Elasticity, plasticity and brittleness

We have looked in the preceding chapters at the atomic structure of solids, described in terms of quantum mechanics. We shall now see how this description explains why solids have the various properties that they exhibit. We begin by looking in this chapter at a group of properties that defines what a solid is as distinguished from a liquid or a gas. The atoms in a solid stay where they are, and do not slide past one another as in a liquid, or fly about in the enclosing volume as in a gas. A solid holds its size and shape, and resists attempts to change them. The ideal solid is a perfect crystal, whose atoms are bound to one another by the electrical forces among their electrons and nuclei. It is as if there were springs connecting each atom to neighbouring atoms. Any stretching or compression produced by an applied force is taken up by the springs, and the solid returns to its original state when the force is removed. The response of a real solid to such a force, however, is more complicated, and I illustrate it with the following experiments.

I take a copper wire and a glass rod, both of the same size. If I grasp the ends of the copper wire and bend it just a little and let go, it springs back to its original shape. If I bend it more than a certain amount and then let go, it stays bent.

I compared our journey through this book to a mountain ramble. The last chapter, about superconductivity, is like a high peak in the landscape of condensed matter physics. I should like to look back at the terrain we have covered to get there. What I have in mind here is not a summary of the preceding chapters, but rather a review of the key elements necessary for our understanding of the properties of condensed matter. These elements belong to three fundamental parts of physics: quantum mechanics, statistical physics, and electromagnetism. We refer to them as theories, but in a very precise sense. They are supported by vast numbers of experiments, and they predict correctly the results of new experiments. The word theory as used here has a significance different from what it has in everyday language as in ‘I have a theory about why it always rains on Saturdays.’

Quantum mechanics is the language in physics that we use to describe the world. It incorporates the fact that everything around us is sometimes particle-like, and sometimes wave-like. This basic duality of nature leads to a kind of graininess: just as matter comes in discrete units like electrons and protons and not fractions thereof, quantities like energy and momentum are quantized and cannot take on any arbitrary value we wish. The wave aspect of nature is an expression of a fundamental indeterminism.

From isolated atom to crystal

A crystal is an orderly arrangement of atoms that are sitting rather close to each other, as we saw in chapter II. We then looked at the basic ideas of quantum mechanics in chapter III. We used these ideas to get a picture of an isolated atom in terms of its nucleus and especially its electrons in chapter IV. The material things that we deal with in our daily life consist not of single atoms or electrons, but rather of very large numbers (like 1021) of particles. We need a statistical approach to cope with such numbers. We saw in chapter V that quantum mechanics, when used to give such a description of assemblies of particles, divides them into two types: fermions and bosons.

The only property that the crystals of chapter II had was their symmetry. A real crystal has in addition many other properties like being a good (or bad) conductor of electric currents, having a colour, and so on. These properties depend on the particular kind of atoms that make up the crystal. For example, silver and silicon are both crystalline, but differ widely in many of their properties. The only essential difference, other than in the mass, between the atoms of any two elements is in the number of electrons surrounding the nucleus. So we can conclude that this difference must play a key role in determining the specific properties of materials.

Some preliminaries

We have, until now, thought of atoms as tiny hard spheres that feel a force of attraction towards one another. We ignored the fact that an atom consists of a nucleus surrounded by one or more electrons. If we stopped at this, we would make little progress towards understanding the properties of solids, other than that the atoms may form crystals. We must therefore go on and take a look at the internal structure of the atom itself, and see how this changes when the atom is surrounded by other atoms in a crystal. We shall then find that we need a description that is somewhat removed from our everyday experience. There is a name for this description, quantum mechanics: mechanics, because it deals with the motion of objects, and quantum, because features of the motion, such as energy, are quantized, meaning that they can only have any one of a set of distinct values, and no other value in between. Imagine for example a car which is so constructed that it can travel only at speeds of 10, or 50, or 100 km per hour, and not at any other speed. Then we could say that the speed of the car is quantized. This idea of quantization is something alien to our usual way of thinking. After all, a car is able to go at any speed we like up to its maximum speed.

What is a superconductor?

