Introduction

An important focus in materials science research has been to discover hitherto unknown materials with properties that might hold the keys to solving the most pressing problems in renewable energy, energy harvesting, or semiconductor power electronics. The augmentation of new materials discovery and the prediction of their properties have been accelerated by the advent of advanced computer algorithms coupled with high-throughput (HT) screening methods1,2,3,4,5,6,7,8,9,10 using accurate quantum mechanical calculations based on density functional theory (DFT). In the recent past, several computational predictions have led to the successful synthesis of new solid-state compounds in a variety of chemistry and structure types in the family of half-Heuslers9,10, double half-Heuslers11, electrides12, AB2X4-based chalcogendes13, and rocksalt-based compounds14.

Crystalline solids with extreme thermal transport properties are technologically important for the efficient management of thermal energy15. While materials with high lattice thermal conductivity (κl) are used in microelectronic devices for heat dissipation, materials with low κl are used in thermal barrier coatings16, thermal data-storage devices17, and high-performance thermoelectrics (TEs)18,19 which can convert heat into electrical energy. The conversion efficiency of the TEs is defined by the figure-of-merit (ZT):

$${\rm{ZT}}={S}^{2}\sigma \mathrm{T}/({\kappa }_{e}+{\kappa }_{l}),$$
(1)

where S, σ, κe, and κl are the Seebeck coefficient, electrical conductivity, electronic thermal conductivity, and lattice thermal conductivity, respectively. Engineering the electronic band structures of crystalline compounds that already possess low κl has emerged to be a very popular strategy to increase the ZT. Therefore, crystalline semiconductors with intrinsically low κl are highly sought after in TEs and other thermal energy management devices.

In pursuit of finding new low κl materials, different classes of crystalline compounds like the perovskites20, half-Heuslers21, full-Heuslers22, and double half-Heuslers11 have been explored using HT computational methods, and subsequently, some of the predicted compounds were experimentally synthesized successfully9,10,11. Despite the existence of a large number of crystalline compounds in various materials databases, e.g., the Inorganic Crystal Structure Database (ICSD)23, Open Quantum Materials Database (OQMD)3,4, Materials Project5, and Aflowlib24, it is quite important to look for hitherto unknown stable and metastable compounds which might exhibit exciting physical and chemical properties. In this work, we present the computational discovery of a large number of stable (628) and low-energy metastable (852) quaternary chalcogenides AMM′Q3 (A = alkali, alkaline earth, post-transition metals; M/M′ = transition metals, lanthanides; Q = chalcogens) that span a huge chemical space across the periodic table. Our results are based on reliable, accurate, and robust HT-DFT calculations where (1) we generated initial AMM′Q3 compositions following the experimentally known AMM′Q3 compounds formation criteria, (2) calculated their energetics in all known seven crystallographic prototypes that are found in this materials family, and (3) performed thermodynamic phase stability analysis of these compositions against all possible competing phases that are present in the OQMD.

About 192 quaternary chalcogenides (see Supplementary Note for a complete list) with the generic formula AMM′Q3 have been synthesized experimentally25,26,27,28,29,30,31,32,33 which reveal that these compounds possess rich chemistries and structure types like the perovskites and Heusler compounds. Koscielski et al.25 noted that these known AMM′Q3 compounds contain no Q–Q bonds and the elements (A, M, M′, Q) balance the charge in their crystal structures with their expected formal oxidation states, making them charge balanced. The AMM′Q3 compounds are further classified into three categories depending on the nominal oxidation states of the three cations A, M, and M′, namely:

  • Type-I (A1+M1+M′4+Q3)

  • Type-II (A2+M1+M′3+Q3)

  • Type-III (A1+M2+M′3+Q3),

where the oxidation states of the cations are indicated with the superscripts. In all cases, we assume that the chalcogen atoms Q (S, Se, and Te) have a nominal 2− charge. Examining the experimentally known AMM′Q3 compounds (see Supplementary Fig. 1), we observe the following: (1) The A-site in these compounds is always occupied by alkali, alkaline earth, or post-transition metals with the only exception of Eu (in a 2+ charge state) which occupies the A-site of some of the Type-II compounds. (2) Whereas only transition metals occupy the M site, the M′ site can be filled either by the transition metals, lanthanides, or actinides. (3) No observed AMM′Q3 compound contains more than one alkali, alkaline earth, or post-transition metals. As detailed later, we will use these criteria in designing our HT workflow for generating the initial crystal structures through prototype decoration. Although the crystal chemistries of these compounds have been characterized somewhat in detail25,26,27,28,29,30,31,32,33, their properties have remained largely unexplored. Recently, it was shown experimentally34 and theoretically34,35,36,37 that many known semiconducting compounds in this crystal family exhibit ultralow κl. In addition, some compounds are shown to possess electronic bands favorable to support high TE performance34,36,37.

From the distribution of the elements forming the known AMM′Q3 compounds, we see that all the cations (A, M, M′) are in their common oxidation states of 1+, 2+, 3+, or 4+ coming from the chemical groups (i.e., alkali, alkaline earth, transition, post-transition metals, lanthanides, actinides) that span a large part of the periodic table. Yet, although a large number (192) of these compounds have been reported experimentally, this number is small compared to the vast number of possible compounds that can be obtained based on charge balanced combinatorial substitutions of the elements in a prototype crystal structure of AMM′Q3. Performing this combinatorial exercise experimentally would require a massive amount of resources and time to discover new AMM′Q3 compounds. However, computational screening can be very helpful in narrowing down the search space of the target compounds that would have higher chances of synthesizability in the laboratory1,8,9,10,11,38,39,40,41,42,43. Here, we have performed HT-DFT calculations followed by accurate ground-state phase stability analysis, and suggest (T = 0 K) thermodynamically stable and metastable AMM′Q3 compounds for experimental synthesis and exploration of their properties. Our calculations of the thermal transport properties of some of the predicted stable compounds using the Peierls–Boltzmann transport equation (PBTE) show that these compounds exhibit innate low κl due to the presence of strong lattice anharmonicity or rattler cations.

