Topological insulators (TIs) constitute a class of modern functional materials regarded as a foundation for spintronics, energy-efficient electronics, thermoelectrics, and quantum technologies. To discover and develop alternative materials with tunable properties, a comprehensive experimental investigation of phase equilibria in the SnTe–PbTe–Bi2Te3 system was conducted for the first time to our knowledge. A solid-phase equilibrium diagram at 500 K, a liquidus surface projection, and some isopleth sections were constructed. The types and coordinates of invariant and monovariant equilibria in the system were identified. Continuous and wide-range solid solution series based on AIVTe·nBi2Te3 homologous compounds with TI properties were identified and characterized. It was established that continuous solid solutions are formed along the SnBi2Te4–PbBi2Te4, SnBi4Te7–PbBi4Te7, and SnBi6Te10–PbBi6Te10 sections. Wide-range solid solutions based on tetradymite-type compounds Sn2Bi2Te5 and Sn3Bi2Te6 were observed along the Sn2Bi2Te5–"Pb2Bi2Te5" and Sn3Bi2Te6–"Pb3Bi2Te6" sections. The unlimited solid solutions of the boundary quasibinary SnTe–PbTe system extend into the composition triangle. A wide homogeneity region is observed based on Bi2Te3. The primary crystallization fields and homogeneity regions of all phases, along with their crystal lattice parameters, were determined. The novel nonstoichiometric phases identified in this study are promising candidates for TIs in low-energy electronics applications.

Phase equilibria in the SnTe-Bi2Te3-Te system were investigated using differential thermal analysis (DTA), X-ray diffraction (XRD), and scanning electron microscopy (SEM). A complete isothermal section at 300 K and a liquidus projection were constructed. The liquidus surface projection comprises seven primary crystallization fields and includes 11 monovariant and 12 invariant equilibria. The types and compositions of both invariant and monovariant equilibria in the system were determined. Polythermal sections were plotted to elucidate the crystallization sequence and the relationships phases. The thermodynamic properties of SnTe-rich alloys were determined using the electromotive force (EMF) method with concentration cells of the type (-) SnTe(s) | KCl-glycerol | Sn-Bi-Te(s) (+). Based on the obtained EMF-temperature dependences, the partial thermodynamic functions of GeTe in the alloys were first calculated, followed by the partial molar functions of germanium. These data were then used to determine the standard thermodynamic functions of Sn2Bi2Te5, Sn3Bi2Te6, and solid solutions based on SnTe using the virtual-cell reaction approach. Obtained results provide a consistent thermodynamic and phase-equilibria description of the Sn-Bi-Te system, essential for understanding and designing layered telluride materials for thermoelectric and topological applications.

The Cu2Se-Ag2Se-GeSe2 system (A) is of considerable interest due to the possibility of forming tunable copper-silver-substituted solid solutions based on Cu8GeSe6 and Ag8GeSe6 compounds, which possess several interesting functional properties, including superionic conductivity. This paper presents the results of a comprehensive study of phase equilibria in the aforementioned system and the thermodynamic properties of Cu8(1-x)Ag8xGeSe6 solid solutions, obtained using differential thermal analysis and X-ray diffraction technique, as well as by measuring the electromotive forces (EMF)of concentration cells with a Cu4RbCl3I2 solid electrolyte. A T-x diagram of the Cu2Se-Ag2Se boundary system, several isoplethal sections, isothermal sections at 300, 800, 1000, and 1100K of the phase diagram, as well as a projection of the system’s liquidus surface (A) have been constructed. The primary crystallization fields of the phases, as well as the types and coordinates of univariant and invariant equilibria, have been determined. It is shown that the system is characterized by the formation of continuous solid solutions between the high-temperature modifications of the Cu8GeSe6 and Ag8GeSe6 compounds and limited solid solutions based on their low-temperature modifications. In the subsolidus region of system (A), phase transformations associated with the polymorphism of binary and ternary compounds have been identified and described. From the EMF measurements, the partial molar functions of copper were determined, based on which the standard thermodynamic formation functions and standard entropies of the above-mentioned solid solutions were calculated.

