Mechanistic insight of KBiQ₂ (Q = S, Se) using panoramic synthesis towards synthesis-by-design

Abstract

Solid-state synthesis has historically focused on reactants and end products; however, knowledge of reaction pathways, intermediate phases and their formation may provide mechanistic insight of solid-state reactions. With an increased understanding of reaction progressions, design principles can be deduced, affording more predictive power in materials synthesis. In pursuit of this goal, in situ powder X-ray diffraction is employed to observe crystalline phase evolution over the course of the reaction, thereby constructing a “panoramic” view of the reaction from beginning to end. We conducted in situ diffraction studies in the K–Bi–Q (Q = S, Se) system to understand the formation of phases occurring in this system in the course of their reactions. Powder mixtures of K₂Q to Bi₂Q₃ in 1 : 1 and 1.5 : 1 ratios were heated to 800 °C or 650 °C, while simultaneously collecting diffraction data. Three new phases, K₃BiS₃, β-KBiS₂, and β-KBiSe₂, were discovered. Panoramic synthesis showed that K₃BiQ₃ serves an important mechanistic role as a structural intermediate in both chalcogen systems (Q = S, Se) in the path to form the KBiQ₂ structure. Thermal analysis and calculations at the density functional theory (DFT) level show that the cation-ordered β-KBiQ₂ polymorphs are the thermodynamically stable phase in this compositional space, while Pair Distribution Function (PDF) analysis shows that all α-KBiQ₂ structures have local disorder due to stereochemically active lone pair expression of the bismuth atoms. The formation of the β-KBiQ₂ structures, both of which crystallize in the α-NaFeO₂ structure type, show a boundary where the structure can be disordered or ordered with regards to the alkali metal and bismuth. A cation radius tolerance for six-coordinate cation site sharing of ∼ 1.3 is proposed. The mechanistic insight the panoramic synthesis technique provides in the K–Bi–Q system is progress towards the overarching goal of synthesis-by-design.

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Introduction

The discovery of new materials has led to revolutions in many scientific and technological fields, such as magnetism,¹ radiation detection,² optics,^3–6 superconductivity,^7,8 lithium-ion batteries,^9,10 thermoelectrics,¹¹ and phase-change memory.¹² Traditionally, materials discovery of inorganic structures involves exploratory synthesis to identify compounds using heuristically determined reaction conditions, where the compounds are isolated and identified upon completion of the reaction. Within this paradigm, it is unclear whether the desired products form upon heating, cooling, or while dwelling at maximum temperature, or if metastable, short-lived products appear and re-dissolve during the reaction. In situ diffraction techniques are a powerful tool that enable the observation of transient phases during the reaction in efforts towards materials discovery. Moreover, observing these transient phases leads to mechanistic understanding of the reaction; thereby advancing efforts to more rationally design or discover solid-state materials.^13–16 In particular, rational design is crucial to the pursuit of structural complexity for many fields, such as in thermoelectrics to achieve materials with lower thermal conductivity¹⁷ and in nonlinear optics for large second harmonic generation.⁵ To fully realize the potential for rational synthetic design, a comprehensive long-term approach to analyze solid-state reactions at all stages of the reaction is required.

A powerful tool that provides full awareness of phase identity and phase evolution during a reaction is in situ powder X-ray diffraction. This technique enables the direct observation of all crystalline phases, including intermediate ones, during a solid-state synthesis.^18,19 This “panoramic” view of the phase space explored during the reaction enables an understanding of the reaction pathway required to form the final products. To this end, in situ powder X-ray diffraction promotes the discovery of new materials and increased mechanistic understanding of solid-state chemistry.

However, simply conducting in situ powder X-ray diffraction on systems in isolation is not sufficient to achieve the generalized understanding of solid-state reaction dynamics necessary to maximize the potential for discovery. Comprehensive studies of multiple closely related compositions and systems are needed to paint a full picture of reactions that can then be extended to new systems. Therefore, the aggregation of mechanistic data describing material formation and progression of intermediate phases is the first step to determining reaction pathways for rational synthesis.^{13,14,20–24} This technique must therefore be applied to a broad, diverse set of systems and phases spaces. Once these reaction pathways of these systems are cataloged and codified, they can be data mined for design principles among similar reaction types.¹³ With reaction types and assembly rules, predictive power of material synthesis can be achieved, leading to intentional synthesis of targeted materials with chosen, desired properties.²⁵

In situ powder X-ray diffraction has been utilized for various interests such as materials discovery^18,19,26 and mechanism elucidation,^25,27–33 as well as the investigation of metathesis reactions,^27,34 self-propagating high temperature syntheses,^14,25,28 phase change,^14,35 decomposition,³⁶ and crystallization.^14,25,28 This approach has shown success in discovering new materials, even in systems that were previously investigated.^18,19 Furthermore, it has been shown that, by using mechanistic insight from in situ studies, the reaction pathway can be influenced.³⁰ Studies focused on building a database of reaction formation and mechanistic understanding for a specific reaction type can be combined in aggregate towards the goal of rational synthesis.

A promising phase space to study is in the A–Pn–Q systems, where A is an alkali metal, Pn is a pnictogen such as bismuth or antimony, and Q is a chalcogenide,^37,38 as many are reported structures and have been of interest for their thermoelectric^39–45 and optical^5,35,46,47 properties. Moreover, a number of the known compounds are isostructural disordered analogues, such as KBiS_2, NaBiS₂, NaBiSe₂, KBiSe₂, and NaSbS₂, while ordered phases contain similar structural motifs.^48–50 Investigation of this family of structures expands the library of known materials, reaction pathways, and understanding of structure–property relationships. Reaction pathways can then be compared within this phase space to understand their design principles. The combination of experimental observation and computational data-mining can accelerate materials discovery and expedite the development of the mechanistic understanding required to achieve rational synthesis of materials.^23,51

Herein, we have applied this in situ diffraction technique using both in-house and synchrotron sources to the K–Bi–S and K–Bi–Se system, by monitoring the reaction between the respective binaries, K₂Q and Bi₂Q₃ (Q = S, Se). Powder mixtures were heated to 800 °C or 650 °C, while diffraction data was simultaneously collected. From this investigation, we found that the two systems share an isostructural intermediate phase, K₃BiQ₃. The presence of this intermediate phase in both reactions suggests K₃BiQ₃ is mechanistically important for KBiQ₂ formation from the binary starting materials. Two additional compounds were discovered: β-KBiS₂ and β-KBiSe₂. These compounds crystallize in the α-NaFeO₂ structure type and are cation-ordered polymorphs of the known rocksalt ternaries. The ability of KBiQ₂ to form both a disordered and ordered phase, while NaBiQ₂ and RbBiQ₂ crystallize in the rocksalt and α-NaFeO₂ type structures, respectively, suggests that there is a cation radius tolerance of 1.33 or greater for high symmetry rocksalt formation. This work is one of the pioneer studies in proposing a tolerance factor for a chalcogenide system.^52–54 We have also performed density functional theory (DFT) calculations to show that the cation-ordered β-KBiQ₂ polymorphs are thermodynamically stable and demonstrated that DFT can be used as a complimentary tool in the panoramic studies to determine the phase stabilities.

