The key instrumentation necessary for photoemission consists of a radiation source, an electron energy analyzer, and a means of preparing and maintaining a clean sample surface in ultrahigh vacuum. Ancillary sample characterization capability, both in situ and ex situ, is also important, especially for non-stoichiometric samples like high-temperature superconductors. We discuss each of these in turn, emphasizing some of the important attributes of each. We do not go into enough depth to allow a reader to construct the instrumentation. We emphasize the crucial characteristics of resolution and signal-to-noise ratio, and how they are limited by instrumentation.

Radiation sources

Work functions may be as high as 5 eV, so light sources for photoemission must emit photons of energies higher than this. Since O2 and N2 absorb photons starting just above 6 eV, the entire optical path must be evacuated to a pressure below about 10–5 Torr in order to ensure significant transmission over light paths of a meter or more. For this reason, the spectral range above about 6 eV is called the vacuum ultraviolet region, Its name changes at around 50–100 eV to the soft x-ray region, for which the optical path must also be in vacuum.

Radiation sources used in the earlier valence-band studies (Berglund and Spicer, 1964b) often were hydrogen discharges. These gave a weak continuum from the H2 molecule and a strong emission line, the Lyman-α line at 10.2 eV, from the H atom (Samson 1967).

We have already mentioned some incremental improvements in photoelectron spectroscopy, increased energy- and angle resolution through improvements in instrumentation. Another is the angle-scanned photoelectron spectroscopy pioneered by Aebi et al. (1994a,b), described in Chapter 5. Lindroos and Bansil (1996) recently pointed out additional advantages of this technique when coupled with suitable calculations. They found not only the usual features produced by direct transitions from initial states on the Fermi surface, but additional features from indirect transitions whose intensities are proportional to the one-dimensional densities of initial states for transitions at a fixed k∥. They illustrated these aspects with calculations for several surfaces of Cu. A re-examination of such spectra for Bi2212 is likely to occur soon. There are several other variations of photoelectron spectroscopy already tested at some level, some of which may play a role in future work on cuprates.

Photoelectron microscopy

The photoemission studies normally carried out collect electrons from an illuminated area usually no smaller than about 100 μm × several hundred μm. There may be lateral inhomogeneities on a scale smaller than this, so photoelectrons from special sites may be averaged with those from the rest of the area sampled. This may be particularly misleading in studies of reactions with overlayers which may take place preferentially at steps. Photoelectron microscopy has been developed over the past fifteen years or so, along with x-ray microscopy, which uses absorption differences for contrast.

Photoelectron spectroscopy, especially XPS, can be used to study some aspects of the surface chemistry of cuprates. Examples of this are the changes seen in the photoelectron spectra of Y123 as a freshly prepared surface deteriorates with time at 300 K in ultrahigh vacuum. In this, the evidence that the changes are due to loss of oxygen is not direct. A better example, to be discussed below, is the study of a metal overlayer on a cuprate. The overlayer may just sit there, making it a candidate for an ohmic contact, or it may disrupt the surface, replacing some of the atoms in the cuprate, which then appear in the surface layer. The appearance or disappearance of core-level peaks can be used to track the depths of constituents, and binding-energy shifts may give clues about oxidation states and location of atoms, i.e., surface or bulk.

Most of the studies of interface reactions have been carried out on the same type of surfaces used in the study of the cuprates themselves, cleaved single crystals or scraped or fractured polycrystalline samples. These cannot be considered technologically important surfaces unless processing in the future is carried out under ultrahigh vacuum conditions. Some of the studies reported below may need to be repeated on air-exposed surfaces, or surfaces handled in inert atmospheres or poorer vacua.

Introduction

Photoemission studies may be said to have begun in 1887 with the observation by Heinrich Hertz that a spark between two electrodes was obtained more easily if the electrodes were illuminated (HeTrtz, 1887), although this occurred before the discovery of the electron. Improved experiments during the next several decades were important to the development of quantum mechanics. The modern era of photoemission spectroscopy arguably may be said to have started in 1964 with the papers of Berglund and Spicer (1964a,b). Although there had been prior experimental work and calculations in which Bloch electrons were assumed and dipole matrix elements calculated, it was these two papers, and papers by Gobeli et al. (1964) and Kane (1964), which stimulated a large amount of work, coming as they did nearly simultaneously with the widespread calculation of accurate electronic structures of many materials and the commercial availability of ultrahigh vacuum components. The first paper of Burglund and Spicer worked out what is called the three-step model for photoemission, and the second applied it to new data on Cu and Ag.

