Introduction

Layered bismuth selenides, especially Bi2Se3, as well as doped phases and solid solutions based on it, are objects of numerous studies as valuable materials exhibiting unique optical, thermoelectric, and other functional properties. This makes them promising for use in various thermoelectric and photoelectrochemical devices, broadband and high-speed optoelectronics, etc. [1,2,3,4,5,6,7,8]. The discovery of a topological insulator (TI), a new quantum state of matter [9, 10], led to a sharp increase in attention to layered narrow-gap semiconductors, in particular, bismuth selenides and tellurides. The results of several studies have shown that these phases exhibit the properties of TI [11,12,13,14,15,16] and can be used in spintronics, quantum computers, medicine, security systems, etc. [17,18,19,20].

The valid data on phase equilibria and thermodynamic functions of the corresponding systems are required for the development of synthesis methods and optimization of the conditions for the preparation of novel materials [20,21,22,23].

The known versions of the phase diagrams of the Bi-Se system (see Fig. S1), constructed in different periods of the last century, differ significantly from each other. The first version of the T-x diagram of this system, given in the handbook [24], is constructed based on the data [25, 26] obtained at the beginning of the last century. According to this diagram (Fig.S1a), the system was characterized by the formation of two compounds—Bi2Se3 and BiSe, melting congruently (979 K) and incongruently (878 K), respectively. The second version (Fig. S1b) was built by Abrikosov et al. [27] based on DTA and microstructure analysis results of samples subjected to long-time (3000–3600 h) thermal annealing at 520 K. This diagram reflects three compounds Bi2Se3, BiSe, and Bi2Se. The former melts congruently at 979 K, the latter two decompose by peritectic reactions at 880 and 741 K, respectively. In addition, it was defined that BiSe is a compound with variable composition and has a wide (41.3–55.5 at.% Se) homogeneity range [27]. In [28], the existence of Bi2Se was disputed and it was shown that the compound richest in bismuth has the composition Bi3Se2, and the homogeneity region of BiSe is 46–56 at.% Se. The results of works [29, 30] do not agree with the data [24, 27]. Thus, in [29], based on the XRD data of selected alloys, the existence of three compounds Bi2Se, BiSe, and Bi2Se3 was shown, between which continuous solid solutions are formed. The same opinion is expressed in [30]. The authors of [31] presented another version of the phase diagram of the Bi-Se system based on DTA, XRD, microstructural analysis, and local X-ray spectral analysis of alloys annealed at 500 K for 1200 h (3000 h for a number of samples). This diagram practically coincides with the data of [27], the only difference that instead of the Bi2Se compound, the formation of the Bi3Se2 compound with incongruent melting at 743 K is indicated, and the homogeneity region of BiSe covers the composition range 42.5–54.5 at.% Se.

With that, a number of works [32,33,34,35,36,37] devoted to the synthesis and study of the crystal structure of several bismuth selenides, including those not reflected in the phase diagram were published. Based on the summary of the available data, Okamoto presented [38] a compiled phase diagram (Fig. S1 c), reflecting the compositions of all known and supposed bismuth selenides. As can be seen from this diagram, the author attributed the peritectic reactions at 743 and 880 K to the Bi3Se2 and Bi4Se5, respectively, and the question regarding the temperature and melting character of other bismuth selenides remained open.

Analysis of the literature data also shows that thermodynamic functions were determined only for Bi2Se3 and BiSe [39,40,41,42,43,44,45]. Some works [46,47,48] present the results of modeling and thermodynamic analysis of the Bi-Se phase diagram, in particular, by the CALPHAD method in the approximation of the associated solution model. However, these studies do not consider the nBi2·mBi2Se3 homologous series as individual compounds.

Therefore, we undertook a repeated comprehensive study of the phase relations and thermodynamic functions of the Bi-Se system. The results of the solid-phase equilibria study in the 50–65 at.% Se composition range and the thermodynamic properties of Bi2Se3, Bi3Se4, Bi8Se9, and BiSe were presented in [49].

The present work aimed to refine the phase diagram of the Bi-Se system in the composition range of 0–60 at.% Se and to study the thermodynamic properties of lower bismuth selenides; Bi8Se7, Bi4Se3, and Bi3Se2.

