Reduction to a regular model, mean field theory

Regular model with “interaction”

The main task of this section is to represent the physical quantities discussed in the preceding chapter in such a form that one could average over the random potential at the beginning of the calculations. One way to do so is the replica trick suggested by Edwards and Anderson (1975) for a study of spin glasses. The first works on the application of field theoretical models to disorder problems were based on this trick (Wegner (1979), Schäfer and Wegner (1980), Efetov, Larkin, and Khmelnitskii (1980)). In the works of Wegner (1979) and Schäfer and Wegner (1980) kinetic quantities were written in terms of functional integrals over conventional numbers, whereas in the work of Efetov, Larkin, and Khmelnitskii (1980) integration over the anticommuting Grassmann variables was used. Then, in both approaches nonlinear σ-models that contained n × n matrices were derived. At the end of calculations one had to take the limit n → 0. The formalism of Efetov, Larkin, and Khmelnitskii (1980) was extended later to include electron–electron interactions (Finkelstein (1983, 1984)) and strong magnetic fields (Levine, Libby, and Pruisken (1984)).

The nonlinear supermatrix σ-model derived later has many common features with the replica σ-model, and it may seem that the calculations within all these models are equivalent.

Small metal particles

General properties

Small clusters of atoms of metallic elements have very unusual physical properties. These objects are not as small as molecules and some of their properties are reminiscent of those of bulk metals. At the same time the electron spectrum is discrete and this has many intriguing consequences. Systems of small metal particles are under very intensive study, and some of their characteristics have various technical applications. However, the physical properties of the metal clusters are no less interesting than possible applications because to describe them one needs modern quantum statistical theories analogous to those developed in nuclear physics. Moreover, some approaches invented studying the metal particles have their applications in nuclear physics and problems of quantum chaos. A complete account of experimental and theoretical results can be found in the review by Halperin (1986). An impression of the state of the art of both the experimental and the theoretical work can be gained from the proceedings of the Fourth International Meeting on Small Particles and Inorganic Clusters (Chapon, Gillet, and Henry (1989)). Some aspects of small particle physics are considered in recent reviews by Staveren, Brom, and de Jongh (1991) and Nagaev (1992).

Following the main idea of this book in this section I want to discuss several important steps in the theoretical understanding of the systems of small metal particles; then in the following sections of this chapter I present results obtained by the supersymmetry technique.

Universal conductance fluctuations

Diffusion modes and conductance fluctuations

During the last 10–15 years study of transport through very small conductors has been a very popular topic of both theoretical and experimental research (for reviews see, e.g., Altshuler, Lee, and Webb (1991); Kirk and Reed (1992); Beenakker and van Houten (1991)). These systems are much larger (typically of the size 100–000 Å) than atomic distances, and, naturally, they cannot be considered as microscopic objects. Then, what can be special in their properties with respect to properties of macroscopic conductors? The answer is related to the quantum interference that proved to be so crucial to the localization problem. Although the conductors may contain internal defects, such disorder does not destroy the coherence of the wave functions and quantum effects can become very important, leading to completely new physical phenomena. The last two chapters were devoted to study of some exotic effects in isolated or almost isolated samples.

The quantum effects in isolated metal particles can be destroyed by inelastic scattering only. It is only at the classical limit that both elastic and inelastic scattering play equivalent roles in transport. Considering inelastic processes one should distinguish between the cases lϕ < L and lϕ > L, where lϕ is the inelastic mean free path and L is the size of the sample.

Brownian motion model

Fokker–Planck equation for the Coulomb gas model

In the preceding chapters the relation between random matrix theory and the zero-dimensional supersymmetric σ-model was emphasized many times. This equivalence makes it possible to use the σ-model for a description of such nontrivial problems as transport through quantum dots in the regime of chaotic dynamics. By changing external parameters such as magnetic field or gate voltage one can study correlations of physical quantities at different values of these parameters. One example has been considered in Section 9.4. One can also study the dependence of average quantities on the external parameters, and the persistent currents considered in Chapter 8 are this type of problem. In many cases the final formulae are quite universal, depending only on the mean level spacing and several parameters characterizing changes in the initial Hamikonian.

