Phase Equilibria in the Ag₂Se–Cu₂SnSe₃ and Ag₈SnSe₆–Cu₂SnSe₃ Systems

Abstract

The phase formation in the sections Ag₂Se–Cu₂SnSe₃ and Ag₈SnSe₆–Cu₂SnSe₃ of the quasi-ternary system Ag₂Se–SnSe₂–Cu₂Sе was studied for the first time by differential thermal analysis (with thermodynamic calculations), X-ray powder diffraction analysis, and microstructural analysis and also by microhardness and density measurements. No new quaternary compounds were detected. It was determined that both sections are quasi-binary and are of simple eutectic type with limited solubility based on the initial selenides. The coordinates of the eutectic points are (40 mol % Ag₂Se, 910 K) and (50 mol % Ag₈SnSe₆, 780 K). The solubility based on Cu₂SnSe₃ was 10 mol % Ag₂Se in the system Ag₂Se–Cu₂SnSe₃ and 15 mol % Ag₈SnSe₆ in the system Ag₈SnSe₆–Cu₂SnSe₃. Single crystals of the compound Cu₂SnSe₃ and the solid solutions (Cu₂SnSe₃)_{1 – х}(Ag₈SnSe₆)_х (х = 0.02–0.15) were grown by the Bridgman–Stockbarger directional crystallization method. It was found that these solutions crystallize in the monoclinic system, and the unit cell parameters increase with increasing Ag₈SnSe₆ content: а = 6.968–6.985 Å, b = 12.051–12.078 Å, с = 6.945–6.968 Å, β = 109.20°–109.30°, space group Сс, Z = 4, and ρ = 5.75–5.86 g/cm³.

INTRODUCTION

Investigation of the interaction between semiconductor compounds, in particular, between silver and copper chalcogenides, which have superionic conduction, is of great scientific and practical interest. According to Ioffe’s concepts [1], the complication of the composition of solid solutions improves a number of physical properties, e.g., the thermoelectric figure of merit. In this context, it was of interest to study the phase equilibria in the systems Ag₂Se–Cu₂SnSe₃ and Ag₈SnSe₆–Cu₂SnSe₃.

Selenides Ag₂Se, Cu₂SnSe₃, and Ag₈SnSe₆ were investigated previously [2–21]. These compounds melt congruently at 1170, 973, and 1017 K, respectively. According to the published data [5–7, 12, 16, 17], Ag₂Sе and Ag₈SnSe₆ undergoes a single polymorphic transition each at 410 and 356 K, respectively. The published data on the crystal structure of the low-temperature polymorph α-Ag₂Se are contradictory [22, 23]. It was reported [24] that the low-temperature polymorph Ag₂Se has a pseudocubic structure with the lattice parameter a = 4.978 Å, and the high-temperature polymorph β-Ag₂Se crystallizes in the orthorhombic system with the unit cell parameters a = 7.067 Å, b = 7.80 Å, and c = 4.34 Å [25, 26], or a = 4.333 Å, b = 7.162 Å, c = 7.764 Å [5, 12] with the space group Р2₁2₁2₁.

The low-temperature polymorph α-Ag₈SnSe₆ crystallizes in the cubic system (a = 11.12 Å), the high-temperature polymorph β-Ag₈SnSe₆ crystallizes in the orthorhombic system (a = 7.9168 Å, b = 7.8219 Å, c = 11.045 Å, space group Рmn2₁, and ρ_calc = 7.072 g/cm³ [15–19]. This compound is isostructural to Сu₈GeS₆ and β-Ag₈GeSе₆ [17]. The average interatomic distance Ag–Se in the structure of Ag₈SnSe₆ is 2.59–2.82 Å.

The compound Cu₂SnSe₃ has a cubic lattice, the period of which varies (a = 5.688–5.696 Å) within the homogeneity region [15, 16, 27]. This compound has a monoclinic structure with the unit cell parameters a = 6.5936 Å, b = 12.1593 Å, c = 6.6084 Å, β = 108.56°, space group Сс [28].