In the following we outline several approaches to the description of photoelectron spectra, starting with the most simple conceptually, then progressing to a more sophisticated picture. As we do this, we point out some assumptions and approximations often made, sometimes tacitly, and whether or not they may be important for photoelectron spectroscopic studies of high-temperature superconductors. Experimental considerations are presented in Chapter 4.

Introduction

Although Y123 and Bi2212 both contain CuO2 planes and have similar values of Tc, there are notable differences which have affected the course of the study of each class of material. We have already mentioned two such differences, the reproducible cleavage of Bi2212 and the stability of the cleaved surface in ultrahigh vacuum. The cleavage plane(s) of Y123 is not really known and may vary from cleave to cleave. Moreover, a single plane is unlikely, and the atomic nature of the exposed surface may change at cleavage steps. Surface reconstruction, or at least relaxation, may be possible, even at low temperature. The surfaces of Y123 generally are not stable in ultrahigh vacuum except at temperatures below about 50 K. However, exceptions to this have been found by several groups, but the reasons for this stability are not yet known. Tc may be varied in Bi2212 by the addition or removal of oxygen and there is an optimum oxygen content at which Tc is a maximum. YBa2Cu3Ox also has variable oxygen stoichiometry, but between x = 6.8 and x = 7, Tc remains very close to its maximum value. In Bi2212 the oxygen content changes occur on or adjacent to the Bi–O planes. In Y123 the changes occur in the Cu–O chains, although the holes provided by the oxygen atoms are believed to reside on the CuO2 planes (Cava et al., 1990). Starting from x = 7, removing oxygen atoms from the chains should lower the hole concentration.

NCCO

Nd2–xCexCuO4–y (NCCO) was the first cuprate superconductor to exhibit n-type conduction. The substitution of Ce4+ for Nd3+ introduces electrons which go to the CuO2 planes. (There are no Cu–O chains.) Tc peaks at 24 K for x = 0.15. The importance of NCCO is that is can be used to address the question of electron–hole symmetry. The electron doping swells slightly the large Fermi surface, while hole doping in other cuprates shrinks it slightly. In the localized picture with its small Fermi surface, how do the electron and hole Fermi surfaces compare? NCCO has a simpler LDA band structure near EF than many other cuprates because there is only one CuO2 plane per unit cell. Fewer bands cross EF. What are the new states in NCCO the doping introduces along with the new electrons, or do those electrons occupy existing states as in the independent-electron picture? Other trivalent rare earths may be substituted for Nd. Why do the magnetic moments of the rare-earth ions not destroy superconductivity as they do in many other superconductors? A brief review of photoemission studies of NCCO was written by Sakisaka (1994).

Most of the studies of NCCO have been carried out on polycrystalline, i.e., sintered, samples. The earliest study on a single crystal (an epitaxial film), that of Sakisaka et al. (1990a,b), was mentioned in Chapter 6. Since then, and since the review of Sakisaka was written, there have been several more studies on single crystals.

The preceding chapter showed how absorbing state transitions arise in catalytic kinetics. Having seen their relevance to nonequilibrium processes, we turn to the simplest example, the contact process (CP) proposed by T.E. Harris (1974) as a toy model of an epidemic (see also §8.2). While this model is not exactly soluble, some important properties have been established rigorously, and its critical parameters are known to high precision from numerical studies. Thus the CP is the ‘Ising model’ of absorbing state transitions, and serves as a natural starting point for developing new methods for nonequilibrium problems. In this chapter we examine the phase diagram and critical behavior of the CP, and use the model to illustrate mean-field and scaling approaches applicable to nonequilibrium phase transitions in general. Closely related models figure in several areas of theoretical physics, notable examples being Reggeon field theory in particle physics (Gribov 1968, Moshe 1978) and directed percolation (Kinzel 1985, Durrett 1988), discussed in §6.6. We close the chapter with an examination of the effect of quenched disorder on the CP.

The model

In the CP each site of a lattice (typically the d-dimensional cubic lattice, Zd) represents an organism that exists in one of two states, healthy or infected. Infected sites are often said to be ‘occupied’ by particles; healthy sites are then ‘vacant.’

The model introduced by Ziff, Gulari, & Barshad (1986) (ZGB) for the oxidation of carbon monoxide (CO) on a catalytic surface has provided a source of continual fascination for students of nonequilibrium phase transitions. This manifestly irreversible system exhibits transitions from an active steady state into absorbing or ‘poisoned’ states, in which the surface is saturated by oxygen (O) or by CO. The transitions attracted wide interest, spurring development of numerical and analytical methods useful for many nonequilibrium models, and uncovering connections between the ZGB model and such processes as epidemics, transport in random media, and autocatalytic chemical reactions.

