Cubic structure

Crystals with cubic structure are of major importance in the fields of electronics and optoelectronics. Indeed, zincblende crystals such as silicon, germanium, and gallium arsenide may be regarded as two face-centered cubic (fcc) lattices displaced relative to each other by a vector (a/4, a/4, a/4), where a is the size of the smallest unit of the fcc structure. Figure 2.1 shows a lattice with the zincblende structure.

A major portion of this book will deal with phonons in cubic crystals. In addition, we will describe the phonons in so-called isotropic media, which are related mathematically to cubic media as explained in detail in Section 7.2. The remaining portions of this book will deal with crystals of würtzite structure, defined in Chapter 3. More specifically, the primary focus of this book concerns phonons in crystalline structures that are dimensionally confined in one, two, or three dimensions. Such one-, two-, and three-dimensional confinement is realized in quantum wells, quantum wires, and quantum dots, respectively. As a preliminary to considering phonons in dimensionally confined structures, the foundational case of phonons in bulk structures will be treated. The reader desiring to supplement this chapter with additional information on the basic properties of phonons in bulk cubic materials will find excellent extended treatments in a number of texts including Blakemore (1985), Ferry (1991), Hess (1999), Kittel (1976), Omar (1975), and Singh (1993).

Ionic bonding – polar semiconductors

As is well known, the crystal structure of silicon is the zincblende structure shown in Figure 2.1.

Introduction

Recent progess in epitaxial growth techniques has promoted the use of semiconductor heterostructures in optoelectronic devices. The physics of these materials relies upon the similarity between the electronic band structures of the different semiconductors. If the bulk band structures are sufficiently similar then changes in composition can be represented primarily as changes in the band splittings and other bulk parameters. In a direct band-gap semiconductor an abrupt change in composition from wide to narrow band gap results in a discontinuity in the conduction and valence band profiles in the growth direction. The heterointerface so formed is Type I or Type II, depending on the band-gap alignments, determined by the conduction band offset (Fig. 6.1).

A quantum well (QW) is made by growing a thin layer – typically a few nanometres (nm) or 10s of nm – of narrower gap material within a wider-gap semiconductor, where the inserted layer is thin enough to cause quantum confinement of the carriers. QWs are similarly classified as Type I or II in direct-gap materials (Fig. 6.2).

In indirect gap materials we need to consider the band-edge discontinuities at different points in the band structure. The AlAs/GaAs heterointerface, for example, is Type I at the Г point but Type II at the X point (Fig. 6.3). The overall band structure of an AlAs/GaAs QW system thus depends on the relative well and barrier widths.

Introduction

High-speed semiconductor devices are key elements in the development of electronic systems for data processing or analogue signal handling at ever-higher frequencies (the state-of-the-art is currently in excess of 100 GHz; a detailed discussion of a wide range of devices is given by Sze (1990)). Most of these systems use circuits based on Si devices prepared by implantation or diffusion, but these fabrication processes are limited in their ability to produce the small-scale device structures required for high-speed operation. However, the advent of highly controllable epitaxial deposition processes such as molecular-beam epitaxy (MBE) and metal-organic vapour-phase epitaxy (MOVPE) has enabled semiconductor structures to be prepared with compositional or dopant properties defined in layers with thicknesses down to the atomic scale. In many cases, this results in the mobile charge carriers being confined in a quasi-two-dimensional sheet, giving rise to a wide range of quantum confinement effects, as described in the earlier chapters of this book. Historically, these heterostructures have been prepared in III–V compound semiconductors, principally GaAs and (Al,Ga)As, although recently much effort is being put into investigating epitaxial Si and SiGe layers. This chapter will review two classes of electronic device, namely those involving transport of charge along the two-dimensional sheets, based on variations on the field-effect transistor (FET) principle, and those using ‘vertical’ transport, i.e. perpendicular to the sheets. It is interesting to note that, with the exception of resonant tunnelling and superlattice structures, the performance characteristics of these devices are not primarily due to quantum confinement effects associated with their two-dimensional nature, although these may play a secondary role in some instances.

Everyone who studies, develops or utilizes modern semiconductor materials must be aware of the importance of low-dimensional structures in optical and electronic devices, crystal growth, semiconductor theory and experiment, and semiconductor material science and chemistry. Virtually every major university in the world has one or several growth facilities dedicated to basic studies of growth processes, the fabrication of heterostructures for device applications, or exploration studies of the properties of new structures. Such facilities reside in physics, chemistry, materials science departments or in electrical, chemical, or mechanical engineering departments. These factors underscore both the inherent interdisciplinary of modern semiconductor science and technology as well as the need for basic textbooks which are appropriate for students across these disciplines.

This book is aimed at the graduate-level student who has completed a first degree in physics, material science, chemistry, or one of the major engineering disciplines. It is based on an advanced Masters-level course which has been given at Imperial College for some years. Like the Masters course itself, the interdisciplinary nature of the subject is reflected in the choice of authors from different departments, colleges and from the University of London Interdisciplinary Research Centre for Semiconductor Materials (now the Centre for Electronic Materials and Devices). Many of the exercises which follow each chapter, and which we regard as an integral part of this book, have been tested by our students either as assignments during the course or as adaptations of examination questions.

Introduction

In Chapter 2, low-dimensional systems were discussed in terms of a single-electron picture, and the behaviour of an electron was examined in the case that it is acted on by various potentials in semiconductors. Those potentials have been supposed to be externally imposed, for instance by a discontinuity in the band gap at an interface between two materials. But an electron will also feel the effect of other electrons in the system in which it finds itself.

