Origins of chalcogenide perovskite instability

Abstract

Chalcogenide perovskites, particularly II–IV ABS₃ compounds, are a promising class of materials for optoelectronic applications. However, these materials frequently exhibit instability in two respects: (1) a preference for structures containing one-dimensional edge- or face-sharing octahedral networks instead of the three-dimensional corner-sharing perovskite framework (polymorphic instability), and (2) a tendency to decompose into competing compositions (hull instability). We evaluate the stability of 81 ABS₃ compounds using density functional theory, finding that only BaZrS₃ and BaHfS₃ are both polymorphically and hull stable, with the NH₄CdCl₃-type structure being the preferred polymorph for 77% of these compounds. Comparison with existing tolerance factor models demonstrates that these approaches work well for known perovskites but overpredict stability for compositions without published experimental results. Polymorphic stability analysis reveals that perovskite structures are stabilized by strong B–S bonding interactions, while needle structures exhibit minimal B–S covalency, suggesting that electrostatic rather than covalent interactions drive the preference for edge-sharing motifs. Hull stability analysis comparing ABS₃ to ABO₃ analogues reveals a weaker inductive effect in sulfides as a possible explanation for the scarcity of sulfides compared with oxides. The relative instability of ABS₃ compounds is further supported by experimental synthesis attempts. These findings provide fundamental insights into the origins of instability in chalcogenide perovskites and highlight the challenges in expanding this promising materials class beyond the few materials that have been reported to date.

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Christopher J. Bartel

Chris Bartel is an Assistant Professor in the Department of Chemical Engineering and Materials Science (CEMS) at the University of Minnesota. Prior to joining CEMS in 2022, he earned a PhD in Chemical Engineering from the University of Colorado and worked as a postdoctoral researcher in Materials Science at UC Berkeley. Chris now leads the “Design of Materials on Computers Lab,” which leverages first-principles calculations, thermodynamic modeling, solid-state chemistry, and machine learning to accelerate the design of solid-state materials for energy-related applications. He has been recognized as a Scialog Fellow in Negative Emissions Science and Sustainable Minerals, Metals, and Materials. Chris grew up near New Orleans and earned a BS in Chemical Engineering from Auburn University. To learn more about the latest work from his research group, please visit https://bartel.cems.umn.edu.

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Introduction

Chalcogenide perovskites have garnered significant attention for optoelectronic applications.¹ These materials exhibit smaller bandgaps than oxides,² greater thermal and chemical stability than hybrid organic–inorganic halides,^3,4 and additional features like high absorption coefficients and defect-tolerant carrier transport that enhance their functionality.^5,6 This class of materials follows the general formula ABX₃, where A and B are cations and X is a chalcogenide anion (S, Se, Te). This work focuses on the most well-studied subset of these materials where A is divalent, B is tetravalent, and X is sulfur, excluding compounds that are radioactive or contain f-electrons due to their limited relevance for optoelectronic applications. While the ideal perovskite structure is cubic, chalcogenide perovskites rarely adopt this high-symmetry structure.⁷ Instead, they typically crystallize in distorted perovskite polymorphs due to the larger, more polarizable chalcogenide anions and non-ideal cation–anion radii ratios. Among these, the GdFeO₃-type orthorhombic perovskite is the most commonly observed.¹ As shown in Fig. 1a, this structure consists of a three-dimensional network of tilted corner-sharing BS₆ octahedra, with the larger A-site cations occupying the interstitial voids. This 3D network plays a crucial role in enabling high and isotropic carrier mobility, direct bandgaps, and other optoelectronic properties.⁸ While most research into ABS₃ materials focuses on the perovskite structure, these compounds often exhibit a tendency to form alternative structures featuring face- or edge-sharing octahedra, which can lead to different electronic and structural properties.^1,9–11

Fig. 1 Common polymorphs of ABS₃ compounds. (a) GdFeO₃-type orthorhombic perovskite, (b) BaNiO₃-type hexagonal and (c) NH₄CdCl₃-type “needle-like” structure. Gray spheres represent A-site cations, blue spheres represent B-site cations, and yellow spheres represent sulfur.

The three most common polymorphs of ABS₃ chalcogenides (including orthorhombic perovskite) are depicted in Fig. 1. The BaNiO₃-type hexagonal structure (Fig. 1b) consists of a one-dimensional network of face-sharing octahedra, while the NH₄CdCl₃-type “needle-like” structure (Fig. 1c) is characterized by a one-dimensional framework of edge-sharing octahedra. These three polymorphs are the focus of this study, but it is worth noting that other structures have been experimentally observed, such as “misfit layered structures” and variants containing mixed edge- and face-sharing octahedra.^12–16

Although numerous ABS₃ compounds within our scope have been synthesized in various structures, only the six compounds spanning A = Ba, Sr, Ca and B = Zr, Hf have been synthesized in the perovskite structure.¹ SrZrS₃ is unique among this group in that it has been synthesized in both the perovskite and needle structures, with perovskite emerging as the higher temperature phase.^17,18 BaZrS₃ is the most well studied for its promising optoelectronic properties and has been synthesized at a number of conditions, including low-temperatures colloidal synthesis.^5,19–23 However, the most common synthetic approach to realize these materials is high-temperature sulfidation of their oxide perovskite analogues,¹⁹ with CaHfS₃ and CaZrS₃ requiring comparatively higher temperatures than the A = Ba, Sr counterparts.^21,24

