Introduction

Spatio-temporal chaos is a feature of nonlinear spatially extended systems with large numbers of degrees of freedom, as described for instance by reaction–diffusion models of activator–inhibitor type. In this chapter we shall study a model system of this type that was introduced in Chapter 2 for layered semiconductor heterostructures, and used for a general analysis of pattern formation in Chapter 3. Here we shall investigate in detail the complex spatio-temporal dynamics including a codimension-two Turing–Hopf bifurcation, and asymptotic and transient spatio-temporal chaos. Chaos control of spatio-temporal spiking by time-delay autosynchronization is applied. The extensive chaotic state is characterized using Lyapounov exponents and a Karhunen–Loève eigenmode analysis.

Spatio-temporal spiking in layered structures

For understanding spatio-temporal chaos it is important to study first the elementary spatio-temporal patterns, which may eventually – in the course of secondary bifurcations – evolve into chaotic scenarios. One such elementary pattern, which has been observed experimentally in various different semiconductor devices exhibiting SNDC, e.g. layered structures such as p–n–p–i–n diodes (Niedernostheide et al. 1992a) and p–i–n diodes (Symanczyk et al. 1991a), or in impurity-impact-ionization breakdown (Rau et al. 1991, Spangler et al. 1992), and in electron–hole plasmas (Aliev et al. 1994), is the spiking mode of current filaments. The properties of localized spiking structures have been studied theoretically in detail by Kerner and Osipov (1982, 1989) in general reaction–diffusion systems. Spiking in a simple chemical reaction–diffusion model – the Brusselator – has been reported by De Wit et al. (1996).

Self-organized pattern formation is closely connected with negative differential conductivity in semiconductors. In this chapter we develop a general framework for the analysis of the formation and stability of patterns such as current filaments, field domains, and fronts. Special emphasis is placed upon the interaction with an external circuit and the resulting global coupling which strongly affects the stability.

Introduction

In semiconductors, spatially homogeneous states of negative differential conductivity are in general unstable against spatio-temporal fluctuations, which may give rise to self-organized pattern formation. If the j (ε) characteristic is N-shaped (NNDC), such as in the Gunn effect, inhomogeneous electric-field profiles in the form of a high-field domain may arise. If the j (ε) characteristic is S-shaped, such as in threshold switching, the current flow may become inhomogeneous over its cross-section and form a current filament. These spatial structures may be static or time-dependent. In the latter case, current oscillations can arise due to domains moving in the direction of the current flow, or filaments “breathing” transversally to the current flow (Schöll 1987).

At this point a word of warning is indicated. First, negative differential conductivity does not always imply instability of the steady state, and positive differential conductivity does not always imply stability. For example, SNDC states can be stabilized by a heavily loaded circuit (and experimentally observed!), and, on the other hand, the Hopf bifurcation of a limit-cycle oscillation can occur on a j (ε) characteristic with positive differential conductivity.

In this chapter, we shall analyse the way self-trapping occurs in real systems. The first issue is whether or not self-trapping is observed in a given material. We have discussed the criteria for self-trapping in Section 3.3, but experiment is the decisive factor. Table 4.1 shows the occurrence of self-trapping of electrons, holes, and excitons in a number of insulators. The absence of self-trapping of electrons and holes can be determined by measuring their mobilities; the absence of self-trapping of excitions is demonstrated by observing freeexciton luminescence. When the interaction of free charge carriers and of excitons with defects or impurities indicates long-range motion, this can imply the absence of self-trapping. Once self-trapping is proven, the atomic and electronic structures of the self-trapped states are needed. The relationship between the self-trapped state and the host atomic and electronic structure is important, especially for self-trapped excitions, since exciton self-trapping provides the energy localisation needed for local modification of the lattice structure and defect formation.

