for the deviation from the classical T^-1 dependence, yielding a temperature scaling of T^-1.1 that closely matches the experimental T^-1.2 behavior. This strong four-phonon scattering originates from the flat optical phonon modes in the 1.2-2.2 THz range, which provide an exceptionally large scattering phase space volume. The low κ_L of Cu₂SnSe₃ relative to other ternary copper-based diamondoid compounds is attributed to the asymmetric potential energy around Cu atoms, which induces large atomic displacement parameters and pronounced anharmonicity in the Cu-related low-frequency optical branches. Furthermore, qualitative analyses of cation disorder and anion substitution show that disorder leads to significant phonon broadening, while replacing Se with S hardens the acoustic branches and shifts the optical modes markedly upward. Our findings clarify the origins of the nonclassical temperature dependence and low κ_L in Cu₂SnSe₃, and provide a general way to design diamondoid thermoelectrics with intrinsically low κ_L.

Supplementary materials

The supplementary material for this article is available at: Supplementary materials.

Authors contribution

Ming H: Investigation, methodology, conceptualization, writing-review & editing.

Lin Z, Xu S: Investigation, formal analysis.

Zheng Y: Conceptualization, supervision.

Luo ZZ: Project administration, conceptualization, supervision, writing-review & editing.

Zou Z: Project administration, supervision.

Conflicts of interest

The authors declare no conflicts of interest.

Ethical approval

Not applicable.

Consent to participate

Not applicable.

Consent for publication

Not applicable.

Availability of data and materials

The data and materials could be obtained from the corresponding author upon request.

Funding

This study is supported in part by the National Natural Science Foundation of China (Grant Nos. 12504032, 52472191, and 52102218), the National Key Research and Development Program of China (Grant No. 2020YFA0710303), and the Fujian Science & Technology Innovation Laboratory for Optoelectronic Information of China (Grant Nos. 2021ZZ127 and 2023RC103). The authors acknowledge the Natural Science Foundation of Fujian Province of China (Grant Nos. 2026J009076, 2024J08290, 2025J010017, and 2025J08029). Yunpeng Zheng is funded by Fuzhou University Testing Fund of precious apparatus (Grant No. 2025T003) and funding (Grant No. XRC-25036-0180-511573). This project is also supported by the State Key Laboratory of New Ceramic Materials Tsinghua University (Grant No. KF202525).

Copyright

© The Author(s) 2026.