The main topic of this book is the physics of solids containing transition elements: 3d − Ti, V, Cr, Mn, … 4d − Nb, Ru, … 5d −Ta, Ir, Pt, … These materials show extremely diverse properties. There are among them metals and insulators; some show metal–insulator transitions, sometimes with a jump of conductivity by many orders of magnitude. Many of these materials are magnetic: practically all strong magnets belong to this class (or contain rare earth ions, the physics of which is in many respects similar to that of transition metal compounds). And last but not least, superconductors with the highest critical temperature also belong to this group (high-Tc cuprates, with Tc reaching ∼ 150 K, or the recently disovered iron-based (e.g., FeAs-type) superconductors with critical temperature reaching 50–60 K).

The main factor determining the diversity of behavior of these materials is the fact that their electrons may have two conceptually quite different states: they may be either localized at corresponding ions or delocalized, itinerant, similar to those in simple metals such as Na (and, of course, their state may be something in between). When dealing with localized electrons, we have to use all the notions of atomic physics, and for itinerant electrons the conventional band theory may be a good starting point.

Until now we have largely been discussing the properties of correlated systems with integer number of electrons; only in a few places, for example in the sections on charge ordering and on the double exchange, did we touch on some properties of doped correlated systems. But in principle the variety of phenomena which can occur in such systems with the change in electron concentration is quite broad – from a strong modification of magnetic properties up to a possibility of obtaining non-trivial, possibly high-temperature superconducting states.

A number of questions arises when we start thinking about doped strongly correlated systems. Would the system be metallic? And if so, would it be a normal metal described by the standard Fermi liquid theory? In effect, even with partially filled bands the electron correlations can still remain strong, with the Hubbard's U (much) bigger than the bandwidth; thus these questions are really nontrivial.

The other question is, what magnetic properties will result when we dope Mott insulators? As we have argued in Chapter 1 and Section 5.2, for partially filled bands the chances of ferromagnetic ordering are strongly enhanced, whereas Mott insulators with integeroccupation of d-shells are typically antiferromagnetic.

One may also expect that some other, new features could appear in strongly correlated systems with partial occupation of d levels.

When dealing with transition metal compounds one has to look at the different degrees of freedom involved and their interplay. These degrees of freedom are charge, spin, and orbitals. And of course all electronic phenomena occur on the background of the lattice, that is one always has to think about the role of the interaction with the lattice, or with phonons.

The electron spins are responsible for different types of magnetic ordering. The orbitals, especially in the case of orbital (or Jahn—Teller) degeneracy, also lead to a particular type of ordering, and the type of orbital occupation largely determines the character of magnetic exchange and of the resulting magnetic structures.

As to charges, the first question to ask is whether the electrons have to be treated as localized or itinerant. We actually started this book by discussing two possible cases: a band description of electrons in solids, in which the electrons are treated as delocalized, and the picture of Mott insulators, with localized electrons.

But even for localized electrons there still exists some freedom, which has to do with charges. In some systems charges may be disordered in one state, for example at high temperatures, and become ordered at low temperatures. This charge ordering (CO) will be the main topic of this chapter. But, to put it in perspective, we will start by discussing different possible types of ordering, connected with charge degrees of freedom.

The history of the development of some of the key concepts discussed in this book is quite interesting and has some rather unexpected twists and turns. In this section we discuss briefly the history of the concepts of Mott insulators, the Jahn–Teller effect, and the Peierls transition.

Mott insulators and Mott transitions

The notion of a Mott insulator as a state conceptually different from the standard band-like insulators and metals can be introduced using two approaches. In the main text, for example in Chapter 1 we described the approach that uses the Hubbard model (1.6) with short-range (on-site) electron-electron repulsion and attributes the insulating nature for strong interaction to the fact that an electron transferred to an already occupied site experiences repulsion from the electron already sitting on that site. This is the picture most often used nowadays to explain the idea of Mott insulators.

But historically these ideas first appeared in a different picture, presented in a paper by Mott published in 1949 (Mott, 1949) – although it already contained some hints about the picture mostly used nowadays, formalized in the Hubbard model. But the main arguments of Mott in this paper rely rather on the long-range character of Coulomb interaction, and the main statement is that, starting from an insulator, one cannot get a metal by exciting as mall number of electrons and holes.

General theory

In several places in this book we have used the language and notions first developed by Landau to describe second-order phase transition, but which are used nowadays in a much broader context. Here we summarize the basics of this theory and illustrate different situations in which it is used. One can find a more detailed description for example in the brilliant original presentation of Landau and Lifshitz (1969), or in Khomskii (2010) (which is more or less followed below).

The original aim of Landau was to describe II order phase transitions – transitions in which a certain ordering, for example ferromagnetic, appears with decreasing temperature at some critical temperature Tc in a continuous manner. But it turned out later that the approach developed has much broader applicability than originally planned.

In thermodynamics and in statistical physics the optimal equilibrium state of a many-particle system is determined by the condition of the minimum of the Helmholtz freeenergy

F(V, T) = E − TS

or of the Gibbs free energy

Φ(P, T) = E − TS + PV

at given temperature and either fixed volume (C.1) or fixed pressure (C.2); more often in reality we are dealing with the second situation. When a certain ordering appears in the system – it may be magnetic ordering, for example ferro- or antiferromagnetic; or ferroelectricity; or an ordering in a structural phase transition – one can introduce a measure of such ordering, different for specific situations, which is called the order parameter; let us denote it η.

