Simulation of light atom diffusion in oxides with orbital degeneration

Fishman A.Ya.1, Ivanov M.A.2, Vykhodets V.B.3, Kurennykh T.E.3 ,Mazurovsky V.L.4

1Institute of Metallurgy, Ural Department of Russian Academy of Sciences,

Amundsen St. 101, 620016 Ekaterinburg, Russia

2Institute of Metal Physics, Ukrainian National Academy of Sciences,

Vernadsky St. 36, Kiev-142, 03680 Ukraine

3Institute of Metal Physics, Ural Department of Russian Academy of Sciences,

S.Kovalevskaya St. 18, 620218 Ekaterinburg, Russia

4College of Judea and Samaria, Science Park, Ariel, 44837 Israel

Introduction

Crystal systems with orbitally degenerate or Jahn-Teller (JT) ions are a suitable object for the analysis of diffusion processes in degenerate condensed systems specifics. The mechanism of many-well potential formation is well investigated for JT ions and it is possible to give a simple microscopic description of inter-center interactions and phase transitions (see e.g. [[1],[2]]). The change of character of degenerate states splitting with temperature or composition can essentially influence the value of diffusion coefficients in such systems. Besides the relaxation times of degenerate centers become very important. Two fundamentally different situations are possible.

The first one - when during the time of diffusion jumps the equilibrium population of vibronic levels (minimums of adiabatic potential) manages to take place on JT centers nearest to the diffusing atom. Then one can concern that the diffusion migration act takes place in the thermodynamically equilibrium lattice configuration. In the second limit situation the equilibrium distribution between the potential energy minimums (vibronic levels) has no time to set in. As a result the potential barrier form (the activation energy for jump frequencies) becomes independent on JT vibronic states random configuration and hence on the corresponding to these states displacements of nearest to saddle point anions from symmetric positions. One can expect that essential qualitative difference can take place in the behavior of diffusion coefficients for these two mentioned regimes of ionic migration in JT systems.

The present work considers both limit cases. The cooperative JT crystals with structure transition of displacement type into the ferroelastic phase are choosed for investigation. The anion diffusion and change of potential barrier parameters in the lattice with cubic symmetry of hightemperature phase [[3]] are analyzed. The simple microscopic description of diffusion coefficients behavior at some types of phase transitions in crystal (structural [[4]], spin-reorienting [[5]], decay [[6]]) is possible in the framework of JT mechanism of inter-particle interaction. As a consequence the use of obtained results is not limited by JT problem and gives the method of qualitative analysis of diffusion experimental data for different condensed systems with indicated phase transitions.

Model of structure phase transitions in crystals with cooperative JT effect

The simplest model of description of structure phase transitions in cooperative JT system was proposed by Kanamory [4]. Let's consider the case of JT ions with two-fold degenerate ground state. Then according to [4] the Hamiltonian of JT subsystem of cubic crystal with structure phase transition of ferro-type (ferroelastic) can be written in the form

, (1)

where sz is the orbital operator defined in the space of wave functions of an orbital doublet, V - is the constant of interaction of JT ions with uniform deformation e º eJT, p- is the parameter of anharmonic interaction, W - is crystal volume per one orbitally degenerate ion, C is the seed elastic module describing the elastic crystal energy for deformation e º eJT for lack of JT subsystem, the summation takes place over all degenerate ions with numbers s. In the case of low-temperature tetragonal phase the JT deformations are described by the expression e º eJT = ezz – (exx + eyy)/2, and the role of module C plays the following combination of cubic crystal elastic constants: C = C11- C12. The free energy of the considered system per one JT ion is described by the following expression in the absence of substitution effects:

, (2)

where kВ is Boltsman constant, T is the absolute temperature, p1 is the parameter characterizing the usual anharmonic interaction in crystal.

The free energy (2) has the minimum value at equilibrium value of uniform deformation eeq, determined by the equation

(3)

Here the relative deformation z = eeq/e0 (e0 = V/(CW) is the deformation at T ® 0 K) and the dimensionless temperature parameter t = kВT/Ve0= CWkВT/V2 are introduced.