Experimental section

Reagents

Chemicals in this work were used as-obtained: bismuth metal (99.99%, American Elements), sulfur, (sublimed, 99.99%, Spectrum), selenium pellets (99.99%, American Elements), potassium metal (99.5%, Aldrich), glassy carbon (99.9%, Aldrich). K₂S and K₂Se were synthesized using stoichiometric amounts of the elements in liquid ammonia as described elsewhere.⁴⁷

Synthesis

Bismuth sulfide, Bi₂S₃. A 1.1226 g amount (0.035 mol) of elemental sulfur was combined with 4.8774 g (0.023 mol) of bismuth shots in a 9 mm (O.D.) fused silica tube. The tube was flame sealed under vacuum (10⁻³ mbar). The sample was heated to 500 °C in 24 hours and soaked for 24 hours before furnace cooling.

Bismuth selenide, Bi₂Se₃. A 2.7104 g amount (0.028 mol) of elemental selenium was combined with 3.8296 g (0.018 mol) of bismuth shots in a 9 mm (O.D.) fused silica tube. The tube was flame sealed under vacuum (10⁻³ mbar). The sample was heated to 850 °C in 8.5 hours and soaked for 12 hours, then cooled to 450 °C in 8 h. The sample was annealed at this temperature for 48 hours to minimize selenium vacancies. The sample was then furnace cooled. The sample was ball milled in a Retsch Mixer Mill MM 200 at 20 rps with 20 balls for an accumulative 2.5 hours.

β-KBiS₂. K₂S (0.071 g, 0.64 mmol) and Bi₂S₃ (0.329 g, 0.64 mmol) were mixed in a mortar and pestle and transferred in a N₂ glovebox to a carbon coated 9 mm (O.D.) fused silica tube, which was flame sealed under vacuum (10⁻³ mbar). The sample was heated to 875 °C in 12 h and soaked for 3 h, then cooled to 775 °C in 16 hours before furnace cooling. Powder X-ray diffraction (PXRD) shows that the sample is bulk phase pure.

K₃BiS₃. K₂S (0.157 g, 1.4 mmol) and Bi₂S₃ (0.243 g, 0.47 mmol) were mixed in a mortar and pestle in a N₂ glovebox. The mixture was transferred to a carbon coated 9 mm (O.D.) fused silica tube and flame sealed under vacuum (10⁻³ mbar). The sample was heated to 450 °C in 6 h and soaked for 192 h before furnace cooling. The sample was rehomogenized using a mortar and pestle in an N₂ glovebox, then subsequently annealed at 450 °C for 48 hours; Rietveld refinement of the sample reveals a 7.7(2) wt% impurity of α-KBiS₂.

β-KBiSe₂. K₂Se (0.077 g, 0.49 mmol) and Bi₂Se₃ (0.0.323 g, 0.49 mmol) were added to a carbon coated 9 mm fused silica tube, which was flame sealed under vacuum (10⁻³ mbar). The sample was heated to 800 °C in 9 h and soaked for 30 minutes before cooling to 575 °C in 72 h. The sample was then cooled to 475 °C in 96 h before furnace cooling. Phase purity confirmed using PXRD.

Physical measurements

In Situ powder X-ray diffraction. A STOE STADI MP high-resolution diffractometer with an oven attachment (STOE HT) were used to collect temperature-dependent data. The diffractometer was equipped with an asymmetric curved germanium monochromator and one-dimensional silicon strip detector (MYTHEN2 1K from DECTRIS). The starting materials were sieved (<45 μm), mixed, and diluted using glassy carbon. Dilutions were required to mitigate sample attenuation of the X-ray beam and, therefore, optimize the signal-to-noise ratio of the diffraction measurement. Samples were then packed into 0.3 mm diameter quartz capillaries, which were flame sealed under vacuum (ca. 3.0 × 10⁻³ mbar). All capillaries were carbon coated to prevent glass attack. Ground fused silica was used as a spacer at the tip of the capillary, where the capillary was re-sealed and bare of carbon coating. Diffraction data was collected every 15 °C using pure-Cu Kα1 radiation (1.54056 Å) operated at 40 kV and 40 mA for the sulfide system and pure-Mo Kα1 radiation (0.70930 Å) operated at 50 kV and 40 mA for the selenide system. Samples were heated to 800 °C or 650 °C with a heating rate of 4°C min⁻¹ and a dwell time at max temperature of 90 minutes. Temperature stability is typically 0.1 °C. Samples were spun during collection. The instrument was calibrated against a NIST Silicon standard (640d) prior the measurement.

Ex situ powder X-ray diffraction. X-ray powder diffraction patterns were collected on a Rigaku Miniflex600 diffractometer with Cu Kα1 radiation (1.5406 Å) operating at 40 kV and 15 mA with a high-speed silicon strip detector. A zero-background silicon sample holder with 0.2 mm × 4 mm well was used. Visualization of the crystal structures was done with Vesta software.⁵⁵

PXRD data used for K₃BiS₃ Rietveld refinement was collected at room temperature on a STOE-STADI-P powder diffractometer equipped with an asymmetric curved germanium monochromator (pure-CuKα1 radiation, λ = 1.54056 Å) and one-dimensional silicon strip detector (MYTHEN2 1K from DECTRIS). The line focused Cu X-ray tube was operated at 40 kV and 40 mA. Powder was packed in a 3 mm metallic mask and sandwiched between two polyimide layers of tape. Intensity data from 3 to 124° 2θ were collected over a period of 45 min. The instrument was calibrated against a NIST Silicon standard (640d) prior to the measurement.

