Introduction

The lattice thermal conductivity (kL) of a given system plays a quite important role in effective heat management for various application scenarios. For example, higher kL is quite desirable to dissipate the excessive heat of electronic devices, while thermoelectric applications favor lower kL to enhance the energy conversion efficiency1,2,3. It is therefore of great interest to give an accurate evaluation of kL, which is, however, very challenging especially when complicated scattering mechanisms are involved. Theoretically speaking, the classical molecular dynamics (CMD) simulations and density functional theory (DFT) calculations are two general approaches to predict kL. However, the accuracy of the former strongly depends on the choice of empirical potentials, while the latter normally requires time-consuming calculations of third-order (or higher) interatomic forces which is thus limited to simple systems.

With substantial advances in data-driven machine learning (ML) techniques, high-throughput predictions of kL can be expected at negligible computational cost. For instance, Seko et al. rapidly evaluated the kL of 54,779 samples from Materials Project database by using the Bayesian optimization algorithm, where the training set of 101 compounds was obtained from first-principles calculations4,5. By adopting the Sure Independence Screening and Sparsifying Operator (SISSO) method, Juneja et al. proposed a descriptor to calculate the kL of bulk systems, where the only input is several basic features such as the cell volume, the average atomic mass, and the maximum phonon frequency6,7. Besides, the so-called machine-learning interatomic potentials (MLIP) have been suggested to reduce the great efforts in the DFT calculations of kL8,9,10,11,12,13,14,15,16. Although considerable progress has been made in the ML prediction of kL, it is still difficult to fulfill the large and reliable training set required17, which definitely affects the accuracy of the predicted values. On the other hand, there are some recent works trying to predict low kL materials purely from structure features18,19. In particular, Liu et al. proposed a new crystallographic parameter (the site occupancy factor) which can be used as an effective indicator to identify a material catalog with glass-like kL19. It should be noted that most of the above-mentioned works are focused on the compounds with integer stoichiometry. Among few studies on the alloyed systems with fractional composition20,21, the virtual crystal approximation22 is generally considered, where the weighted averages of the fundamental features (such as the atomic mass, radius, and electronegativity) may lead to poor results. To date, it remains a great challenge to evaluate the kL of alloyed systems with arbitrary stoichiometry, both rapidly and accurately.

In this work, we propose a reliable approach to quickly predict the kL of alloyed compounds from the perspective of configurational entropy. We choose the tetradymites as a prototypical class of examples since they have attracted wide attention in the fields of thermoelectric, photovoltaic, and topological materials23,24,25,26,27,28,29,30,31,32. Using first-principles calculations combined with the MLIP, we first compute the kL of limited tetradymites with integer stoichiometry. On top of these results, the kL of alloyed systems at arbitrary composition can be quickly obtained in terms of their configurational entropy. The strong predictive power of our approach is validated by higher Pearson correlation coefficient between the calculated and experimental values. In principle, such a conceptually sound strategy can be applicable to any other material systems, as further demonstrated by good agreement between the predicted and measured kL of alloyed half-Heusler compounds.

Results

Thermal conductivities of tetradymites with integer stoichiometry

As shown in Fig. 1a, the tetradymite compounds consist of group-VA (As, Sb, Bi) and group-VIA atoms (S, Se, Te), which, respectively, occupy the cation (A, B) and anion (C, D, E) sites in the crystal structure. The lattice can be viewed as five covalently bonded atomic layers held together by weaker van der Waals (vdW) forces, known as the quintuple layers (QLs). By mutation of possible atoms at each site, one can in principle find 35 = 243 tetradymites with integer stoichiometry. Combining the DFT calculations, MLIP, and phonon BTE discussed above, we can obtain the kL of all these compounds without too much computational efforts, as summarized in Supplementary Table 1. Note that we focus on 300 K in the present work, and our conclusions should be applicable for any other temperatures. We see that the in-plane kL of these tetradymites are obviously larger than the out-of-plane values, as generally found in many layered systems. For simplicity, the average kL along three directions is considered in the following discussions.

Fig. 1: Crystal structures.
Fig. 1: Crystal structures.
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a The tetradymite compounds can be viewed as quintuple layers (QLs) held together by van der Waals (vdW) forces, where five atomic sites (A, B, C, D, E) are marked. b Four configurations and their mole contents for the ternary tetradymite \({{{\mathrm{Bi}}}}_{1.2}{{{\mathrm{Sb}}}}_{0.8}{{{\mathrm{Te}}}}_3\).

