Phase States of the Cooperative Jahn-Teller Systems with Nonequivalent Crystallographic

Phase States of the Cooperative Jahn-Teller Systems with Nonequivalent Crystallographic Sublattices

A.Ya.Fishman, A.S.Lahtin

Institute of Metallurgy of UB RAS, Ekaterinburg, Russia

N.K.Tkachev

Institute of High-Temperature Electrochemistry of UB RAS, Ekaterinburg, Russia

Introduction

It is known from the theory of phase equilibria that phase transforms of the first or second order (structural, magnetic, spin-reorientation, and other types) in multicomponent systems are accompanied by appearance of two-phase regions on the temperature concentration phase diagram[[1]-[2]]. For example, at any substitutions in a system with the first-order structural phase transition the region of coexistence phases with high and lowered symmetries is immediately formed. The phase diagrams of mixed condensed systems with restricted solubility of components are characterized by great topological variety. The problem of their microscopic description and identification of the mechanisms responsible for the characteristic features of the diagrams remains topical to this day. In this connection, crystalline systems containing Jahn-Teller (JT) centers are ideal objects for modeling the various phase transformations in systems with cooperative interactions. The JT mechanism of interparticle interaction is responsible for different types of the phase transitions [[3],[4],[5],[6],[7]]. It can be considered established that stabilization effects due to the splitting of degenerate states by JT interactions will lead to structural, spin-reorientation-like, and decomposition phase transformations in mixed condensed systems. It was shown that the JT model of cooperative inter-center interactions allows for recounting wide class of immiscibility phase diagrams for quasi-binary systems [[8],[9]].

The problem addressed in the present study is to describe the specific features of the phase transitions in cooperative JT systems, in which the cations under substitutions can occupy the sites in nonequivalent crystallographic positions. The spinel-type systems with structural JT transition in ferro-distortive phase are chosen as the object for our analysis. We consider systems in which JT ions are replaced by orbitally nondegenerate ions. Similar to our recent studies we use the model proposed by Kanamori for the first- and second-order structural phase transitions in JT crystals. Parameters of JT interactions can be found from the temperature dependence data of low-symmetry deformation for pure compound. Usefulness of this approach has been proved for several JT crystals as Mn3-xAlxO4 and Mn3-xCrxO4 [[10]]. It will be shown, that JT interactions essentially influence the energy of preference and the cation distribution in low symmetry phase, and, consequently, topology of equilibrium and metastable phase states boundaries.

Structural phase transitions (cooperative JT effect)

The Hamiltonian and free energy. Let us confine the consideration to structural phase transitions in JT ferroelastic with the doubly degenerate ions. The Hamiltonian of this system in Kanamori model [5] has the form:

(1)

where N is the number of elementary cells in crystal; V2 and V3 are the parameters of anharmonic interaction of the doubly degenerate term with uniform JT deformations eEJ and eEe ; index s labels JT ions.

This model describes the structure phase transitions of the second order in absence of anharmonic interactions, while transitions of the first order are depicted when these terms are taken into account. The cubic crystals with spinel structure, containing JT ions of the iron group (Mn3+, Cu2+, ets.) in octa-sites, can serve as an example of such ferroelastics. For clarity we shall consider the spinel systems of the following type: . It is proposed that JT ions of Mn3+ can occupy only the octahedral sites.

The JT contribution to the Helmholtz free energy of the above system is the sum of terms due to splitting of degenerate levels, the configuration entropy contribution corresponding to distribution of mixed cations over the crystal sub-lattices (octa- and tera- sites) and the term describing the preference of solute cations to choose tetrahedral neighborhood:

(2)

where Ep is the energy of the preference.

In contrast to quasi-binary system that we have considered earlier, thus formulated model is characterized by two order parameters. As usual, the role of the order parameter for JT structural phase transition is played by the deformation eEJ, which is defined by the equation

(3)

where value (2-2c-2h) º CJT is the concentration of JT ions and for the simplicity the Kanamori constant V3 is considered to be zero.

The concentration of substitution atoms in tetrahedral sites is the second order parameter of this system. Both parameters e and h can be found easily by the free energy minimization.

The structural phase transition can be described analytically only when the anharmonic parameter is small. Afterward the phase transitions are nearly of the second order and can be characterized by the following values of the transition temperature TD and JT deformation e(TD)

For the arbitrary values of the anharmonic parameter p the analysis of the mixture phase states can be performed only numerically. Typical evaluated temperature dependences of JT strain are presented at Fig. 1. It is rather obviously that the drop-like behavior of JT deformation from e(TD) to zero at temperature TD must be accompanied with the jumping-type dependence of the parameter h over temperature.

Mixture at T®0K. Let us consider in more detail, the equilibrium state (the values of order parameters) of the mixed system for T®0K. The value of JT strain is determined by the expression:

(4)

One can see that the crucial dependence of JT strain on parameter h takes place. The fraction h of B2+ cations in tetrahedral positions is balanced by the relation between JT energy FJT(T®0K) and the preference energy hEp . One can introduce some boundary value of concentration c0(Ep), which determines the type of the mixture ground state (see Fig.2). If c < c0(Ep) the ground phase state with maximum possible values h and correspondingly cJT must take place and vice versa for c > c0(Ep) the ground state is characterized by minimum possible values of h and cJT for given concentration c. Thus, the qualitative change of ground state is typical for c = c0(Ep).

Fig. 1. Temperature dependences of the reduced JT strain eEq/eEq(T®0K) in low-symmetry tetragonal phase (relative units, cJT =1). / Fig. 2. The dependence of the infill of tetrahedral sites by B2+- cations at T®0K on the relation Ep/|EJT(T®0K)|, p=-0.2.

