The optical properties of low-dimensional systems are put to wide practical use, the semiconductor laser being an obvious example. In this chapter we shall expand the general results derived in Chapter 8 and apply them to low-dimensional structures.

First, the general theory needs to be developed further. A surprising result is that the real and imaginary parts of the complex dielectric function or conductivity are not independent functions, but can be derived from one another. This relies on the principle of causality, that a response should follow its stimulus, embodied in the Kramers–Kronig relations. Other important results follow, such as the f-sum rule that controls the total absorption of a material integrated over all frequencies.

Although transport properties often rely on one kind of carrier alone, this is not true of optical processes. We must therefore treat the valence band of semiconductors in detail, a task that we have long postponed. We shall do this with the celebrated Kane model, an extension of the k · p theory developed earlier. We also need to consider the full wave function within the effective-mass approximation. Usually we neglect the Bloch part and study only the slowly varying envelope, but both must be included in the matrix elements and each makes a contribution to the selection rules. Quite different results emerge for transitions between bands and those within the same band.

Electric and magnetic fields are among the most valuable probes of an electronic system. An obvious use of an electric field is to drive a current through a conductor; we studied conduction due to tunnelling in the previous chapter and will consider the opposite case of freely propagating electrons weakly scattered by impurities or phonons in later chapters. It is more surprising that useful information or practical applications can be obtained by applying an electric field to an insulator. An example of this is a change in optical absorption near a band edge caused by a strong electric field, the Franz–Keldysh effect, which we shall calculate in Section 6.2.1. This becomes even more useful when the electrons and holes are confined in a quantum well, and is used as an optoelectronic modulator.

A magnetic field has remarkable effects on a low-dimensional system. For example, the continuous density of states of a two-dimensional electron gas splits into a discrete set of δ-functions called Landau levels. This is reflected in the longitudinal conductivity as the Shubnikov–de Haas effect, giving a distinct signature of two-dimensional behaviour. The Hall effect is a widely used tool in semiconductors, and the combination with Landau levels in a two-dimensional electron gas gives the integer quantum Hall effect, where the Hall conductance is an exact multiple of e2/h.

This book is about low-dimensional semiconductors, structures in which electrons behave as though they are free to move in only two or fewer free dimensions. Most of these structures are really heterostructures, meaning that they comprise more than one kind of material. Before we can investigate the properties of a heterostructure, we need to understand the behaviour of electrons in a uniform semiconductor. This in turn rests on the foundations of quantum mechanics, statistical mechanics, and the band theory of crystalline solids. The first two chapters of this book provide a review of these foundations. Unfortunately it is impossible to provide a full tutorial within the space available, so the reader should consult one of the books suggested at the end of the appropriate chapter if much of the material is unfamiliar.

This first chapter covers quantum mechanics and statistical physics. Some topics, such as the theory of angular momentum, are not included although they are vital to a thorough course on quantum mechanics. The historical background, treated at length in most textbooks on quantum mechanics, is also omitted. There is little attempt to justify quantum mechanics, although the rest of the book could be said to provide support because we are able to explain numerous experimental observations using the basic theory developed in this chapter.

Low-dimensional systems have revolutionized semiconductor physics. They rely on the technology of heterostructures, where the composition of a semiconductor can be changed on the scale of a nanometre. For example, a sandwich of GaAs between two layers of AlxGa1−xAs acts like an elementary quantum well. The energy levels are widely separated if the well is narrow, and all electrons may be trapped in the lowest level. Motion parallel to the layers is not affected, however, so the electrons remain free in those directions. The result is a two-dimensional electron gas, and holes can be trapped in the same way.

Optical measurements provide direct evidence for the low-dimensional behaviour of electrons and holes in a quantum well. The density of states changes from a smooth parabola in three dimensions to a staircase in a two-dimensional system. This is seen clearly in optical absorption, and the step at the bottom of the density of states enhances the optical properties. This is put to practical use in quantum-well lasers, whose threshold current is lower than that of a three-dimensional device.

Further assistance from technology is needed to harness low-dimensional systems for transport. Electrons and holes must be introduced by doping, but the carriers leave charged impurities behind, which limit their mean free path. The solution to this problem is modulation doping, where carriers are removed in space from the impurities that have provided them.

I joined the Department of Electronics and Electrical Engineering at Glasgow University some ten years ago. My research was performed in a group working on advanced semiconducting devices for both electronic and optical applications. It soon became apparent that advances in physics and technology had left a gap behind them in the education of postgraduate students. These students came from a wide range of backgrounds, both in physics and engineering; some had received extensive instruction in quantum mechanics and solid state physics, whereas others had only the smattering of semiconductor physics needed to explain the operation of classical transistors. Their projects were equally diverse, ranging from quantum dots and electro-optic modulators to Bloch oscillators and ultrafast field-effect transistors. Some excellent reviews were available, but most started at a level beyond many of the students. The same was true of the proceedings of several summer schools. I therefore initiated a lecture course with John Barker on nanoelectronics that instantly attracted an enthusiastic audience. The course was given for several years and evolved into this book.

It was difficult to keep the length of the lecture course manageable, and a book faces the same problem. The applications of heterostructures and low-dimensional semiconductors continue to grow steadily, in both physics and engineering. Should one display the myriad ways in which the properties of heterostructures can be harnessed, or concentrate on their physical foundations?