Results

Structural prototypes

The experimentally known AMM′Q3 compounds crystallize in seven structure types25,26,27,28,29,30,31,32,33: KCuZrSe3 (space group (SG): Cmcm, #63), Eu2CuS3 (SG: Pnma, #62), BaCuLaS3 (SG: Pnma, #62), Ba2MnS3 (SG: Pnma, #62), NaCuTiS3 (SG: Pnma, #62), BaAgErS3 (SG: C2/m, #12), and TlCuTiTe3 (SG: P21/m, #11). All these structure types are visualized in their extended unit cells in Fig. 1, where the conventional unit cell is outlined with black lines. Among these, five structure types (KCuZrSe3, Eu2CuS3, Ba2MnS3, NaCuTiS3, TlCuTiTe3) are layered where the rows of Am+ cations stack alternatively with the [MM′Q3]m− layers and interact through electrostatic interactions35,37. The strength of the interactions vary with the charges on the cations (i.e., m+) and induce modifications in the structures as well as in their properties36. When the interactions between the layers increase significantly, the atoms from the neighboring [MM′Q3]m− layers interact, giving rise to the three-dimensional channel structures (BaCuLaS3 and BaAgErS3). In BaCuLaS3 (Fig. 1c) and BaAgErS3 (Fig. 1f), two rows of the A-site cations occupy the empty spaces inside the channels formed by the M, M′, and Q atoms. Figure 1h shows that 71% of the known AMM′Q3 compounds crystallize in the KCuZrSe3 structure followed by 16% of the compounds crystallizing in the Eu2CuS3 structure type. The rest of the known AMM′Q3 compounds (13%) crystallize in the other five structure types. It is worth noting that in Eu2CuS3, the Eu atoms have mixed oxidation states (Eu2+CuEu3+S3) and sit in two different sites in the crystal structure. Similarly, the Ba atoms in Ba2MnS3 also occupy two different sites. We have used all these structure types in our HT-DFT design and discovery of new AMM′Q3 compounds. We note that the KCuZrSe3 and TlCuTiTe3 structure types have 12 atoms in their primitive unit cells and the rest of the structure types have 24 atoms.

Fig. 1: Crystallographic prototypes of the AMM′Q3 compounds.
figure 1

a–g Seven structural prototypes in the family of experimentally known AMM′Q3 compounds. Most of them have layered crystal structures where the layers of Am+ cations are sandwiched between the covalently bonded layers of [MM′Q3]m− anion sublattice, and interact with the later through electrostatic interactions. The black solid lines in each figure indicate the conventional unit cell of each compound. h Distributions of the experimentally known AMM′Q3 compounds in the seven structure types.

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Materials design strategy

The discovery of new compounds through HT-DFT method often starts with the decoration of prototype crystal structures with chemically similar elements from the periodic table to generate their initial crystal structures. DFT calculations are then performed on these newly decorated compounds followed by rigorous thermodynamic phase stability analysis to screen for stable and metastable compounds. Rather than generating the input crystal structures in a brute-force manner by substituting every element of the periodic table at all atomic sites in the prototype structures, in this work we restricted ourselves by the following screening criteria which are derived by examining the experimentally known AMM′Q3 compounds: (1) we substitute alkali, alkaline earth, or post-transition metal elements at the A-site. The M and M′ sites are populated with the transition metals and lanthanides. Three chalcogens, i.e., S, Se, and Te are substituted at the Q site. (2) We choose A-site cations with nominal oxidation states of 1+ or 2+, and M/M′-site cations with nominal oxidation states of 1+, 2+, 3+, or 4+. The elements that are chosen for substitutions at the A, M, M′ and Q sites along with their oxidation states are shown in Fig. 2. (3) We only consider compound compositions that are charge balanced. (4) We exclude any radioactive elements during the prototype decorations although some of the known AMM′Q3 compounds contain them. Adhering to these preconditions helps us narrow down our search space of compound exploration, reduces the computational cost, and most importantly it increases the success rates of stable compound prediction through HT-DFT calculations which will be evident later.

Fig. 2: Elemental distributions of the experimentally known and theoretically decorated AMM′Q3 compounds.
figure 2

The AMM′Q3 compounds are categorized into three types depending on the nominal oxidation states of the cations: Type-I (A1+M1+M’4+Q3), Type-II (A2+M1+M’3+Q3), and Type-III (A1+M2+M’3+Q3). The color-coded periodic table indicates the group of elements (whose oxidation states are written in blue color) that are used for prototype decorations in our HT-DFT calculations. Elements that are denoted with the check marks constitute the experimentally known AMM′Q3 compounds. Salmon, cyan, and yellow colors represent elements that occupy A, M/M′, and Q sites, respectively. The gray color represents the elements that were not used for prototype decoration. Eu has mixed oxidation states and occupies both A and M′ site. Although Ba occupies A/M’ sites in Ba2MnS3, we have kept Ba fixed only to the A-site in this work during prototype decoration. All experimental and decorated compounds are charge balanced.

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First, we generate the crystal structures of 4659 AMM′Q3 compositions (see Table 1) using the KCuZrSe3 structure as (1) it is the most prevalent structure type in this family and (2) all experimentally known AMM′Q3 compounds have low energies (within 50 meV/atom above the convex hull) in this structure type. After performing DFT calculations for all these compositions followed by T = 0 K phase stability analysis, we kept only those compounds (~1700) that have an energy within 50 meV/atom of the ground-state convex hull and discarded the rest from our search space. The DFT relaxed structures of all these 1700 compounds retain the KCuZrSe3 structure type. In the next step, we take these 1700 compositions and regenerate their crystal structures in each of six additional structure types to perform DFT calculations of 6 × 1700 = 10,200 AMM′Q3 compounds. Next, we perform T = 0 K thermodynamic phase stability analysis of those 1700 compositions considering all seven structure types and their competing phases that are available in the OQMD. In the final step, we obtain 628 thermodynamically stable and 852 metastable hitherto unknown AMM′Q3 compounds after performing a total number of 4659 + 10,200 = 14,859 DFT calculations. The stable 628 compounds include 69 Type-I, 231 Type-II, and 328 Type-III compounds, and among them, a total number of 570 compounds possess finite band gaps. A schematic of the HT-DFT flowchart is shown in Fig. 3a, and a summary is given in Table 1.

