When a crystalline solid is terminated by a surface new types of energy bands can form that are localized at or near the surface. A geometrically perfect surface may have “intrinsic” surface states with energies lying within the band-gap region, above or below the bulk energy bands. These surface bands have wavefunctions that decrease exponentially with increasing distance into the crystal. The further the energy of the surface state is from the bulk band-edge energy, the more rapidly its wavefunction decreases with distance from the surface. Localized surface states associated with defects such as oxygen vacancies can also occur. Such surface bands and states can play an important role in chemisorption and catalysis in transition metal oxides.

In this chapter we will review the theoretical concepts that underlie the formation of surface bands and defect states based on our empirical LCAO model. The material is essential to the understanding of more fundamental and accurate calculation methods. A comprehensive review of the experiments on transition metal-oxide surfaces is available [1], but only those relevant to doped insulating perovskites and metallic NaxWO3 will be discussed here.

Metal oxides having the cubic (or nearly cubic), ABO3 perovskite structure constitute a wide class of compounds that display an amazing variety of interesting properties. The perovskite family encompasses insulators, piezoelectrics, ferroelectrics, metals, semiconductors, magnetic, and superconducting materials. So broad and varied is this class of materials that a comprehensive treatise is virtually impossible and certainly beyond the scope of this introductory text. In this book we treat only those materials that possess electronic states described by energy band theory. However, a chapter is devoted to the quasiparticle-like excitations observed in high-temperature superconducting metal oxides. Although principally dealing with the cubic perovskites, tetragonal distortions and octahedral tilting are discussed in the text. Strong electron correlation theories appropriate for the magnetic properties of the perovskites are not discussed. Discussions of the role of strong electron correlation are frequent in the text, but the development of the many-electron theory crucial for magnetic insulators and high-temperature superconductors is not included.

This book is primarily intended as an introductory textbook. The purpose is to provide the reader with a qualitative understanding of the physics and chemistry that underlies the properties of “d-band” perovskites. It employs simple linear combinations of atomic orbitals (LCAO) models to describe perovskite materials that possess energy bands derived primarily from the d orbitals of the metal ions and the p orbitals of the oxygen ions.

As we have seen in Chapter 6 the δ function (sometimes called the Dirac delta function) is a useful mathematical tool. In this Appendix we derive formulae for the representation of the delta functions employed in Chapter 6.

The δ function is defined by its properties:

where f(x) and its derivative are continuous, single-valued functions and the integral is over any range containing x0. The result, ∫ δ(x – x0) dx = 1, follows from (B.2) for f(x) = 1. Another property, δ(x – x0) → ∞ as x → x0, is implied by (B.1) and (B.2). Clearly if (B.1) holds, the δ function must be arbitrarily large at x0 if (B.2) is valid.

There are numerous analytical representations of the delta function. We shall use a frequently employed representation wherein δ(x – x0) is the limit of a particular function:

where “ℑM” indicates the imaginary part of the quantity and λ is a small positive number. In using this representation there is an implied order of doing things. The limiting process λ → 0 (λ > 0) is to be performed last. This means one must calculate the imaginary part first, then take the limit as λ → 0. This limiting process is often indicated by using the symbol 0+ as we did in Chapter 6.

The imaginary part of (B.3) is

It is easy to show that the delta function defined by (B.4) satisfies the equations (B.1) and (B.2).

The LCAO method described in the previous chapter forms the basis for a number of empirical or qualitative models. In such models the LCAO matrix elements are treated as “fitting” parameters to be determined from experiment or in some empirical way. Such models have provided a great deal of physical insight into the electronic properties of molecules and solids.

One of the first and simplest LCAO models was used by Hückel [1] to discuss the general qualitative features of conjugated molecules. Later, Slater and Koster [2] introduced an LCAO method for the analysis of the energy bands of solids. The Slater–Koster LCAO model has been used extensively as an interpolation scheme.