The two-dimensional electron gas (2DEG) trapped at a doped heterojunction is the most important low-dimensional system for electronic transport. It forms the core of a field-effect transistor, which goes by many acronyms including modulation-doped field-effect transistor (MODFET) and high electron mobility transistor (HEMT). I shall use the first, which emphasizes the close relation to the silicon MOSFET but with MODulation doping. The silicon MOSFET is perhaps the most common electronic device, with electrons or holes trapped in an inversion layer at an interface between Si and SiO2. Many of the ideas in this chapter were originally derived for the MOSFET but it has been almost completely superseded in physics experiments by the MODFET because of the enormous improvement in the mobility of electrons and holes. The highest mobility of electrons in a MOSFET is around 4 m2 V−1 s−1, whereas values over 1000 m2 V−1 s−1 have been achieved in a MODFET. These mobilities are measured at low temperature, where they are limited by scattering from impurities, defects, and interfaces rather than phonons. The almost perfect crystalline quality of III–V heterostructures and the ability to separate carriers from the impurities that provide them by modulation doping mainly account for this huge difference.

First we shall study the electrostatics of modulation-doped layers to get estimates of important quantities such as the density of electrons, and then we shall develop models for the energy level and wave function of a two-dimensional electron gas.

This chapter provides a review of the general properties of heterostructures, semiconductors composed of more than one material. Variations in composition are used to control the motion of electrons and holes through band engineering. Knowledge of the alignment of bands at a heterojunction, where two materials meet, is essential but has proved difficult to determine even for the best-studied junction, GaAs–AlxGa1−xAs. Although effort was initially concentrated on materials of nearly identical lattice constant, current applications require properties that can be met only by mismatched materials, giving strained layers.

A huge variety of devices has been fabricated from heterostructures, for both electronic and optical applications, and we shall survey these before studying them in greater detail in later chapters. Finally, we shall look briefly at the effective-mass approximation. This is a standard simplification that allows us to treat electrons as though they are free, except for an effective mass, rather than using complicated band structure. It means, for example, that an electron in a sandwich of GaAs between Al0.3Ga0.7As can be treated as the elementary problem of a potential well.

We shall neglect the random nature of alloys such as AlxGa1−xAs, where there is assumed to be no ordering of the Ga and Al ions over the cation sites of the lattice. In principle Bloch's theorem does not apply to such materials because they lack translational invariance from cell to cell.

In Chapter 4 we looked at how electrons could be trapped in various examples of potential wells and made to behave as though they were only two-dimensional (or less). In this chapter we shall look at free electrons that encounter barriers or other obstacles as they travel. Again, most of the potential profiles will be one-dimensional and we need only solve the Schrödinger equation in this dimension, although the other dimensions enter into the calculation of the current. We shall use the general tool of T-matrices, which can simply be multiplied together to yield the transmission coefficient for an arbitrary sequence of steps and plateaus. Two particular applications are to resonant tunnelling through a double barrier and to an infinite, regularly spaced sequence of barriers, a superlattice. Two barriers show a narrow peak in the transmission when the energy of the incident electron matches that of a resonant or quasi-bound state between the barriers (Section 5.5). This peak broadens into a band in the superlattice, and Section 5.6 shows how band structure and Bloch's theorem emerge for a specific example.

Many low-dimensional structures cannot simply be factorized into one-dimensional problems but have many leads, each with several propagating modes. These will be treated in Section 5.7 and we shall derive one of the famous results of low-dimensional systems, the quantized conductance.

Fermi's golden rule is one of the most important tools of quantum mechanics. It gives the general formula for transition rates, the rates at which particles are ‘scattered’ from one state to another by a perturbation. ‘Scattered’ is in quotation marks because it is a much more general concept than one might guess. An obvious example is provided by impurities in a crystal, which scatter an electron from one Bloch state to another. They change its momentum but not its energy. Similarly, phonons (vibrations of the lattice) also scatter electrons, but in this case they change the energy of the electrons as well as their momentum. A less obvious example is the absorption of light, which can be viewed as a scattering process in which an electron collides with a photon. The converse process also occurs, where an electron loses energy to a photon, and gives rise to spontaneous and stimulated emission. Thus scattering is a remarkably general concept.

The examples suggest that there are two broad classes of scattering processes that we should treat:

1. (i) potentials that are constant in time, such as impurities in a crystal, which do not change the energy of the particle being scattered;

2. (ii) potentials that vary harmonically in time as cos ωqt, such as phonons and photons, which change the energy of the particle by ±ħωq.

Few low-dimensional systems are periodic (superlattices provide an obvious exception), but they all consist of relatively large scale structures superposed on the structure of a host. This may be a true crystal such as GaAs or a random alloy such as (Al,Ga)As; we shall ignore the complications introduced by the alloy and treat it as a crystal ‘on average’. We must understand the electronic behaviour of the host before treating that of the superposed structure.

This chapter deals first with one-dimensional crystals, followed by three-dimensional materials. The final section is devoted to phonons, lattice waves rather than electron waves, which also have a band structure imposed by the periodic nature of the crystal. Photons are the third kind of wave that we shall encounter, and structures that display band structure for light have recently been demonstrated. Their behaviour can be described with a similar theory but we shall not pursue this.

Band Structure in One Dimension

The potential energy in a real crystal is clearly far more complicated than the systems that we have studied in the previous chapter. In Section 5.6 we shall solve the simple example of a square-wave potential in detail, but the most important results follow from the qualitative feature that the potential is periodic. In one dimension this means that V(x + a) = V(x), where a is the lattice constant, the size of each unit cell of the crystal.