The Cu2S-Ag2S-GeS2 (A) system is of interest for the preparation of copper–silver-substituted solid solutions based on ternary compounds possessing a number of interesting functional properties. This paper presents the results of a detailed experimental study of phase equilibria in this system using the DTA/DSC and X-ray diffraction methods. A new refined version of the T-x diagram of the Ag2S-GeS2 boundary system is presented. A new picture of phase equilibria in the system (A) is obtained, including diagrams of solid-phase equilibria at 300 K and 750 K, a projection of the liquidus surface of the system, as well as some polythermal sections and isothermal sections of the phase diagram at 1200, 1100 and 1000 K. It is established that the quasi-binary system Ag2S-GeS2 is characterized by the formation of two intermediate compounds, Ag8GeS6 and Ag2GeS3. The first melts congruently at 1220 K and undergoes a polymorphic transformation at 495 K. The second is formed as a result of a peritectic reaction at 895 K. System (A) is characterized by the formation of continuous solid solutions between the high-temperature cubic modifications of Cu8GeS6 and Ag8GeS6, limited solid solutions based on their low-temperature modifications, and Cu2GeS3 and Ag2GeS3 compounds. In the subsolidus region of system (A), phase transformations associated with the polymorphism of binary and ternary compounds are observed.

This work presents the results of a study on solid-phase equilibria within the FeSe2–FeIn2Se4–In2Se3–Se compositional region of the system Fe-In-Se and the thermodynamic properties of FeIn2Se4 and FeSe2 compounds using X-ray diffraction analysis (XRD), differential thermal analysis (DTA) and electromotive force (EMF) methods. It was established that, at room temperature, this subsystem comprises two three-phase regions: FeSe2–FeIn2Se4–Se and FeIn2Se4–In2Se3–Se. Based on the EMF measurement data of concentration cells of the type (-) Fe(s) | glycerol + KCl + FeCl2 | (Fe in alloy) (+) in the temperature range of 300–450 K, linear equations for the temperature dependence of the EMF were obtained for alloys from the above-mentioned three-phase regions and for the two-phase FeSe + Se alloys. From these data, the relative partial molar thermodynamic functions of iron in these alloys at 298 K were calculated. Based on the phase diagram, equations of virtual cell reactions were established, which allowed the calculation of standard thermodynamic functions of formation (ΔfG°, ΔfH°, ΔfS°) and standard entropies (S°) for the FeSe2 and FeIn2Se4 compounds. Both thermodynamically characterized compounds are of great practical interest as potential environmentally friendly multifunctional materials possessing magnetic, photovoltaic, optical, and other properties.

This work presents the first determination of thermodynamic functions for the layered tetradymite-type compounds MnBi2Te4, MnBi4Te7, and MnBi6Te10using the electromotive force (EMF) method. These compounds belong to the MnBi2nTe3n+1 homologous series, which has attracted considerable interest due to its tunable magnetic ordering and topological electronic properties. EMF measurements were conducted in the temperature range of 300-450 K using glycerol-based electrolytes in galvanic cells with defined Mn activity. The EMF of the investigated concentration cells exhibited linear temperature dependencies over this range. The Gibbs energies, enthalpies, and entropies of formation were calculated from the temperature-dependent EMF data, with reference to MnTe and higher homologues. A solid-phase equilibrium diagram of the MnTe-Bi2Te3-Te system was constructed using X-ray diffraction results for equilibrated alloys to identify phase boundaries and establish the relevant heterogeneous phase regions for the target compounds. The obtained thermodynamic functions are internally consistent and in agreement with established phase diagram data. These results fill a critical gap in the thermodynamic characterization of Mn-Bi-Te magnetic topological insulators and provide a foundation for modeling phase equilibria and optimizing synthesis conditions within this layered material system.

