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.

The states of electrons and holes in semiconductors and insulators can be divided into two classes: delocalized electron states in the conduction band (holes in the valence band) and localized states. Being in a delocalized state, the charge carrier, electron or hole, contributes to the conductivity. The electrons and holes localized in impurities or defects do not participate in the conduction (they can participate in hopping conduction but its mobility is comparatively much lower). The transition of an electron or a hole from a localized state to a delocalized one or the creation of an electronhole pair is called generation, the inverse process is called recombination. The term ‘trapping’ is also used when an electron or a hole is captured by an impurity. A comprehensive review of the physics of nonradiative recombination processes is presented in the book by Abakumov, Perel' & a Yassievich (1991).

Since the elementary generation and recombination processes are random, the number of charge carriers, i.e., electrons or holes in delocalized states, fluctuates around some mean value which determines the mean conductance of the specimen. The fluctuations of the charge carriers' number produce fluctuations of the resistance and, consequently, of current and/or voltage if a nonzero mean current is passing through the specimen. This noise is called generation-recombination noise (G-R noise). It is, perhaps, the most important mechanism of modulation noise, i.e., noise produced by random modulation of the resistance.

Introduction

In 1925 J.B. Johnson, studying the current fluctuations of electronic emission in a thermionic tube with a simple technique, found, apart from the shot noise whose spectral density was independent of frequency and was in agreement with the Schottky formula (1.5.10) (Schottky, 1918), also a noise whose spectral density increased with decreasing frequency f (Johnson, 1925). Schottky (1926) suggested that this last noise arises from slow random changes of the thermocathode's surface, and proposed for this kind of noise the name ‘flicker effect’, or ‘flicker noise’. The same type of current noise spectrum was found also in carbon microphones (Christensen & Pearson, 1936), and later, in the 1940s and 1950s, in various semiconductors and semiconductor devices. It has become evident that the flicker noise is a very often encountered, if not universal, phenomenon in conductors.

Up to the present, measurements of the current noise spectra have been performed on a vast number of semiconductors, semiconductor devices, semimetals, metals in normal state, superconductors and superconductor devices, tunnel junctions, strongly disordered conductors etc. One observes, in practically all cases, an increase of the spectral density of current noise with decreasing frequency f approximately proportional to 1/f, down to the very lowest frequencies at which the measurements of the spectral density have been performed. Therefore this current noise is usually called 1/f noise (or of 1/f type). The term proposed by Schottky (flicker noise) and the term ‘excess noise’ are now more rarely used.

One of the most important parts of the physics of fluctuations in solids is the physics of fluctuations in solid-state plasma, i.e., in a gas of charge carriers, electrons and/or holes. The spectral density of electric fluctuations in an equilibrium conductor (zero bias voltage and current) is given by the Nyquist fluctuation–dissipation relation (Sec. 2.2) in terms of the dissipative part of the complex impedance Z(f). It means that the problem of calculation and measurement of noise in an equilibrium state is reduced to the problem of, respectively, calculation and measurement of the complex resistance. This problem is usually considered to be simpler. In an equilibrium system, the spectral density of noise does not contain any information other than that contained in Z(f). However, the fluctuation–dissipation relation holds only in the equilibrium state. For conductors that are in nonequilibrium states, for instance, for conductors with hot electrons in strong electric fields, there is no general relation between the spectral density of noise, on one hand, and any characteristics of the response, frequency dependent or static, of the current or voltage, on the other hand. Thus, the calculation of current noise can not be reduced to the calculation of mean current (e.g., its dependence on voltage) and is an individual problem.

This chapter is devoted mainly to fluctuations in a gas of hot charge carriers (for brevity we call them ‘hot electrons’). As is well known, electrons become hot in strongly biased conductors.

Introduction

In a ballistic conductor the charge carriers are moving between the electrodes without being scattered at impurities or phonons. Obviously, the length of the conductor has to be smaller than the free path length of the charge carriers, hence it has to be smaller than, say, 1 b.mum. Before the 1980s the only real solid state ballistic conductors were some very pure monocrystalline metals at liquid helium temperatures and microcontacts (microbridges) between bulk metals (Fig. 5.1). The latter were used in point contact spectroscopy of metals (Yanson, 1974). The resistance of the entire point contact is determined by the transverse dimensions and length of the narrow part (constriction). The conduction is ballistic if these dimensions are smaller than the electrons' freepath length. Since in metals the de-Broglie wavelength of electrons at the Fermi surface, λF, is of the order of interatomic distances, the microcontact's dimensions are inevitably many times greater than λF, and the ballistic motion of electrons in such microcontacts is always quasi-classical.

The recent progress in semiconductor technology made it possible to fabricate ballistic devices, the dimensions of which are of the same order as the wavelength of the charge carriers. These devices are called quantum ballistic systems, or quantum point contacts. They are made by giving a definite shape to the two-dimensional electron gas (2DEG).

The electrons in a 2DEG are confined to a two-dimensional quantum well. For instance, in a semiconductor heterostructure GaAlAs-GaAs-GaAlAs electrons in the GaAs layer are confined between two GaAlAs layers which serve as barriers.

Many metals and alloys become superconductive at low temperatures owing to mutual attractive interaction between electrons and to the sharpness of the Fermi distribution edge at the Fermi energy Ef. Qualitatively, the electron system in the superconductive state may be viewed as being composed of electrons bound in electron pairs (Cooper pairs) each of which contains two electrons with opposite momenta and spins. The dimensions of the pairs are greater than the mean interelectron distance, i.e., the pairs strongly overlap. This state of paired electrons is called ‘condensate’. It is superconductive, that is, current flows without any resistance. At temperatures T lower than the critical temperature TC, the free energy of the condensate is lower than that of an unpaired electron gas of the same density. Therefore, at T =Tc the metal, undergoes a phase transition into the superconductive state.

The properties of a superconductor are determined by the binding energy of electron pairs in the condensate, 2Δ. This quantity is called also the superconductive energy gap, because just this energy is required to break an electron pair in the condensate and create two quasi-particles that are able, like electrons in a normal metal, to dissipate the current. The energy Δ decreases with increasing temperature T and becomes zero at T = Tc.

In any physical system the dependence of the fluctuations' correlation function on time or, equivalently, the frequency dependence of the spectral density, on one hand, and the response of the same system to external perturbation, on the other hand, are governed by the same kinetic processes, and one can expect that there is some relationship between the two kinetic characteristics of the system. For example, the velocity correlation function of a Brownian particle t ψ(t1 – t2) decays exponentially with t1–t2 (Sec. 1.9). The corresponding relaxation time τ depends on the viscosity of the liquid, on the mass and linear dimensions of the particle. If the Brownian particle is brought into motion by an external perturbation (e.g., by an electric field if the particle is charged) the particle's stationary velocity and the time of its acceleration and deceleration after switching off the force are determined by the same parameters and, consequently, by the same relaxation time τ.

Such qualitative considerations are usually true for any physical system. However, for equilibrium systems an exact relationship holds between the spectral density of fluctuations at any given frequency f and that part of the linear response of the same system to an external perturbation of the same frequency f, which corresponds to the dissipation of the power of the perturbation. This fundamental relation is called the fluctuation–dissipation relation (FDR), or theorem (FDT).