Number mismatch between cations and anions as an indicator for low lattice thermal conductivity in chalcogenides

Abstract

Thermal conductivity is one of the most fundamental properties of materials with the value being determined by nearly all-scale structural features and multiple physical processes. Rapidly judging material’s thermal conductivity is extremely important but challenging for the applications. The material genome paradigm offers a revolutionary way to efficiently screen and discover materials with designed properties by using accessible indicators. But such a performance indicator for thermal conductivity is quite difficult to propose due to the existence of multiple mechanisms and processes, especially for the materials with complex structures such as chalcogenides. In this study, the number mismatch between cations and anions is proposed as a practical performance indicator for lattice thermal conductivity in complex copper and silver chalcogenides, which can be used to explain the observed experimental data and find new low thermal conductivity materials. Such a number mismatch brings about rich phenomena to affect thermal conductivity including the complication of the unit cell and the creation of chemical hierarchy, point defects, rattling modes and lone-pair electrons. It is expected that this rich-connotation performance indicator can be also extended to other complex materials to discover designed thermal conductivities.

Introduction

Thermal conductivity (κ) is a material’s basic property that is used to characterize its heat conducting capability. High-κ materials are widely used in heat dissipating applications while low-κ ones are usually used in thermal insulation. In particular, thermal conductivity is one of the key parameters in thermoelectric materials. It also plays an important role in other energy conversion and storage materials such as solar cells and battery materials. Understanding thermal conductivity is thus of great significance in various disciplines. For semiconductors, heat conduction comes mainly from the vibration of lattice (phonons), i.e., the lattice thermal conductivity (κ_L)¹. It is determined by heat capacity C_V, sound speed v and phonon relaxation time τ through the simplified relationship^2,3 κ_L ∝ C_Vv²τ. The structures on atomic and/or unit cell levels basically decide the phonon dispersion, elastic properties, and the intensity of phonon–phonon normal and Umklapp scattering etc.^4,5, while the microstructures in all scales contribute additional boundaries, defects, and interfaces as phonon scattering centers⁶. The physical mechanisms on different levels have been developed to explain material’s thermal conductivity observed in experiment. Particularly, a few conceptual directions have been proposed towards low κ_L, such as large cell with heavy atoms^1,7, small elastic parameters⁸, and the existence of lone-pair electrons^9,10, soft bonds⁸, rattling atoms¹¹, and liquid-like ions¹².

For the fields of thermoelectrics, design and discovery of new materials with low thermal conductivity is a key task. Materials genome initiative (MGI)¹³ offers a revolutionary scenario to efficiently find new materials with desired properties¹⁴, which is particularly suitable for thermal conductivity. MGI requires simple yet straightforward performance indicators for initial but fast high-throughput screening to pick up possible candidates. For lattice thermal conductivity, intuitive parameters such as average atomic mass (M) and primitive unit cell volume (V_PUC) are available in the database, but they are far below the requirements for materials with complex crystal structures and lattice dynamics. For example, in chalcogenides, chemical bonding inequality, and hierarchy are diverse, leading to various factors affecting thermal transport such as abundant intrinsic defects, loose bonding, lone-pair, and rattling atoms. They are likely to coexist and interact with each other. Thus, it is quite difficult to find simple performance indicators to quickly guide the thermal conductivity.

In this work, we report that the number mismatch (δ) between cations (N_cation) and anions (N_anion), δ = (N_cation − N_anion)/N_anion, can be regarded as the simple performance indicator to quickly and directly explain and discover materials with low lattice thermal conductivity (κ_L) in complex chalcogenides. Furthermore, new low-κ_L materials such as Cu₂Sn₄S₉ are screened by the proposed large number mismatch between cations and anions, and confirmed by experiment (Fig. 1).

Fig. 1: Lattice thermal conductivity (κ_L) at room temperature varying with number mismatch between cations and anions (δ) in ternary Cu- and Ag-based chalcogenides.

The red line is a guide to the eyes. The green square is Cu₄Sn₇S₁₆, which is used for the case study. Cu₂Sn₄S₉ and Cu₄SnS₄ (red circle) are the screened low κ_L materials based on proposed number mismatch between cations and anions. The raw data and the source for each material can be found in Supplementary Table 1.

