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.

Overview and background

Recent progress in crystal growth and microfabrication technologies have allowed us to explore a new field of semiconductor device research. The quantum-mechanical wave-nature of electrons is expected to appear in mesoscopic semiconductor structures with sizes below 100 nm. Instead of conventional devices, such as field effect transistors and bipolar transistors, a variety of novel device concepts have been proposed based on the quantum mechanical features of electrons. The resonant tunnelling diode (RTD), which utilises the electron-wave resonance in multi-barrier heterostructures, emerged as a pioneering device in this field in the mid-1970s. The idea of resonant tunnelling (RT) in finite semiconductor superlattices was first proposed by Tsu and Esaki in 1973 shortly after molecular beam epitaxy (MBE) appeared in the research field of compound semiconductor crystal growth. A unique electron tunnelling phenomenon was predicted for an AlGaAs/GaAs/AlGaAs double-barrier heterostructure, based on electron-wave resonance, analogous to the Fabry–Perot interferometer in optics. In the particle picture, each electron is constrained inside the GaAs quantum well for a certain dwell time before escaping to the collector region. The bias dependence of the tunnelling current through the double-barrier structure shows negative differential conductance (NDC) as a result of RT. Experimental results reported in the early days showed only weak features in current–voltage (I–V) characteristics at low temperatures and did no more than confirm the theoretical prediction of resonant tunnelling.

Tunneling is perhaps the oldest example of mesoscopic transport. Single-barrier tunneling has found widespread applications in both basic and applied research. The latest example is scanning tunneling microscopy which has made it possible to image on an atomic scale. However, our purpose in this chapter is not to discuss single-barrier tunneling; the field is far too large and well-developed. Instead we will focus on what is called a double-barrier structure, consisting of two tunneling barriers in series. Since the pioneering work of Chang, Esaki and Tsu (Appl. Phys. Lett. 24, (1974) 593) much research has been devoted to the study of such structures. Two important paradigms of mesoscopic transport have emerged from this study, namely, resonant tunneling and single-electron tunneling. At the same time, the current–voltage characteristics of these structures exhibit useful features at room temperature and high bias, unlike most other mesoscopic phenomena which are limited to the low temperature linear response regime.

We start in Section 6.1 with a discussion of current flow through a double-barrier structure, assuming that transport is coherent. The current can then be obtained by calculating the coherent transmission through the structure from the Schrödinger equation. In Section 6.2 we discuss how scattering processes inside the well affect the peak current and the valley current.

In Chapter 2 we have tried to establish that there exists a useful quantity called the transmission function in terms of which one can describe the current flow through a conductor. In this chapter we address the question of how the transmission function can be calculated for actual mesoscopic conductors. As we might expect, this chapter is somewhat mathematical and familiarity with matrix algebra is required. It could be skipped on first reading since it is not essential to know how to calculate the transmission function in order to appreciate mesoscopic phenomena, just as it is not necessary to understand the microscopic theory of diffusion or mobility in order to appreciate bulk transport phenomena. However, we will occasionally (especially in Chapter 5) use some of the concepts introduced here.

If the size of the conductor is much smaller than the phase-relaxation length then transport is said to be coherent and one can calculate the transmission function starting from the Schrödinger equation. A large majority of the theoretical work in this field is centered around this coherent transport regime where we can relate the transmission function to the S-matrix as discussed in Section 3.1.

When dealing with a large conductor it is often convenient to divide it conceptually into several sections whose S-matrices are determined individually. We discuss in Section 3.2 how the S-matrices of successive sections can be combined assuming complete coherence, complete incoherence or partial coherence among the sections.

So far in this book we have described the effect of electron–phonon or electron–electron interactions in phenomenological terms, through a phase-relaxation time. In this chapter we will describe the non-equilibrium Green's function (NEGF) formalism which provides a microscopic theory for quantum transport including interactions. We will introduce this formalism using simple kinetic arguments based on a one-particle picture that are only slightly more difficult than those used to derive semiclassical transport theories like the Boltzmann equation. This heuristic description is not intended as a substitute for the rigorous descriptions available in the literature [8.1–8.8]. Our intention is simply to make the formalism accessible to readers unfamiliar with the language of second quantization. We will restrict our discussion to steady-state transport as we have done throughout this book.

The NEGF formalism (sometimes referred to as the Keldysh formalism) requires a number of new concepts like correlation functions which we introduce in Sections 8.1 and 8.2. We then describe the formalism in Sections 8.3–8.6. In Section 8.7 we relate it to the Landauer–Büttiker formalism which, as we have seen, has been very successful in describing mesoscopic phenomena. For non-interacting transport the two are equivalent, and the added conceptual complexity of the NEGF formalism is not necessary. The real power of this formalism lies in providing a general approach for describing quantum transport in the presence of interactions.

In contrast to the preceding chapters, which concentrated mainly on the physics of RTDs, this chapter reviews some applications of RTDs and related three-terminal devices. As briefly described in the first chapter, RTDs have two distinct features over other semiconductor devices from an applications point of view: namely, their potential for very-high-speed operation and their negative differential conductance. The former feature arises from the very small size of the resonant tunnelling structure along the direction of carrier transport; because of the short distance through which carriers must travel, RTDs can be designed to have very high cut-off frequencies. As a result, oscillation in submillimetre wave frequencies has been reported. Besides this highspeed potential, the negative differential conductance makes it possible to operate RTDs as so-called functional devices, which enables circuits to be designed on different principles than conventional devices. For example, signal processing circuits with a significantly reduced number of devices and multiple-valued memory cells using RTDs have been proposed and demonstrated. These functional applications are highly promising since RTDs, with their simple structure and small size, can be easily integrated with conventional devices such as field effect transistors (FETs) and bipolar transistors.

In Section 5.1, high-speed applications, including high-frequency signal generation and high-speed switching, are discussed. Functional applications, such as a one-transistor static random access memory (SRAM) and a multi-valued memory circuit, are described in Section 5.2.