Introduction

In the eigenvalue problem of a Hamiltonian in quantum mechanics, the eigenfunctions of the Hamiltonian are classified in terms of the unitary irreducible representations (unirreps) of the symmetry group G of the Hamiltonian. In constructing approximate eigenfunctions by LCAO-MOs of a molecule belonging to a certain symmetry group G, the corresponding problem is to find the irreducible basis sets of G constructed by the linear combinations of the atomic orbitals belonging to the equivalent atoms of the molecule. Such a set is called a set of symmetry-adapted linear combinations (SALC) of the equivalent basis functions or equivalent orbitals. A standard method for such a problem is the generating operator method introduced in Section 6.9: it generates the desired basis set from an appropriate basis function. This method is very general and powerful but it is often extremely laborious to use; Cotton (1990). It is so very formal that one has little feeling until one arrives at the final result, which often could simply be obtained by inspection.

For point groups and their extensions, there exists a simple direct method of constructing the SALC belonging to a unirrep of a symmetry group G. The method requires knowledge of the basis functions of a space vector r = (x, y, z) in three dimensions belonging to the unirrep. The basis sets are well known for all point groups; e.g., those for the point groups Tp and D3p are reproduced in Tables 7.1 and 7.2, respectively, from the Appendix.

This book is written for graduate students and professionals in physics, chemistry and in particular for those who are interested in crystal and magnetic crystal symmetries. It is mostly based on the papers written by the author over the last 20 years and the lectures given at Temple University. The aim of the book is to systematize the wealth of knowledge on point groups and their extensions which has accumulated over a century since Schönflies and Fedrov discovered the 230 space groups in 1895. Simple, unambiguous methods of construction for the relevant groups and their representations introduced in the book may overcome the abstract nature of the group theoretical methods applied to physical chemical problems.

For example, a unified approach to the point groups and the space groups is proposed. Firstly, a point group of finite order is defined by a set of the algebraic defining relations (or presentation) through the generators in Chapter 5. Then, by incorporating the translational degree of freedom into the presentations of the 32 crystallographic point groups, I have determined the 32 minimum general generator sets (MGGSs) which generate the 230 space groups in Chapter 13. Their representations follow from a set of five general expressions of the projective representations of the point groups given in Chapter 12. It is simply amazing to see that the simple algebraic defining relations of point groups are so very far-reaching.

Let G be a finite group and H be a subgroup of G. Then a representation of the group G automatically describes a representation of the subgroup H of G. Such a representation is called a representation of H subduced by a representation of G. Conversely, from a given representation of a subgroup H of G we can form a representation of the group G. Such a representation is called a representation of G induced by a representation of its subgroup H. The problem is to form the irreducible representations (irreps in short) of G from the irreps of its subgroup H. If the group G is finite and solvable (see Section 8.4.1), the problem of forming the irreps of G may be solved by a step-by-step procedure from the trivial irrep of the trivial identity subgroup. This method is possible, for example, for a crystallographic point group. An alternative approach is via the induced irreps of G from the so-called small representations of the little groups of the irreps of H. As a preparation, we shall discuss subduced representations first.

Subduced representations

Let G = {g} be a group and H = {h} be a subgroup of G. Let Γ(G) = {Γ(g); g ∈ G} be a representation of G, then it provides a representation of H by {Γ(h); h ∈ H}. This representation is called the subduced representation of Γ(G) onto H or the representation of H subduced by the representation Γ(G).

There are several reasons this book was written. One was to provide a survey of all of the information obtained to date on cuprates by photoelectron spectroscopy, a technique that has been one of the more productive techniques for providing information on the electronic structure of cuprates. Thus the book serves as a review of research, but such reviews soon become dated in a rapidly moving field. Another reason was to provide a textbook, albeit of a limited nature, for persons entering the field of photoelectron spectroscopy. This aspect of the book should be useful for a longer time. We have limited our discussion to only the techniques used in photoemission on cuprates, excluding other applications of photoelectron spectroscopy, but, in fact, not much has been omitted. Finally, we hope that experienced theorists and experimentalists from fields other than photoelectron spectroscopy will learn something of the difficulties with the photoemission experiments and problems with the interpretation of the data. Photoemission data are widely quoted and we hope to provide a better understanding of the phenemona and the experimental difficulties for those who use such data in the future. In the past few years there has been much activity in the study of complex oxides, e.g., manganates, nickelates, ruthenates, often by photoelectron spectroscopy. We hope those whose interests lie with these materials rather than cuprates will find much of value in this book.

Introduction

In this chapter we present photoelectron spectra of several materials. We present valence-band data for all of them, and in most cases core-level spectra are discussed. Na is a simple metal, about as simple a metal there is. If the photoelectron spectra of Na cannot be understood it is difficult to be comfortable with interpretations of photoelectron spectra of the cuprates. Cu and Ni are more complicated, having 3d bands near and spanning the Fermi level, respectively. Then we present results on NiO, Cu2O, and CuO which introduce new effects. The photoelectron spectra of NiO and especially CuO bear some resemblance to those of the cuprates. The studies reported on the three oxides do not represent all of the work on these materials. Studies on related materials, e.g., Cu- and Ni halides, have been useful, for one can see the influence on the photoelectron spectra of changes in ionicity, number, type, and disposition of nearest neighbors, and distances between nearest neighbor transition metal atoms. Studies of a series of 3d transition-metal oxides are also helpful, but we do not discuss these here. Another approach, also not discussed here, is to start with a clean single-crystal surface of a transition metal and let it oxidize in stages in the experimental chamber, taking spectra and LEED patterns at various stages. In some cases, e.g., NiO, the final stage may be several layers of the same oxide that is studied in bulk samples.