Results and discussion

We go back to check the room-temperature lattice thermal conductivity of 73 complex ternary copper and silver chalcogenides (Fig. 1 and Supplementary Table 1). It ranges from 0.18 W m⁻¹ K⁻¹ in Ag₈GeTe₆ to 6.8 W m⁻¹ K⁻¹ in CuInTe₂, a quite large range. For each material, the δ value is calculated, and the room temperature κ_L varying with δ is shown in Fig. 1. Despite the diversity in composition and crystal structure, a clear decreasing trend of κ_L is observed when δ departs from 0, no matter whether the compounds are cation or anion rich. Particularly, all the materials with |δ| ≥ 0.5 have the κ_L values smaller than 0.5 W m⁻¹ K⁻¹. As shown in Supplementary Fig. 1, the significant, negative correlation between κ_L and |δ| is quantified by the Spearman correlation coefficient ρ_s = −0.79, which is closer to −1 (totally negative relation) than either κ_L-M or κ_L-V_PUC. Furthermore, for each specific family of Cu-based materials, the decrease of normalized κ_L with |δ| is more straightforward (see Supplementary Fig. 2). These results strongly indicate that δ can be an effective performance indicator for low lattice thermal conductivity in Cu- and Ag-based complex chalcogenides. The detailed physical mechanisms for the low κ_L can be included in the simple parameter δ that can be immediately obtained from the chemical formula.

Cu₄Sn₇S₁₆: a case study

Here we pick up Cu₄Sn₇S₁₆ as a case study to show how the simple parameter δ can be a performance indicator for the low κ_L. Complex semiconductor Cu₄Sn₇S₁₆ is a largely distorted derivative of diamond-like compound with δ = −0.3125¹⁵. As shown in Fig. 2a, Cu₄Sn₇S₁₆ exhibits a low κ_L < 1 W m⁻¹ K⁻¹ above room temperature, and the dependence of κ_L on T is quite weak, which is comparable with multiple-filled skutterudites¹⁶ and Type-I clathrates¹⁷.

Fig. 2: Thermal transport properties and crystal structure for Cu₄Sn₇S₁₆.

a κ_L as a function of temperature for various Cu-based chalcogenides. Data are taken from the followings: CuFeS₂ (ref.²²), Cu₂SnS₃ (this work), skutterudites (ref.¹⁶), clathrates (ref.¹⁷), Cu₁₂Sb₄S₁₃ (ref.²³), Cu₂Se (ref.¹²), Cu₃SbSe₃ (ref.²⁴), and Cu₄Sn₇S₁₆ (this work, Supplementary Fig. 4). b C_p/T versus T² for Cu₄Sn₇S₁₆. The black squares are the experimental data. The red solid line represents the fitted curve by taking account the electronic, one Debye and two Einstein modes, which are shown by the dashed lines. Fitting parameters are given in Supplementary Table 2. c Phonon scattering mechanisms in Cu₄Sn₇S₁₆. U, B, PD, and RES denote the Umklapp process, grain-boundary scattering, point-defect scattering and resonant scattering, respectively. Fitting parameters are given in Supplementary Table 3. d Crystal structure of Cu₄Sn₇S₁₆ and local coordination of three types of Cu atoms.

There are a few reasons for such an observed low κ_L. Firstly and apparently, Cu₄Sn₇S₁₆ possesses a complex crystal structure. In this structure with the R$\overline {\mathrm{3}}$m (Z = 3) symmetry, sulfur atoms S1 (18 h), S2 (18 h), S3 (6c), and S4 (6c) constitute triangular, tetrahedral, and octahedral frameworks, accommodating Cu and Sn atoms¹⁵ (Fig. 2d). This complex structure with diverse local structure units leads to a very large-cell volume (1695 ų) with 81 atoms per unit cell^15,18. The primitive unit cell volume V_PUC is 565 ų for Cu₄Sn₇S₁₆, which is considerably larger than typical diamond-like materials (V_PUC is 150–250 ų). Intuitively, a large and complex cell corresponds to numerous optical phonon modes (see Fig. 3d). These optical modes store a majority of heat (~78% at 20 K as shown in Fig. 2b) but exhibit a low group velocity while the acoustic modes contribute only ~20%. In addition, the optical modes may also suppress and even interact with the acoustic modes, thus suppressing the thermal transport.

