1 Phase diagram calculation in the TeBiSb ternary system J. Cholinski, M. Lasocka, H. Matyja To cite this version: J. Cholinski, M. Lasocka, H. Matyja. Phase diagram calculation in the TeBiSb ternary system. Revue de Physique Appliquee, 1977, 12 (1), pp.15. < /rphysap: >. <jpa > HAL Id: jpa Submitted on 1 Jan 1977 HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
2 even The Chalcogenide On On by 2.1 lack the REVUE DE PHYSIQUE APPLIQUÉE Classification Physics Abstracts PHASE DIAGRAM CALCULATION IN THE TeBiSb TERNARY SYSTEM J. CHOLINSKI (*), M. LASOCKA and H. MATYJA Institute of Materials Science and Engineering, Warsaw Technical University Narbutta 85, Warszawa, Poland (Reçu le 9 octobre 1975, révisé le 23 avril 1976, accepté le 25 août 1976) 2014 Résumé. calcule le diagramme de l équilibre liquidesolide dans le système ternaire TeBi Sb par une théorie de solution régulière associée, utilisée par Laugier Abstract. the basis of the theory of regular associated solutions the equilibrium phase diagram of the ternary system TeBiSb was calculated by Laugier s method. 1. Introduction. systems, among which tellurium alloys represent the most numerous group, are increasingly often employed, mainly in electronics, both in the crystalline and noncrystalline states . They show the interesting property of the dependence of the energy gap on alloy composition, which has been intensively investigated for the TeSn Pb system [1, 4, 5]. Control of the composition of the alloy permits gaining control of its electronic properties. Since different techniques for growing alloy crystals have already been developed , it is now of interest to determine the dependence of the concentration on temperature. This can be achieved either by means of time and laborconsuming measurements which permit plotting the equilibrium diagram , or more frequently means of theoretical calculations of equilibrium curves on the basis of the thermodynamics of nonideal solutions [10, 11, 12]. There is no general calculation method being common for various systems. However, in the case of some types of systems, the application of wellfounded assumptions and simplifications affords results consistent with the experimental ones [9, 13]. Consequently, these methods are utilized also for systems which have not been experimentally studied. In this manner equilibrium diagrams have been determined for a number of binary systems. Recently, the methods has been applied also to many ternary systems [e. g. 7,10,12]. 2. Method of calculation. SYSTEM TeBiSb. aim of the present study was to calculate the equilibrium diagram for the TeBiSb system which is homologous to the system TeSnPb. There is a simila (*) Institute of Industrial Chemistry, Rydygiera 8,01793 Warszawa, Poland. rity in the shape of the equilibrium curves in the partial binary systems (TeBi, TeSb, BiSb and TeSn, TePb, SnPb, respectively ), and in the pseudobinary systems formed by the stoichiometric compounds (Bi2Te3Sb2Te3  and SnTePbTe [11, 17], respectively). Both systems exhibit a continuous change in the composition of the pseudobinary compound (BixSb1x)2 Te3  and (SnxPb1x)Te , which is associated with a continuous change in electronic properties . Both systems, TeBiSb and TeSnPb, satisfy the conditions for application of the theory of regular associated solutions to calculation of the equilibrium diagram. These conditions are as follows [7, 910] : of solubility in the binary systems TeA~(Te Bi, TeSb) and TeAIV(TeSn, TePb), and complete solubility in the binary systems BiSb and Sn Pb , pseudobinary systems formed by the stoichiometric compounds Bi2Te3Sb2Te3 and SnTePbTe [16, 17] exhibit complete miscibility in both liquid and solid states, and form a continuous series of solid solutions satisfying Vegard s law, while the liquidus and solidus curves follow a monotonic course and are situated at a short distance from each other. Since the above conditions are satisfied, the liquid state can be considered an equilibrium mixture of five kinds of species : Te, Bi, Sb, Bi2Te3, Sb2Te3 in the system TeBiSb, and Te, Sn, Pb, SnTe, PbTe in the system TeSnPb, respectively, and the solid state can be regarded as an equilibrium mixture of two kinds of species : Bi2Te3, Sb2Te3 in the system TeBiSb, and SnTe, PbTe in the system TeSnPb, respectively. The respective relationships between the total num Article published online by EDP Sciences and available at