High resolution synchrotron powder diffraction data were collected using beamline 11-BM at the Advanced Photon Source (APS), Argonne National Laboratory using an average wavelength of 0.4577 Å. Discrete detectors covering an angular range from 2 to 24° 2θ are scanned over a 6° 2θ range, with data points collected every 0.001° 2θ and scan speed of 0.01° s⁻¹.

Pair distribution function measurements and analysis. Kapton capillaries (0.8 mm diameter) were filled with sieved powder samples (<45 μm) inside a glovebox. X-ray scattering data for α-KBiQ₂ (Q = S, Se) samples were collected at Sector 11-ID-B of the Advanced Photon Source at Argonne National Laboratory using an incident wavelength of 0.143 Å, up to a Q_max of 36.4 Å⁻¹. The sample-to-detector distance was approximately 175 mm for the sulfide, resulting in a Q_max of 36.4 Å⁻¹. The sample-to-detector distance for the selenide was approximately 250 mm, resulting in a Q_max of 29.1 Å⁻¹.

GSAS-II was used to calibrate the sample-to-detector distance, detector alignment, and instrumental resolution parameters using a NIST CeO₂ standard and to integrate the data (ref. 56). The empty Kapton capillary signal was subtracted from each scattering data pattern using PDFgetX3 ⁵⁷ prior to obtaining the Pair Distribution Functions (PDFs). PDFgui⁵⁸ was used to fit PDFs in the range 2 Å < r < 30 Å. Subgroups were generated using the Bilbao Crystallographic Server;^59–61 space groups investigated included R3̄m, Pm3̄m, C2/m, P3̄m1, I4/mmm, Pn3̄m, P3₂21.

Optical measurements. Optical diffuse reflectance measurements were made at room temperature with a Shimadzu UV-3600 PC double-beam, double-monochromator spectrophotometer. The measurement of diffuse reflectivity can be used to obtain values for the band gap from a powder sample at room temperature. BaSO₄ powder was used as a reference. The reflectance data was converted to absorbance using the Kubelka–Munk equation: α/S = (1 − R)2/2R. R is the reflectance at a given wavelength, α is the absorption coefficient, and S is the scattering coefficient. Valence band maximum energy (VBM) was measured by photoemission spectroscopy in air on a Riken-Keiki PESA AC-2. Samples are scanned under ambient conditions using UV light (4.20–6.20 eV). The number of photoelectrons generated at each excitation energy is measured and the VBM is the onset of the PESA spectra, as photoelectrons are only generated when the photon energy is higher than the VBM energy. Conduction band minimum energies are found by subtracting the optical band gap from the VBM energy.

Differential thermal analysis. Differential thermal analysis (DTA) was performed using a Netzsch STA 449 F3 Jupiter simultaneous thermal analysis (STA) instrument. Samples were loaded into carbon-coated fused silica tube and flame-sealed under vacuum (ca. 3 × 10⁻³ mbar). Samples were heated to 900 °C, cooled to 200 °C, then heated once more to 900 °C, before cooling to room temperature. The rate for cooling and heating was set to 7.5 °C min⁻¹.

Computational methods

All DFT calculations are performed using the projector augmented wave (PAW) method,^62,63 as implemented in the Vienna Ab initio Simulation Package (VASP).^64,65 The Perdew–Burke–Ernzerhof (PBE) exchange–correlation functional⁶⁶ and a plane wave basis set with a cutoff energy of 520 eV were used. A Γ-centered k-mesh with ≈8000 k-points per reciprocal atom (KPPRA) was used to sample the Brillouin zone. The Open Quantum Material Database (OQMD)^51,67 was used for convex hull construction. To calculate the formation energy of the cation-disordered KBiQ₂Fm3̄m structures, a 32-atom special quasi-random structure (SQS)⁶⁸ was generated by using the Monte Carlo algorithm (mcsqs) as implemented in the Alloy Theoretic Automated Toolkit (ATAT).^69,70 The cluster correlations used to define the SQS were specified using a distance-based cutoff of all 2-, 3-, and 4-body clusters with a maximum distance of 5–6 Å. To calculate the effective band structure of the K/Bi disordered cubic phase (Fm3̄m), a 2 × 2 × 2 supercell was generated based on the perfect rocksalt structure with bismuth and potassium alternatively stacking along the [111] direction. The potassium and bismuth atoms are then randomly distributed in the supercell to simulate the cation-disordered KBiQ₂Fm3̄m structures. Effective band structures were recovered from the supercell calculations using the band unfolding technique proposed by Medeiros et al.^71,72

Results and discussion

Panoramic synthesis KBiS₂

The synthesis of KBiS₂ has been previously reported using potassium carbonate, elemental bismuth, and elemental sulfur.^48,73 KBiS₂ crystallizes in the simple rocksalt structure,^48,73 where potassium and bismuth have equivalent occupancy in the sodium site and sulfur is in the chloride site. Moreover, the K–Bi–S phase space is of interest for thermoelectric materials⁷⁴ and photovoltaics,⁷⁵ in efforts towards rational design within this phase space, we began with this simple structure.

To monitor the reaction progression in the K–Bi–S system, panoramic syntheses were conducted using a commercial STOE high-temperature furnace (HT) attachment for a STOE STADI-MP powder X-ray diffractometer. In previously reported syntheses, an excess of potassium starting material was used to form KBiS₂, therefore a ratio of 1.5 K₂S : 1 Bi₂S₃ is used.⁷³ The reaction has a maximum temperature of 800 °C. The overlaid diffraction patterns from the panoramic synthesis are shown in Fig. 1A. Phases are identified primarily through pattern matching. A schematic of the reaction pathway with relative mole fractions of the phases is shown in Fig. 1B. The relative mole fraction of the starting material amorphous K₂S was determined according to the experimental loading of the material. It is useful to regard the reaction between K₂S and Bi₂S₃ as a Lewis acid–base reaction in which K₂S is the base and Bi₂S₃ is the acid. In this context, basic activity increases with the K₂S fraction and acidic activity increases with the Bi₂S₃. The end crystalline products are KBiS₂ and cristobalite, a high temperature polymorph of SiO₂. The ground amorphous silica, which was used as a spacer to protect the glass tip of the capillary, crystallized after reaching temperatures above 700 °C. To avoid this, all subsequent panoramic syntheses in this system have a maximum temperature of 650 °C. An intermediate phase is observed during the reaction, appearing at 270 °C and disappearing at 510 °C as the product, KBiS₂, emerges. The intermediate phase was isolated ex situ as described in the Experimental section and determined to be K₃BiS₃via Rietveld refinement (Fig. S1 and Tables S1–S4†).