In the absence of anharmonicity (at при р = p1 = 0) such a model describes the structure phase transition of the second kind that is the transition takes place without the jump change of entropyDS (and enthalpy DH), and also of tetragonal JT deformation Dz (De). When р¹0 the phase transition is of the type of first order transition where the jumps of indicated values are larger for large absolute value of anharmonicity parameter. The model (see Fig.1) shows the existence of large enough region of metastable "overheat" tetragonal state with z ¹ 0 in the range of temperatures higher than the critical one (temperature of structure transition).

Fig. 1. Temperature dependences of tetragonal JT strains; p = 0 (1), 0.2 (2), 0.25 (3).

The generalization of Kanamory model on the case of solid solutions with JT ions is not difficult. At that it is important to note that all these systems with CEJT are characterizes by strong tendency; to decay on phases with high and low content of JT ions.

The potential barrier in quickly relaxing systems with CEJT

The dependence of diffusing atom energy in the main E0 and Ea upon JT deformation e º eJT can be presented in the form [[7]]

(4)

where are the constants of interaction between diffusing atom and deformations in the ground and activated states. It is supposed that states of diffusing atom in the ground and saddle positions are non-degenerate and that one can neglect the change of volume at JT deformations in crystal. If the energy of vacancy formation Ef enters the activation energy of diffusion then the corresponding additional contribution to the part of activation energy depending on the deformation Q(eij) = Ea(eij) + Ef(eij) - E0(eij) appears

(5)

Let's note that in the high-symmetry positions of diffusing atom (especially in positions with octahedral or tetrahedral coordination in the ground state) the linear interaction with JT deformations eJT. is absent. Only the term Ea(eij) connected with low-symmetrical activated states of diffusing atom can give contribution to the diffusion activation energy appropriate to these deformations The typical microscopic deformations (eJT) of cubic crystal at JT structure phase transitions are within the limits (10-2 – 10-1). As a result when parameter 105K the contribution of JT deformations into the value of Q should have the order of value ~ 0.1 eV.

In the high-symmetrical cubic phase of JT systems that is at temperatures T which are higher than structure transition temperature TD the diffusion coefficients tensor Dij has an isotropic form Dij = D0dij. In the low-symmetrical phase (T < TD) the anisotropy of diffusion coefficients caused by JT deformations should appear. In the case of tetragonal distortion of crystal lattice we have for the components of diffusion coefficients tensor: Dzz ¹ Dxx = Dyy At that the theory of random walks gives for corresponding values Dii the following expressions [[8]]

(6)

where xk , yk , zk are projections of diffusing atom displacements on coordinate axes at a jump of k type, Gk is the frequency of these jumps.

In the case of large enough JT deformations the effect of anisotropy of diffusion coefficients depends exponentially upon the value of JT deformation and it can turn to be rather sizeable.

In the framework of Landau 's theory of phase transitions at p1 one can obtain the following expressions for TD and e1T(TD)

(7)

где e0 is JT deformation atT ® 0 K. Then foe the value we have

. (8)

It can be seen that at typical values and magnitudes of anharmonicity parameter ïpï ~ 0.1 the essential change both of diffusion coefficients and their anisotropy should take place in the point of structure phase transition. With the increase of anharmonicity parameter when the character of temperature dependence of JT deformation approaches to theta-function the value of effects under investigation should also essentially rise. Let's note that such stepped form of JT deformation dependence on temperature is typical for quite a number of spinel systems with CEJT [[9]].

Diffusion of anions in elastic model of potential barriers

Three types (neglecting JT deformations) of anions jumps between nearest positions which have different type of kation environment of saddle point (jumps 1-3 on Fig. 2) can be chosen in spinel crystal lattice. As a result the expression for diffusion coefficient (6) has in the high-symmetrical phase the following form

(9)

where a is spinel lattice parameter. It is supposed that each jump frequency Gk (k =1-3) in the system with CEJT has the arrenius-like dependence upon temperature

,

where is the pre-exponential factor and is the value of potential barrier.

Fig. 2. A fragment of a base plane (001) in a crystal lattice with spinel structure.

Cations in tetra-positions are accordingly above the considered plane, or under it on distances a/8, where a is the lattice parameter in high symmetry phase.