Thermal conductivities of alloyed tetradymites

We now focus on the alloyed tetradymites with fractional stoichiometry, which in principle can be viewed as solid solutions of those having integer composition. For example, the ternary \({{{\mathrm{Bi}}}}_{1.2}{{{\mathrm{Sb}}}}_{0.8}{{{\mathrm{Te}}}}_3\) shown in Fig. 1b can be obtained by alloying four different configurations, namely, the \({{{\mathrm{BiBiTeTeTe}}}}\), \({{{\mathrm{SbSbTeTeTe}}}}\), \({{{\mathrm{BiSbTeTeTe}}}}\), and \({{{\mathrm{SbBiTeTeTe}}}}\) (atoms arranged in the sequence of ABCDE mentioned above). The mole content (\({\it{w}}_i\)) of each component is, respectively, 36%, 16%, 24%, and 24%, which can be determined by the compositions of the constituent atoms (details are given in the Supplementary Material). As a further example, we summarize in Supplementary Table 2 the mole contents of the 32 different configurations for the quaternary \({{{\mathrm{Bi}}}}_{1.9}{{{\mathrm{Sb}}}}_{0.1}{{{\mathrm{Te}}}}_{2.7}{{{\mathrm{Se}}}}_{0.3}\). Unlike those found in the \({{{\mathrm{Bi}}}}_{1.2}{{{\mathrm{Sb}}}}_{0.8}{{{\mathrm{Te}}}}_3\) with comparable \({\it{w}}_i\), we see that the mole content of the \({{{\mathrm{BiBiTeTeTe}}}}\) is significantly higher (~66%) compared with other components, which can be attributed to much large compositions of the Bi and Te atoms in the \({{{\mathrm{Bi}}}}_{1.9}{{{\mathrm{Sb}}}}_{0.1}{{{\mathrm{Te}}}}_{2.7}{{{\mathrm{Se}}}}_{0.3}\) compound.

In order to quantitatively characterize the disorder caused by many different configurations in the alloyed systems, we adopt the concept of configurational entropy (\({\Delta}S\))33,34,35,36,37 that is defined as:

$${\Delta}S = - k_{\rm{B}}\mathop {\sum}\limits_{i = 1}^N {{\it{w}}_i\ln (w_i)} ,\mathop {\sum}\limits_{i = 1}^N {w_i} = 1$$
(1)

where the wi is the mole content of the i-th configuration discussed above, kB is the Boltzmann constant, and the summation is over all the N configurations. For instance, the ΔS of the \({{{\mathrm{Bi}}}}_{1.2}{{{\mathrm{Sb}}}}_{0.8}{{{\mathrm{Te}}}}_3\) shown in Fig. 1b is calculated by:

$$\begin{array}{l}{\Delta}S = - k_{\rm{B}}\left[36\% \times \ln (36\% ) + 16\% \times \ln (16\% )\right. \\ \qquad\,\,\,\,\,\,\left. + \,24\% \times \ln (24\% ) + 24\% \times \ln (24\% )\right]\end{array}$$
(2)

for the quaternary \({{{\mathrm{Bi}}}}_{1.9}{{{\mathrm{Sb}}}}_{0.1}{{{\mathrm{Te}}}}_{2.7}{{{\mathrm{Se}}}}_{0.3}\) mentioned before, the ΔS can be similarly obtained by summation over all the 32 configurations listed in Table 1. As a general case, we plot in Fig. 2 the configurational entropy of alloyed tetradymites \(({{{\mathrm{Bi}}}}_x{{{\mathrm{Sb}}}}_{1 - x})_2({{{\mathrm{Se}}}}_y{{{\mathrm{Te}}}}_{1 - y})_3\) with respect to arbitrary stoichiometry of the x and y. It is clear that the tent-like ΔS increases with x and y and reaches the maximum at the center. Specifically, the value approaches zero at the four corners that represent the well-known binary tetradymites \({{{\mathrm{Bi}}}}_2{{{\mathrm{Te}}}}_3\), \({{{\mathrm{Sb}}}}_2{{{\mathrm{Te}}}}_3\), \({{{\mathrm{Bi}}}}_2{{{\mathrm{Se}}}}_3\), and \({{{\mathrm{Sb}}}}_2{{{\mathrm{Se}}}}_3\), which is reasonable since each of them has a single component so that the mole content (Wi) is 100%. When the x and y are both 0.5, we obtain a maximum ΔS of 3.47kB which corresponds to the largest disorder of the alloyed tetradymites. Generally speaking, a higher configurational entropy means enhanced phonon scatterings which will lead to lower lattice thermal conductivity36,38,39. Accordingly, the kL of any alloyed tetradymites can be derived as:

$$\kappa _L = a\frac{{\mathop {\sum}\limits_{i = 1}^N {(w_i \times \kappa _i)} }}{{{\Delta}S{{{\mathrm{/}}}}k_{\rm{B}}}} + b$$
(3)

here, the ki refers to the lattice thermal conductivity of the i-th configuration with integer stoichiometry. The weighted summation in the numerator reveals the coexistence of N different configurations with mole content of wi, as discussed above. The scaling constants a and b are determined by fitting several experimentally measured kL of alloyed tetradymites, which are 0.21 and 0.39, respectively. Note that the configurational entropy enters Eq. (3) in terms of dimensionless \({\Delta}S{{{\mathrm{/}}}}k_{\rm{B}}\) so that we have a correct unit (Wm–1 K–1) for the lattice thermal conductivity. It should be mentioned that we have also checked the linear regression by adopting the ML approach SISSO6. In the derived descriptor, the scaling constants are almost the same as our fitted values, suggesting strong reliability of the proposed Eq. (3).

Fig. 2: Configurational entropy.
Fig. 2: Configurational entropy.
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The calculated ΔS of the alloyed \(({{{\mathrm{Bi}}}}_x{{{\mathrm{Sb}}}}_{1 - x})_2({{{\mathrm{Se}}}}_y{{{\mathrm{Te}}}}_{1 - y})_3\) as a function of x and y.

As the state-of-the-art thermoelectric materials around the room temperature, tetradymite compounds exhibit intrinsically lower lattice thermal conductivity, which can be further reduced by alloying with isoelectronic elements. Unfortunately, reliable predictions on the kL of alloyed tetradymites are prohibitively expensive and time-consuming. Such a fundamental important problem can be effectively addressed by using our approach represented by Eq. (3), which only requires the lattice thermal conductivities of limited configurations and associated configurational entropy. Based on the kL of the 32 tetradymites with integer stoichiometry (Supplementary Tables 1 and 2), the kL of numerous alloyed compounds \(({{{\mathrm{Bi}}}}_x{{{\mathrm{Sb}}}}_{1 - x})_2({{{\mathrm{Se}}}}_y{{{\mathrm{Te}}}}_{1 - y})_3\) can be readily obtained, as shown in Fig. 3a. Due to the inverse correlation between the kL and ΔS, we observe an inverted tent which provides a simple map to rapidly screen the alloyed tetradymites with desired lattice thermal conductivity. Compared with those tetradymites having integer stoichiometry, we see that the kL of the alloyed systems can be obviously reduced even with a small content of alloying elements. In particular, a minimum kL of 0.53 Wm−1 K−1 can be reached at x = 0.70 and y = 0.38, which is quite favorable to further enhance the thermoelectric performance of tetradymites.

Fig. 3: The lattice thermal conductivities of alloyed tetradymites.
Fig. 3: The lattice thermal conductivities of alloyed tetradymites.
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a The predicted kL of the alloyed \(({{{\mathrm{Bi}}}}_x{{{\mathrm{Sb}}}}_{1 - x})_2({{{\mathrm{Se}}}}_y{{{\mathrm{Te}}}}_{1 - y})_3\), plotted as a function of x and y. b The intuitive linear correlation between the predicted and experimental kL of several alloyed tetradymites, where the shaded area indicates the range of the mean absolute error.

Checking the predictive power

In order to check the reliability of our proposed strategy, we show in Fig. 3b the experimentally measured kL for a series of alloyed tetradymites (Supplementary Table 3)40,41,42,43,44,45,46,47, as compared with our predicted results. We see that all the samples are located around the solid line that represents equality, where the corresponding Pearson correlation coefficient is as high as 0.91. Moreover, according to the shaded area in Fig. 3b, the mean absolute error (MAE) between the predicted and experimental kL is as small as 0.04. Both findings demonstrate the strong predictive power of our approach. It should be mentioned that we have only considered four group-VA and group-VIA atoms (Bi, Sb, Te, Se) in the above discussions. If all the six elements are taken into account (As, Bi, Sb, S, Se, Te), we can obtain an even complicated alloyed tetradymite with a nominal formula of \(({{{\mathrm{As}}}}_x{{{\mathrm{Bi}}}}_y{{{\mathrm{Sb}}}}_{1 - x - y})_2({{{\mathrm{S}}}}_u{{{\mathrm{Se}}}}_v{{{\mathrm{Te}}}}_{1 - u - v})_3\). In principle, such a multi-component system can be viewed as a solid solution of 243 different configurations with integer stoichiometry (Supplementary Table 1). As a consequence, the kL of any alloyed compound \(({{{\mathrm{As}}}}_x{{{\mathrm{Bi}}}}_y{{{\mathrm{Sb}}}}_{1 - x - y})_2({{{\mathrm{S}}}}_u{{{\mathrm{Se}}}}_v{{{\mathrm{Te}}}}_{1 - u - v})_3\) can be similarly obtained from Eq. (3) with neglectable computational efforts.