The initial compounds Ag₂Se, Ag₈SnSe₆, and Cu₂SnSe₃ attract researchers’ attention as promising materials for using in nonlinear optical devices and photoelements [29–34], in view of which the investigation of the interaction between them are of both theoretical and practical importance.

EXPERIMENTAL

The initial compounds and a set of alloys were prepared for the experiments. Ag₂Se, Ag₈SnSe₆, and Cu₂SnSe₃ were synthesized from high-purity selenium, high-purity electrical copper, and silver (>99.999%). The compounds were obtained by alloying elemental components weighed with an accuracy of 0.0001 g. The alloying was carried out in sealed evacuated (to 10^–3 mm Hg) quartz ampules. The temperature was increased slowly with 1-h-long holdings at the melting points of Ag₂Se, Ag₈SnSe₆, and Cu₂SnSe₃ (1170, 1017, and 973 K, respectively). After the completion of the synthesis, a 2–3-h-long holding was performed at 1060 K for Ag₂Se, 920 K for Ag₈SnSe₆, and 900 K for Cu₂SnSe₃. The cooling was carried out in a turned -off furnace. To study the phase equilibria in each of the systems Ag₂Se–Cu₂SnSe₃ and Ag₈SnSe₆–Cu₂SnSe₃, 11 alloys of various compositions at an interval of 5–10 mol % were synthesized from the obtained initial compounds by alloying in sealed evacuated ampules with 40–45-min-long holding at temperatures 50–60°C above the melting points. The alloys were quenched in water and then were subjected to homogenizing annealing for 250–340 h at 740 K for the system Ag₈SnSe₆–Cu₂SnSe₃ and 850 K for the system Ag₂Se–Cu₂SnSe₃.

The alloys were studied by differential thermal analysis, X-ray powder diffraction analysis, and metallography. The differential thermal analysis was performed with a Kurnakov pyrometer using a chromel–alumel thermocouple. The heating and cooling curves were recorded for each sample. The heating rate was 8–10 deg/min, the cooling was carried out in a turned-off furnace. The solidus points were determined at the beginning of the deviation of the differential curve in the heating curve; and the solidus points, at the beginning of the deviation of the differential curve in the cooling curve. The liquidus temperature found from the cooling curves was controlled by the maximum deviation of the differential curve during heating, and the solidus temperature obtained from the heating curves was controlled by the maximum deviation during cooling. The sample weight was 2–3 g.

The X-ray powder diffraction analysis was made with a Bruker D2 РILSENER (CuK_α radiation, Ni filter, scan angle range 10° ≤ 2θ ≤ 90°). The microhardness of the samples was measured with a PMT-3 microhardness meter (the optimal load was 0.02 kg), and the microstructure of samples was investigated with a MIM-7 microscope with diluted aqueous nitric acid solution as an etchant. The error of measuring the heats was ±2°, the error of calculating the unit cell parameters was ±0.001 Å, and the error of measuring the microhardness was ±0.005 MPa.

RESULTS AND DISCUSSION

The T–x diagrams of the systems Ag₂Se–Cu₂SnSe₃ and Ag₈SnSe₆–Cu₂SnSe₃ were constructed using the differential thermal analysis data (Figs. 1, 2). As Fig. 1 shows, the section Ag₂Se–Cu₂SnSe₃ is of the eutectic type. The coordinates of the eutectic point are (40 mol % Ag₂Se, 910 K). The solubility based on Cu₂SnSe₃ at the eutectic temperature is 15 mol % Ag₂Se, and that at 300 K is 10 mol % Ag₂Se. The solubility based on silver selenide was not virtually detected. The difference of the ionic and atomic radii of Cu⁺ and Ag⁺ is 8.87%, which is within 15% dictated by the Hume-Rothery rule [35]. Although the geometric parameters are close, the mutual solubility range is narrow, which is likely to be related to the difference between the crystal structures of Ag₂Se (cubic) and Cu₂SnSe₃ (monoclinic). The liquidus of the section comprises the branches of primary crystallization of β-Ag₂Se and γ-solid solutions based on Cu₂SnSe₃. The phase transition α-Ag₂Se $\rightleftarrows$ β-Ag₂Se occurs at 410 K. The phase formation in the section Ag₈SnSe₆–Cu₂SnSe₃ is similar. As Fig. 2 shows, the section is a quasi-binary section of the quasi-ternary system Ag₂Se–SnSe₂–Cu₂Sе and is of the eutectic type. The coordinates of the eutectic point is (50 mol % Cu₂SnSe₃, 780 K). The mutual solubility between the components is limited: 15 mol % Ag₈SnSe₆ based on Cu₂SnSe₃ and 5 mol % Cu₂SnSe₃ based on cubic Ag₈SnSe₆. The temperature of the phase transition α-Ag₈SnSe₆ $\rightleftarrows$ β-Ag₈SnSe₆ decreases with increasing Cu₂SnSe₃ content; i.e., the transformation is eutectoid.