For simplicity sake let's confine ourselves by elastic model of diffusion potential barrier forming that is we shall assume that jump activation energy is determined by the work connected with deformation of the medium (cavity) and diffusing atom when it goes through the saddle point. Then not depending on the details of activation energy calculations (J.D.Eshelby [[10]], Y.A.Bertin [[11]], A.J.Ferro [[12]] and others) one can assume that it is proportional to the square of the difference of cavity and diffusing atom sizes:

(11)

Fig 3 shows an example of local anion environment in one of the saddle points corresponding to jump frequencies in XY plane.

Fig. 3. The nearest environment of the first type saddle point. The cavity for diffusing atom is shaded. / Fig 4. Deformation dependence of activation energy for the case of the first type saddle point.

Fig.4 presents some of the calculated dependences of relative activation energy change DEk(eJT)/DEk(0) for these jumps upon the value of JT deformation eJT. The following assumptions were used at the same time

(12)

The jump frequencies , in planes XZ and YZ are calculated similarly.

One can see that JT deformations eJT ~ 0.1 can essentially change the activation energy of jumps and hence the diffusion activation energy. Thus the led analysis gives an opportunity to estimate change of diffusion coefficients and their anisotropy due to CEJT in the systems under consideration. Typical temperature dependencies of activation energy and diffusion coefficients at different parameters of CEJT (deformation) are presented on Fig. 5.The case of diffusion along OX-axis when jumps of one type (G1 or G3) give the main contribution is considered. At that the asymptotic high-temperature behavior of curves describes also diffusion along OZ-axis.

Thus it is shown that at relatively small discrepancies between the sizes of cavity and diffusing atom even small lattice deformations can essentially change elastic contribution to the diffusion activation energy. At that it can occur that when crystal goes from high- to low-temperature phase the diffusion coefficients can rise but the inverse effect is more traditional. Change of the value of diffusion coefficients can be also so large as they are at phase transitions with change of crystal structure type (e.g. FCCÛBCC) that is when there is an opportunity of experimental check of the effect assumed in this work.

/ Fig. 5. Temperature dependence of diffusion coefficient Dxx. The following values of parameter p are used: p= 0.05(1); 01(2); 015(3); 0.2(4).

The obtained results can be easily extended on the case of mixed systems where JT ions are changed to other JT ions or orbitally non-degenerate ions. At that one can consider effects connected with suppression of structural phase transition and decay of system into phases with high and low concentrations of JT ions (see Fig. 6).

In much the same using Hamiltonian of the type of (1) it is possible to investigate the behavior of diffusion coefficients in JT systems with competing anisotropy [5], which arises at differing signs of JT deformations for two types of mixed JT ions (that is at opposite signs of anharmonicity parameters p in Hamiltonian (1) of these ions).

Fig. 6. Concentration dependence of diffusion coefficient Dxx. A continuous curve describes the diffusion without the account of disintegration processes and dotted curve is constructed in view of these processes (p=0.3).

The phase diagram of such systems is characterized by presence of region with intermediate or “oblique” phase and correspondingly by phase transitions of spin-reorienting type [2,5]. Thus the analyzed model of diffusion in the systems with JT inter-particle interaction allows to describe change of diffusion coefficients at different phase transitions. However results may be different for systems with other, non-ferrodistortional type of orbital ordering.

Diffusion in systems with slowly relaxing JT centers

The present part deals with the analysis of anions diffusion coefficients in supposition that during the time of diffusion jump the equilibrium population of degenerate states has no time to be settled [[13]]. In this case the symmetry of anion environment of diffusing atom in the saddle point is determined by random configuration of vibronic states of the nearest JT centers (see Fig. 7-9).

Let the JT center (JT cation plus anion octahedron - Fig. 1) be characterized by three lowest vibronic states corresponding to stretching of octahedron along one of the coordinate axes and compression along two other axes [1]. Then the position (displacement) of each anion is determined by the set of vibronic wave functions set on three nearest to it JT octa-cations (Fig.8). Altogether there are nine such states of anion. At that if JT deformations are accompanied by insignificant alteration of volume the anion displacement uJT along the connection line towards the JT cation in two of three possible vibronic states of this cation turns out to be half of its value in the case when it moves away from cation in the third state.

It is obvious that at random character of JT states during the diffusion jump the corresponding random distribution of activation energies for frequencies of these jumps should also take place.