Previous computational studies have evaluated the thermodynamic stability of various subsets of chalcogenide perovskites.^12,25–29 For example, Huo et al. used density functional theory (DFT) with the PBE functional³⁰ to study 168 ABX₃ compounds (including X = O, S, Se) in four different crystal structures: cubic perovskite, orthorhombic perovskite, hexagonal, and needle.²⁸ They found that ∼30% adopted a perovskite ground state, including sulfides with A = Ba, Ca and B = Hf, Zr. Brehm et al. used LDA³¹ to study ABS₃ compounds with d⁰ configurations¹² (including I–V and II–IV valences for A–B) and found none had a perovskite ground state; instead, BaZrS₃ was calculated in that study to prefer the needle structure. Their finite-temperature analysis, however, showed that perovskites are stabilized at elevated temperatures for systems with A = Ba, Sr, Sn, Pb and B = Zr. In addition to polymorphic stability, some studies have also considered the stability of ABX₃ compounds against decomposition into competing phases.^25,26 For example, Körbel et al. computed ternary phase diagrams for a broad range of ABX₃ compounds (not limited to chalcogenides), identifying BaHfS₃, BaZrS₃, and BaZrSe₃ as the only chalcogenide perovskites within 25 meV per atom of stability according to their calculated convex hull.²⁵

In parallel to thermodynamic analyses with DFT, tolerance factor models have long been used to assess the tendency for ABX₃ materials to crystallize in perovskite or non-perovskite structures (in this work, perovskite refers to materials comprised of corner-sharing BX₆ octahedra surrounding an A-site cation). The most well-known of these models is the Goldschmidt tolerance factor,³² which relies only on geometric considerations and the ionic radii of constituent elements, usually Shannon's ionic radii calibrated for oxides.³³ While this model and subsequent refinements³⁴ have worked well for oxide and halide perovskites, its accuracy declines when applied to non-oxide chalcogenides.^1,35,36

Recent efforts have sought to improve tolerance factor models for chalcogenide perovskites. In 2022, Jess et al. introduced a modified tolerance factor, recognizing that the significantly lower electronegativity of sulfur compared to oxygen results in greater covalent bonding.³⁵ To account for this, their model introduced an electronegativity-weighted correction to the Goldschmidt tolerance factor:

where Δχ represents the electronegativity differences between the indicated sites, and r_A, r_B and r_X are the ionic radii for the A, B, and X sites, respectively. This electronegativity-modified tolerance factor successfully grouped 9 experimentally known chalcogenide perovskites within 0.94 < t* < 1.12 (BaZrS₃, BaHfS₃, SrZrS₃, SrHfS₃, CaZrS₃, CaHfS₃, EuZrS₃, EuHfS₃, and BaUS₃) with no known non-perovskites falling within this interval.

In 2024, Turnley et al. introduced another tolerance factor based approach,³⁶ noting a key limitation in previous models: the ionic radii data used in traditional tolerance factors (Shannon's dataset) was empirically derived from metal oxides and fluorides, assuming ionic bonding. However, the lower electronegativity of sulfides yields a tendency for stronger covalent interactions, making the oxide-calibrated radii less applicable to chalcogenide perovskites. Shannon recognized this issue and created a separate dataset of metal-sulfide-derived radii, although this dataset remains less extensive.³⁷ Turnley et al. used this sulfide-derived radii dataset (extrapolated where necessary) to establish a refined perovskite stability criterion. Their screening method involves three metrics. First, materials were screened using the octahedral factor (μ = r_B/r_X) to determine whether B and X-site ions were appropriately sized to form stable octahedra. Materials passing the octahedral factor test (0.414 < μ < 0.732) were then evaluated based on a modified Goldschmidt tolerance factor using sulfide-derived radii (t^S), constrained within 0.865 ≤ t^S ≤ 0.965 along with an electronegativity-based metric, χ_diff = 1/5(3χ_S − χ_A − χ_B), requiring χ_diff ≥ 1.025 for perovskite stability. In their study, Turnley et al. applied this multi-step screening method to 100 compounds from the Materials Project database of DFT calculations,³⁸ 24 of which are experimentally known chalcogenide perovskites. Among those, their tolerance factor model correctly identified 18 as lying within the proposed stability window, including f-electron and radioactive elements, as well as materials belonging to different valence families (e.g., III–III compounds). When restricting the comparison to the six experimentally reported chalcogenide perovskites within our scope, the Turnley model successfully places all six within the predicted perovskite stability window.

While tolerance factors are a valuable tool for assessing perovskite stability, they intrinsically focus only on polymorphic stability and do not probe the tendency for decomposition into competing compositions (hull stability).³⁹ Both polymorphic and hull stability at 0 K can be evaluated using DFT calculations, though they involve slightly different energetic comparisons. Polymorphic stability is determined by comparing total energies across different polymorphs at fixed chemical composition, with the lowest-energy structure defined as the ground state. The energy difference between a given polymorph and the ground state is denoted as ΔE_gs. Hull stability, assessed via the convex hull formalism, requires computing formation energies for all ground-state structures in a given chemical space (e.g., AS, BS₂, etc., for A–B–S in ABS₃). A material is considered hull-stable if its formation energy is negative and lower than any linear combination of competing phases in that chemical space. The decomposition energy (ΔE_d) quantifies the energy difference relative to the most stable phase-separated state, with stability requiring ΔE_d ≤ 0. Materials with ΔE_d > 0 are computed to be unstable with respect to decomposition but may still be synthetically accessible metastable phases if ΔE_d and ΔE_gs are not too large.⁴⁰ Such metastable phases remain of high interest. For example, SrZrS₃ is thermodynamically stable in the needle phase, but its perovskite polymorph has been synthesized under high-temperature conditions.^17,18 Likewise, BaZrSe₃ and alloyed variants have been successfully synthesized and retained in the perovskite structure^41,42 despite their thermodynamic instability.²⁹ In the Results and discussion section, we examine ΔE_gs and ΔE_d of previously synthesized metastable materials in the context of prior studies^40,43 that have proposed heuristic thresholds and statistical trends relating stability and synthesizability.