Fig. 1 (A) Heat map of 1.5 K₂S + 1 Bi₂S₃. Asterisks indicate observed peaks from cristobalite formation. (B) Reaction map of 1.5 K₂S + 1 Bi₂S₃. Relative compositions are shown; dashed lines represent an amorphous phase.

The intermediate phase, K₃BiS₃, shown in Fig. 2, crystallizes in the cubic Na₃AsS₃ structure type (space group P2₁3; a = b = c = 9.4425(1) Å) and is isostructural to K₃SbS₃ and Na₃SbS₃.⁷⁶ The formation of an intermediate phase of this composition is surprising because it implies a reaction of K₂S and Bi₂S₃ in a 3 : 1 ratio, which is a more basic condition than what is being used in this reaction, where the ratio is 3 : 2. This suggests that K₃BiS₃ is a kinetic product.

Fig. 2 (A) Ball and stick representation of K₃BiS₃ (left) and an illustration of the three different octahedra of KS₆ and all BiS₃ units (right). (B) Isolated representations of the KS₆ distorted octahedra.

K₃BiS₃ has two structural motifs: trigonal-pyramidal BiS₃ units and distorted octahedral KS₆ units. The isolated pyramidal molecule of [BiS₃]³⁻ makes the structure of K₃BiS₃ essentially molecular. We synthesized this intermediate ex situ using the conditions identified by the panoramic synthesis experiments and refined its structure using Rietveld refinement (Fig. S1†). The trigonal-pyramidal units of BiS₃ have bond lengths and bond angles of 2.593(7) Å and 99.2(3)°, respectively. The distorted KS₆ octahedra exhibit three different coordination environments, shown in different colors in Fig. 2B. The distortion of the octahedra give rise to a range of bond lengths between 3.11(2) and 3.53(1) Å. The distorted octahedra of K1, shown in blue, is capped by a bismuth and shares a corner with the three other BiS₃ units in the structure. K2, in turquoise, is edge-sharing with three BiS₃ units, while K3, shown in purple, is exclusively corner-sharing with the BiS₃ polyhedra.

By comparing the structures in the 1.5 K₂S + Bi₂S₃ reaction, shown in Fig. 3, the role of K₃BiS₃ as an intermediate phase in the formation of KBiS₂ is examined. The coordination environment of potassium can be described as a gradual progression from tetrahedral to octahedral. In the starting material, the potassium cation begins in the K₂S anti-fluorite structure consisting of KS₄ tetrahedra with bond lengths of 3.191 Å,⁷⁷ and over the course of the reaction expands to the distorted KS₆ octahedra in K₃BiS₃. By the final product, potassium forms ideal KS₆ octahedra in KBiS₂. Of note, the K–S bond lengthens in K₃BiS₃, before shortening to 3.02 Å in the final Fm3̄m structure.⁷⁸ In contrast, the bismuth coordination sphere does not undergo a gradual progression. In fact, bismuth undergoes an unexpected decrease in coordination number from square pyramidal and distorted octahedral to trigonal pyramidal, before transitioning to octahedral in KBiS₂. In K₃BiS₃, bismuth stereochemically expresses its lone pair into a void created by the potassium cations, then no longer expresses its lone pair in the final product, KBiS₂. The bismuth precursor, Bi₂S₃, contains square-pyramidal BiS₅ units and distorted BiS₆ octahedra, as illustrated in Fig. 3. Three different Bi–S bond lengths are present in the BiS₅ units. The bond lengths in the basal plane are divided equally between 2.74(1) and 2.96(1) Å, while the apical Bi–S in BiS₅ is the shortest bond length in the structure at 2.56(1) Å.⁷⁷ The BiS₆ octahedra has two bond lengths at 2.67(1) Å, another two at 2.96(1) Å, as well as one at 3.05(1) and 2.69(1) Å. The Bi–S bond length shortens to 2.62(1) Å in K₃BiS₃, before increasing to 3.02 Å in KBiS₂. A concurrent shortening of the Bi–S bonds and lengthening of the K–S bonds in K₃BiS₃ is likely a result of the concomitant separation of the Bi₂S₃ chains and incorporation of the KS₄ units. Owing to its utility as a structural transition framework, K₃BiS₃ therefore can be considered a structural intermediate for the formation of KBiS₂.

Fig. 3 Reaction progression of the starting materials, K₂S and Bi₂S₃, to KBiS₂ through the intermediate phase K₃BiS₃. Unique cation coordination environment is shown for each structure. A bismuth atom in K₃BiS₃ is shown within the cavity formed by the KS₆ distorted octahedra.

To explore if the formation of rocksalt KBiS₂ persists, an increase in amount of Bi₂S₃ relative to K₂S was investigated. The overlaid diffraction patterns and reaction map of the 1 K₂S : 1 Bi₂S₃ panoramic synthesis are shown in Fig. 4A and B, respectively. As before, K₃BiS₃ forms first and appears between 240–250 °C, which is similar to the formation temperature of 270 °C in the K₂S-rich reactions featuring a ratio of 1.5 : 1. The continued formation of K₃BiS₃ before the crystallization of KBiS₂, despite a different final product, corroborates that this kinetic intermediate is mechanistically important for the formation of the rocksalt structure. The KBiS₂ rocksalt structure emerges at ca. 435 °C. The rocksalt phase persists through the maximum temperature and, on cooling, a third phase is observed starting between 465–480 °C. KBiS₂ persists until approximately 300 °C on cooling, where the third phase is the sole final product.

Fig. 4 (A) Overlaid diffraction patterns of 1 K₂S + 1 Bi₂S₃. (B) Reaction map of 1 K₂S + 1 Bi₂S₃. Nominal compositions are shown; dashed lines represent a noncrystalline phase. (C) Structure solution for β-KBiS₂. (D) Structural representation of the ordered rocksalt, β-KBiS₂.