3 The 2 bers of moles of the components A, B and C of the ternary system (ni, n2, n3, respectively) and the numbers of moles of the species A2 C3, B2 C3, A, B, C are : Expressions for the molar fractions related to the respective numbers of moles are as follows : An analogous formula can be written for B2 C3. The term in the last bracket is small and can be neglected. Consequently : From (4), (5), (7) we obtain : 2.2 SYSTEM OF EQUATIONS FOR THE SOLIDUS AND LIQUIDUS CURVES. dependence of the compositions of the solid state and liquid state on the temperature was derived according to , with consideration of the changes upon transition to a system forming associates of type A2 C3 and B2 C3 instead of type AC and BC. At the solid stateliquid state equilibrium the following relationships between chemical potentials J" AZC3(~"B2c3) in the solid state and chemical potentials lia, pc (MB, pc) in the liquid state hold true : Expressions for the dissociation constants Ki and K2 are as follows : The chemical potentials of A2 C3, B2 C3 in the solid state are expressed as follows : From these expressions we obtain : From (1), (2), (3), we obtain : where y is the mole fraction of A2 C3 in solid state, ms o, A2C is the chemical potential of pure A2 C3 in solid state. The following formula developed by  expresses ~Â c3 as a function of the chemical potentials of A, C in the stoichiometric liquid solution Jl1, pà, the melting point of A2 C3 Tml, the entropy of fusion of A2 C3 I1Sm1 and the specific heat difference of AZ C3 in the solid and the liquid states I1CP1 : The degrees of dissociation are defined as follows : Using (1), (2), (3), (11), (12) one can express XÂ2C3 xb2~3 in terms of dl, d2, Xl, X2 : Now we introduce the parameters ~3~, P2 representing the values of dl, d2 for the stoichiometric liquid. Thus, from (9) we obtain :
4 Comparison 3 FIG. 1. of the calculated and the expérimental liquidus curves for parts of the binary systems TeAv. Analogous expressions can be written for B2 C3. To obtain expressions for XA2C3> XB2C,5 we assume very strong association, i. e. ~1 l, ~32 1. In this case the right side of equation (14) vanishes and we obtain the following expressions for x A2C3 xb2~3 : However, upon acceptance of the above assumption, no expressions for the respective molar equations are obtained when X This is so, because at X3 > 0.6, fli, ~32 1 is equivalent to xa, x~ N 0, whereas, at X3 0.6, ~il, ~2 1 is equivalent to xc Eé 0. Thus, by assuming very strong association (~1, ~2 negligibly small) we obtain at x3 > 0.6 two constraints, which enables us to find xazc3, xb2c3. At x3 0.6 we have only one constraint, xc N 0, and thus only xa2c3 + xb2c3 can be determined. The next problem is to choose a form of ya2c3, 7B2C3 as a function of composition. We assume an empirical form of In ya2c3 (ln "ib2c3) in the binary solution (quadratic function) : As can be seen from figure 1, this function fits the binary liquidus quite well. The expression for In yaz~3 (ln y12c3) in the ternary solution (which in the particular case of a binary solution takes the form presented in (16)), satisfies the boundary conditions (17) : is as follows : When substituting (18) and the expression for XÂZC3 XB2C3 calculated from (13) in (10), we obtain :
5 4 From (8), (15), (19) we obtain the ternary solidusliquidus equations (the solid solution is assumed to be regular, For the pseudobinary liquidus : xl = 0.4 x ; x2 = 0.4(1 x) ; di = Pl ; d2 = P2. X,42CII XB1C3 are expressed by (13). As we assume ~1, ~2 1, we can write lj6i~l~2~1~;l+4~i~l+4j82=l+4~. Hence we obtain : From (8), (15), (19), (21), the pseudobinary solidusliquidus equations are : 2.3 THERMODYNAMIC DATA AND PARAMETERS REQUIRED FOR CALCULATION OF PHASE DIAGRAM. Data on the melting entropy of the compounds A2 C3 and B2 C3, as well as on the respective melting temperatures were obtained from ref. . The phase diagrams for the systems BiTe and SbTe were taken from  and for the