In this study, we used DFT calculations at the meta-GGA level of theory⁴⁴ to assess the polymorphic and hull stability of 81 ABS₃ chalcogenides and probed the two recently introduced tolerance factors in the context of our DFT-calculated thermodynamics. Our study highlights the sparsity of thermodynamically stable chalcogenide perovskites and proposes thermodynamic and chemical origins for their instability.

Methods

Computational details

All DFT calculations were performed using the Vienna Ab Initio Simulation Package^45,46 (VASP, version 6.4.1) and the projector augmented wave (PAW) method.^47,48 A plane-wave energy cutoff of 520 eV, and Γ-centered k-point grid with a minimum spacing between k-points of 0.22 Å⁻¹ (KSPACING in VASP), were used for all calculations. The convergence criteria were set as 10⁻⁶ eV for electronic optimization and 0.03 eV Å⁻¹ for ionic relaxation. To assess the thermodynamic stability of various ABS₃ compounds, the r²SCAN meta-generalized gradient approximation (meta-GGA) density functional was applied.⁴⁴ Although r²SCAN offers only modest improvements over GGA for decomposition energies, it significantly improves formation energy accuracy—often reducing errors by a factor of 1.5 to 3.⁴⁹ This makes it particularly well-suited for our polymorphic stability analysis, which depends on precise total energy differences. These improvements have been demonstrated across transition metal compounds, main group compounds, and chalcogenides.

The initial structures for the 81 ABS₃ compounds investigated in this study were generated by applying atomic substitutions to three prototype structures queried from the Materials Project database:³⁸ GdFeO₃ (orthorhombic perovskite), BaNiO₃ (hexagonal), and SrZrS₃ (needle-like), which are representative of stable polymorphs in their respective structure types. After substitution, all structures were optimized using the calculation settings mentioned above. Polymorphic stability was assessed by comparing DFT total energies at 0 K across the three polymorphs for each composition. Hull stability was evaluated by querying all available stable competing phases in each A–B–S phase diagram from the Materials Project database³⁸ and recomputing all formation energies using the aforementioned calculation settings. As our analysis is restricted to 0 K ground-state energetics, it does not capture phases that may become thermodynamically stable at elevated temperatures or pressures.

The LOBSTER package⁵⁰ was used for COHP⁵¹ analysis. Charge analysis was performed via the Bader method.^52,53 Structures and charge densities were analyzed using VESTA.⁵⁴ Pymatgen was used to prepare and analyze DFT and LOBSTER calculations.^55,56

For tolerance factor analysis, ionic radii were taken directly from the supplementary data of the respective studies (Jess et al., Turnley et al.) and used as reported^35,36 with the exception of Pb⁴⁺ and Si⁴⁺, whose sulfide-derived radii were not reported in the Turnley dataset. For Pb⁴⁺, the radius was extrapolated from available coordination numbers. For Si⁴⁺, the oxide-derived radius was used instead.

Experimental details

Reactions targeting the synthesis of CaTiS₃ and BaZrS₃ were carried out by the attempted boron-assisted sulfidation of the corresponding oxide perovskites, CaTiO₃ and BaZrO₃.^57,58 All manipulations were performed in air unless explicitly specified. The oxide precursors were prepared by mixing stoichiometric amounts of the alkaline earth carbonates, CaCO₃ (Sigma-Aldrich 99%) and BaCO₃ (J. T. baker 99%), with the corresponding transition metal oxides, ZrO₂ (Sigma-Aldrich 99%) and TiO₂ (Aldrich Chemical 99.9%), in an agate mortar with a small amount of isopropanol for 15 min and evaporated to dryness. The powders were then pelletized in a 0.5 in die to 2 tons of force, placed in an alumina crucible and heated in a tube furnace to 1100 °C at a ramp rate of 10 °C min⁻¹ and held at temperature in air for 24 h. For the sulfidation reaction, stoichiometric amounts of oxide perovskite precursor and sulfur (NOAH Technology Corporation 99.5%), with an excess 2.2 molar ratio of boron (Aldrich Chemistry 95%) targeting AMO₃ + 2.2B + 3S, were ground in an agate mortar for 15 min and loosely packed within an alumina crucible that was held in an extruded quartz ampule. These ampules (16 mm O.D.; 14 mm I.D.) were flame sealed under vacuum (∼15 mTorr) and placed into a muffle furnace. The reaction with CaTiO₃ initially showed significant reaction with the SiO₂ ampule. As such, the reaction was repeated in a stainless-steel tube with 3/8 in inner diameter (no Al₂O₃ crucible). The tube was crimpled closed and then welded using an MRF Arc Melt Furnace under ∼0.85 atm of argon. All samples were then heated at a ramp rate of 5 °C min⁻¹ to 300 °C and 600 °C with holding times of 5 h each, then finally to 1000 °C for a holding time of 36 h. The samples were let to furnace cool and opened in air for analysis.

Powder X-ray diffraction (PXRD) measurements were conducted using a Cu Kα X-ray source and a LynxEye XE-T position-sensitive detector on a Bruker D8 Discover diffractometer to verify phase purity. Before measurement, the samples were placed on a “zero diffraction” Si wafer and immobilized with petroleum jelly. Quantitative phase analysis employing the Rietveld method was performed using TOPAS v6.