Fig. 1.

Phase diagram of the system Ag₂Se–Cu₂SnSe₃.

Fig. 2.

Phase diagram of the system Ag₈SnSe₆–Cu₂SnSe₃.

For the microscopic examination, flat samples were cut from the synthesized alloys. The surface of the samples was initially ground and then polished with a diamond paste. The polished layer was removed from the surface by etching with 50% HNO₃ + 50% H₂O to open the microstructure of the material. The micrographs of the surfaces of each sample were taken with a MIM-7 reflecting-type metallographic microscope. The micrographs of the surfaces of the (Cu₂SnSe₃)_{1 – х}(Ag₈SnSe₆)_х samples (Fig. 3) demonstrate that all the obtained alloys are single-phase and have large-block structures without any inclusions. The structure, sizes, and positions of grains differ in different samples. Single crystals of the solid solutions (Cu₂SnSe₃)_{1 – х}(Ag₈SnSe₆)_х (х = 0.02 and 0.15) were grown by the Bridgman–Stockbarger directional crystallization method. For this purpose, the synthesized polycrystalline samples with the composition Cu_1.98Ag_0.02SnSe₃ and Cu_1.85Ag_0.15SnSe₃ were placed in 12-cm-long ampules 0.5 cm i.d.. The evacuated (to 10^–3 mm Hg) ampules with the samples were moved within a vertical furnace with two temperature zones. In the upper zone, the temperature (T₁) was maintained 50–60 K above the melting point of the sample with a certain composition, and in the lower zone, the temperature (T₂) was kept 50–60 K below this melting point. In the upper zone, the sample in the ampule are in the molten state. Once the tip of the ampule passes through the melting point, in the lower zone, Cu_1.98Ag_0.02SnSe₃ Cu_1.85Ag_0.15SnSe₃ crystal nuclei form. The motion of the ampule at a velocity of 0.2–0.3 cm/h was suitable for growing the formed crystal nuclei of the samples. Figure 4 shows the thus grown single crystals of Cu_1.98Ag_0.02SnSe₃ and Cu_1.85Ag_0.15SnSe₃, which are suitable for single-crystal X-ray diffraction and physical studies. The formation of a limited solubility region in the system Ag₈SnSe₆–Cu₂SnSe₃ was also confirmed by the X-ray powder diffraction data (Fig. 5). It was determined that the X-ray powder diffraction patterns of the alloys containing ≤15 mol % Ag₈SnSe₆ are qualitatively identical to that of Cu₂SnSe₃; i.e., these alloys are substitutional solid solutions based on this compound (α-phase), whereas the X-ray powder diffraction patterns of the alloys containing ≤5 mol % Cu₂SnSe₃ are similar to those of pure Ag₈SnSe₆ (β-phase). The X-ray powder diffraction patterns of the alloys containing 5–85 mol % Cu₂SnSe₃ comprise the lines of the α- and β-phases, which is consistent with the phase diagram (Fig. 2). The X-ray powder diffraction patterns of the α-phase were indexed in the monoclinic system (Table 1). Within the homogeneity region of the α-phases, the concentration dependences of their unit cell parameters are virtually linear.

Fig. 3.

Microstructures of (a) Cu_1.95Ag_0.05SnSe₃, (b) Cu_1.85Ag_0.15SnSe₃, and (c) Ag_7.6Cu_0.4SnSe₆ (×360).

Fig. 4.

