The Simulation of the properties of metal, oxide and other Materials with different types of ENERGY oR orientational degeneracy on the base of JahN-teller system behaviour

B.S.Tsukerblat

Ber-Sheva University, ISRAEL

L.D.Falkovskaya, A.Ya.Fishman, V.Ya.Mitrofanov

Institute of Metallurgy of UB RAS, Ekaterinburg, Russia

M.A.Ivanov

Institute of Metal Physics of NAN, Kiev, Ukraine

V.I.Tsidilkovski

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

1. Introduction. There is a large class of systems containing centers, whose ground state is degenerate, quasi-degenerate or orientationally degenerate. It is enough to mention centers with charge transfer between nearest to non-isovalent substitution ions, splitting (dumbell-like) configurations, off-center ions, impurity pairs consisting of substitution and interstitial ions in high symmetry lattices, systems with orbital degeneracy etc. Distinctive feature of all these centers is multy-well profile of their potential energy. The physicochemical properties of such systems are determined by splitting of energy levels in equivalent potential wells due to the tunneling effects, external fields, random crystal fields and molecular fields of various types.Our interest to these systems is stipulated by the following circumstances.

1) The anomalously large contribution of such centers to different properties of condensed systems and the unusual character of these properties in comparison with the non-degenerate systems.

2) The strong dependence of properties upon the distribution of random crystal fields removing degeneracy on the considered ions. On one hand these fields decrease the susceptibility of the system to external fields and on the other hand they can form the properties of the system e.g. the low temperature - heat capacity. In the diluted degenerate systems one of the main sources of such fields can be just the multy-well potential centers.

Jahn-Teller (JT) ions occupy the special position among the multy-well potential centers [[1],[2],[3],[4],[5],[[6]],[7],[8]]. The microscopic nature of multy-well potential is known for the JT systems (unlike other ones) and the exhibitions of multy-well potential in different properties (magnetic, thermodynamic, spectral and other) are very diverse. The JT centers can have simultaneously spin and orbital degeneracy, the dipole moment, that is why as a degenerate subsystem they make available the high susceptibility of crystal to the various external actions. Besides, JT states are extended enough widely, e.g., in oxides with cubic lattice one of the 3d ion charge state can be orbitally degenerate. Here, we’ll try to illustrate the peculiarity of structural, thermodynamic, magnetic, spectral and other properties of degenerate systems on the example of JT ions or molecules in crystals.

2. Jahn-Teller states - simple example. The JT theorem claims that for orbitally degenerate ions and molecules the symmetric configuration of environment is unstable with respect to deformations. As a result the adiabatic potential of such ions or molecules (systems with pseudo-degeneracy) turns out to be of multy-well type (see Fig. 1).

/ Fig.1. Three-well potential at JT ions with cubic E-term in the ground state in the absence of tunnelling splitting.

This potential W() can be described by the following function of JT strains e(E)=ezz–(exx+eyy)/2 and e(E) = 3(exx – eyy)/2:

W() = W0cos3, ctg = e(E)/e(E) . (1)

The characteristic displacements of nearest to JT center anions in states corresponding to minimums of adiabatic potential ( -2/3, 0, 2/3) are presented on Fig. 2.

/ Fig. 2.The displacement of nearest to JT center anions at minimums of adiabatic potential.

3. The behavior of the dilute systems with degenerate centers in random fields. The low-symmetry crystal fields may be the dominant mechanism of removing degeneracy at the JT ion. In this case the character of random field distribution function determines the peculiarities of low-temperature behaviour of thermodynamic, magnetic and other properties of JT systems. The distinctive property of these functions is their many-component character.For centers with cubic E-term we have the following Hamiltonian H of interaction with random fields, described by the two-component distribution function [[9]],

(2)

where h ,s () are the components of random field at the JT center with index s, U,s are orbital operators. The explicit form of these functions may be sometimes obtained if the character of random field source and the law of low-symmetry field decrease with distance from a source are given. For example, the random field distribution function of deformational nature f(h,h) for a small concentrations of these field sources xi < 1 has the following form

