General Remarks

Evidently, the advent of mesoscopic layered semiconductor structures generated a need for a simple analytic description of the confinement of electrons and phonons within a layer and of how that confinement affected their mutual interaction. The difficulties encountered in the creation of a reliable description of excitations of one sort or another in layered material are familiar in many branches of physics. They are to do with boundary conditions. The usual treatment of electrons, phonons, plasmons, excitons, etc., in homogeneous bulk crystals simply breaks down when there is an interface separating materials with different properties. Attempts to fit bulk solutions across such an interface using simple, physically plausible connection rules are not always valid. How useful these rules are can be assessed only by an approach that obtains solutions of the relevant equations of motion in the presence of an interface, and there are two types of such an approach. One is to compute the microscopic bandstructure and lattice dynamics numerically; the other is to use a macroscopic model of long-wavelength excitations spanning the interface. The latter is particularly appropriate for generating physical concepts of general applicability. Examples are the quasi-continuum approach of Kunin (1982) for elastic waves, the envelope-function method of Burt (1988) for electrons, and the wavevector-space model of Chen and Nelson (1993) for electromagnetic waves and excitons.

This book has grown out of my own research interests in semiconductor multilayers, which date from 1980. It therefore runs the risk of being far too limited in scope, of prime interest only to the author, his colleagues and his research students. I hope that this is not the case, and of course I believe that it will be found useful by a large number of people in the field; otherwise I would not have written it. Nevertheless, knowledgeable readers will remark the lack of such fashionable topics as the quantum-Hall effect, Coulomb blockade, quantized resistance, quantum tunnelling and any physical process that can be studied only in the millikelvin regime of temperature. This has more to do with my own ignorance than any lack of feeling that these phenomena are important. My research interests have not lain there. My priorities have always been to try to understand what goes on in practical devices, and as these work more or less at room temperature, the tendency has been for my interest to cool as the temperature drops.

Introduction

An electron in a quantum-well subband can be scattered to another state in the same subband or into a state in another subband. Intrasubband and intersubband scattering rates have to be calculated separately since different wavefunction symmetries are involved in the two cases, and this implies correspondingly different symmetries of the optical mode. For simplicity we will assume that the electrons are completely confined within the well and that the interaction is with polar optical modes. In the case of LO modes in a polar material this interaction is via a scalar potential. However, as we will see, it is possible in the unretarded limit (velocity of light is infinite) to replace the vector potential of the electromagnetic interface wave with a scalar potential via a unitary transformation (not a gauge transformation) and treat the IP mode on the same footing as an LO mode, but with a frequency-dependent scalar potential. We assume the TO mode has no interaction.

No fewer than four different scattering sources exist, in general. Two of these are associated with well modes, two with barrier modes. In general, the LO band of frequencies in either material does not span the range between the LO and TO zone-centre frequencies, ωLO and ωTO.

Electron in a strong magnetic field

General remarks

Some aspects of electron motion in a two-dimensional (2D) disordered metal have been considered in Chapter 5. These were effects that could be studied by using the perturbation theory in diffusion modes. The renormalization group scheme is a way to sum up a certain class of graphs and is a straightforward extension of the simple perturbation theory. Although the first quantum correction, Eq. (5.23), contains many interesting effects that can be and have been confirmed by numerous experiments, some other interesting effects cannot be obtained in this simple manner.

One of the most interesting phenomena occurring in two dimensions is the quantum Hall effect (von Klitzing, Dorda, and Pepper (1980)). Since its discovery a lot of theoretical and experimental activity has been devoted to studying this effect and related properties of 2D electron gases. It makes no sense to review this direction of research here because many interesting and comprehensive reviews and books already exist (see, e.g., Prange and Girvin (1990), Büttiker (1992), Stone (1992), Aoki (1986), Janßen et al. (1994)). The aim of this section is to demonstrate only how the supersymmetry technique can help in studying electron motion in a 2D disordered metal.

In fact, in discussing theoretical aspects of the quantum Hall effect one should distinguish between the integer quantum Hall effect (occurring at integer filling factor vf) and the fractional effect discovered by Tsui, Stormer, and Gossard (1982).