Results and discussion

To better understand previous experimental and computational results, we considered a subset of the materials evaluated by Jess et al.³⁵ We selected 6 ABS₃ compounds predicted by this tolerance factor to be perovskite as well as 39 ABS₃ compounds that lie near the borders of perovskite stability with respect to this tolerance factor. An additional 36 ABS₃ compounds were computed to consider all A/B combinations spanned by these 45 compounds. For each of these 81 compounds, we performed DFT calculations in the distorted perovskite, needle, and hexagonal polymorphs using the r²SCAN meta-GGA functional.⁴⁴ In Fig. 2, we show the DFT-predicted thermodynamic stability of the analyzed compounds. These stabilities were assessed in two ways: ΔE_gs indicates the difference in energy between the perovskite structure and the corresponding ground-state polymorph; ΔE_d indicates the decomposition energy for the ground-state polymorph and is calculated by considering all stable competing phases available in the Materials Project database³⁸ (recomputed with r²SCAN in this work). This ΔE_d should be interpreted as a lower bound, as additional competing phases not currently present in the Materials Project database could further destabilize a given compound. For example, we did not systematically investigate other perovskite-related compounds (e.g., Ruddlesden–Popper) in our hull analysis unless these compounds were already present in Materials Project. All ΔE_gs, ΔE_d and calculated ground-state structures are listed in Table S1, along with the predicted decomposition reactions and number of stable compounds in each A–B–S phase diagram. An ABS₃ compound is thermodynamically stable in the perovskite structure if ΔE_gs and ΔE_d are both ≤0. The magnitude of thermodynamic instability for a perovskite-structured ABS₃ is then given by the sum of ΔE_gs and ΔE_d. Compounds that have been previously synthesized in any polymorph are highlighted in blue, and the shape of the marker indicates the DFT-calculated ground-state structure.

Fig. 2 Heatmap summarizing the thermodynamic stability of 81 selected ABS₃ compounds. Each square corresponds to a unique ABS₃ compound, with A-site elements on the y-axis and B-site elements on the x-axis. Each lower left triangle shows the polymorphic instability, ΔE_gs, for the perovskite structure (a value of 0 indicates that perovskite is the ground-state structure). Each upper right triangle indicates the decomposition energy, ΔE_d, relative to the A–B–S convex hull (a value of 0 indicates that the ground-state structure is thermodynamically stable with respect to decomposition into competing compounds). Marker shapes indicate the DFT-predicted ground-state polymorph, and blue-filled markers denote compounds that have been previously synthesized (in any structure). Markers outlined in white indicate compounds for which the experimentally reported structure does not match the DFT-predicted ground-state structure.

Our calculations show that all 24 of the previously synthesized ABS₃ materials (in any polymorph) have ΔE_d < 90 meV per atom with 20/24 having ΔE_d < 30 meV per atom. We should note that in some cases specialized synthetic approaches, such as high total pressure (≥2 GPa), were used to access the materials (as is the case with ASnS₃, where A = Ba, Pb, Sr).^14,59 Among the materials considered that have not yet been synthesized, seven have ΔE_d ≤ 90 meV per atom, including three with ΔE_d < 61 meV per atom (CaTiS₃, CaSnS₃ and SrVS₃). For context, a previous study analyzing over 29 000 materials in the Materials Project database found that 90% of synthesized metastable materials are within 67 meV per atom of the hull.⁴⁰ While these numbers provide a helpful reference for assessing metastability, it is important to emphasize that being below this threshold does not guarantee synthesizability. It is possible that there are no thermodynamic conditions under which a specific material is thermodynamically accessible, even if it has a weakly positive ΔE_d. These results highlight the sparsity of ABS₃ compounds that are thermodynamically stable or weakly unstable with respect to decomposition into competing phases in each A–B–S chemical space.

As for ΔE_gs, our calculations predict that 12 of 81 compounds have perovskite ground states (ΔE_gs = 0). This includes 4 of the 6 compounds that have been previously synthesized in the perovskite structure (BaHfS₃, BaZrS₃, CaHfS₃ and CaZrS₃). The two remaining experimentally known perovskites within our scope are SrZrS₃ and SrHfS₃, which have marginal polymorphic instability (ΔE_gs of 6 and 8 meV per atom, respectively). Importantly, many of these compounds with perovskite ground states exhibit hull instabilities (as discussed later). Of ABS₃ compounds that have been synthesized in non-perovskite polymorphs, the least unstable perovskite is SnVS₃ with ΔE_gs = 31 meV per atom. Of the 24 ABS₃ compounds that have been synthesized in any structure, 16 have been synthesized in the DFT-calculated ground state structure. Six of the remaining eight compounds (ABS₃ with A = Pb, Sn and B = V, Nb, Ti) have only been reported in the misfit layered structure, which is not included in our calculations. Among the remaining two, one is SrHfS₃, where the experimentally observed structure (perovskite) is calculated to be just 8 meV per atom above the calculated ground-state structure (needle). The other is SrTiS₃, which has been reported to form in a hexagonal structure or a related hexagonal misfit structure.^{14,20,60–62} However, our calculations predict the needle structure as the ground state, with the hexagonal structure significantly higher in energy (81 meV per atom above needle). Interestingly, several other computational studies have also identified the needle structure as the ground state,^12,28,29 a notable discrepancy between theoretical predictions and experimental observations. As with ΔE_d, it is possible that ABS₃ compounds with ΔE_gs > 0 could be synthesized in the perovskite structure. A previous study proposed using the energy of the amorphous phase as a physically meaningful upper bound for ΔE_gs,⁴³ but this relatively generous upper bound varies widely from compound to compound. In the absence of exotic synthetic approaches, the probability of accessing a perovskite-structured ABS₃ should decrease substantially for compounds with larger ΔE_gs, and it is notable that the largest ΔE_gs of any previously reported perovskite is only 8 meV per atom (SrHfS₃).

