The phase diagram of the quasiternary system Ga2kHg3 − 3kSe3lTe3 − 3l

Journal of Alloys and Compounds 391 (2005) 42–48

The phase diagram of the quasiternary system Ga2kHg3 − 3kSe3lTe3 − 3l Martina Kerkhoff, Volkmar Leute∗ Institut f¨ur Physikalische Chemie der Universit¨at M¨unster, Corrensstrasse 36, 48149 M¨unster, Germany Received 30 July 2004; accepted 25 August 2004

Abstract The quasiternary system Ga2k Hg3 − 3k Se3l Te3 − 3l was investigated by X-ray phase analysis and electron microprobe analysis. The system yields both ordered regions and spinodal miscibility gaps. Along the quasibinary edges, extended regions of solid solutions occur. The phase diagram can be modeled by a Gibbs energy function for a sub-regular system with ordering tendencies. © 2004 Elsevier B.V. All rights reserved. Subject classification: Ga2 Se3 ; Ga2 Te3 ; HgSe; HgTe Keywords: Quasiternary systems; Thermodynamics; Ordering processes; Solid solutions; Lattice constants; Chalcogenides

1. Introduction The quasiternary system Ga2k Hg3 − 3k Se3l Te3 − 3l seems to be very simple because all four binary components HgSe, HgTe, Ga2 Te3 and Ga2 Se3 crystallize, at least at high temperatures, in the cubic zinc-blende structure. Moreover, the quasibinary edge systems Hg(Sel Te1 − l ) [1] and Ga2 (Sel Te1 − l )3 [2] show complete solid solubility. But, considering the stoichiometry of the Ga chalcogenides, the situation will become more complicated because of the structural vacancies in their cationic sublattice, i.e. only two-thirds of the cation sites of the zinc-blende lattice are occupied by Ga. Including these structural vacancies as V into the formulas, the two quasibinary edge systems with a mixed cation sublattice have to be written as (Ga2k Vk Hg3 − 3k )Se3 [3] and (Ga2k Vk Hg3 − 3k )Te3 [4]. From the investigation of these two systems, we know that the structural vacancies cause a tendency towards ordering near k = 3/8 and 3/4. The super structures at k = 3/8 occur only at lower temperatures (T < 800 K), where the annealing times for equilibration are very long. Therefore, we restricted the experiments to T = 900 K. At this temperature, as a consequence of the ordering tendencies, only the chalcopyrite ∗

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structure around k = 3/4 should be formed. Thus, it is to be expected that, at 900 K the field of complete solid solubility in the phase diagram is intersected by a super structure phase with the stoichiometry Ga2 VHg(Sel Te1 − l )4 that is neighbored on both sides by miscibility gaps. 2. X-ray measurements The experimental data for the phase diagram were gathered by X-ray diffraction phase analysis (XRD) and electron microprobe analysis (EMA). Fig. 1 shows the overall compositions of all quasiternary samples that were prepared from the binary components to detect the one- and two-phase regions. The methods of preparation are described in [2–4]. The X-ray experiments at samples with k = 3/4 show that, in contrast to the expectation, there is no complete solid solubility between HgGa2 VSe4 and HgGa2 VTe4 . The telluriderich chalcopyrite phase extends only from l = 0 up to l ≈ 0.3 and the selenide-rich one from l = 1 down to l ≈ 0.9 (cf. Fig. 1). 2.1. The experimental determination of lattice constants The homogeneous one-phase samples were used to determine the structures and their lattice parameters. Besides

M. Kerkhoff, V. Leute / Journal of Alloys and Compounds 391 (2005) 42–48

Fig. 1. The phase square with the samples investigated by X-ray diffraction. The sites of the following symbols correspond to the overall compositions of samples that were either homogeneous or heterogeneous after equilibration, with a cubic set of reflections for Hg chalcogenide-rich samples (◦, ) and for Ga chalcogenide-rich samples (
), with a tetragonal set of reflections for selenide-rich samples (+) and for telluride-rich samples (×). The superpositions of two or three of these symbols at one site indicate two-phase or three-phase samples; the superpositions of two cubic symbols indicate a spinodal demixing.