Magnetic topological insulators based on Mn-containing bismuth tellurides require precise control of phase formation, solid-solution ranges, and crystallization conditions. In this work, the high-temperature phase relations of the MnTe–SnTe–Bi2Te3 pseudoternary system were investigated as a basis for the targeted design of Mn/Sn-substituted layered telluride materials. More than 55 alloys were prepared from pre-synthesized MnTe, SnTe, and Bi2Te3 compounds using melt quenching followed by prolonged annealing. Differential thermal analysis, powder X-ray diffraction, and scanning electron microscopy were used to determine phase-transition temperatures, phase compositions, lattice parameters, and microstructural features. Based on these data, the liquidus surface projection of the MnTe–SnTe–Bi2Te3 system was constructed, and 12 primary crystallization fields were identified. Five isopleth sections were analyzed to establish crystallization sequences, invariant and monovariant reactions, and subsolidus phase relations. Continuous Mn/Sn-substituted solid solutions were confirmed along the MnBi2Te4–SnBi2Te4, MnBi4Te7–SnBi4Te7, and MnBi6Te10–SnBi6Te10 sections. The linear increase in lattice parameters with Sn content confirms the compositional tunability of these homologous layered phases. The obtained phase equilibrium diagram defines composition-temperature regions suitable for preparing single-phase solid solutions and controlled multiphase assemblages, providing a practical guide for materials design in the Mn–Sn–Bi–Te system.

The phase equilibria in the Mn-Sb-Bi-Te system play an important role in designing layered magnetic topological insulator (MTI) phases. This study reports, for the first time, the phase equilibria of the MnTe-Sb2Te3-Bi2Te3 system to aid in designing MTI phases. The solid-phase equilibrium diagram at 500 degrees C, the liquidus surface projection, and the phase diagrams of the five isopleth sections have been constructed using experimental data of differential thermal analysis, X-ray diffraction analysis, and scanning electron microscopy techniques. The primary crystallization fields of phases, types, and coordinates of invariant and monovariant equilibria were determined. Complete solid solutions were detected in the MnSb2Te4-MnBi2Te4 and MnSb2Te4-MnBi2Te4 vertical sections, with no miscibility gap. The lattice parameters of a continuous series of solid solutions were refined via Le Bail methods and compared with values predicted by Vegard's law, confirming the compositional dependence of the lattice constants. There are also noticeable homogeneity regions based on the other homologous members of the (MnBi2Te4)(Bi2Te3) m family, with m = 2,,6. Besides, aliovalent cation-substituted solid solubility dominates the Sb2Te3-Bi2Te3 side of the diagram. The obtained tetradymite-type layered solid solutions in this study demonstrate tunable magnetic properties compared to the parent compounds, providing a pathway for optimizing their functional characteristics in MTIs.

The GeSb2Te4–Sb2Te3–Te system was investigated in the temperature range of 300–450 K using X-ray diffraction (XRD) and electromotive force (EMF) measurements on reversible galvanic cells of the type: (−) GeTe (solid) │ glycerol + KCl (electrolyte) │ Ge–Sb–Te (solid) (+). The results indicate that, within the studied temperature range, elemental tellurium establishes tie lines with all telluride phases present in the system. Based on the EMF measurement data, linear equations describing the temperature dependence of the electromotive force for electrode alloys corresponding to different phase regions of the system were established. These equations were used to calculate the partial thermodynamic functions of GeTe in the alloys. By combining the obtained data with the integral thermodynamic functions of GeTe, the partial molar functions of germanium in the alloys were calculated. Based on these data and the solid-phase equilibrium diagram of the GeSb2Te4–Sb2Te3–Te system, as well as corresponding thermodynamic functions of Sb2Te3, the standard Gibbs free energy of formation, enthalpy of formation, and standard entropy of GeSb2Te4, GeSb4Te7, GeSb6Te10, and Sb2Te3-based solid solutions were calculated.

