Michael D. Kaplan

Cooperative phenomena in Jahn-Teller crystals

A comprehensive study of the microscopic theory of structural phase transitions and the cooperative Jahn-Teller effect, for scientists, engineers, and students interested in any aspect of phase transformation. The volume contains six chapters: the Jahn-Teller effect; interaction of Jahn-Teller cente

The Jahn—Teller Effect

In solving the Schrodinger equation, when several wave functions ϕ i correspond to the same energy E, such a state is called degenerate. Degeneracy is always associated with the existence of some symmetry element. The three p functions of the hydrogen atom serve as an example. Their degeneracy is due to the fact that such a system has spherical symmetry: Rotation about any axis through the nucleus leaves the Hamiltonian invariant, while transforming the p functions into each other. Such threefold degeneracy persists even if the free atom is in an external field of cubic symmetry created, for example, by six point charges forming an octahedral pattern around the nucleus. In fact, it is readily verified that cubic group operations transform the surrounding charges into each other, i.e., they leave the Hamiltonian invariant while once again transforming the three p functions into each other. At the same time, the distortion of the octahedron along the z axis (extension or compression) decreases the symmetry from cubic to tetragonal, partially lifting the degeneracy: E(p z ) ≠ E(p x ) = E(p y ). In turn, the twofold degeneracy remaining in the tetragonal group can be lifted by orthorhombic perturbations.

The Dynamics of Jahn—Teller Crystals

This chapter deals with the spectrum of elementary excitations of Jahn—Teller elastics; in general they represent a set of dynamically coupled electron-phonon (or, as they are sometimes called, vibron-phonon) modes active in structural phase transition. The notion of elementary excitations in a crystal, one of the most fundamental precepts in solid-state physics, plays an extremely important role in the problem of phase transitions. This is attributable to the relationship of this notion to the soft-mode concept [1–3], according to which the frequency of one of the elementary excitations of the crystal tends to zero as the system approaches the phase transition temperature, thus accounting for the instability against the creation of a new phase.

Mutual Influence of Distortive, Magnetic, and Electric Dipole Orderings in Jahn—Teller Elastics

One of the distinctive attributes of Jahn—Teller crystals is the fact that not only the interaction responsible for the ordering of local distortions and hence for the corresponding phase transition, but also interactions conducive to the ordering of magnetic moments, electric dipoles, or local distortions having a different symmetry can play a major role. The important mutual influence of these interactions is attributable to the presence of a group of lower energy levels (degenerate or nondegenerate), typical of Jahn—Teller ions, on whose basis the corresponding operators are defined. According to the way the interactions are related (magnitudes, commutative relations for the corresponding operators), phase transitions of one order or another or a phase with corresponding orderings of a different physical nature occurs in crystals. The present chapter deals with interactions between the orderings and their mutual influence. We note that this interaction is considered here for arbitrary relations between the interacting quantities. However, even when one of them prevails and a single phase transition occurs, interstitial correlations of some other nature have a strong effect on it. Another distinctive feature of Jahn—Teller crystals, namely the onset of various combinations of significant anomalies of various properties, is a consequence of the above interaction of distortive, electric dipole, and magnetic structures. In what follows we show that, in addition to the peculiar elastic anomalies considered in the preceding chapter, Jahn—Teller compounds are also characterized by very specific magnetic and dielectric properties. Combinations of the above properties depend on the particular type of structural ordering, whether of the ferro, antiferro, or ferri type. Various attendant physical situations are also discussed in this and subsequent chapters.

