Abstract
The development of efficient thermal energy management devices such as thermoelectrics and barrier coatings often relies on compounds having low lattice thermal conductivity (κ_{l}). Here, we present the computational discovery of a large family of 628 thermodynamically stable quaternary chalcogenides, AMM′Q_{3} (A = alkali/alkaline earth/posttransition metals; M/M′ = transition metals, lanthanides; Q = chalcogens) using highthroughput density functional theory (DFT) calculations. We validate the presence of low κ_{l} in these materials by calculating κ_{l} of several predicted stable compounds using the Peierls–Boltzmann transport equation. Our analysis reveals that the low κ_{l} originates from the presence of either a strong lattice anharmonicity that enhances the phononscatterings or rattler cations that lead to multiple scattering channels in their crystal structures. Our thermoelectric calculations indicate that some of the predicted semiconductors may possess high energy conversion efficiency with their figureofmerits exceeding 1 near 600 K. Our predictions suggest experimental research opportunities in the synthesis and characterization of these stable, low κ_{l} compounds.
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 highthroughput (HT) screening methods^{1,2,3,4,5,6,7,8,9,10} 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 solidstate compounds in a variety of chemistry and structure types in the family of halfHeuslers^{9,10}, double halfHeuslers^{11}, electrides^{12}, AB_{2}X_{4}based chalcogendes^{13}, and rocksaltbased compounds^{14}.
Crystalline solids with extreme thermal transport properties are technologically important for the efficient management of thermal energy^{15}. While materials with high lattice thermal conductivity (κ_{l}) are used in microelectronic devices for heat dissipation, materials with low κ_{l} are used in thermal barrier coatings^{16}, thermal datastorage devices^{17}, and highperformance thermoelectrics (TEs)^{18,19} which can convert heat into electrical energy. The conversion efficiency of the TEs is defined by the figureofmerit (ZT):
where S, σ, κ_{e}, and κ_{l} 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 κ_{l} has emerged to be a very popular strategy to increase the ZT. Therefore, crystalline semiconductors with intrinsically low κ_{l} are highly sought after in TEs and other thermal energy management devices.
In pursuit of finding new low κ_{l} materials, different classes of crystalline compounds like the perovskites^{20}, halfHeuslers^{21}, fullHeuslers^{22}, and double halfHeuslers^{11} have been explored using HT computational methods, and subsequently, some of the predicted compounds were experimentally synthesized successfully^{9,10,11}. 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 Project^{5}, and Aflowlib^{24}, 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 lowenergy metastable (852) quaternary chalcogenides AMM′Q_{3} (A = alkali, alkaline earth, posttransition 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 HTDFT calculations where (1) we generated initial AMM′Q_{3} compositions following the experimentally known AMM′Q_{3} 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′Q_{3} have been synthesized experimentally^{25,26,27,28,29,30,31,32,33} 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′Q_{3} 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′Q_{3} compounds are further classified into three categories depending on the nominal oxidation states of the three cations A, M, and M′, namely:

TypeI (A^{1+}M^{1+}M′^{4+}Q_{3})

TypeII (A^{2+}M^{1+}M′^{3+}Q_{3})

TypeIII (A^{1+}M^{2+}M′^{3+}Q_{3}),
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′Q_{3} compounds (see Supplementary Fig. 1), we observe the following: (1) The Asite in these compounds is always occupied by alkali, alkaline earth, or posttransition metals with the only exception of Eu (in a 2+ charge state) which occupies the Asite of some of the TypeII 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′Q_{3} compound contains more than one alkali, alkaline earth, or posttransition 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 detail^{25,26,27,28,29,30,31,32,33}, their properties have remained largely unexplored. Recently, it was shown experimentally^{34} and theoretically^{34,35,36,37} that many known semiconducting compounds in this crystal family exhibit ultralow κ_{l}. In addition, some compounds are shown to possess electronic bands favorable to support high TE performance^{34,36,37}.