I have noticed that physicists are rarely given to excesses of exuberance in talking about their subject, even among themselves. So when they use a prefix like super for a material, it must be something quite extraordinary. I shall first tell you what the phenomenon is, and then explain why it is extraordinary. Suppose I take a wire of the metal lead which has a resistance of one ohm at room temperature, about 290 K, and measure its electrical resistance at different temperatures as I cool it down. I show the result in fig. XII-1. The resistance drops smoothly as the temperature drops. Near 7 K, the resistance drops abruptly by a factor of more than 1020 below its value at room temperature, and remains so at all lower temperatures. Put differently, the wire now conducts electric currents at least 1020 times better than it did before. In fact, we can only set an upper limit to this number because of the limits on the sensitivity of the measuring instruments. For all we know, the wire in this condition may be a perfect conductor, which means that a current can pass through it with zero voltage applied to it. Well, super is a bit less than perfect, and so we are content to say that lead has become superconducting, and to call the phenomenon itself superconductivity.

When you use a plastic spoon to stir hot coffee, the handle does not get hot, but if you use a silver spoon, it does. A sheet of glass is transparent so that you can see right through it, but a much thinner sheet of aluminium foil is opaque and lets no light through. You can bend a piece of copper wire back and forth without breaking it, but a glass rod is brittle and will break if you try to bend it. A magnet will attract an iron screw, but not a brass screw. A piece of copper is reddish in colour, while silver is – well, silvery. The electric current flowing through the tungsten filament of a light bulb makes it so hot that it gives off light. This current also flows through the connecting copper wires but does not make them even warm, and it cannot flow at all through the rubber insulation that covers the wires. The filament, the wires, and the insulation behave quite differently from one another as far as the electric current is concerned.

These examples illustrate the basic point I want to make: we are in our everyday life surrounded by objects that display a wide variety of properties. All these objects are made up of tiny things called atoms. A piece of silver is composed of silver atoms that are all exactly alike but are different from the atoms of, say, copper.

Introduction

I give in this chapter the quantum-mechanical description of an atom. I look in some detail at the hydrogen atom, which is the simplest atom we have, with just one proton and one electron. From the picture that emerges, we shall be able to see how to describe all other atoms. Before continuing, I need to explain what I mean when I say that the proton and electron have equal and opposite electric charges. It is because of these charges that the atom is stable, and does not spontaneously break up into the two separate particles.

Electric charge and potential energy

If I take a ball in my hand and then let go, it drops to the floor. This is because of the gravitational interaction between the ball and the earth. The ball had gravitational potential energy when I held it, and when I release it, this energy is progressively converted to kinetic energy as it falls. Gravitation makes two bodies move towards each other, and not away from each other: the gravitational force is always attractive.

As a result of many experiments done over the years, we find that there is another kind of force in nature, somewhat analogous to, but in one aspect different from, gravitation. I take two idealized examples, and use them to show the features of this force. First I consider two electrons that I imagine to be pinned down a distance r apart.

Some preliminaries

A sheet of glass is transparent to light, whereas a much thinner sheet of aluminium foil is opaque: materials can be divided into those which let light go through, and others which block it. Examples of the first kind are glass, clear plastics, crystals like sugar and common salt. Metals on the other hand are opaque to light. We know that all these materials are made up of atoms. We saw in chapter IV that the atom itself is mostly empty space, because its nucleus and electrons occupy a tiny fraction of the volume of the atom. We might think therefore that light, which can travel freely through empty space, should be able to go right through all materials, but this clearly is not so. The transparency or opacity of materials results from what happens to the quantum energy states of the electrons when the atoms come together to form a solid.

Colour is another property of materials which must have something to do with how light interacts with them. Metallic copper for example is red and gold is yellow, in daylight. Sapphire is blue and ruby is red. The explanation again lies in the electronic structure of such solids, together with the quantum nature of light, namely that it consists of photons with energy equal to the frequency multiplied by the Planck constant.