Experimental

Synthesis

The alloys for the study were synthesized by fusion of high-purity elemental bismuth (99.999%) and selenium (99.999%) purchased from Alfa Aesar (Germany) in evacuated (~ 10–2 Pa) quartz ampoules.

When developing the synthesis methodology of samples, we proceeded from the results of numerous studies that the bulk samples of layered phases obtained by the widely used fusion method do not reach an equilibrium state even after a prolonged (2000–3000 h) thermal annealing [27, 28, 31, 50, 51]. This is apparently because, unlike conventional bulk samples, Van der Waals phases obtained in non-equilibrium crystallization conditions (i.e., ordinary cooling of the melt) practically do not undergo any changes during further heat treatment due to very low diffusion between layers.

Taking this into account, to ensure a high dispersion of samples, after alloying, some of them (series I) were quenched by dropping ampoules into ice water from a liquid state (1000 K), followed by annealing at 700 K (1000 h) and 400 K (20 h). For comparison, some alloys with selective compositions (series II) were synthesized by the traditional fusion method and annealed under similar conditions.

Methods

Studies were carried out by the differential thermal (DTA), powder X-ray phase diffraction (PXRD), scanning electron microscopy (SEM), and electromotive force (EMF) methods.

DTA was performed using a NETZSCH 404 F1 Pegasus differential scanning calorimeter. The crystal structure was analyzed by a powder X-ray diffraction (PXRD) technique at room temperature using a Bruker D8 diffractometer (CuKα radiation) in the range of 2θ = 10–70°. High-resolution SEM images were recorded using a TESCAN VEGA3 SBH instrument.

For the thermodynamic studies, the EMF of the concentration cells of the type

$$( - ){\text{Bi}}({\text{Solid}}){\text{|ionicliquid + Bi}}^{{3 + }} |{\text{Bi-Se (Solid) (+)}}$$
(1)

was measured in the 300–450 K temperature range.

Various modifications of the EMF method and different electrolytes (solid and liquid) are successfully used for the thermodynamic study of inorganic systems [52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68]. When studying metal sulfides and selenides, EMF measurements should be carried out at temperatures below the solidus of the corresponding system. The results of a number of works [44, 55, 56, 63,64,65,66] have shown that for such measurements, the most suitable electrolytes were glycerol solutions of halides of alkali and alkaline earth metals. We have shown that in such low-temperature measurements, ionic liquids can also be used as electrolytes [67, 68].

Alloys with the composition of 42–48 at.% Se of series (I) were used as right (positive) electrodes, while the elementary bismuth was used as a negative electrode in the cells of the type (1). The prepared alloys consisted of the following two-phase mixtures: Bi8Se7 + BiSe, Bi4Se3 + Bi8Se7 or Bi3Se2 + Bi4Se3. Note that, despite the fact that we failed to obtain a homogeneous sample of the Bi3Se2 compound, the alloys with the compositions 42.0 and 42.7 at.% Se were two-phase (Bi3Se2 + Bi4Se3) and did not contain traces of elemental bismuth.

An ionic liquid (morpholine formate) with the addition of BiCl3 was used as an electrolyte. In [69], the conductivities of the morpholinium cation-based protic ionic liquids were measured in the 0–100 ºC temperature range and it was shown that morpholine formate has a fairly high (9.92 mS cm−1 at 60  ºC and 29 mS cm−1 at 100 ºC) pure ionic conductivity. To obtain the ionic liquid, we used morpholine (CAS No.110–91-8), formic acid (CAS No. 64–18-6), and anhydrous BiCl3 (CAS No.7787–60-2) purchased from Alfa Aesar (Germany). A detailed description of the method for obtaining an ionic liquid is given in [68, 69]. We have assembled electrochemical cells, the designs of which are described in detail in the works [67, 68].

The first equilibrium EMF values were obtained after holding the concentration cell at ~ 350 K for ~ 40–60 h, and the subsequent ones every 3–4 h after a certain temperature were established. In the entire temperature range of measurements, reproducible EMF values were obtained. The EMF values ​​that did not differ from each other during repeated measurements at a given temperature by more than 0.2 mV, regardless of the direction of temperature change, were recorded as equilibrium values. During the experiment, the EMF of each sample was measured three times at two constant temperatures. For example, for an alloy from the Bi8Se7 + BiSe phase region at 350.6 K, EMF values were 90.95, 91.99, and 91.29 mV Table 1 presents the averaged value of 91.11 mV.