It is clear now that study of parametric correlations leads to new and very interesting and unexpected results. Again, much important work has been performed in the last 1–2 years, and therefore the content of this chapter reflects only the present state of the art.

I want to present first what was known about parametric correlations in random matrix theory in the past. The most important work in this area is that of Dyson (1962b), who proposed to use the idea of Brownian motion, well known in kinetic theory (Chandrasekhar ((1943), Uhlenbeck and Ornstein (1930), Wang and Uhlenbeck (1945), Isihara (1971)), to describe parametric variations of physical quantities.

Although oxides have been the subject of active research for many years, they have attracted an increasing interest in the last decade. One reason for this interest is the discovery of superconductivity in copper oxide based materials, in 1986, with critical temperatures higher than the temperature of liquid nitrogen. Simple oxides have also been more thoroughly studied and a detailed analysis of their surface properties has been undertaken, thanks to several technological advances made during this period. The success in compensating charging effects, for example, has allowed spectroscopic measurements to be performed. Photoemission, x-ray absorption, Auger spectroscopy and low-energy electron diffraction now yield quantitative information, as they do for semi-conductors and metals. Topographic images of insulating surfaces can be recorded with an atomic force microscope. On the theoretical side, advanced numerical codes have been developed, which solve the electronic structure, optimize the geometry, and start accounting for dynamical effects in an ab initio way. The results presently available allow a first synthesis of the field.

The interest and the richness of the field of oxide surfaces lies in its inter-disciplinary nature and in the diversity of questions it raises, both on a fundamental and on an applied level. For example, geophysicists and geologists consider in detail the surface properties and porosity of the rocks of our earth, made of complex oxides whose properties are, to a large extent, controlled by the grain boundaries and internal surfaces.

In the surface layers, the breaking of anion–cation bonds and the modifications of inter-atomic distances, induced by relaxation, rumpling or reconstruction effects, perturb the electrostatic potentials and the orbital hybridization. The surface electronic structure is thus modified and presents specific features compared to the bulk characteristics. In this chapter, we will discuss various aspects of these changes, relevant for planar semi-infinite systems, thin films and defected surfaces. We will restrict ourselves to the results obtained on single crystals, prepared under controlled conditions and studied in ultra-high vacuum (Henrich, 1983; 1985; Henrich and Cox, 1994). Instead of focusing on specific properties of this or that oxide, we will try to extract the general trends concerning the density of states, the gap width, the charge densities, etc., and, whenever this is possible, we will point out the physical origin of the differences found between the bulk and the surface electronic structure.

Experimental and theoretical studies

The study of the surface electronic structure requires specific tools. For example, in spectroscopic experiments, in order to enhance the surface signal with respect to the bulk one, to obtain information on the outer layers, one has to send or detect particles with small mean free paths, which mainly sample the few outer layers. Similarly, special care has to be taken in the numerical approaches. We will quickly review some aspects of this question, both from the experimental and the theoretical sides.

Introduction

The research field of metal–oxide interfaces is very active, partly because of their important technological applications. For example, in heterogeneous catalysis, oxide powders or porous compounds, such as zeolites, are used as supports for transition metal clusters, because they provide a large – external or internal – specific area of contact with the metal. In many cases, it is also recognized that they modify the cluster reactivity (Dufour and Perdereau, 1988). Oxide surfaces, such as those of MgO or SrTiO3, whose quality and planarity are well controlled, have been used as substrates for the deposition of thin superconductor films. This has been particularly important since the discovery that some copper oxide based compounds remain superconductors above liquid nitrogen temperature. Thin metallic films are also deposited on various oxides in the fabrication of optical devices, or on glass in the fabrication of mirrors.

Oxides are often chosen as insulation materials, for example as sheaths for resistive heaters, due to their low electrical and thermal conductivity. In MOS transitors (MOS = Metal–Oxide–Semi-conductor), a thin SiO2 layer is deposited between a doped silicon substrate and the metallic gate to control the channel conductivity. In more complicated electronic devices, with several integration levels, SiO2 is also used to make insulating dielectric layers.