Micrographs of single crystals of (a) Cu_1.98Ag_0.02SnSe₃ and (b) Cu_1.85Ag_0.15SnSe₃ that were grown by directional crystallization of melt.

Fig. 5.

X-ray powder diffraction patterns of the system Ag₈SnSe₆–Cu₂SnSe₃; (1) Ag₈SnSe₆, (2) 5 mol % Cu₂SnSe₃, (3) 85 mol % Cu₂SnSe₃, (4) 90 mol % Cu₂SnSe₃, (5) 95 mol % Cu₂SnSe₃, and (6) Cu₂SnSe₃.

Table 1.   Crystallographic data of solid solutions (Cu₂SnSe₃)_{1 – х}(Ag₈SnSe₆)_х, monoclinic system, Z = 4, space group Сс

THERMODYNAMIC ANALYSIS

The boundaries of the solid solutions formed in the quasi-binary sections Ag₂Se–Cu₂SnS₃ and Ag₈SnSe₆–Cu₂SnSe₃ were refined using the dependence of the Gibbs free energy on temperature and concentration. An analysis of the published data showed that the compounds Ag₂Se, Cu₂SnS₃, and Ag₈SnSe₆ differ significantly in composition, structure, and unit cell parameters. Therefore, in the Gibbs–Helmholtz equation, an asymmetric variant of the model of regular solid solutions of nonmolecular compounds was used [36–39]. In particular, the Gibbs–Helmholtz equations for the solid solutions (1 – x)Ag₈SnSe_{6 – x}Cu₂SnSe₃ and (1 – x)Ag₂Se–xCu₂SnSe₃ have the form

$$\begin{gathered} \Delta G_{T}^{0} = \left( {1 - x} \right)\left[ {\Delta H_{{298}}^{0}\left( {{\text{A}}{{{\text{g}}}_{8}}{\text{SnS}}{{{\text{e}}}_{6}}} \right)} \right. \\ - \,\,\left. {T\Delta S_{{298}}^{0}\left( {{\text{A}}{{{\text{g}}}_{8}}{\text{SnS}}{{{\text{e}}}_{6}}} \right)} \right] \\ + \,\,x\left[ {\Delta H_{{298}}^{0}\left( {{\text{C}}{{{\text{u}}}_{2}}{\text{SnS}}{{{\text{e}}}_{3}}} \right) - T\Delta S_{{298}}^{0}\left( {{\text{C}}{{{\text{u}}}_{2}}{\text{SnS}}{{{\text{e}}}_{3}}} \right)} \right] \\ + \,\,\left( {a + bT} \right){{x}^{m}}{{(1 - x)}^{n}} \\ + \,\,RT\left[ {\left( {1 - x} \right)\ln {{f}_{1}}\left( x \right) + x\ln {{f}_{2}}\left( x \right)} \right], \\ \end{gathered} $$

(1)

$$\begin{gathered} \Delta G_{T}^{0} = \left( {1 - x} \right)\left[ {\Delta H_{{298}}^{0}\left( {{\text{A}}{{{\text{g}}}_{2}}{\text{Se}}} \right) - T\Delta S_{{298}}^{0}\left( {{\text{A}}{{{\text{g}}}_{2}}{\text{Se}}} \right)} \right] \\ + \,\,x\left[ {\Delta H_{{298}}^{0}\left( {{\text{C}}{{{\text{u}}}_{2}}{\text{SnS}}{{{\text{e}}}_{3}}} \right) - T\Delta S_{{298}}^{0}\left( {{\text{C}}{{{\text{u}}}_{2}}{\text{SnS}}{{{\text{e}}}_{3}}} \right)} \right] \\ + \,\,\left( {a + bT} \right){{x}^{m}}{{(1 - x)}^{n}} \\ + \,\,RT\left[ {\left( {1 - x} \right)\ln {{f}_{1}}\left( x \right) + x\ln {{f}_{2}}\left( x \right)} \right]. \\ \end{gathered} $$

(2)