From the distribution of the elements forming the known AMM′Q_{3} 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, posttransition 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′Q_{3}. Performing this combinatorial exercise experimentally would require a massive amount of resources and time to discover new AMM′Q_{3} 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 laboratory^{1,8,9,10,11,38,39,40,41,42,43}. Here, we have performed HTDFT calculations followed by accurate groundstate phase stability analysis, and suggest (T = 0 K) thermodynamically stable and metastable AMM′Q_{3} 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 κ_{l} due to the presence of strong lattice anharmonicity or rattler cations.
Results
Structural prototypes
The experimentally known AMM′Q_{3} compounds crystallize in seven structure types^{25,26,27,28,29,30,31,32,33}: KCuZrSe_{3} (space group (SG): Cmcm, #63), Eu_{2}CuS_{3} (SG: Pnma, #62), BaCuLaS_{3} (SG: Pnma, #62), Ba_{2}MnS_{3} (SG: Pnma, #62), NaCuTiS_{3} (SG: Pnma, #62), BaAgErS_{3} (SG: C2/m, #12), and TlCuTiTe_{3} (SG: P2_{1}/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 (KCuZrSe_{3}, Eu_{2}CuS_{3}, Ba_{2}MnS_{3}, NaCuTiS_{3}, TlCuTiTe_{3}) are layered where the rows of A^{m+} cations stack alternatively with the [MM′Q_{3}]^{m−} layers and interact through electrostatic interactions^{35,37}. 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 properties^{36}. When the interactions between the layers increase significantly, the atoms from the neighboring [MM′Q_{3}]^{m−} layers interact, giving rise to the threedimensional channel structures (BaCuLaS_{3} and BaAgErS_{3}). In BaCuLaS_{3} (Fig. 1c) and BaAgErS_{3} (Fig. 1f), two rows of the Asite 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′Q_{3} compounds crystallize in the KCuZrSe_{3} structure followed by 16% of the compounds crystallizing in the Eu_{2}CuS_{3} structure type. The rest of the known AMM′Q_{3} compounds (13%) crystallize in the other five structure types. It is worth noting that in Eu_{2}CuS_{3}, the Eu atoms have mixed oxidation states (Eu^{2+}CuEu^{3+}S_{3}) and sit in two different sites in the crystal structure. Similarly, the Ba atoms in Ba_{2}MnS_{3} also occupy two different sites. We have used all these structure types in our HTDFT design and discovery of new AMM′Q_{3} compounds. We note that the KCuZrSe3 and TlCuTiTe_{3} structure types have 12 atoms in their primitive unit cells and the rest of the structure types have 24 atoms.
Materials design strategy
The discovery of new compounds through HTDFT 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 bruteforce 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′Q_{3} compounds: (1) we substitute alkali, alkaline earth, or posttransition metal elements at the Asite. 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 Asite 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′Q_{3} 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 HTDFT calculations which will be evident later.
First, we generate the crystal structures of 4659 AMM′Q_{3} compositions (see Table 1) using the KCuZrSe_{3} structure as (1) it is the most prevalent structure type in this family and (2) all experimentally known AMM′Q_{3} 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 groundstate convex hull and discarded the rest from our search space. The DFT relaxed structures of all these 1700 compounds retain the KCuZrSe_{3} 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′Q_{3} 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′Q_{3} compounds after performing a total number of 4659 + 10,200 = 14,859 DFT calculations. The stable 628 compounds include 69 TypeI, 231 TypeII, and 328 TypeIII compounds, and among them, a total number of 570 compounds possess finite band gaps. A schematic of the HTDFT flowchart is shown in Fig. 3a, and a summary is given in Table 1.
Phase stability analysis
We now present a detailed analysis of the T = 0 K groundstate phase stability of all (known and predicted) AMM′Q_{3} compounds. We begin our assessment with the phase stability analysis of the experimentally known AMM′Q_{3}. Out of 192 known compounds, we find only 119 compounds (TypeI: 39, TypeII: 30, and TypeIII: 50) in OQMD before we performed any new calculations from this work. We designate these 119 compounds as SetI. As the initial crystal structures of the experimentally known compounds in OQMD mostly come from the ICSD, the DFT calculations of SetI 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 (TypeI: 1, TypeII: 60, and TypeIII: 12) as SetII, which were not present in OQMD due to (1) the absence of their structures in the ICSD and (2) no previous HTDFT calculations were performed based on the prototype decorations in this AMM′Q_{3} family. However, our HTDFT calculations of all the decorated AMM′Q_{3} compositions include the experimentally known AMM′Q_{3} compounds in SetII. Hence, we will first analyze the phase stability of the SetI compounds to see if our DFT calculations are able to correctly capture the energetics of the known compounds in SetI and then we will utilize the phase stability data of SetII compounds to validate the reliability of our approach for the discovery of new stable AMM′Q_{3} compounds based on prototype decoration through HTDFT calculations.