Fig. 3: Lattice dynamics of Cu₄Sn₇S₁₆.

a–c Molecular dynamics trajectories for Cu₄Sn₇S₁₆; d Phonon dispersion and projected phonon density of states.

Secondly, Cu1 atoms randomly occupy half of the 6c sites¹⁵. Therefore, high-density intrinsic vacancies are present and strongly scatter high-frequency phonons, reducing the κ_L by one order of magnitude (from 11.1 to 1.68 W m⁻¹ K⁻¹) at room temperature (Fig. 2c).

Thirdly, our ab initio molecular dynamics (AIMD) simulations (Fig. 3a–c) and crystallographic data^19,20 show that Cu2 and Cu3 atoms exhibit a noticeably large vibration amplitude along c-axis and within ab-plane like rattling modes, respectively. These rattling-like behaviors lead to low-lying optical modes which start from as low as 2 meV and are intensely overlapped with the acoustic modes as indicated in Fig. 3d. In addition to altering the acoustic phonon dispersion near the boundary of Brillouin zone (particularly along Γ-A/L direction), the low-lying optical modes also introduce resonant scattering on acoustic phonons²¹, further suppressing κ_L to 0.78 W m⁻¹ K⁻¹ at 300 K as shown in Fig. 2c. The reduction of κ_L is more directly demonstrated by the spectra data shown in Supplementary Fig. 3, where areas II and III are the contributions by point-defect scattering and resonant scattering, respectively.

As shown above, the large |δ| value in Cu₄Sn₇S₁₆ compound is unambiguously related to the low κ_L through the detailed physical mechanisms of large cell, rattling modes, and point defects. In fact, such a relationship can be easily extended to other materials. For example, Cu₃SbSe₃ has a large δ value of 0.33 and a low κ_L of 0.5–0.6 W m⁻¹ K⁻¹ at 300 K with the mechanisms of Cu rattling^11,25 and/or lone-pair electrons⁹. Similarly, tetrahedrite Cu₁₂Sb₄S₁₃ has a δ of 0.23 and a low κ_L about 0.65 W m⁻¹ K⁻¹ at room temperature^26,27. All these detailed mechanisms of suppressing κ_L can be included in the simple performance indicator δ, which is schematically depicted in Fig. 4a, b and will be expanded in the next part.

Fig. 4: Mechanisms of ionic number mismatch influencing thermal transport.

a, b Schematic depiction of the effects of number mismatch between cations and anions on thermal transport in Cu-based chalcogenides. Ionic number mismatch varying with c primitive cell volume for Cu- and Ag-based compounds and d local bond distortion for Cu-based compounds.

Effects of δ on thermal transport

δ and large-cell effect

In chalcogenides, the relatively simple crystal structure cannot persist when |δ| is large enough. Thus, a larger cell with more atoms and lower symmetry tends to be adopted in order to accommodate the diverse bonding. In Fig. 4c, when |δ| is large, the primitive cell volume is large too. Although the data are scattered, the positive correlation between |δ| and V_PUC is quite clear. For large cells, the lattice complexity shrinks the first Brillouin zone and induces more optical modes. As a consequence, a majority of heat is damped in slowly propagating optical waves, and the acoustic branches are smothered or even interacted by the optical modes⁴. All these factors lead to the great reduction of κ_L.

δ and vacancies

Large δ means that certain crystallographic positions are not fully occupied. As an extreme case of point defects, vacancies have zero mass and free space to introduce large lattice fluctuation to scatter phonons. In fact, phonon scattering by point defects is a well-known strategy to reduce κ_L, which have been widely employed for a few dozens of years.