(3)

where NJT isthe number of JT ions in crystal, VEis the parameter of vibronic interaction of JT ion with JT active deformations,  is the volume of the unit cell,  is the Poisson coefficient. At high concentrations of random field sources xi ~ 1 we have for f(h,h)

(4)

where the dispersion isequal to the random field value at the nearest to a source of such field position, h (h,h).

density of states, heat capacity. The important property of degenerate system is density of states for the excitation energies g(E):

(5)

where Ei,exited(h,h) and E0(h,h) are the energies of excited and ground states of JT center in random fields correspondingly. We consider the case when the dispersion  of random fields is larger than the parameter tun, describing tunnelling between the minimums of adiabatic potential. Then approximately in some region of energies E constant density of states g(E) ~ NJT / takes place. As a result the linear temperature dependence of heat capacity occurs in the crystals with JT centers

CJT~ Tg(E=0) ~ NJTT/ , kBT . (6)

Because of the fact that at small concentration of random field sources xi < 1 dispersion ~xii, the heat capacity CJT is proportional to NJTT(xii)-1. In the case when just the degenerate centers are the source of random field, the density of states and heat capacity in corresponding regions of energies and temperatures become independent upon xJT. However the width of the region, where this effect takes place, depends linearly upon xJT.

Similarly the modification of the other thermodynamic property behaviour of crystals with the degenerate centers (elastic modules, thermal expansion coefficient, magnetic and dielectric susceptibility) always takes place at temperatures.

These results have rather common character and can be generalized easily on the case of concrete degenerate systems. So, the first examination of obtained results was performed for the analysis of heat capacity of the inert gases crystals with molecular impurities (for example, Kr: 14N2) having the identical system of lowest excited states [[10]]. Other exotic object for application of obtained results is weakly doped fullerites AxC60 (x<1), where the considered peculiarities of low temperature properties can be also observed [[11]].

The dipole centers R3+-OH in oxide proton conductors АIIВIV1-xRIIIxO3-ycan serve one moreexample for illustration [[12]]. Protons are attached to the oxygen ions forming OH centers, which themselves possess dipole moment and form more complex dipoles with dopant ions R3+. Four equivalent potential minima for protons exist at every oxygen and respectively six minima for OH centers near the dopant in the perovskite lattice. both types of multy-well states without regard for interactions between different “dipoles” and their interactions with other imperfections are orientationally degenerate. At low temperatures, when hydrogen migration between oxygen atoms is impossible practically, it is enough to consider the four-level states of R3+-OHcenter. These tunnelling states of hydrogen have symmetry A1+E+B1 of point group C4v and energy values:

E(A1) = tun, E(E) = 0 , E(B1) = tun ,

where tun is the tunneling splitting parameter. Temperature dependencies of the contribution of the indicated centers to the heat capacity С and dielectric susceptibilityare shown in a Fig. 3-4. We used the relative temperature scale with t = kBT/tun. It is seen, that heat capacity Сis a linear function of temperature in low-temperature region kBT. For this system the value -1/2N(OH)exp{-(tun/)2}/plays the role of density of states g(0). As a result the coefficient at linear temperature term is maximum, when /tun = 2.


/ Fig.3. Calculated temperature dependences of heat capacity C.
/tun= 0(1), 0.5(2), 1(3), 5(4).
Fig.4.Calculated temperature dependences of dielectric susceptibility .
/tun= 0(1),0.5(2),1(3),4(4)

It is important to underline, that completely similar results take place for a heat capacity and magnetiс susceptibility of dilute JT systems with zircon structure and JT ions Tm3+, Tb3+, Dy3+[8]. In this case tetragonal crystal field is the analog of tunnel splitting parameter for hydrogen and ensures the identical systematisation of terms in four-level system.

4.Dilute systems with cooperative interactions between degenerate centers. In JT systems with xJT< 1 interaction between JT ions is stipulated mainly by interchanging through phonons. It decreases with distance as R3, and its angular dependence has alternating character. As a result, when this interaction is dominating, the state of low-temperature phase of a system can not be characterized by the long-range order parameter different from zero. Nevertheless, the low-temperature phase of a glass type can take place in such system. The analysis of thermodynamic properties of a system with interacting JT ions has shown that at low temperatures new type of state - JT glass may occur.