Table 1 A table summarizing the design and discovery of materials through prototype decoration.
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Fig. 3: A brief workflow and summary of high-throughout DFT calculations.
figure 3

a A schematic flowchart of the HT-DFT calculations for the discovery of new AMM′Q3 compounds. b The distribution of 628 predicted stable compounds in the seven structure types. c 570 out of 628 compounds possess finite band gaps. Histogram plot for the DFT-calculated band gaps of these 570 semiconductors.

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Phase stability analysis

We now present a detailed analysis of the T = 0 K ground-state phase stability of all (known and predicted) AMM′Q3 compounds. We begin our assessment with the phase stability analysis of the experimentally known AMM′Q3. Out of 192 known compounds, we find only 119 compounds (Type-I: 39, Type-II: 30, and Type-III: 50) in OQMD before we performed any new calculations from this work. We designate these 119 compounds as Set-I. As the initial crystal structures of the experimentally known compounds in OQMD mostly come from the ICSD, the DFT calculations of Set-I compounds in OQMD were performed based on their experimental crystal structures taken from the ICSD. We designate the rest of the known 192 − 119 = 73 compounds (Type-I: 1, Type-II: 60, and Type-III: 12) as Set-II, which were not present in OQMD due to (1) the absence of their structures in the ICSD and (2) no previous HT-DFT calculations were performed based on the prototype decorations in this AMM′Q3 family. However, our HT-DFT calculations of all the decorated AMM′Q3 compositions include the experimentally known AMM′Q3 compounds in Set-II. Hence, we will first analyze the phase stability of the Set-I compounds to see if our DFT calculations are able to correctly capture the energetics of the known compounds in Set-I and then we will utilize the phase stability data of Set-II compounds to validate the reliability of our approach for the discovery of new stable AMM′Q3 compounds based on prototype decoration through HT-DFT calculations.

As detailed in the “Methods” section and as well as in other references1,3,4,5,6,24,41,44, the hull distance (hd) is a metric of the thermodynamic stability of a compound. If the formation energy of the compound breaks the convex hull, then it is considered to be thermodynamically stable with hd = 0, indicating the likelihood of its synthesizability. On the other hand, compounds with a small positive hd (typically within a few tens of meV/atom) are called metastable and may also be in some cases experimentally synthesized39,40. According to these criteria, all experimentally known AMM′Q3 compounds should possess zero or small positive hd’s. Our analysis reveals that in Set-I, all (39) Type-I compounds and all but one (29) Type-II compounds have hd = 0, which is in line with our expectations. The one Type-II compound that has a small positive hd is Eu2+CuEu3+S3 (hd = 37 meV/atom). Also, all but three (47) Type-III compounds of Set-I have hd = 0. These three Type-III compounds are CsCoYbS3 (hd = 192 meV/atom), CsCoYbSe3 (hd = 151 meV/atom), CsZnYbSe3 (hd = 76 meV/atom). So, 115 of 119 experimentally synthesized compounds in Set-I are thermodynamically stable in the OQMD. Hence, stability is an excellent metric for the synthesizability of the predicted compounds. From this analysis, we also find that one of the Type-II compounds in Set-I, BaAgErS3, possesses a small positive hd of 12 meV/atom when its calculation is performed on the experimental crystal structure having C2/m SG (#12), indicating that it is metastable at 0 K in this structure. Interestingly, our HT-DFT calculations reveal that BaAgErS3 is stable T = 0 K in the KCuZrSe3 structure type (see Supplementary Fig. 2).

Since the 73 compounds in Set-II did not exist in OQMD before our HT-DFT calculations, we generate their crystal structures through prototype decorations as mentioned before to perform DFT calculations and T = 0 K phase stability analysis. As these compounds have already been synthesized experimentally, our DFT calculations and phase stability analysis provide a key test of our methodology. It also gives us an opportunity to examine how reliably our calculations can predict hitherto unknown stable AMM′Q3 compounds. After performing the thermodynamic stability analysis, we found that among the 73 compounds in Set-II (type-I: 1, type-II: 60, type-III: 12), 64 compounds have hd = 0, and only four compounds (SrCuYbS3: hd = 84 meV/atom, BaCuYbTe3: hd = 62 meV/atom, EuCuYbS3: hd = 72 meV/atom, PbCuYbS3: hd = 104 meV/atom) of Type-II and five compounds (CsMnYbSe3: hd = 8 meV/atom, CsZnYbS3 : hd = 163 meV/atom, CsZnYbTe3: hd = 71 meV/atom, RbZnYbSe3: hd = 102 meV/atom, RbZnYbTe3: hd = 82 meV/atom) of Type-III have positive hd’s.

Thus, our HT-DFT calculations based on the decorated structures successfully capture the stability of all but 13 (out of 192) of the experimentally known AMM′Q3 compounds. The 13 compounds which are experimentally observed but with hd > 0 all contain Eu and Yb, and the calculated energetics of these compounds may originate from the choice of incorrect pseudopotentials as discussed later. These results and analysis give us strong confidence in designing and discovering new AMM′Q3 compounds using the HT-DFT calculations and thermodynamic T = 0 K stability as a metric for synthesizable compounds. We have provided the phase stability data of all 192 experimentally known AMM′Q3 compounds in the Supplementary information.

Discovery of AMM′Q3 compounds

After performing the T = 0 K stability analysis of all newly decorated AMM′Q3 compounds for which HT-DFT calculations are performed in all seven structure types, we discover a large number of 628 (type-I: 69, type-II: 231, type-III:328) thermodynamically stable compounds that exclude the experimentally known 192 compounds. To put this number into perspective, the OQMD (containing more than 900,000 entries as of September 2020) has stable (hd = 0) 1161 full-Heuslers (SG: Fm\(\bar{3}\)m, #225), 618 half-Heuslers (SG: F\(\bar{4}\)3m, #216), 353 cubic (SG: Pm\(\bar{3}\)m, #221) perovskites, 242 orthorhombic (SG: Pnma, #62) perovskites. Similar to the experimentally known AMM′Q3 compounds, the KCuZrSe3 and Eu2CuS3 structure types are the most common, which constitute 41% and 25% of the predicted stable compounds, respectively (Fig. 3b). Among the other structure types, NaCuTiSe3 and TlCuTiTe3 are quite common, which constitute 19% and 9% of the predicted stable compounds. The rest 6% compounds have the BaCuLaS3 and BaAgErS3 structure types. However, we found no new stable compound in the Ba2MnS3 structure type. Our analysis shows that 570 out of 628 compounds possess finite bands that range from 0.2 to 5.34 eV, among which most of the compounds have band gaps within 1.5 eV (Fig. 3c). This is not surprising since all the decorated compositions are charge balanced. In addition, we found 852 potentially synthesizable metastable compounds (Type-I: 59, Type-II: 282, Type-III: 511) with small positive hds (i.e., 0 < hd ≤ 50 meV/atom). A summary of the HT-DFT calculations is shown in Table 1, and the lists of all predicted stable and metastable compounds are given in the Supplementary information.