Although self-trapping is induced as a consequence of coupling with the lattice, existing defects play a role in self-trapping in two ways: extrinsic self-trapping and defect-perturbed self-trapping. Extrinsic self-trapping occurs by the interaction with the lattice and defects in materials in which the interaction with the lattice is not sufficient to induce intrinsic self-trapping, while defect-perturbed self-trapping occurs by the interaction between intrinsic self-trapping and defects (Fig. 4.1). Trapping by defects without self-trapping does not involve large lattice distortion.

Amorphous materials lack long-range order in one of several possible ways. What happens in an amorphous solid following electronic excitation may differ from what occurs when a crystalline solid is similarly excited. In this chapter, we compare the behaviours following excitation of crystalline and amorphous materials. In amorphous materials, electronic excitation induces photo-luminescence and generates defects, as in crystals. Excitation of amorphous materials can lead both to distinctive gradual structural changes and to changes of macroscopic properties. In this chapter, we compare the processes induced by the excitation of specific materials in their crystalline and amorphous forms. Photo-induced processes in amorphous materials have been reviewed by Shimakawa et al. (1995) and by Morigaki (1999).

There are two major complications in referring to amorphous materials (see Stoneham (1987b), pp 4–6). The first issue concerns whether a solid is amorphous or not. Normally, an experimenter will call a solid ‘amorphous’ when there are diffraction rings, rather than spots. A theorist will say it is amorphous when there is no long-range order. These two descriptions are not equivalent. For example, quasi-crystalline and para-crystalline materials may show diffraction spots without having long-range order. Likewise, diffraction rings may indicate a powder or micro-crystalline material. The second complication is that whereas one can be sure what is meant by a crystalline system in thermal equilibrium, an amorphous material is not so well defined.

General comparisons: Switching between states and motion

For wide-bandgap ionic solids, electron–hole recombination energies can exceed defect formation energies. In semiconductors, the gap is smaller, and usually significantly less than the energy needed to create a defect. Defect production is predicted to occur only in special cases, as in porous silicon (Caldas et al. 1997) and perhaps in other nanocrystals. Most wide-gap ionic solids have strong electron–phonon coupling, so that self-trapping of excitons and certain carriers is common. In semiconductors, the lower electron–phonon coupling makes self-trapping unlikely. Even excitonic complexes (e.g. Kuvalovskii et al. 1986) are relatively ineffective, since the electrons and holes tend to recombine one at a time. The role of electronic excitation takes a different form in semiconductors from that observed in wide-gap ionic solids. In semiconductors, it is processes involving impurities, or pre-existing defects such as dislocations, which are affected. The major consequences of excitation are enhanced processes like diffusion, or the production of metastable, non-equilibrium states of defects. The ways in which defect processes are influenced by excitations are often identified only after frustrations. A common example might be that devices survive for long times in storage, but degrade rapidly when used because of dislocation climb.

Discussions of materials modification imply modification controlled in space or time, or with selected defect states. Control in space is common, e.g. lithography and photography.

Electronic excitation is a means to change materials properties. It offers both novel types of change and better-controlled changes. The new developments in laser technologies, in electron beam methods, and also in the use of particle beams have led to a wide range of new opportunites, both scientific and technological.

The basic ideas are not new. Photography is over 150 years old. Photochromics have been on sale for decades. Photolithography, electron lithography, and ablation are standard tools. Photosynthesis is nearly as old as life, and incorporates some of the ideas we shall describe. What has changed in the last few years has been the gradual growth in insight in areas which are not usually connected, and a vast increase in the technology available to manipulate materials by excitation in a controlled way. Our aim is to analyse the important features of the changes induced by electronic excitation, to identify what is critical, and to provide a basis from which materials modification can be developed successfully.

We shall emphasise the mechanisms underlying materials modifications in inorganic non-metals following electronic excitation. Metals and organics are mentioned mainly when they show interesting parallels with the inorganic insulators which we emphasise. It would take at least another book to cover the photochemistry of organic and biological systems. We shall not (and could not) cover every aspect of electronic excited states, which would need to include vast areas of photonics.

