Introduction

In Chapter 4, we established general expressions for the probability of quantum diffusion Pd (r, r′, ω) and for the structure factor Γω(r, r). With the aid of these quantities, which are solutions of the integral equations (4.24) and (4.25), we can describe all the physical phenomena studied in this book. It is therefore useful to present these solutions for commonly encountered geometries. Moreover, we have shown that for an infinite medium, in the regime of slow variations, Pd and Γω are solutions of a diffusion equation. In this chapter, we shall study the solutions of this equation for certain geometries. The validity of the diffusion equation will be discussed for the cases of infinite and semi-infinite media in Appendices A5.1 and A5.3.

We shall take particular interest in the Laplace transform Pγ (r, r′) of the probability P(r, r′, t). This measures the sum of the contributions to the probability from multiple scattering trajectories between r and r′ for times less than 1/γ. From this quantity we shall define a characteristic time, the recurrence time, which describes the total time spent around an arbitrary point in the medium. It depends on the space dimensionality and on the geometry of the system.

For a finite system, typically a cube of side L, there is a natural characteristic time scale τD defined by L2 = DτD. It separates the short time regime, where the role of boundary conditions may be neglected, from the long time regime where these conditions become essential. The inverse of this time defines the so-called Thouless frequency 1/τD or the Thouless energy Ec = ħ/τD.

Wave propagation in random media has been the subject of intense activity for more than two decades. This is now an important area of research, whose frontiers are still fuzzy, and which includes a variety of problems such as wave localization (weak and strong), mesoscopic physics, effects of electron–electron interactions in metals, etc. Moreover, since many disorder effects are not truly specific to a given kind of wave, various approaches have been developed independently in condensed matter physics, in optics, in atomic physics and in acoustics.

A large number of monographs or review articles already exist in the literature and they cover in detail various aspects of the field. Our aim is rather to present the basic common features of the effects of disorder on wave propagation and also to provide the non-specialist reader with the tools necessary to enter and practice this field of research.

Our first concern has been to give a description of the basic physical effects using a single formalism independent of the specific nature of the waves (electrons, electromagnetic waves, etc.). To this purpose, we have started with a detailed presentation of “single-particle” average quantities such as the density of states and elastic collision time using the framework of the so-called “Gaussian model” for the two most important examples of waves, namely Schrödinger and scalar Helmholtz wave equations. We have tried, as much as possible, to make precise the very basic notion of multiple scattering by an ensemble of independent effective scatterers whose scattering cross section may be obtained using standard one-particle scattering theory.

Introduction

Phase coherence is at the basis of the interference effects which lead to weak localization in electronics. This phase coherence also has important consequences in optics. Moreover, using an incident laser beam, it is possible in optics to study the angular behavior of both transmitted and reflected waves. This is difficult in electronic devices, where electrons are injected and collected from reservoirs and do not have an accessible angular structure. In this chapter, we study the intensity of the light reflected by a diffusive medium and we show that it has an angular structure that is due to the coherent effects associated with the Cooperon. We also show that it is possible to single out and analyze the contribution of multiple scattering paths as a function of their length. This leads to a kind of “spectroscopy” of diffusive trajectories.

The issue of wave scattering in disordered media has a long history. At the turn of the twentieth century, a purely classical approach to the description of radiative transfer of electromagnetic waves through the atmosphere, based on the Boltzmann equation, had already been proposed by Schuster [206]. This problem was subsequently extended to include the related domains of turbulent media, meteorology and liquids. It was only during the 1980s, however, that the possibility of phase coherent effects in the multiple scattering of waves in random media was raised. The interest surrounding this question is certainly related to new developments in similar questions in the quantum theory of scattering [207–209]. Asystematic description of coherent effects emphasizing the role of the Cooperon was initially proposed in references [210] and [211].

In this chapter we denote by v0, the average density of states per spin direction and by ρ0 = v0/Ω, the density of states per unit volume. The characteristic energy Δ = 1/(ρ0Ω) = 1/v0is the average spacing between energy levels per spin direction. We set ħ = 1.

Introduction

In the previous chapters, we have considered different signatures of the phase coherence on average values of transport quantities such as the electrical conductivity or the albedo. In order to observe such coherent effects, it is necessary to couple the disordered medium to the outside world, such as electric wires for the conductivity or the free propagation medium for the albedo. However, it is also possible to characterize isolated disordered systems by measuring their spectral properties.

A disordered medium can be viewed as a complex system whose energy or frequency spectrum cannot be described by means of a given deterministic series of numbers. The nature of the spectrum and of its correlations reflects different physical properties. For example, the spectrum of a disordered metal is quite different if it is a good or a bad conductor. The understanding of these properties is essential for the description of thermodynamic properties such as orbital magnetism or persistent current, which we shall study later and which express the sensitivity of the spectrum to an applied magnetic field or to an Aharonov–Bohm flux (Chapter 14).

The purpose of this chapter is to describe the spectral properties of a disordered metal by means of statistical methods.