This first chapter summarizes the main bulk characteristics of insulating oxides, as a prerequisite to the study of surfaces. The foundations of the classical models of cohesion are first recapitulated, and the distinction between charge-transfer oxides and correlated oxides is subsequently established. Restricting ourselves to the first family, which is the subject of this book, we analyse the mixed iono–covalent character of the anion–cation bonding and the peculiarities of the bulk electronic structure. This presentation will allow us to introduce various theoretical and experimental methods – for example, the most common techniques of band structure calculation – as well as some models – the partial charge model, the alternating lattice model – which will be used in the following chapters.

Classical models of cohesion

Ionic solids are made up of positively and negatively charged ions – the cations and the anions, respectively. The classical models postulate that the outer electronic shells of these ions are either completely filled or empty, so that the charges have integer values: e.g. O–– (2p6 configuration) or Mg++ (3s0 configuration). The strongest cohesion is obtained when anions and cations are piled up in an alternating way – the anions surrounded by cations and vice versa –, a stacking which minimizes the repulsion between charges of the same sign.

The hard-sphere model

In the first models, due to Born and Madelung, the ions are described as hard spheres, put together in the most compact way (Kittel, 1990).

When a crystal is cut along some orientation, the atoms located in the few outer layers experience non-zero forces which are induced by the breaking of oxygen–cation bonds. Generally, they do not remain at the positions fixed by the three-dimensional lattice. Point or extended defects may result, as well as lattice distortions. This chapter analyses the structural features of oxide surfaces, which is also a useful step, before starting the discussion of the surface electronic properties. Yet, conceptually, this presentation is not fully satisfactory, because the structural and electronic degrees of freedom are coupled and both determine the ground state configuration. Despite a rich literature, the structural properties of oxide surfaces are not fully elucidated. It is often difficult to prepare stoichiometric and defect-free surfaces, and the characterization is hindered by charging effects and by an uncertainty about the actual crystal termination.

Preliminary remarks

We will make some preliminary remarks concerning the designation of the surfaces, their polar or non-polar character and their structural distortions – relaxation, rumpling and reconstruction.

Notations

A plane in a crystal, is identified by three integers (h, k, l), called the Miller indices, which are in the same ratio as (1/x, 1/y, 1/z), the inverses of the coordinates of the intersections of this plane with the crystallographic axes (van Meerssche and Feneau-Dupont, 1977). Notations with four indexes (h, k, –(h + k), l) are used in hexagonal structures, such as α-quartz, corundum α-alumina, or the wurtzite ZnO structure.

A semi-infinite crystal supports specific excitations which are not present in the bulk material. In oxides, these new modes result from the breaking of anion–cation bonds at the surface, whose effects on the electronic and atomic structure have been the subject of the two previous chapters. Here, we will describe the excitations associated with the atomic and electronic degrees of freedom. The phonons, which are the quantized modes of vibrations of the atoms, have small characteristic energies, of the order of a few tens of milli-electron volts (meV). The electron–hole pairs and plasmons, which are characteristic of the electronic degrees of freedom, have much higher energies, of the order of a few electron-volts. Due to the different time scales involved, a decoupling between these two types of excitations takes place (Born–Oppenheimer decoupling). The electrons are able to follow ‘instantly’ the atomic displacements, while the atoms have no time to move at the time scale of the electron delocalization and excitations.

Surface phonons

We have seen in Chapter 2 that bond-breaking on a surface greatly modifies the atomic energy levels. The same is true for the atomic vibrations around the equilibrium positions. In this section, we will review the main experimental and theoretical results concerning surface phonons on oxide surfaces.

Experimental and theoretical approaches

High-resolution electron energy loss (HREELS = high-resolution electron energy loss spectroscopy) experiments and inelastic scattering of helium atoms allow a determination of the surface vibration modes and of the phonon dispersion curves.