Interfaces control a variety of technological applications of materials, and their characterisation, synthesis, and modification determine how well these interfaces perform. Interfaces have several roles of direct interest to us. First, interfaces provide sites where electronic excitation energy can be localised. This makes it possible to modify interfaces without affecting bulk properties. Secondly, at an interface atoms from the materials on both sides of the interface meet. Bond rearrangements at the interface can affect microscopic structures. Thirdly, interfaces include a variety of defects which do not exist in the bulk. The structures of these defects are also susceptible to electronic excitation. Some of these defects are associated with mismatch in lattice parameter between the media on either side of the interface. Fourthly, a wealth of new interfaces is being created by methods such as ion-beam sputtering. These include tens or hundreds of alternating layers, for example of Si/SiO2, A12O3/SiO2, Ta2O5/SiO2, TiO2/SiO2, HfO2/SiO2 or MgF2/SiO2 (see, e.g., Howe (1999), or Reichling et al. (1998a)). Reichling et al. suggest that most induced laser damage occurs at the substrate/film boundary. These multilayers can be crystalline or amorphous, and can be constructed so that layer thicknesses match a particular wavelength (typically, λ/4 layers with a λ/2 cap might be used). In some cases, self-organisation has been seen. Since these systems are used in laser applications, their performance under excitation is important. Finally, we note that interfaces can evolve in interesting ways when the interfacial free energy is lowered by mixing.

Basic ideas about the localisation of charge and energy

Our theme is excited state phenomena in materials, and the way that excitation can affect the nature and performance of materials. The emphasis on excited states means that we are primarily concerned with non-metals, although certain of the interesting features do not depend on the existence of a bandgap. It is natural that excited states evolve with time as equilibrium is reestablished. Our concern includes the ways in which the direction of this evolution can be controlled, for example, so as to produce chosen defect populations. How this happens, and the degree of control which is possible, depends on how the excitation energy is provided as well as on the intrinsic features of the excited states.

The components of materials modification by excitation can be classified in a few categories, although the mechanisms may be very varied. First, there are the transport processes: energy transport and charge transport. These are the processes by which excitation reaches specific regions of a solid. Control of which zones are affected and which are unaffected is important in any proposed application. Secondly, there are the localisation processes: energy localization and charge localization. Energy localisation can often be described as the trapping or self-trapping of an exciton. In some cases, it will be preceded by charge localisation, as in the trapping or self-trapping of a hole followed by capture of an electron.

Energy absorption near surfaces

Electronic excitation at solid surfaces can be induced in two main ways. The first is electronic excitation in the bulk, followed by direct energy transfer or by the diffusion of excitons or of electron–hole pairs to the surface. The second route involves the excitation of surface states or surface defect states. Such states are especially important when the excitation energy is less than the bulk bandgap. However, the division into two classes is not clear-cut, for there will usually be a space-charge region of a few nanometres in thickness; energy absorption in this region may make an important contribution to near-surface excitation. De-excitation processes at surfaces may follow the same pattern as in the bulk. The modification of local surface structures will often be simpler and easier than in the bulk. What will be different are reactions involving specific surface species, including adsorbed species, and processes which involve the emission of atoms, ions, or molecules. Our emphasis on the modification of the underlying material continues. We shall therefore note, but say little about, processes such as the laser removal of silicon chloride species during plasma etching (C C Cheng et al. 1994).

Several terms are used for the emission of atoms or ions, depending on the rate and the application: desorption; sputtering; ablation. In this book, desorption usually refers to reemission of adsorbates on surfaces, whereas sputtering is the emission of atoms or ions constituting the host material.