the cubic zinc-blende structure, only the tetragonal structure of the defect chalcopyrite could be detected. As this is a superstructure of the zinc-blende lattice, a so-called pseudo-cubic lattice constant, defined by a∗ = (a2 c/2)1/3 , will be attributed to this lattice. Considering the composition dependences of the lattice constants for the quasibinary edge systems Ga2 VSe3l Te3 − 3l [2], Hg3 − 3k Ga2k Vk Se3 [3], Hg3 − 3k Ga2k Vk Te3 [4], Hg(Sel Te1 − l ) [5] and including the cubic and pseudo-cubic lattice constants of the quasiternary samples, as determined for the present paper, the composition dependence of the lattice constant in the quasiternary system can be described by a (k, l)/pm = 645.94 − 57.16k − 8.03k2 + 9.62k3 − 10.72kl + 14.30k2 l − 11.36k3 l − 37.44l. (1) The highest deviations of the values calculated by use of Eq. (1) from the above-mentioned literature data for the quasibinary edge systems and from the X-ray data for the quasiternary samples do not exceed ±0.4 pm. Fig. 2 shows a net of ‘iso lattice constant lines’ in steps of 5 pm calculated from Eq. (1). 2.2. The domain structure The homogeneous one-phase samples with k = 3/4 that were annealed at 900 K show, in addition to the reflections of the normal chalcopyrite structure, many sharp reflections with rather low intensity. Corresponding X-ray patterns can also be observed when Hg is substituted by Cd [6]. It is very astonishing that after quenching to room temperature, homogeneous samples that were annealed at high

43

Fig. 2. Projection of lines with constant values of the cubic or pseudo-cubic lattice constant, a∗ (k, l) = const, onto the square of the quasiternary system; the ‘iso lattice constant lines’ are plotted in steps of 5 pm.

temperatures (900 K) yield X-ray patterns with much more reflections then belonging to the chalcopyrite structure that is realized when the same samples are annealed at lower temperatures. Though ordering should decrease with increasing temperature, the high temperature pattern looks like one of a higher ordered lattice, but no single superstructure of the zinc-blende lattice could be found that allows explaining all reflections. As at high enough temperatures the energy difference between similar superstructures gets more and more negligible, it seems to be possible that at such high temperatures domains of several different superstructures of the zinc-blende lattice could exist simultaneously because of energy fluctuations. We think that by quenching from the high annealing temperature (900 K) down to room temperature, big enough domains of some of such superstructures are preserved in addition to the energetically most favorable chalcopyrite structure. Practically, each of the observed additional reflections can be assigned to one of these possible superstructures. Therefore, we call the structure yielding this reflection-rich X-ray pattern the ‘domain structure’. Using only the reflections belonging to the tetragonal chalcopyrite structure we calculated the lattice constants of this phase in dependence on the mole fraction l. Fig. 3a shows that the lattice constant a changes linearly with l, whereas c (Fig. 3b) at first is constant and then decreases linearly for l > 0.08. The c/a-ratio (Fig. 3c) starts at l = 0 with the ideal value of two. This means that the lattice of the pure HgGa2 VTe4 is undistorted, whereas the lattice becomes more and more distorted when Te is successively substituted by Se. This distortion is connected with a splitting of characteristic reflections (e.g. (0 0 4), (0 2 4), (1 1 6)) of the undistorted tetragonal chalcopyrite structure into doublets. According to Gastaldi [7], the increasing lattice distortion of the Ga-containing chalco-

44

M. Kerkhoff, V. Leute / Journal of Alloys and Compounds 391 (2005) 42–48

Fig. 3. Lattice constants of the telluride-rich part of the section at k∗ = 3/4 for the tetragonal chalcopyrite structure; (a) lattice constant a, (b) lattice constant c, (c) c/a-ratio of the lattice constants.

genides is coupled with an increasing ordering in their cation sublattice.