Table 1 Nonvariant equilibria in the Bi-Bi0.4Se0.6 system
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Results and discussion

Phase relations

The study of both series alloys has shown that samples of series (I) are closer to the equilibrium state than samples of series (II).

The results of XRD analysis of alloys of series (I) in the 50–60 at.% Se composition range including compounds Bi2Se3, Bi3Se4, Bi8Se9, BiSe are presented and discussed in [49]. The PXRD patterns of some alloys with high bismuth contents are shown in Figs. 1 and 2. As can be seen, alloys with the compositions Bi8Se7 and Bi4Se3 (Fig. 1) are single-phase, while the alloy with the Bi3Se2 composition (Fig. 2a) is three-phase and in addition to Bi3Se2 contains Bi4Se3 and elemental bismuth. We have also found that intermediate alloys between stoichiometric compounds are two-phase. As an example, Fig. 2b, c shows powder diffraction patterns of samples with the compositions 42 and 45 at.% Se. As can be seen, they consist of two-phase mixtures Bi3Se2 + Bi4Se3 and Bi8Se7 + Bi4Se3, respectively. Note that, the absence of other phases in these alloys is an additional feature of their equilibria.

Fig. 1
figure 1

The PXRD patterns of Bi8Se7 and Bi4Se3 compounds

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Fig. 2
figure 2

The PXRD patterns of alloys of the Bi-Se system with the compositions 40 (a), 42 (b) and 45 at.% Se (c)

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The types and parameters of crystal lattices of bismuth selenides (Table S1) determined based on analysis of diffraction patterns using the TOPAS 4.2 software are in good agreement with the literature data [31,32,33,34,35,36,37].

The results of SEM analysis were in agreement with the XRD data. As an example, Fig. 3 shows the SEM patterns of alloys with 42.9 and 46.7 at.% Se contents, corresponding to the stoichiometry of Bi4Se3 and Bi8Se7. As can be seen, both samples are single-phase and have alike layered structures.

Fig. 3
figure 3

SEM patterns of alloys with 42.9 and 46.7 at.% Te contents

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Based on the DTA and XRD data, we constructed a phase diagram of the Bi-Se system in the 0–60 at.% Se composition range (Fig. 4). This diagram reflects seven bismuth selenides: Bi2Se3 melts congruently at 978 K, the other six compounds melt with decomposition by peritectic reactions (Table 1).

Fig. 4
figure 4

Phase diagram of the Bi-Se system in the 0–60 at.% Se composition range

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Comparison of Fig. 4 with the previously constructed versions (Fig. S1) of the Bi-Se phase diagram shows that all three compounds (Bi3Se2, BiSe и Bi2Se3) indicated in [28, 31] (Fig. S1,b) were confirmed in this work. Moreover, the temperature of the peritectic decomposition of Bi3Se2 (742 K) practically coincides with the literature data. The peritectic temperature for BiSe (866 K) determined by us is somewhat lower than that given in [27, 28, 31] (873–879 K). A wide homogeneity region of BiSe has not been confirmed by us. Instead, in the “homogeneity” region (42–55.5 at.% Se) of this compound, four individual phases (Bi4Se3, Bi8Se7, BiSe, and Bi8Se9) were revealed (Fig. S1, Table 1).

The DTA heating curves of some samples of series I are presented in Fig. 5 and series II in Fig. 6, while the cooling curves are shown in Fig. 7. As can be seen in Fig. 5, the heating curves of this series of alloys are in full agreement with the T-x diagram (Fig. 4). Endothermic effects at 829, 866, and 881 K are related to the temperatures of peritectic decomposition of Bi4Se3, BiSe, and Bi8Se9, respectively. The intermediate effect at 890 K (Fig. 5,c) corresponds to the peritectic decomposition, while at 928 K to the liquidus.

Fig. 5
figure 5

DTA heating curves of some samples of series I

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Fig. 6
figure 6

DTA heating curves of some samples of series II

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Fig. 7
figure 7

DTA cooling curves of some samples

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DTA heating curves of alloys of series II with the same composition (Fig. 6) have a slightly different form. For example, in Fig. 6, thermal effects are corresponding to the eutectic melting (540 K) and the peritectic decomposition of Bi3Se2, which do not exist in Fig. 5a. In addition, the weakest endothermic effect in the 823–853 K range is in better agreement with the phase diagram given in Fig. S-1b. The heating curves (Fig. 6) also contain several overlapping endothermic effects, that are absent in the curves in Figs. 5 b,c which indicates that alloys of series II are in a non-equilibrium state.