A friend once remarked ruefully to me, ‘Whenever I hear someone talk about a well-rounded person, I think of a spherical object with no distinguishing features on it.’ A closer examination of such a person does reveal some features; these, however, are likely to be concentrated in areas that are labelled literature, art, music, philosophy, history and others generally called the liberal arts or the humanities. The area called physics, on the other hand, usually tends to be blank, although physics is as impressive an achievement of the mind as any of the others. A reason for this state of affairs is contained in the response I often get from someone who hears that I am a physicist: ‘Physics was my most difficult subject in school. I never understood it.’ Instead of giving up, I try to explain to the person some problem I am working on at the moment, maybe in superconductivity. I use words, analogies, pictures drawn on a paper serviette: anything but mathematics that is more advanced than simple arithmetic. I am then, even if not always, rewarded by the person with a dawning of interest and a desire to learn more. Having enjoyed this activity during all these years that I have myself been learning physics through teaching and research, I decided to gather and organize these ideas into a book for people (and I believe there are many) who are curious about the physicist's picture of the world, especially the part which forms our immediate surroundings.

Introduction

Any object that we see around us, even though it may be very small like a grain of sand or a pin, consists of a very large number of atoms, about 1021. When the atoms are relatively far apart, as in a gas, the electronic structure, meaning the wave functions, of each atom is practically unaffected by the presence of the other atoms. When the atoms come close together in a solid, the wave functions become different from what they were for the isolated atoms. In crystals of metallic elements like silver and copper this change is such that the electrons in the highest energy state of the isolated atom are no longer bound to stay near the corresponding nucleus, but are free to wander around in the whole crystal. This freeing of the outer electrons happens with each atom in the crystal. So we end up with a crystal in which a large number of electrons, 1021 or more, are wandering about. These electrons are responsible for many of the metallic properties of copper or silver: the shiny appearance and the ability to carry electric current and to conduct heat easily, for example. We shall see later how this comes about. For the present we need to find a way to describe the properties and behaviour of such large numbers of identical entities like electrons or atoms. We do this with statistics.

The phase interference between two distinct electron (or hole) waves was treated in the Introduction to this book. Whereas the past few chapters dealt largely with the quasi-ballistic transport of these waves through mesoscopic systems, in this chapter we want to begin to treat systems in which the transport is more diffusive than quasi-ballistic. How do we distinguish between these two regimes? Certainly the existence of scattering is possible in both regimes, but we distinguish the diffusive regime from the quasi-ballistic regime by the level of the scattering processes. In the diffusive regime, we assume that scattering dominates the transport to a level such that there are no “ballistic” trajectories that extend for any significant length within the sample. That is, we assert that l = εF τ ≪ L, where L is any characteristic dimension of the sample. Typically, this means that the material under investigation is characterized by a relatively low mobility, certainly not the mobility of several million that can be obtained in good modulation-doped heterostructures. In a sense, the transport is now considered to be composed of short paths between a relatively large number of impurity scattering centers. Thus, we deal with the smooth diffusion of particles through the mesoscopic system. To be sure, the Landauer formula does not distinguish ballistic from diffusive transport, but its treatment in multimode waveguides is more appropriately considered a ballistic transport. To illustrate the difference, consider the Aharonov-Bohm effect and the presence of weak localization. In the former, the wavefunction (particles) splits into two parts that propagate around opposite sides of a ring “interferometer,” as illustrated in Fig. 1.4.

This book has grown out of our somewhat disorganized attempts to teach the physics and electronics of mesoscopic devices over the past decade. Fortunately, these have evolved into a more consistent approach, and the book tries to balance experiments and theory in the current understanding of mesoscopic physics. Whenever possible, we attempt to first introduce the important experimental results in this field followed by the relevant theoretical approaches. The focus of the book is on electronic transport in nanostructure systems, and therefore by necessity we have omitted many important aspects of nanostructures such as their optical properties, or details of nanostructure fabrication. Due to length considerations, many germane topics related to transport itself have not received full coverage, or have been referred to by reference. Also, due to the enormity of the literature related to this field, we have not included an exhaustive bibliography of nanostructure transport. Rather, we have tried to refer the interested reader to comprehensive review articles and book chapters when possible.

The Introduction of Chapter 1 gives a general overview of the important effects that are observable in small systems that retain a degree of phase coherence. These are also compared to the needs that one forsees in future small electron devices. Chapter 2 provides a general introduction to quantum confined systems, and the nature of quasi-two-, quasione- and quasi-zero-dimensional systems including their dielectric response and behavior in the presence of an external magnetic field. It concludes with an overview of semi-classical transport in quantum wells and quantum wires including the relevant scattering mechanisms in quantum confined systems.