As detailed in the “Methods” section and as well as in other references^{1,3,4,5,6,24,41,44}, 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 synthesized^{39,40}. According to these criteria, all experimentally known AMM′Q_{3} compounds should possess zero or small positive hd’s. Our analysis reveals that in SetI, all (39) TypeI compounds and all but one (29) TypeII compounds have hd = 0, which is in line with our expectations. The one TypeII compound that has a small positive hd is Eu^{2+}CuEu^{3}+S_{3} (hd = 37 meV/atom). Also, all but three (47) TypeIII compounds of SetI have hd = 0. These three TypeIII compounds are CsCoYbS_{3} (hd = 192 meV/atom), CsCoYbSe3 (hd = 151 meV/atom), CsZnYbSe_{3} (hd = 76 meV/atom). So, 115 of 119 experimentally synthesized compounds in SetI 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 TypeII compounds in SetI, BaAgErS_{3}, 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 HTDFT calculations reveal that BaAgErS_{3} is stable T = 0 K in the KCuZrSe_{3} structure type (see Supplementary Fig. 2).
Since the 73 compounds in SetII did not exist in OQMD before our HTDFT 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′Q_{3} compounds. After performing the thermodynamic stability analysis, we found that among the 73 compounds in SetII (typeI: 1, typeII: 60, typeIII: 12), 64 compounds have hd = 0, and only four compounds (SrCuYbS3: hd = 84 meV/atom, BaCuYbTe_{3}: hd = 62 meV/atom, EuCuYbS_{3}: hd = 72 meV/atom, PbCuYbS_{3}: hd = 104 meV/atom) of TypeII and five compounds (CsMnYbSe_{3}: hd = 8 meV/atom, CsZnYbS_{3} : hd = 163 meV/atom, CsZnYbTe_{3}: hd = 71 meV/atom, RbZnYbSe_{3}: hd = 102 meV/atom, RbZnYbTe_{3}: hd = 82 meV/atom) of TypeIII have positive hd’s.
Thus, our HTDFT calculations based on the decorated structures successfully capture the stability of all but 13 (out of 192) of the experimentally known AMM′Q_{3} 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′Q_{3} compounds using the HTDFT 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′Q_{3} compounds in the Supplementary information.
Discovery of AMM′Q_{3} compounds
After performing the T = 0 K stability analysis of all newly decorated AMM′Q_{3} compounds for which HTDFT calculations are performed in all seven structure types, we discover a large number of 628 (typeI: 69, typeII: 231, typeIII: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 fullHeuslers (SG: Fm\(\bar{3}\)m, #225), 618 halfHeuslers (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′Q_{3} compounds, the KCuZrSe_{3} and Eu_{2}CuS_{3} structure types are the most common, which constitute 41% and 25% of the predicted stable compounds, respectively (Fig. 3b). Among the other structure types, NaCuTiSe_{3} and TlCuTiTe_{3} are quite common, which constitute 19% and 9% of the predicted stable compounds. The rest 6% compounds have the BaCuLaS_{3} and BaAgErS_{3} structure types. However, we found no new stable compound in the Ba_{2}MnS_{3} 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 (TypeI: 59, TypeII: 282, TypeIII: 511) with small positive hds (i.e., 0 < hd ≤ 50 meV/atom). A summary of the HTDFT 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 Cucontaining compounds in TypeI and TypeII categories compared to Ag or Au. In TypeIII, there are more Mn compounds than other M elements with a 2+ oxidation state. (3) The number of stable compounds increases from TypeI (69) to TypeII (231) to TypeIII (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 TypeI to TypeIII 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 Licontaining compounds in sulfides, selenides, and tellurides of TypeI compounds, respectively. Similarly, the number of compounds that have Sr increase from sulfides (7) to selenides (20) to tellurides (34) in TypeII.