Checking the transferability

Beyond the tetradymite family, it is interesting to note that our theoretical strategy should be applicable to any other material classes. As a further test, we select the half-Heusler compounds which exhibit a cubic structure and are composed of three different atoms occupying the X, Y, and Z sites, as shown in Fig. 4a. Compared with the tetradymites, the half-Heusler compounds generally possess much larger lattice thermal conductivity. On top of the reported kL of several half-Heusler compounds with integer stoichiometry (Supplementary Table 4)48,49,50,51, we can rapidly predict those of alloyed systems by using Eq. (3) where the scaling constants are fitted to be a = 0.08 and b = 3.44. For instance, the \({{{\mathrm{CoTi}}}}_{{{{\mathrm{0}}}}{{{\mathrm{.9}}}}}{{{\mathrm{Zr}}}}_{{{{\mathrm{0}}}}{{{\mathrm{.1}}}}}{{{\mathrm{Sb}}}}\) can be viewed as a solid solution of the \({{{\mathrm{CoTiSb}}}}\) and \({{{\mathrm{CoZrSb}}}}\) (atoms arranged in the sequence of XYZ mentioned above) with mole contents of 90% and 10%, respectively. The corresponding configurational entropy \({\Delta}S = 0.33k_{{{\mathrm{B}}}}\) and the lattice thermal conductivity is calculated to be 8.43 Wm−1 K−1. Furthermore, we see from Fig. 4b that the predicted kL of several alloyed half-Heusler compounds agree well with those measured previously (Supplementary Table 4)52,53,54,55. The higher Pearson correlation of 0.90 and lower MAE of 1.20 (note that the half-Heusler compounds have much larger kL compared with the tetradymites) confirms again the effectiveness of Eq. (3). Collectively speaking, although the values of kL investigated in the present work span over three orders of magnitude, we can still find good agreement between the predicted and experimental results, which substantiates strong reliability and transferability of our approach in predicting the lattice thermal conductivity of various alloyed systems.

Fig. 4: Generalization to half-Heusler compounds.
Fig. 4: Generalization to half-Heusler compounds.
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a The crystal structure of half-Heusler compounds and the three atomic sites (X, Y, Z) are marked. b The intuitive linear correlation between the predicted and experimental kL of several alloyed half-Heusler compounds, where the shaded area indicates the range of the mean absolute error.

Discussion

In summary, we propose a simple and reliable strategy that allows fast prediction on the lattice thermal conductivity of alloyed compounds at arbitrary stoichiometry. The approach is inherently rooted in the concept of configurational entropy, considering the fact that the alloyed systems can be in principle viewed as solid solutions of those having integer composition. Compared with prohibitively expensive first-principles calculations, our approach only requires the lattice thermal conductivity of a few configurations, and is able to predict those of countless alloyed systems at negligible computational efforts. The strong predictive power and transferability of our approach are demonstrated by checking the well-known alloyed tetradymites and half-Heusler compounds with either lower or higher lattice thermal conductivity. Our work therefore offers accelerated discovery of novel materials with specific lattice thermal conductivity to meet various application requirements.

Methods

The kL of a few tetradymites with integer stoichiometry are calculated via the full iterative solution of the phonon Boltzmann transport equation (BTE), as implemented in the ShengBTE program56. We apply the finite displacement approach with the 4 × 4 × 4 supercells to derive the harmonic interatomic force constants (IFCs), as embedded in the PHONOPY package57. The anharmonic IFCs can be obtained via the moment tensor potentials (MTPs) trained by using the MLIP code58,59. The data set is collected from the trajectories of four separate ab-initio molecular dynamics (AIMD) simulations at 50, 300, 500, and 700 K for 1000 steps with a time step of 1 fs. Besides, the ninth nearest neighbors and a fine q-mesh of 21 × 21 × 21 are considered to achieve converged kL. The involved DFT calculations are performed by using the projector-augmented wave (PAW) which is implemented in the Vienna ab-initio Simulation Package (VASP)60,61,62,63,64,65. The exchange-correlation energy is in the form of Perdew–Burke–Ernzerhof (PBE) with the generalized gradient approximation (GGA)66,67, where the plane wave cutoff energy is set at 500 eV. The effect of vdW interactions is explicitly included in our calculations by adopting the optB86b-vdW functional68. In terms of the configurational entropy determined by their respective compositions, the kL of any alloyed tetradymites with fractional stoichiometry can be readily obtained.