We further examine the predicted stable compounds within each type in terms of their chemistries (sulfides, selenides, and tellurides) and the cations (A, M, M′). The results are displayed as bar charts in Fig. 4, where the bar corresponding to an element represents the number of stable compounds that contain it. We see that: (1) the elements that form the stable compounds constitute almost the entire periodic table. (2) There are more Cu-containing compounds in Type-I and Type-II categories compared to Ag or Au. In Type-III, there are more Mn compounds than other M elements with a 2+ oxidation state. (3) The number of stable compounds increases from Type-I (69) to Type-II (231) to Type-III (328). This is not surprising given that the number of elements that can occupy the M and M′ sites (satisfying the charge neutrality criteria) increase from Type-I to Type-III compounds (Table 1). A general trend that is noticeable across three types is that as we go from sulfides to selenides to tellurides, the number of compounds with smaller A cations decrease whereas the number increases with larger A cations. For example, there are two, two, and no Li-containing compounds in sulfides, selenides, and tellurides of Type-I compounds, respectively. Similarly, the number of compounds that have Sr increase from sulfides (7) to selenides (20) to tellurides (34) in Type-II.

Fig. 4: Elemental distributions of the predicted stable AMM′Q3 compounds.
figure 4

Distributions of the cations (A, M, M′) forming the 628 predicted stable AMM′Q3 compounds in the three categories: Type-I, Type-II, and Type-III. The bar corresponding to an element represents the number of stable compounds containing it. Total number of sulfides, selenides, and tellurides are mentioned in each panel. These compounds exclude the experimentally known 192 AMM′Q3 compounds.

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Lattice thermal transport properties

We now focus on exploring the lattice thermal transport properties of the predicted stable AMM′Q3 compounds. An accurate estimation of κl of a compound within a first-principles DFT framework is computationally very expensive45. Hence, the calculations of κl for all the predicted stable compounds would require a massive amount of computational resources. However, to demonstrate the thermal transport properties of our newly predicted stable compounds, we randomly select a handful of compounds with some criteria. We first screen for non-magnetic and semiconducting compounds, where the lattice contribution dominates the total thermal conductivity. Next, we search for those compounds which have the KCuZrSe3 structure type as it possesses the highest crystal symmetry (SG: Cmcm, #63) and the smallest unit cell (12 atoms). Finally, we randomly select ten compounds for κl calculations. The selected compounds, which include sulfides, selenides, and tellurides, are: CsCuZrS3, BaCuScSe3, BaCuScTe3, BaCuTbSe3, BaAgGdSe3, CsZnYS3, CsZnGdS3, CsZnScSe3, CsZnScTe3, CsCdYTe3. We note that this list also includes Type-I (the first in the list), Type-II (the next 4) as well as Type-III (the last 5) compounds. The electronic structures and phonon dispersions of these compounds are given in the Supplementary information.

We calculate the κl of these ten compounds using the PBTE (see “Methods” section) and present the results in Fig. 5. We see that all these compounds exhibit very low κl where the in-plane \(({\kappa }_{l}^{\perp })\) and the cross-plane \(({\kappa }_{l}^{\parallel })\) components are lower than 3 Wm−1 K−1 and 1.2 Wm−1 K−1, respectively, for T ≥ 300 K. Here, \({\kappa }_{l}^{\perp }\) is perpendicular to the stacking direction of the layers in the crystal structure of the AMM′Q3 compounds and \({\kappa }_{l}^{\parallel }\) is parallel to it. As a reference, we compare our calculated κl with that of a prototypical TE material SnSe, which was experimentally shown to possesses low κl that leads to a very high TE figure-of-merit46. The measured47\({\kappa }_{l}^{\perp }\) and \({\kappa }_{l}^{\parallel }\) for single-crystalline stoichiometric samples of SnSe are 1.9 and 0.9 Wm−1 K−1, respectively, at 300 K, which become much lower in the off-stoichiometric polycrystalline samples46. Further examination of the results in Fig. 5 reveals that in terms of the anisotropy of the \({\kappa }_{l}^{\perp }\) and \({\kappa }_{l}^{\parallel }\) components, Type-I and Type-III compounds are quite similar, but Type-II compounds are different from the rest i.e., \({\kappa }_{l}^{\perp }/{\kappa }_{l}^{\parallel }\) (Type-I/III) > \({\kappa }_{l}^{\perp }/{\kappa }_{l}^{\parallel }\) (Type-II). The difference in the anisotropy of the properties arises from the fact that in Type-II compound the electrostatic attractions between A2+ and [MM′Q3]2− layers are stronger than that of between A1+ and [MM′Q3]1− layers in Type-I/III compounds. The stronger interlayer interactions give rise to a shorter interlayer distance in Type-II, which make \({\kappa }_{l}^{\perp }\) and \({\kappa }_{l}^{\parallel }\) less anisotropic.

Fig. 5: Lattice thermal transport properties of the AMM′Q3 compounds.
figure 5

a–j The calculated temperature-dependent lattice thermal conductivity (κl) of ten predicted stable AMM′Q3 compound which are non-magnetic and semiconducting. \({\kappa }_{l}^{\perp }\) (blue disk) and \({\kappa }_{l}^{\parallel }\) (orange pentagon) are the two components of κl that are perpendicular and parallel to the stacking directions in the crystal structure, respectively. k Predictive statistical model of κl. A linear regression model, f(x), is fitted to the average (Av.) sound velocity vs average κl data (at 300 K) of those ten compounds, which is used to predict κl of other AMM′Q3 compounds based on their average speed of sounds.