Lattice thermal transport properties
We now focus on exploring the lattice thermal transport properties of the predicted stable AMM′Q_{3} compounds. An accurate estimation of κ_{l} of a compound within a firstprinciples DFT framework is computationally very expensive^{45}. Hence, the calculations of κ_{l} 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 nonmagnetic and semiconducting compounds, where the lattice contribution dominates the total thermal conductivity. Next, we search for those compounds which have the KCuZrSe_{3} 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 κ_{l} calculations. The selected compounds, which include sulfides, selenides, and tellurides, are: CsCuZrS_{3}, BaCuScSe_{3}, BaCuScTe_{3}, BaCuTbSe_{3}, BaAgGdSe_{3}, CsZnYS_{3}, CsZnGdS_{3}, CsZnScSe_{3}, CsZnScTe_{3}, CsCdYTe_{3}. We note that this list also includes TypeI (the first in the list), TypeII (the next 4) as well as TypeIII (the last 5) compounds. The electronic structures and phonon dispersions of these compounds are given in the Supplementary information.
We calculate the κ_{l} 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 κ_{l} where the inplane \(({\kappa }_{l}^{\perp })\) and the crossplane \(({\kappa }_{l}^{\parallel })\) 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′Q_{3} compounds and \({\kappa }_{l}^{\parallel }\) is parallel to it. As a reference, we compare our calculated κ_{l} with that of a prototypical TE material SnSe, which was experimentally shown to possesses low κ_{l} that leads to a very high TE figureofmerit^{46}. The measured^{47}\({\kappa }_{l}^{\perp }\) and \({\kappa }_{l}^{\parallel }\) for singlecrystalline stoichiometric samples of SnSe are 1.9 and 0.9 Wm^{−1} K^{−1}, respectively, at 300 K, which become much lower in the offstoichiometric polycrystalline samples^{46}. 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, TypeI and TypeIII compounds are quite similar, but TypeII compounds are different from the rest i.e., \({\kappa }_{l}^{\perp }/{\kappa }_{l}^{\parallel }\) (TypeI/III) > \({\kappa }_{l}^{\perp }/{\kappa }_{l}^{\parallel }\) (TypeII). The difference in the anisotropy of the properties arises from the fact that in TypeII compound the electrostatic attractions between A^{2+} and [MM′Q_{3}]^{2−} layers are stronger than that of between A^{1+} and [MM′Q_{3}]^{1−} layers in TypeI/III compounds. The stronger interlayer interactions give rise to a shorter interlayer distance in TypeII, which make \({\kappa }_{l}^{\perp }\) and \({\kappa }_{l}^{\parallel }\) less anisotropic.
To gain a deeper understanding, we examine the lattice dynamics and thermal transport properties of BaCuScTe_{3} (TypeII) and CsCdYTe_{3} (TypeIII) in detail. Our analysis reveals that the underlying physical principles governing the low κ_{l} in TypeII compounds are different from the TypeI/III compounds. We start our analysis with the phonon dispersion of BaCuScTe_{3} (Fig. 6a), which shows the presence of very lowfrequency acoustic (<45 cm^{−1}), and optical (~15 cm^{−1} along XSR directions) phonon modes, which give rise to low phonon group velocities. The phonon dispersion and the phonon density of states (Fig. 6b) of BaScCuTe_{3} 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 lowsound 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 κ_{l} (\({\kappa }_{l}^{\perp }\) = 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 CsCdYTe_{3} (Fig. 6d) features nearly dispersionless optical phonon branches along the XSRA 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 κ_{l} of CsCdYTe_{3} becomes ultralow with \({\kappa }_{l}^{\perp }\) and \({\kappa }_{l}^{\parallel }\) being 0.82 and 0.25 Wm^{−1} K^{−1}, respectively, at 300 K.
