A crystal is made up of a regular arrangement of atoms in a pattern that repeats itself in all three spatial dimensions. This structural feature has the practical result that most properties of crystals are anisotropic, that is, different values are obtained for the property when measured along the different directions of a crystal. This anisotropy distinguishes crystals from non-crystalline materials (glasses) or from random polycrystalline aggregates, both of which show isotropic properties.

The purpose of this book is to show how to analyze the anisotropic properties of crystals in terms of the (tensor) nature of the properties and of the symmetry of the crystals. In the first two chapters, we focus on the nature of the various properties with which we will be dealing. We will see how the tensor character of a property helps to define its variation with orientation. Questions of crystal symmetry will then be dealt with, starting from Chapter 3.

Definition of crystal properties

For a crystal, regarded as a thermodynamic system (i.e. in equilibrium with its surroundings), any physical property can be defined by a relation between two measurable quantities. For example, the property of crystal density is simply the ratio of mass to volume, both measurable quantities; similarly, elastic compliance is the ratio of mechanical strain to stress. Often, one of the measurable quantities can be regarded as a generalized ‘forces’ and the other as a response to that force.

k-states in disordered metals

The simplest amorphous metals are probably the liquid non-transition metals such as liquid sodium or liquid zinc. The simplicity arises partly because no d-electrons are involved and partly because the liquids are of a single component whereas by contrast all metallic glasses involve at least two components. So let us see how far we can understand the electron transport of these amorphous metals before we tackle systems with the additional complication of two or more components.

As soon as we confront the problem of electron transport in highly disordered systems like liquids or glasses several questions spring to mind. How useful is the concept of a k-state when we are so far from having translational symmetry? How valid is the concept of a Fermi surface? The answers depend, not surprisingly, on the degree of scattering involved. Thus it is not just the degree of disorder involved but also the strength of the individual scattering processes. If the mean free path of the electrons is l and the electron wavelength at the Fermi level is λ we require that l be much greater than λ. More commonly we choose the almost equivalent condition kFl ≫ 1 where kF = 2π/λ. In a highly disordered system the mean free path tends to be only weakly temperature dependent so that the condition itself is essentially temperature independent. Experience with other systems of limited mean free path, for example, calculations on concentrated random binary crystalline alloys which give results in accord with experiment, suggests that a Fermi surface and its associated k-vectors for the conduction electrons are useful and satisfactory concepts provided that kFl≫ 1.

The source of electrical resistance

In trying to understand the electron transport properties of metallic glasses – properties such as electrical conductivity, Hall coefficient and thermopower – we shall start by using conventional theories that have been successful in accounting for the corresponding properties in crystalline metals and alloys and see how far these theories are successful in describing the properties of metallic glasses. I will explain what I mean by ‘conventional’ theories as we go along.

The starting point in understanding the electrical conductivity, σ, or resistivity, ρ(= 1/σ) of metals is the fact that the de Broglie waves which represent the conduction electrons can propagate without attenuation through a perfectly periodic lattice, such as that formed by the positive ions of an ideally pure and perfect crystalline metal at absolute zero. There is thus no electrical resistivity. More strictly, the electrons are scattered by the ions but only coherently as in Bragg reflections from the lattice planes. Such coherent scattering alters the way the electrons respond to applied electric and magnetic fields but does not cause electrical resistance. Such resistance comes about through random, incoherent scattering of the electron waves; this occurs only when the periodicity of the lattice and its associated potential is upset.

This means that, if you now add to your pure and perfect crystal chemical impurities randomly distributed, they will disturb the perfect periodicity and cause resistance to the flow of the electric current. Likewise, physical imperfections such as vacancies, dislocations or grain boundaries will produce electrical resistance even at the absolute zero. These imperfections and chemical impurities upset the perfect periodicity and so scatter the electrons that carry the electric current.

The purpose of this book is to explain in physical terms the many striking electrical properties of disordered metals or alloys, in particular metallic glasses. The main theme is that one central idea can explain many of the otherwise puzzling behaviour of these metals, particularly at low temperatures and in a magnetic field. That idea is that electrons in such metals do not travel ballistically between comparatively rare scattering events but diffuse through the metal. These new effects are not large but they are so universal in high-resistivity metals, so diverse and qualitatively so different from anything to be expected in metals where the electrons have a long mean free path, that they cry out for an explanation.

The book is not a critical research review; the motivation is mainly to explain. In interpreting theory there are always the dangers of overinterpretation, misinterpretation and failure to interpret and I do not expect to have escaped these completely. Nonetheless, our new understanding of disordered metals and alloys constitutes a substantial addition to conventional Boltzmann theory and deserves to be more widely known and appreciated.

