The attraction between electrons

Superconductivity was long considered the most extraordinary and mysterious of the properties of metals; but the theory of Bardeen, Cooper and Schrieffer—the BCS theory—has explained so much that we can say that we now understand the superconducting state almost as well as we do the ‘normal’ state. We shall not attempt here

to cover the whole of a large and rapidly advancing subject; the emphasis will be on the atomic processes from which so many unusual macroscopic phenomena arise.

The whole effect springs from a small force of attraction between any two electrons which have nearly the same energy. We usually assume that free electrons repel one another, through their Coulomb interaction, although, as shown in §5.8, this field is considerably reduced at long distances by the screening effect of the ‘Other’ electrons. But in a lattice an electron tends to pull towards itself the positive ions, so that it is surrounded by a region where the lattice is slightly denser than usual. Another electron coming into the vicinity will be drawn towards this region; it will look as if it were attracted towards the first electron. The two particles can, so to speak, gain by sitting close together in the same depression of the mattress.

This new edition is meant still to conform to the principles expounded in the above Preface. But eight years is about the doubling time of modern scientific knowledge and solid state physicists have not been idle in the interval. Most of the original text still stands, but several new sections have been added, to cover topics that have come into greater prominence lately or where there has been a significant shift of understanding or emphasis. I have also attempted to make reference in passing to a number of phenomena or fields of study that are relevant to the basic theory, even if they cannot be discussed in detail. In this way, the general scope of the book has been widened, to include, for example, something about magnetic and non-magnetic impurities, F-centres, surfaces, tunnelling, junctions, and type II superconductivity. But the general level of mathematical sophistication has not been raised, even though the technical formalism of advanced quantum theory is now becoming more commonplace in this field.

I am most grateful to many colleagues—especially to Bob Chambers here in Bristol and to Federico Garcia-Moliner in Madrid—for a number of detailed comments of which I have tried to take account in the new text. Bob Evans helped greatly by preparing a new index. And lest the reader may feel that absence from Cambridge has been a long period of exile, may I simply add that Bristol, too, is just as good a Ole to go to.

The Frontiers of Knowledge (to coin a phrase) are always on the move. Today's discovery will tomorrow be part of the mental furniture of every research worker. By the end of next week it will be in every course of graduate lectures. Within the month there will be a clamour to have it in the undergraduate curriculum. Next year, I do believe, it will seem so commonplace that it may be assumed to be known by every schoolboy.

The process of advancing the line of settlements, and cultivating and civilizing the new territory, takes place in stages. The original papers are published, to the delight of their authors, and to the critical eyes of their readers. Review articles then provide crude sketch plans, elementary guides through the forests of the literature. Then come the monographs, exact surveys, mapping out the ground that has been won, adjusting claims for priority, putting each fact or theory into its place.

Finally we need textbooks. There is a profound distinction between a treatise and a textbook. A treatise expounds; a textbook explains. It has never been supposed that a student could get into his head the whole of physics, nor even the whole of any branch of physics. He does not need to remember what he can easily discover by reference to monographs, review articles and original papers. But he must learn to read those references: he must learn the language in which they are written : he must know the basic experimental facts, and general theoretical principles, upon which his science is founded.

Lattice dynamics

The simplest solid is probably solid argon, which is a regular array of neutral atoms, with tightly bound closed shells of electrons, held together by van der Waals forces, which act mainly between nearest neighbours in the lattice. The physics of such a crystal is the thermal motion of the atoms about their idealized equilibrium positions.

To discuss this motion the most elementary idea is the Einstein model, in which each atom vibrates like a simple harmonic oscillator in the potential well of the force fields of its neighbours. This field can never be precisely a quadratic well with spherical symmetry, but this is a reasonable approximation on the average. The excitation spectrum of the crystal then consists of levels spaced a distance ħVE apart, where vE is the Einstein frequency, i.e. the frequency of oscillation of each atom in its potential well.

This model is useful in some contexts where a very crude account of thermal vibrations is sufficient—especially at relatively high temperatures, when the assumption that the various atoms vibrate independently is justified. But we can see at once that if two or more atoms move in unison the forces between them, tending to restore each of them to its equilibrium position, will be reduced, so that the energy required to excite a quantum may be rather less. There is a tendency for the motion of the adjacent atoms to be correlated.

