Prologue

In Chapter 2, we constructed the Fermi sphere of free-electrons with the radius kF in reciprocal space. It represents the distribution of the quantized electronic states at absolute zero, in which the states in k≤kF are all occupied but those in k>kF are vacant. At finite temperatures, thermal energy would excite some electrons in the range k≤kF into the range k>kF. The redistribution of electrons will occur so as not to violate the Pauli exclusion principle. As noted in Section 2.7, the Fermi energy in typical metals is of the order of several eV and is equivalent to ∼ 10000K on the temperature scale. Hence, only electrons near the Fermi surface can be excited at temperatures below ∼ 1000K. The aim of the present chapter is to formulate first the Fermi–Dirac distribution function, which determines the distribution of electronic states or the Fermi surface at finite temperatures, and then to deduce the temperature dependence of various physical properties due to conduction electrons by calculating relevant quantities involving the Fermi–Dirac distribution function.

Fermi–Dirac distribution function (I)

We know that the velocity of dilute gas molecules obeys the Maxwell–Boltzmann distribution law. Unfortunately, however, classical statistics cannot be applied to the conduction electron system in metals because of an extremely high electron density of the order of 1028–1029/m3. As emphasized in the preceding chapter, an electron carries a spin of ½ and particles with a half-integer spin should obey the Pauli exclusion principle. In addition, they are indistinguishable from each other. Our first objective in this section is to deduce the statistical distribution function under these two conditions imposed by quantum mechanics.

Prologue

We have so far discussed the electronic structure of metals and semiconductors existing as elements in the periodic table and have assumed a crystal to be ideally perfect without containing any defects like impurities, vacancies, dislocations and grain boundaries. In reality, no metal is perfectly free from such defects, which certainly disturb the periodicity of the lattice and cause scattering of the Bloch electron. Foreign elements can be intentionally added to a given metal, resulting in the formation of an alloy. When the amount of the added element is dilute, the added atoms may be treated as impurities. But when its concentration exceeds several atomic %, the interaction among the added atoms is no longer neglected. In this chapter, we discuss first the effect of an impurity atom on the electronic structure of a host metal and then move on to discuss the electronic structure of concentrated alloys.

Impurity effect in a metal

Let us consider first a perfect metal crystal consisting of the atom A with the valency Z1. All atoms become positive ions with the valency +Z1 by releasing the outermost Z1 electrons per atom to form the valence band. As a result, conduction electrons carrying negative charges are uniformly distributed over any atomic site with equal probability densities and maintain charge neutrality with the array of ions with positive charges. Now we replace the atom A at a given lattice site by the atom B with valency Z2 (Z2>Z1). Effectively, a point charge equal to ΔZ=Z2-Z1 is formed at the atom B and the uniform charge distribution is disrupted.

Prologue

We learned in Chapter 10 that lattice vibrations in a metal always give rise to a finite resistivity and that it disappears only when the metal resumes perfectly periodic ion potentials at absolute zero. However, there are many metals the resistivity of which completely vanishes at finite temperatures. Kamerlingh Onnes from the Netherlands, is famous for his success in liquefying helium for the first time in 1908. During his extensive studies on the electrical resistivity of various metals by immersing them in liquid helium, he happened to discover in 1911 that the resistivity of mercury suddenly drops to zero at 4.2K, the boiling point of liquid helium at 1 atmospheric pressure. This is the superconducting phenomenon discovered three years after his helium liquefaction.

Since then, superconductivity has been discovered in many metals, alloys and compounds. As listed in Table 12.1, the value of the superconducting transition temperature Tc of elements in the periodic table is always less than 10K. Many researchers have attempted to synthesize new superconductors with as high a Tc value as possible. In 1986, Bednorz and Müler revealed that the electrical resistivity of La–Ba–Cu–O sharply dropped at about 35K and vanished below about 13K and pointed out the possibility of synthesizing a new high-Tc superconducting oxide. Their work opened up a new era for the research of high-Tc superconductors and the Nobel Prize in physics was awarded to them in 1987 for their discovery.

What is the electron theory of metals?

Each element exists as either a solid, or a liquid, or a gas at ambient temperature and pressure. Alloys or compounds can be formed by assembling a mixture of different elements on a common lattice. Typically this is done by melting followed by solidification. Any material is, therefore, composed of a combination of the elements listed in the periodic table, Table 1.1. Among them, we are most interested in solids, which are often divided into metals, semiconductors and insulators. Roughly speaking, a metal represents a material which can conduct electricity well, whereas an insulator is a material which cannot convey a measurable electric current. At this stage, a semiconductor may be simply classified as a material possessing an intermediate character in electrical conduction. Most elements in the periodic table exist as metals and exhibit electrical and magnetic properties unique to each of them. Moreover, we are well aware that the properties of alloys differ from those of their constituent elemental metals. Similarly, semiconductors and insulators consisting of a combination of several elements can also be formed. Therefore, we may say that unique functional materials may well be synthesized in metals, semiconductors and insulators if different elements are ingeniously combined.

A molar quantity of a solid contains as many as 10 atoms. A solid is formed as a result of bonding among such a huge number of atoms. The entities responsible for the bonding are the electrons. The physical and chemical properties of a given solid are decided by how the constituent atoms are bonded through the interaction of their electrons among themselves and with the potentials of the ions.

