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.

The Hall coefficient of simple liquid metals is, for the most part, freeelectron- like. This is true, for example, of liquid Na, K, Rb, Cs, Al, Ga, In, Zn, Ge, Sn, and of liquid Cu, Ag and Au. This is also true of a wide range of non-transition metal glassy alloys of which some examples (out of many) are given in Table 9.1. Departures from the free-electron value are, however, found, for example in Ca-Al alloys and, as the table shows, in metallic glasses containing transition metal elements, which can show positive Hall coefficients in circumstances where hole conduction can scarcely be involved. To explain these positive values thus poses a problem; it emphasises still more the importance of the transition metal alloys. We must therefore look at the theory of the Hall coefficient, in particular that of alloys containing transition metals, whose behaviour forces us to recognise that the Ziman theory cannot be the whole story.

Conventional theory

Let us first consider the predictions of conventional theory for the Hall coefficient of a metal. In a crystalline metal the Hall coefficient can be difficult to calculate because it depends in a fairly complicated way on the shape of the Fermi surface and on how the electron velocities and relaxation times vary over the surface.

The obvious success of the Ziman theory does not extend to the liquid transition metals and, as we shall see in the next chapter, the Hall coefficient of a number of glasses containing a substantial proportion of transition metal is positive, thereby posing a powerful challenge to conventional theories. Before we try to compare theory and experiment, however, let us look at some of the important properties of transition metals and their alloys, in both crystalline and glassy forms.

Crystalline transition metals

A transition metal is one whose atoms have incomplete d-shells, such as iron or tungsten. Typically in the free atom there are also s electrons from a higher electron shell, for example, there may be two 3d-electrons and one 4s. In the solid state the wavefunctions overlap and the single states of the free atom spread out into bands whose electrons can therefore take part in the conduction process. The s-levels broaden much more than the d-levels as the atoms get closer. This is because the s-electrons come from the outer reaches of the atom with wavefunctions that overlap strongly with those of their neighbours in the solid. The d-electrons by contrast are more tightly bound within the atom and so in the solid form a much narrower band whose electrons tend to have much lower velocities.

The validity of the independent electron picture

We have treated the electrons as effectively independent particles subject to occasional scattering processes even though we know that there are strong Coulomb forces between electrons and between electrons and ions. This picture certainly has some validity which can be partly understood in the following way. First of all, as we have seen in Chapter 4, the range of the Coulomb interaction between electrons is screened out over a distance of the order of the interionic separation because the conduction electrons are attracted to the neighbourhood of the positive ions and so produce electrical neutrality when viewed from a short distance away. Thus the cross-section for scattering of an electron is of the same order as that of an ion, i.e. of atomic dimensions.

Second, the Pauli exclusion principle drastically reduces the number of processes by which conduction electrons can interact and be scattered by other conduction electrons. We can see this from the following argument. Consider an electron gas at absolute zero with all the states up to E0 filled and those above empty. Assume that we give one electron a small amount of energy ε above E0. It can only be scattered by another electron in the Fermi sea if, after the collision, both particles have empty states of the right energy to go to. This means that, since energy is conserved in the collision, the initial state of the second electron must lie within an energy range ε of the Fermi level; otherwise the collision could not raise its energy above E0 where there are empty states.

Having looked at some of the ideas in terms of which the electrical conductivity of metals has conventionally been interpreted, we now look at the conductivity of metallic glasses to see how far we can understand it in terms of what we have learned. The broad features of the conductivity of glasses made from simple metals have been interpreted in terms of the Ziman model (as established for simple metal liquids). Those that contain a substantial proportion of at least one transition metal have properties that cannot, for the most part, be so interpreted and indeed it was soon recognised that even simple metal alloys require an extension of the theory. Because all these materials we are considering are highly disordered, we can be sure that their electrical resistivity will be large at all temperatures and will not vary a great deal with temperature; its precise magnitude will of course depend on the specific constituents of the alloy.