Phase equilibria in the MnSe-Ga2Se3-In2Se3 system were studied by differential thermal analysis, X-ray diffraction analysis, and scanning electron microscopy techniques. A number of polythermal and isothermal sections of the phase diagram, as well as a projection of the liquidus surface were constructed. The fields of primary crystallization of seven phases, as well as, the types and coordinates of in-and monovariant equilibria in the system were determined. It was shown that wide solid solution areas formed on the MnGa2Se4-MnIn2Se4 section expand significantly towards the boundary system Ga2Se3-In2Se3 and result in large homogeneous areas. The homogeneity regions based on Ga2Se3 extend significantly-over 40 mol%-into the composition triangle, forming a broad homogeneity area. As part of this study, the boundary quasi-binary system MnSe-In2Se3 was reinvestigated, and a new phase diagram, markedly different from that reported in the literature, was constructed. It was found that the system forms a congruently melting MnIn2Se4 compound at 1193 K and an incongruently melting Mn2In2Se5 compound at 1196 K via a peritectic reaction. MnIn2Se4 has a wide homogeneity area. Based on the powder diffraction patterns, the crystal structures of both compounds were determined and their lattice parameters were refined by the Rietveld method. The study of the temperature dependence of electrical conductivity and the volt-ampere characteristics of the MnGaInSe4 compound revealed that the current flow in the nonlinear region of the current-voltage curve corresponds to Frenkel's theory of thermoelectric ionization. The concentration of ionized centers in this crystal (Nt =9 & sdot;1014 cm-3), the mean free path of charge carriers (lambda 1= 7.3 & sdot;10-5 cm) and the shape of the potential well of traps were determined. The activation energy of the charge carriers was determined from the temperature dependence of the electrical conductivity of the MnGaInSe4 crystal (E = 0.30 eV).

A new complete picture of phase equilibria in the Cu-As-Se system was obtained by experimentally studying carefully crystallized alloys by prolonged thermal annealing using differential thermal analysis and powder X-ray diffraction methods, as well as considering information found in the literature. The liquidus surface projection, isothermal section at 300 K and some vertical sections of the phase diagram are presented and discussed. The types and coordinates of nonvariant and monovariant phase equilibria are determined. It is established that the liquidus surface consists of 15 regions corresponding to the primary crystallization of three initial components, seven binary and five ternary compounds. The presented phase diagram reflects the compounds Cu3AsSе4, Cu3AsSе3, Cu7As6Sе13, CuAsSе2 and CuAsSе, which can be considered as synthetic analogues of natural copper-arsenic minerals. The ternary compounds Cu6As4Se9 and Cu4As2Se5, indicated in the literature, have not been confirmed by us. The thermodynamic properties of ternary copper-arsenic selenides were studied using the EMF method with the solid electrolyte Cu4RbCl3I2. From the EMF measurement data in the corresponding three-phase regions, linear equations of the temperature dependence of the EMF were obtained, from which partial thermodynamic functions of copper in the alloys were calculated. Based on these data and the solid-phase equilibrium diagram of the Cu-As-Se system, using the corresponding thermodynamic functions of copper, As2Se3, AsSe and Cu3AsSe4, the standard Gibbs free energy of formation and the enthalpy of formation, as well as the standard entropy of the ternary compounds Cu3AsSе3, Cu7As6Sе13, CuAsSе2 and CuAsSе, were calculated.

An Erratum to this paper has been published: https://doi.org/10.1134/S0036024425010017

Complex chalcogenides based on transition elements, in particular ternary compounds of the АВ2Х4 type (M = Mn, Fe, Co, Ni; B = Ga, In, Sb, Bi; X = S, Se, Te) are among the important functional materials. Compounds of this class exhibit the phenomena of electronically or optically controlled magnetism and are very promising for the creation of lasers, light modulators, photodetectors, and other functional devices controlled by a magnetic field. Recent studies demonstrated that these compounds can also find application in photocatalysis, photovoltaics, and thermoelectric converters. The study presents new data on phase equilibria in the MnSe–In2Se3 system, obtained by differential thermal analysis, X-ray phase analysis, and scanning electron microscopy. Two ternary compounds, MnIn2Se4 with congruent melting at 1193 K and Mn2In2Se5, melting incongruently at 1196 K, were formed in the system. The first is a phase of variable composition and has a 5–6 mol. % homogeneity region towards an excess of In2Se3. Based on powder diffraction data, the Rietveld method was used to refine the crystal structures and lattice parameters of both ternary compounds