The third and final phase in this reaction is a cation-ordered polymorph of KBiS₂ in the rhombohedral α-NaFeO₂ structure type (R3̄m, a = b = 4.12983(3) Å, c = 22.09484(17) Å). The structure of the KBiS₂ polymorph, hereafter named β-KBiS₂, was determined using Rietveld refinement on synchrotron PXRD data (Fig. 4C). β-KBiS₂ is an ordered rocksalt that is isostructural to RbBiS₂.⁷⁹ In this structure type, cation trigonal anti-prisms are connected through edge-sharing as in the disordered rocksalt structure (Fig. 4D).⁸⁰ The layers alternate between CdCl₂-type [BiS₂]⁻ layers and potassium cations. It is notable that the ordered R3̄m forms only when stoichiometric amounts of bismuth and potassium precursors are used; an excess of just 5% of K₂S yields the rocksalt structure (Fig. S3†). The original synthesis for the cation-disordered Fm3̄m KBiS₂ (now called α-KBiS₂) also uses an excess of the potassium precursor, K₂CO₃.⁷³ This suggests that an excess of potassium may have a role in destabilizing the lower symmetry structure.

Thermal analysis data of β-KBiS₂ shown in Fig. S5† and Density Functional Theory (DFT) calculations, described further below, corroborate the experimental evidence that cation-ordered β-KBiS₂ is the thermodynamically stable phase. The endothermic peak at ca. 480 °C upon heating and broad exothermic peak at ca. 405 °C upon cooling in the thermal analysis data indicate β-KBiS₂ undergoes a phase transition. The temperature of these peaks corresponds to the transition temperature for the α-KBiS₂ to β-KBiS₂ transition in Fig. 4A. This, in conjunction with the identity of the post-thermal analysis product as β-KBiS₂, confirms these signals result from phase transitions from a low to high temperature phase. Moreover, DFT calculations show that cation-ordered β-KBiS₂ sits on the convex hull and, thus, is the thermodynamically stable phase, while cation-disordered α-KBiS₂ sits at 75 meV per atom above the convex hull (i.e. above β-KBiS₂).

Panoramic synthesis of KBiSe₂

KBiSe₂ is reported to crystallize in the rocksalt structure and was hypothesized to proceed through a similar mechanism to α-KBiS₂ owing to their isostructural nature. The panoramic synthesis on 1 K₂Se + 1 Bi₂Se₃ was conducted and is illustrated in Fig. 5. Different colors are used to highlight different phases that form during the reaction, beginning with the starting materials in blue. At 195 °C, again the analogous material, K₃BiSe₃, forms (Fig. S6;†P2₁3; a = b = c = 9.773(2) Å).⁴⁹ The appearance of K₃BiSe₃ (indicated in green) as an intermediate phase also in the selenide analogue reaction supports that this molecular salt phase may be important mechanistically to the reaction path.

Fig. 5 (A) Overlaid diffraction pattern for K₂Se + Bi₂Se₃ panoramic synthesis. Asterisks indicate cristobalite (SiO₂) crystallization. (B) Reaction map of 1 K₂Se + 1 Bi₂Se₃. Nominal compositions are shown; dashed lines represent a noncrystalline phase.

The reaction selectivity of K₃BiQ₃ (Q = S, Se) directly proceeding the formation of the KBiQ₂ ternary is postulated to be structural in origin as opposed to energetic. In situ studies observing the phase formation found the composition of the first transient phase to be that which yielded the largest energy release⁸¹ or the fastest energy release.³⁰ The former does not apply for these chalcogenide systems as the formation of K₃BiQ₃ is less energetically favorable than the β-KBiQ₂ phases. Specifically, the formation energies of K₃BiS₃ and K₃BiSe₃ are calculated to be −48 meV per atom and −28 meV per atom below the convex hull, respectively, while those of β-KBiQ₂ are −68 meV per atom and −60 meV per atom below the hull for the sulfide and selenide, respectively. The alpha phases have positive formation energies relative to the convex hull according to DFT calculations and are, therefore, considered unstable. K₃BiQ₃ also does not provide the fastest energy release as diffusion must occur for this phase to form as compared to KBiQ₂. The precursor loadings in this work are 3 K₂S : 2 Bi₂S₃ (e.g. 1.5 : 1) and 1 K₂Q : 1 Bi₂Q₃ (Q= S, Se). The precursors were sieved and thoroughly mixed, such that the average contact between particles are expected to mirror that of the loading ratio. Correspondingly, no notably K₂Q-rich or poor areas are expected. The fastest energy release would therefore be the formation of KBiQ₂, however, K₃BiQ₃ reproducibly forms first in both chalcogenide system regardless of loading ratio. The formation of the K₃BiQ₃ structural intermediate therefore cannot be easily rationalized by current thermodynamic or kinetic theories. However, the formation of the potassium-rich intermediate, persistent even in relatively potassium-poor reactant loading, underscores that this compound is necessary for the formation of the KBiQ₂ ternary in these conditions.

The structural intermediate, K₃BiSe₃, shown in green in Fig. 5, persists until a new phase emerges at 285 °C, shown in purple. This new phase disappears at 555 °C before reappearing at 525 °C on cooling as the final product. This phase is isostructural to β-KBiS₂ and hereafter called β-KBiSe₂ (R3̄m, a = b = 4.26442 (1) Å, c = 23.0290(1) Å). When β-KBiSe₂ disappears, the known rocksalt structure-type α-KBiSe₂ begins to emerge. Upon cooling, β-KBiSe₂ reappears by 500 °C. In order to isolate α-KBiSe₂, the product was quenched at 650 °C and annealed. Attempts to synthesize α-KBiSe₂ using the binaries without quenching, even with an excess of K₂Se, were unsuccessful.