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To gain a deeper understanding, we examine the lattice dynamics and thermal transport properties of BaCuScTe3 (Type-II) and CsCdYTe3 (Type-III) in detail. Our analysis reveals that the underlying physical principles governing the low κl in Type-II compounds are different from the Type-I/III compounds. We start our analysis with the phonon dispersion of BaCuScTe3 (Fig. 6a), which shows the presence of very low-frequency acoustic (<45 cm−1), and optical (~15 cm−1 along X-S-R directions) phonon modes, which give rise to low phonon group velocities. The phonon dispersion and the phonon density of states (Fig. 6b) of BaScCuTe3 show that a strong hybridization exists between the phonon branches up to 100 cm−1, where Ba and Te atoms have large contributions. Soft acoustic phonon branches give rise to very low-sound velocities and a strong hybridization between the phonons at low energies enhances the phonon scattering phase space. Both these factors help suitably to give rise to a very low κl (\({\kappa }_{l}^{\perp }\) = 1.7 Wm−1 K−1 and \({\kappa }_{l}^{\parallel }\) = 0.76 Wm−1 K−1 at 300 K). On the other hand, the phonon dispersion of CsCdYTe3 (Fig. 6d) features nearly dispersion-less optical phonon branches along the X-S-R-A directions in the Brillouin zone at low energies, which are the characteristics of rattler vibrations in the crystal structure. In addition, it also has soft acoustic phonon branches (<35 cm−1). The calculated κl of CsCdYTe3 becomes ultralow with \({\kappa }_{l}^{\perp }\) and \({\kappa }_{l}^{\parallel }\) being 0.82 and 0.25 Wm−1 K−1, respectively, at 300 K.

Fig. 6: Harmonic and anharmonic lattice dynamical properties of the AMM′Q3 compounds.
figure 6

Harmonic phonon dispersions, atom-resolved phonon density of states, anharmaonic frequency-cumulative κl, and their derivatives with respect to the frequency for BaCuScTe3 (a–c) and CsCdYTe3 (d–f). The phonon dispersions are color coded with the participation ratios of the phonon modes (indicated in the color bar). \({\kappa }_{l}^{\perp }\) (solid blue line) and \({\kappa }_{l}^{\parallel }\) (solid red line) are perpendicular and parallel to the stacking directions in the crystal structure of the AMM′Q3 compound, respectively. \({{\kappa }_{l}^{\prime}}^{\perp }\) (dotted blue line) and \({{\kappa }_{l}^{\prime}}^{\parallel }\) (dotted red line) are the first-order derivatives of \({\kappa }_{l}^{\perp }\) and \({\kappa }_{l}^{\parallel }\), respectively, with respect to the frequency, which are in arbitrary units in the figures.

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Rattler phonon modes are highly localized, which strongly inhibit the transport of phonons, giving rise to ultralow κl in many crystalline solids such as TlInTe248, CsAg5Te349, etc. It was shown that the filler atoms in clathrates50 and skutterudites51 act as ideal rattlers, which give rise to dispersion-less phonon branches where the phonon frequencies remain highly localized having very small participation ratio (PR) values ~0.2 (see “Methods” section). The PRs of the phonon modes of BaCuScTe3 and CsCdYTe3 are color coded in Fig. 6a, d, respectively. We see that while most of the low-energy phonon modes (<100 cm−1) of BaCuScTe3 have PR values close to 1, signifying the absence of phonon localization, the low-energy dispersion-less phonon branches of CsCdYTe3 have small PRs (<0.2), indicating the highly localized nature of the phonon modes. The phonon density of states (Fig. 6e) also reveals that these localized phonons primarily arise from the Cs atoms (confined in 25–55 cm−1) that act as rattlers. Thus, our analysis shows that the ultralow κl in CsCdYTe3 is primarily caused by the localized vibrations of the rattling phonons.

We further investigate the origin of such poor thermal transport properties in BaCuScTe3 in terms of a more fundamental material quantity, the lattice anharmonicity. Strong lattice anharmonicity is one of the important factors that induces very low κl in compounds like SnSe52, NaSbTe253, etc. To estimate the strength of the intrinsic anharmonicity, we calculate the macroscopic average Gruneisen parameter,

$$\gamma =\mathop{\sum}\limits_{i}{\gamma }_{i}{C}_{\mathrm{v},i}/\mathop{\sum}\limits_{i}{C}_{\mathrm{v},i},$$
(2)

where γi and Cv,i are the Gruneisen parameter and specific heat capacity at constant volume for the ith phonon mode. The calculated γ of BaCuScTe3 (1.5) is larger than that of CsCdYTe3 (1.2), signifying the presence of stronger anharmonicity in the former. These γ values are comparable to that of NaSbSe2 (1.7), NaSbTe2 (1.6), NaBiTe2 (1.5), etc. compounds which are experimentally53 shown to possess ultralow κl. Thus, we see that while the low κl in Type-II compounds is caused by the low-sound velocities and a stronger lattice anharmonicity, the presence of rattler cations in Type-III as well Type-I compounds (see Supplementary Fig. 4) are primarily responsible for inducing low κl in them.

To examine which phonons are primarily responsible for the conduction of heat in BaCuScTe3 and CsCdYTe3, we plot the cumulative-κl and their first-order derivatives (\({\kappa }_{l}^{\prime}\)) with respect to the frequency at T = 300 K. We see from Fig. 6c that while the acoustic and low-energy optical phonons up to 90 cm−1 mainly contribute to \({\kappa }_{l}^{\perp }\), \({\kappa }_{l}^{\parallel }\) is primarily contributed by the phonons up to 45 cm−1 in BaCuScTe3. On the other hand, Fig. 6f shows that while the acoustic as well as the optical phonons up to 100 cm−1 have large contributions toward \({\kappa }_{l}^{\perp }\), only the acoustic phonons (up to 35 cm−1) primarily carry heat for \({\kappa }_{l}^{\parallel }\) in CsCdYTe3. We also notice that the anisotropy (\({\kappa }_{l}^{\perp }/{\kappa }_{l}^{\parallel }\) = 3.3 at T = 300 K) in CsCdYTe3 is much larger that of BaCuScTe3 (\({\kappa }_{l}^{\perp }/{\kappa }_{l}^{\parallel }\) = 2.2 at T = 300 K). The origin of this anisotropy can be attributed to the contrasting interlayer and intralayer interactions in BaCuScTe3 and CsCdYTe3. For example, the analysis of the interatomic force constants (IFCs) reveals that interlayer interactions in CsCdYTe3 are much weaker (IFC(Cs − Te) = − 0.333 eV/Å2) compared to BaCuScTe3 (IFC(Ba − Te) = − 1.204 eV/Å2), which makes the transport of optical phonons (above 25 cm−1) along the stacking direction (i.e., \({\kappa }_{l}^{\parallel }\)) of CsCdYTe3 less effective. On the other hand, the intralayer interactions in CsCdYTe3 are much stronger (IFC(Cd − Te) = − 3.923 eV/Å2, IFC(Y − Te): −2.216 eV/Å2) than those of BaCuScTe3 (IFC(Cu − Te) = −2.396 eV/Å2, IFC(Sc − Te) = −1.890 eV/Å2). As a result, while phonons up to 90 cm−1 mainly carry the heat for \({\kappa }_{l}^{\perp }\) in BaCuScTe3, in CsCdYTe3 they are carried by phonons with frequencies up to 100 cm−1 very effectively.