Rattler phonon modes are highly localized, which strongly inhibit the transport of phonons, giving rise to ultralow κ_{l} in many crystalline solids such as TlInTe_{2}^{48}, CsAg_{5}Te_{3}^{49}, etc. It was shown that the filler atoms in clathrates^{50} and skutterudites^{51} act as ideal rattlers, which give rise to dispersionless 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 BaCuScTe_{3} and CsCdYTe_{3} are color coded in Fig. 6a, d, respectively. We see that while most of the lowenergy phonon modes (<100 cm^{−1}) of BaCuScTe_{3} have PR values close to 1, signifying the absence of phonon localization, the lowenergy dispersionless phonon branches of CsCdYTe_{3} 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 κ_{l} in CsCdYTe_{3} is primarily caused by the localized vibrations of the rattling phonons.
We further investigate the origin of such poor thermal transport properties in BaCuScTe_{3} in terms of a more fundamental material quantity, the lattice anharmonicity. Strong lattice anharmonicity is one of the important factors that induces very low κ_{l} in compounds like SnSe^{52}, NaSbTe_{2}^{53}, etc. To estimate the strength of the intrinsic anharmonicity, we calculate the macroscopic average Gruneisen parameter,
where γ_{i} and C_{v,i} are the Gruneisen parameter and specific heat capacity at constant volume for the ith phonon mode. The calculated γ of BaCuScTe_{3} (1.5) is larger than that of CsCdYTe_{3} (1.2), signifying the presence of stronger anharmonicity in the former. These γ values are comparable to that of NaSbSe_{2} (1.7), NaSbTe_{2} (1.6), NaBiTe_{2} (1.5), etc. compounds which are experimentally^{53} shown to possess ultralow κ_{l}. Thus, we see that while the low κ_{l} in TypeII compounds is caused by the lowsound velocities and a stronger lattice anharmonicity, the presence of rattler cations in TypeIII as well TypeI compounds (see Supplementary Fig. 4) are primarily responsible for inducing low κ_{l} in them.
To examine which phonons are primarily responsible for the conduction of heat in BaCuScTe_{3} and CsCdYTe_{3}, we plot the cumulativeκ_{l} and their firstorder derivatives (\({\kappa }_{l}^{\prime}\)) with respect to the frequency at T = 300 K. We see from Fig. 6c that while the acoustic and lowenergy 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 BaCuScTe_{3}. 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 CsCdYTe_{3}. We also notice that the anisotropy (\({\kappa }_{l}^{\perp }/{\kappa }_{l}^{\parallel }\) = 3.3 at T = 300 K) in CsCdYTe_{3} is much larger that of BaCuScTe_{3} (\({\kappa }_{l}^{\perp }/{\kappa }_{l}^{\parallel }\) = 2.2 at T = 300 K). The origin of this anisotropy can be attributed to the contrasting interlayer and intralayer interactions in BaCuScTe_{3} and CsCdYTe_{3}. For example, the analysis of the interatomic force constants (IFCs) reveals that interlayer interactions in CsCdYTe_{3} are much weaker (IFC_{(Cs − }_{Te)} = − 0.333 eV/Å^{2}) compared to BaCuScTe_{3} (IFC_{(Ba − Te)} = − 1.204 eV/Å^{2}), which makes the transport of optical phonons (above 25 cm^{−1}) along the stacking direction (i.e., \({\kappa }_{l}^{\parallel }\)) of CsCdYTe_{3} less effective. On the other hand, the intralayer interactions in CsCdYTe_{3} are much stronger (IFC_{(Cd − Te)} = − 3.923 eV/Å^{2}, IFC_{(Y − Te)}: −2.216 eV/Å^{2}) than those of BaCuScTe_{3} (IFC_{(Cu − Te)} = −2.396 eV/Å^{2}, IFC_{(Sc − Te)} = −1.890 eV/Å^{2}). As a result, while phonons up to 90 cm^{−1} mainly carry the heat for \({\kappa }_{l}^{\perp }\) in BaCuScTe_{3}, in CsCdYTe_{3} they are carried by phonons with frequencies up to 100 cm^{−1} very effectively.