Assessing polymorphic and hull stability together, ABS₃ compounds where perovskites could plausibly be synthesized are indicated by black or nearly black squares in Fig. 2. Six of the 81 compounds we calculated have been previously synthesized in the perovskite structure (BaZrS₃, BaHfS₃, CaZrS₃, CaHfS₃, SrZrS₃, SrHfS₃). BaZrS₃ and BaHfS₃ are the only compounds in our study calculated to be ground-state perovskites (ΔE_gs = 0) and stable against competing phases (ΔE_d ≤ 0). CaZrS₃ and CaHfS₃ are also calculated to have perovskite ground states but are slightly above the convex hull (ΔE_d = 25 and 27 meV per atom, respectively). SrZrS₃ and SrHfS₃ are calculated to have needle ground states that lie on the convex hull (ΔE_gs = 6 and 8 meV per atom, respectively). Considering all 81 compounds in our study, two key observations emerge: (1) the needle structure is overwhelmingly preferred as the ground-state structure, accounting for 77% of the 81 materials analyzed, and (2) thermodynamically stable perovskites are sparse with only two of 81 compounds having ΔE_gs = 0 and ΔE_d ≤ 0.

Given the limited number of stable perovskites and the challenges in synthesizing those calculated to be unstable, it is natural to ask whether tolerance factors can reliably predict perovskite formability in chalcogenides beyond the previously synthesized materials used to develop these models. In Fig. 3a, we show a comparison between the Jess tolerance factor³⁵ (x-axis) and the DFT-calculated ΔE_gs (y-axis). The gray-shaded region marks the range where this tolerance factor predicts a stable perovskite (0.92 < t* < 1.10). For clarity, this figure shows a zoomed-in view of the data with the full range available in Fig. S1. This comparison shows that Jess tolerance factor predictions largely deviate from the DFT predictions for many hypothetical compounds. For example, SnGeS₃ strongly favors the needle structure (ΔE_gs = 131 meV per atom) but is predicted to be a stable perovskite by the tolerance factor (t* = 0.99).

Fig. 3 (a) DFT-predicted energy difference between the perovskite structure and the corresponding ground-state structure (ΔE_gs) plotted against the Jess tolerance factor (t*), where points at ΔE_gs = 0 indicate a perovskite ground state. The gray region denotes the t* range where this tolerance factor predicts perovskite stability. (b) The electronegativity difference, χ_diff, plotted against the Turnley tolerance (t^S). The color bar corresponds to the same DFT-calculated ΔE_gs. The gray region denotes the t^S range above a certain χ_diff where the Turnley tolerance factor predicts stable perovskites. In both plots, blue-outlined data points denote previously synthesized compounds (in any structure), and marker shapes indicate the DFT-predicted ground-state structure: diamonds for perovskite, circles for needle, and squares for hexagonal.

A similar comparison is shown in Fig. 3b for the Turnley tolerance factor,³⁶ highlighting only the 45 compounds we assessed that pass the octahedral factor filter (0.414 < μ < 0.732). As was done in ref. 36, each of these compounds are shown with respect to the sulfide-adjusted Goldschmidt tolerance factor t^S (x-axis) and the electronegativity difference, χ_diff = 1/5(3χ_S − χ_A − χ_B) (y-axis). Points are colored by their calculated ΔE_gs with lighter points being less stable in the perovskite structure. Turnley et al., identified a perovskite stability window in the range 0.865 < t^S < 0.965 and χ_diff > 1.025 (shaded gray in Fig. 3b). Five materials with DFT-predicted perovskite ground-states from our study fall within this window, along with four other compounds calculated to have non-perovskite ground states. There appears to be a relationship between electronegativity and energetic ordering, where compounds with a lower ΔE_gs tend to cluster at higher χ_diff values. There is no clear trend along the t^S axis, reinforcing that this modified Goldschmidt model also fails to capture a trend with energetic ordering. Calculated values for both tolerance factors and their predicted stable structures are provided in Table S1, along with the ionic radii, μ, and χ_diff values used in the calculations.

Together, these analyses highlight the need for a more comprehensive framework to accurately predict chalcogenide perovskite stability. In both cases, there is limited correlation between the tolerance factor and ΔE_gs as we would expect ΔE_gs to be minimal in the gray shaded region and increase as t* or t^S deviates from the perovskite-stable region. Furthermore, while both models perform well for previously synthesized materials, their boundaries were empirically defined based on these previously synthesized perovskites. Our results suggest that even electronegativity-adjusted tolerance factors often overpredict chalcogenide perovskite stability, emphasizing that the interactions in these systems are more nuanced than those captured by these models. Motivated by this, we aimed to uncover the factors that dictate polymorphic and hull (in)stability.

Analysis of polymorphic instabilities

To go beyond geometric considerations and better understand why chalcogenides favor non-perovskite motifs (e.g., edge-sharing octahedra), we turn to the role of covalency and chemical bonding—particularly how the lower electronegativity of sulfur compared to oxygen may contribute to destabilizing the corner-sharing network that is so common in oxides. Previous studies have shown that even in oxide perovskites, increased covalency leads to destabilization of the perovskite, which is further exacerbated by oxide ion polarization.⁶³ Here, crystal orbital Hamilton populations (COHP)⁵¹ were used as an analytical tool to observe trends in covalent interactions among ABS₃ polymorphs. We mainly compare the perovskite and needle structures due to the strong preference for the needle structure exhibited by the compounds studied in this work.