Introduction

So far, we have concentrated on situations in which the number of excitons is low, so that behaviour can be analysed in terms of individual electrons or holes, or excitons, or perhaps biexcitons or small exciton complexes. This type of description holds to modest levels of excitation. Thus Tanimura & Itoh (1988b, 1990) discussed exciton fusion, in which a free exciton and a self-trapped exciton in RbI transform to either an F–H pair or a luminescent singlet selftrapped exciton. Yet there are important cases for which this class of description is dubious. The numbers of excitons, or electrons and holes, can even be of the order of the number of atoms. Such a massive degree of excitation needs a different description. For example, the host interatomic forces will be changed severely. If, as a result, the host is unable to resist shear, a description in terms of a liquid state or a plasma is tempting. The energy per atom can easily be large enough to cause evaporation or exoelectron emission, with associated electric fields. In the cooling process, the system may evolve into a different solid structure, possibly a non-equilibrium form. In this chapter we discuss some of these situations. It is hardly surprising that we cannot give a detailed dynamical and microscopic picture of the behaviour. However, for practical applications, we can make do with less information.

The linking of electronic and nuclear dynamics is of major importance in understanding the ways that electronic excitation can lead to materials modification. In particular, the exchange of energy between electronic excitation and lattice vibrational energy is crucial in the dynamics of excited states. This chapter summarises these unifying features, and provides a framework for fuller descriptions.

Basics of electron-lattice coupling

Some relatively simple ideas form the basis for understanding lattice vibrations. They can be generalised, so as to apply to transient, non-equilibrium systems and in other situations important to us.

Adiabatic approximations

The first basic assumption is used to separate the electronic and nuclear motions. The Born–Oppenheimer approximation asserts that the light-mass electrons respond so rapidly to the sluggish motion of the nuclei that one can assume the nuclei to be at rest when calculating the electronic wavefunctions. There is then an electronic term in the energy which determines the motion of the nuclei.

This approximation is excellent in many circumstances (for fuller references, see Stoneham (1975)). Some of its generalisations are extensions of the same idea, for instance when there is electronic degeneracy, and the Jahn–Teller effect is important. In other circumstances, as in non-radiative transitions, the breakdown of the adiabatic assumption is the driving force for the transitions. In other cases, the same ideas are exploited but in different ways.

In Chapter 4, we discussed the relaxation of excitons and of electron–hole pairs to form self-trapped excitons. In halides, these relaxations may involve asymetrical atomic displacements, including the [Se, Sh] self-trapped exciton. In Chapter 5, we discuss how these asymmetric displacements relate to the asymmetric displacements which lead to defect formation. In the most important case, the displacement creates a pair of neutral defects on the halogen sublattice. This defect pair comprises a halogen atom vacancy (F centre) and a halogen atom interstitial (H centre). The F–H pair is sometimes referred to as a Frenkel pair, but this is misleading, since the term is rightly used for a defect pair comprising a halogen ion vacancy (α centre, X– missing) and a halogen ion interstitial (I centre, Xi– rather than Xi0, differing in charge, site, and symmetry from the H centre).

Almost all of the discussion in this chapter relates to the alkali halides and alkaline-earth fluorides, in which excitons can be self-trapped. The arguments would apply in similar ways to many other halides, but not necessarily to those for which self-trapping does not occur (like the thallous and cuprous halides). Since the self-trapped exciton has a substantial latent energy, only a small energy is needed to transform a self-trapped exciton into an interstitial-vacancy pair on the halogen sublattice. The exciton–phonon coupling which causes exciton self-trapping also assists the creation of defect pairs.