Discussion
We notice that in SetI and SetII, all the known compounds that have positive hd’s have either Yb^{3+} or Eu^{3+} cations in them. Given the fact OQMD used Yb_2 and Eu_2 PPs which are intended for compounds having Yb^{2+} and Eu^{2+} cations, those energetic results are somewhat suspect. However, we note that our HTDFT calculations predict the stability of other rareearth elements containing known compounds in this family correctly. Hence, the newly predicted compounds containing Eu^{3+} and Yb^{3+} 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 Eu^{3+} and Yb^{3+} 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 facilities^{54}, 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 DFTcalculated band gaps, γ, and average κ_{l} calculated at 300 K. These calculated quantities along with κ_{l} can be compared with the experimentally measured values of these materials. Finally, we note that κ_{l} for each compound has been calculated using only threephonon scattering processes. The inclusion of additional fourphonon scattering rates^{55,56} and grainboundary^{57} limited phonon scattering mechanisms could further lower the calculated κ_{l} 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.
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 (BaAgGdSe_{3}, CsZnScTe_{3}, and CsCdYTe_{3}) 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 × 10^{19} 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 κ_{l} on other predicted stable AMM′Q_{3} compounds. To achieve this, we have made a predictive statistical model, f(x), of κ_{l} using the linear regression technique (see Fig. 5k and “Methods” section for details) which can be used to predict κ_{l} of any AMM′Q_{3} 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 κ_{l} at 300 K. These compounds and their predicted κ_{l} are: NaZnHoSe_{3} (1.24 Wm^{−1} K^{−1}), BaCuTbTe_{3} (1.17 Wm^{−1} K^{−1}), CsCuHfS_{3} (2.20 Wm^{−1} K^{−1}), SrCuHoSe_{3} (1.69 Wm^{−1} K^{−1}), BaAgPrTe_{3} (1.25 Wm^{−1} K^{−1}), KCdTbS_{3} (1.73 Wm^{−1} K^{−1}), NaZnGdSe_{3} (2.02 Wm^{−1} K^{−1}), CsCdLaSe_{3} (1.00 Wm^{−1} K^{−1}), RbZnYTe_{3} (0.95 Wm^{−1} K^{−1}), RbCuZrSe_{3} (1.82 Wm^{−1} K^{−1}). From these results, we see that the predicted AMM′Q_{3} compounds generically exhibit very low κ_{l}.
In summary, we use HTDFT calculations to discover a large number of 628 thermodynamically stable quaternary chalcogenides (AMM′Q_{3}). 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′Q_{3} compounds exhibit intrinsically very low κ_{l}, and the anisotropy in κ_{l} is much smaller in TypeII compounds compared to TypeI/III compounds. Our analysis further reveals that low κ_{l} in this family originates either due to the presence of rattling cations (in TypeI/III compounds) or stronger lattice anharmonicity (TypeII 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 κ_{l}. In addition, there exists a strong coupling between the acoustic and lowenergy optical phonon modes in TypeII compounds which increases the phonon scattering rates of the heatcarrying phonons. A detailed TE characterization of some of the predicted compounds show the potential of the AMM′Q_{3} 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 κ_{l} 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 Package^{58} and utilizing the projectoraugmented wave^{59} potentials with the Perdew–Burke–Ernzerhof^{60} generalized gradient approximation to the exchangecorrelation functional. The atomic positions and other cell degrees of freedom of the compounds were fully relaxed and spinpolarized calculations were performed for compounds that contain partially filled d or fshells elements with a ferromagnetic arrangement of spins in accordance with the HT framework as laid out in the qmpy suite of tools^{3,4}. 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 laboratory^{9,10,11}. To assess the T = 0 K thermodynamic stability of the compounds, we calculate their formation energies (ΔH_{f}) utilizing the DFT total energy (groundstate) of each compound using the formula:
where E is the DFT total energy (at 0 K) of an AMM′Q_{3} compound in a crystal structure denoted by σ, μ_{i} is the chemical potential of element i with its fraction n_{i} in that compound. For each composition, we used a number of prototype crystal structures, σ, based on structural prototypes of known AMM′Q_{3} 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 AMM′Q phase space) for every AMM′Q_{3} 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 DFTcalculated 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′Q_{3} compound, it is necessary to identify the set of phases (elemental, binary, ternary as well as quaternary) in the fourdimensional composition space of AMM′Q that have the lowest formation energies at their compositions. In OQMD, we present the convex hull of a quaternary compound through a fourdimensional phase diagram which is represented as an isometric shot of the Gibbs’ tetrahedron (Fig. 7a). Each face of the tetrahedron represents a threedimensional 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′Q_{3} compound is given by the difference (which is defined as the hd) between the calculated formation energy (ΔH_{f}) of an AMM′Q_{3} compound under consideration and its hull energy (ΔH_{e}) at that composition:
The hull energy is defined as the energy at the convex hull at that AMM′Q_{3} 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 literature^{10,61,62} we term those AMM′Q_{3} 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 laboratory^{39,40}.
