More than a dozen years after my book on Nonequilibrium Phase Transitions in Semiconductors – Self-organization Induced by Generation and Recombination Processes appeared, the subject of nonlinear dynamics and pattern formation in semiconductors has become a mature field. The aim of that book had been to link two hitherto separate disciplines, semiconductor physics and nonlinear dynamics, and advance the view of a semiconductor driven far from thermodynamic equilibrium as a nonlinear dynamic system. It focussed on one particular class of instabilities related to nonlinear processes of generation and recombination of carriers in bulk semiconductors, and was essentially restricted to either purely temporal nonlinear dynamics, or nonlinear stationary spatial patterns. Within the past decade extensive research, both theoretical and experimental, has elaborated a great wealth of complex self-organized spatio-temporal patterns in various semiconductor structures and material systems. Thus semiconductors have been established as a model system with several advantages over the classical systems in which self-organization and nonlinear dynamics have been studied, viz. hydrodynamic, optical, and chemically reacting systems. First, semiconductor structures nowadays can be designed and fabricated by modern epitaxial growth technologies with almost unlimited flexibility. By controlling the vertical and lateral dimensions of those structures on an atomic length scale, systems with specific electric and optical properties can be tailored. Second, the dynamic variables describing nonlinear charge-transport properties are directly and easily accessible to measurement as electric quantities.

This book deals with complex nonlinear spatio-temporal dynamics, pattern formation, and chaotic behavior in semiconductors. Its aim is to build a bridge between two well-established fields: The theory of dynamic systems, and nonlinear charge transport in semiconductors. In this introductory chapter the foundations on which the theory of semiconductor instabilities can be developed in later chapters will be laid. We will thus introduce the basic notions and concepts of continuous nonlinear dynamic systems. After a brief introduction to the subject, highlighting dissipative structures and negative differential conductivity in semiconductors, the most common bifurcations in dynamic systems will be reviewed. The notion of deterministic chaos, some common scenarios, and the particularly challenging topic of chaos control are introduced. Activator–inhibitor kinetics in spatially extended dynamic systems is discussed with specific reference to semiconductors. The role of global couplings is illuminated and related to the external circuits in which semiconductor elements are operated.

Introduction

Semiconductors are complex many-body systems whose physical, e.g. electric or optical, properties are governed by a variety of nonlinear dynamic processes. In particular, modern semiconductor structures whose structural and electronic properties vary on a nanometer scale provide an abundance of examples of nonlinear transport processes. In these structures nonlinear transport mechanisms are given, for instance, by quantum mechanical tunneling through potential barriers, or by thermionic emission of hot electrons that have enough kinetic energy to overcome the barrier. A further important feature connected with potential barriers and quantum wells in such semiconductor structures is the ubiquitous presence of space charge.

In this chapter we study vertical high-field transport in semiconductor superlattices. Depending upon the circuit conditions and the material parameters, e.g. the mean doping density ND, either stable stationary domains (for high ND), or self-sustained oscillations of the domains (for intermediate ND) are found. We shall see that this behavior is strongly affected by growth-related imperfections such as small fluctuations of the doping density, or the barrier and quantum-well widths, and that weak disorder on microscopic scales can be quantitatively detected in the global macroscopic current–voltage characteristics. The bifurcations which occur and the roles of the various realizations of the microscopic disorder are discussed, as is the dynamics of domain formation.

Introduction

In Section 2.2.1 (Fig. 2.5) it was mentioned that vertical high-field transport in GaAs/AlAs superlattices is associated with NNDC and field-domain formation induced by resonant tunneling between adjacent quantum wells. This was observed experimentally by many groups (Esaki and Chang 1974, Kawamura et al. 1986, Choi et al. 1987, Helm et al. 1989, Helgesen and Finstad 1990, Grahn et al. 1991, Zhang et al. 1994, Merlin et al. 1995, Kwok et al. 1995, Mityagin et al. 1997). Those domains are the subject of the present chapter. The field domains may be either stationary, leading to characteristic sawtooth current–voltage characteristics (Esaki and Chang 1974), or traveling, associated with self-sustained current oscillations (Kastrup et al. 1995, Hofbeck et al. 1996). In strongly coupled superlattices, i.e. superlattices with small barrier widths, oscillations above 100 GHz at room temperature (Schomburg et al. 1998, 1999) have been realized experimentally, whereas in weakly coupled superlattices the frequencies are many orders of magnitude lower (Kastrup et al. 1997).

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.