The iron-gallium chalcogenides are promising materials for application in sustainable energy. This work aimed to investigate the solid-phase relations in the Fe-Ga-Se system in the FeSe-Ga2Se3-Se compositions area and the thermodynamic properties of ternary phases. Based on the experimental results and using literature data on boundary systems, the diagram of solid-phase equilibria of the above pointed composition range at room temperature was constructed. It was obtained that the solid solutions based on Ga2Se3 and FeGa2Se4 are in stable tie-lines with elemental selenium.For the thermodynamic study, the following reversible concentration cells (-) Fe(s.) | glycerol + KCl+FeCl2 | (Fe in alloy) (+) were assembled and their electromotive forces (EMF) were measured between 300–450 K temperature interval.The relative partial molar Gibbs energy, enthalpy, and entropy of iron in the alloys were calculated using the linear equations of the temperature dependences of the EMF in the various phase areas of the FeSe-Ga2Se3-Se system that were obtained from these EMF observations. The standard thermodynamic functions of formation and standard entropies of the FeGa2Se4 compound and (FeSe)1-х(Ga2Se3)х (x¯ = 0,55; 0,8; 0,9) solid solutions were computed from the given partial molar quantities using thermodynamic data for Ga2Se3 and the solid-phase equilibria diagram.

The Ag2S-Ag8SiS6-Ag8SnS6 system was studied using DTA/DSC, diffraction of X-ray, as well as SEM methods. The T-x phase diagram of the Ag8SiS6-Ag8SnS6 boundary system, several vertical and isothermal sections of the phase diagram, as well as a projection of the liquidus surface, were plotted, and the thermodynamic functions of polymorphic transitions of the Ag8SiS6 compound and Ag8Si1-xSnxS6 solid solutions were calculated. It was obtained that the Ag8SiS6-Ag8SnS6 boundary system is quasi-binary and is characterized by the formation of a continuous solid solutions during low-temperature orthorhombic and high-temperature cubic modifications of the initial compounds. It is shown that investigated system is a quasi-ternary plane of the Ag–Si–Sn–S concentration tetrahedron. The liquidus surface consists of two fields corresponding to the primary crystallization of high-temperature Ag8Si1-xSnxS6 solid solutions based on HT-Ag2S. Below the solidus, phase transformations associated with the polymorphism of the initial compounds and phases based on them were observed. Based on DSC data, the temperatures, enthalpies, and entropies of phase transitions of the argyrodite phases from low-temperature orthorhombic modification to high-temperature cubic modification were calculated. It was determined that the heats and entropies of tranformations of above phases have anomalously high values ​​compared to ordinary polymorphic transitions. In addition, it was established that the entropies of phase transformations of solid solutions are practically equal to the sum of the corresponding functions of the original compounds. This indirectly indicates the quasi-ideality of solutions between both modifications of these compounds.

The paper analyzes the works of the last decade on the study of ionic conductivity, thermoelectric, photovoltaic, photocatalytic, optical, etc. properties of the argyrodite family compounds (prototype Ag 8 GeS 6 ), as well as phases and composite materials based on them. Considered works allow us to characterize them as valuable environmentally friendly functional materials with great potential for practical application. The main approaches which have been used in these works to enhance functional properties, as well as to improve the application capabilities of this class of substances have been analyzed. The importance of further systematic investigations on their design based on the “composition-structure-property” relationship has been noted. In the present review, special attention is paid to the analysis of works on phase equilibria in the corresponding systems since the information accumulated in phase diagrams is extremely important for optimizing the properties of compounds through targeted variation of composition and structure. Available data on the phase equilibria and thermodynamic properties of ternary and quaternary systems forming Cu/Ag argyrodite compounds and solid solutions based on them are presented and analyzed. It has been shown that anionic and Cu ↔ Ag substitutions in compounds lead to a strong decrease in their polymorphic phase transition temperatures and an expansion of the temperature-composition ranges of the existence of high-temperature ion-conducting phases up to room temperature and below. The possibility of replacing part of the chalcogen atoms in the compounds with halogens, and the Cu(Ag) atoms with elements of the zinc subgroup, which expands the range of argyrodite phases is also shown. The significance of expanding studies of phase equilibria and thermodynamic properties of these systems, especially five-component and more complex systems possessing a big possibility of the formation of high-entropy argyrodite phases, which have thermodynamic stability in a wide temperature-composition range and better applied characteristics has also been specified.