Phonon participation ratio
Phonon dispersions have been calculated using 2 × 2 × 1 supercell of the primitive unit cell using Phonopy^{63}. 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 formula^{50,65}:
where e_{i}(ω_{q}) is the eigenvector of the phonon mode at wavevector, q with frequency ω, M_{i} is the mass of the ith atom in the unit cell containing a total number of N atoms. The value of P(ω_{q}) ranges between 0 and 1. In an ordered crystal, when P(ω_{q}) 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 (Ba_{8}Si_{46} and Ba_{8}Ga_{16}Ge_{30})^{50,65}, where the filler atoms act as ideal rattlers that induce ultralow κ_{l} in them.
Thermal conductivity calculations
We calculate the κ_{l} utilizing the phonon lifetimes obtained from the thirdorder IFCs^{67,68,69}, which was shown to reproduce κ_{l} within 5% of the experimentally measured κ_{l} in this AMM′Q_{3} family of compounds^{34,35}. We constructed the thirdorder IFCs of each compound based on DFT calculations of displaced supercell configurations by limiting the cutoff distance (r_{c}) 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.py^{45} utility for the calculation of IFCs. Using the second and thirdorder IFCs in the ShengBTE code^{45}, we calculate the temperaturedependent phonon scattering rates and κ_{l} utilizing a full iterative solution to the PBTE for phonons using a 12 × 12 × 12 qpoint mesh. The calculated κ_{l} generally depends on the r_{c} which accounts for the maximum range of interaction in the thirdorder IFCs^{52}. It was shown that good convergence of κ_{l} was obtained by limiting r_{c} to the third nearest neighbor within the crystal structure in this family of compounds^{35}.
Calculation of thermoelectric properties
We calculate the TE properties of the four compounds in Table 2, i.e., BaCuScTe_{3}, BaAgGdSe_{3}, CsZnScTe_{3}, CsCdYTe_{3} to assess their energy conversion efficiencies. We have calculated their electrical transport properties i.e., the Seebeck coefficient (S), power factor (S^{2}σ), where σ is the electrical conductivity using the BoltzTrap code^{70} within the constant relaxation time approximation, which has been widely used to estimate the themoelectric performance of semiconductors. We have estimated the TE figureofmerit (ZT) using Eq. (1). While κ_{l} is calculated using the PBTE, we have determined κ_{e} 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 kmesh 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: outofplane direction which is parallel (∥) to the stacking direction of the layers in the crystal structure and inplane direction, which is perpendicular (⊥) to the stacking direction. Furthermore, we plot the properties as a function of electron (i.e., ntype doping) and hole (i.e., ptype 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 firstprinciples quantum mechanical methods are computationally very expensive. Although τ has been calculated for elemental metal^{71}, binary^{72}, or even ternary^{73} systems with relatively smaller unit cells through determination of the electronphonon scattering matrix elements using ab initio methods, such calculations become computationally prohibitive for quaternary systems. Hence, the determination of τ using firstprinciples methods for compounds with quaternary chemistry, such as the family of AMM′Q_{3} 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 × 10^{19} to 6.5 × 10^{19} cm^{−3}. On the other hand, it was theoretically shown^{73} that τ varies from 108 to 6 fs between 300 and 900 K in BaAu_{2}P_{4}. Hence, to get an estimate of the TE performance of the AMM′Q_{3} 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 (BaCuScTe_{3}, BaAgGdSe_{3}, CsZnScTe_{3}, and CsCdYTe_{3}) 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 electronphonon 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 electronphonon matrix elements, which are generally computationally very expensive^{71}, and hence falls outside the scope of the present work.