The thermodynamic functions of JT subsystem can be retrieved by a method of virial expansion at temperatures exceeding transition temperature Tg to a low-temperature phase of JT glass. In the case of JT ions, whose ground state is the orbital doublet, the following expressions take place [[13]] for transition temperature Tg, heat capacityC, generalized susceptibility (magnetic or dielectric susceptibility and elastic constant-module, describing interaction of degenerate term with strains)

(7)

where  is the density of matrix, sis the velocity of a sound, p0 is the electrical dipole moment of the degenerated center, CJT0 is the elastic module of a matrix in the absence of interaction between JT ions and strains. It is obvious, that the value of Tg/T is the parameter of virial expansionand the phase transition into the low-temperature glass phase at temperature Tg is possible. As a result quite definite frozen configuration of strains takes place at the degenerate centers (at given random field distribution in a system) for TTg[13,[14]]. However, the average value of strains in crystal remains equal to zero. Experimentally such states were observed in mixed JT systems with zircon structure (TmxY1-xVO4, TmxY1-xPO4) [[15],[16]].

Thus, JT model enables to describe the low-temperature thermodynamics of degenerate system both at a dominating role of random fields, and in the case of cooperative interaction between the degenerate states.

5. Peculiarities of dilute magnetic systems with JT ions.Magnetic anisotropy and magnetostriction. In comparison with the non-magnetic crystals splitting of orbitally degenerate state in magnetics depends essentially on the direction of exchange fields at JT ion. E.g. the Hamiltonian of magnetic anisotropy in view of three lowest states of JT ion with cubic E-term can be written as

(8)

where D is the single-ion anisotropy constant, S is spin, n is the unit vector of magnetization.

The role of the anisotropy energy is played by the splitting energy of three lowest vibronic states by the spin-orbit interaction in the second order. As a result the angular dependence of the free energy of magnetic anisotropy Fanis has quite non-traditional form at temperature kBTD(see also Fig. 5)

(9)

Such unusual angle dependence leads to stepwise change of Fanis (anisotropy of “light axis”-type) with the magnetization direction (see Fig. 5).

/ Fig. 5. The light axes of JT subsystem

The anomalously large value of this contribution to the magnetic crystallographic anisotropy energy is caused by the fact that the constants of single-ion anisotropy for orbitally degenerate centers are proportional to  or 2cub, where cub is the cubic crystal field parameter. For usual non-degenerate ions the anisotropy constants are proportional to 4cub3 in cubic crystals.

This peculiarity of the behavior of JT subsystem free energy leads to the appearance of oblique phases in systems with competing anisotropy (due to the JT term proportional to ni2 in contrast to ni4 in magnetic anisotropy energy of matrix).

The magnetostriction properties of these systems are also original. At a given direction of magnetization the spin-orbit interaction removes the degeneracy from the directions of local JT deformations which were earlier equivalent and this leads to the magnetostriction distortion of the crystal. The JT ions contribution to the magnetostriction constants in the absence of random fields is equal to xJTeJT, where eJTis the characteristic local JT deformation (eJT ~ 10-1-10-2).

Only tetragonal deformations can take place at the centers under consideration and each of the three energy levels turns to be the lowest in the definite region of magnetization angles. The transfer of magnetization through the boundary of these regions is accompanied at low temperatures by the change of the type of tetragonal deformation at JT ions, that is the JT ion contribution to magnetostriction. Such effects were observed in the system YIG: Mn3+ [[17]].

The low-symmetry random fields which are always present in real systems remove partly or absolutely the degeneracy at JT center. If random field dispersion D, then the resulting contribution of JT centers to magnetoelastic characteristics diminishes D/ times (when kBT). However, even in this case the values of magnetostriction constants largely surpass the corresponding values which are typical for orbitally non-degenerate ions.