There are circumstances in which these many-electron effects can be ignored, for example, in an undoped semiconductor with very few free charges. But in many cases, effects due to the presence of other electrons can be extremely important. Some of the most interesting low-dimensional systems involve many charges: there can be many free electrons, and there will often be in addition some distribution of fixed charges (space charge). To study such systems properly, we must discuss how to take into account the presence of such charges. The problem is one of self-consistency because we are trying to predict the behaviour of electrons (or holes), while that behaviour will itself depend upon those charges whose behaviour we are trying to predict: in other words, the problem itself depends upon the solution to the problem.

Many-body Effects

The Hartree Approximation

Consider the reaction of conduction electrons to the presence of a potential well V0(z) (we suppose this to be an externally determined well, e.g. a finite square well, which restricts electrons into a two-dimensional region).

Introduction

Epitaxial growth is a process during which a crystal is formed on an underlying crystalline surface as the result of deposition of new material onto that surface. The study of this process dates back over 150 years, but it was not until the work of Louis Royer in the 1920s that the systematics of epitaxial growth began to be revealed (Royer, 1928). Royer carried out an extensive study of the growth of ionic crystals on one another and on mica, mainly from aqueous solution and, using optical microscopy, summarized his observations with a set of rules based on crystal structure. These rules led Royer to coin the term ‘epitaxy’, which is a combination of the Greek words epi, meaning ‘upon’, and taxis, meaning ‘order’, to convey the notion of growing a new crystal whose orientation is determined by a crystalline substrate and to distinguish epitaxial growth from polycrystalline and amorphous growth. A review of the history of epitaxial growth has been given by Pashley (1956).

The modern era of the epitaxial growth of semiconductors began with the work of Henry Theurer at Bell Telephone Laboratories in Murray Hill, New Jersey (Theurer, 1961). Motivated by the need to reduce the base resistance of discrete bi-polar transistors, Theurer demonstrated that thin epitaxial silicon layers could be grown on a silicon substrate. The idea that epitaxial structures could lead to new electronic and optical phenomena was founded on a suggestion in the late 1960s by Leo Esaki and Raphael Tsu (1970), then working at the IBM Research Laboratories in Yorktown Heights, New York.

Introduction

Traditional solid-state physics is based on the concept of the perfect crystalline solid, sometimes with a relatively low density of defects. This perfect crystallinity has played a crucial role in the development of the subject, with Bloch's theorem providing the central conceptual base. Concepts that arise from this theorem, such as bands, Brillouin zones, vertical transitions, effective mass and heavy and light holes, are really only well-defined in a perfect infinite crystal. In the absence of crystallinity none of these concepts is strictly valid, though in some cases it provides a useful starting point. In general, however, a new approach is required to characterize electrons and phonons in disordered solids.

When we consider low-dimensional structures Bloch's theorem may or may not be valid. There is nothing intrinsic to low dimensionality which invalidates it. Many of the simple examples in quantum mechanics and solid-state physics textbooks are, in fact, one-dimensional (e.g. the particle in a box, the Kronig–Penney model). Indeed, in a quantum well prepared by any of the standard growth methods (Chapter 1), much of the physics can be understood by using first-year undergraduate quantum mechanics and the effective mass approximation (Chapter 2). This is because a region of adjacent GaAs layers in AlxGa1−xAs can, for many purposes, be regarded as a perfect potential well. By doping the AlGaAs, the electrons in the well can be spatially separated from the scattering due to the ionized donor atoms (Chapter 3).

Introduction

The semiconductor industry is a massive business and has a seemingly inexhaustible appetite for new devices, new materials and new applications. A single type of device can open up markets worth hundreds of millions of dollars if it can fill a suitable gap or demonstrate superior performance to those currently available. The rewards are even greater if the same device can then be integrated with many others. The high-electron mobility transistor, or HEMT (Chapter 10), is a good example of such a device. Conceived in the late seventies (Dingle et al., 1978), it was the focus of substantial research and development during the early 1980s (Mimura et al., 1980) and is now arguably the most important element in highspeed, low-noise communications systems such as those used in direct broadcast satellite television.

With such a powerful driving force it is not surprising that so much effort is devoted to researching new semiconductor device technologies. Low-dimensional systems have received particular attention during the last ten or fifteen years and as a result some elegant physics has emerged. It is therefore natural for physicists and engineers to explore ways in which the unique properties of low-dimensional structures might be exploited in devices of the future. Because these structures are usually in the size regime which lies somewhere between the microscopic world of atoms and the macroscopic world we live in, they are often lumped together under the single title of mesoscopic devices.

Introduction

Electronics has progressed from the large to the small. In the process, intuitive ideas gained from the ordinary world of classical physics have had to give way to those of the entirely different world of quantum mechanics. When electronic processes are taking place in structures with sizes of the order of centimetres, or even microns, one can describe what is happening in a continuous way. By changing conditions very slightly, one can expect the results to show only very slight changes. When the physical size of the system becomes smaller, however, quantum mechanical effects become important. Typically, a small enough system will be able to have only a few discrete energies. This is the obvious difference between the continuous possibilities that appear to be available to the electrons making up a current in a wire, and the picture one must use to describe what an individual electron must be doing when it is bound to an individual atom making up that wire. In the latter case, one must talk about the possible discrete energy levels which are permitted.

This used to be a clear-cut division. Nature makes the quantum world, while people manufacture the wires. What is new, however, is that people can now make materials and structures which may be large on the scale of atoms, but which are small enough that the graininess that comes with quantum effects is crucial to understanding their behaviour.