The LCAO parameters are determined by choosing the model parameters to give results that approximate those of more accurate numerical energy band calculations at a few points in the Brillouin zone. Once the parameters are determined the LCAO model gives approximate energies at any point in the Brillouin zone.

LCAO models have been remarkably useful for ordered solids and molecules having a high degree of symmetry. The reason for this is that in many cases the electronic structure is qualitatively determined by symmetry or group theoretical considerations. The group theoretical properties of a system are preserved in LCAO models and therefore they are able to correctly represent the general features of the electronic states.

The majority of perovskites are not cubic, but many of the non-cubic structures can be derived from the cubic (aristotype) structure by small changes in the ion positions. Several types of distortions occur among the perovskites. The most important types are those involving (a) ion displacements in which, for example, the B ion or A ion (or both) moves off its site of symmetry; (b) rotations or tilting of the BO6 octahedra; and (c) both tilting and displacements.

Departure from the cubic perovskite structure will occur whenever distortions lead to a lower total energy. The lowering of the total energy in most cases is small, typically of the order of a tenth of an eV/cell and dependent upon the temperature. The additional stabilization energy of one structure over another depends in a subtle manner upon the competition between a number of electronic factors including changes in the Coulomb interactions (Madelung potentials), changes in the degree of covalent bonding, and the number of electrons occupying the antibonding conduction bands. In many cases changes in the A–O covalent bonding is thought to play a key role in determining the distorted structure.

Clearly it is not possible to predict structures based on the simple LCAO model we have been studying. However, given a particular structure that is close to the cubic structure one can examine the changes in the electronic states with the goal of understanding why the distorted structure is more stable.

Background

In 1986 Bednorz and Müller [1] made the surprising discovery that the insulating, ceramic compound, La2CuO4, was superconducting at low temperature when suitably doped with divalent ions. In fact, all of the members of the class of copper oxides, La2–xMxCuO4 (where M is Ba2+, Sr2+, or Ca2+ ions), were found to be superconducting for x in the range, 0.1 ≲ x ≲ 0.25. At optimal doping of x ≈ 0.15, the superconducting transition temperature, Tc, of La2–xSrxCuO4 was about 38 K.

Since 1986 there has been an enormous scientific effort focused on copper oxides with similar structures. Over 18 000 research papers were published in four years following Bednorz and Müler's report. As work progressed around the world, new compounds were discovered with higher Tc's. In 1987 doped samples of YBa2Cu3O7 (“YBCO”) were found to be superconducting at 92 K, thus becoming the first superconductors with Tc higher than the boiling point of liquid nitrogen (77 K). Recent studies [2] on mercuric cuprates report Tc in excess of 165 K.

A very brief list of some of the most studied high-temperature superconductors (HTSC) and their transition temperatures is given in Table 11.1. In column two “alias” refers to the frequently used name of the undoped, “parent” composition. For example, Tl1223 refers to: one Tl, two Ba, two Ca, and three Cu atoms. The tetragonal structure for the La2CuO4, is shown in Fig. 11.1(a).

In Chapter 1 a qualitative description of the energy bands of perovskites was developed starting from a simple ionic model. It was argued that the essential electronic structure of the perovskites is derived from the BO3 ions of the ABO3 compound. The A ion was shown to be important in determining the ionic state of the B ion. Also, the A ion contributes to the electrostatic potentials. However, the energy bands associated with the outer s orbitals of the A ion were found to be far removed in energy from the lowest empty d bands and were unoccupied. As a result of these considerations the electronic structure of the A ion can be neglected in discussing the principal features of perovskite energy bands.

The energy bands that are electronically and chemically active are derived from the d orbitals of the transition metal (B) cation and the 2p orbitals of the oxygen anions. The deep core states of the ions produce atomic-like levels at energies far below the valence bands and may also be omitted in our discussion.

One of the first energy band calculations for perovskites was carried out by Kahn and Leyendecker [1] for SrTiO3. They employed a semiempirical approach based on the method of Slater and Koster [2] (described in Section 3.2). The model presented in this chapter follows closely the work of Kahn and Leyendecker.