The book is aimed at those who know little of the subject such as students starting work in this field or those outside the field who wish to know of developments in it. There is no attempt at rigorous derivations; the aim is to present the physics as clearly as possible so that readers can think about the subject for themselves and be able to apply their thinking in new contexts.

The results that we have just obtained are rather formal but do give us some important insights into the nature of electrical conduction in metals. We see from equation (3.35) that this conductivity depends entirely on the properties of the conduction electrons at the Fermi level. Moreover, the properties involved are of two distinct kinds: the first kind relates to the dynamics of the electrons, as represented by the distribution of electron velocities over the Fermi surface. The second kind represented by τ is concerned with scattering, our theme in this chapter.

As we have already noted, scattering in a metal arises from anything that upsets the periodicity of the potential: disorder of the ionic positions, which is paramount in metallic glasses; random changes in chemical composition, which are of great importance in random alloys; impurities, physical imperfections, thermal vibrations, random magnetic perturbations and so on. Let us therefore see how the scattering from some of these can be treated.

In this chapter we deal in section 4.1 with some basic ideas about scattering theory; in 4.2 there is a very brief discussion of Fourier transforms because they appear so frequently in scattering problems; in 4.3 we look at the influence of scattering angle on resistivity; in 4.4 at the effect of the Pauli exclusion principle on scattering; in 4.5 we consider electron screening in metals because the mobile electrons can markedly alter the scattering potential by electrical screening of the scatterer; and in 4.6 the pseudopotential because this has been used very successfully to represent the scattering potential in some simple amorphous metals.

Definitions

The origin of the thermoelectric effects is very simple. They arise because an electric current in a conductor carries not only charge but also heat. Consequently when an electric current flows through the junction of one conductor with another, although the charge flow is exactly matched, there is in general a mismatch in the associated heat flow; the difference is made manifest as the Peltier heat. If the current flows through a conductor in which there is a temperature gradient the heat shows up as the Thomson heat which is the heat that must be added to or subtracted from the conductor to maintain the temperature gradient unchanged; the electric current behaves as if it were a fluid with a heat capacity (either positive or negative). The third manifestation of thermoelectricity is the Seebeck effect which is the inverse of the other two. In this a heat current is established by means of a temperature gradient and this produces an electric current. However this cannot be done with a single material since in such a closed circuit the current induced in one part would cancel that in the other. Instead two materials are needed; moreover it is more convenient to measure not the circulating current that results but the emf that arises when the electrical circuit is broken. More explicitly, if conductor A is connected to conductor B at its two ends and the two junctions are maintained at different temperatures, an emf appears in the circuit.

Introduction

Ordered crystalline metals

Our understanding of the electrical conductivity of metals began almost a century ago with the work of Drude and Lorentz, soon after the discovery of the electron. They considered that the free electrons in the metal carried the electric current and treated them as a classical gas, using methods developed in the kinetic theory of gases.

A major difficulty of this treatment was that the heat capacity of these electrons did not appear in the experimental measurements. This difficulty was not cleared up until, in 1926, Pauli applied Fermi–Dirac statistics to the electron gas; this idea, developed by Sommerfeld and his associates, helped to resolve many problems of the classical treatment. The work of Bloch in 1928 showed how a fully quantal treatment of electron propagation in an ordered structure could explain convincingly many features of the temperature dependence of electrical resistance in metals. In particular it showed that a pure, crystalline metal at absolute zero should show negligible resistance.

From these beginnings followed the ideas of the Fermi surface, band gaps, Brillouin zones, umklapp processes and the development of scattering theories: the scattering of electrons by phonons, impurities, defects and so on. By the time of the Second World War, calculations of the resistivity of the alkali metals showed that the theory was moving from qualitative to quantitative success.

In 1950, the recognition that the de Haas–van Alphen effect provided a measure of the extremal cross-section of the Fermi surface normal to the applied magnetic field made possible a big advance in the experimental study of Fermi surfaces.

Introduction

There is a further effect that arises in systems in which there is heavy elastic scattering of the conduction electrons; it shows itself at low temperatures through the unusual temperature and magnetic field dependence of the electrical resistance and since its contribution can be confused with that from weak localisation it is important to describe its consequences before we try to complete the survey of that effect.

The localisation effect described in the last chapter involves single electrons and would exist even if these electrons did not interact with each other. By contrast this new effect, sometimes called the Coulomb anomaly, arises ultimately from the interaction of one electron with another. Hence its rather uninformative alternative name ‘the interaction effect’, which does however emphasise that it could not occur with noninteracting electrons. The ‘enhanced interaction effect’ is perhaps a better name.