Energetic particles in solids lose their energy partly to atoms and ions, by elastic collisions, and partly to electronic excitations. Most of the present chapter describes the way in which energetic particle beams generate electronic excitation. We shall begin by considering the interaction of charged particles with solids. We include the effects of X-ray, γ-ray and neutron irradiation, since these lead to secondary electrons or ions. In discussing the basics of particle beam interactions with solids, our emphasis is on identifying situations where there is reason to believe that electronic excitation is an essential feature. We shall say relatively little about those situations which are described by elastic collisions, and for which the electrons merely provide a potential to define the interactions between atoms. Thus we shall not discuss certain areas of major industrial significance, such as the doping of semiconductors by ion implantation, for which electronic excitation is a secondary feature. However, we should be aware of the energies and doses which are used in these applications, since they will influence the extent to which excitation is an issue. In ion implantation, the critical parameters are chosen to produce a specific concentration profile for a specific need. For example, doping polysilicon uses high doses (typically 1016 ions/cm2) whereas the adjustment of threshold voltages uses low doses (perhaps 1012 ions/cm2), even though both use similar energies, perhaps 50 keV.

Basic phenomena

Oxides and halides: Basics and similarities

Our discussion of halides showed the central importance of self-trapped excitons and of the processes which compete with self-trapping. When we turn to oxides, we have to ask whether the same classes of process occur. Moreover, we cannot confine ourselves to ideal single crystals, but should assess the situation for oxide ceramics, since these polygrannular materials are typical of functional materials in technology. In this chapter we emphasise materials modification in oxides due to excitation, rather than knock-ons, and attempt to understand the mechanisms. For many technological materials, the mechanisms are not always clear. Most probably, there are further complications, such as diffusion, chemical reactions, and changes of phase.

Excitons are self-trapped in some oxides. Yet there seem to be no cases of defect formation as a consequence of a single bandgap electronic excitation, even when the exciton is self-trapped. Why is this? One possibility is simply that there is a potential barrier between the self-trapped exciton and the defect pair. If this potential barrier could be overcome or bypassed, electronic mechanisms for defect formation would be possible in oxides. If this were the explanation, then the differences between halides and oxides would be differences of degree only. However, there are several other common differences between oxides and halides, and these may be important.

Aims of materials modification

Most of our book aims at a coherent and (where possible) quantitative understanding of materials modification following electronic excitation. Yet this understanding is only one part of the picture. Another major component is practical: how can the consequences of this controllable excitation be exploited? We may categorise most areas of major practical importance under a small number of headings:

Removal of surface material, especially in a controlled way;

Modification of the surface layers;

Modification of the bulk in a selected way;

Altering the absolute rate of a bulk or surface process;

Switching from one process to another: altering branching ratios. In this category, we might also include the manipulation of excitation through materials design, even though the material itself is not modified by this excitation. For example, the xerographic process relies on the manipulation of electron and hole excitations in Se. The photochemical splitting of water using colloidal particles (typically TiO2/RuO2 with Pt) (e.g. Grätzel 1981) relies on the appropriate use of the energy stored following optical excitation, which ideally leaves the particle itself unaltered.

Creation and control of damage

Damage may be intentional or unintentional. Damage avoidance is achieved by the minimisation of those processes by which excitation causes harmful modifications, or simply by arranging for the damage to occur only in unimportant regions.

Separation of variables method

We will assume that the solution for solute concentration C(y,t) can be separated into two functions, one depending only on position, the other on time, i.e.

If we substitute this into Fick's Second Law then

We then separate the y- and t-dependent terms,

For the first equality in this expression to be valid, each side must be constant. We set this constant to be (—λ2). We now solve each side of the equation separately. The time-dependent equation

can be integrated to yield

where ⊖0 is a constant of integration. The spatial equation

is integrated twice, giving

where A′ and B′ are integration constants. Combining these expressions gives

where A = A′⊖0 and B′ = B′⊖0 . In general, there are a multitude of solutions to Fick's Second Law, all of which have this form, but with different values of A, B and λ. Therefore, the most general solution is given by the sum of these

This solution is generally valid for diffusion in one dimension. Similar expressions can be derived for cylindrical or spherical symmetry.

In order to proceed further we need to define the initial and boundary conditions appropriate to the problems at hand. We will consider only one example, namely diffusion out of a plate with an initially uniform concentration, and the concentration fixed at the surface of the plate.