δ and local bonding distortion, rattling and lone pair

Large δ also brings about various local bonding distortions including the change in bond lengths and angles. Particularly, some atoms can be loosely bonded along certain directions or within certain planes, showing rattling-like vibrations to dramatically alter heat storage and transport. This effect has been demonstrated in the filled skutterudites^16,28, Cu₃SbSe₃¹¹, Cu₁₂Sb₄S₁₃²⁶, and Ag-based materials^8,29,30. An extreme case of bonding inequality is the lone-pair electrons, which can further enhance the lattice anharmonicity^9,10. Here we defined the microscopic distortion index Δ to qualitatively describe the degree of local chemical bond distortion, which is simply assumed as the sum of distortions in bond length Δ_l and bond angle Δ_θ. For most Cu-based ternary chalcogenides, the cations are tetrahedrally or octahedrally bonded with anions in an ideal or distorted way. For the i^th cation coordination, it is defined

$$\Delta _l^{i}\,=\,\left( {l_{\text{max}}^{i}\,-\,l_{\text{min}}^{i}} \right)/l_{\text{ideal}}^{i}$$

(1)

$$\Delta _\theta ^i\,=\,\left( {\theta _{\text{max}}^{i}\,-\,\theta _{\text{min}}^{i}} \right)/\theta _{\text{ideal}}^{i},$$

(2)

where l_max and l_min are the largest and smallest bond lengths, respectively, and θ_max and θ_min are the largest and smallest bond angles, respectively. Here the data of ideally coordinated bonds are taken as the standard values. The detailed calculation method is described in Supplementary Fig. 5. For example, θ_ideal are 109.5° and 90° for tetrahedral and octahedral bonding, respectively. The total distortion degree of a material is the sum of the ones for each cation divided by the cation number:

$$\Delta\,=\,\mathop {\sum}\limits_i {\left( {\mathop {\Delta }\nolimits_l ^{\mathrm{i}}\,+\,\mathop {\Delta }\nolimits_{\theta} ^{\mathrm{i}}} \right)/N_{{\mathrm{cation}}}}.$$

(3)

For the case of lone pairs, it is assumed that the distance between the center cation and the lone pair is half of the normal bond length.

We calculated the Δ values based on the crystal structure of 19 Cu-based materials. Only 16 materials have the value of κ_L (Supplementary Table 4). As shown in Supplementary Fig. 6a, the lattice thermal conductivity exhibits a clear decreasing trend with Δ. The correlation between Δ and δ is relatively weak and scattered at the first sight (Supplementary Fig. 6b). Nonetheless, we found that this poor relativity is mainly caused by the treatment of vacancy-containing materials, in which the bonding between the anions and the virtue void is not included. After removing the five vacancy-containing materials (Cu₄Sn₇S₁₆, Cu₂SnSe₄, Cu₂GeS₄, Cu₄SnS₆, and CuSn_3.75S₈), the positive correction is clearly seen in Fig. 4d between the microscopic chemical bonding distortion parameter and the macroscopic ionic number mismatch. Thus, when δ deviates from 0, the degree to local chemical bond distortion is increased to greatly suppress κ_L.

κ _L-δ diagram in Cu–Sn–S family

Parameter δ is an effective indicator for κ_L in the materials with structural complexity and local bonding distortions. Large absolute value of δ means low κ_L. Thus the δ criterion can be used to discover low κ_L materials. Here we still take Cu–Sn–S family as the “material base” from the reported phase diagram³¹. It includes a variety of low-cost and environmentally friendly compounds and part of them can be potential candidates for various applications like thermoelectric, solar cells, and electrode materials. As shown in Fig. 5, the available experimental data (green symbols) of this family well follow the “δ criterion”, further confirming the validity of this simple performance indicator. For the compounds³¹ (the short vertical lines at the bottom of Fig. 5) without κ_L values being reported in the previous studies, the “δ criterion” suggests that Cu₂Sn₂S₉, CuSn_3.75S₈, Cu₂Sn₄S₉, Cu₉Sn₂S₉, and Cu₅SnS₄ are likely to exhibit low lattice thermal conductivities because of their large |δ| values. In experiment, we selected the nominally valence-state-balanced compound Cu₂Sn₄S₉ (δ = −0.33) to check its κ_L. A low value of 1.0 W m⁻¹ K⁻¹ is observed at room temperature, well consistent with our prediction. In addition, the compound Cu₄SnS₄ has a large δ of 0.25, but there are some discrepancies on the reported κ_L^32,33. Particularly, Goto et al. reported a high value of 4 W m⁻¹ K⁻¹³³, which is in contradictive with the “δ criterion”. We rechecked this compound and found its κ_L at room temperature is 0.36 W m⁻¹ K⁻¹, an extremely low value in chalcogenides. Furthermore, the κ_L data of the two compounds show weak dependences on temperature (Supplementary Fig. 14), which are in close analogy to Cu₄Sn₇S₁₆ with similarly large δ while in sharp contrast to Cu₂SnS₃. All the data have substantiated the effectiveness of the concise δ criterion in screening low-κ_L materials.