There is one generalisation that can be made at the outset. Experimental data show that, as we would expect, the residual resistivity, ρo? of a glass is comparable to that of the corresponding liquid and indeed its resistance looks like the natural continuation of that of the liquid to low temperatures. This is illustrated in Figure 11.1 for Ni60Nb40 and Pd81S19, which also shows that the crystalline form at low temperatures with its much higher degree of order has a much lower resistivity. All this is reassuring.

Introduction

We have now seen in some detail how weak localisation and the interaction effect can modify the electron transport properties of electrons in metallic glasses or, more specifically, of electrons that are subject to strong elastic scattering, whether this be in the crystalline or the amorphous phase. What this survey shows is that many of the qualitative features to be expected are indeed observed in the resistivity, magnetoresistance and Hall coefficient of metallic glasses. The final question is: how far do the theories provide a quantitative account of the experiments?

It is at once clear, I think, why it is difficult to answer this question unequivocally. There are so many parameters that can influence the behaviour of these properties that unless some can be controlled or eliminated there are too many adjustable quantities to make possible a convincing comparison between theory and experiment.

One common way to overcome this problem is to make measurements of a range of properties so that a given specimen is very well characterised and as few as possible of the relevant parameters are left undetermined. So let us decide what quantities we know or can deduce with some reliability from experiment.

We can measure the low-temperature heat capacity of the metallic glass to find the term linear in temperature, which allows us to deduce the density of states at the Fermi level. In order to interpret the thermopower we would like to know the electron–phonon enhancement factor in the alloy; if it is a superconductor we can derive this from our knowledge of its superconducting properties.

This chapter is intended as an introduction to the fundamental physics of resonant tunnelling diodes (RTDs). The idea of global coherent tunnelling is introduced in order to provide an intuitive and clear picture of resonant tunnelling. The theoretical basis of the global coherent tunnelling model is presented in Section 2.2. The Tsu–Esaki formula, based on linear response theory, is adopted and combined with the transfer matrix method to calculate the tunnelling current through double-barrier resonant tunnelling structures (Section 2.2.1). The global coherent tunnelling model is improved by taking Hartree's selfconsistent field (Section 2.2.2) into account. A more analytical transfer Hamiltonian formula is also presented (Section 2.2.3). Section 2.3 introduces the electron dwell time, which is one of the important quantities required to describe the high-frequency performance of RTDs. The effects of quantised electronic states in the emitter are then studied in Section 2.4. Section 2.5 describes resonant tunnelling through double-well structures. Finally, Section 2.6 discusses the idea of incoherent resonant tunnelling induced by phase-coherence breaking scattering. The problem of collision-induced broadening is then discussed in terms of the peak-to-valley (P/V) current ratio of RTDs by using a phenomenological Breit–Wigner formula.

Resonant tunnelling in double-barrier heterostructures

Let us start with a simple discussion of resonant tunnelling through the double-barrier heterostructure depicted in Fig. 2.1 (a). A resonant tunnelling diode (RTD) typically consists of an undoped quantum well layer sandwiched between undoped barrier layers and heavily doped emitter and collector contact regions.

According to Ohm's law, the resistance of an array of scatterers increases linearly with the length of the array. This describes real conductors fairly well if the phase-relaxation length is shorter than the distance between successive scatterers. But at low temperatures in low-mobility samples the phase-relaxation length can be much larger than the mean free path. The conductor can then be viewed as a series of phase-coherent units each of which contains many elastic scatterers. Electronic transport within such a phase-coherent unit belongs to the regime of quantum diffusion which has been studied by many authors since the pioneering work of Anderson (P. W. Anderson (1958), Phys. Rev.109, 1492). In this regime, interference between different scatterers leads to a decrease in the conductance. For a coherent conductor having a overall conductance much greater than ~ (e2/h) or 40 μΩ-1, the decrease in the conductance is approximately (e2/h). Such a conductor is said to be in the regime of weak localization (Section 5.2). This effect is easily destroyed by a small magnetic field (typically less than 100 G), so that it can be identified experimentally by its characteristic magnetoresistance (Section 5.3). This is a very important effect, because unlike most other transport phenomena it is sensitive to phase relaxation and not just to momentum relaxation. Indeed the weak localization effect is often used to measure the phase-relaxation length.