3. The phase diagram of the quasiternary system Ga2k Hg3 − 3k Se3l Te3 − 3l 3.1. The determination of tie lines and three-phase fields by use of the electron microprobe analysis As shown by Fig. 1, the X-ray patterns of most samples that were annealed at 900 K yield two or even three phases in equilibrium. For each of these phases, the cubic or pseudocubic lattice constant can be determined, but in quasiternary systems the knowledge of the lattice constants yields only the corresponding ‘iso lattice constant lines’ (cf. Fig. 2), but it is not sufficient to determine the boundaries of the miscibility gaps. For this purpose, we investigated additionally the compositions of the separated phases by electron microprobe analysis. Because of the restricted lateral resolution of this method, the composition of these phases can only be determined with sufficient accuracy, if the precipitations of the different phases are larger than about 5 ␮m. If, as in most samples, this condition is not fulfilled, one can still measure the local mean composition at many points statistically scattered over the surface of the sample. In two-phase samples, the compositions corresponding to these points must be situated in the phase diagram along the tie line between the equilibrium compositions of the two phases. Thus, the cloud of these composition points can be

approximated by a straight line indicating the corresponding tie line. The two points of section of this straight line with the ‘iso lattice constant lines’ belonging to the two phases mark the compositions of the end points of the equilibrium tie line (Fig. 4a). In three-phase samples, the cloud of composition points measured by EMA extend over the whole region of the three-phase triangle with highest density near that corner that is nearest to the point marking the overall composition of the sample (Fig. 4b). Fig. 5 shows the phase diagram of the quasiternary system as determined by X-ray phase analysis and electron microprobe analysis. The cubic one-phase field extends only along the edges of the phase square. It is interrupted by miscibility gaps (Z2–Z5, Z7) caused by the tetragonal superstructure near k = 3/4. Moreover, there are 2 two-phase fields (Z1, Z6) between two cubic regions. The 3 three-phase fields (D1–D3) represent always two cubic and one tetragonal phase in equilibrium. 3.2. Theoretical consideration To understand the experimentally determined phase diagram, we will try to calculate it from data that we know from the quasibinary edge systems with as few as possible additional ternary parameters. The calculation of quasiternary phase diagrams of systems represented by a square such as Hg3 Te3 /Ga2 Te3 / Ga2 Se3 /Hg3 Se3 can be done according to the formalism given in [8,9]. For this purpose, the mean molar Gibbs energy g for the given temperature should be

M. Kerkhoff, V. Leute / Journal of Alloys and Compounds 391 (2005) 42–48

45

Fig. 5. The phase diagram as derived from measured tie lines.

gm = TR ln[kk (1 − k)(1−k) ll (1 − l)(1−l) ].

(4)

The excess term can be split into five terms: one for each of the four quasibinary systems and one for the quasiternary interactions. E E E gE = l g((Hg,Ga)Se) + (1 − l) g((Hg,Ga)Te) + k g(Ga(Se,Te)) E + (1 − k) g(Hg(Se,Te)) + k (1 − k) R G0

(5)

3.2.1. The standard Gibbs energy of the exchange reaction The last term in Eq. (5) represents the ternary interaction and considers interactions between components of the two sublattices. R G0 is the standard Gibbs energy for the solidstate reaction between the four pure binary components [8]. Fig. 4. Composition points (a) within a two-phase field, (b) within the threephase fields as derived from electron microprobe analysis; ♦, 夽, ,
,

= overall compositions of the samples. The straight lines characterize the tie lines, their end points (the same symbols as above, but smaller) were determined according to the method as described in the text. To determine the corners of the triangles, the X-ray data of the three-phase samples were also evaluated.

known as function of the independent mole fractions k and l. Generally, the mean molar Gibbs energy of such systems can be described by the expression: g = g0 + gm + gE

(2)

The standard term g0 , representing the Gibbs energy of the pure independent components, reads: g0 = k µ0 (Ga2 Se3 ) + (l − k) µ0 (Hg3 Se3 ) + (1 − l) µ0 (Hg3 Te3 ).

