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).

This book is an introduction to the physics of electronic noise and fluctuation phenomena in solids. It is written for physicists and electrical engineers interested in investigation of electric and magnetic noise and in development of sensitive solid-state devices. It can be used also in graduate schools of the Departments of Physics and Electrical Engineering.

Fluctuations or noise are spontaneous random (stochastic) variations of physical quantities in time or, more precisely, random deviations of these quantities from some mean values that are either constant or vary nonrandomly in time. Fluctuations are a manifestation of the thermal motion of matter and discreteness of its structure. The introduction of the concept of fluctuations and development of the physics of fluctuations is one of the greatest achievments of twentieth-century physics. The theory of Brownian motion developed by A. Einstein and by M. von Smoluchowski in the first decade of this century and its experimental proof by J.B. Perrin and T. Svedberg was a strong and, perhaps, the final argument in favor of the molecular-kinetic concept of heat. Hence, the physics of fluctuations is of great conceptual importance.

The applications of the physics of fluctuations stem from the fact that the ultimate accuracy of measurement of any physical quantity is limited just by fluctuations of this quantity, and the ultimate sensitivity of many devices is also limited by fluctuations.

An enormous variety of conductors can be called strongly disordered. First of all, these are conductors prepared of macroscopically randomly inhomogeneous materials, i.e., various composites: dielectrics (polymers, plastics) into which in some proportion fine particles of metal or graphite are introduced, say, before hardening. Such material becomes conductive due to contacts between the particles of the conducting substance forming continuous chains between the electrodes attached to the whole sample. One may attribute to such conductors also cermets, polycrystalline solids in which each crystallite is strongly anisotropic (the material as a whole may be isotropic owing to random orientation of the crystallites), metallic films subjected to sandblasting (sand particles randomly remove some parts of the film), etc. In the simplest case the conducting particles are homogeneous within their boundaries, their dimensions are many times larger than the free-path length of the charge carriers, and the contacts between them are perfect. Real disordered materials, of course, are usually much more complex, and the simplest case described above can be considered merely as a model.

Semiconductors with hopping conductivity may also be attributed to strongly disordered conductors. The charge carriers are hopping between impurity centers by tunneling. Owing to the random distribution of impurity centers and strong, exponential, dependence of the tunneling probability on the distance between the impurities, the current paths are extremely intricate, as in randomly inhomogeneous media.

The static conductivity of disordered mixtures, one of the components of which is an insulator, tends to zero as the fraction p of the conductive component approaches from above a critical nonzero value pc. This phenomenon is a kind of metal-insulator transition.

In the 15 years since the first edition was published, the subject of composite materials has become broader and of greater technological importance. In particular, composites based on metallic and ceramic matrices have received widespread attention, while the development of improved polymer-based systems has continued. There have also been significant advances in the understanding of how composite materials behave. Furthermore, the wider range of composite types has led to greater interest in certain properties, such as those at elevated temperature. We therefore decided to produce a major revision of the book, covering a wider range of topics and presenting appreciably deeper treatments in many areas. However, because the first edition has continued to prove useful and relevant, we have retained much of its philosophy and objectives and some of its structure. Throughout the book, emphasis is given to the principles governing the behaviour of composite materials. While these principles are applicable to all types of composite material, examples are given illustrating how the detailed characteristics of polymeric-, metallic- and ceramic-based systems are likely to differ.

The first chapter gives a brief overview of the nature and usage of composite materials. This is followed by two chapters covering, firstly, the types of reinforcement and matrix materials and, secondly, geometrical aspects of how these two constituents fit together. The next three chapters are concerned with the elastic deformation of composites. Chapter 4 deals with material containing unidirectionally aligned continuous fibres, loaded parallel or transverse to the fibre axis.

Polymer composites

There are many commercial processes for the manufacture of PMC components. These may be sub-divided in a variety of ways, but broadly speaking there are three main approaches to the manufacture of fibre-reinforced thermosetting resins and two distinct production methods for thermoplastic composites. These are briefly covered below under separate headings. A simple overview of the starting materials and approaches adopted to their incorporation into components is given in Fig. 11.1. In most cases, the main microstructural objectives are to ensure that the fibres are well wetted, uniformly distributed and correctly aligned. Practical considerations relating to capital cost, speed of production and component size and shape capability are often of paramount importance.

A book on composite materials which is fully comprehensive would embrace large sections of materials science, metallurgy, polymer technology, fracture mechanics, applied mechanics, anisotropic elasticity theory, process engineering and materials engineering. It would have to cover almost all classes of structural materials from naturally occurring solids such as bone and wood to a wide range of new sophisticated engineering materials including metals, ceramics and polymers. Some attempts have been made to provide such an over-view of the subject and there is no doubt that the interaction between different disciplines and different approaches offers a fruitful means of improving our understanding of composite materials and developing new composite systems.

This book takes a rather narrower view of the subject since its main objective is to provide for students and researchers, scientists and engineers alike, a physical understanding of the properties of composite materials as a basis for the improvement of the properties, manufacturing processes and design of products made from these materials. This understanding has evolved from many disciplines and, with certain limitations, is common to all composite materials. Although the emphasis in the book is on the properties of the composite materials as a whole, a knowledge is required of the properties of the individual components: the fibre, the matrix and the interface between the fibre and the matrix.

The essence of composite materials technology is the ability to put strong stiff fibres in the right place, in the right orientation with the right volume fraction.

Elastic deformation of anisotropic materials

Hooke's law

A review of some basic points about stress and strain is appropriate.