DTA cooling curves provide very useful information (Fig. 7). Note that, as expected, they were the same for alloys of both series. On the other hand, there are several exothermic effects on the cooling curves (Fig. 7) after the onset of crystallization, which do not agree with the data (Fig. 1b) about the presence of a wide range of solid solutions based on BiSe. These curves are in better agreement with the heating curves for samples of series II (Fig. 6).

Thus, a comparison of Figs. 5–7 shows that the data presented in Fig. 5 are close to equilibrium, while Figs. 6 and 7 reflect non-equilibrium states and processes. Thermal effects on the cooling curves reflect the series of peritectic formation reactions of the compounds indicated in the phase diagram (Fig. 4). Since, when the samples are cooled, these reactions do not proceed completely and pass into other lower-temperature reactions, including absent in the equilibrium diagram. The closeness of the character of these curves to the heating curves of the series II samples shows that despite prolonged annealing, samples of this series are far from the equilibrium state. This confirms the data [50, 51] on the inefficiency of thermal annealing of coarse-crystalline mixtures of phases with a layered structure.

In conclusion, it should be noted that a comparison of the constructed phase diagram (Fig. 4) with the compiled phase diagram [38] shows that 8 out of 15 compounds indicated in this diagram were not confirmed by us. It should also be noted that according to [38], the Bi3Se2 and Bi3Se4 compounds melt incongruently at 743 and 880 K, as well as the character and melting point of other compounds were not indicated at all. In the T-x phase diagram constructed by us, the peritectic temperature of Bi3Se4 is slightly higher (890 K), and the horizontal at 881 K refers to Bi8Se9.

Thermodynamic properties

EMF measurements data of cells of type (1) were in agreement with the constructed phase diagram (Fig. 4). This allows using them for thermodynamic calculations. The EMF measurements for alloys in the 50–65 at.% Se composition range are presented and processed in [49].

E and T pairs of values ​​for the alloys from the 42–48 at.% Se composition range are shown in Table 2, and the corresponding graphs of temperature dependences of EMF are shown in Fig. 8. Taking into account the linearity of these dependencies, they were processed using the Microsoft Office Excel 2003 software by the least squares method. The linear equations obtained are given in Table 3 in the form recommended in [53, 54]

$$E = a + bT \pm t\left[ {\frac{{\delta _{\text E}^{2} }}{n} + \delta _{\text b}^{2} (T - \overline{T} )^{2} } \right]^{{1/2}}$$
(2)
Table 2 Primary data of EMF measurements for alloys from the Bi8Se7 + BiSe Bi4Se3 + Bi8Se7 and Bi3Se2 + Bi4Se3 phase regions
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Fig. 8
figure 8

Temperature dependences of the EMF of concentration cells (Eq. 1) for cathode phase assemblages a Bi8Se7 + BiSe, b Bi4Se3 + Bi8Se7, and c Bi3Se2 + Bi4Se3

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Table 3 Temperature dependences of the EMF of chains type (1) for alloys of the Bi-Se system in the 300–450 K temperature range
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In Eq. (2), a and b are coefficients, n is the number of pairs of E and T values; \(\overline{T}\) is the average temperature, K; t Student’s t test, and T is the temperature, K. \(\delta _{\text E}^{2}\) and \(\delta _{\text b}^{2}\) are dispersions of individual EMF values and the constant b. Considering the number of experimental points, n = 30, at a confidence level of 95%, the Student’s test is t ≤ 2.