We illustrate in Fig. 4 how distinct covalent interactions emerge in these two structure types by presenting CaHfS₃ (which has a perovskite ground state) as a representative example. In Fig. 4a and c, we compare the COHP of CaHfS₃ in the perovskite (a) and needle structures (c). Both structures exhibit similar S–S antibonding interactions near the Fermi level, arising from the relatively large electron cloud of the sulfide anion, overlapping within and between octahedra in each structure. In contrast, the Hf–S bonding in the perovskite structure is markedly stronger with a large peak in –COHP > 0 (bonding interactions), whereas the needle structure exhibits effectively no Hf–S covalency. In Fig. 4b and d, we compare the real space charge densities for this same energy interval in each structure. Within the chosen 2D slices of the charge density, the perovskite structure exhibits a linked network of tilted corner-sharing octahedra with substantial covalent Hf–S bonding. In contrast, the edge-sharing octahedra of the needle structure have very little charge density lying between Hf and S ions, instead favoring high density on each S ion.

Fig. 4 COHP analysis and charge density visualization for CaHfS₃ in perovskite and needle structures. (a) and (c) COHP curves plotted with respect to the Fermi energy (E_F) for perovskite and needle structures, respectively. Purple curves represent Hf–S interactions; red curves represent S–S interactions. COHP values are normalized per formula unit. (b) and (d) Two-dimensional slices of the DFT-calculated electron density as viewed along [0 0 1] and [−1 1.9 −2.5] directions for the perovskite and needle structures, respectively. Contours delineate regions of equal density. The needle phase exhibits significantly weaker B–S bonding interactions and reduced charge density between B and S sites compared to the perovskite structure.

Analysis of our COHP results suggests that covalent interactions tend to preferentially stabilize the perovskite structure over the needle structure, which is surprising given the overwhelming stability of the needle structure. To assess whether there is a broader trend, we compare integrated COHP (ICOHP) of needle vs. perovskite across the ABS₃ materials within our scope. The bonding trends observed in CaHfS₃ generalize across the materials class. As shown in Fig. 5, covalent B–S interactions differ substantially between the two structure types. Compounds that adopt the perovskite ground state (and those with low ΔE_gs) typically exhibit strong B–S covalency within their BS₆ octahedral units. In contrast, compounds that crystallize in the needle structure often show little to no B–S covalency, and consistently less than their perovskite counterparts. For clarity, a zoomed-in version of Fig. 5 is provided in Fig. S2 to better distinguish individual scatter points. Due to the relatively large radius of sulfur (compared to oxygen), both structure types display considerable S–S overlap. As shown in Fig. S3, many compounds exhibit similar S–S antibonding interactions near the Fermi level in both the perovskite and needle structures.

Fig. 5 Parity plot comparing integrated COHP (ICOHP) values for B–S bonding interactions in the needle and perovskite structures, normalized per formula unit and integrated from 0 to −7 eV below the Fermi level. More negative values indicate more bonding interactions. Points are colored according to the difference in energy from the perovskite to the needle structures. Points outlined in blue correspond to previously synthesized materials. Diamond-shaped markers represent materials that have a perovskite ground state, while circles represent a needle ground state (according to our calculations). Because we compare needle and perovskite only, we have excluded materials with a hexagonal ground state. Here, we have only considered materials with differing A- and B-site atoms. The units for ICOHP are arbitrary.

This analysis reveals that the small number of perovskites that are lower in energy than their needle counterparts are stabilized by B–S covalent bonding interactions. Recalling that the needle structure is by far the most common ground state among ABS₃ compounds (77% of compounds analyzed in this work), our results rule out covalency as the origin of this preference for the needle structure. It is instead plausible that the needle structure offers a more favorable electrostatic arrangement. This notion is supported by previous work from Young and Rondinelli on halide perovskites (e.g., CsPbI₃), where electrostatic interactions were found to be the primary origin of stability in the needle structure.⁶⁴

Analysis of hull instabilities

While our COHP analysis sheds light on how covalent interactions influence polymorphic preference, ΔE_gs alone does not provide a complete picture of stability. Fig. 2 revisits this point, illustrating how some materials, despite being polymorphically stable, remain thermodynamically unstable due to their tendency to decompose into competing compounds in a given A–B–S chemical space. Much of the perovskite literature, particularly studies involving tolerance factors, tends to focus exclusively on polymorphic stability while neglecting stability with respect to competing compounds. This oversight often leads to false positives in theoretical predictions, where materials appear stable in the perovskite structure (ΔE_gs = 0) but are thermodynamically unstable due to highly positive decomposition energies (e.g., CaNbS₃ in Fig. 2).

To test the validity of some of these calculations, we performed experimental synthesis reactions of materials predicted to be stable and unstable. An effective method for the synthesis of these materials is to react mixtures of binary oxides or ternary oxides with boron and sulfur.⁵⁷ The formation of low-melting and separable B₂O₃ (via solution chemistry or via vapor transport) assists in facilitating both the kinetics and thermodynamics of the reaction, as has been shown for refractory actinide sulfides.⁶⁵ BaZrS₃, predicted to be stable, is easily synthesized by this method (Fig. S3a). CaTiS₃, predicted to lie above the convex hull (ΔE_d = 52 meV per atom), instead phase separates to CaS, TiS₃, and TiS₂; we also observe the formation of CaB₂O₄ (Fig. S3b). Our computed decomposition reaction predicts that CaTiS₃ should decompose into CaS and TiS₂, consistent with the appearance of these two impurity phases after synthesis. The additional formation of CaB₂O₄ leads to Ti excess relative to Ca and the additional formation of a TiS₃ byproduct, which is calculated to be a stable competing compound in the Ti–S phase diagram. These results emphasize that it is important to understand the origin of the thermodynamic instability of ternary chalcogenides, particularly considering the stability of their oxide analogues (e.g., CaTiO₃ vs. CaTiS₃).