One-dimensional σ-model

One-dimensional σ-model for disorder problems

Most of the results of Chapters 6–10 have been obtained by using the zero-dimensional (0D) σ-model and we have seen that many rather different physical problems can be treated within this approximation successfully. Nevertheless, for some problems that can be described by the supermatrix σ-model the 0D version is not applicable. For example, localization in two-dimensional disordered metals was discussed in Chapter 5 and it had to be described by the two-dimensional (2D) σ-model. The problem is rather complicated and the solution presented there is not complete.

There are many problems of disordered metals and quantum chaos that can be reduced to a study of the one-dimensional (ID) σ-model. In contrast to those with higher dimensions, the ID σ-model can be studied exactly by the transfer matrix technique, and very often it is possible to get explicit final results although the calculations are somewhat more difficult than in the 0D case. By now, the procedure of computation of different correlation functions with the ID σ-model is well worked out, but before presenting it in detail let us discuss physical problems that can be treated in this way.

It is natural to suppose that the ID σ-model describes one-dimensional disordered metals. However, when using the term one-dimensional one should distinguish between truly one-dimensional chains and microscopically three-dimensional metals with a one-dimensional geometry of the sample.

What is supermathematics?

All supersymmetric theories are based on the use of anticommuting classical variables first introduced by Grassmann in the last century. At first glance, these objects look very artificial and seem to have no relation to the real world. There is a certain threshold for physicists to start using the Grassmann anticommuting variables for calculations because one expects the game to have very unusual rules. Surprisingly, it is not true, and provided proper definitions are given, one can simply generalize conventional mathematical constructions so that it is possible to treat both commuting and anticommuting variables on an equal footing. Sometimes the corresponding branch of mathematics is called supermathematics.

Of course, the main purpose of this book is to consider different physical results obtained with the use of the Grassmann variables, and therefore one could try to demonstrate how these variables work while making some concrete calculations. However, it seems to be more reasonable to present the basic formulae of supermathematics in one place, first, because it may be the best way to get used to the anticommuting variables, and, second, because one can see that practically all the rules of operating with “superobjects” are quite standard.

Today the mathematical analysis and algebra of functions of both commuting and anticommuting variables are very well developed.

Historical remarks

The last 15–20 years has witnessed spectacular progress in the study of disordered metals and semiconductors. These systems are interesting not only from the point of view of different technical applications but also because they reveal new unusual physical properties that are very different from what one would expect in clean regular materials. Although very often thermodynamic characteristics are already quite influenced by disorder, the most remarkable effects are observed in kinetics. Of course, in many cases one may use the classical transport theory based on the Boltzmann equation for a description of electron motion. However, if the disorder is strong or temperature is low, quantum effects become important, and to construct a theory in this situation one has to start from the Schrödinger equation in a potential that is assumed to be random.

To get information about physical properties of the system one has to solve the Schrödinger equation for an arbitrary potential, calculate a physical quantity, and, at the end, average over the random potential. Sometimes it is important to have information not only about the average but also about fluctuations. In this case one has to calculate moments of the physical quantities and even an entire distribution function. Needless to say, generally speaking, this program cannot be carried out exactly even in the absence of electron–electron interaction and one should use different approximation schemes.

Random matrix theory

General formulation

According to the basic principles of quantum mechanics the energy spectrum of a particle in a limited volume is discrete. The positions of the energy levels and the spacings between them depend on the boundary conditions and interactions in the system. In the simplest cases these quantities can be calculated exactly or approximately. However, often the interactions are so complicated that calculations for the levels become impossible. On the other hand, the complexity and variety of interactions lead to the idea of a statistical description in which information about separate levels is neglected and only averaged quantities are studied. Density of states, energy level and wave function correlations, and the like, can be so considered. The analogous approach is used in statistical physics, where information about separate particles is neglected and only averages over large number of particles are calculated.

The idea of statistical description of the energy levels was first proposed by Wigner (1951, 1958) for study of highly excited nuclear levels in complex nuclei. In such nuclei a large number of particles interact in an unknown way and it is plausible to assume that all the interactions are equally probable. Of course, the first question one can ask is what the characterization “equally probable” means and therefore one should introduce a measure for averaging.