In Eqs. (1) and (2), x is the mole fraction of Cu₂SnSe₃ in the solid solutions (1 – x)Ag₈SnSe₆–xCu₂SnSe₃ and (1 – x)Ag₂Se–xCu₂SnSe₃; R = 8.314 J mol^–1 K^–1; and $\Delta H_{{298}}^{0}$ and $\Delta S_{{298}}^{0}$ are the standard enthalpies and entropies of formation of compounds, respectively. Their values were taken from the literature [40, 41]: $\Delta H_{{298}}^{0}$(Ag₈SnSe₆) = –207.6 kJ mol^–1, $\Delta H_{{298}}^{0}$(Cu₂SnSe₃) = –320.4 kJ mol^–1, $\Delta S_{{298}}^{0}$(Ag₈SnSe₆) = –31.33 J mol^–1 K^–1, and $\Delta S_{{298}}^{0}$(Cu₂SnSe₃) = 100.35 J mol^–1 K^–1. ax_m(1 – x)ⁿ represents the enthalpies of mixing of solid solutions in the asymmetric variant of the model of regular solid solutions. The constants in this formula were found by a multipurpose genetic algorithm [42]. Iterations were made under the following conditions together with the experimental data of the differential thermal analysis:

$$\begin{gathered} x = 0{\kern 1pt} - {\kern 1pt} 1;\,\,\,\,a > 0;\,\,\,\,b > 0;\,\,\,\,m > n > 1; \\ 325 < T < 780\,\,{\text{K and }}410 < T < 910\,\,{\text{K}}. \\ \end{gathered} $$

The last terms in Eqs. (1) and (2) represent the configurational entropy of mixing (Т∆S) of the solid solutions with taking into account the nonmolecular nature of the compounds Ag₈SnSe₆, Cu₂SnSe₃, and Ag₂Se. The stoichiometric coefficients of unlike atoms in the molecules of the compounds are included as constants in the functions f₁(x) [43].

In particular, Eq. (1) after substituting the values of the thermodynamic quantities and the parameters of the model of regular solid solutions takes the form

$$\begin{gathered} \Delta G_{T}^{0}\left( {{\text{kJ/mol}}} \right) = x\left( { - 207.6 + 0.03133T} \right) \\ + \,\,\left( {1 - x} \right)\left( { - 320.4 - 0.10035T} \right) + 8.314T/1000 \\ \times \,\,\left( {1 - x} \right)\left( {8\ln \left( {1 - x} \right) + 2x\ln \left( x \right)} \right) + 31{\kern 1pt} 000{{x}^{4}}{{\left( {1 - x} \right)}^{5}}. \\ \end{gathered} $$

(3)

Figure 6 illustrates the dependences of the Gibbs free energy of formation of the solid solutions (1 – x)Ag₈SnSe₆– xCu₂SnSe₃ on the composition at temperatures of 325 and 780 K. Figure 6 suggests that, at medium concentrations, the solid solutions are unstable and break up into two phases. The homogeneity regions were determined from the position of the common tangent in Fig. 6 based on the equality of the chemical potentials of a component in the heterogeneous mixture of the solid solutions of two different types (Figs. 1 and 2). The calculations were performed and visualized using the software OriginLab2019 and www.matematikam.ru.

Fig. 6.

Dependences of the Gibbs free energy of formation of solid solutions on the composition at various temperatures (Eq. (3)).

CONCLUSIONS

The T–x phase diagrams of the quasi-binary sections Ag₂Se–Cu₂SnSe₃ and Ag₈SnSe₆–Cu₂SnSe₃ of the quasi-ternary system Ag₂Se–SnSe₂–Cu₂Sе were constructed for the first time. It was determined that their state diagrams are of the eutectic type with limited solubility based on the components. The dependences of the Gibbs free energy of alloys on temperature and concentration were characterized, which enabled one to refine the boundaries of the solid solutions (1 – x)Ag₈SnSe₆–xCu₂SnSe₃: 15 mol % Ag₈SnSe₆ based on Cu₂SnSe₃ and 12 mol % Cu₂SnSe₃ based on orthorhombic Ag₈SnSe₆.

The constructed phase diagrams were used for choosing the composition of melt solutions growing single crystals of the α-phase (solid solution based on Cu₂SnSe₃) and the β-phase (solid solution based on Ag₈SnSe₆) of a given composition by directional crystallization.