Introduction

One of the central themes of this book is how reducing the size of semiconductor structures down to mesoscopic and smaller scales brings the quantum wave nature of electrons into play, resulting in electronic and optical properties which are markedly different from those of bulk semiconductors. Chapters 6–10 describe how these properties can be exploited in device applications. But one of the key challenges facing physicists and engineers is how to make devices operate at room temperature. The main obstacle to achieving this goal is the unavoidable presence of phonons, the quantum vibrations of atoms making up a solid, and their ability to scatter electrons.

Phonons would thus appear to occupy an uncomfortable position in the study of low-dimensional structures, with an understanding of their properties required solely for the purpose of finding ways to reduce their interaction with electrons. As we shall see in this chapter, however, the physics of phonons in low-dimensional structures is sufficiently fundamental and non-trivial to be of interest in its own right. Just as for electrons, phonons can be confined within heterostructures and we would like to know how the dynamics of low-dimensional phonons differs from that of bulk phonons. We would also like to understand the effects of dimensionality on the electron–phonon interaction and hence such electron transport properties as the phonon-scattering-limited mobility.

In fact, finding ways to reduce the electron–phonon interaction is not the only reason for investigating phonons in low-dimensional structures.

Introduction

Previous chapters in this book have described in detail how low-dimensional structures affect the optical properties of semiconductor materials. It should therefore be no surprise to readers to find that the main applications of low-dimensional materials have been in optical devices which emit light – particularly the semiconductor laser. The semiconductor laser, even without the use of low-dimensional structures, has become the most common form of laser and new operating wavelengths; new characteristics and new applications appear at an amazing rate. This chapter could not hope to provide a comprehensive review of all these developments. Specialist texts (Agrawal, 1986; Zory, 1993; Coldren and Corzine, 1995) will do that far more effectively. Instead we hope to introduce some of the key concepts, presented in the context of developments in semiconductor physics, which will lead the reader to the more advanced texts. Consequently, many of the sources quoted are review papers and not the original texts.

Many of the advances in low-dimensional semiconductors have been motivated by the fascinating new range of physical phenomena which arise when electrons and holes are confined in very small dimensional structures (Bastard, 1988; Schmitt-Rink et al., 1989; Weisbuch and Vinter, 1991). Advances in semiconductor lasers have driven this fascination, but there has also been a clear focus for the development. The most important of these, to date, has been the need to develop very high-performance semiconductor lasers for optical-fibre-based communication systems (Koch and Koren, 1990).

Prologue

The electron theory of metals pursues the development of ideas that lead to an understanding of various properties manifested by different kinds of materials on the basis of the electronic bondings among constituent atoms. Here the concept of the energy band plays a key role and is introduced in Section 2.2. Condensed matter is often classified in terms of bonding mechanisms; metallic bonding, covalent bonding, ionic bonding and van der Waals bonding. After their brief introduction in Section 2.3, we focus on metallic bonding and discuss the Sommerfeld free-electron model in Sections 2.4–2.6. The construction of the Fermi sphere is discussed in Section 2.7.

Concept of an energy band

Let us first briefly consider the electron configurations in a free atom. The central-field approximation is useful to describe the motion of each electron in a many-electron atom, since the repulsive interaction between the electrons can be included on an average as a part of the central field. Because of the spherical symmetry of the field, the motion of each electron can be conveniently described in polar coordinates r, θ and ϕ centered at the nucleus. All three variables r, θ and ϕ are needed to describe electron motion in three-dimensional space. In quantum mechanics, the three degrees of freedom lead to three different quantum numbers, by which the stationary state or the quantum state of an electron is specified; the principal quantum number n, which takes a positive integer, the azimuthal or orbital angular momentum quantum number ℓ, which takes integral values from zero to n–1, and the magnetic quantum number m, which can vary in integral steps from −ℓ to ℓ, including zero.

Prologue

Electron transport properties can be investigated by measuring the response of conduction electrons to a temperature gradient or to external fields such as an electric field, a magnetic field, or a combination of these applied to a specimen. In this chapter, we study basic transport properties with subsequent derivation of the Boltzmann transport equation. The electrical conductivity is then formulated in the light of the Boltzmann transport equation. The temperature-dependent electrical resistivity expression known as the Baym resistivity formula is derived by taking into account the electron–phonon interaction and is applied to obtain the well-known Bloch–Grüneisen law for a crystal metal. The remaining transport properties including the thermal conductivity, the thermoelectric power, the Hall effect and the magnetoresistance will be discussed in Chapter 11.

The Drude theory for electrical conductivity

In Chapter 2, we learned that electrons on the Fermi surface in a metal like Na carry a Fermi wave number kF of the order of a few 10 nm-1, which is converted to a Fermi velocity vF of approximately 106 m/s through the relation vF=ħkF/m. In spite of such high velocities of electrons on the Fermi surface, no electrical current can flow, unless an electric field is applied. The reason for this is that the Fermi surface is always symmetric with respect to the origin in reciprocal space and that there always exists an electron with -vF for the electron with vF. Obviously, a current flows only when the applied field displaces the Fermi sphere from the origin and its symmetrical geometry breaks.

This book is an English translation of my book on the electron theory of metals first published in two parts in 1995 and 1996 by Uchida Rokakuho, Japan, the content of which is based on the lectures given for advanced undergraduate and graduate students in the Department of Applied Physics and in the Department of Crystalline Materials Science, Nagoya University, over the last two decades. Some deletions and additions have been made. In particular, the chapter concerning electron transport properties is divided into two in the present book: chapters 10 and 11. The book covers the fundamentals of the electron theory of metals and also the greater part of current research interest in this field. The first six chapters are aimed at the level for advanced undergraduate students, for whom courses in classical mechanics, electrodynamics and an introductory course in quantum mechanics are called for as prerequisites in physics. It is thought to be valuable for students to make early contact with original research papers and a number of these are listed in the References section at the end of the book. Suitable review articles and more advanced textbooks are also included. Exercises, and hints and answers are provided so as to deepen the understanding of the content in the book.

It is intended that this book should assist students to further their training while stimulating their research interests. It is essentially meant to be an introductory textbook but it takes the subject up to matters of current research interest. I consider it to be very important for students to catch up with the most recent research developments as soon as possible.

Prologue

There exists a family of solids, in which the electron–electron interaction plays so substantial a role that the one-electron approximation fails. This is known as the strongly correlated electron system. Historically, De Boer and Verwey were the first to point out, as early as in 1937, that NiO in the NaCl structure should be metallic, since the Fermi level falls in the middle of the Ni-3d band. This already posed serious difficulty in the one-electron band calculations at that time, since NiO is known to exist as a transparent insulator having a band gap of a few eV. Peierls noted in the same year that this difficulty stemmed from the neglect of the repulsive interaction between the electrons and that the electron–electron interaction must be treated beyond the Hartree–Fock one-electron approximation.

Various transition metal oxides, including NiO and various layered perovskite cuprates, the latter being known to undergo a transition to the superconducting state upon carrier doping, have now been recognized as solids typical of a strongly correlated electron system. Their electronic structures and electron transport properties have been extensively studied in the last ten years, i.e., the 1990s. In this chapter, we introduce first the concept of the Fermi liquid theory, which justified the one-electron approximation for electrons near the Fermi level in ordinary metals and alloys, and then extend our discussion to cases where the one-electron approximation fails because of the electron–electron interaction. The Hubbard model is introduced as a model appropriate to describe the short-range motion of electrons transferring from one atomic site to another in competion with the on-site repulsive Coulomb interaction.

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.