Part IV presents advanced techniques and methods that are useful for understanding phase transitions in materials. The emphasis is on aspects of free energy, energy, entropy, and kinetic processes, and less on specific phase transformations. The chapters are far from a complete set of advanced topics, however, and other topics can be argued to be just as important. The topics in Part IV have proved their value, though, and appear in the literature with some frequency. The reader is warned that some of the presentations assume a higher level of mathematics or physics than the other sections in the book, and some important results are stated without proof.

The chapters in Part IV follow no natural sequence, and may be selected for interest or need. Some topics on energy (Chapter 21), entropy (Chapter 24), and atom movements (Chapter 23) are continuations of content in Chapters 6, 7, 9 of Part II. Chapter 19 presents analyses of phase boundaries at low and high temperatures, and Chapter 20 presents techniques for analyzing thermodynamics and physical properties very close to a critical temperature.

This chapter analyzes the thermodynamic stability of “static concentration waves.” The idea is that an ordered structure can be described as a variation of chemical composition from site to site on a crystal lattice, and this variation can be written as a wave, with crests denoting B-atoms and troughs the A-atoms, for example. The wave does not propagate, so it is called a “static” concentration wave. Another important difference from conventional waves is that the atom sites are exactly on the tops of crests or at the bottoms of troughs, so we do not consider the intermediate phases of the concentration wave, at least not in our main examples. A convenient feature of this approach is that an ordered structure can be described by a single wavevector, or a small set of wavevectors. The disordered solid solution has no such periodicity, so the amplitude of the concentration wave, η, serves as a long-range order parameter.

This chapter begins with a review of how periodic structures in real space are described by wavevectors in k-space, and then explains the “star” of the wavevector of an ordered structure. A key step for phase transitions is writing the free energy in terms of the amplitudes of static concentration waves.

Chapter 3 derived the diffusion equation with the assumption of random atom jumps. Solutions to the diffusion equation were presented, but the reader was warned that these solutions require a constant diffusion constant D, and this is rarely true as an alloy evolves during a phase transformation. There are other risks in using the diffusion equation when atom motions occur by the vacancy mechanism, where a mobile vacancy rearranges atoms in its wake. This chapter explains the nonrandomness of atom jumps with a vacancy mechanism, and these nonrandom characteristics occur even when the vacancy itself moves by random walk. Furthermore, in an alloy with chemical interactions strong enough to cause a phase transformation, the vacancy frequently resides in energetically favorable locations, so any assumption of random walk may be seriously in error.

When materials with different diffusivities are brought into contact, their interface is displaced with time because the fluxes of atoms across the interface are not equal in both directions. Other phenomena such as stresses and voids may develop during interdiffusion. An applied field can bias the diffusion process towards a particular direction, and such a bias can also be created by chemical interactions between atoms. Chapter 9 ends with two other topics of diffusion – one is atom diffusion that occurs in parallel with atom jumps forced without thermal activation, and the second is a venerable statistical mechanics model of diffusion that has components used today in many computer simulations of diffusion.

Nanostructured materials are of widespread interest in science, engineering, and technology. For the purpose of thermodynamics, it is useful to define nanomaterials as materials with structural features of approximately 10 nm or smaller, i.e., tens of atoms across. Important physical properties of nanomaterials originate from one or two basic features:

1. • Nanomaterials have a high surface-to-volume ratio, and a large fraction of atoms located at, or near, surfaces.

2. • Nanomaterials confine electrons, phonons, or polarons to relatively small volumes, altering their energies. The confinement of structural defects such as dislocations or internal interfaces alters their energies and interactions, too.

A practical question is whether nanostructures are adequately stable at modest temperatures. A more basic question is how the thermodynamics of nanostructured materials differs from conventional bulk materials. In short, their internal energy is raised by the surfaces, interfaces, or composition gradients in nanostructures. Chapter 16 discusses the thermodynamics of interfaces, but Sections 6.6 and 11.2 covered important aspects of surface energy, including surface relaxation and reconstruction processes that are driven by chemical energy. Some basic issues for the confinement of electrons in nanostructures are presented here.

The free energy of nanostructured materials is altered by the entropy from the configurations of nanostructural degrees of freedom and their excitations. These entropy contributions tend to stabilize a nanomaterial at finite temperatures.

Section 1.1 put phase transitions in materials into a broader context of phase transitions in general. Most of this book has been on how atoms arrange themselves at different T and P, and how these arrangements change abruptly through a phase transition. Atoms in solids tend to be a bit sluggish in their movements, however, and their arrangements can be slow to attain states of thermodynamic equilibrium. Diffusion and nucleation, which retard, redirect, or even arrest the paths to equilibrium, are kinetic phenomena of interest and importance. Those nonthermodynamic phenomena are essential to the full life cycle of a phase transformation, but they obscure the singularities in the free energy function or its derivatives that underlie the thermodynamics of a phase transition.

The more general field of phase transitions often places rigorous emphasis on thermodynamic equilibrium, even at temperatures that are very low, or at temperatures very near a critical temperature where atomic structures may not attain equilibrium in reasonable times. Liquid–gas transitions and magnetic transitions are often better candidates for studies of phase transitions for their own sake. Nevertheless, concepts from the broader field of phase transitions do help our understanding of phase transformations in solid materials. Much of the interest in the basic physics of phase transitions is in how a system behaves very close to the critical temperature.