The overall reaction pathway in the selenide and sulfide systems is similar. Both systems proceed through a K₃BiQ₃ (Q = S, Se) intermediate. The coordination sphere of potassium increases from an anti-fluorite tetrahedron to distorted octahedra in K₃BiSe₃, as it does in the sulfide system. The coordination sphere of bismuth in the selenide system, however, decreases during the intermediate phase from an octahedron in Bi₂Se₃, shown in Fig. S7,† to trigonal pyramidal in the discrete ions in K₃BiSe₃, before returning to an octahedron in both the highly dense ordered and the disordered rocksalt type KBiSe₂. The formation temperature of the selenide ternaries is observed lower than that of the sulfide likely as a result of the lower melting points of the selenide binaries, the higher basicity of K₂Se than K₂S, and the more polarizable anionic framework of the selenium ternary, which facilitates ion mobility.⁸² Similar to the KBiS₂ system, thermal analysis (Fig. S8†) and DFT calculations show that the ordered, α-NaFeO₂ type polymorph of KBiSe₂ is the thermodynamically stable phase. Comparable to its sulfur analogue, cation-disordered α-KBiSe₂ sits 76 meV per atom above the ordered, stable β-KBiS₂ phase.

Investigation of local structure

Local off-centering from ns² lone pairs has been demonstrated previously for materials containing heavy main-group elements as for example the emphanisis effects in PbQ.^83–88 The structure of the titular KBiQ₂ compounds is therefore expected to be locally distorted as a result of stereochemical activity of the Bi³⁺ lone pair, while the global structure remains in the rocksalt or α-NaFeO₂ structure type for the alpha and beta phases, respectively. Pair distribution function (PDF) analysis confirms that the data are well fit using Fm3̄m for α-KBiS₂ and α-KBiSe₂ at long r ranges (Fig. 6A and E). However, the split of the first PDF peak is not well fit, as shown in Fig. 6C and G. The Bilbao Crystallographic Server^59–61 was used to generate possible subgroups of space group Fm3̄m assuming the Wyckoff positions 4a and 8c. Those subgroups were used to fit the PDF data. The short range (r < 6 Å) were better fit using the lower symmetry space group P2₁3 with partial site occupations in a cubic lattice cell as shown in Fig. 6B, D, F, and H. The split in the first coordination shell was better modeled by off-centering the atoms from the ideal rocksalt positions as illustrated in Fig. 6I and J.

Fig. 6 PDF fits for α-KBiS₂ using a (A) Fm3̄m model and (B) P2₁3 model. These models are zoomed in to the short range (r < 8 Å), highlighted in grey, for the Fm3̄m model (C) and for the P2₁3 model (D). In the PDF fits, red is the calculated PDF pattern, blue is the experimental data, and green is the difference pattern. PDF fits for α-KBiSe₂ using a (E) Fm3̄m model and (F) P2₁3 model; zoomed in region shown for the Fm3̄m model (G) and for the P2₁3 model (H). (I) Shows a 2 × 2 × 2 unit cell representation of KBiQ₂ and (J) highlights the off-centering of bismuth in the [BiS₆]³⁻ and [BiSe₆]³⁻ octahedra.

Optical properties of α- and β-KBiQ₂

As the ratio of K₂Q to Bi₂Q₃ becomes stoichiometric in the syntheses, the final product transitions from α-KBiQ₂ to β-KBiQ₂ with band gap energies of 0.84(2) eV and 1.09(2) eV, respectively, for the selenides (red, Fig. 7A) and 1.23(2) eV and 1.55(2) eV for the sulfides (blue, Fig. 7A). As the compounds undergo a decrease in symmetry moving from the disordered α-KBiQ₂ to the ordered β-KBiQ₂, we observe a subsequent increase in band gap. The dimensional reduction from the ordering of β-KBiQ₂ into layers of [BiQ₂]⁻ and K⁺ (Fig. 7B) yields a blue shift in the experimental band gap, as similarly observed for other semiconductors.^84,89Fig. 7C shows the energies of the valence band maxima for α-KBiQ₂ and β-KBiQ₂ measured using a Kelvin Probe (see Experimental section) as well as for four well-known semiconductors (Si, CdTe, methylammonium lead iodide (MAPbI₃), and TiO₂) for reference. The energies of the conduction band minima are included and were extrapolated using the valence band maxima energies and experimental or known band gaps.

Fig. 7 (A) Experimental band gaps for the known α-KBiQ₂ (Fm3̄m) and discovered β-KBiQ₂ phases (R3̄m). (B) Illustration of dimension reduction from the 3D covalent framework in the high temperature (HT) phase, α-KBiQ₂, to the layered 2D covalent framework in the low temperature (LT) phase, β-KBiQ₂. For both structures, only covalent bonds (e.g. Bi–Q bonds) are shown. The site distribution of bismuth and potassium in α-KBiQ₂ was randomized, while constrained to maintain equivalent amounts of the cations. The inset for α-KBiQ₂ tilts the indicated portion of the structure to highlight the 3D connectivity through the plane. (C) Depiction of valence band maxima, conduction band minima, and band gap energies for α-KBiQ₂, β-KBiQ₂, Si,^90,91 CdTe,⁹² MAPbI₃,⁹³ and TiO₂ (anatase).^94,95

Calculation of thermodynamic stability and electronic band structure

DFT calculations were performed to determine the thermodynamic stabilities and electronic structures of the α-KBiQ₂ and β-KBiQ₂ materials. When compiling the phase diagram for the K–Bi–S and K–Bi–Se phase spaces in Fig. 8A and B, all the related compounds stored in the Open Quantum Materials Database⁵¹ are considered and the grand canonical linear programming (GCLP) is used to construct the thermodynamic stability convex hull. The energy calculation for the structure of KBi_6.33S₁₀ and its partial occupancies³⁹ is computationally intensive; thus, the compound is included in the phase diagram assuming it has T = 0 K stability. Notably, all ternary phases discovered to date in these phase diagrams lay along the K₂Q–Bi₂Q₃ composition line, including the titular compounds. The intermediate phases, K₃BiS₃ and K₃BiSe₃, are also on the convex hull, and hence are considered thermodynamically stable ground state phases at T = 0 K. Our calculations show that the ordered phases, β-KBiS₂ and β-KBiSe₂, are on the convex hull, while the disordered α-KBiS₂ and α-KBiSe₂ structures are 75 and 76 meV per atom above the hull, respectively. This indicates the beta phases are the thermodynamically stable polymorphs, while the alpha phases are metastable. This agrees with the experimental observations as the beta phases are stable at room temperature, while the alpha phases form at high temperature.

Fig. 8 Phase diagram of (A) K–Bi–S and (B) K–Bi–Se phase space. The band structure and partial density of states (PDOS) of (C) β-KBiS₂ (R3̄m) and (D) β-KBiSe₂ (R3̄m).