Ternary oxides are more commonly observed than their sulfide counterparts. For example, >250 ABO₃ materials have been synthesized in the perovskite structure,³⁴ compared with ∼25 ABS₃ perovskites (including all possible A/B sites).¹ To quantify the hull instability of ABS₃ compounds (in any structure) compared with their analogous ABO₃ compounds, we queried Materials Project³⁸ for the 81 ABO₃ oxides generated from the 81 sulfides studied in this work. Among these, 51 were available in Materials Project, of which 31 were found to lie on the convex hull. In contrast, only 15 of the 81 sulfides from our calculations are hull-stable. Table S2 lists all available values of ΔE_d for this analysis. Even for those ABS₃ compounds that are thermodynamically stable (on the convex hull), their stability is marginal compared with their ABO₃ analogues. In Fig. 6, we illustrate how the thermodynamic driving force to form ternary sulfides is generally weaker than the driving force to form the analogous oxide by showing 0 K DFT reaction energies for the formation of six ABX₃ compounds from AX + BX₂ reactants. In all cases, there is a larger driving force for ABO₃ compared with ABS₃ formation. One might attribute this to the inherent stability of binary sulfides, but this argument alone is insufficient, as oxides also form highly stable binary compounds, usually with higher melting points than their sulfide analogues. This suggests that additional mechanisms must be at play in stabilizing ternary oxides compared with sulfides.

Fig. 6 DFT reaction energies for AX + BX₂ → ABX₃ reactions for 6 pairs of orthorhombic perovskites: BaHfX₃, BaZrX₃, SrZrX₃, CaHfX₃, CaSnX₃ and CaZrX₃ (where X = O or S). The vertical position of each bar indicates its reaction energy (ΔE_rxn), with more negative values indicating a larger driving force to form the ABX₃ compound.

One possible mechanism is the inductive effect, which has been explored in previous studies as a potential explanation for the stabilization of ternary compounds compared with their binary counterparts. The inductive effect can be understood as an electron pressure exerted when two electropositive cations are bonded to opposite sides of an anion in an A–X–B bonding environment.⁶⁶ If A is significantly more electropositive than B, it donates additional electron density to the anion (X), which in turn strengthens the B–X bond by making it more covalent. This effect has been documented in nitride systems (among others), such as CaNiN, where the introduction of a highly electropositive Ca cation stabilizes the compound relative to its unstable Ni₃N binary.⁶⁷

Ternary oxides likely benefit more from inductive stabilization than their sulfide counterparts due to the higher electronegativity of oxygen relative to sulfur. Previous studies have largely attributed the inductive effect to differences in cation electronegativity.^66,68,69 Extending this reasoning, one can infer that the strength of the inductive effect is also influenced by the electronegativity of the anion (X). We posit that an anion with lower electronegativity will be less effective at attracting electron density from the electropositive cation, thereby diminishing the stabilizing influence of the inductive effect. To test this hypothesis, in Table 1, we show the change in Bader charge for the A- and B-site cations (Δδ_A and Δδ_B) calculated as the difference between the cation charge in the ternary compound (ABX₃) and in its corresponding binary (AX or BX₂). We selected a subset of the compounds shown in Fig. 6 for this analysis. Separate values are presented for both sulfides and oxides. An indicator of the inductive effect is an increased charge (Δδ_A > 0) of the A-site cation going from binary (e.g., CaO) to ternary (e.g., CaHfO₃) and a decreased charge of the B-site cation (Δδ_B < 0) as the A–X bonds become more ionic and the B–X bonds become more covalent. Good examples to test this are CaHfX₃, CaZrX₃, and CaSnX₃. Both CaHfX₃ and CaZrX₃ have orthorhombic perovskite ground states as both oxides and sulfides, yet CaHfS₃ and CaZrS₃ exhibit slight hull instabilities (ΔE_d = 27 meV per atom and 25 meV per atom, respectively), while their oxide counterparts are hull-stable. CaSnS₃, in contrast, is both polymorphically and hull unstable in the perovskite structure (ΔE_gs = 44 meV per atom; ΔE_d = 61 meV per atom) whereas its oxide counterpart, CaSnO₃, is a stable perovskite.

Table 1 Bader charge differences and reaction energies of selected perovskite oxides and sulfides. The Bader charge difference for the A-site (Δδ_A) and B-site (Δδ_B) is calculated as the charge in the ternary chalcogenide minus that in the corresponding binary compound. For example, Δδ_A = δ(Ca in CaHfS₃) − δ(Ca in CaS); Δδ_B = δ(Hf in CaHfS₃) − δ(Hf in HfS₂). Also shown are the reaction energies (ΔE_rxn, in meV per atom) for the formation of each ternary from its binary precursors (e.g., CaS + HfS₂ → CaHfS₃)

	ΔδA	ΔδB	ΔErxn
CaS + HfS2 → CaHfS3	0.05	0.02	27
CaO + HfO2 → CaHfO3	0.08	−0.05	−92
CaS + ZrS2 → CaZrS3	0.05	0.02	25
CaO + ZrO2 → CaZrO3	0.08	−0.03	−63
CaS + SnS2 → CaSnS3	0.05	0.00	104
CaO + SnO2 → CaSnO3	0.09	−0.05	−91

---

The data in Table 1 reveals consistent trends in charge redistribution when going from binary to ternary compounds. In all systems, the A-site cation shows a larger increase in Bader charge when going from the binary to the ternary in oxides than in sulfides, indicating more electropositive behavior and stronger electron donation in oxides. Meanwhile, the B-site cation in oxides shows a clear decrease in charge from the binary to the ternary, consistent with increased covalency in the B–O bond due to the inductive effect. In contrast, sulfides show a smaller change or even an increase in B-site charge when forming the ternary. This contrasting behavior supports the notion of a weaker inductive effect in sulfides than oxides, potentially contributing to their differing thermodynamic stabilities and the relative sparsity of ABS₃ compared with ABO₃ compounds.