In this study, the thermodynamic properties of the Bi2Te3-rich manganese-bismuth tellurides were determined using an electromotive force (EMF) method with a liquid electrolyte in a 300-450 K temperature interval. Solid-phase equilibrium diagram of the Bi2Te3-rich corner of the MnTe-Bi2Te3-Te system at 300 K was constructed using literature data and X-ray diffraction (XRD) analysis of synthesized electrode alloys. Functions (, and) corresponding to linear equations were calculated for different phase regions using EMF values. The relative partial molar functions of manganese in alloys were calculated with the help of these data along with the corresponding thermodynamic functions of MnTe and Bi2Te3. The standard thermodynamic functions of formation and standard entropies of the Bi2Te3-rich manganese bismuth tellurides - MnBi8Te13, MnBi10Te16, MnBi12Te19, and MnBi14Te22 compounds, and solid solutions based on Bi2Te3 have been obtained.

Complex copper-based chalcogenides are among the most important functional materials in modern engineering and technology due to their diverse physical and physicochemical properties, environmental safety and availability. The development of new similar materials and the improvement of the applied characteristics of known compounds is largely associated with the use of approaches based on the physicochemical analysis and, in particular, the “composition-structure-property” relationship. This review summarizes the available data on phase equilibria in ternary systems Cu-Tl(BIV, BV)-X (BIV-Si, Ge, Sn; BV-As, Sb, Bi; X-S, Se, Te) and the thermodynamic properties of their intermediate phases. Similar data are also considered for more complex systems forming solid solutions of various types of substitution based on known ternary copper chalcogenides. A significant part of the presented sets of mutually consistent data on phase equilibria and thermodynamic properties of the considered systems was obtained by our group over the past 10-15 years. Although these data cover only a small part of the systems described above, they provide great possibilities for manipulation of composition and structure, including entropic engineering strategies. The authors consider it extremely important to further develop fundamental research on phase equilibria and thermodynamic properties of complex copper chalcogenides and use their results widely in selecting alloy compositions for physical measurements

Here, a complete phase equilibria picture in the Cu-As-S system was obtained by experimental study of carefully crystallized via long-term thermal annealing alloys by means of methods of differential thermal analysis and powder x-ray diffraction, as well as using the available literature data. The projection of the liquidus surface, the isothermal section at 300 K, and some vertical sections of the phase diagram are presented and discussed. The fields of primary crystallization of phases, types, and coordinates of invariant and monovariant phase equilibria are determined. The presented phase diagram reflects four ternary compounds Cu3AsS4, Cu12As4S13, Cu6As4S9, and CuAsS, which are synthetic analogues of natural copper-arsenic sulfide minerals. Particular attention is paid to the Cu2S-As2S3 section. It is shown that this section, in contrast to the literature data, is not quasi-binary. The thermodynamic data for copper-arsenic sulfides, previously obtained by the authors by the electromotive force method with Cu4RbCl3I2 solid electrolyte, have also been revised. Experimental data on the partial thermodynamic functions of copper in some phase regions of the Cu-As-S system were processed taking into account the constructed new version of the solid-phase equilibria diagram and updated data on the standard thermodynamic functions of formation and standard entropies of the ternary compounds Cu3AsS4, Cu12As4S13, Cu6As4S9, and CuAsS were obtained.

Complex copper-tin and copper-antimony chalcogenides are of great interest for the development of new environmentally friendly and inexpensive thermoelectric materials. Recently, these compounds have been drawing more interest due to the possibility of increasing their thermoelectric performance with various cationic and anionic substitutions. In this article, we continued the study of multi-component systems based on the copper chalcogenides and presented the results of the study of phase equilibria in the Cu2SnSe3–Sb2Se3–Se system. The study was conducted using differential thermal analysis and powder X-ray diffraction. Based on the experimental data, a projection of the liquidus surface and three polythermal cross sections of the phase diagram were plotted. We determined the regions of primary crystallisation of the phases and the nature and temperatures of non-variant and monovariant equilibria. It was established that the liquidus surface consisted of two primary crystallisation regions based on Cu2SnSe3 and Sb2Se3 phases. The primary crystallisation region of elementary selenium was degenerate. A large immiscibility region of two liquid phases was found in the system.