The literature on surface reactions continues to expand as variants of the ZGB scheme are explored. In this chapter we do not attempt to give even a partial survey; we define the model, examine its phase diagram, and describe mean-field and simulation methods used to study it.

The Ziff–Gulari–Barshad model

To begin, let us describe some facts about the oxidation of CO, a catalytic process of great technological importance; see Engel & Ertl (1979). (An immediate poison is converted into a global one!) The reaction, which is catalyzed by various platinum-group metals, proceeds via the Langmuir—Hinshelwood mechanism: to react, both species must be chemisorbed. CO molecules adsorb end-on, and require a relatively small area.

The ordinary kinetic versions of the Ising model may be modified to exhibit steady nonequilibrium states. This is illustrated in chapter 4 where a conflict between two canonical mechanisms (diffusion and reaction) drives the configuration away from equilibrium. A more systematic investigation of this possibility, when the conflict is between different reaction processes only, is described here. We focus on spin systems evolving by a superposition of independent local processes of the kind variously known as spin flips, birth/death or creation/annihilation. The restriction to spin flip dynamics does not prevent the systems in this class from exhibiting a variety of nonequilibrium phase transitions and critical phenomena. Their consideration may therefore help in developing nonequilibrium theory. In addition, they have some practical interest, e.g., conflicting dynamics may occur in disordered materials such as dilute magnetic systems, and some of these situations can be implemented in the laboratory.

The present chapter describes some exact, mean-field and MC results that together render an intriguing picture encouraging further study. It is argued in §7.1 that some of the peculiar, emergent, macroscopic behavior of microscopically disordered materials may be related to diffusion of disorder. This provides a physical motivation for the nonequilibrium random-field Ising model (NRFM).

The subject of this book lies at the confluence of two major currents in contemporary science: phase transitions and far-from-equilibrium phenomena. It is a subject that continues to attract scientists, not only for its novelty and technical challenge, but because it promises to illuminate some fundamental questions about open many-body systems, be they in the physical, the biological, or the social realm. For example, how do systems composed of many simple, interacting units develop qualitatively new and complex kinds of organization? What constraints can statistical physics place on their evolution?

Nature, both living and inert, presents countless examples of nonequilibrium many-particle systems. Their simplest condition — a nonequilibrium steady state — involves a constant flux of matter, energy, or some other quantity (de Groot & Mazur 1984). In general, the state of a nonequilibrium system is not determined solely by external constraints, but depends upon its history as well. As the control parameters (temperature or potential gradients, or reactant feed rates, for instance) are varied, a steady state may become unstable and be replaced by another (or, perhaps, by a periodic or chaotic state). Nonequilibrium instabilities are attended by ordering phenomena analogous to those of equilibrium statistical mechanics; one may therefore speak of nonequilibrium phase transitions (Nicolis & Prigogine 1977, Haken 1978, 1983, Graham 1981, Cross & Hohenberg 1993).

When we extend the basic CP, a host of intriguing questions arise: How do multi-particle rules and diffusion affect the phase diagram? When should we expect a first-order transition? Can multiple absorbing configurations or conservation laws change the critical behavior? In this chapter we examine a diverse collection of models whose behavior yields some insight into these issues.

Multiparticle rules and diffusion

In the CP described in chapter 6, the elementary events (creation and annihilation) involve single particles. What happens if one of the elementary events involves a cluster of two or more particles? Consider pairwise annihilation: in place of • → ○ (as in the CP), • • → ○ ○. (In other words, a pair of particles at neighboring sites can annihilate one another, but there is no annihilation of isolated particles.) In fact, we have already seen one instance of pairwise annihilation: in the ZGB model (chapter 5), adsorption of O2 destroys a pair of vacancies. The transition to the absorbing O-saturated state corresponds to (and belongs to the same class as) the CP. Another example is a version of the CP (the ‘A2’ model), in which annihilation is pairwise (Dickman 1989a,b). Here again the transition to the absorbing state is continuous, and the critical exponents are the same as for the CP.

Chapter 2 contains an essentially phenomenological description of the DLG. We now turn to theoretical descriptions, a series of mean-field approximations, which yield analytical solutions for arbitrary values of the driving field E and the jump ratio Γ. We find that mean-field approximations are more reliable in the present context than in equilibrium. For example, a DLG model exhibits a classical critical point for Γ, E → ∞ (§3.4), and simulations provide some indication of crossover towards mean-field behavior with increasing Γ (§2.3). On the other hand, the methods developed here can be used to treat the entire range of interest: attractive or repulsive interactions, and any choice of rate as well as any E and Γ value. As illustrated in subsequent chapters, these methods may be generalized to many other problems.