This chapter contains a description of essential concepts and tools which will be used throughout the book. We take ħ = 1.

The average Green function describes the evolution of a plane wave in a disordered medium, but it does not contain information about the evolution of a wave packet. For optically thick media, or for metals, most physical properties are determined not by the average Green function, but rather by the probability P(r, r′, t) that a particle will move from some initial point r to a point r′, or possibly return to its initial point.

In this chapter, starting from the Schrödinger equation (or the Helmholtz equation), we establish a general expression describing the quantum probability for the propagation of a particle, i.e., a wave packet, from one point to another. When this probability is averaged over a random potential we can identify, in the weak disorder limit kle ≫ 1, three principal contributions:

• the probability of going from one point to another without any collision;

• the probability of going from one point to another by a classical process of multiple scattering;

• the probability of going from one point to another by a coherent process of multiple scattering.

We will show that the last two processes, called respectively Diffuson and Cooperon, satisfy a diffusion equation in certain limits.

Definition

Starting from the Schrödinger equation, we want to determine the probability of finding a particle of energy ∈0 at point r2 at time t, if it was initially at r1 at t = 0.

In this chapter v0is the average density of states per spin direction and ρ0 = v0/Ω is the density of states per unit volume. The energy Δ = 1/(ρ0Ω) = 1/v0is the average level spacing per spin direction. Most results will be presented in the CGS system, in the form most commonly found in the literature. Unless specified, we take ħ = 1.

Introduction

Up to this point, electron–electron interaction has been neglected in the description of spectral properties and electronic transport. Although electrons interact through the Coulomb interaction, the free electron model is a very good approximation for the description of many physical properties. This is due to the screening of the Coulomb interaction which occurs on a length of the order of the average distance between electrons. However, the electron–electron interaction has important physical consequences which can be classified in two categories.

• Each electron is sensitive not only to the disorder potential but also to the electronic density fluctuations induced by other electrons. As a result, the energy levels are shifted and the thermodynamic and transport properties are modified, particularly the density of states and the conductivity. The change in density of states is maximum around the Fermi level, thus constituting a direct signature of the interaction. Moreover, this change is important since it affects the orbital magnetism of the electron gas and the persistent current (Chapter 14). The change in conductivity is of the same order of magnitude as the weak localization correction, but its nature is quite different. In particular, it does not depend on the magnetic field, making it more difficult to observe.

• […]

Interference and disorder

Wave propagation in a random medium is a phenomenon common to many areas of physics. There has been a recent resurgence of interest following the discovery, in both optics and quantum mechanics, of surprising coherent effects in a regime in which disorder was thought to be sufficiently strong to eliminate a priori all interference effects.

To understand the origin of these coherent effects, it may be useful to recall some general facts about interference. Although quite spectacular in quantum mechanics, their description is more intuitive in the context of physical optics. For this reason, we begin with a discussion of interference effects in optics.

Consider a monochromatic wave scattered by an obstacle of some given geometry, e.g., a circular aperture. Figure 1.1 shows the diffraction pattern on a screen placed infinitely far from the obstacle. It exhibits a set of concentric rings, alternately bright and dark, resulting from constructive or destructive interference. According to Huygens' principle, the intensity at a point on the screen may be described by replacing the aperture by an ensemble of virtual coherent point sources and considering the difference in optical paths associated with these sources. In this way, it is possible to associate each interference ring with an integer (the equivalent of a quantum number in quantum mechanics).

Let us consider the robustness of this diffraction pattern. If we illuminate the obstacle by an incoherent source for which the length of the emitted wave trains is sufficiently short that the different virtual sources are out of phase, then the interference pattern on the screen will disappear and the screen will be uniformly illuminated.

Materials development and crystal growth techniques

This chapter outlines the nature and importance of semiconductors. The industrially important semiconductors are tetrahedrally coordinated, diamond and related structure IVB, III-V and related materials. The sp3 tetrahedral covalent bonding is stiff and brittle, unlike the metallic bond, which merely requires closest packing to minimize the energy. The atomic core structures of extended defects in semiconductors depend on this stiff, brittle bonding and in turn give rise to the electrical and optical properties of defects.

The semiconductors' closely related adamantine (diamond-like) crystal structures and energy band diagrams are outlined. There are a large number of families of such semiconducting compounds and alloys, some of which are non-crystalline. However, only a few have been developed to the highest levels of purity and perfection so that single crystal wafers are available. Instead, with modern epitaxial growth techniques, thin films, quantum wells, wires and dots and artificial superlattices can be produced. This can be done with many semiconductor materials, including alloys of continuously variable composition, with the necessary quality on one of the few available types of wafer. These epitaxial materials have ‘engineered’ energy band structures and hence electronic and optoelectronic properties and can be designed for incorporation into devices to meet new needs. It is largely to this field that materials development has moved, except for the occasional development of an additional material like GaN.

The chapter closes with a brief account of the way that competitive materials development, responding to economic demand, determines which materials enter production.