Introduction

Surface topographic changes occur as a result of all particle–solid interactions. For large doses > 1017 particles/cm2 of energetic (tens of kilo-electronvolts) particles, these changes are often visible with the naked eye. Many mechanisms give rise to these topographic features, for example bombardment-induced defects in a solid or differential sputtering yields across a surface, due to grain boundaries or impurity inclusions. Electron micrographs of such surfaces reveal a wide variety of features such as etch pits, ridges, facets, ripples, cones and pyramids. The article by Carter, Navinšek and Whitton (1983) illustrates some of the features that have been observed, but many examples can be found also in most recent issues of the Journal of Vacuum Science and Technology. Features that develop can be an unwanted artefact of the ion bombardment technique, for example, they can be responsible for considerable uncertainty in the depth resolution of surface analytical techniques such as dynamic SIMS (secondary ion mass spectroscopy).

On the other hand, surface engineering seeks to etch well-defined patterns on surfaces. Ion beam lithography uses focused beams to etch patterns directly into a substrate but for most technological applications broad beams with masks are used. The modelling of the development of surface shape as a result of particle bombardment is therefore important from the point of view of understanding the basic physical processes involved. In addition it also has important engineering applications.

The general features of crystals

Materials in the crystalline state are commonplace and they play an important part in everyday life. The household chemicals salt, sugar and washing soda; the industrial materials, corundum and germanium; and the precious stones, diamonds and emeralds, are all examples of such materials.

A superficial examination of crystals reveals many of their interesting characteristics. The most obvious feature is the presence of facets and well-formed crystals are found to be completely bounded by flat surfaces – flat to a degree of precision capable of giving high-quality plane-mirror images. Planarity of this perfection is not common in nature. It may be seen in the surface of a still liquid but we could scarcely envisage that gravitation is instrumental in moulding flat crystal faces simultaneously in a variety of directions.

It can easily be verified that the significance of planar surfaces is not confined to the exterior morphology but is also inherent in the interior structure of a crystal. Crystals frequently cleave along preferred directions and, even when a crystal is crudely fractured, it can be seen through a microscope that the apparently rough, broken region is actually a myriad of small plane surfaces.

Another feature which may be readily observed is that the crystals of a given material tend to be alike – all needles or all plates for example – which implies that the chemical nature of the material plays an important role in determining the crystal habit.

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. On the other hand, no controversy attaches to acoustic waves.

Introduction

This chapter deals with several topics. There is considerable interest in fabricating quasi-2D structures in which the electron–phonon interaction is reduced. Optical-phonon engineering is in its infancy, but already there have been investigations of the effect of incorporating monolayers and conducting layers. One of the first quasi-2D systems to be studied was the thin ionic slab, yet there are still problems connected with the description of optical modes in such structures. The increasing sophistication of microfabrication techniques has led to the creation of quasi-one-dimensional (quantum wires) and quasi-zero-dimensional (quantum dots) structures that are expected to have interesting physical properties. It is important to establish the mode structure, both electron and vibrational, in these systems. In this chapter we consider some of these topics briefly.

Monolayers

The study of short-period superlattices in electronic and optical devices has received considerable attention and there are several reasons why this has been so. Ease of growth and reduction of interface roughness and residual impurities make for more perfect structures. Replacing random alloys, such as AlxGa1 – x As, with their ordered superlattice counterparts (GaAs)m/(AlAs)n eliminates alloy scattering. In the AlxGa1–x As system there is the added advantage of avoiding the troublesome DX center. The replacement of random alloys by equivalent superlattices in bandgap engineering is unproblematic.

Charged-Impurity Scattering

Introduction

Scattering of electrons by charged impurity atoms dominates the mobility at low temperatures in bulk material and is usually very significant at room temperature (Fig. 9.1). The technique of modulation doping in high-electron-mobility field-effect transistors (HEMTs) alleviates the effect of charged-impurity scattering but by no means eliminates it. It remains an important source of momentum relaxation (but not of energy relaxation because the collisions are essentially elastic). Though its importance has been recognized for a very long time, obtaining a reliable theoretical description has proved to be extremely difficult.

There are many problems. First of all there is the problem of the infinite range of the Coulomb potential surrounding a charge, which implies that an electron is scattered by a charged impurity however remote, leading to an infinite scattering cross-section for vanishingly small scattering angles. Intuitively, we would expect distant interactions with a population of charged impurities to time-average to zero, leaving only the less frequent, close collisions to determine the effective scattering rate. This intuition motivated the treatment by Conwell and Weisskopf (1950) in which the range of the Coulomb potential was limited to a radius equal to half the mean distance apart of the impurities. Setting an arbitrary limit of this sort was avoided by introducing the effect of screening by the population of mobile electrons as was done by Brooks and Herring (1951) for semiconductors, following the earlier approach by Mott (1936).

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.