Implications for selenide and telluride perovskites

Although the broader class of chalcogenide perovskites includes selenides and tellurides, this study focuses on sulfide compounds due to their greater potential for optoelectronic applications. While a detailed analysis of selenides and tellurides is beyond the scope of this work, trends observed in our sulfide study allow us to speculate on their behavior. From a geometric perspective, the larger ionic radii of Se (1.98 Å) and Te (2.21 Å) compared to S (1.84 Å) decreases the number of A/B cation combinations that fit the corner-sharing perovskite geometry. Of the 81 A/B combinations analyzed, 17 ABS₃ compounds fall within the perovskite-stable region of the Jess tolerance factor whereas only 6 ABSe₃ and 5 ABTe₃ materials fall within that region. These selenide and telluride counts are based on a slightly narrower Jess tolerance factor range (1.01–1.12),³⁵ which was defined using known selenide perovskites. In the absence of known tellurides, the same range was applied for ABSe₃ and ABTe₃ compounds.

According to the Turnley tolerance factor, initial filtering using the μ criterion allows 45 sulfides, 27 selenides, and 0 tellurides to pass, implying a substantially smaller pool of selenides and tellurides capable of forming BSe₆ or BTe₆ octahedra. Electronegativity trends further support the sparsity of telluride perovskites. In Fig. 3b, we observe that instability with respect to other polymorphs (ΔE_gs) grows with decreasing χ_diff. Se is only slightly less electronegative than S (2.55 vs. 2.58), but Te is significantly less (2.10), placing all telluride compositions below the χ_diff threshold of 1.025 defined by the Turnley tolerance factor. As a result, no ABTe₃ compounds and only 6 ABSe₃ compounds satisfy the Turnley conditions for perovskite stability, compared to 9 ABS₃ compounds.

In our polymorphic instability analysis, we found that increased covalency in B–X bonding enhances the energetic stability of the perovskite relative to the needle structure. Because selenides have similar electronegativities to sulfides, they likely exhibit comparable covalent stabilization. Tellurides, on the other hand, may experience stronger B–X bonding interactions, which could theoretically favor perovskite formation by reducing competition with the needle structure. While increased covalency might promote polymorphic stability, the larger ionic radius of Te makes it more difficult to form stable BX₆ octahedra, as discussed previously.

In terms of thermodynamic stability with respect to phase separation, sulfide phase diagrams are the most populated with 196 stable competing compounds in the 81 A–B–S chemical spaces. There are 170 and 148 stable competing compounds in the corresponding A–B–Se and A–B–Te chemical spaces, respectively. This may reflect either weaker competition for ternary compound formation or indicate a relative lack of exploration within the Materials Project database for these less common anions. Moreover, as we discussed in our hull stability analysis, a decrease in electronegativity leads to diminished inductive stabilization. This could help explain the lack of experimental synthesis reports of ABTe₃ compounds. In total, our analysis suggests that ABSe₃ perovskites face comparable stability challenges as ABS₃ perovskites, while ABTe₃ perovskites are significantly less likely to form.

Conclusion

This study probes the thermodynamic stability of chalcogenide perovskites and reveals that very few are thermodynamically stable. By assessing these instabilities with respect to polymorphism and decomposition into competing compounds, we identify chemical and thermodynamic factors governing the scarcity of stable chalcogenide perovskites. In terms of polymorphism, the needle structure was calculated to be the ground-state polymorph for 77% of the 81 compounds analyzed in this study. Of the few compounds with perovskite ground states, covalent B–S bonding was found to be a significant driver for stability. In terms of stability with respect to decomposition, we propose that a weak inductive effect for ternary sulfide formation leads to weaker thermodynamic driving force for ABS₃ formation compared to analogous ABO₃ compounds. Our findings demonstrate that recently proposed tolerance factors adapted for chalcogenide perovskites tend to overpredict perovskite stability as we identify several hypothetical ABS₃ that are unstable in the perovskite structure yet lie within the bounds of these tolerance factors where perovskites are expected. This underscores that accurate prediction of chalcogenide perovskite stability requires consideration of multiple factors beyond geometry. While few ABS₃ perovskites are calculated to be thermodynamically stable, several hypothetical materials are computed to be weakly unstable and therefore plausibly accessible under carefully chosen synthetic conditions. In total, this work leads to new understanding of the thermodynamics and crystal chemistry of this compelling class of materials.

Conflicts of interest

There are no conflicts to declare.

Data availability

All data presented in this work is available in the SI tables as CSV files.

Tabulated information pertaining to the thermodynamic analysis, tolerance factor analysis, and previous reports of synthesis for the studied compounds as well as supplementary figures supporting the analyses of chemical bonding and phase identification following experimental synthesis within this work. See DOI: https://doi.org/10.1039/d5tc02282g

Acknowledgements

This work was supported by the National Science Foundation Division for Materials Research Award No. 2433203. The authors acknowledge the Minnesota Supercomputing Institute (MSI) at the University of Minnesota for providing resources that contributed to the research results reported herein. The authors also acknowledge Lauren Borgia and Yi-Ting Cheng for helpful discussions during the development of this work.

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