The Boltzmann equation

The carriers in a metal or semiconductor can be affected by external fields, and by temperature gradients. They also suffer scattering from impurities, lattice waves, etc. These effects have to be balanced against each other—we have to consider situations in which the electron is accelerated by a field, but loses its extra energy and momentum by scattering. In this chapter we shall consider the ‘Ordinary’ transport properties, such as are observed when constant fields are applied.

Much the simplest approach to this problem, in general, is to set up the transport equation or Boltzmann equation. We study a quantity fk(r), the local concentration of carrier in the state k in the neighbourhood of the point r in space. Strictly speaking, this quantity can only be defined in terms of fine-grained distributions, ensemble averages, density matrices, etc. There is a considerable literature upon this point—but it belongs more to the formal theory of quantum statistical mechanics than to the theory of solids.

Free electrons

Consider solid Na. Each atom contains 11 electrons. But 10 of these are in states which are tightly bound to the nucleus to form an ion of net positive charge + |e|. In the free atom the final electron moves in an orbital around this ion. When the atoms are brought together into a solid, the orbitais overlap and interact. It is argued that the overlap is so extensive that the quantization scheme in which each electron is localized on its own atom must break down. It must be replaced by a scheme in which the wave-function of each electron is a solution of the Schrödinger equation for motion in the potential of all the ions. Thus, we distinguish at the outset between the core electrons, which are treated as almost completely localized, and the valence electrons or conduction electrons, which are assumed to go into Bloch states—states extended throughout the whole crystal.

In this chapter we shall use the one-electron model, where we ignore any interactions between the valence electrons due to their Coulomb repulsion. These effects, of correlation and exchange, are discussed in Chapter 5. Such effects in the core, or between core and valence electrons, are usually treated as part of the potential V(r) seen by each valence electron as it moves through the crystal. Thus, V(r) is supposed to be calculated as a Hartree or Hartree–Fock potential of the ion.

Translational symmetry

A theory of the physical properties of solids would be practically impossible if the most stable structure for most solids were not a regular crystal lattice. The N-body problem is reduced to manageable proportions by the existence of translational symmetry. This means that there exist basic vectors, a1; a2, a3such that the atomic structure remains invariant under transtation through any vector which is the sum of integral multiples of these vectors.

In practice, this is only an ideal. A laboratory specimen is necessarily finite in size, so that we must not carry our structure through the boundary. But the only regions where this matters are the layers of atoms near the surface, and in a block of Ar atoms these constitute only about N⅔ atoms—say 1 atom in 108 in a macroscopic specimen. Most crystalline solids are also structurally imperfect, with defects, impurities and dislocations to disturb the regularity of arrangement of the atoms. Such imperfections give rise to many interesting physical phenomena, but we shall ignore them, except incidentally, in the present discussion. We are mainly concerned here with the perfect ideal solid, and with the properties it shows; the phenomena which are associated with the solid as the matrix, vehicle, or background for little bits of dirt, or tiny cracks and structural flaws, belong to a different realm of discourse.

Perturbation formulation

The theory of electronic structure, as presented in Chapter 3, is a one-electron model; each electron is treated as an independent particle, moving in a well-defined potential, and the interactions between conduction electrons are ignored. But we know that these interactions are strong, and are of long range, being the Coulomb force between the charges and the so-called exchange force associated with the antisymmetry of the wave-functions.

Naively we assume that these interactions can be taken care of by a Hartree or Hartree–Fock self-consistent calculation that adjusts the atomic potentials for the charge distribution of the valence electrons (as well, of course, as the electrons in the closed shells of the ion cores). But this is not easy to do properly—and we may have to fall back on some assumption, like that used by Wigner and Seitz (§4.3), that the electron sees the potential of a singly charged ion in the cell where it happens to be, but that neighbouring cells are electrically neutral.

In recent years, therefore, a lot of effort has been expended on the many-body problem of a gas of electrons interacting via their Coulomb potential, and the basic effects of the interaction are now well understood. Much of the theory is expressed in complicated formal language; the main results are surprisingly simple, and can be derived by elementary arguments.