Jahn—Teller Crystals in External Fields: Phase Diagrams and Properties

In the previous chapters we have shown that Jahn—Teller crystals are systems with very unique anomalous elastic, dielectric, and magnetic properties. These compounds are unique in that they can be characterized by a combination of the anomalies. At the same time, an especially valuable for practical applications property of Jahn—Teller elastic materials is the possibility of affecting these anomalies by means of fairly weak external fields, which are easily obtained in experiment. This is because the rare-earth compounds discussed below, in particular, are characterized by low critical structural phase transition temperatures. Accordingly, the Jahn—Teller molecular fields responsible for phase transitions and the splittings of electron levels by these fields are also not too large. The most pronounced changes of the crystal properties occur specifically in this case when splittings of electron levels by external fields turn out to be of the same order as those produced by the molecular field.

The Elastic Properties of Crystals with Jahn—Teller Structural Phase Transitions

In the first two chapters we have described how the Jahn—Teller effect gives rise to local distortions around a degenerate center and how the degenerate states of different Jahn—Teller sites interact. We have shown that interstitial interaction makes the ground state of the crystal correspond to a certain packing of “frozen” (in contrast with the case of an isolated center) local Jahn—Teller distortions. With increasing temperature the correlations of the Jahn—Teller distortions become weaker, and a transition to a disordered phase occurs. This structural phase transition associated with the cooperative Jahn—Teller effect is naturally determined by the lowest electronic state of the lattice sites and by the character of their vibronic coupling. In this chapter we discuss the main types of conceptually different situations produced by the cooperative Jahn—Teller effect and examine the elastic anomalies accompanying this structural phase transition.

Interaction of Jahn—Teller Centers

In Chapter 1 we have briefly considered the fundamentals of the Jahn—Teller effect and have discussed some of its manifestations for isolated Jahn—Teller centers. One of the main conclusions of Chapter 1 is that the Jahn—Teller effect does not induce static distortions. A distorted nuclear configuration does actually correspond to a certain electronic state of the degenerate term, but such distorted equilibrium configurations (potential energy minima) are not always unique, and dynamic averaging over them restores the initial high symmetry. On the other hand, it has been noted that even slight low-symmetry perturbations (external fields, random strains, etc.) in Jahn—Teller situations can tend to localize the system at one of the minima and bring about significant distortions. In this situation it would be interesting to know whether interaction between Jahn—Teller centers, as in the case, e.g., of a crystal with an ionic sublattice whose ground state is degenerate, can lead to localization and a net distortion. In such a multicenter Jahn—Teller situation it is also reasonable to expect phase transitions similar to those accompanying spin ordering, the only difference is that pseudospins specified in a basis of orbital states should be addressed in this case. The investigation of such phase transitions due to ordering of the orbital components of degenerate electronic states of lattice ions is the main purpose of this book. For phase transitions to occur, there must be interaction between electronic states of different sites. In this chapter we shall discuss the mechanisms of such interaction.

Cooperative Pseudo Jahn-Teller Effect as Microscopic Mechanism of (OMT)-(LTT) Structural Phase Transition in La 2−x Ba x CuO 4 Superconductors

A microscopic mechanism for the structural phase transition from the orthorhombic mediate temperature (OMT) phase into the low temperature tetragonal (LTT) one is suggested on the basis of the cooperative pseudo Jahn-Teller effect. The local distortions mixing the ground and the first excited electronic states are ordered antiferrodistortively and are connected in part, with the oxygen octahedra rotations around the [100] axis. The results are in agreement, with the neutron scattering experiments data.

Microscopic Mechanism of Stripe Pairing Phase Formation

Stripe phase formation in manganite compounds with the colossal magnetoresistance is discussed on the basis of the cooperative Jahn-Teller effect theory. It is shown that the stripes are the result of a peculiar structural phase transition. The microscopic mechanism of this structural ordering is considered and it is found that the interference of two symmetry types of distortion ordering takes place, that of totally symmetrical and Jahn-Teller distortions. The quadratic electron-phonon interaction after two-mode canonical transformation leads to a three center interaction responsible for the stability of a combination of local distortions around the nearest neighbor Mn3+-Mn4+-Mn3+ ions as a structural unit of a stripe. The described combination of the local distortions is in agreement with the experimentally observed electron diffraction pattern.