The present chapter is organized as follows. §3.1 contains a description of the method and of the approximations involved. The one-dimensional case is solved in §3.2 for arbitrary values of n, E, and Γ, and for various rates. A hydrodynamic-like equation and transport coefficients are derived in §3.3. In §§3.4–3.6 we deal with two- (and, eventually, three-) dimensional systems. In particular, the limiting case for Γ, E → ∞ studied in van Beijeren & Schulman (1984) and in Krug et al. (1986) is generalized in §3.4 by combining the one-dimensional solution of §3.2 with an Ω-expansion, to obtain explicit equations for finite fields and for the two limits Γ → ∞ and Γ → 0. A two-dimensional model is solved in §3.5.

A class of biochemical processes is modeled by an open system into which reactants are continuously fed, and out of which products are continuously removed to maintain a steady nonequilibrium state. Studying such idealized models helps in understanding the formation and stability of complex spatial and temporal structures in biological systems. In a model with this aim, one needs ultimately to assume that molecules also diffuse. Systems that are governed by an interplay of reaction and diffusion processes are relevant to many problems in physics and other disciplines (Turing 1952). This chapter describes kinetic lattice gases whose dynamics involve competition between reaction and diffusion. The pioneering study of such stochastic models was due both to mathematicians and physicists; the more recent activity described below was motivated by theoretical as well as practical interest in related nonequilibrium situations.

The chapter is organized as follows. In §4.1, the nonlinear reaction–diffusion equation, a hydrodynamic-like description in which mean values and fluctuations have both spatial and temporal variations, is introduced. §4.2 presents a lattice model with competition between creation/annihilation processes at temperature T, and diffusion at temperature T′. The relation between the hydrodynamic and microscopic levels of description is examined in §4.3, and the implications of the resulting macroscopic equations are discussed in §4.4.

Nature provides countless examples of many-particle systems maintained out of thermodynamic equilibrium. Perhaps the simplest condition we can expect to find such systems in is that of a nonequilibrium steady state; these already present a much more varied and complex picture than equilibrium states. Their instabilities, variously described as nonequilibrium phase transitions, bifurcations, and synergetics, are associated with pattern formation, morphogenesis, and self-organization, which connect the microscopic level of simple interacting units with the coherent structures observed, for example, in organisms and communities.

Nonequilibrium phenomena have naturally attracted considerable interest, but until recently were largely studied at a macroscopic level. Detailed investigation of phase transitions in lattice models out of equilibrium has blossomed over the last decade, to the point where it seems worthwhile collecting some of the better understood examples in a book accessible to graduate students and researchers outside the field. The models we study are oversimplified representations or caricatures of nature, but may capture some of the essential features responsible for nonequilibrium ordering in real systems.

Lattice models have played a central role in equilibrium statistical mechanics, particularly in understanding phase transitions and critical phenomena. We expect them to be equally important in nonequilibrium phase transitions, and for similar reasons: they are the most amenable to precise analysis, and allow one to isolate specific features of a system and to connect them with macroscopic properties.

Generating nonequilibrium steady states by conflicting dynamics, i.e., superposition of spin flip processes in Ising-like systems has a precedent in the consideration of spatially nonuniform distributions of temperature (Garrido & Marro 1987). It was introduced as a drastic simplification of more complex practical situations such as systems with different temperatures at opposing boundaries (Creutz 1986, Harris & Grant 1988). This has in turn suggested studying spins that suffer competing action of several baths, e.g., two independent thermal baths with respective probabilities p and 1 — p, and the case in which different sites of the lattice, either a sublattice or else a set of sites chosen at random, are at different temperatures. After some effort, it appears that these systems involving several temperatures probably have the critical behavior that characterizes the original system from which they derive, at least when all the involved temperatures are finite, and no essential symmetry is broken. In spite of such apparent simplicity, phase diagrams and other thermodynamic properties are varied and interesting, and some questions remain unresolved.

Our concern in this chapter is a class of closely related systems that are less well understood. We discuss a (rather arbitrary) selection of stochastic, interacting particle systems of the kind introduced in chapter 7, namely, a nonequilibrium Ising glass (§8.1 and §8.2), and the cases of bond dilution and very strong bonds (§8.1), invasion and voting processes (§8.3), and kinetic versions of systems in which interactions extend beyond nearest neighbors (§8.4).