Introduction

As soon as the cuprates were discovered, measurements of all types were made on the samples then at hand. Photoelectron spectroscopy was employed early to learn something of the electronic structure of these materials. As there were many more XPS spectrometers than ultraviolet, many XPS papers appeared in the first year of the “cuprate era.” One of the early goals was to seek evidence of Cu3+ ions in those cuprates that were superconducting, thereby demonstrating that the holes introduced by doping resided, at least partially, on the Cu sites. Cu 2p spectra existed for a series of compounds in which the Cu ions had a known valence, mostly +1 and +2, but at least one, NaCuO2, in which it was trivalent. The Cu 2p apparent binding energy increased as the valence increased, so the binding energy could be used to determine the valence to about one significant figure. (The shift is not linear with valence. The binding energy difference between Cu2+ and Cu3+ is rather small (Karlsson et al., 1992).) The search was not successful. Core-level spectra from all the other elements in the cuprates were studied, not just the Cu 2p. Valence-band spectra were measured for comparison with densities of states from band calculations. Considerable agreement was found, but not enough to say definitively that LDA calculations did or did not give an accurate picture of the electronic structure of the cuprates.

Between 1911 and 1986, superconductivity was strictly a low-temperature phenomenon. The highest critical temperature Tc for any superconductor was 23.2 K for Nb3Ge. For any possible applications, the only useful refrigerants were liquid helium and liquid hydrogen. For much of this period, an understanding of the microscopic origin of superconductivity was lacking. Bardeen, Cooper, and Schrieffer published the BCS theory in 1957, and the more general Eliashberg theory soon followed. Theoretical predictions of the highest Tc one could hope for were not very useful for finding new materials with Tc above 23.2 K. In 1986, Bednorz and Müller discovered that La2CuO4 went superconducting when Ba substituted for some of the La. Later doping studies with Ca, Sr, and Ba showed that in this system Tc reached 38 K. Soon thereafter, Wu et al. found that suitably doped YBa2Cu3O7 had a Tc of 92 K, well above the boiling point of liquid nitrogen. In the intervening years several other classes of high-temperature superconductors were found. All of these had several properties in common. All contained one or more planes of Cu and O atoms per unit cell, all had structures which were related to the cubic perovskite structure, and all were related to a “parent compound” which was antiferromagnetic and insulating. Doping this parent compound (in one of several ways) produced a metal which was superconducting and, for some optimum doping, Tc could be very high, usually above the boiling point of liquid nitrogen.

Several techniques not closely related to photoelectron spectroscopy give results similar to some of those obtained from photoelectron spectroscopy. They may also give other types of information as well. In the following we describe a few of them, but only with respect to data that can be compared with photoelectron spectroscopy. Other useful results from these techniques are not mentioned. We finish this chapter with some results from the spectroscopies closely related to photoelectron spectroscopy: electron energy-loss spectroscopy, soft x-ray absorption, and soft x-ray emission.

Infrared spectroscopy

Infrared spectroscopy probes the sample with photons of energies below about 1 eV, down to perhaps 1 meV. The interaction Hamiltonian is the same as for photoexcitation, proportional to A · p + p · A, and similar selection rules apply. The electronic excitations usually are divided into intraband excitations, often called the Drude contribution, and interband excitations. Because infrared photons have such small wave vectors, an additional scattering mechanism, phonons or impurities, is required in the Drude contribution, leading to a second-order process, and often to a strong temperature dependence. However, this can still lead to intense absorption for systems with metallic electron densities. The Drude term is the high-frequency analog of the electrical conductivity. In a superconductor, one then expects a delta function in the optical conductivity at zero energy and, perhaps, zero absorption until an energy equal to the gap energy 2Δ, is reached.

Photoemission studies generally give information on the electronic structure of a material, not the geometrical structure, although some surface structural information can be obtained by photoelectron spectroscopy. Knowledge of the crystal structures of the high-Tc materials is important in photoemission studies in several ways. First, crystal structures are required to determine the crystal potential for the calculation of the electronic structures. Even if a simple model of the crystal is being used, as in cluster calculations, the model should resemble part of the full crystal structure. Second, angle-resolved photoemission provides information in reciprocal space: energies as a function of angle or wave vector. The Bravais lattice of the crystal determines the Brillouin zone, hence the “space” in which theory and experiment often are compared. Third, the amount and nature of the anisotropy of the crystal structure, although difficult to quantify, are helpful in determining the anisotropy one might expect in physical properties, including photoelectron spectra. Orienting single crystals for angle-resolved photoemission studies also requires some structural knowledge. Fourth, there is an important experimental consideration. Most photoelectron spectroscopy is carried out on clean surfaces produced by cleaving in ultrahigh vacuum. It often is crucial to know what the cleavage surface is. Core-level photoelectron spectroscopy can help identify the atomic character of a cleaved surface, but knowledge of the crystal structure is needed and some idea of interplanar bonding is helpful. Fortunately, a structure determination is usually one of the first studies made on any new material.