Behavior of reorientable centers containing a 3d-ion of mixed valencein cubic magnets. The indicated results (besides JT systems) have the direct relation to magnetic properties of cubic magnets with mixed valence (MV) centers stipulated by non-isovalent substitutions or vacancies (see, for example, [[18]])). Such centers of several 3d-ions with the charge, localized on them, arise in the case of non-isovalent substitutions and in the presence of anion or cation vacancies. The MV complexes in addition to traditional properties of JT centers have peculiarities stipulated by reorientation of an extra charge (electron or hole) between ions of a complex, when the degeneracy is taken out at the expense of external perturbations or cooperative interactions.

/ Fig. 6. The trigonal MV center near anion vacancy
V - vacancy, C - cation, А - anion, h - extra hole

First of all it concerns to occurrence of essential electrical dipole moment at MV center. The example of MV center with С3v symmetry is presented in Fig. 6 for the case of cubic crystal lattice (in particular for spinel structure). It consists of three cations (3d-ions) with localized t2g -hole on them.

In the case of small hopping integrals b of an excess charge (bDor ) reorientation of MV center (the transfer of an extra charge) is similar to tunneling of JT ion between minimums of adiabatic potential. The similar systems are characterized by anomalously strong contribution to constants of magnetic anisotropy and magnetostriction, nonconventional angular dependence of the free energy of a magnetic anisotropy and effects, connected with competing anisotropy [[19]]. An example of the typical phase diagram (relative temperature - concentration) for systems, where the competition of a magnetic anisotropy of a matrix and MV centers takes place is presented in Fig. 7.

/ Fig. 7. Magnetic phase diagram of a system with mixed valence centers Cr4+-Cr3+ (Cr2+-Cr3+).
Continuous and dashed curves represent the lines of first - and second - order phase transitions, respectively.

7.Resonance properties of degenerate systems.As a rule the character of resonance spectrums of degenerate centers is determined by the dominating mechanism of splitting of the ground and lowest excited states. Such spectrums are very sensitive to any quantitative and qualitative changes of the system state. As a result the degenerate center spectrums can be of the convenient source of information about the main crystal matrix.

JTions or mixed-valence centers of the 3d group have a substantial influence on the different resonance spectrums (ultrasonic attenuation, ESR, FMR, NMR and other) of crystals. Let us consider the distinctive features of the observed NMR spectrum of such centers in magnetic cubic crystals. In this case the hyperfine fields at the nuclei of degenerate centers have a substantial anisotropic component and therefore depend strongly on the state of the orbital subsystem. In the case of doubly degenerate JT ions with octahedral (Mn3+, Cu2+) or tetrahedral coordination (Fe2+) the Hamiltonian of hyperfine interaction Hhf has the following form [[20], [21]]. (10)

.

Here S and I are the electron and nuclear spins of JT ion, A1 and A2are parameters of the isotropic and anisotropic hyperfine interactions. We are omitting rather small quadrupole interactions to streamline equations. Then according to the expression (10) the NMR frequency of the JT ion is equal to

(11)

where means the quantum-statistical or quantum-mechanical expectation value. It is easy to see that the spectral distribution of NMR frequencies strongly depends on the quantum-mechanical average values of orbital operators UE> and <UE> or in other words on the dominant mechanism of the orbital degeneracy removal (one-ion anisotropy field, random and local low-symmetry crystal fields). It is obvious that in contrast to pure spin 3d ions with the resonance frequencies depending on one parameter of the isotropic hyperfine interaction A1, the NMR spectrum of JT ions is highly anisotropic. The anisotropy factor is determined by the ratio which can be of the order of unity or greater.

In the case when the main contribution to the splitting of the orbital doublet brings in the one-ion anisotropy, the NMR spectrum at low temperatures contains only one line. The position of this line depends on the direction of magnetization. The character of the NMR spectrum is drastically changed when random crystal fields bring in the dominant contribution to the splitting of the orbital doublet. At > 1and low temperatures (kBT) the distribution of orientations of pseudo-spin moments UE and <UE> becomes practically uniform and the absorption band of width thus arises. The maximal width of the NMR spectrum takes place when the direction of magnetizations is parallel to [100]-axes. In this case the absorption peaks in the static limit of the JT effect correspond to the frequencies [20-22]