Similar behavior is embodied in high temperature region as well. As a result, the investigated JT system can be considered as quasi-binary only for very limiting cases, namely, when êEp/EJT(T®0K)ê>1. In particular, for Ep > 0 the substitutions are provided in octahedral sites until c£ 1, i.e. in the whole region of JT phase existing.

The phase states of JT mixture in case of Ep =0. Let us consider firstly the specific features of phase transitions for the most interesting case when Ep =0. The splitting of degenerate levels by JT cooperative interaction then presents single mechanism of all phase transitions. The structural phase transition appears when the free energies of the cubic and tetragonal phases are equal F(c,T)ºF(c,T,h(c,T),e(c,T)), but the distribution of B2+ ions between nonequivalent crystal sublattices in these phases has to be unlike in case of equilibrium

(5)

where the equilibrium value htet in low-symmetry phase is equal to c for c1 or htet=1 for c1 and the value hcub(c) is determined by the equation: 2hcub3+hcub2(1-4c)+hcub(1+2c2)-c2=0.

Thus, concentration dependence of TD(c) for c < 1 is associated only with the distribution hcub(c) of B2+ ions in cubic phase in this model, because the concentration of JT ions in octahedral positions of lattice does not changed. The phase diagram of the system with Ep =0 in the absence of the immiscibility phase curves is presented at Fig. 3.

Fig. 3. Phase diagram of JT systems in the absence of the decomposition effect. / Fig. 4. Decomposition phase diagram.

For the comparison, possible dependences of TD(c) are presented by dashed and dotted curves. They correspond to the cases when phase transition takes place at fast cooling (dash) or heating (dotted) from equilibrium high or low temperatures. One can readily seen that for considered systems the transition temperature does not have single meaning and significantly depends on the sample fore-history and the rate of the process.

If we take into account the decomposition the equilibrium phases of the investigated model system are described by the diagram, presented at Fig. 4. The bold curves are the binodals and the dashed line presents dependence of TD(c) in the absence of the decomposition effect. It is reasonable that the equilibrium is hardly ever can be realized at low temperatures. Besides, the investigated spinel-like systems should be oxidized additionally while cooling within another mechanism of crystal structure adjust.

It is necessary to mention that the broad region of two-phase coexistence with c < 1 realizes in such mixture, and it is unusual that this system does not have the absolute instability (spinodal) region. Some typical concentration dependences of the free energy, which illustrate this conclusion and demonstrate how the absolute instability region is formed are presented in Fig. 5. The possibility of metastable states increases drastically for the mixtures (at least for those with c1) because chemical diffusion along reverse direction with respect to concentration gradient is absent.

Fig. 5a,b. The concentration dependences of the Helmholtz free energy at various temperatures.

The phase states of JT mixture in case of Ep ¹0. It is obvious that for systems with negative Ep the situation does not change qualitatively in comparison with already considered example. With increasing the absolute value of Ep the corresponding rise of the concentration h must take place in high symmetry phase. As a result the mismatch of h values in cubic and tetragonal phases has to be decreasing and the concentration dependence of TD(c) for c1 should become less important. In the limit of çEp箥 the system turns to be quasi-binary, whose phase diagram is presented in Fig. 6.

Fig. 6a. Phase diagram of JT systems with value of -Ep® ¥ (in the absence of the decomposition effect). / Fig. 6b. Decomposition phase diagram of JT systems with value of -Ep® ¥.

The systems in which cooperative JT effect and the preference energy are competing interactions (Ep > 0) are much more interesting. One can see (from the results presented at Fig. 2) that value eºEp/|EJT(T®0K)| determines the region of the tetragonal state existences. If we neglect the decomposition effects, then for e < 1 the increasing of e leads to narrowing of this region in interval 1c2. For e1 the tetragonal phase can take place only in this interval c1, but additional phase transition must appear associated with the attribute change of the tetragonal phase parameter in this region (e) as well as h. The corresponding illustrations are presented at Fig. 7a. Typical values of strain e in two tetragonal phases can be obtained in low temperature limit from expression (4), when h = c in phase I and h = 0 in phase II. With increasing of the parameter e the region of the phase II amplifies (dilates) and phase I reduces. The last phase must vanish entirely for e³6 (see Fig. 2) and such system can be considered as quasi-binary, where the presence of cations B2+ in tetrahedral sites is not actual one.

Fig. 7a. Structural phase boundaries of JT systems with e=1.6.

Fig.7b. Decomposition phase diagram (e=1.6).

The boundary of cubic phase withh=0 is shown by the dotted curve.

The performed calculations allow to carry out the analysis of some typical fragments of the experimental phase diagram with the substituted JT systems (for example, systems on the basis of Mn3O4 [[11]]). It is possible to show, that the recognized properties of the phase states can explain certain peculiarities of immiscibility phase diagrams (equilibrium and metastable) in JT systems with nonequivalent crystallographic sublattices.

Conclusion

The calculations of phase equilibriums in considered multi component JT system have shown that the Jahn-Teller interactions can essentially influence the distribution of cations in non-equivalent crystallographic sub-lattices and, accordingly, topology of equilibrium and metastable phase states. It is important to emphasize that real phase states in such systems are determined to a great extent by means of the way that the mixture made of from high or low temperatures and some of them appear to be metastable in wide region of experimental times.

The work was supported by Russian Foundation for Basic Research (grant 02-03-32877) and International Program “Fusion” (contract №Б0035/1533).

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