The band structures of β-KBiS₂ and β-KBiSe₂ are shown in Fig. 8C and D, respectively. These two materials are indirect band gap semiconductors with calculated band gaps of 1.45 and 1.12 eV for β-KBiS₂ and β-KBiSe₂, respectively, using PBE functional method. These are consistent with the experimental band gaps of 1.55(2) eV and 1.09(2) eV as well as the expectation of the more electronegative sulfur analogue having a wider band gap than the selenium. For both β-KBiQ₂ phases, the conduction band minimum is at the Z point of the Brillouin zone, while the valence band maxima are in the region along Γ–L, Z–B and Γ–X directions. The conduction band maximum is primarily comprised of the bismuth 6p_z orbital with contribution from the sulfur 3s orbital (Fig. S10†). The bismuth 6s and sulfur 3p orbitals form the valence band maxima regions (Fig. S10†). The Γ–L and Γ–X direction have more contribution from the sulfur 3p_x orbital and the bismuth 6s, sulfur 3p_x and 3p_y orbitals, respectively. Overall, the conduction band is far more dispersive than the top of the valence band as shown in Fig. 8C and D, indicating that, in the beta phase, the charge transport of electrons would be far more efficient than that of holes. The partial density of states (PDOS) shows that the relatively flat valence band maxima is a consequence of the localized S/Se p orbitals that make up this band.

In order to understand the electronic structures of the alpha phases, the effects of bismuth off-centering and K/Bi disorder must be separated. The band gaps of the rocksalt-ordered KBiQ₂ as a function of coherent bismuth off-centering are shown in Fig. 9A. As the off-centering of bismuth from its ideal positions increases, the band gaps of α-KBiS₂ and α-KBiSe₂ are calculated to increase. The disordering of α-KBiQ₂ is simulated by randomly arranging potassium and bismuth atoms in supercells and the calculated band gaps using the same PBE functional method are 0.77 and 0.58 eV for α-KBiS₂ and α-KBiSe₂, respectively. The band structures of the alpha phases are shown in Fig. 9B and C. Both dimensional reduction⁸⁹ and stereochemical lone pair expression,⁸⁴ manifesting as cation ordering (Fig. 7B) and presenting as the off-centering of the bismuth atom, increase the band gap. Therefore, the band gaps of these compounds observed at finite temperature is a combination of these two effects (i.e. increasing the temperature yields a reduction in cation ordering and, on the other hand, an increase in off-centering). The comparative lower value for the experimental band gaps of disordered α-KBiQ₂ than those of the ordered β-KBiQ₂ materials indicates the influence of ordering has a larger effect than that of off-centering on the band gap, which is consistent with our calculations.

Fig. 9 (A) The DFT-calculated change in band gap as bismuth is increasingly off-centered in the K–Bi–K direction from its ideal position. As described in the Experimental section, the band gaps for α-KBiQ₂ (Q = S, Se) are calculated by stacking the rocksalt structure along the [111]-direction to form a supercell and subsequently distributing the cations randomly. While the site distribution is randomized, the distribution is constrained to maintain equivalent amounts of cations. The unfolded band structure of (B) α-KBiS₂ and (C) α-KBiSe_2, which is unfolded to the Brillouin zone of R3̄m structures. The color bar indicates the band intensity.

Comparison of A–Bi–Q

When looking at ABiQ₂ (A = alkali metal, Q = S, Se, Te) compounds, simultaneous trends of decreasing symmetry and dimensionality are observed as the size of the alkali metal increases relative to bismuth (Fig. 10). As we move across the sulfides in Fig. 10D, the [BiQ₂]¹⁻ framework is progressively broken up by the increasingly larger alkali metal illustrated in Fig. 10A–C. This is consistent with the so-called “counterion effect,” where, given an anionic framework, a countercation can induce a specific structural change as result of steric effects as a function of its size.^96–99 Previous work^5,97 has shown that the smaller the countercation, the higher the dimensionality and the higher coordination numbers of the metal in the covalent framework.

Fig. 10 Representative illustrations of ABiQ₂ (A = alkali metal, Q = S, Se, Te) structures that crystallize in the (A) rocksalt structure type (Fm3̄m), (B) α-NaFeO₂ structure type (R3̄m), and (C) CsSbS₂ structure type (P2₁/c). Monovalent cation shown in blue, trivalent cation in black, and chalcogenide in yellow. (D) Chart of ABiQ₂ compounds reported in literature^{47–49,79,100,101} and in this paper.

As the alkali metal increases in size, there is a decrease in dimensionality in agreement with the counterion effect that results from the end of site-sharing for the alkali metal and pnictogen.^96,97 The previously reported lithium, sodium, and potassium ABiQ₂ compounds crystallize in the rocksalt structure with the alkali metal and bismuth sharing the same octahedral cation site, shown in Fig. 10A. As observed in this work, KBiS₂ and KBiSe₂ can be directed to crystallize in the β-phase (Fig. 10B), which is in the α-NaFeO₂ structure type. DFT mixing entropy calculations, shown in Fig. S13,† corroborate these experimental results and underscore that the rocksalt structure cannot be stabilized by finite temperature compared to the α-NaFeO₂ structure type. This structure consists of [BiS₂]⁻ layers stacked perpendicular to the c-axis and separated by K⁺ cations (Fig. 10B). The potassium ternary marks an inflection point of site-sharing of bismuth and the alkali metal, as the rubidium and cesium ternaries also do not exhibit site-sharing of the two cations. The framework here transitions from a three dimensional covalent framework to a two dimensional layered framework, as highlighted previously in Fig. 7B. RbBiS₂ crystallizes in the α-NaFeO₂ structure type, similar to β-KBiS₂ and β-KBiSe₂, where bismuth and potassium separately occupy sites in alternate layers. RbBiSe₂ was synthesized using a 1.5 Rb₂Se : 1 Bi₂Se₃ ratio and is indexed to be in the R3̄m space group. CsBiS₂ crystallizes in the CsSbS₂ structure type (P2₁/c), shown in Fig. 10C. This structure exemplifies the transition from a 2D covalent framework to a 1D framework as it comprises extended, corrugated [BiS₂]⁻ chains separated by Cs⁺ atoms.