From equations (Table 3) using thermodynamic relationships [52, 53]

$$\overline{{\Delta G}} _{\text {Bi}} = - zFE$$
(3)
$$\overline{{\Delta S}} _{\text {Bi}} = zF\left( {\frac{{\partial E}}{{\partial T}}} \right)_{\text P} = zFb$$
(4)
$$\overline{{\Delta H}} _{\text {Bi}} = - zF\left[ {E - T\left( {\frac{{\partial E}}{{\partial T}}} \right)_{\text P} } \right] = - zFa$$
(5)

the partial molar Gibbs free energy, enthalpy, and entropy of bismuth in alloys were calculated (Table 4). Because, Bi8Se7, Bi4Se3, and Bi3Se2 compounds have an almost constant composition, the above-mentioned partial molar values are thermodynamic functions of the following potentialforming reactions (the physical state of substances is crystalline) [53, 54]:

$${\text{Bi}} + {\text{7BiSe}} = {\text{Bi}}_{{\text{8}}} {\text{Se}}_{{\text{7}}}$$
(6)
$${\text{Bi}} + 0.{\text{75 Bi}}_{{\text{8}}} {\text{Se}}_{{\text{7}}} = {\text{1}}.{\text{75Bi}}_{{\text{4}}} {\text{Se}}_{{\text{3}}}$$
(7)
$${\text{Bi}} + {\text{2Bi}}_{{\text{4}}} {\text{Se}}_{{\text{3}}} = {\text{3Bi}}_{{\text{3}}} {\text{Se}}_{{\text{2}}}$$
(8)
Table 4 Relative partial functions of bismuth in the alloys of the Bi-Se system at 298 K
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From relationships (6)–(8) according to the expressions

$$\Delta _{\text f} Z^{0} (Bi_{8} Se_{7} ) = \overline{{\Delta Z_{\text {Bi}} }} + 7\Delta _{\text f} Z^{0} (BiSe)$$
(9)
$$\Delta _{\text f} Z^{0} (Bi_{4} Se_{3} ) = \frac{4}{7}\overline{{\Delta Z_{\text {Bi}} }} + \frac{3}{7}\Delta _{\text f} Z^{0} (Bi_{8} Se_{7} )$$
(10)
$$\Delta _{\text f} Z^{0} (Bi_{3} Se_{2} ) = \frac{1}{3}\overline{{\Delta Z_{\text {Bi}} }} + \frac{2}{3}\Delta _{\text f} Z^{0} (Bi_{4} Se_{3} )$$
(11)

the standard Gibbs free energy and the enthalpy of formation and from

$$S^{0} \,(Bi_{8} Se_{7} ) = \Delta \bar{S}_{\text {Bi}} + S^{0} (Bi) + 7S^{0} (BiSe)$$
(12)
$$S^{0} \,(Bi_{4} Se_{3} ) = \frac{4}{7}\Delta \bar{S}_{\text {Bi}} + \frac{4}{7}S^{0} (Bi) + \frac{3}{7}S^{0} (Bi_{8} Se_{7} )$$
(13)
$$S^{0} \,(Bi_{3} Se_{2} ) = \frac{1}{3}\Delta \bar{S}_{{Bi}} + \frac{1}{3}S^{0} (Bi) + \frac{2}{3}S^{0} (Bi_{4} Se_{3} )$$
(14)

the standard entropies of bismuth selenides were determined. The data obtained are shown in Table 5. In the calculations, we used the data of [40] on the standard entropy of bismuth (56.7 ± 0.5 J mol−1 K−1) and selenium (42.1 ± 0.2 J mol−1 K−1), as well as the standard integral thermodynamic functions of BiSe [49]. Errors were determined by the error accumulation method.

Table 5 Standard integral thermodynamic functions of bismuth selenides at 298 K
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Table 5, in addition to the results of the present contribution, also shows the data obtained by us [49] for other bismuth selenides. It should be noted that a detailed comparative analysis of our and published data for the Bi2Se3 and BiSe compounds is given in [49].

Conclusions

Using the DTA, XRD, and SEM methods, as well as measurements of the EMF of concentration chains relative to the bismuth electrode, we obtained a new set of mutually consistent data on the phase relationships and thermodynamic functions of the Bi-Se system. The constructed phase diagram represents the formation of the compounds Bi2Se3, Bi3Se4, Bi8Se9, BiSe, Bi8Se7, Bi4Se3, and Bi3Se2 with almost stoichiometric compositions. Except for Bi2Se3 with congruent melting at 978 K, the above-mentioned compounds melt with decomposition by peritectic reactions at 890, 881, 866, 853, 829, and 742 K, respectively. The partial Gibbs free energy, enthalpy, and entropy of bismuth in alloys, standard integral thermodynamic functions of formations, and standard entropies of bismuth selenides were calculated from the EMF measurements at equilibrium conditions.