Introduction

In recent years there has been growing interest in sources of radiation whose frequencies occupied the region of the electromagnetic spectrum between, roughly, 300GHz and 30THz. The numerous applications include their use in medicine, molecular spectroscopy, communication, and security. The frequency range is covered by free-electron lasers, but there is an obvious need for more portable and adaptable sources, and this need has focussed attention on the properties of semiconductors and semiconductor multilayers. Existing semiconductor microwave generators such as the Gunn diode run out of power above 100GHz and IMPATTs above 300GHz. A new generation of devices that can extend the wavelength range beyond the familiar millimetre wave regime and into the sub-millimetre regime is required, and there has been substantial progress in recent years, much of it associated with the exploitation of the femtosecond Ti-sapphire laser and various non-linear optical processes. Semiconductors have also been used as THz detectors, notably, Ge and Si bolometers operating at liquid helium temperatures. The photon energy at 1THz is 4meV, and photon energies of room temperature radiation are around 25meV, so the problem of detection of THz radiation is that it is always against a competing, pervading background. Mention in Section 14.6 has already been made of cascade lasers operating in the near infrared.

The Wurtzite Lattice

Wurtzite is zinc oxide. Its lattice is made up of tetrahedrally bonded atoms, as it is in zinc blende, but it differs in the stacking sequence: in zinc blende the structure has face-centred cubic symmetry, in wurtzite the structure has hexagonal symmetry. The wurtzite lattice is the preferred thermodynamically stable phase of a number of binary semiconductors, among them the nitrides, AlN, GaN, and InN, which have become of considerable technical importance in recent years. Among the applications are high-power microwave transistors, light-emitting diodes, and highly efficient solar cells. If the paradigm of the cubic III-V semiconductors is GaAs, the paradigm of the hexagonal nitrides is GaN. Being familiar with the properties of GaAs, we need to know how the lower symmetry of GaN affects the band structure of electrons and holes, the phonon spectrum and the electron–phonon interaction. We focus on these topics in this chapter. Fuller accounts of the properties of the nitrides can be found in a number of excellent reviews (e.g. Ambacher, 1998; Ambacher et al., 2002; Piprek (ed.) Nitride Semiconductor Devices, 2007).

The unit cell of the wurtzite lattice is depicted in Fig. 13.1. It consists of stacked hexagonal planes so that were the atoms to be spherical with the same radius, the lattice would be a close-packed hexagonal structure.

Introduction

We now turn to the problem of describing long-wavelength lattice vibrations in multilayered structures. At a microstructural level this problem is solved by using the techniques of lattice dynamics requiring intense numerical computation. This approach is inconvenient, if not impracticable, at the macroscopic level, where the kinetics and dynamics of large numbers of particles need to be described. In this theoretical regime it is necessary to obtain models that transcend those at the level of individual atoms in order to describe electron and hole scattering, and all the energy and momentum relaxation processes that underlie transport and optical properties of macroscopic structures. The basic problem is to make a bridge between the atomic crystal lattice and the classical continuum. We know from the theory of elasticity and the theory of acoustic waves, which hark back to the nineteenth century, that continuum theory works extremely well for long-wave acoustic waves. The case of optical vibrations is another matter. Here, the essence is one atom vibrating against another in a primitive unit cell and it is by no means obvious that a continuum approach can work in this case. This has been highlighted by controversy concerning the boundary conditions that long-wave optical vibrations obey at each interface of a multilayer structure.

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 on 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.

Single Heterostructures

In bulk material, neutralization of the enormous electric fields produced by the spontaneous polarization can be achieved by a relatively modest adjustment of the population of surface states. This is no longer possible in the case of inhomogeneous material exhibiting inhomogeneous spontaneous polarization, such as the AlGaN/GaN heterostructure. A layer of AlGaN grown epitaxially on GaN has not only a spontaneous polarization larger than that of GaN but it has, in addition, a piezoelectric polarization associated with the elastic strain inevitably present as a consequence of the lattice mismatch. No straightforward adjustment of the population of the surface states can eliminate the resultant fields. As a result, nitride heterostructures differ uniquely from heterostructures of cubic semiconductors in exhibiting large, built-in electric fields, and these result in the spontaneous formation of a quasi-2D electron gas at the AlGaN/GaN interface. In the cubic case such a gas must be produced by doping, thereby, ineluctably, introducing charged-impurity scattering even when the system is modulation doped. Doping is unnecessary in nitride structures so, in principle, enormous lowtemperature mobilities are possible and will, no doubt, be achieved as growth techniques evolve.

An estimate of the density of the polarization-induced electron population at the AlGaN/GaN interface can be obtained assuming the existence of a common branch-point Fermi energy and simple electrostatics.

This book had its origin in a graduate course in statistical mechanics given by Professor W. C. Schieve in the Ilya Prigogine Center for Statistical Mechanics at the University of Texas in Austin.

The emphasis is quantum non-equilibrium statistical mechanics, which makes the content rather unique and advanced in comparison to other texts. This was motivated by work taking place at the Austin Center, particularly the interaction with Radu Balescu of the Free University of Brussels (where Professor Schieve spent a good deal of time on various occasions). Two Ph.D. candidate theses at Austin, those of Kenneth Hawker and John Middleton, are basic to Chapters 3 and 4, where the master equations and quantum kinetic equations are discussed. The theme there is the dominant and fundamental one of quantum irreversibility. The particular emphasis throughout this book is that of open systems, i.e. quantum systems in interaction with reservoirs and not isolated. A particularly influential work is the book of Professor A. McLennan of Lehigh University, under whose influence Professor Schieve first learned non-equilibrium statistical mechanics.

An account of relatively recent developments, based on the addition in the Schrödinger equation of stochastic fluctuations of the wave function, is given in Chapter 13. These methods have been developed to account for the collapse of the wave function in the process of measurement, but they are deeply connected as well with models for irreversible evolution.

The first six chapters of the present work set forth the theme of our book, particularly extending the entropy principle that was first introduced by Boltzmann, classically. These, with equilibrium quantum applications (Chapters 7, 8, 9 and possibly also Chapters 14 and 15), represent a one-semester advanced course on the subject.

As frequently pointed out in the text, quantum mechanics introduces special problems to statistical mechanics. Even in Chapter 1, written by the coauthor of this work, Professor Lawrence P. Horwitz of Tel Aviv, the idea of a density operator is required which is not a probability distribution, as in the classical case.