Discussion

We notice that in Set-I and Set-II, all the known compounds that have positive hd’s have either Yb3+ or Eu3+ cations in them. Given the fact OQMD used Yb_2 and Eu_2 PPs which are intended for compounds having Yb2+ and Eu2+ cations, those energetic results are somewhat suspect. However, we note that our HT-DFT calculations predict the stability of other rare-earth elements containing known compounds in this family correctly. Hence, the newly predicted compounds containing Eu3+ and Yb3+ should be taken with caution. We note that none of our predicted stable compounds contain these cations. However, 11 and 7 of the predicted metastable compounds contain Eu3+ and Yb3+ cations, respectively.

Concerning the experimental validation of our prediction, we note that the experimental synthesis of a large number of compounds is a daunting task. While automated synthesis of a large number of compounds is now possible through HT experimental facilities54, here, we suggest only four compounds (Table 2) that can be immediately picked up for experimental verification of our prediction. These compounds contain no toxic element and their calculated lattice thermal conductivity is very low. In Table 2, we provide their DFT-calculated band gaps, γ, and average κl calculated at 300 K. These calculated quantities along with κl can be compared with the experimentally measured values of these materials. Finally, we note that κl for each compound has been calculated using only three-phonon scattering processes. The inclusion of additional four-phonon scattering rates55,56 and grain-boundary57 limited phonon scattering mechanisms could further lower the calculated κl in this family of compounds. Also, as the electronic structure of the compounds features nearly flat bands and multiple peaks near the valence/conduction band extrema (Supplementary Fig. 3), some of these compounds are expected to exhibit good potential for TE applications as well.

Table 2 Suggested compounds for experimental synthesis and characterization.
Full size table

To estimate the usefulness of these materials as TEs, we perform detailed TE calculations (see “Methods” section for details) on the four compounds in Table 2. Our analysis shows that ZT of three compounds (BaAgGdSe3, CsZnScTe3, and CsCdYTe3) exceeds 1 at 600 K, indicating their high TE performance. To highlight the TE efficiency of these four compounds, in Table 2, we provide the Seebeck coefficient, power factor and ZT (at 600 K) at a nominal carrier concentration (2 × 1019 cm−3). We provide detailed characterization of TE properties of these compounds in the Supplementary Information (Supplementary Discussion and Supplementary Figs. 5–8).

Finally, we comment on the general trend of κl on other predicted stable AMM′Q3 compounds. To achieve this, we have made a predictive statistical model, f(x), of κl using the linear regression technique (see Fig. 5k and “Methods” section for details) which can be used to predict κl of any AMM′Q3 compound based on its average speed of sound and without explicitly doing PBTE calculations. To this end, we randomly choose ten compounds (predicted, stable) that are different from Fig. 5, and calculate their average speed of sounds using which we predict their average κl at 300 K. These compounds and their predicted κl are: NaZnHoSe3 (1.24 Wm−1 K−1), BaCuTbTe3 (1.17 Wm−1 K−1), CsCuHfS3 (2.20 Wm−1 K−1), SrCuHoSe3 (1.69 Wm−1 K−1), BaAgPrTe3 (1.25 Wm−1 K−1), KCdTbS3 (1.73 Wm−1 K−1), NaZnGdSe3 (2.02 Wm−1 K−1), CsCdLaSe3 (1.00 Wm−1 K−1), RbZnYTe3 (0.95 Wm−1 K−1), RbCuZrSe3 (1.82 Wm−1 K−1). From these results, we see that the predicted AMM′Q3 compounds generically exhibit very low κl.

In summary, we use HT-DFT calculations to discover a large number of 628 thermodynamically stable quaternary chalcogenides (AMM′Q3). As all compositions in this family are charge balanced, our analysis shows that 570 of 628 compounds possess finite band gaps which vary between 0.2 and 5.34 eV. Our calculations of the thermal transport properties show that AMM′Q3 compounds exhibit intrinsically very low κl, and the anisotropy in κl is much smaller in Type-II compounds compared to Type-I/III compounds. Our analysis further reveals that low κl in this family originates either due to the presence of rattling cations (in Type-I/III compounds) or stronger lattice anharmonicity (Type-II compound). While the rattler cations give rise to localized phonon modes that inhibit the propagation of phonons, a stronger lattice anharmonicity enhances the phonon scattering phase space, leading to a low κl. In addition, there exists a strong coupling between the acoustic and low-energy optical phonon modes in Type-II compounds which increases the phonon scattering rates of the heat-carrying phonons. A detailed TE characterization of some of the predicted compounds show the potential of the AMM′Q3 compounds to be efficient TEs. Our work is thus interesting not just from the perspective of materials discovery but also for finding the presence of low κl in them, which hold promises for further research and possible applications in energy materials, particularly in TEs and related devices.