With potassium acting as an inflection point in the ABiQ₂ system, there is a region of stability for six-coordinate site-sharing of bismuth and alkali metal corresponding to a cation radii ratio of 1.33 or less, using Shannon ionic radii.¹⁰² For ratios higher than 1.33, the cation-ordered structure forms (e.g. RbBiS₂, RbBiSe₂, and CsBiS₂). A similar cation radii tolerance is found in the ALnS₂⁵⁴ and AAn₂Q₆¹⁰³ structures (A = alkali metal or Tl, Ln = lanthanide, An = Th, U, Np), thus supporting the plausibility of this proposed site-sharing tolerance. Inspection of antimony analogues corroborates the cation radii tolerance for metal site sharing. NaSbS₂ forms in both the rocksalt structure⁵⁰ and in the KSbS₂ structure type.¹⁰⁴ The ratio is 1.3, which within approximation agrees with the previous trend for bismuth. KSbS₂ crystallizes only in a cation-ordered structure and is not observed in a disordered rocksalt phase; the radius ratio for potassium and antimony in KSbS₂ is 1.8, thereby following the proposed cation radii tolerance.¹⁰⁵

To demonstrate the robustness of the tolerance factor, the total energies of the three prototypes (Fig. 10A–C) were calculated for all reported structures in Fig. 10D and their accuracy were subsequently compared to the room-temperature structures. The DFT-calculated predictions found that at T = 0 K all structures are most stable in the ordered α-NaFeO₂ structure type. Notably, the energy difference for CsBiS₂ between the rhombohedral and monoclinic structures, as shown in Table S13,† is quite small (≤10 meV per atom) at 0 K and, thus, the monoclinic CsSbS₂ (P2₁/c) structure type is very likely stabilized by temperature. The transition temperature between the rocksalt and α-NaFeO₂ structure types, and therefore their temperature dependence, were investigated by introducing ideal mixing entropy (i.e. the configurational entropy) to the T = 0 K energies and fitting the transition temperatures using the experimental transition temperature of KBiS₂. As shown in Table S14,† the calculated transition temperatures follow the disorder-to-order trend predicted by the tolerance factor. The lithium ternaries have a transition temperature below room temperature, indicating that the rocksalt phase would be present at room temperature. The sodium ternaries have similarly low transition temperatures, while the potassium, rubidium, and cesium ternaries have a transition temperature of over ∼700 K, thus confirming the robustness of the proposed tolerance factor.

Understanding and employing this proposed cation radii tolerance profitably impacts the numerous fields using ternary chalcogenide semiconductors. More specifically, alkali metal pnictogen chalcogenide ternaries are of interest for thermoelectrics and non-linear optics. In the field of thermoelectrics, binary chalcogenide semiconductors (i.e. GeQ, SnQ, PbQ) that crystallize in the rocksalt structure type have historically been of great interest. Current studies now capitalize on structure–property relationships to optimize properties by exploring solid solutions of rocksalt materials.^44,45 These solid solutions of rocksalt structures reduce the lattice thermal conductivity in the thermoelectric material through phonon scattering as well as exploit band gap engineering. The proposed cation radii ratio , therefore, provides guidance for the compositional requirements to form a cation-disordered APnQ₂ ternary (A = alkali metal, Pn = Sb, Bi, Q = S, Se) that may then be mixed with another rocksalt structure. For non-linear optical materials, noncentrosymmetry is a structural requirement. The probability of achieving a noncentrosymmetric space group greatly increases with cation ordering as well as expression of the pnictogen ns² lone pair. Therefore, the proposed cation radii ratio can similarly be used to guide cation selection criterion for non-linear optical materials.

Conclusion

In this work, panoramic synthesis is shown to be a valuable tool for materials discovery and mechanistic insight into structure formation of KBiS₂ and KBiSe₂. Three new compounds were identified, K₃BiS₃, β-KBiS₂, and β-KBiSe₂. The KBiQ₂ however do not form directly. The appearance of K₃BiQ₃ (Q = S, Se) preceding the formation of KBiQ₂ suggests that this compound is mechanistically important as an intermediate in the reaction pathway. K₃BiQ₃ serves as a transition point for the potassium coordination environment and for cation intermixing. The observation of the KBiQ₂ as both a rocksalt and α-NaFeO₂ type structure, while NaBiQ₂ and RbBiQ₂ crystallize in the Fm3̄m and R3̄m space groups, respectively, suggests that there is a cation radius tolerance of 1.33 or greater for high symmetry rocksalt formation. Thermal analysis data and DFT calculations show that the cation-ordered polymorphs are the thermodynamically stable phases, while PDF analysis shows that the structures have local off-centering from stereochemically active 6s² lone pair expression on the bismuth. DFT calculations also corroborate the increase in band gap from α-KBiQ₂ to β-KBiQ₂ as a result of dimensional reduction. By continuing to develop and catalog reactions using in situ diffraction techniques, we can work to expand our mechanistic understanding and predictive power towards the ultimate goal of synthesis-by-design.

Author contributions

The idea for this project was conceived by M. G. K. and R. M. The manuscript was written by R. M. and M. G. K. with input from all authors. R. M. conducted all syntheses, structure solutions, and optical characterizations. C. D. M. assisted with optimizing the in situ experimental set-up, conducted the Pair Distribution Function (PDF) experiments, and provided fruitful conversations for refinements. J. S. and J. H. ran all DFT calculations under the guidance of C. W. The PDF analyses were completed by G. B. G. The overall project was supervised by M. G. K.

Conflicts of interest

The authors declare no competing financial interest.

Acknowledgements

This work was supported by the National Science Foundation through the MRSEC program (NSF-DMR 1720139) at the Materials Research Center. Use was made of the IMSERC Facility at Northwestern University, which has received support from the Soft and Hybrid Nanotechnology Experimental (SHyNE) Resource (NSF ECCS-1542205), the State of Illinois, and International Institute for Nanotechnology (IIN). Use of the computational resources were provided by the Quest high performance computing facility at Northwestern University. The authors thank Dr Kyle M. McCall for helpful discussions. Use of the Advanced Photon Source (11-BM-B and 11-ID-B beamline) at Argonne National Laboratory was supported by the U. S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under contract no. DE-AC02-06CH11357.

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Footnote

† Electronic supplementary information (ESI) available. CCDC 2003740–2003742. For ESI and crystallographic data in CIF or other electronic format see DOI: 10.1039/d0sc04562d

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