Hg3 Te3 + Ga2 Se3 → Hg3 Se3 + Ga2 Te3

(6)

Principally, the value of R G0 could be calculated from the standard chemical potentials at the equilibrium conditions, but even if these data were known, their errors of measurement are generally so high that in most cases even the sign of R G0 would be uncertain. Thus, this term has to be determined by fitting the calculated phase diagram to the experimental results. 3.2.2. The interaction parameters If the quasibinary systems are treated as behaving subregularly, i.e. if the interaction parameters depend linearly on composition, and if excess entropy is allowed too, then the normal parts of the excess terms [10] read as: E,n g((Hg,Ga)Se) = k (1 − k) [(α((Hg,Ga)Se) − T σ((Hg,Ga)Se)

+ k (β((Hg,Ga)Se) − T τ((Hg,Ga)Se) )]

(7)

(3)

The mixing term considers the configurational entropy due to an ideal mixing of the components on each sublattice:

E,n g((Hg,Ga)Te) = k (1 − k) [(α((Hg,Ga)Te) − T σ((Hg,Ga)Te) )

+ k (β((Hg,Ga)Te) − T τ((Hg,Ga)Te) )]

(8)

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M. Kerkhoff, V. Leute / Journal of Alloys and Compounds 391 (2005) 42–48

E,n g(Ga(Se,Te)) = l (1 − l) [(α(Ga(Se,Te)) − T σ(Ga(Se,Te)) )

+ l (β(Ga(Se,Te)) − T τ(Ga(Se,Te)) )]

(9)

E E E + (1 − l) g((Hg,Ga)Te) + k g(Ga(Se,Te)) gE = l g((Hg,Ga)Se)
E,ord E + (1 − k) g(Hg(Se,Te)) + k (1 − k) R G0 + gk∗ (l) k∗

E,n g(Hg(Se,Te)) = l (1 − l) [(α(Hg(Se,Te)) − T σ(Hg(Se,Te)) )

+ l (β(Hg(Se,Te)) − T τ(Hg(Se,Te)) )]

(10)

3.2.3. The ordering parameters If, as in the present system, ordering tendencies are involved that arise only from the formation of superstructures of the zinc-blende structure, then the whole quasiternary system can formally be treated as one single phase. In this case, the excess Gibbs energy terms of those quasibinary subsystems, i, showing ordering tendencies have to be extended by excess terms gE,ord considering the thermodynamic effects of ordering in dependence on composition and temperature.
E,ord giE = giE,n + (11) gi,k∗ k∗

In previous papers [3,4,6,10] it was shown that in quasibinary systems for each stoichiometric composition k* , where superstructures are formed, these ordering terms can be described by a function of the Gaussian type:   ∗ )2 (k − k E,ord gi (12) = −Tr G0k∗ exp − 2σk2∗ with Tr G0k∗ =



In the present case for k∗ = 3/4 the ordering parameters must be given explicitly as functions of l. Provided, the transformation Gibbs energy has an extreme value at the position (lextr |Tr Gextr ), the function Tr G(l) can be approximated by branches of two parabolic functions Tr G0k∗ ,0 (l) = a0 + b0 l + c0 l2

0

−T

Tr Sk0∗

for T < TTr,k∗ for T ≥ TTr,k∗

This set of ordering equations holds for both the system (Ga2k Vk Hg3 − 3k )Se3 and the system (Ga2k Vk Hg3 − 3k )Te3 . At T = 900 K, only one type of ordered structures at k∗ = 3/4 occurs. Tr G0k∗ describes the difference between the Gibbs energy of the totally disordered state and the ordered state of the superstructure with the exact stoichiometry k∗ . Tr Hk0∗ and Tr Sk0∗ are the molar enthalpy and entropy for the transition from the fully ordered state to the totally disordered state of the pure stoichiometric compound. The exponential expression with its characteristic term σk∗ describes, how fast the ordering contribution decreases with increasing distance k − k* from the exact stoichiometric composition. Generally, the ordering parameters at k∗ depend on the mole fraction l too. The first two terms of the sum in Eq. (5) would give rise to a linear dependence on l for the whole ordering term of Eq. (12). However, if it is to be assumed that ordering tendencies do not change linearly with l, but should show, for example, an extreme value at an intermediate mole fraction l, then it is better to add a separate term of the form gkE,ord to Eq. (5) instead of including the ordering terms into ∗ the quasibinary terms as in Eq. (11):

for

0 < l < lextr

(14)

for

1 > l > lextr

(15)

and Tr G0k∗ ,1 (l) = a1 + b1 l + c1 l2 with a0 = Tr G((Hg,Ga)Te) , −2(Tr G((Hg,Ga)Te) − Tr Gextr ) , lextr

b0 =

(Tr G((Hg,Ga)Te) − Tr Gextr )

c0 =

(16)

lextr 2

and a1 = Tr G((Hg,Ga)Se) − b1 =

Tr Hk0∗

(13)

(Tr Gextr − Tr G((Hg,Ga)Se) ) (1 − lextr )2

(2lextr − 1),

2 (Tr Gextr − Tr G((Hg,Ga)Se) ) lextr

c1 = −

(1 − lextr )2 (Tr G((Hg,Ga))Se − Tr ) (1 − lextr )2

,

.