Methods

DFT calculations

We performed DFT calculations using the Vienna Ab initio Simulation Package58 and utilizing the projector-augmented wave59 potentials with the Perdew–Burke–Ernzerhof60 generalized gradient approximation to the exchange-correlation functional. The atomic positions and other cell degrees of freedom of the compounds were fully relaxed and spin-polarized calculations were performed for compounds that contain partially filled d or f-shells elements with a ferromagnetic arrangement of spins in accordance with the HT framework as laid out in the qmpy suite of tools3,4. For more details on the calculation parameters, we refer to refs. 3,4. T = 0 K phase stability analysis often serves as an excellent indicator for the possibility of synthesizability of a predicted compound in the laboratory9,10,11. To assess the T = 0 K thermodynamic stability of the compounds, we calculate their formation energies (ΔHf) utilizing the DFT total energy (ground-state) of each compound using the formula:

$${{\Delta }}{H}_{f}(\sigma )={E}(\sigma )-\mathop{\sum}\limits_{i}{n}_{i}{\mu }_{i},$$
(3)

where E is the DFT total energy (at 0 K) of an AMM′Q3 compound in a crystal structure denoted by σ, μi is the chemical potential of element i with its fraction ni in that compound. For each composition, we used a number of prototype crystal structures, σ, based on structural prototypes of known AMM′Q3 compounds. To determine the thermodynamic stability of a compound, we need to compare its formation energy against all its competing phases, not only against other compounds at the same composition. To this end, we generated the quaternary phase diagram (i.e., the T = 0 K convex hull of the A-M-M′-Q phase space) for every AMM′Q3 compound considering all elemental, binary, ternary, and quaternary phases present in the OQMD, which (as of September 2020) corresponded to nearly 900,000 entries of DFT-calculated energies. The calculated convex hd (explained in the next section) then serves as a metric to determine whether a compound is stable (i.e., hd = 0), metastable (small positive hd), or unstable (large positive hd).

Convex hull construction

To construct the convex hull of an AMM′Q3 compound, it is necessary to identify the set of phases (elemental, binary, ternary as well as quaternary) in the four-dimensional composition space of A-M-M′-Q that have the lowest formation energies at their compositions. In OQMD, we present the convex hull of a quaternary compound through a four-dimensional phase diagram which is represented as an isometric shot of the Gibbs’ tetrahedron (Fig. 7a). Each face of the tetrahedron represents a three-dimensional phase diagram of a ternary composition that is represented as the Gibbs’ triangle (Fig. 7b). The vertices in Fig. 7a, b represent the elements constituting the quaternary and ternary phase space, respectively, and the edges connecting any two vertices represent the binary composition axis between those two elements. Any node within Fig. 7a, b represents a stable compound, which is denoted by a cyan disk. The metastable and unstable compounds are not shown in these figures for clarity which fall off the nodes. The stability of an AMM′Q3 compound is given by the difference (which is defined as the hd) between the calculated formation energy (ΔHf) of an AMM′Q3 compound under consideration and its hull energy (ΔHe) at that composition:

$$hd={{\Delta }}{H}_{f}-{{\Delta }}{H}_{e}.$$
(4)
Fig. 7: Visualization of quaternary and ternary phase diagrams.
figure 7

a The four-dimensional phase diagram (T = 0 K) of the Li-Cu-Zr-S quaternary system, which is the isometric shot of a Gibbs’ tetrahedron. b One of the faces of the tetrahedron represents the three-dimensional phase diagram of the Cu-Zr-S ternary system which is presented as the Gibbs’ triangle. Each cyan node in a and b represents a stable compound. For clarity, we do not show any metastable/unstable compounds and mark only few stable compounds in this figure. LiCuZrS3 is one of the predicted stable compounds in this work, which and its competing phases are denoted with dashed arrows in a.

Full size image

The hull energy is defined as the energy at the convex hull at that AMM′Q3 composition. By definition, the hd of a stable compound is zero, whereas for metastable and unstable compounds they are real positive numbers. In keeping with the heuristic conventions used in literature10,61,62 we term those AMM′Q3 compounds to be metastable whose hd’s lie within 50 meV/atom above the convex hull (i.e., 0 < hd ≤ 50 meV/atom). The metastable compounds are also potentially synthesizable in the laboratory39,40.

Phonon participation ratio

Phonon dispersions have been calculated using 2 × 2 × 1 supercell of the primitive unit cell using Phonopy63. The high symmetry paths in the Brillouin zones were adopted following the conventions used by Setyawan et al.64. To examine the extent of localization of the phonon modes, we calculate their phonon PR using the formula50,65:

$$P({\omega }_{q})={\left(\mathop{\sum }\limits_{i}^{N}\frac{{{| e}_{i}({\omega }_{q})| }^{2}}{{M}_{i}}\right)}^{2}{\Large/}N\mathop{\sum }\limits_{i}^{N}\frac{{{| e}_{i}({\omega }_{q})| }^{4}}{{M}_{i}^{2}},$$
(5)

where ei(ωq) is the eigenvector of the phonon mode at wavevector, q with frequency ω, Mi is the mass of the ith atom in the unit cell containing a total number of N atoms. The value of P(ωq) ranges between 0 and 1. In an ordered crystal, when P(ωq) becomes close to 1, it indicates that the phonon mode is propagative where all the atoms in the unit cell participate. On the other hand, very low values of PR (~0.2)65,66 indicate the strong localization of the phonon modes (e.g., rattling phonons) where only a few atoms in the unit cell participate in the vibrations. Examples of rattler atoms containing compounds include filled clathrates (Ba8Si46 and Ba8Ga16Ge30)50,65, where the filler atoms act as ideal rattlers that induce ultralow κl in them.

Thermal conductivity calculations

We calculate the κl utilizing the phonon lifetimes obtained from the third-order IFCs67,68,69, which was shown to reproduce κl within 5% of the experimentally measured κl in this AMM′Q3 family of compounds34,35. We constructed the third-order IFCs of each compound based on DFT calculations of displaced supercell configurations by limiting the cut-off distance (rc) up to the third nearest neighbor. We used 2 × 2 × 1 supercell (containing 48 atoms) of the primitive unit cell (with 12 atoms) using thirdorder.py45 utility for the calculation of IFCs. Using the second and third-order IFCs in the ShengBTE code45, we calculate the temperature-dependent phonon scattering rates and κl utilizing a full iterative solution to the PBTE for phonons using a 12 × 12 × 12 q-point mesh. The calculated κl generally depends on the rc which accounts for the maximum range of interaction in the third-order IFCs52. It was shown that good convergence of κl was obtained by limiting rc to the third nearest neighbor within the crystal structure in this family of compounds35.