Predictive statistical modeling of κ _{l}
The creation of predictive and transferable model of κ_{l} 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 highquality model requires a large set of highquality data which is quite scare as the generation of the κ_{l} data using PBTE or any other firstprinciples method is computationally prohibitive.
Here, we have made a linear regression model which can be used to predict the κ_{l} of any AMM′Q_{3} 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 κ_{l} using the PBTE (Fig. 5). Utilizing B and G, we calculated the longitudinal (v_{L}) and transverse (v_{T}) speed of sounds:
where ρ is the density of a compound. Next, we calculate the average speed of sound (v_{av}) using the formula^{75}:
We plot the directionally averaged κ_{l} (at 300 K) against the average speed of sound (v_{av}) for those ten compounds in Fig. 5k. It can be seen that the average κ_{l} shows almost linear correlation with the average κ_{l}. 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 R^{2} value of 0.71, which implies that given the average speed of sound of any AMM′Q_{3} compound, this model can predict its κ_{l} 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 κ_{l} of another four compounds (BaCuScS_{3}, BaAgYS_{3}, TlCuHfSe_{3}, BaCuYTe_{3}) that belong to the validation set using the DFT and PBTE. Next, we predict their κ_{l} using the above regression model, compare them with their actual calculated values (at T = 300 K), and estimate the error in prediction: BaCuScS_{3} (κ_{l} (actual) = 2.87 Wm^{−1} K^{−1}, κ_{l} (predicted) = 3.14 Wm^{−1} K^{−1}, error = 9.4%), BaAgYS_{3} (κ_{l} (actual) = 1.94 Wm^{−1} K^{−1}, κ_{l} (predicted) = 2.29 Wm^{−1} K^{−1}, error = 18.0%), TlCuHfSe_{3} (κ_{l} (actual) = 1.29 Wm^{−1} K^{−1}, κ_{l} (predicted) = 1.45 Wm^{−1} K^{−1}, error = 12.4%), and BaCuYTe_{3} (κ_{l} (actual) = 1.75 Wm^{−1} K^{−1}, κ_{l} (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, κ_{l} data of the compounds in the training, validation, and test sets in the Supplementary Information in Supplementary Tables 1–3, respectively.
Data availability
The data that support the findings of the work are in the manuscript and Supplementary Information. The structures and energetics of the predicted compounds would be made available through the Open Quantum Materials database (OQMD) in a future release. Additional data will be available upon reasonable request.
Code availability
Opensource codes are used throughout this work.
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Acknowledgements
K.P. and C.W. acknowledge support from the U.S. Department of Energy under Contract No. DESC0014520 (thermal conductivity calculations) and the Center for Hierarchical Materials Design (CHiMaD) and from the U.S. Department of Commerce, National Institute of Standards and Technology under Award No. 70NANB14H012 (HTDFT calculations). J.S. and J.H. acknowledge support from the National Science Foundation through the MRSEC program (NSFDMR 1720139) at the Materials Research Center (phase stability). Y.X. acknowledges support from Toyota Research Institute (TRI) through the Accelerated Materials Design and Discovery program (lattice dynamics). Y.L. and M.G.K. were supported in part by the National Science Foundation Grant DMR2003476. K.P. sincerely thanks Sean Griesemer for useful discussion on the abundance of various crystallographic prototypes in the OQMD. We acknowledge the computing resources provided by (1) the National Energy Research Scientific Computing Center (NERSC), a U.S. Department of Energy Office of Science User Facility operated under Contract No. DEAC0205CH11231, (2) Quest highperformance computing facility at Northwestern University which is jointly supported by the Office of the Provost, the Office for Research, and Northwestern University Information Technology, and (3) the Extreme Science and Engineering Discovery Environment (National Science Foundation Contract ACI1548562).
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K.P. conceived and designed the project. K.P. performed calculations and analysis with help and suggestions from Y.X., J.S., J.H., Y.L., M.G.K., and C.W. C.W. supervised the whole project. All authors discussed the results, provided comments, and contributed to writing the manuscript.
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Pal, K., Xia, Y., Shen, J. et al. Accelerated discovery of a large family of quaternary chalcogenides with very low lattice thermal conductivity. npj Comput Mater 7, 82 (2021). https://doi.org/10.1038/s4152402100549x
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