(17)

As to the parameter σk∗ in gkE,ord (l), we will treat it here as ∗ linearly depending on the mole fraction l. 3.3. The calculation of the phase diagram Provided any ternary interaction between the cation and anion sublattices could be neglected and the ordering enthalpy and entropy of the chalcopyrite phase could be linearly interpolated between l = 0 and 1, then the phase diagram could be calculated using only the interaction and ordering parameters of the quasibinary edge systems. Tables 1 and 2 show these parameters as taken from previous papers. Table 1 Interaction parameters System

Lit.

α (J/mol)

α (J/mol)

β (J/mol)

τ (J/mol)

(Ga2k Vk Hg3 − 3k )Se3 (Ga2k Vk Hg3 − 3k )Te3 Ga2 V(Se3l Te3 − 3l ) Hg3 (Se3l Te3 − 3l )

[3] [4] [2] [5]

1060 −5000 11500 9000

3.5 0.0 0.0 0.0

360 −7000 −8000 −4500

1.2 0.0 0.0 0.0

M. Kerkhoff, V. Leute / Journal of Alloys and Compounds 391 (2005) 42–48

47

Table 2 Ordering parameters System

Lit. k∗

(Ga2k Vk Hg3 − 3k )Se3 (Ga2k Vk Hg3 − 3k )Te3

[3] [4]

Tr Hk0∗ (J/mol)

0.75 5000 0.75 2350

Tr Sk0∗ (J/mol)

TTr,k∗ (K)

σk

3.94 1.30

1269 1808

0.03/0.03 0.04/0.02

Table 3 Ternary parameters R G0 (J/(mol K))

Tr G0extr (J/mol)

lextr

40,000

50

0.75

Fig. 6a shows that, for this case, the miscibility gaps are restricted to the neighborhood of the ordered region and that the chalcopyrite phase extends from the telluride edge up to the selenide edge of the phase square in contradiction to the experiments. To obtain more extended miscibility gaps, obviously there must be within the ternary region also a tendency to spinodal demixing. Such behaviour is connected with a positive ternary interaction parameter. To demonstrate this effect, we neglected the ordering tendencies and used a value of R G0 = 40 kJ/mol. Fig. 6b shows that under these conditions a broad closed spinodal miscibility gap exists within the phase square. The tie lines show a positive slope. If we now assume that both the tendency to demixing due to the ternary interaction parameter and the tendency to ordering, resulting in the formation of the chalcopyrite phase occur simultaneously, we could expect to get a phase diagram describing the real behaviour. But there is still a discrepancy in so far as these calculations yield a complete miscibility of HgGa2 VSe4 and HgGa2 VTe4 . Obviously, we have to assume that the ordering tendency does not change linearly with the mole fraction l, but decreases distinctly by mixing Se and Te in the anion sublattice. Choosing Tr Gextr = 50 J/mol and lextr = 0.75 as the coordinates for the minimum of the Gibbs energy of transformation we calculated a phase diagram as given in Fig. 7.

Fig. 7. The phase diagram as calculated with the parameters of Tables 1–3.