If, in an ordered metal, an electron in a plane wave state of wave vector k is scattered into state k′, we must have k′ = k + q where q is a Fourier component of the scattering potential. In a disordered metal, however, there is an uncertainty in k because of the scattering; this uncertainty is of order 1/l, where l is the relevant mean free path. Thus the above relation will break down if the scattering vector q is less than 1/l. This suggests that any unusual effects will occur at small q and that our interest will focus on states for which ql ≤ 1; this means that the smaller the mean free path involved, the greater the range of q-vectors that can contribute to the effects.

What are metallic glasses?

The word ‘glass’ as we normally use it refers to window glass. As we all know, this is a brittle, transparent material with vanishingly small electrical conductivity. It is in fact a material in which the constituent molecules are arranged in a disordered fashion as in a liquid but not moving around; that is to say, each molecule keeps its same neighbours and the glass behaves like a solid. Most of the solids that physicists have hitherto dealt with are crystalline i.e. their atoms or molecules are arranged in strictly ordered arrays. This is the essential difference between a so-called ‘glass’ and a crystal: a glass has no long-range order. Although the word ‘glass’ was originally used to designate only window glass it has now taken on this generalised meaning of what we may call an amorphous solid.

Electrically insulating glasses have been studied for a long time and it was generally thought that in order to form a glass by cooling a liquid it was necessary to have a material composed of fairly complicated molecules so that, on cooling through the temperature range at which crystallisation would be expected to occur, the molecules would have difficulty in getting into their proper places and could be, as it were, frozen in a disordered pattern at lower temperatures without the thermal energy necessary to get into their ordered positions. This general picture is correct and helpful although the expectations based on it have proved in some respects wrong. It was thought that because metals and alloys are usually of simple atoms, it would be impossible to form a glass from such constituents.

Qualitative picture

First of all we concentrate on the transverse magnetoresistance in which the magnetic field is applied normal to the current direction. The calculation of the magnetoresistance of a crystalline material is very difficult unless there are simplifying features. In the metallic glasses fortunately there are indeed such features. If we make the same assumptions as in our first derivation of the Hall coefficient we find zero magnetoresistance. The effect of the magnetic field is so perfectly compensated by the transverse electric field (the Hall field) that the resultant current is completely unperturbed and so there is no change in resistance i.e. no magnetoresistance.

In the alloys of non-transition metals there is only one type of charge carrier and no obvious source of anisotropy so the magnetoresistance due to conventional mechanisms must be vanishingly small.

If there is to be a non-zero magnetoresistance some additional feature has to come into the story. One example of such a feature is the presence of the two different types of charge carrier that we postulated for transition metal alloys.

Two-band model

If we assume that there are two kinds of carrier, we can perhaps understand the physics of this type of magnetoresistance in macroscopic terms as we did for the Hall effect.

Magnetoresistance in the particle–hole channel

High fields

When a magnetic field is applied to the material, the relative phase of the two electrons in the processes we discussed in section 13.2.2 is not changed by the flux that passes through their common orbit because the electrons execute this in the same sense (not in opposite senses as in weak localisation) and so the effect on the phase is the same for both. On the other hand if the two electrons have antiparallel spins the magnetic field B changes their relative energy by the Zeeman splitting gμB. Here μ is the Bohr magneton and g is the splitting factor, which looks after any change in the magnetic moment of the electron introduced by its environment. In fact we are here considering electron–hole pairs so that since the hole has a spin opposite in sign to that of an electron the triplet state occurs when the electron and hole have antiparallel spins and the singlet state when they are parallel.

If we refer to Table 11.2 (p. 126), we see that in the triplet state only two out of the three spin wavefunctions involve parallel spins; the third is a composite state, which, like the singlet state, involves antiparallel spins. Thus the two components of the electron–hole pair with parallel spins are the ones that are split in energy by the magnetic field. The frequencies of the electron and the hole are correspondingly altered and for this reason dephasing occurs.

Normal modes in glasses

Thermal energy causes the ions in a metallic glass, as in a crystal, to vibrate about their mean positions; in a glass there may be additional ionic motion in which ions actually shift between two or more sites but we ignore this for the present. The complex vibrational motion can, as a first approximation, be resolved into a superposition of normal modes, each of which is to this approximation a harmonic motion independent of all the other modes. This ignores anharmonicity and tunnelling modes, which can be very important in glasses. For our present purposes we take the normal mode description as adequate but bear in mind its limitations. These modes introduce into the solid changes in charge density that are periodic in time and cause corresponding changes to the potential seen by the conduction electrons. These changes scatter the electrons.

When such harmonic motions are quantised we associate with each mode phonons in accordance with the intensity of the particular mode. In disordered materials the normal modes of vibration exist although they are not necessarily extended waves; some may be localised to the neighbourhood of particular ions. As long as the vibrations are quasiharmonic, however, phonons are a valid concept in disordered materials although it may not be possible to assign to them a well-defined wave vector if the mode is strongly localised.