Prologue

When a crystal is melted, the periodic lattice is destroyed and the atomic distribution is randomized. This causes the Bloch theorem to fail in liquid metals. The discussion of the conduction electron in liquid metals dates back as early as 1936 when the Mott and Jones book on the electron theory of metals was first published. Since the 1970s, new melt-quenching techniques have been developed and amorphous alloys stable at room temperature have become available in ribbon form. A bundant production of thermally stable amorphous alloys has enabled us to study their electron transport properties down to very low temperatures and has certainly widened the research field in the electron theory of a non-periodic system.

In 1984, Shechtman et al. revealed that the electron diffraction pattern of the melt-quenched Al86Mn14 alloy exhibits two-, three- and five-fold symmetries incompatible with the translational symmetry of a crystal and suggested that this material belongs to a new family of substances different from crystals. Since then a number of solids of this class have been discovered along with progress in theoretical studies. They are indeed new solids classified, in crystallographic terms, as quasicrystals. Because of the possession of five-fold symmetry incompatible with the translational symmetry, the Bloch theorem breaks down. Hence, they are grouped together with liquid metals and amorphous alloys into the category of non-periodic systems in spite of the possession of a high degree of ordering, evidenced from very sharp Bragg reflections.

The fundamental understanding of the atomic structure of quasicrystals has been deepened through the construction of a quasiperiodic Penrose lattice obtained by projecting lattice points in a selected region of a six-dimensional lattice onto a three-dimensional physical space.

The theory of electric transport in semiconductors describes how charge carriers interact with electric and magnetic fields, and move under their influence. In general, we will consider the regime far from thermodynamic equilibrium, under which linear relations between current and voltage do not hold, and more sophisticated modeling of the microscopic charge-transport processes is required. There are various levels at which semiconductor transport can be modeled, depending upon the specific structures under consideration as well as the operating conditions. In this chapter we will survey a hierarchy of approaches to nonlinear charge transport and discuss the regimes of validity. This hierarchy of models will form the physical basis for the instabilities and spatio-temporal pattern formation processes to be discussed in the subsequent chapters. It will also serve to introduce some systematics and give guidance concerning the confusing variety of nonlinear dynamic models that have been developed and studied in the field of semiconductor instabilities within the recent past.

Introduction

With the advent of modern semiconductor-growth technologies such as molecular beam epitaxy (MBE) and metal–organic chemical vapor deposition (MOCVD), artificial structures composed of different materials with layer widths of only a few nanometers have been grown, and additional lateral patterning by electron-beam lithography or other lithographic or etching techniques (ion-beam, X-ray, and scanning-probe microscopies) can impose lateral dimensions of quantum confinement in the 10 nm regime. Alternatively, lateral structures can be induced by the Stranski–Krastanov growth mode in strained material systems, which leads to the self-organized formation of islands (“quantum dots”) a few nanometers in diameter (Bimberg et al. 1999).

It has been found experimentally that an external magnetic field applied perpendicular to the electric field can sensitively affect the spatio-temporal instabilities in the regime of impurity-impact ionization. For instance, it has been observed that even a relatively weak magnetic field can induce complex chaotic current oscillations. In this chapter we study the nonlinear and chaotic dynamics of carriers in crossed electric and magnetic fields. We present a general framework for the description of a dynamic Hall instability, and apply it in particular to the conditions of low-temperature impurity breakdown. We show that chaos control by time-delayed feedback can stabilize those chaotic oscillations. Furthermore, we discuss the complex spatio-temporal dynamics of current filaments in a Hall configuration, resulting either in lateral motion or in a deformation of the filamentary patterns.

Introduction

Ever since E. H. Hall discovered in 1879 the effect which bears his name, galvanomagnetic phenomena in solids have received significant attention. The Hall effect has been used as an important probe of material properties in many branches of solid-state physics, and more than 200 million components of devices successfully utilize the Hall effect (Chien and Westgate 1980).

Along with the development of practical uses of the Hall effect, the theoretical foundation of galvanomagnetic phenomena has been established (Madelung 1957). Interest has recently been revived strongly by the extension of the classical Hall effect into the regime of the integral (von Klitzing 1990) or fractional (Eisenstein and Störmer 1990) quantum Hall effect, and by the discovery of the negative Hall effect and chaotic dynamics in lateral superlattices (Fleischmann et al. 1994, Schöll 1998b).

In this chapter we discuss a model system that has been studied thoroughly both experimentally and theoretically within the last decade: Impurity impact-ionization breakdown at low temperatures. This system exhibits a variety of temporal and spatiotemporal instabilities ranging from first- and second-order nonequilibrium phase transitions between insulating and highly conducting states via current filamentation and traveling waves to various chaotic scenarios. There are several models that can account for periodic and chaotic current self-oscillations and spatio-temporal instabilities. Here we focus on a model for low-temperature impurity breakdown that combines Monte Carlo simulations of the microscopic scattering and generation–recombination processes with macroscopic nonlinear spatio-temporal dynamics in the framework of continuity equations for the carrier densities coupled with Poisson's equation for the electric field. A period-doubling route to chaos, traveling-wave instabilities, and the dynamics of nascent and fully developed current filaments are discussed including two-dimensional simulations for thin-film samples with various contact geometries.

Introduction

Impact ionization of charge carriers is a widespread phenomenon in semiconductors under strong carrier heating. It is a process in which a charge carrier with high kinetic energy collides with a second charge carrier, transferring its kinetic energy to the latter, which is thereby lifted to a higher energy level. Impact-ionization processes may be classified as band–band processes or band–trap processes depending on whether the second carrier is initially in the valence band and makes a transition from the valence band to the conduction band, or initially at a localized level (trap, donor, or acceptor) and makes a transition to a band state (Landsberg 1991).

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).