4. Conclusions 4.1. Clusters and domains The topology of the calculated phase diagram in Fig. 7 is identical with that of the experimentally determined phase diagram shown in Fig. 5. Obviously the complexity of this quasiternary system is caused by the superposition of a tendency to spinodal demixing and a trend to form an ordered structure at k∗ = 3/4. At first sight this seems to be inconsistent, but the spinodal demixing is determined by the interaction parameters that depend on the first coordination sphere, whereas the ordering tendencies are caused by farther reaching interactions including neighbors of second or higher coordination shells. The interaction parameters being used to describe the thermodynamic behaviour of sub-regular systems can be explained in a good approximation by the population probabilities of tetrahedrally coordinated clusters [11]. For example, the occurrence of a spinodal miscibility gap, as shown in Fig. 6b, is favored if the probabilities of the clusters describing

Fig. 6. (a) The phase diagram for the case of R G0 = 0 and for linear interpolation of the ordering Gibbs energy at k∗ = 3/4; (b) the phase diagram for the case of R G0 = 40 kJ/mol and neglecting the ordering tendencies at k∗ = 3/4.

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the pure binary components like (HgHgHgHg)Te and Hg(TeTeTeTe) for HgTe or (GaGaGaV)Se, (GaGaVV)Se, Ga(SeSeSeSe) and V(SeSeSeSe) for Ga2 VSe3 are increased and the probabilities of mixed clusters like Hg(SeSeTeTe) or Ga(SeSeTeTe) are decreased relative to the ideal statistical distribution of these clusters. In solid solutions, the probabilities of these clusters are determined by their energies and in the quasibinary edge systems they can be described on principle as functions of only the interaction parameters. At special stoichiometric compositions, the agglomeration of several tetrahedrally coordinated clusters to so-called ‘domains’ can be connected with a gain of energy. The difference between the molar energy of such a domain and the sum of the molar energies of the individual clusters is the transformation energy, in our example for the ordering process at k∗ = 3/4. A domain of four clusters of type (HgGa2 V)X with X = Se or Te forms the smallest building unit from which the tetragonal unit cell of the realized chalcopyrite structure (space ¯ as described by the lattice constants of group No. 82, I4), Fig. 3, can be constructed. If the four clusters were connected in a little different way, one would obtain the domain from ¯ m) which the stannite structure (space group No. 112, I42 can be constructed. The body centered unit cells of each of these two very similar structures are formed in both cases from two such four-cluster domains. To construct a primitive unit cell of the same dimensions (space group No. 121, ¯ P42c), one needs a single domain composed of eight clusters of the same type. If enough domains of the same type are fitted together, the X-ray patterns will show the reflections of this superstructure. Because at high temperatures the energy differences between such differently ordered structures become negligible, all possible superstructures will exist simultaneously and after quenching to room temperature the X-ray pattern of this so-called ‘domain structure’ will yield a collection of reflections of such differently ordered structures with highest intensities for the reflections of that structure that is most stable at room temperature, i.e. of the chalcopyrite structure. It is unknown, whether Se and Te in the anion sublattice of the chalcopyrite structure, at 0 < l < 1, are randomly distributed or can also be ordered at special stoichiometric compositions. But we know from the calculation of the phase diagram that, when the anion sublattice is mixed, the tendency of clusters of type (HgGa2 V)X to agglomerate to energetically more favorable domains decreases and reaches its minimum at l = 3/4.

The parameter σk∗ in Eq. (12) describes the ability of the ordered structures to include clusters with deviating composition without destroying the order. Thus, a superposition of spinodal demixing and ordering processes is possible. We have seen that at the one side the deviation of the solid solution from ideal behaviour can be described by the individual cluster probabilities and at the other side the ordering processes by the tendency of special clusters to form domains. In terms of the Gibbs energy, these are additive properties, i.e. demixing and ordering can occur simultaneously in one and the same sample. 4.2. The solid-state reaction The positive slope of the tie lines in the two-phase fields means that the equilibrium of the solid-state reaction described in Eq. (6) is shifted to the left side. Thus, we can conclude from the phase diagram (Figs. 5 and 7) that a reaction between HgTe and Ga2 Se3 will yield a relatively simple reaction path with only one interface between a HgTe-rich and a Ga2 Se3 -rich cubic phase. In this case, only a small diffusion exchange, but no separated product layer will occur, i.e. the original interface will be stable. But especially when the interdiffusion between Hg and Ga is distinctly faster than the chalcogen exchange in the anion sublattice, the reverse reaction between HgSe and Ga2 Te3 will result in a complicated multi-phase layer with tetragonal chalcopyrite phases and cubic HgTe-rich and Ga2 Se3 -rich reaction products.

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