Fig. 5: κ_L-δ diagram for the Cu–Sn–S compounds.

The squares are the experimental data. The red line is fitted to guide the eyes. The short vertical line donates the mismatch of various Cu–Sn–S compounds from the reported phase diagram³¹ whose lattice thermal conductivity has not been reported. Materials with large |δ| values are expected to exhibit low κ_L.

Limitations of δ

There are certainly some limitations for this simple indicator. Fundamentally, the case of δ = 0 is pre-assumed to be the tetrahedrally bonded diamond-like semiconductors. The real case is more complicated. A typical example for this deviation is the Sb/Bi-containing compounds (e.g., CuBiS₂ and AgSbS₂), where the lone-pair electrons of Sb/Bi make the structure largely different from the chalcopyrite despite the same 1–1–2 composition. This is the main reason why the κ_L data are largely scattered for δ = 0 (Fig. 1). In addition, the mismatch criterion does not consider the difference in atomic mass. Therefore, this indicator is more effective and reliable for homologs, i.e., a family of compounds with the same elements but different compositions. A good demonstration has been shown above for the Cu–Sn–S materials.

In real applications, the total thermal conductivity is of greater interest. We found that most Cu/Ag-based ternary chalcogenides have extremely low κ_e (<5% of κ_total) at room temperature. Also, there seems no clear correlation between κ_e and |δ|. However, things can be more complicated when κ_e is nonnegligible in other systems, and the possible relationship between κ_e and |δ| needs further investigation.

In summary, this work proposed that the mismatch (δ) between the number of cations and anions is a practical performance indicator for low κ_L in ternary Cu- and Ag-based chalcogenides. When the mismatch exists, multiple phenomena and effects such as large cell, point defects, chemical bonding distortion and inequality, rattling atoms and lone pair may be induced to lower κ_L. This criterion and the underlying mechanisms have been illustrated in the Cu₄Sn₇S₁₆ compound and well extended to other ternary Cu- and Ag-based chalcogenides. Guided by the κ_L-δ relationship, low-κ_L Cu₂Sn₄S₉ compound was predicted and confirmed. Such an intuitive indicator provides a unique perspective for understanding thermal transports and is powerful for efficiently screening and designing low κ_L materials for energy conversion and thermal management applications.

Methods

Synthesis

Cu₄Sn₇S₁₆, Cu₂SnS₃, Cu₄SnS₄, and Cu₂Sn₄S₉ polycrystalline samples were synthesized by different methods with the details described in Supplementary Information.

Characterization

The phase purity and crystal structure were detected by X-ray diffraction with Cu K_α X-ray sources (XRD, Rigaku D/max 2550 V). The elemental distribution was characterized by scanning electron microscopy (ZEISS supra 55) and energy dispersive X-ray spectroscopy (Oxford) with accelerate voltage set as 20 kV. Samples’ phase purity and homogeneity are shown in Supplementary Figs. 7–13. High-temperature thermal conductivity was derived from the formula κ = ρC_pλ, where ρ is the density, C_P is the heat capacity calculated by Dulong–Petit law, and λ is thermal diffusivity measured between 300 and 700 K by using the laser flash method (LFA 457, NETZSCH). Low-temperature C_P and κ_L were measured by Physical Property Measurement System (PPMS Quantum Design). The uncertainty for κ_L is within 10%.

Computation

The AIMD was performed for Cu₄Sn₇S₁₆ at 300 K. A large supercell, 3 × 3 × 1 of the 81-atom unit cell (729 atoms in total), was constructed for the AIMD. The simulation time was over 18 ps with a step of 2 fs. The AIMD was carried out by the Vienna ab initio simulation package³⁴, with the projector augmented wave method³⁵ and Perdew–Burke–Ernzerhof generalized gradient approximation functional³⁶. The temperature dependent effective potential method^37,38 was adopted to extract the harmonic interatomic force constants and phonon dispersions from the AIMD trajectories and forces.