pseudobinary system BiZTe3 Sb2Te3 from . The numerical data are recorded in table I. TABLE 1 Values of entropy of melting and melting temperatures for the systems Bi2Te3 and SbZTe~ The parameters ai, ai, (X2, a2, j5i, P2 were determined by the least squares method, to obtain a set of equations (20) giving a best fit to the liquidus curves for the binary systems. The interaction parameter w was determined on the basis of the set of equations (22). These parameters are presented in table II. 3. Results of phase diagram calculations. The phase diagram for the TeBiSb system calculated by the method described above is recorded in figure 2 ; the experimental results obtained for this system  are also shown in the figure. As can be seen in figure 2, the experimental and calculated results are consistent. The calculated values of parameters seem to justify the abovementioned assumptions. The value of the interaction parameter w in the solid state is small. This confirms the validity of considering the solid state to be a regular solution. The small values of the dissociation parameters in the liquid state justify the acceptance of the assumption of very strong association. As regards the activity coefficient in the liquid state, the parameter values occurring in the expression for 1 c, of the compound cannot be compared with the TABLE II Parameters used in calculations
6 Phase Thanks The 4. Summary. The phase diagram for the BiSbTe system has been calculated using the approach of the theory of regular associated solutions, with a modification of the system of equations applied in ref. . The necessary parameters were determined by the least squares method. Comparison of the calculated values with the experimental results points to their fairly good agreement. 5 FIG. 2. diagram calculated for the system BiSbTe, with indication of the experimental values (dashed curves). respective values obtained for other systems (forming compounds AC), on account of the different form of this expression. However, the fit this expression is rather good for binary systems. Acknowledgement. authors are grateful to the U.S. National Science Foundation for its support of this work under Grant GF Note. to the reviewer s kind help, most recent Szapiro s  extension of the RAS model to ternary systems came to our attention. Although both Laugier s  and Szapiro s  treatments refer to systems forming compounds AC and ~C and are based on the same SLE equations, their equations for the activity coefficients in the liquid phase are different. Furthermore, Szapiro s mole fractions xa~ and xoc are established numerically, while Laugier s are calculated analytically by assuming very strong association. Szapiro has accounted for the variation of the dissociation parameters with temperature. References  HILSUM, C., in : Electronic Materials, Eds. N. B. Hannay, U. Colombo (Plenum Press, New York) 1973, Chapters 3 and 7.  BOSNELL, J. R., THOMAS, C. B., Phil. Mag. 27 (1973) 665.  LASOCKA, M., MATYJA, H., Postepy Fiz. 25 (1974) 373.  MELNGAILIS, I., J. Physique 29 (1968) 84.  SELIMOVA, L. E., ABRIKOSOV, N. Ch., Izv. AN SSSRNeorg. Mat. 9 (1973)  LINDEN, K. J., KENNEDY, C. A., J. Appl. Phys. 40 (1969)  LAUGIER, A., CADOZ, J., FAURE, M., MOULIN, M., J. Crystal Growth 21 (1974) 235.  ABRIKOSOV, N. Ch., PORECKAJA, L. V., Izv. AN SSSR Neorg. Mat. 1 (1965) 503.  PANISH, M. B., ILLEGEMS, M., Progr. SolidState Chem. 7 (1972) 39.  LAUGIER, A., Revue Phys. Appl. 8 (1973) 259.  HARRIS, J. S., LONGO, J. T., GERTNER, E. R., CLARKE, J. E., J. Crystal Growth 28 (1975) 334.  SZAPIRO, S., J. Electron Mat. 5 (1976) 223.  STRINGFELLOW, G. B., J. Phys. Chem. Solids 34 (1973)  HANSEN, M., ANDERKO, K., Constitution of Binary Alloys (McGrawHill, New York) 1958, 2nd ed.  ELLIOTT, R. P., Constitution of Binary Alloys, First Supplement (McGrawHill, New York)  MCSMITH, J., KNIGHT, R. J., SPENCER, C. W., J. Appl. Phys. 33 (1962)  STEIIVINGER, J., J. Appl. Phys. 41 (1970)  ABRIKOSOV, N. Ch., MAKAREEVA, E. G., Izv. AN SSSR Neorg. Mat. 9 (1973)  VIELAND, L. J., Acta Metal. 11 (1963) 137.  HOWLETT, B. W., MISRA, S., BEVER, M. B., Trans. AIME230 (1964) 1367.
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