Calculation of thermoelectric properties

We calculate the TE properties of the four compounds in Table 2, i.e., BaCuScTe3, BaAgGdSe3, CsZnScTe3, CsCdYTe3 to assess their energy conversion efficiencies. We have calculated their electrical transport properties i.e., the Seebeck coefficient (S), power factor (S2σ), where σ is the electrical conductivity using the BoltzTrap code70 within the constant relaxation time approximation, which has been widely used to estimate the themoelectric performance of semiconductors. We have estimated the TE figure-of-merit (ZT) using Eq. (1). While κl is calculated using the PBTE, we have determined κe employing the Wiedemann–Franz law. BoltzTrap code employs the Boltzmann transport equation (BTE) for electrons to calculate the electrical transport properties. We use a dense 20 × 20 × 20 k-mesh to obtain the electronic band energy which are used in the BoltzTrap code while solving the BTE. Since the compounds in Table 2 have layered crystal structures, we have plotted all TE properties along two crystallographic directions: out-of-plane direction which is parallel (∥) to the stacking direction of the layers in the crystal structure and in-plane direction, which is perpendicular (⊥) to the stacking direction. Furthermore, we plot the properties as a function of electron (i.e., n-type doping) and hole (i.e., p-type doping) carrier concentrations. We provide the figures showing the TE properties of those four compounds in the Supplementary Information (Supplementary Figs. 5–8).

The determination of realistic values of the power factor and ZT crucially depends on the electronic relaxation time (τ) and its accurate determination using first-principles quantum mechanical methods are computationally very expensive. Although τ has been calculated for elemental metal71, binary72, or even ternary73 systems with relatively smaller unit cells through determination of the electron-phonon scattering matrix elements using ab initio methods, such calculations become computationally prohibitive for quaternary systems. Hence, the determination of τ using first-principles methods for compounds with quaternary chemistry, such as the family of AMM′Q3 compounds, remains an important yet challenging future work.

It is known that τ in doped semiconductors and metals can vary from few fs (1 fs = 10−15 s) to a few tens of fs and shows strong dependence on temperature as well as carrier concentration. For example, from the fit to the experimentally measured electrical conductivity data, Hao et al.74 showed that τ of SnSe varies from 27 to 4 fs as temperature increases from 300 to 800 K for hole concentrations that change from 4 × 1019 to 6.5 × 1019 cm−3. On the other hand, it was theoretically shown73 that τ varies from 108 to 6 fs between 300 and 900 K in BaAu2P4. Hence, to get an estimate of the TE performance of the AMM′Q3 compounds considered in this work, here, we take three conservative choices of relaxation times, namely 5, 10, and 20 fs, and evaluate the power factor and ZT of the four compounds (BaCuScTe3, BaAgGdSe3, CsZnScTe3, and CsCdYTe3) at two different temperatures (300 and 600 K). We present the results in Table 2 highlighting the TE performance of these compounds and present their detailed figures (Supplementary Figs. 5–8) in the Supplementary Information.

Finally, we note that the constant relaxation time approximation used in this study assumes that (1) τ for both holes and electrons is independent of the wave vector and energy and (2) all the details of electron scattering (of which the electron-phonon processes are usually the most important) are lumped into the constant, τ. Hence, a large degree of uncertainty may be involved in our assumed values of τ’s. As a result, the calculated TE properties, particularly, the power factor and ZT’s of the compounds should be taken with caution. A reliable method of estimation of τ would be through the calculations the full electron-phonon matrix elements, which are generally computationally very expensive71, and hence falls outside the scope of the present work.

Predictive statistical modeling of κ l

The creation of predictive and transferable model of κl is an active field of research. Such a model in principle can be made using machine learning methods such as transfer learning or deep learning. However, like any accurate machine learning model, training a high-quality model requires a large set of high-quality data which is quite scare as the generation of the κl data using PBTE or any other first-principles method is computationally prohibitive.

Here, we have made a linear regression model which can be used to predict the κl of any AMM′Q3 compound without explicitly doing PBTE calculations. To this end, we calculate bulk (B) and shear (G) moduli of the ten compounds for which we already calculated κl using the PBTE (Fig. 5). Utilizing B and G, we calculated the longitudinal (vL) and transverse (vT) speed of sounds:

$$\begin{array}{l}{\mathrm{v}}_{L}=\sqrt{(B+\frac{4}{3}G)/\rho }\\ {\mathrm{v}}_{T}=\sqrt{G/\rho },\end{array}$$
(6)

where ρ is the density of a compound. Next, we calculate the average speed of sound (vav) using the formula75:

$$3{{{\mathrm{v}}}_{av}}^{-3}={{\mathrm{v}}_{L}}^{-3}+2{{\mathrm{v}}_{T}}^{-3}.$$
(7)

We plot the directionally averaged κl (at 300 K) against the average speed of sound (vav) for those ten compounds in Fig. 5k. It can be seen that the average κl shows almost linear correlation with the average κl. Next, we train a linear regression model on these data which form the training set.

We fit a linear regression model, f(x), to the data in Fig. 5k which yields an R2 value of 0.71, which implies that given the average speed of sound of any AMM′Q3 compound, this model can predict its κl with 71% accuracy. Given the very small size of the training data, the performance of the model can be considered to be quite good. To validate this model, we calculate the average speed of sounds and κl of another four compounds (BaCuScS3, BaAgYS3, TlCuHfSe3, BaCuYTe3) that belong to the validation set using the DFT and PBTE. Next, we predict their κl using the above regression model, compare them with their actual calculated values (at T = 300 K), and estimate the error in prediction: BaCuScS3 (κl (actual) = 2.87 Wm−1 K−1, κl (predicted) = 3.14 Wm−1 K−1, error = 9.4%), BaAgYS3 (κl (actual) = 1.94 Wm−1 K−1, κl (predicted) = 2.29 Wm−1 K−1, error = 18.0%), TlCuHfSe3 (κl (actual) = 1.29 Wm−1 K−1, κl (predicted) = 1.45 Wm−1 K−1, error = 12.4%), and BaCuYTe3 (κl (actual) = 1.75 Wm−1 K−1, κl (predicted) = 1.59 Wm−1 K−1, error = 9.1%). Hence, the average error in prediction is 12.3% which falls within the error of the trained regression model. We provide the bulk modulus, shear modulus, speed of sound, κl data of the compounds in the training, validation, and test sets in the Supplementary Information in Supplementary Tables 1–3, respectively.