When the lattice gas was introduced in statistical physics around 1985 (see Frisch, Hasslacher and Pomeau, 1986), it was originally constructed as a physical model for hydrodynamics. In fact, the concept of the lattice gas is as much a physical concept – and we shall indeed start with intuitive physical ideas – as it is a mathematical concept, as a more formal definition can also be given. We first present the point of view of the physicist (Section 1.1), then we describe the lattice gas automaton from the mathematical viewpoint (Section 1.2), and in Section 1.3 we discuss the two aspects.

The physicist's point of view

A lattice gas can be viewed as a simple, fully discrete microscopic model of a fluid, where fictitious particles reside on a finite region of a regular Bravais lattice. These fictitious particles move at regular time intervals from node to node, and can be scattered by local collisions according to a node-independent rule that may be deterministic or non-deterministic. Thus, time, space coordinates and velocities are discrete at the microscopic scale, that is, at the scale of particles, lattice nodes and lattice links.

The stationary states in statistical equilibrium and thus the large-scale dynamics of a lattice gas will crucially depend on its conservation properties, that is, on the quantities preserved by the microscopic evolution rule of the system.

In Chapter 2, we established the equations governing the microscopic dynamics of the lattice gas. This microscopic dynamics provides the basics of the procedure for constructing automata simulating the behavior of fluid systems (see Chapter 10). However, in order to use LGAs as a method for the analysis of physical phenomena, we must go to a level of description which makes contact with macroscopic physics. Starting from a microscopic formalism, we adopt the statistical mechanical approach.

We begin with a Liouville (statistical) description to establish the lattice Liouville equation which describes the automaton evolution in terms of configuration probability in phase space. We then define ensemble-averaged quantities and, in particular, the average population per channel whose space and time evolution is governed by the lattice Boltzmann equation (LBE), when we neglect pre-collision correlations as well as post-collision re-correlations (Boltzmann ansatz). The LBE will be seen to play in LGAs the same crucial role as the Boltzmann equation in continuous fluid theory; an H-theorem follows from which the existence of a Gibbsian equilibrium distribution is established when the semi-detailed balance is satisfied.

The Liouville description

In Chapter 2 we introduced the notion of Boolean field ni(r*, t*) to obtain a complete microscopic description of the time-evolution of lattice gases. We also introduced the phase space Γ which is the set of all possible Boolean configurations of the whole lattice.

In Chapter 4, we gave a statistical mechanical analysis of lattice gases, and discussed the equilibrium properties; we described uniform uncorrelated statistical equilibria, and we showed that they have a Fermi–Dirac probability distribution. The existence of these uniform equilibrium solutions can be established without recourse to the Boltzmann approximation; the only required conditions are the semi-detailed balance and the existence of local invariants. These equilibrium solutions – the analogue of global equilibria in usual statistical mechanics – are uniform by construction: the macroscopic variables, i.e. the ensemble-averages of the local invariants, are space-and time-independent.

Now, we address the problem of space-and time-varying macrostates in the ‘hydrodynamic limit’, that is, macrostates for which macroscopic variables vary over space and time scales much larger than the characteristic microscopic scales (lattice spacing and time-step duration). This scale separation between microscopic and macroscopic variables is a crucial physical ingredient in the theory of lattice gas hydrodynamic equations, and it is at the core of the forthcoming derivation, which rests upon a discrete version of the well-known ‘Chapman–Enskog method’ (see Chapman and Cowling, 1970).

Although the principles of the method described hereafter apply to a wider class of models, we are led, at a certain point of the forthcoming algebraic procedure, to particularize our study to the (still wide) class of ‘single-species thermal models’ defined in Chapter 4, Section 4.5.2. The simpler case of nonthermal models is also treated, but in a less detailed manner (see Section 5.7).

The object of the present chapter is small deviations from local equilibrium which are triggered by spontaneous fluctuations. In real fluids these fluctuations which temporarily disturb the system from local equilibrium are such that a fluid at global equilibrium can be viewed as a reservoir of excitations extending over a broad range of wavelengths and frequencies from the hydrodynamic scale down to the range of the intermolecular potential. Non-intrusive scattering techniques are used to probe these fluctuations at the molecular level (neutron scattering spectroscopy) and at the level of collective excitations (light scattering spectroscopy) (Boon and Yip, 1980). The quantity measured by these scattering methods is the power spectrum of density fluctuations, i.e. the dynamic structure factor S(k, ω) which is the space- and time-Fourier transform of the correlation function of the density fluctuations. The spectral function S(k, ω) is important because it provides insight into the dynamical behavior of spontaneous fluctuations (or forced fluctuations in non-equilibrium systems). Whereas the fluctuations extend continuously from the molecular level to the hydrodynamic scale, there are experimental and theoretical limitations to the ranges where they can be probed and computed. Indeed, no theory provides a fully explicit analytical description of space-time dynamics establishing the bridge between kinetic theory and hydrodynamic theory. Scattering techniques have limited ranges of wavelengths over which fluctuation correlations can be probed.

Since 1985, lattice gas automata have become a widely and actively explored field. Academic groups and industrial laboratories around the world have invested considerable effort in their research activities to drive the subject in various new and promising directions. As a result, the literature on lattice gases and related topics has grown so rapidly and has become so voluminous that an exhaustive list and a detailed review of all relevant lattice gas publications would practically make a book by itself.

Here we select research areas where lattice gases have played an important role, and, for each of them, we quote articles considered as representative, historically or presently. Our review is by no means complete and our choices are certainly selective; unavoidably we haven't done justice to those whose work escaped our selection. Nevertheless we have attempted to cover a range of works expanding over various aspects of the subject in the available literature. Our goal will be reached if the reader finds this chapter a helpful tool for exploring the subject beyond the scope of this book.

The historical ‘roots’

Discrete Kinetic theory

In the early sixties, the problem of shock waves in dilute gases was a subject of increasing research effort, for fundamental as well as industrial reasons.

As LGAs are constructed as model systems where point particles undergo displacements in discrete time steps and where configurational transitions on the lattice nodes represent collisional processes, one can view the lattice gas as a discretized version of a hard sphere gas on a regular lattice where particles are subject to an exclusion principle instead of an excluded volume. The advantage with LGAs is that, starting from exact microdynamical equations, statistical mechanical computations can be conducted rather straightforwardly in a logical fashion with well controlled assumptions to bypass the many-body problem. This is well exemplified by the development in Chapter 4 leading to the lattice Boltzmann equation. For the moment we shall consider the lattice gas automaton as a bona fide statistical mechanical model with extremely simplified dynamics. Nevertheless we may argue that the lattice gas exhibits two important features:

1. (i) it possesses a large number of degrees of freedom;

2. (ii) its Boolean microscopic nature combined with stochastic microdynamics results in intrinsic fluctuations.

Because of these spontaneous fluctuations and of its large number of degrees of freedom, the lattice gas can be considered as a ‘reservoir of thermal excitations’ in much the same way as a real fluid. Now the question must be raised – as for the hard sphere model in usual statistical mechanics – as to the validity of the lattice gas automaton to represent actual fluids. In Chapters 5 and 8 we consider full hydrodynamics and macroscopic phenomena.

The story of lattice gas automata started around 1985 when pioneering studies established theoretically and computationally the feasibility of simulating fluid dynamics via a microscopic approach based on a new paradigm: a fictitious oversimplified micro-world is constructed as an automaton universe based not on a realistic description of interacting particles (as in molecular dynamics), but merely on the laws of symmetry and of invariance of macroscopic physics. Imagine point-like particles residing on a regular lattice where they move from node to node and undergo collisions when their trajectories meet at the same node. The remarkable fact is that, if the collisions occur according to some simple logical rules and if the lattice has the proper symmetry, this automaton shows global behavior very similar to that of real fluids.

In this chapter, we develop the ‘microdynamic formalism’, which describes the instantaneous microscopic configuration of a lattice gas and its discrete-time evolution. This exact description of the microscopic structure of a lattice gas is the basis for all further theoretical developments, in particular for the prediction of large-scale continuum-like behavior of LGAs.

We first introduce the basic tools and concepts for a general instantaneous description of the microscopic configurations (Section 2.1). The time evolution of the lattice gas is then given in terms of the ‘microdynamic equations’ (Section 2.2). Thereafter, we define microscopic characteristics (e.g. various forms of reversibility), which have a crucial incidence on the macroscopic behavior of the gas, and therefore on its suitability to simulate real physical situations (Section 2.3). The last section is devoted to special rules needed to handle boundary problems (obstacles, particle injections, etc.).

This chapter deals with rather abstract concepts which will find their application in Chapter 3, where lattice gas models are described at the microscopic level.

Basic concepts and notation

The lattice and the velocity vectors

One of the most important features of lattice gases is the underlying Bravais lattice structure which gives a geometrical support to the abstract notion of a cellular automaton. Strictly, a Bravais lattice is by definition infinite. We consider that the cellular automaton only occupies a connected subset ℒ of the D-dimensional underlying Bravais lattice.